Calculo Diferencial em R

❈❤r✐st✐❛♥ ❏♦sé ◗✉✐♥t❛♥❛ P✐♥❡❞♦ ❈➪▲❈❯▲❖ ❉■❋❊❘❊◆❈■❆▲ ❊▼ R ❈❤r✐st✐❛♥ ❏♦sé ◗✉✐♥t❛♥❛ P✐♥❡❞♦ ❈➪▲❈❯▲❖ ❉■❋❊❘❊◆❈■❆▲ ❊▼ R ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ■❙❇◆ 978 − 85 − 8236 − 040 − 8 ❈♦♣②r✐❣❤t ❝ ❊❞✉❢❛❝ 2017✱ ❈❤r✐st✐❛♥ ❏♦sé ◗✉✐♥t❛♥❛ P✐♥❡❞♦ ❊❞✐t♦r❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❝r❡ ✲ ❊❞✉❢❛❝ ❘♦❞✳ ❇❘ 364✱ ❑♠ 04 ♦ ❉✐str✐t♦ ■♥❞✉str✐❛❧ 69920 − 900 ♦ ❘✐♦ ❇r❛♥❝♦ ♦ ❆❝r❡ ❉✐r❡t♦r ❏♦sé ■✈❛♥ ❞❛ ❙✐❧✈❛ ❘❛♠♦s ❈❖◆❙❊▲❍❖ ❊❉■❚❖❘■❆▲ ❆❞❛✐❧t♦♥ ❞❡ ❙♦✉s❛ ●❛❧✈ã♦✱ ❆♥t♦♥✐♦ ●✐❧s♦♥ ●♦♠❡s ▼❡sq✉✐t❛✱ ❇r✉♥♦ P❡r❡✐r❛ ❞❛ ❙✐❧✈❛✱ ❈❛r❧❛ ❇❡♥t♦ ◆❡❧❡♠ ❈♦❧t✉r❛t♦✱ ❉❛♠✐á♥ ❑❡❧❧❡r✱ ❊✉stáq✉✐♦ ❏♦sé ▼❛❝❤❛❞♦✱ ❋❛❜✐♦ ▼♦r❛❧❡s ❋♦r❡r♦✱ ❏❛❝ó ❈és❛r P✐❝❝♦❧✐✱ ❏♦sé ■✈❛♥ ❞❛ ❙✐❧✈❛ ❘❛♠♦s✱ ❏♦sé ▼❛✉r♦ ❙♦✉③❛ ❯❝❤ô❛✱ ❏♦sé P♦r✜r♦ ❞❛ ❙✐❧✈❛✱ ▲✉❝❛s ❆r❛ú❥♦ ❈❛r✈❛❧❤♦✱ ▼❛♥♦❡❧ ❉♦♠✐♥❣♦s ❋✐❧❤♦✱ ▼❛r✐❛ ❆❧❞❡❝② ❘♦❞r✐❣✉❡s ❞❡ ▲✐♠❛✱ ❘❛✐♠✉♥❞❛ ❞❛ ❈♦st❛ ❆r❛r✉♥❛✱ ❙✐♠♦♥❡ ❞❡ ❙♦✉③❛ ▲✐♠❛✱ ❚✐❛❣♦ ▲✉❝❡♥❛ ❞❛ ❙✐❧✈❛✱ ❨✉r✐ ❑❛r❛❝❝❛s ❞❡ ❈❛r✈❛❧❤♦✳ ❊❞✐t♦r❛ ❞❡ P✉❜❧✐❝❛çõ❡s ❏♦❝í❧✐❛ ❖❧✐✈❡✐r❛ ❞❛ ❙✐❧✈❛ ❉❡s✐❣♥ ❊❞✐t♦r✐❛❧ ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈❛♣❛ ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❘❡✈✐sã♦ ❚❡①t✉❛❧ ❖r♠✐❢r❛♥ P❡ss♦❛ ❈❛✈❛❧❝❛♥t❡ ❉❛❞♦s ■♥t❡r♥❛❝✐♦♥❛✐s ❞❡ ❈❛t❛❧♦❣❛çã♦ ♥❛ P✉❜❧✐❝❛çã♦ ✭❈■P✮ ❊❞✉❢❛❝ 2016 ❉✐r❡✐t♦s ❡①❝❧✉s✐✈♦s ♣❛r❛ ❡st❛ ❡❞✐çã♦✿ ❊❞✐t♦r❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❝r❡ ✭❊❞✉❢❛❝✮✱ ❈❛♠♣✉s ❘✐♦ ❇r❛♥❝♦✱ ❇❘ 364✱ ❑♠ 4✱ ❉✐str✐t♦ ■♥❞✉str✐❛❧ ✲ ❘✐♦ ❇r❛♥❝♦✲❆❈✱ ❈❊P 69920 − 900 68. 3901 2568 ✲ ❡✲♠❛✐❧✿❡❞✉❢❛❝✳✉❢❛❝❣♠❛✐❧✳❝♦♠ ❊❞✐t♦r❛ ❆✜❧✐❛❞❛✿ ❋❡✐t♦ ❉❡♣ós✐t♦ ▲❡❣❛❧ P649❝ P✐♥❡❞♦✱ ❈❤r✐st✐❛♥ ❏♦sé ◗✉✐♥t❛♥❛ ❈á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❡♠ ❘ ✴ ❈❤r✐st✐❛♥ ❏♦sé ◗✉✐♥t❛♥❛ P✐♥❡❞♦✳ ✲ ❘✐♦ ❇r❛♥❝♦✿ ❊❞✉❢❛❝✱ 2017✳ 390 ♣✳✿ ✐❧✳ ■♥❝❧✉✐ ❜✐❜❧✐♦❣r❛✜❛✳ ■❙❇◆✿ 978 − 85 − 8236 − 040 − 8 1. ▼❛t❡♠át✐❝❛✳ 2. ❈á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧✳ ■✳ ❚ít✉❧♦✳ ❈❉❉✿ 517.2 ❇✐❜❧✐♦t❡❝ár✐❛ ▼❛r✐❛ ❞♦ ❙♦❝♦rr♦ ❞❡ ❖✳ ❈♦r❞❡✐r♦ ✲ ❈❘❇ 11/667 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ❆ ♠❡✉s ✜❧❤♦s✿ ▼✐❧❛❣r♦s✱ ❆♥❞ré✱ ◆②❦♦❧❛s✱ ❑❡✈②♥✱ ❡ ❈❡❝í❧✐❛✱ ✈✐ R ♣❡❧❛s ❡t❡r♥❛s ❧✐çõ❡s ❞❡ ✈✐❞❛✳ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ◆❖❚❆➬Õ❊❙ ❙❡çã♦ ··· N Z Q R C ∪ ≈ ∩ ∅ < > 6= ∀ ∃ /. ⇒ ≤ abc ≥ [|x|] n P ai s✐❣♥✐✜❝❛✿ ❝♦♥t✐♥✉❛r s✉❝❡s✐✈❛♠❡♥t❡ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ✉♥✐ã♦ ❞❡ ❝♦♥❥✉♥t♦s s✐❣♥✐✜❝❛✿ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✐♥t❡rs❡çã♦ ❞❡ ❝♦♥❥✉♥t♦s ❝♦♥❥✉♥t♦ ✈❛③✐♦ r❡❧❛çã♦ ❡str✐t❛♠❡♥t❡ ♠❡♥♦r q✉❡ ✳✳✳ r❡❧❛çã♦ ❡str✐t❛♠❡♥t❡ ♠❛✐♦r q✉❡ ✳✳✳ ♥ã♦ é ✐❣✉❛❧ ❛✳ ✳ ✳ q✉❛♥t✐✜❝❛❞♦r ✉♥✐✈❡rs❛❧ ✭♣❛r❛ t♦❞♦ ✮ q✉❛♥t✐✜❝❛❞♦r ❡①✐st❡♥❝✐❛❧ ✭❡①✐st❡ ✮ t❛✐s q✉❡✳ ✳ ✳ ✐♠♣❧✐❝❛ ✭❡♥tã♦ ✮ r❡❧❛çã♦ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛✳✳✳ ♥ú♠❡r♦ ❢♦r♠❛❞♦ ♣♦r três ❛❧❣❛r✐s♠♦s r❡❧❛çã♦ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛✳✳✳ ♣❛rt❡ ✐♥t❡✐r❛ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ x ❄❄ ✶✳✷ ✶✳✷ ✶✳✷ ✶✳✷ ✶✳✷ ✶✳✷ ✶✳✷ ✶✳✷ ✶✳✷ ✶✳✷ ✶✳✷ ✶✳✷✳✶ ✶✳✷✳✶ ✶✳✷✳✶ ✶✳✷✳✶ ✶✳✺ ✶✳✺ ✶✳✷ ✶✳✻ ✶✳✸ s♦♠❛✿ a1 + a2 + a3 + · · · + an−1 + an ✶✳✸ n! ⇔ ⊂ m|n m ∤! n s✐❣♥✐✜❝❛ ♦ ♣r♦❞✉t♦✿ 1 × 2 × 3 × · · · × (n − 1) × n ❜✐❝♦♥❞✐❝✐♦♥❛❧ ✭s❡✱ ❡ s♦♠❡♥t❡ s❡✮ ✐♥❝❧✉sã♦ ♣ró♣r✐❛ ❞❡ ❝♦♥❥✉♥t♦s m é ❞✐✈✐s♦r ❞❡ n m ♥ã♦ é ❞✐✈✐s♦r ❞❡ n ✶✳✸ ✶✳✹✳✸ ✶✳✻ ✶✳✽✳✶ ✶✳✽✳✶ ⊆ A×B ✐♥❝❧✉sã♦ ❞❡ ❝♦♥❥✉♥t♦s ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❞♦s ❝♦♥❥✉♥t♦s A ❝♦♠ B i=1 n k s✐❣♥✐✜❝❛✿ n! k!(n − k)! ✶✳✽✳✸ ✈✐✐ ✷✳✶ ✷✳✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✈✐✐✐ R 09/02/2021 ❙❯▼➪❘■❖ ◆♦t❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✈✐✐ ■❞❡♥t✐❞❛❞❡s ❉✐✈❡rs❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ①✐✈ P❘❊❋➪❈■❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ①✐① ✶ ❙■❙❚❊▼❆ ❉❖❙ ◆Ú▼❊❘❖❙ ❘❊❆■❙ ✶✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❙✐st❡♠❛ ❞♦s ♥ú♠❡r♦s r❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶ ❆❞✐çã♦ ❡ ▼✉❧t✐♣❧✐❝❛çã♦ ❝♦♠ ♥ú♠❡r♦s r❡❛✐s ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✶✲✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❘❡❧❛çã♦ ❞❡ ♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✶✲✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ❉❡s✐❣✉❛❧❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹✳✶ ■♥❡q✉❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹✳✷ ■♥t❡r✈❛❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹✳✸ ❆ r❡t❛ ❛♠♣❧✐❛❞❛✳ ■♥t❡r✈❛❧♦s ✐♥✜♥✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✶✲✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❱❛❧♦r ❛❜s♦❧✉t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✶✲✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✻ ❆①✐♦♠❛ ❞♦ s✉♣r❡♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✼ ■♥❞✉çã♦ ♠❛t❡♠át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✽ Pr♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✽✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✽✳✷ ▼á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✳ ▼í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ✶✳✽✳✸ ◆ú♠❡r♦s ♣r✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✶✲✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼✐s❝❡❧â♥❡❛ ✶✲✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❋❯◆➬Õ❊❙ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✷ ✺ ✶✸ ✶✼ ✷✼ ✷✾ ✷✾ ✸✵ ✸✵ ✸✼ ✹✶ ✹✺ ✹✼ ✹✽ ✺✹ ✺✹ ✺✺ ✺✻ ✺✾ ✻✶ ✻✺ ✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✐① ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✷✳✷ ❘❡❧❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ ❉♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ❞❡ ✉♠❛ r❡❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✷ ❘❡❧❛çõ❡s ❞❡ R ❡♠ R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✷✲✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✶ ●rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✷ ❉❡✜♥✐çã♦ ❢♦r♠❛❧ ❞❡ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✸ ❉♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ❞❡ ✉♠❛ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✹ ❖❜t❡♥çã♦ ❞♦ ❞♦♠í♥✐♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✺ ❈♦♥str✉çã♦ ❞♦ ❣rá✜❝♦ ❝❛rt❡s✐❛♥♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ✷✳✸✳✻ ❋✉♥çã♦✿ ■♥❥❡t✐✈❛✳ ❙♦❜r❡❥❡t✐✈❛✳ ❇✐❥❡t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✼ ❋✉♥çã♦ r❡❛❧ ❞❡ ✈❛r✐á✈❡❧ r❡❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✷✲✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ❋✉♥çõ❡s ❡s♣❡❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✶ ❋✉♥çã♦ ❛✜♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✷ ❋✉♥çã♦ ❝♦♥st❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✸ ❋✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ ❡♠ R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✹ ❋✉♥çã♦ ❧✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✺ ❊q✉❛çã♦ ❞❡ ✉♠❛ r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✻ ❋✉♥çã♦ ♠❛✐♦r ✐♥t❡✐r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✼ ❋✉♥çã♦ r❛✐③ q✉❛❞r❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✽ ❋✉♥çã♦ s✐♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✾ ❋✉♥çã♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❡ x ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✶✵ ❋✉♥çã♦ q✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✶✶ ❋✉♥çã♦ r❛❝✐♦♥❛❧ ✐♥t❡✐r❛ ♦✉ ♣♦❧✐♥ô♠✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✶✷ ❋✉♥çã♦ r❛❝✐♦♥❛❧ ❢r❛❝✐♦♥ár✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✶✸ ❋✉♥çõ❡s ❞❡ ♦❢❡rt❛ ❡ ❞❡♠❛♥❞❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✷✲✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺ ❖♣❡r❛çõ❡s ❝♦♠ ❢✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺✳✶ ❈♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺✳✷ ❋✉♥çã♦ ✐♥✈❡rs❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺✳✸ ❘❡❧❛çã♦ ❡♥tr❡ ♦ ❣rá✜❝♦ ❞❡ f ❡ ❞❡ f −1 ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✷✲✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻ ❖✉tr♦s t✐♣♦s ❞❡ ❢✉♥çõ❡s r❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻✳✶ ❋✉♥çõ❡s ✐♠♣❧í❝✐t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻✳✷ ❋✉♥çã♦ ♣❡r✐ó❞✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻✳✸ ❋✉♥çã♦ ❛❧❣é❜r✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻✳✹ ❋✉♥çã♦ ♣❛r✳ ❋✉♥çã♦ í♠♣❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ R ✻✻ ✻✼ ✻✽ ✼✸ ✼✺ ✼✺ ✼✻ ✼✻ ✼✼ ✼✽ ✽✵ ✽✷ ✽✾ ✾✶ ✾✶ ✾✶ ✾✷ ✾✷ ✾✸ ✾✺ ✾✻ ✾✻ ✾✻ ✾✼ ✾✼ ✾✽ ✾✾ ✶✵✸ ✶✵✼ ✶✵✽ ✶✶✶ ✶✶✸ ✶✶✼ ✶✷✶ ✶✷✶ ✶✷✶ ✶✷✷ ✶✷✸ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✷✳✻✳✺ ✷✳✻✳✻ ✷✳✻✳✼ ❋✉♥çã♦ ♠♦♥♦tô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❋✉♥çã♦ ❧✐♠✐t❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❋✉♥çã♦ ❡❧❡♠❡♥t❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✷✲✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼ ❋✉♥çõ❡s tr❛♥s❝❡♥❞❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼✳✶ ❆ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ❜❛s❡ a ✳ ✳ ✳ ✳ ✷✳✼✳✷ ❋✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✷✲✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼✳✸ ❋✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼✳✹ ❋✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ✐♥✈❡rs❛s ✳ ✳ ✳ ✷✳✼✳✺ ❋✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✷✲✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼✐s❝❡❧â♥❡❛ ✷✲✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▲■▼■❚❊❙ ✶✷✹ ✶✷✺ ✶✷✼ ✶✷✾ ✶✸✸ ✶✸✸ ✶✸✺ ✶✸✾ ✶✹✶ ✶✹✽ ✶✺✶ ✶✺✸ ✶✺✼ ✶✻✶ ✸✳✶ ❱✐③✐♥❤❛♥ç❛ ❞❡ ✉♠ ♣♦♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ▲✐♠✐t❡ ❞❡ ✉♠❛ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✸✲✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❞♦s ❧✐♠✐t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✸✲✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸ ▲✐♠✐t❡s ❧❛t❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹ ▲✐♠✐t❡s ❛♦ ✐♥✜♥✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✸✲✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✺ ▲✐♠✐t❡s ✐♥✜♥✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✻ ▲✐♠✐t❡ ❞❡ ❢✉♥çõ❡s tr❛♥s❝❡♥❞❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✻✳✶ ▲✐♠✐t❡s tr✐❣♦♥♦♠étr✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✻✳✷ ▲✐♠✐t❡s ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ✐♥✈❡rs❛s ✸✳✻✳✸ ▲✐♠✐t❡ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❡ ❧♦❣❛rít♠✐❝❛ ✳ ❊①❡r❝í❝✐♦s ✸✲✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼✐s❝❡❧â♥❡❛ ✸✲✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❖◆❚■◆❯■❉❆❉❊ ✶✻✶ ✶✻✷ ✶✻✾ ✶✼✶ ✶✼✾ ✶✽✸ ✶✽✺ ✶✾✶ ✶✾✺ ✷✵✵ ✷✵✵ ✷✵✷ ✷✵✹ ✷✶✶ ✷✶✺ ✷✶✼ ✹✳✶ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✹✲✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✷ ❈♦♥t✐♥✉✐❞❛❞❡ ❡♠ ✐♥t❡r✈❛❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✷✳✶ ❋✉♥çõ❡s ❝♦♥tí♥✉❛s ❡♠ ✐♥t❡r✈❛❧♦s ❢❡❝❤❛❞♦s ❊①❡r❝í❝✐♦s ✹✲✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼✐s❝❡❧â♥❡❛ ✹✲✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ①✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✽ ✷✷✺ ✷✷✾ ✷✸✶ ✷✸✾ ✷✹✸ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✺ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ❉❊❘■❱❆❉❆❙ ✷✹✺ ✺✳✶ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✷ ❉❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✷✳✶ ❘❡t❛ t❛♥❣❡♥t❡✳ ❘❡t❛ ♥♦r♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✸ ❉❡r✐✈❛❞❛s ❧❛t❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✹ ❉❡r✐✈❛❜✐❧✐❞❛❞❡ ❡ ❝♦♥t✐♥✉✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✹✳✶ ❘❡❣r❛s ❞❡ ❞❡r✐✈❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✹✳✷ ❉❡r✐✈❛❞❛ ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✹✳✸ ❉❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ ✐♥✈❡rs❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✹✳✹ ❘❡❣r❛ ❞❛ ❝❛❞❡✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✹✳✺ ❉❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ✐♠♣❧í❝✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✺✲✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✺ ❉❡r✐✈❛❞❛ ❞❡ ❢✉♥çõ❡s tr❛♥s❝❡♥❞❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✺✳✶ ❉❡r✐✈❛❞❛ ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✺✳✷ ❉❡r✐✈❛❞❛ ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ✐♥✈❡rs❛s ✳ ✳ ✳ ✳ ✳ ✺✳✺✳✸ ❉❡r✐✈❛❞❛ ❞❛s ❢✉♥çõ❡s✿ ❊①♣♦♥❡♥❝✐❛❧ ❡ ❧♦❣❛rít♠✐❝❛ ✳ ✳ ✳ ✳ ✺✳✺✳✹ ❉❡r✐✈❛❞❛ ❞❛s ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✺✲✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✻ ❆♣r♦①✐♠❛çã♦ ❧♦❝❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✻✳✶ ❋✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✺✳✻✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✻✳✸ ❙✐❣♥✐✜❝❛❞♦ ❣❡♦♠étr✐❝♦ ❞♦ ❞✐❢❡r❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✼ ❚❡♦r❡♠❛ s♦❜r❡ ❢✉♥çõ❡s ❞❡r✐✈á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✼✳✶ ■♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ t❡♦r❡♠❛ ❞❡ ❘♦❧❧❡ ✳ ✳ ✳ ✳ ✳ ✺✳✼✳✷ ■♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦✳ ✳ ❊①❡r❝í❝✐♦s ✺✲✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼✐s❝❡❧â♥❡❛ ✺✲✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❆P▲■❈❆➬Õ❊❙ ❉❆❙ ❉❊❘■❱❆❉❆❙ ✻✳✶ ❱❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛✳ ❆❝❡❧❡r❛çã♦ ✐♥st❛♥tâ♥❡❛✳ ✻✳✶✳✶ ❱❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✶✳✷ ❆❝❡❧❡r❛çã♦ ✐♥st❛♥tâ♥❡❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✻✲✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✷ ❊st✉❞♦ ❞♦ ❣rá✜❝♦ ❞❡ ❢✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✷✳✶ ❋✉♥çã♦ ❝r❡s❝❡♥t❡✳ ❋✉♥çã♦ ❞❡❝r❡s❝❡♥t❡ ✳ ✳ ✻✳✷✳✷ ❆ssí♥t♦t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①❡r❝í❝✐♦s ✻✲✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✸ ❋♦r♠❛s ✐♥❞❡t❡r♠✐♥❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ①✐✐ ✷✹✻ ✷✹✼ ✷✺✵ ✷✺✸ ✷✺✺ ✷✺✼ ✷✻✷ ✷✻✸ ✷✻✺ ✷✻✻ ✷✻✾ ✷✼✸ ✷✼✸ ✷✼✺ ✷✼✽ ✷✼✾ ✷✽✶ ✷✽✺ ✷✽✻ ✷✽✽ ✷✽✽ ✷✾✵ ✷✾✺ ✷✾✽ ✸✵✶ ✸✵✺ ✸✵✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵✽ ✸✵✾ ✸✶✵ ✸✶✸ ✸✶✺ ✸✶✺ ✸✷✸ ✸✸✺ ✸✸✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✻✳✸✳✶ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ❋♦r♠❛s ✐♥❞❡t❡r♠✐♥❛❞❛s r❡❞✉tí✈❡✐s à ❢♦r♠❛ ❊①❡r❝í❝✐♦s ✻✲✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✹ ❆♣❧✐❝❛çõ❡s ❞✐✈❡rs❛s 0 0 ♦✉ ∞ ∞ R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺✶ ❊①❡r❝í❝✐♦s ✻✲✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻✶ ▼✐s❝❡❧â♥❡❛ ✻✲✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻✸ ❘❡❢❡rê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻✼ ❮♥❞✐❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻✽ ①✐✐✐ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ■❞❡♥t✐❞❛❞❡s ❛❧❣é❜r✐❝❛s ❈♦♥s✐❞❡r❛r a, b ∈ R ❡ m, n ∈ Z✱ ❡♥ ❣❡r❛❧ t❡♠✲s❡✿ • a a =a • • (am )n = amn • • (ab)m = am bm • • m n m+n a
m am = m, b b • b 6= 0 √ √ am/n = n am = ( n a)m , a > 0 √ √ √ n ab = n a · n b, a > 0, b > 0 p √ √ m n a = mn a, a > 0 r √ n a a n = √ , a > 0, b > 0 n b b • am = am−n an • a−n = • (a + b)2 = a2 + 2ab + b2 • (a + b)3 = a3 + 3a2 b + 3ab2 + b3 • (a − b)2 = a2 − 2ab + b2 • (a − b)3 = a3 − 3a2 b + 3ab2 − b3 • a3 − b3 = (a − b)(a2 + ab + b2 ) • a3 + b3 = (a + b)(a2 − ab + b2 ) • an − bn = (a − b)(an−1 + an−2 b + an−3 b2 + · · · + abn−2 + bn−1 ) • an + bn = (a + b)(an−1 − an−2 b + an−3 b2 − · · · − abn−2 + bn−1 ) 1 , an a 6= 0 q✉❛♥❞♦ n✲í♠♣❛r ■❞❡♥t✐❞❛❞❡s tr✐❣♦♥♦♠étr✐❝❛s ❈♦♥s✐❞❡r❛r α, β ∈ R✳ • cos(−α) = cos α sen2 α + cos2 α = 1 • senα · csc α = 1 • tan2 α + 1 = sec2 α • cos α · sec α = 1 • cot2 α + 1 = csc2 α • tan α · cot α = 1 • sen2 α = 1 − cos 2α 2 • cos2 α = • sen2α = 2senα · cos α • cos 2α = cos2 α − sen2 α • sen(α + β) = senα cos β + senβ cos α • cos(α + β) = cos α cos β − senαsenβ • tan α + tan β tan(α + β) = 1 − tan α · tan β • tan(2α) = • 2senαsenβ = cos(α − β) − cos(α + β) • tan α = • 2senα cos β = sen(α+β)+sen(α−β) • 2 cos α cos β = cos(α+β)+cos(α−β) • sen(−α) = −senα • ①✐✈ 1 + cos 2α 2 2 tan α 1 − tan2 α 1 − cos2α sen2α = sen2α 1 + cos 2α 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ■❞❡♥t✐❞❛❞❡s ❣❡♦♠étr✐❝❛s ✶✳ A❂ár❡❛✱ P ❂ ♣❡rí♠❡tr♦✱ l❂ ◗✉❛❞r❛❞♦ r ❧❛❞♦✱ ❂ r❛✐♦ ❘❡tâ♥❣✉❧♦ l ❈ír❝✉❧♦ a l b A = l2 A = πr2 A=b×a P = 2πr P = 2(a + b) P = 4l ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✷✳ A❂ár❡❛✱ P ❂ ♣❡rí♠❡tr♦✱ c❂ ❤✐♣♦t❡♥✉s❛✱ a ❡ b ❂ ❝❛t❡t♦s✱ h ❂ ❛❧t✉r❛✱ r ❂ r❛✐♦✱ α ❂ â♥❣✉❧♦ ❝❡♥tr❛❧✱ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s c ✟ ✟ ✟ ✟ ✟ ✟ ✟✟ b c2 = a2 + b2 L ❂ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡t♦r ❝✐r❝✉❧❛r ❚r✐â♥❣✉❧♦ ❙❡t♦r ❝✐r❝✉❧❛r ✑ ✑ ✑ ❆ a ✑ h ❆ ✑ ❆ ✑ ❆ ✑ c a b 1 A= b×h 2 P =a+b+c 1 A = r2 α 2 P = αr ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✸✳ A❂ár❡❛✱ P ❂ ♣❡rí♠❡tr♦✱ B ❂ ❜❛s❡ R ❂ r❛✐♦ ♠❛✐♦r✱ r ❂ r❛✐♦ ♠❡♥♦r✱ P❛r❛❧❡❧♦❣r❛♠♦ ♠❛✐♦r✱ b ❚r❛♣❡③ó✐❞❡ ❂ ❜❛s❡ ♠❡♥♦r✱ h ❂ ❛❧t✉r❛✱ ❈♦r♦❛ ❝✐r❝✉❧❛r b h h b A=b×h B ❅ ❅ ❅ 1 A = (B + b)h 2 A = π(R2 − r2 )h P = 2π(R + r) ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ①✈ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✹✳ A❂ár❡❛✱ P ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ❂ ♣❡rí♠❡tr♦✱ S❂ s✉♣❡r❢í❝✐❡ t♦t❛❧✱ V ❂ ✈♦❧✉♠❡✱ h ❂ ❛❧t✉r❛✱ r R ❂ r❛✐♦ ❚r✐â♥❣✉❧♦ ❊q✉✐❧át❡r♦ P❛r❛❧❡❧❡♣✐♣❡❞♦ r❡t♦ ❈✐❧✐♥❞r♦ ❅ ❅ l h❅l c ♣♣♣♣♣♣♣· · · · · · · · · b a ❅ l √ 3 2 l 4 √ 3 l h= 2 A= V =a×b×c S = 2(a + b)c + 2ab V = πr2 h S = 2πrh + 2πr2 ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✺✳ V ❂ ✈♦❧✉♠❡✱ h ❂ ❛❧t✉r❛✱ ❚r✐â♥❣✉❧♦ c ✑ ✑ ✑ ✑ r ❂ s✉♣❡r❢í❝✐❡ ❈♦♥❡ ❝✐r❝✉❧❛r r❡t♦ ✑ ✑ ✑ ❆ ❚r♦♥❝♦ ❞❡ ❝♦♥❡ ❆a ❆ ❆ b p A = p(p − a)(p − b)(p − c) p= S ❂ r❛✐♦✱ 1 V = πr2 h 3 √ S = πr r2 + h2 a+b+c 2 1 V = π(R2 + rR + r2 )h 3 ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✻✳ V ❂ ✈♦❧✉♠❡✱ h ❂ ❛❧t✉r❛✱ r ❂ r❛✐♦✱ S ❂ s✉♣❡r❢í❝✐❡ ❊s❢❡r❛ Pr✐s♠❛ 4 V = πr3 3 S = 4πr2 V =B×h B ❂ ár❡❛ ❞❛ ❜❛s❡ ①✈✐ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ■❞❡♥t✐❞❛❞❡s ♣❛r❛ ❞❡r✐✈❛❞❛s ❙❡❥❛♠ C❂ n ∈ Q✱ ♥❡♣❡r✐❛♥♦✱ logb x a ∈ R✱ ❝♦♥st❛♥t❡✱ Lnx❂❧♦❣❛r✐t♠♦ f (x), g(x) ❂ ❢✉♥çõ❡s✱ α❂â♥❣✉❧♦✱ ❂ ❧♦❣❛r✐t♠♦ ♥❛t✉r❛❧ ♥❛ ❜❛s❡ b✳ • Dx C = 0 • Dx (f + g) = Dx f + Dx g • Dx (f · g) = f · Dx g + g · Dx f • g · Dx f − f · Dx g f Dx ( ) = g g2 • Dx f (g(x)) = Dx f (g(x)) · Dx g • Dx [f ]n = n · Dx [f ]n−1 • Dx [ef (x) ] = ef (x) · Dx [f (x)] • Dx af = af · Dx f · Lna, • Dx (Lnf ) = • Dx (logb f ) = • Dx senx = cos x • Dx tan x = sec2 x • Dx cos x = −senx • Dx cot x = − csc2 x • Dx sec x = sec x tan x • Dx csc x = − csc x cot x • Dx arcsenx = √ • Dx arccos x = − √ • Dx arctan x = • 1 Dx arcsecx = √ x x2 − 1 1 · Dx f, f f 6= 0 1 1 − x2 1 1 + x2 a>0 1 ·Dx f, f · Lnb f 6= 0 1 1 − x2 ■❞❡♥t✐❞❛❞❡s ❞✐✈❡rs❛s • b, c ∈ R+ ✱ m ∈ Q t❡♠✲s❡✿ logb a = N ⇔ a = bN ✳ ▲♦❣♦✿ ✭✐✮ logb (a · c) = logb a + logb c✱ ✭✐✐✮ logb (a/c) = logb a − logb c✱ ✭✐✐✐✮ logb am = m logb a✱ ✭✐✈✮ logc a = logb a · logc b • P❛r❛ ♥ú♠❡r♦s ♥❛ ❜❛s❡ ❞❡❝✐♠❛❧✿ • ❊q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❣r❛✉s s❡①❛❣❡s✐♠❛✐s ❡ r❛❞✐❛♥♦s✳ ❙✉♣♦♥❤❛♠♦s α ❣r❛✉s 0o 300 45o 60o 90o α an an−1 · · · a1 a0 = 10n an +10n−1 an−1 +· · ·+10a1 +a0 senα cos α tan α cot α sec α csc α 0 0 π 6 π 4 π 3 π 2 1 √2 2 √2 3 2 1 1 √ 3 √2 2 2 1 2 0 0 √ 3 3 − √ 3 1 √ − 1 1 √ 3 3 0 r❛❞✐❛♥♦s ①✈✐✐ √ 3 − 2 3 3 √ 2 2 − 2 √ 2 √ 2 3 3 1 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❋♦r♠❛s ❞❡t❡r♠✐♥❛❞❛s ❡ ❋♦r♠❛s ✐♥❞❡t❡r♠✐♥❛❞❛s ✭❄✮ lim f (x) = lim g(x) = h(x) = lim h(x) = ❞❡ ♠♦❞♦ s✐♠❜ó❧✐❝♦ ±∞ ±∞ f (x) + g(x) ±∞ ±∞ ± ∞ = ±∞ 3 +∞ 4 −∞ K∈R x→ 1 2 5 +∞ x→ +∞ f (x) + g(x) −∞ −∞ f (x) · g(x) −∞ (+∞) · (−∞) = −∞ f (x) · g(x) −∞ (+∞) · K = −∞ f (x)/g(x) 0 ±∞ f (x)/g(x) ❄ K/ ± ∞ = 0 0+ f (x)/g(x) +∞ 7 +∞ K>0 8 +∞ K<0 9 ±∞ 0 f (x) · g(x) f (x) · g(x) f (x) · g(x) ±∞ ±∞ +∞ +∞ ❄ +∞ + K = +∞ −∞ + K = −∞ (+∞) · (+∞) = +∞ (+∞) · K = +∞ ±∞ · 0 = ❄ ±∞/ ± ∞ = ❄ K/0+ = +∞ 12 K>0 13 +∞ 0+ f (x)/g(x) +∞ +∞/0+ = +∞ 14 K>0 0− f (x)/g(x) 15 +∞ 0− f (x)/g(x) −∞ K/0+ = −∞ 16 0 0 f (x)/g(x) −∞ +∞/0− = −∞ g(x) ❄ 00 = ❄ ∞∞ = ❄ ❄ 17 0 0 [f (x)] 18 ∞ ∞ [f (x)]g(x) ❄ [f (x)]g(x) ❄ ∞ ∞ 0 [f (x)]g(x) ❄ ∞ [f (x)]g(x) ❄ 19 20 21 ❙❡❥❛ 0 1 K ∈ R✱ ◆♦ ❧✐♠✐t❡✿ ♥ã♦ ❡①✐st❡♠ ❡♠ lim x→0 ❄ K∈R +∞ 11 (+∞) − (+∞) = +∞ +∞ K ❄ f (x) + g(x) 6 10 f (x) − g(x) +∞ x→ 1 = ±∞, x 0/0 = ❄ 0∞ = ❄ ∞0 = ❄ 1∞ = ❄ K K , 00 , ✳ 0 ∞ 1 lim = 0, lim xx = 1 x→±∞ x x→0 R✿ ①✈✐✐✐ 09/02/2021 P❘❊❋➪❈■❖ ❖ ♣r♦♣ós✐t♦ ❞❡ ✉♠❛ ♣r✐♠❡✐r❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ é ❡♥s✐♥❛r ❛♦ ❡st✉❞❛♥t❡ ❛s ♥♦çõ❡s ❜ás✐❝❛s ❞❛ ❞❡r✐✈❛❞❛ ❛ss✐♠ ❝♦♠♦ ❛s té❝♥✐❝❛s ❡ ❛♣❧✐❝❛çõ❡s ❡❧❡♠❡♥t❛r❡s q✉❡ ❛❝♦♠✲ ♣❛♥❤❛♠ t❛✐s ❝♦♥❝❡✐t♦s✳ ❊st❛ ♦❜r❛ r❡♣r❡s❡♥t❛ ♦ ❡s❢♦rç♦ ❞❡ sí♥t❡s❡ ♥❛ s❡❧❡çã♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣r♦❜❧❡♠❛s q✉❡ s❡ ❛♣r❡s❡♥t❛ ❝♦♠ ❢r❡q✉ê♥❝✐❛✱ q✉❛♥❞♦ ✉♠ ❡st✉❞❛♥t❡ ❞❛ ❣r❛❞✉❛çã♦ ❝♦♠❡ç❛ ❛ ❡st✉❞❛r ❝á❧❝✉❧♦ ♥♦ ✐♥í❝✐♦ ❞♦ s❡✉s ❡st✉❞♦s✳ ❖ ♦❜❥❡t✐✈♦ ❞❡st❡ ❧✐✈r♦ é ✐♥tr♦❞✉③✐r ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ❞♦ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❡ s✉❛s ❛♣❧✐❝❛çõ❡s ❝♦♠❡ç❛♥❞♦ ❝♦♠ ✉♠❛ r❡✈✐sã♦ ❞❛ ♠❛t❡♠át✐❝❛ ❜ás✐❝❛✱ ❛ss✐♠ ❝♦♠♦ ♦r✐❡♥t❛r ❛ ♠❡t♦❞♦❧♦❣✐❛ ♣❛r❛ q✉❡ ♦ ❧❡✐t♦r ♣♦ss❛ ✐❞❡♥t✐✜❝❛r ❡ ❝♦♥str✉✐r ✉♠ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ ❡ ❧♦❣♦ r❡s♦❧✈ê✲❧♦✳ ❈❛❞❛ ❝❛♣ít✉❧♦ s❡ ✐♥✐❝✐❛ ❝♦♠ ♦s ♦❜❥❡t✐✈♦s q✉❡ s❡ ♣r❡t❡♥❞❡ ❛❧❝❛♥ç❛r❀ ❛ ❢❛rt❛ ✈❛r✐❡❞❛❞❡ ❞♦s ❡①❡♠♣❧♦s ❡ ❡①❡r❝í❝✐♦s ❛♣r❡s❡♥t❛❞♦s ❡stã♦ ❝❧❛ss✐✜❝❛❞♦s ❞❡ ♠❡♥♦r ❛ ♠❛✐♦r ❞✐✜❝✉❧❞❛❞❡✳ ❆ ✈❛r✐❡❞❛❞❡ ❞♦s ♣r♦❜❧❡♠❛s ❡ ❡①❡r❝í❝✐♦s ♣r♦♣♦st♦s ♣r❡t❡♥❞❡ tr❛♥s♠✐t✐r ♣❛rt❡ ❞❡ ♠✐♥❤❛ ❡①♣❡r✐ê♥❝✐❛ ♣r♦✜ss✐♦♥❛❧ ❞✉r❛♥t❡ ♠✉✐t♦s ❛♥♦s ❞❡ ❡①❡r❝í❝✐♦ ❝♦♠♦ ♣r♦❢❡ss♦r ❞❡ ❡♥s✐♥♦ s✉♣❡r✐♦r ❛ss✐♠✱ ❝♦♠♦ ❈♦♥s✉❧t♦r ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛✱ ❝♦♠ ❛t✉❛çã♦ ♥❛ ❣r❛❞✉❛çã♦ ❡ ♣ós✲ ❣r❛❞✉❛çã♦ ❞❛ ❞♦❝ê♥❝✐❛ ✉♥✐✈❡rs✐tár✐❛✳ ❋✐❝♦ ♣r♦❢✉♥❞❛♠❡♥t❡ ❣r❛t♦ ❝♦♠ ♦s ❡st✉❞❛♥t❡s ❞♦s ❞✐✈❡rs♦s ❝✉rs♦s ♦♥❞❡ ❞✐❢✉♥❞✐ ❛s ✐❞❡✐❛s ❡ ♦ ❝♦♥t❡ú❞♦ ❞❛s ♥♦t❛s ❞❡st❡ tr❛❜❛❧❤♦✳ ❚❛♠❜é♠ ❛❣r❛❞❡ç♦ ❛s ❝♦♥tr✐❜✉✐çõ❡s ❡ s✉❣❡stõ❡s ❞♦s ❧❡✐t♦r❡s✱ ❡♠ ♣❛rt✐❝✉❧❛r ❞♦s ♠❡✉s ❝♦❧❡❣❛s✱ ♣❡❧❛ s✉❛ ❝♦♥st❛♥t❡ ❞❡❞✐❝❛çã♦ ♣❛r❛ ❛ r❡✈✐sã♦ ❡ ❞✐s❝✉ssã♦ ❞♦s ♣r♦❜❧❡♠❛s ♣r♦♣♦st♦s✳ ❆t✉❛❧♠❡♥t❡ ❡stá ❡♠ ❝♦♥str✉çã♦ ♦ ❧✐✈r♦ ✏ ❙✉♣❧❡♠❡♥t♦ ❞❡ ❈á❧❝✉❧♦ I ✑✱ ♦♥❞❡ s❡ ❡♥❝♦♥tr❛ ❛ s♦❧✉çã♦ ❞❡ t♦❞♦s ♦s ❡①❡r❝í❝✐♦s ♣r♦♣♦st♦s ♥❡st❡ ❧✐✈r♦ ❡ ♣♦❞❡ s❡r ♦❜t✐❞♦ s♦❧✐❝✐t❛♥❞♦ ✉♠❛ ❝ó♣✐❛ ❛♦ ❛✉t♦r ❡♠✿ ❝❤r✐st✐❛♥❥q♣❅②❛❤♦♦✳❝♦♠✳❜r ✳ ❈❤r✐st✐❛♥ ◗✉✐♥t❛♥❛ P✐♥❡❞♦✳ P❛❧♠❛s ✲ ❚❖✱ ▼❛rç♦ ❞❡ ①✐① 2020 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❆ ♠❛t❡♠át✐❝❛ ❛♣r❡s❡♥t❛ ✐♥✈❡♥çõ❡s tã♦ s✉t✐s q✉❡ ♣♦❞❡rã♦ s❡r✈✐r ♥ã♦ só ♣❛r❛ s❛t✐s❢❛③❡r ♦s ❝✉r✐♦s♦s ❝♦♠♦✱ t❛♠❜é♠ ♣❛r❛ ❛✉①✐❧✐❛r ❛s ❛rt❡s ❡ ♣♦✉♣❛r tr❛❜❛❧❤♦ ❛♦s ❤♦♠❡♥s✑✳ ✏ ❘✳ ❉❡s❝❛rt❡s (1596 − 1650) ◆ã♦ ❛❞✐❛♥t❛ t❡r ✉♠ ♠❛r ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s✱ ❝♦♠ ❛ ♣r♦❢✉♥❞❡③❛ ❞❡ ✉♠ ♠✐✲ ❧í♠❡tr♦✳✑ ✏ ❈❤✳ ◗✳ P✐♥❡❞♦ (1954−) Pr♦❢❡ss♦r❡s t❡♥❞❡♠ à ❡t❡r♥✐❞❛❞❡❀ ♥✉♥❝❛ ♣♦❞❡rã♦ s❛❜❡r ♦♥❞❡ t❡r♠✐♥❛ s✉❛ ✐♥✢✉❡♥❝✐❛✳✑ ✏ ✶ ❍❡♥r② ❆❞❛♠s (1838 − 1918) ✶ ❍❡♥r② ❇r♦♦❦s ❆❞❛♠s ✭1838 − 1918✮✱ ❢♦✐ ✉♠ ❡st❛❞✉♥✐❞❡♥s❡ ❤✐st♦r✐❛❞♦r✱ ❥♦r♥❛❧✐st❛ ❡ ♥♦✈❡❧✐st❛✳ ①① 09/02/2021 ❈❛♣ít✉❧♦ ✶ ❙■❙❚❊▼❆ ❉❖❙ ◆Ú▼❊❘❖❙ ❘❊❆■❙ ❊r❛tóst❡♥❡s ♥❛s❝❡✉ ❡♠ ❈✐r❡♥❡ ✭276 a.C. − 197 a.C.✮✱ ♦ q✉❡ ❤♦❥❡ é ❛ ▲í❜✐❛✳ ❉❡♣♦✐s ❞❡ ❡st✉❞❛r ❡♠ ❆❧❡①❛♥❞r✐❛ ❡ ❆t❡♥❛s✱ ❡❧❡ s❡ t♦r♥♦✉ ❞✐r❡t♦r ❞❛ ❢❛♠♦s❛ ❇✐❜❧✐♦t❡❝❛ ❞❡ ❆❧❡①❛♥❞r✐❛✳ ❊❧❡ tr❛❜❛❧❤♦✉ ❝♦♠ ❣❡♦♠❡tr✐❛ ❡ ♥ú♠❡r♦s ♣r✐♠♦s✳ ❊r❛tóst❡♥❡s é ♠❛✐s ❝♦♥❤❡❝✐❞♦ ♣❡❧♦ s❡✉ ❝r✐✈♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ✭♦ ✏❈r✐✈♦ ❞❡ ❊r❛tóst❡♥❡s✑✮✱ ♦ q✉❛❧✱ ❝♦♠ ❛❧❣✉♠❛s ♠♦❞✐✜❝❛çõ❡s✱ ❛✐♥❞❛ é ✉♠ ✐♥str✉♠❡♥t♦ ✐♠♣♦rt❛♥t❡ ❞❡ ♣❡sq✉✐s❛ ♥❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✳ ❊r❛tóst❡♥❡s t❛♠❜é♠ ❢❡③ ✉♠❛ ♠❡❞✐çã♦ ❡①tr❡♠❛♠❡♥t❡ ♣r❡❝✐s❛ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❛ ❚❡rr❛✱ ❝♦♠♣❛r❛♥❞♦ ❛s s♦♠❜r❛s ♣r♦❞✉③✐❞❛s ♣❡❧♦ ❙♦❧ ❞♦ ♠❡✐♦✲❞✐❛✱ ♥♦ ✈❡rã♦✱ ❡♠ ❙✐❡♥❛ ❡ ❆❧❡①❛♥❞r✐❛✳ ❊r❛tóst❡♥❡s ❝❛❧❝✉❧♦✉ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❛ ❚❡rr❛ ❡♠ tâ♥❝✐❛ ❛té ♦ ❙♦❧ ❡♠ à ▲✉❛ ❡♠ 780.000 804.000.000 250.000 ❊❧❡ ✶ ❡stá❞✐♦s ✱ ❛ ❞✐s✲ ❡stá❞✐♦s ❡ ❛ ❞✐stâ♥❝✐❛ ❞❛ ❚❡rr❛ ❡stá❞✐♦s ✳ ❊r❛tóst❡♥❡s t❛♠❜é♠ ♠❡❞✐✉ ❛ ✐♥❝❧✐♥❛çã♦ ❞♦ ❡✐①♦ ❞❛ ❚❡rr❛ ❝♦♠ ❣r❛♥❞❡ ♣r❡❝✐sã♦✱ ❡♥❝♦♥tr❛♥❞♦ ♦ 23 ❣r❛✉s✱ 51′ 15′′ ✳ ❚❛♠❜é♠ ♦r❣❛♥✐③♦✉ ✉♠ ❝❛tá❧♦❣♦ ❛str♦♥ô♠✐❝♦✱ ❝♦♥t❡♥❞♦ 675 ❡str❡❧❛s✳ ✈❛❧♦r ❞❡ ❊r❛tóst❡♥❡s ✜❝♦✉ ❝❡❣♦ ❡♠ ✐❞❛❞❡ ❛✈❛♥ç❛❞❛ ❡ ❞✐③✲s❡ q✉❡ t❡r✐❛ ❝♦♠❡t✐❞♦ s✉✐❝í❞✐♦✱ r❡❝✉s❛♥❞♦✲s❡ ❛ ❝♦♠❡r ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ♠♦rr❡♥❞♦ ❞❡ ✐♥❛♥✐çã♦✳ ❆ ♣❛❧❛✈r❛ ✏❝r✐✈♦✑ s✐❣♥✐✜❝❛ ♣❡♥❡✐r❛✳ ❖ q✉❡ ❊r❛tóst❡♥❡s ✐♠❛❣✐♥♦✉ ❢♦✐ ✉♠❛ ✏♣❡♥❡✐r❛✑ ❝❛♣❛③ ❞❡ s❡♣❛r❛r ♦s ♥ú♠❡r♦s ♣r✐♠♦s ❞♦s ❝♦♠♣♦st♦s✳ ♥ú♠❡r♦ ♣r✐♠♦ p ❆ ✐❞❡✐❛ ❞♦ ❊r❛tóst❡♥❡s ❢♦✐ ❛ s❡❣✉✐♥t❡✿ ❥á q✉❡ ✉♠ é ❛q✉❡❧❡ q✉❡ s♦♠❡♥t❡ ♣♦ss✉✐ ❞♦✐s ❞✐✈✐s♦r❡s ✐♥t❡✐r♦s ✲ ♦ 1 ❡ ♦ ♣ró♣r✐♦ p ✲ ♣♦❞❡r✐❛ ❤❛✈❡r ✉♠❛ ♣❡♥❡✐r❛ q✉❡ ♣✉❞❡ss❡ s❡♣❛r❛r ❡st❡s ♥ú♠❡r♦s ✭q✉❡ só tê♠ ❞♦✐s ❞✐✈✐s♦r❡s✱ ❡ ♣♦rt❛♥t♦ sã♦ ♣r✐♠♦s✮ ❞♦s ♦✉tr♦s✱ q✉❡ ♣♦ss✉❡♠ ♠❛✐s ❞❡ ❞♦✐s ❞✐✈✐s♦r❡s ✭❡ sã♦ ❝❤❛♠❛❞♦s ❞❡ ✏❝♦♠♣♦st♦s✑✮✳ ✶✳✶ ■♥tr♦❞✉çã♦ ❆ ♠❛t❡♠át✐❝❛ ❛ s❡r ❡st✉❞❛❞❛ ♥♦s ♣r✐♠❡✐r♦s ❝✉rs♦s ❞❛ ❣r❛❞✉❛çã♦ ❡stá ✐♥s♣✐r❛❞❛ ❡♠ ❞✉❛s ❢♦♥t❡s✿ • ❆ ♣r✐♠❡✐r❛ é ❛ ✏ ❧ó❣✐❝❛ ♠❛t❡♠át✐❝❛ ✑❀ ❡❧❛ s❡ ❞❡s❡♥✈♦❧✈❡ ♣♦r ♠❡✐♦ ❞❡ ♣r♦♣♦s✐çõ❡s ✭❢r❛✲ s❡s✮✱ às q✉❛✐s ♣♦❞❡♠♦s ❛tr✐❜✉✐r ✉♠ ✈❛❧♦r ❧ó❣✐❝♦ ❞❡ ✈❡r❞❛❞❡ ♦✉ ❞❡ ❢❛❧s✐❞❛❞❡ ✭s♦♠❡♥t❡ ✶ ❊stá❞✐♦ ❡r❛ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ♥❛ ●r❡❝✐❛✱ ❡q✉✐✈❛❧❡♥t❡ ❛ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✶✽✵♠ ❞❡ ❝♦♠♣r✐♠❡♥t♦✳ ✶ ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✉♠ ❞❡st❡s ✈❛❧♦r❡s✮✳ P♦r ❡①❡♠♣❧♦✿ ✕ ❆ t❡rr❛ t❡♠ ❛ ❢♦r♠❛ ❛rr❡❞♦♥❞❛❞❛ ✭ ✈ ✕ ❆ t❡rr❛ é ❞❡ ❢♦r♠❛ q✉❛❞r❛❞❛ ✭ ❢ ❂ ✈❡r❞❛❞❡✮✳ ❂ ❢❛❧s♦✮ ◆❛ ❧ó❣✐❝❛ ♠❛t❡♠át✐❝❛✱ ❛ ♥❡❣❛çã♦ ❞❡ ✉♠❛ ♣r♦♣♦s✐çã♦ ♥ã♦ ✐♠♣❧✐❝❛ ♥❛ ❛✜r♠❛çã♦ ❞♦ ❝♦♥trár✐♦✳ • ❆ s❡❣✉♥❞❛ é ♦ ✏❝á❧❝✉❧♦ ✑❀ ✐st♦ s❡rá ♦❜❥❡t♦ ❞❡ ♥♦ss♦ ❡st✉❞♦✳ ❖ ❡st✉❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❞♦ ❝á❧❝✉❧♦ ❡stá ♦r✐❡♥t❛❞♦ s♦❜ ❝♦♥❝❡✐t♦s ❞❡ ❞✐❢❡r❡♥❝✐❛çã♦✱ ✐♥t❡✲ ❣r❛çã♦ ❡ s✉❛s ❛♣❧✐❝❛çõ❡s ❡♠ ❞✐✈❡rs♦s ❝❛♠♣♦s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦✳ P♦r ❡①❡♠♣❧♦✿ • ❯♠ ❢❛❜r✐❝❛♥t❡ ❞❡ ❝❛✐①❛s ❞❡ ♣❛♣❡❧ã♦ ❞❡s❡❥❛ ♣r♦❞✉③✐r ❝❛✐①❛s s❡♠ t❛♠♣❛✱ ✉s❛♥❞♦ ♣❡❞❛✲ ç♦s q✉❛❞r❛❞♦s ❞❡ ♣❛♣❡❧ã♦ ❝♦♠ 40 cm ❞❡ ❧❛❞♦✱ ❝♦rt❛♥❞♦ q✉❛❞r❛❞♦s ✐❣✉❛✐s ♥♦s q✉❛tr♦ ❝❛♥t♦s ❡ ✈✐r❛♥❞♦ ✈❡rt✐❝❛❧♠❡♥t❡ ✭♣❛r❛ ❝✐♠❛✮ ♦s q✉❛tr♦ ❧❛❞♦s✳ ❆❝❤❛r ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦s ❧❛❞♦s ❞♦s q✉❛❞r❛❞♦s ❛ s❡r❡♠ ❝♦rt❛❞♦s ❛ ✜♠ ❞♦ ♦❜t❡r ✉♠❛ ❝❛✐①❛ ❝♦♠ ♦ ♠❛✐♦r ✈♦❧✉♠❡ ♣♦ssí✈❡❧✳ • ❯♠ ❞✐str✐❜✉✐❞♦r ❛t❛❝❛❞✐st❛ t❡♠ ✉♠ ♣❡❞✐❞♦ ❞❡ • ❙✉♣♦♥❤❛♠♦s q✉❡ ✉♠ t✉♠♦r ♥♦ ❝♦r♣♦ ❞❡ ✉♠ ♣♦r❝♦ t❡♥❤❛ ❢♦r♠❛ ❡s❢ér✐❝❛✳ ◗✉❛♥❞♦ ♦ 30.000 ❝❛✐①❛s ❞❡ ❧❡✐t❡ q✉❡ ❝❤❡❣❛♠ ❛ 5 s❡♠❛♥❛s✳ ❆s ❝❛✐①❛s sã♦ ❞❡s♣❛❝❤❛❞❛s ♣❡❧♦ ❞✐str✐❜✉✐❞♦r ❛ ✉♠❛ r❛③ã♦ ❝♦♥st❛♥t❡ ❞❡ 1.800 ❝❛✐①❛s ♣♦r s❡♠❛♥❛✳ ❙❡ ❛ ❛r♠❛③❡♥❛❣❡♠ ♥✉♠❛ s❡♠❛♥❛ ❝✉st❛ ❘$ 0, 05 ♣♦r ❝❛✐①❛ ✳ ◗✉❛❧ é ♦ ❝✉st♦ t♦t❛❧ ❞❡ ♠❛♥✉t❡♥çã♦ ❞♦ ❡st♦q✉❡ ❞✉r❛♥t❡ 10 s❡♠❛♥❛s ❄ ❝❛❞❛ r❛✐♦ ❞♦ t✉♠♦r é ❞❡ 0, 5 cm✱ 0, 0001 cm ✐♥st❛♥t❡ t0 ❄ ❛ t❛①❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞♦ r❛✐♦ é ❞❡ ◗✉❛❧ é ❛ t❛①❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞♦ ✈♦❧✉♠❡ ❞♦ t✉♠♦r ❡♠ ❛❧❣✉♠ ♣♦r ❞✐❛✳ ❖s ♣r♦❜❧❡♠❛s ❞♦ ❡①❡♠♣❧♦ ❛❝✐♠❛ ♣♦❞❡♠ s❡r r❡s♦❧✈✐❞♦s ❝♦♠ té❝♥✐❝❛s ❡ ♦♣❡r❛çõ❡s ❝♦♠ ♥ú♠❡r♦s r❡❛✐s✳ P❛r❛ ❝♦♠♣r❡❡♥❞❡r ❜❡♠ ❛s ♦♣❡r❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s ❞♦ ❝á❧❝✉❧♦✱ ❡st✉❞❛r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ❜❡♠ ❝♦♠♦ ❛s ♦♣❡r❛çõ❡s ♣❡r♠✐t✐❞❛s ❝♦♠ ♦s ♠❡s♠♦s✳ ✶✳✷ ❙✐st❡♠❛ ❞♦s ♥ú♠❡r♦s r❡❛✐s ❯♠ ♥✉♠❡r❛❧ é ✉♠ sí♠❜♦❧♦ ♦✉ ❣r✉♣♦ ❞❡ sí♠❜♦❧♦s q✉❡ r❡♣r❡s❡♥t♦✉ ✉♠ ♥ú♠❡r♦ ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ✐♥st❛♥t❡ ❞❛ ❡✈♦❧✉çã♦ ❞♦ ❤♦♠❡♠✳ ❊♠ ❛❧❣✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❡s❝r✐t❛ ♦✉ é♣♦❝❛✱ ♦s ♥✉♠❡r❛✐s ❞✐❢❡r❡♥❝✐❛r❛♠✲s❡ ❞♦s ♥ú♠❡r♦s✱ ❞♦ ♠❡s♠♦ ♠♦❞♦ q✉❡ ❛s ♣❛❧❛✈r❛s s❡ ❞✐❢❡r❡♥❝✐❛r❛♠ ❞❛s ❝♦✐s❛s às q✉❡ s❡ r❡❢❡r❡♠✳ ❖s sí♠❜♦❧♦s ✏ 12✑✱ ✏❞♦③❡✑ ❡ ✏ XII ✑ ✭❞♦③❡ ❡♠ ▲❛t✐♠✮ sã♦ ♥✉♠❡r❛✐s ❞✐❢❡r❡♥t❡s r❡♣r❡s❡♥t❛t✐✈♦s ❞♦ ♠❡s♠♦ ♥ú♠❡r♦✱ ❛♣❡♥❛s ❡s❝r✐t♦ ❡♠ ✐❞✐♦♠❛s ❡ é♣♦❝❛s ❞✐❢❡r❡♥t❡s✳ ✷ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❖s ♥ú♠❡r♦s r❡♣r❡s❡♥t❛♠ ♣❛♣❡❧ ✈✐t❛❧ ♥ã♦ só ♥❛ ♠❛t❡♠át✐❝❛✱ ❝♦♠♦ ♥❛ ❝✐ê♥❝✐❛ ❞❡ ✉♠ ♠♦❞♦ ❣❡r❛❧✱ ❡ ♥❛ ♥♦ss❛ ✈✐❞❛ ❞✐ár✐❛✳ ❱✐✈❡♠♦s ❝❡r❝❛❞♦s ❞❡ ♥ú♠❡r♦s✱ ❤♦rár✐♦s✱ t❛❜❡❧❛s✱ ❣rá✲ ✜❝♦s✱ ♣r❡ç♦s✱ ❥✉r♦s✱ ✐♠♣♦st♦s✱ ✈❡❧♦❝✐❞❛❞❡s✱ ❞✐stâ♥❝✐❛s✱ t❡♠♣❡r❛t✉r❛s✱ r❡s✉❧t❛❞♦s ❞❡ ❥♦❣♦s✱ ❡t❝✳ ❆ ♠❛✐♦r ♣❛rt❡ ❞❛s q✉❛♥t✐❞❛❞❡s ❡st✉❞❛❞❛s ♥❡st❛s ♥♦t❛s ✭ár❡❛s✱ ✈♦❧✉♠❡s✱ t❛①❛s ❞❡ ✈❛r✐✲ ❛çã♦✱ ✈❡❧♦❝✐❞❛❞❡s✱ ✳ ✳ ✳ ✮ sã♦ ♠❡❞✐❞❛s ♣♦r ♠❡✐♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s ❡ ♥❡ss❡ s❡♥t✐❞♦ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ♥♦ss♦ ✏❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧✑ s❡rá tr❛❜❛❧❤❛❞♦ ♥♦ s✐st❡♠❛ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳ ❖ ❡st✉❞♦ ❞♦ s✐st❡♠❛ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♣❡❧♦ ♠ét♦❞♦ ❛①✐♦♠át✐❝♦✱ ❝♦♥s✐st❡ ❡♠ ❞❡✜♥✐r ❡st❡ ✏ s✐st❡♠❛ ♥✉♠ér✐❝♦ ✑ ♠❡❞✐❛♥t❡ ✉♠ ❣r✉♣♦ ❞❡ ❛①✐♦♠❛s✱ ❞❡ ♠♦❞♦ q✉❡ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s✿ ♥❛t✉r❛✐s✱ ✐♥t❡✐r♦s✱ r❛❝✐♦♥❛✐s ❡ ✐rr❛❝✐♦♥❛✐s s❡❥❛♠ ❢♦r♠❛❞♦s ♣♦r s✉❜❝♦♥❥✉♥t♦s ♣ró♣r✐♦s ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s R✳ ❍❛ ♦✉tr♦ ♠♦❞♦ ❞❡ s❡ ❡st✉❞❛r ♦s ♥ú♠❡r♦s r❡❛✐s❀ ♣♦❞❡♠♦s ❞❡✜♥✐✲❧♦s ❡♠ t❡r♠♦s ❞❡ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ ✉s❛♥❞♦ ♦s ❝❧áss✐❝♦s ❝♦rt❡s ❞❡ ❉❡❞❡❦✐♥❞ P♦ré♠✱ ♣❛r❛ ♦ ♥♦ss♦ ❡st✉❞♦ ❞♦ ✲ ✏❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✷ ✸ ♦✉ ❛s s✉❝❡ssõ❡s ❞❡ ❈❛✉❝❤② ✳ R✑ ✲ é s✉✜❝✐❡♥t❡ ✐♥tr♦❞✉③✐r ♦ s✐st❡♠❛ ♣❡❧♦ ♠ét♦❞♦ ❛①✐♦♠át✐❝♦✳ ❈♦♥s✐❞❡r❡♠♦s ♦s s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s✿ N ❂ Z ❂ {0, 1, 2, 3, 4, 5, · · · , n · · · , } ♥❛t✉r❛✐s✳ { · · · , −3, −2, −1, 0, 1, 2, 3, 4, · · · } a Q ❂ { /. a, b ∈ Z, b 6= 0} b 5 11 3 Q = { · · · , −2, · · · − , · · · , −1, 0, 1, , 3, , · · · } 2 2 4 √ √ √ 3 I = {± 2, ±π, ±e, ± 7, 5, · · · } S R=Q I √ C = {a + bi; a, b ∈ R ♦♥❞❡ i = −1 } ✐♥t❡✐r♦s✳ r❛❝✐♦♥❛✐s✳ r❛❝✐♦♥❛✐s✳ ✐rr❛❝✐♦♥❛✐s✳ r❡❛✐s✳ ❝♦♠♣❧❡①♦s C = {1 + 2i, 3 + 2i, 5 − 4i, −1 − i, i, 2, 8i, 7, · · · } ❝♦♠♣❧❡①♦s ◗✉❛❧q✉❡r ♥ú♠❡r♦ r❡❛❧ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ♦✉ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧✳ ❊st❡s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❝♦♥s✐st❡♠ ❞♦s s❡❣✉✐♥t❡s✿ ❛✮ ❖s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✱ ♥❡❣❛t✐✈♦s ❡ ♦ ③❡r♦✿ · · · − 6, −5, −4, · · · , −1, 0, 1, 2, 3, · · · , 12, 13, 14, · · · ✳ ❜✮ ❆s ❢r❛çõ❡s ♣♦s✐t✐✈❛s ❡ ♥❡❣❛t✐✈❛s✿ ✷ ❘✐❝❤❛r❞ ❉❡❞❡❦✐♥❞ ✭1831 − 1916✮ ❢♦✐ ❛❧✉♥♦ ❞❡ ❈❛r❧ ❋✳ ●❛✉ss ✭1777 − 1855✮ ❡ ❉✐r✐❝❤❧❡t (1805 − 1859)✳ ❊st✉❞♦✉ ♦ ♣r♦❜❧❡♠❛ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✱ ❡ é ♠❛✐s ❝♦♥❤❡❝✐❞♦ ♣❡❧♦ s❡✉ tr❛❜❛❧❤♦ ♥♦s ❢✉♥❞❛♠❡♥t♦s ❞♦ s✐st❡♠❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s✳ ✸ ❆✉❣✉st✐♥ ❈❛✉❝❤② ✭1789 − 1857✮ ❢♦✐ ♦ ❢✉♥❞❛❞♦r ❞❛ ❛♥á❧✐s❡ ♠♦❞❡r♥❛✱ ❛♣♦rt♦✉ ✐♠♣♦rt❛♥t❡s r❡s✉❧t❛❞♦s ❡♠ ♦✉tr❛s ár❡❛s ❞❛ ♠❛t❡♠át✐❝❛✳ ❆❧é♠ ❞❡ s✉❛s ❛t✐✈✐❞❛❞❡s ♣♦❧ít✐❝❛s ❡ r❡❧✐❣✐♦s❛s✱ ❡s❝r❡✈❡✉ 759 tr❛❜❛❧❤♦s ❡♠ ♠❛t❡♠át✐❝❛✳ ✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 8 1 96 8 13 ··· − , ··· − , ··· , ··· , , , ···✳ 5 2 15 5 14 ❝✮ ❖s ♥ú♠❡r♦s ❞❡❝✐♠❛✐s ❧✐♠✐t❛❞♦s ✭♣♦s✐t✐✈♦s ❡ ♥❡❣❛t✐✈♦s✮✿ 5, 37 = ❞✮ 537 ✱ 100 −3, 2841 = − 32.841 ✱ 10.000 0, 528 = 528 1.000 ❖s ♥ú♠❡r♦s ❞❡❝✐♠❛✐s ✐❧✐♠✐t❛❞♦s ✭♣♦s✐t✐✈♦s ❡ ♥❡❣❛t✐✈♦s✮✿ 3 745 0, 333333 · · · ≈ ✱ −3, 745745745 · · · ≈ −3 − 2, 5858585858 · · · ≈ 9 999 9 58 8, 9999999 · · · ≈ 8 + 2+ ✱ 99 9 ❖ sí♠❜♦❧♦ ≈ s✐❣♥✐✜❝❛ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✳ ❖❜s❡r✈❡✿ 9 ❙❡ ❝♦♥s✐❞❡r❡♠♦s ❛ r❡❧❛çã♦ 0, 999999 · · · = = 1 ✐st♦ é ✉♠ ❛❜s✉r❞♦ ❥á q✉❡ ♦ ♥ú♠❡r♦ 1 9 é ✐♥t❡✐r♦ ❡ 0, 999999 · · · é ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ❞❡❝✐♠❛❧ ❝♦♠ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ❛❧❣❛r✐s♠♦s 9 ♥♦✈❡✳ ❆ss✐♠ é ♠❡❧❤♦r ❡♥t❡♥❞❡r q✉❡ 0, 999999 · · · ≈ = 1 9 • ❖s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s sã♦ ❛q✉❡❧❡s ♥ú♠❡r♦s ❞❡❝✐♠❛✐s ♥ã♦ ♣❡r✐ó❞✐❝♦s✳ P♦r ❡①❡♠♣❧♦✿ √ √ 5 = 2, 2360679774997896 · · · ❀ 19 = 4, 35889894354067 · · · √ 3 π = 3, 14159265358979323846 · · · ❀ ✲ 28 = −3, 03658897187 · · · ❆ ❋✐❣✉r❛ ✭✶✳✶✮ ♠♦str❛ ♠❡❞✐❛♥t❡ ❞✐❛❣r❛♠❛s ❞❡ ❱❡♥♥✹ ❛ r❡❧❛çã♦ ❞❡ ✐♥❝❧✉sã♦ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s✳ C I ✬ ✫ ✬ ✩ Z ✪ ✫ R ✩ Q N ✪ ❋✐❣✉r❛ ✶✳✶✿ ❈♦♥❥✉♥t♦ ◆✉♠ér✐❝♦ ◆♦t❛çõ❡s✿ N+ = N − {0} = { 1, 2, 3, 4, 5, · · · , n, · · · } Z+ = {1, 2, 3, 4, 5, · · · } ➱ ✐♠♣♦rt❛♥t❡ ❞❡st❛❝❛r q✉❡ ♦ ♥ú♠❡r♦ ③❡r♦ ♥ã♦ é ♥ú♠❡r♦ ♣♦s✐t✐✈♦ ♥❡♠ ♥❡❣❛t✐✈♦✳ ❙✉♣♦♥❤❛ t❡♠♦s q✉❡ r❡❛❧✐③❛r ♦♣❡r❛çõ❡s ❛r✐t♠ét✐❝❛s ❡❧❡♠❡♥t❛r❡s ✭❛❞✐çã♦✱ s✉❜tr❛çã♦✱ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❞✐✈✐sã♦✱ ♣♦t❡♥❝✐❛çã♦ ❡ r❛❞✐❝❛çã♦✮ ❝♦♠ ❞♦✐s ♥ú♠❡r♦s q✉❛✐sq✉❡r ❞❡ ✉♠ s✉❜✲ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ❡ ❞❡s❡❥❛♠♦s q✉❡ ♦ r❡s✉❧t❛❞♦ ♣❡rt❡♥ç❛ ❛♦ ♠❡s♠♦ s✉❜❝♦♥❥✉♥t♦✳ ❖❜s❡r✈❡✱ ❝♦♠ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s 4 ❡ 7 ♥ã♦ é ♣♦ssí✈❡❧ ❡❢❡t✉❛r ❛ ♦♣❡r❛çã♦ 4 − 7 ✭s✉❜tr❛çã♦✮✱ ♣♦✐s s❛❜❡♠♦s q✉❡ 4 − 7 = −3 ♥ã♦ ♣❡rt❡♥❝❡ ❛♦ ❝♦♥❥✉♥t♦ N✳ ❆ss✐♠✱ ❡♠ ❣❡r❛❧ t❡♠♦s q✉❡ ♥♦ ❝♦♥❥✉♥t♦ ♥✉♠ér✐❝♦✿ ❡♠ ✹ ❏♦❤♥ ❱❡♥♥ ✭1834 − 1923✮ ♣✉❜❧✐❝♦✉ ✏▲ó❣✐❝❛ ❙✐♠❜ó❧✐❝❛✑ ❡♠ 1889✳ 1881 ❡✱ ✏❖s Pr✐♥❝í♣✐♦s ❞❡ ▲ó❣✐❝❛ ❊♠♣ír✐❝❛✑ ❖ s❡❣✉♥❞♦ ❞❡st❡s é ♠❡♥♦s ♦r✐❣✐♥❛❧ q✉❡ ♦ ♣r✐♠❡✐r♦✱ ♣♦ré♠ é ❞❡s❝r✐t♦ ❝♦♠♦ ♦ tr❛❜❛❧❤♦ ♠❛✐s ❞✉r❛❞♦✉r♦ ❡♠ ❧ó❣✐❝❛✳ ✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R N s♦♠❡♥t❡ é ♣♦ssí✈❡❧ ❡❢❡t✉❛r ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦✳ Z s♦♠❡♥t❡ é ♣♦ssí✈❡❧ ❡❢❡t✉❛r ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦✱ s✉❜tr❛çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦✳ Q é ♣♦ssí✈❡❧ ❡❢❡t✉❛r ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦✱ s✉❜tr❛çã♦ ✱ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❞✐✈✐sã♦ ✭❞❡s❞❡ q✉❡ ♦ ❞✐✈✐s♦r ♥ã♦ s❡❥❛ ③❡r♦✮✳ I R é ♣♦ssí✈❡❧ ❡❢❡t✉❛r ♦♣❡r❛çõ❡s ❞❡ ♠♦❞♦ r❡str✐t♦✳ ♣♦❞❡♠♦s ❡❢❡t✉❛r ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦✱ s✉❜tr❛çã♦✱ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❞✐✈✐sã♦ ✭❞❡s❞❡ q✉❡ ♦ ❞✐✈✐s♦r ♥ã♦ s❡❥❛ ③❡r♦✮✳ C é ♣♦ssí✈❡❧ ❡❢❡t✉❛r ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦✱ s✉❜tr❛çã♦✱ ❞✐✈✐sã♦ ✭❝♦♠ ❞✐✈✐s♦r ♥ã♦ ③❡r♦✮✱ ♠✉❧✲ t✐♣❧✐❝❛çã♦✱ ♣♦t❡♥❝✐❛çã♦ ❡ r❛❞✐❝❛çã♦✳ C ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ♥ú♠❡r♦s r❡❛✐s ❞♦ ❝♦♥❥✉♥t♦ R✳ R✳ t❡♠ ♠❛✐s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ◆♦ss♦ ♦❜❥❡t✐✈♦ ♥❡st❡ ❝❛♣ít✉❧♦ s❡rá ❡st✉❞❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ✐♠♣♦rt❛♥t❡s ▼♦str❛✲s❡ q✉❡✱ Q T I = ∅✳ x ∈ R é ♣♦ssí✈❡❧ ❛ss♦❝✐❛r ✉♠ ♣♦♥t♦ ❞❡ ✉♠❛ r❡t❛✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ❡st❡ ♥ú♠❡r♦ r❡❛❧ x ❝♦rr❡s♣♦♥❞❛ ✉♠✱ ❡ s♦♠❡♥t❡ ✉♠✱ ú♥✐❝♦ ♣♦♥t♦ P ❝♦♠♦ ✐♥❞✐❝❛ ❛ ❋✐❣✉r❛ ✭✶✳✷✮✳ ❆♦s ❡❧❡♠❡♥t♦s ❞❡ ✛ −∞ ··· r −4 r −3 r −2 r −1 r 0 r 1 r r r R x 2 3 ··· ✲ +∞ ❋✐❣✉r❛ ✶✳✷✿ ❘❡t❛ ♥✉♠ér✐❝❛ ❉❡✜♥✐çã♦ ✶✳✶✳ ❙✐st❡♠❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s✳ ❉✐③❡♠♦s ✏ s✐st❡♠❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s✑ ❛♦ ❝♦♥❥✉♥t♦ R✱ ♥♦ q✉❛❧ ❡stã♦ ❞❡✜♥✐❞❛s ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ✭+✮✱ ♠✉❧t✐♣❧✐❝❛çã♦ ✭⋆✮✱ ✉♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠ ✭< ✮ q✉❡ s❡ ❧ê ✏ ♠❡♥♦r q✉❡✑ ❡ ♦ ❛①✐♦♠❛ ❞♦ s✉♣r❡♠♦✳ ❖ s✐st❡♠❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s ♣♦❞❡ s❡r ❞❡♥♦t❛❞♦ ❝♦♠♦ ✭R, ❡s❝r❡✈❡✲s❡ +, ⋆, < ✮ ♦✉ s✐♠♣❧❡s♠❡♥t❡ R✳ ❖✉tr❛ ♥♦t❛çã♦ ♣❛r❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ é ✉♠ ♣♦♥t♦✳ ❆ss✐♠✱ ♣♦r ❡①❡♠♣❧♦✱ s❡ t❡♠♦s ✶✳✷✳✶ a·b s✐❣♥✐✜❝❛ ♠✉❧t✐♣❧✐❝❛çã♦ ✭♣r♦❞✉t♦ ✮ ❞♦s ♥ú♠❡r♦s a ❡ b✳ a, b ∈ R✱ ❆❞✐çã♦ ❡ ▼✉❧t✐♣❧✐❝❛çã♦ ❝♦♠ ♥ú♠❡r♦s r❡❛✐s ❆❞✐çã♦ é ✉♠❛ ❞❛s ♦♣❡r❛çõ❡s ❜ás✐❝❛s ❞❛ ❛r✐t♠ét✐❝❛✳ ◆❛ s✉❛ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s✱ ❛❞✐çã♦ ❝♦♠❜✐♥❛ ❞♦✐s ♥ú♠❡r♦s ✭t❡r♠♦s✱ s♦♠❛♥❞♦s ♦✉ ♣❛r❝❡❧❛s✮ ❡♠ ✉♠ ú♥✐❝♦ ♥ú♠❡r♦ ❝❤❛♠❛❞♦ ✏ ❛ s♦♠❛ ✑✳ ❆❞✐❝✐♦♥❛r ♠❛✐s ♥ú♠❡r♦s ❝♦rr❡s♣♦♥❞❡ ❛ r❡♣❡t✐r ❛ ♦♣❡r❛çã♦✳ ✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R P♦❞❡ t❛♠❜é♠ s❡r ✉♠❛ ♦♣❡r❛çã♦ ❣❡♦♠étr✐❝❛✱ ❛ ♣❛rt✐r ❞❡ ❞♦✐s s❡❣♠❡♥t♦s ❞❡ r❡t❛ ❞❛❞♦s ❞❡t❡r♠✐♥❛r ✉♠ ♦✉tr♦ ❝✉❥♦ ❝♦♠♣r✐♠❡♥t♦ s❡❥❛ ✐❣✉❛❧ à s♦♠❛ ❞♦s ❞♦✐s ✐♥✐❝✐❛✐s✳ ❉❡✜♥✐çã♦ ✶✳✷✳ ❙❡❥❛ A ▲❡✐ ❞❡ ❝♦♠♣♦s✐çã♦ ✐♥t❡r♥❛✳ s✉❜❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ❝♦♥❥✉♥t♦ A R✳ ▲❡✐ ❞❡ ❝♦♠♣♦s✐çã♦ ✐♥t❡r♥❛ s♦❜r❡ ✉♠ é ✉♠❛ r❡❧❛çã♦ ❡♠ q✉❡✱ ❛ ❝❛❞❛ ♣❛r ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ♦✉tr♦ ❡❧❡♠❡♥t♦ c ∈ A✳ a, b ∈ A ❝♦rr❡s♣♦♥❞❡ ❊①❡♠♣❧♦ ✶✳✶✳ • ❉❛❞♦s 8, 9 ∈ N • ❉❛❞♦s 12, 4 ∈ R • ❉❛❞♦s 18, 7 ∈ N t❡♠♦s 8 + 9 = 17 ∈ N✳ t❡♠♦s 12 = 3 ∈ R✳ 4 t❡♠♦s ✐♥t❡r♥❛✳ ❆q✉✐✱ ❛ ❧❡✐ ❞❡ ❝♦♠♣♦s✐çã♦ ✐♥t❡r♥❛ é ❛ ❛❞✐çã♦✳ ❆q✉✐✱ ❛ ❧❡✐ ❞❡ ❝♦♠♣♦s✐çã♦ ✐♥t❡r♥❛ é ❛ ❞✐✈✐sã♦✳ 7 − 18 = −11 ∈ / N✳ ❆q✉✐✱ ♥ã♦ ❡①✐st❡ ❧❡✐ ❞❡ ❝♦♠♣♦s✐çã♦ ❈♦♥s✐❞❡r❡♠♦s ❞♦✐s ❛①✐♦♠❛s ❞❡✜♥✐❞♦s ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s R✳ ❊st❡s ❛①✐♦♠❛s ❞❡✜♥✐❞♦s ♣❡❧❛s ❧❡✐s ❞❡ ❝♦♠♣♦s✐çã♦ ✐♥t❡r♥❛ sã♦✿ ❆①✐♦♠❛ ❞❛ ❆❞✐çã♦ ✭❙♦♠❛✮✿ P❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ a ❡ b✱ t❡♠♦s q✉❡ a + b é ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ❆①✐♦♠❛ ❞❛ ▼✉❧t✐♣❧✐❝❛çã♦ ✭Pr♦❞✉t♦✮✿ P❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ a ❡ b✱ t❡♠♦s q✉❡ a · b é ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ❖♥❞❡ ❡st❡s ❛①✐♦♠❛s ❞❛ ❛❞✐çã♦ ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s ❝✉♠♣r❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ ❆✶ ❆✷ ❆✸ ❆✹ P✶ P✷ P✸ P✹ ❉✶ ❉✷ ∀ a, b ∈ R ✭❝♦♠✉t❛t✐✈❛✮ a+b=b+a ∀ a, b, c ∈ R ∃ 0 ∈ R /. a + 0 = 0 + a = a ∀ a ∈ R, ∀a ∈R ∃ − a ∈ R /. a + (−a) = (−a) + a = 0 ∀ a, b ∈ R ✭❛ss♦❝✐❛t✐✈❛✮ (a + b) + c = a + (b + c) ✭♥❡✉tr♦✮ ✭✐♥✈❡rs♦ ❛❞✐t✐✈♦✮ a.b = b.a ✭❛ss♦❝✐❛t✐✈❛✮ ∀ a, b, c ∈ R (a.b).c = a.(b.c) ∃ 1 ∈ R /. a.1 = 1.a = a ∀ a ∈ R ∀ a ∈ R, a 6= 0, ✭❝♦♠✉t❛t✐✈❛✮ ✭♥❡✉tr♦✮ ∃ a−1 ∈ R /. a.a−1 = a−1 .a = 1 ✭✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✮ ∀ a, b, c ∈ R a.(b + c) = a.b + a.c ✭❞✐str✐❜✉t✐✈❛✮ ∀ a, b, c ∈ R (a + b).c = a.c + b.c ✭❞✐str✐❜✉t✐✈❛✮ Pr♦♣r✐❡❞❛❞❡ ✶✳✶✳ P❛r❛ t♦❞♦s ♦s ♥ú♠❡r♦s r❡❛✐s a, b, c t❡♠♦s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s ✿ ✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✶✳ ❖s ❡❧❡♠❡♥t♦s ♥❡✉tr♦✱ ✐♥✈❡rs♦ ❛❞✐t✐✈♦ ❡ ♠✉❧t✐♣❧✐❝❛t✐✈♦ sã♦ ú♥✐❝♦s✳ ✷✳ a = −(−a)✳ ✸✳ ❙❡ ✹✳ a.0 = 0.a = 0✳ ✺✳ −a = (−1).a✳ ✻✳ a.(−b) = (−a).b = −(a.b) ✼✳ (−a).(−b) = a.b ✽✳ a+c = b+c ✾✳ ❙❡ a 6= 0 ✶✵✳ a.b = 0 ✶✶✳ a2 ❂ b2 a = (a−1 )−1 ✳ ❡♥tã♦ a.c = b.c R s❡✱ ❡ s♦♠❡♥t❡ s❡ c 6= 0✱ ❡ ❡♥tã♦ a = b✳ a = 0 s❡✱ ❡ s♦♠❡♥t❡ s❡ s❡✱ ❡ s♦♠❡♥t❡ s❡ a = b✳ a = b ♦✉ ♦✉ ❉❡♠♦♥str❛çã♦✳ ✭✷✮ b = 0✳ a = −b✳ ∀ a ∈ R ❡①✐st❡ −a ∈ R q✉❡ ❝✉♠♣r❡ ❛ ✐❣✉❛❧❞❛❞❡ a + (−a) = (−a) + a = 0✳ ❆ss✐♠ ♣❛r❛ t♦❞♦ (−a) ∈ R ❡①✐st❡ −(−a) ∈ R q✉❡ ❝✉♠♣r❡ ❛ ✐❣✉❛❧❞❛❞❡ (−a) + (−(−a)) = (−(−a)) + (−a) = 0✳ ❊♥tã♦ a + (−a) + (−(−a)) = (−(−a)) + a + (−a)❀ ✐st♦ é a = −(−a)✳  P❡❧♦ ❆①✐♦♠❛ ❆✹✱ t❡♠♦s✿ ❉❡♠♦♥str❛çã♦✳ ✭✹✮ a.0 = a(0 + 0)❀ ♣♦✐s 0=0+0 ▲♦❣♦✱ ♣❡❧♦ ❆①✐♦♠❛ ❉✶ s❡❣✉❡ a.0 = a · (0 + 0) = a.0 + a.0✱ ❡♥tã♦ a.0 = 0  ❉❡♠♦♥str❛çã♦✳ ✭✺✮ a + (−1)a = 1.a + (−1).a ❂ [1 + (−1)].a ❂ 0 ✐st♦ ❞❡ ❞✐str✐❜✉t✐✈✐❞❛❞❡ ❡♥tã♦✱ ❛♣❧✐❝❛♥❞♦ ♦ ❆①✐♦♠❛ ❆✹ ♣❛r❛ ❉❡♠♦♥str❛çã♦✳ ✭✾✮ −1 a = a(c.c ✮ ❂ (a.c).c−1 = (b.c).c−1 ❂ b(c.c−1 ) = b a = 1.a [1 + (−1)] = 0 ❡ a.0 = 0 a✱ s❡❣✉❡ (−1)a = −a ✐st♦ ❞❡ a = a.1 ❡ 1 = c.c−1  ♣♦✐s ♣♦r ❤✐♣ót❡s❡✳ c · c−1 = 1 ❡ c 6= 0  b·1=b ❉❡♠♦♥str❛çã♦✳ ✭✶✵✮ ❙✉♣♦♥❤❛♠♦s a=0 ♦✉ b = 0✳ ❊♥tã♦ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✶✳✶✮✲(4) s❡❣✉❡ a · b = 0✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s✉♣♦♥❤❛✳ a · b = 0 ❡ q✉❡ a 6= 0✳ ❊♥tã♦ a−1 (a.b) = a−1 .0 = 0✱ ✐st♦ é ✭a−1 · a) · b = b = 0✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ s✉♣♦♥❤❛ q✉❡ b 6= 0✳ ▲♦❣♦ a = 0✳ ❙✉♣♦♥❤❛♠♦s 1 · b = 0❀ ❧♦❣♦ ✼ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❡✜♥✐çã♦ ✶✳✸✳ ❆ ❞✐❢❡r❡♥ç❛ ❡ ♦ q✉♦❝✐❡♥t❡ ❞❡ ❞♦✐s ♥ú♠❡r♦s r❡❛✐s é ❞❡✜♥✐❞♦ ♣♦r✿ ✶✳ ✷✳ a − b = a + (−b) a = a.b−1 b s❡ ❞✐❢❡r❡♥ç❛✳ b 6= 0 q✉♦❝✐❡♥t❡ Pr♦♣r✐❡❞❛❞❡ ✶✳✷✳ P❛r❛ t♦❞♦s ♦s ♥ú♠❡r♦s r❡❛✐s a, b, c, d✱ t❡♠♦s✿ ✶✳ a − b = −(b − a). ✷✳ a − b = c ✱ ❡♥tã♦ a = b + c✳ ✸✳ a.(b − c) = a.b − a.c✳ ad + bc a c + = . ✹✳ ❙❡ b 6= 0 ❡ d 6= 0✱ ❡♥tã♦ b ✺✳ ❙❡ b 6= 0 ✻✳ ❙❡ a 6= 0 ❡ ❡ ❉❡♠♦♥str❛çã♦✳ ✭✶✮ d bd a c ad − bc d 6= 0✱ ❡♥tã♦ − = . b d bd c−b . ax + b = c ✱ ❡♥tã♦ x = a a ❡ b ♥ú♠❡r♦s r❡❛✐s✱ ❡♥tã♦ a − b é ✉♠ −(a − b)✳ ❆ss✐♠ (a − b) + (−(a − b)) = 0✳ ❙❡♥❞♦ ❛❞✐t✐✈♦ P❡❧❛ ❉❡✜♥✐çã♦ ♥ú♠❡r♦ r❡❛❧✳ ▲♦❣♦ ❡①✐st❡ s❡✉ ♦♣♦st♦ ✭✶✳✸✮ s❡❣✉❡✿ (a − b) − (a − b) = 0 a + (−b) − (a − b) = 0 ♦✉ ✭✶✳✶✮ −(b − a) é ✉♠ ♥ú♠❡r♦ r❡❛❧✱ ❧♦❣♦ ❡①✐st❡ s❡✉ ✐♥✈❡rs♦ ❛❞✐t✐✈♦ −[−(b − a)]✱ ❧♦❣♦ −(b − a) + {−[−(b − a)]} = 0✳ ❆ss✐♠ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✶✳✶✮✲(2) t❡♠♦s ✿ −(b − a) + (b − a) = 0 ❡♥tã♦ P♦r ♦✉tr♦ ❧❛❞♦✱ −(b − a) + b + (−a) = 0 ✭✶✳✷✮ (a + (−b) − (a − b)) + (−(b − a) + b + (−a)) = 0✱ ✐st♦ é −(a − b) + (−(b − a)) = 0❀ ♦♥❞❡ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✶✳✶✮✲(8) ❞♦ ♦♣♦st♦ ❛❞✐t✐✈♦ ❞❡ (a − b) r❡s✉❧t❛ −(b − a)) = (a − b)✳  ❉❡ ✭✶✳✶✮ ❡ ✭✶✳✷✮ t❡♠♦s ❉❡♠♦♥str❛çã♦✳ ✭✻✮ ❙❡❥❛♠ a 6= 0 ❡ ax + b = c✱ ❡♥tã♦ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ❡ ❉❡✜♥✐çã♦ ✭✶✳✸✮✲✭2✮ r❡s✉❧t❛ ax = c − b✳ a 6= 0 t❡♠♦s a−1 (ax) = a−1 (c−b) ❡✱ ♣❡❧♦ ❆①✐♦♠❛ c−b x=  a P❡❧♦ ♦♣♦st♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❞♦ ♥ú♠❡r♦ P3 ✭✶✳✷✮✲(2) ❝♦♥❝❧✉í♠♦s ✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❡♠♦♥str❛çã♦✳ ✭2✮ ✲ ✭5✮ ❊①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❉❡✜♥✐çã♦ ✶✳✹✳ ◆ú♠❡r♦ ♣❛r✳ ❉✐③❡♠♦s q✉❡ a ∈ Z é ♥ú♠❡r♦ ♣❛r s❡ ❡①✐st❡ β ∈ Z t❛❧ q✉❡ a = 2β ✳ ❊♠ Z✱ t♦❞♦ ♥ú♠❡r♦ q✉❡ ♥ã♦ é ♣❛r✱ é ❞❡♥♦♠✐♥❛❞♦ ♥ú♠❡r♦ í♠♣❛r✳ ❖❜s❡r✈❛çã♦ ✶✳✶✳ ✶✳ ❚♦❞♦ ♥ú♠❡r♦ í♠♣❛r b ∈ Z é ❞❛ ❢♦r♠❛ b = 2α + 1✱ ♣❛r❛ α ∈ Z✳ ✷✳ ❙❡❣✉♥❞♦ ♥♦ss❛ ❞❡✜♥✐çã♦ ❞❡ ♥ú♠❡r♦ ♣❛r✱ ♦ ③❡r♦ é ♣❛r✳ ❉❡✜♥✐çã♦ ✶✳✺✳ ❉✐✈✐s♦r ❝♦♠✉♠✳ ❙❡❥❛♠ ♦s ♥ú♠❡r♦s a, b, d ∈ Z s❡✱ d ❞✐✈✐❞❡ ❛ a ❡ b✱ ♦ ♥ú♠❡r♦ d é ❝❤❛♠❛❞♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b✳ Pr♦♣r✐❡❞❛❞❡ ✶✳✸✳ ❉❛❞♦s ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s a ❡ b✱ ❡①✐st❡ ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❛ ❢♦r♠❛ d = ax + by ♣❛r❛ ❛❧❣✉♠ x, y ∈ Z❀ ❡✱ t♦❞♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b ❞✐✈✐❞❡ ❡st❡ d✳ ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❉❡✜♥✐çã♦ ✶✳✻✳ ▼á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✳ ❖ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞♦s ♥ú♠❡r♦s a ❡ b ♥ã♦ ♥✉❧♦s ✭❡s❝r✐t♦s ❝♦♠♦ ♣r♦❞✉t♦ ❞❡ ❢❛t♦r❡s ♣r✐♠♦s✮ ❞❡♥♦t❛❞♦ mdc{a, b} é ♦ ♣r♦❞✉t♦ ❞♦s ❢❛t♦r❡s ❝♦♠✉♥s ❛ ❡❧❡s✱ ❝❛❞❛ ✉♠ ❡❧❡✈❛❞♦ ❛♦ ♠❡♥♦r ❡①♣♦❡♥t❡✳ ❉❡✜♥✐çã♦ ✶✳✼✳ ▼í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠✳ ❖ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞♦s ♥ú♠❡r♦s a ❡ b ♥ã♦ ♥✉❧♦s ✭❡s❝r✐t♦s ❝♦♠♦ ♣r♦❞✉t♦ ❞❡ ❢❛t♦r❡s ♣r✐♠♦s✮ ❞❡♥♦t❛❞♦ mmc{a, b} = ab ✳ mdc{a, b} ❉❡✜♥✐çã♦ ✶✳✽✳ ◆ú♠❡r♦s ♣r✐♠♦s✳ ❙❡❥❛ n ∈ Z✱ ❞✐③❡♠♦s q✉❡ n é ♥ú♠❡r♦ ♣r✐♠♦✱ s❡ n > 1 ❡ s❡✉s ú♥✐❝♦s ❞✐✈✐s♦r❡s ♣♦s✐t✐✈♦s sã♦ 1 ❡ ♦ ♣ró♣r✐♦ n✳ ❙❡ n ♥ã♦ é ♥ú♠❡r♦ ♣r✐♠♦ ❡♥tã♦ é ❝❤❛♠❛❞♦ ❞❡ ♥ú♠❡r♦ ❝♦♠♣♦st♦✳ ✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✶✳✷✳ • ❙ã♦ ♥ú♠❡r♦s ♣r✐♠♦s✿ • ◆ã♦ sã♦ ♥ú♠❡r♦s ♣r✐♠♦s✱ sã♦ ♥ú♠❡r♦s ❝♦♠♣♦st♦s✿ 2, 3, 7, 11 13, 17, 19 4, 6, 8, 10, 16, 24✳ Pr♦♣r✐❡❞❛❞❡ ✶✳✹✳ n>1 ❚♦❞♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ é ♥ú♠❡r♦ ♣r✐♠♦ ♦✉ ♣r♦❞✉t♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✳ ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ Pr♦♣r✐❡❞❛❞❡ ✶✳✺✳ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✳ P❛r❛ q✉❛✐sq✉❡r ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♥ã♦ ♥✉❧♦s a ❡ b✱ ❡①✐st❡♠ ✐♥t❡✐r♦s ú♥✐❝♦s q ❡ r✱ ❞❡♥♦♠✐♥❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ♦✉ r❡sí❞✉♦✱ t❛✐s q✉❡✿ 0≤r<b a = bq + r, ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❊①❡♠♣❧♦ ✶✳✸✳ ✭❛✮ −2805 = −24(119) + 51 ✭❜✮ 758 = 3(242) + 32 ✭❝✮ 780 = −16(−48) + 12 ✭❞✮ 826 = 33(25) + 1 ❉❡✜♥✐çã♦ ✶✳✾✳ ◆ú♠❡r♦s r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s✳ ❉✐③❡♠♦s q✉❡ ❞♦✐s ♥ú♠❡r♦s a, b ∈ Z sã♦ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s✱ s❡ ♦ mdc{ a, b } = 1✳ ❊①❡♠♣❧♦ ✶✳✹✳ ❖s s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s✱ sã♦ ❞❡ ♥ú♠❡r♦s r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s✿ A = {8, 9} ❊①❡♠♣❧♦ ✶✳✺✳ ❊♠♣r❡st❡✐ ♦s B = {86, 25} 2 3 5 6 ❞♦s ❞♦s 3 5 C = {32, 49} D = {18, 19} ❞❡ ✉♠ ❞✐♥❤❡✐r♦ q✉❡ t✐♥❤❛ ❡ ❛✐♥❞❛ t❡♥❤♦ ❞❡ ✉♠ 1 5 ❞❡ ♠✐❧❤ã♦ ❞❡ r❡❛✐s✳ ◗✉❡ q✉❛♥t✐❞❛❞❡ ❞❡ ❞✐♥❤❡✐r♦ ❡♠♣r❡st❡✐ ❄ ❙♦❧✉çã♦✳ ❖ s✐❣♥✐✜❝❛❞♦ ♠❛t❡♠át✐❝♦ ❞❛s ♣❛❧❛✈r❛s ✏ ❞♦s ✑✱ ✏ ❞❛s ✑✱ ✏ ❞♦ ✑✱ ✏ ❞❡ ✑✱ ♣♦❞❡♠♦s ❡♥t❡♥❞❡r ❝♦♠♦ s❡ ❢♦r ♦ ♦♣❡r❛❞♦r ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ t✐♥❤❛ x r❡❛✐s✳ ❊♠♣r❡st❡✐ 2 5 3 ( )( )( )x✱ 3 6 5 ❧♦❣♦ t❡♥❤♦ 1 2 5 3 x − ( )( )( )x = ( )(1.000.000) 3 6 5 5 ✶✵ 1 ( )(1000, 000)✳ 5 ❆ss✐♠✿ ⇒ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ x − ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R 1 2 x = 200.000 ⇒ x = 200.000 ⇒ x = 300.000 3 3 P♦rt❛♥t♦✱ t✐♥❤❛ 300.000 r❡❛✐s ❡ ❡♠♣r❡st❡✐ ❘$100.000✳ ❊①❡♠♣❧♦ ✶✳✻✳ ❆♦ ❝❤❡❣❛r ❛ ♠✐♥❤❛ ❝❛s❛ ❡♥❝♦♥tr❡✐ ✈ár✐❛s ❛r❛♥❤❛s ❡ ❜❛r❛t❛s✱ ❞❡♣♦✐s ❞❡ ♠❛t❛r ❡st❡s ✐♥s❡t♦s✱ ❝♦♥t❡✐ ♦ ♥ú♠❡r♦ ❞❡ ♣❛t❛s ❡ ♦❜s❡r✈❡✐ q✉❡ ❡r❛♠ 108✳ 16 ❈❛❧❝✉❧❛r q✉❛♥t❛s ❜❛r❛t❛s ❡ ❛r❛♥❤❛s ❡♥❝♦♥tr❡✐ ❛♦ ❝❤❡❣❛r ❛ ❝❛s❛✳ ❙♦❧✉çã♦✳ ➱ s✉✜❝✐❡♥t❡ s❛❜❡r♠♦s ♦ ♥ú♠❡r♦ ❞❡ ♣❛t❛s q✉❡ ❝❛❞❛ ✐♥s❡t♦ ♣♦ss✉✐✱ ❡ ❡♠ s❡❣✉✐❞❛ ❛♥❛❧✐s❛r ♦s ❞❛❞♦s ❡ ♦ q✉❡ s❡ ♣❡❞❡ ♥♦ ♣r♦❜❧❡♠❛✳ ❙✉♣♦♥❤❛✱ q✉❡ ❡①✐st❛♠ b ❜❛r❛t❛s ❡ (16 − b) ❛r❛♥❤❛s✳ ❈♦♠♦✱ ❝❛❞❛ ❜❛r❛t❛ t❡♠ 6 ♣❛t❛s ❡ ❝❛❞❛ ❛r❛♥❤❛ t❡♠ 8 ♣❛t❛s✱ t❡♠♦s q✉❡✿ 6b + 8.(16 − b) = 108✳ ▲♦❣♦✱ b = 10✳ P♦rt❛♥t♦✱ ♦ t♦t❛❧ ❞❡ ❜❛r❛t❛s q✉❡ ❡♥❝♦♥tr❡✐ ❢♦r❛♠ 10 ❡ ❛s ❛r❛♥❤❛s t♦t❛❧✐③❛r❛♠ s❡✐s✳ ❊①❡♠♣❧♦ ✶✳✼✳ ❯♠ ❢❛❜r✐❝❛♥t❡ ❞❡ ❧❛t❛s✱ ❞❡s❡❥❛ ❢❛❜r✐❝❛r ✉♠❛ ❧❛t❛ ❡♠ ❢♦r♠❛ ❞❡ ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r r❡t♦ ❝♦♠ 10 cm ❞❡ r❛✐♦ ❡ 6283, 2 cm3 ❞❛ ❝❛♣❛❝✐❞❛❞❡✳ ❉❡t❡r♠✐♥❡ s✉❛ ❛❧t✉r❛✳ ❙♦❧✉çã♦✳ ❙❛❜❡♠♦s q✉❡ ♦ ✈♦❧✉♠❡ V ✱ ❞♦ ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r r❡t♦ ❞❡ r❛✐♦ r ❡ ❛❧t✉r❛ h é ❞❛❞♦ ♣❡❧❛ ❢ór♠✉❧❛ V = πr2 h✳ P❡❧♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛ t❡♠♦s r = 10 cm✱ V = 6283, 2 cm3 ✳ ❆ss✐♠ ♥❛ ❢ór♠✉❧❛ 6.283, 2 cm3 = π(10cm)2 .h ⇒ 6.283, 2 cm3 = (3, 1416)(100 cm2 ).h ⇒ 6.283, 2 cm3 = (314, 16 cm2 ).h ⇒ h = ⇒ 6.283, 2 cm3 = 20 cm 314, 16cm2 P♦rt❛♥t♦ ❛❧t✉r❛ ❞♦ ❝✐❧✐♥❞r♦ ❞❡✈❡rá ♠❡❞✐r 20 cm✳ ❊①❡♠♣❧♦ ✶✳✽✳ ❆ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞❡ 8 ♥ú♠❡r♦s é ❊♥tã♦ ❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞❡ss❡s 14 6❀ ❥á ❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞❡ ♦✉tr♦s 6 ♥ú♠❡r♦s é 8✳ ♥ú♠❡r♦s é✿ ❙♦❧✉çã♦✳ ❙✉♣♦♥❤❛♠♦s t❡♠♦s ♦s ♥ú♠❡r♦s a1 , a2 , a3 , · · · , a7 , a8 ❡ b1 , b2 , b3 , · · · b5 , b6 ✳ P❡❧♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛ t❡♠♦s q✉❡✿ a1 + a2 + · · · + a7 + a8 = 6 8 ❡ b1 + b2 + · · · + b5 + b6 = 8 6 ❊♥tã♦✱ a1 + a2 + · · · + a7 + a8 = (8)(6) ❡ b1 + b2 + · · · + b5 + b6 = (6)(8)✱ ❧♦❣♦✿ [a1 + a2 + · · · + a7 + a8 ] + [b1 + b2 + · · · + b5 + b6 ] = (8)(6) + (6)(8) = 96✳ ✶✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R [a1 + a2 + · · · + a7 + a8 ] [b1 + b2 + · · · + b5 + b6 ] 96 + = = 6, 84✳ 8+6 8+6 14 P♦rt❛♥t♦✱ ❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞❡ss❡s 14 ♥ú♠❡r♦s é 6, 84✳ ▲♦❣♦✱ ❊①❡♠♣❧♦ ✶✳✾✳ 10 ❧✐tr♦s ❞❡ ✉♠❛ ❝♦♥t❡♥❤❛ 25% ❞❡ ó❧❡♦❄ ◗✉❛♥t♦s ❧✐tr♦s ❞❡ ó❧❡♦ ❞❡✈❡♠ s❡r ❛❞✐❝✐♦♥❛❞♦s ❛ 15% ❞❡ ó❧❡♦✱ ♣❛r❛ ♦❜t❡r ♦✉tr❛ ♠✐st✉r❛ q✉❡ ♠✐st✉r❛ q✉❡ ❝♦♥té♠ ❙♦❧✉çã♦✳ ❙✉♣♦♥❤❛ q✉❡ ♥❛ ♠✐st✉r❛ ♦r✐❣✐♥❛❧ t❡♥❤❛♠♦s q✉❡ ❛❞✐❝✐♦♥❛r x ❧✐tr♦s ❞❡ ó❧❡♦✳ ❖❜s❡r✈❛♥❞♦ ❛ ❋✐❣✉r❛ ✭✶✳✸✮✱ t❡♠♦s✿ 10( 15 25(10 + x) )+x= 100 100 ❘❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ t❡♠♦s q✉❡ P♦rt❛♥t♦✱ t❡r❡♠♦s q✉❡ ❛❞✐❝✐♦♥❛r 4 3 x✲ 4 x= ✳ 3 10 ❧✐tr♦s ❞❡ ó❧❡♦✳ ó❧❡♦ ❅ ■ ❅ ❅ ✒ ❅ ❘ ❅ 15%✲ 25% ✠ ó❧❡♦ ❊①❡♠♣❧♦ ✶✳✶✵✳ ▲❛♥ç❛♠✲s❡ ❞♦✐s ❞❛❞♦s ♥ã♦✲t❡♥❞❡♥❝✐♦s♦s✳ ◗✉❛❧ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ s♦♠❛ ❞♦s ♣♦♥t♦s s❡r ✐❣✉❛❧ ❛ 7 ❋✐❣✉r❛ ✶✳✸✿ ❄ ❙♦❧✉çã♦✳ ❈♦♠ ♦s ❞❛❞♦s D1 ❡ D2 ♣♦❞❡♠♦s ♦❜t❡r ♦s ❝♦♥❥✉♥t♦ ❞❡ ❝❛s♦s ♣♦ssí✈❡✐s✿ D1 × D2 = {(1, 1), . . . , (1, 6), (2, 1), . . . , (2, 6), (3, 1), . . . , (4, 1), . . . , (5, 1), . . . (6, 5), (6, 6)} ❖s ❝❛s♦s ❢❛✈♦rá✈❡✐s sã♦ P♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ❞♦s q✉❛✐s ❛ s♦♠❛ ✈❛❧❡ 7✳ { (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) }✳ ❤á 6 × 6 = 36 r❡s✉❧t❛❞♦s ♣♦ssí✈❡✐s ✐❣✉❛❧♠❡♥t❡ ♣r♦✈á✈❡✐s✱ ❆ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ s♦♠❛ ❞♦s ♣♦♥t♦s s❡r ✐❣✉❛❧ ❛ 7 é P = 6 1 = 36 6 ❡♠ 6 ✳ ❖❜s❡r✈❛çã♦ ✶✳✷✳ ❙❡♥❞♦ a, b, c três ❛❧❣❛r✐s♠♦s ❡s❝r❡✈❡r❡♠♦s abc ♣❛r❛ ✐♥❞✐❝❛r q✉❡ abc = 100a + 10b + c ❡♠ ❣❡r❛❧✱ s❡ an , an−1 , an−2 , · · · a1 a0 sã♦ ❛❧❣❛r✐s♠♦s✱ an an−1 an−2 · · · a1 a0 = 10n an + 10n−1 an−1 + 10n−2 an−2 · · · 10a1 + a0 é ❝❤❛♠❛❞❛ ✏❞❡❝♦♠♣♦s✐çã♦ ♣♦❧✐♥ô♠✐❝❛ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛ ❜❛s❡ ❞❡❝✐♠❛❧✳✑ ✶✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✶✲✶ ✶✳ ❙❡❥❛♠✱ N ♦ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ Z ♦ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❉❡✲ t❡r♠✐♥❡ q✉❛✐s ❞❡♥tr❡ ❛s s❡❣✉✐♥t❡s ♣r♦♣♦s✐çõ❡s é ✈❡r❞❛❞❡✐r❛ ✭✈✮ ❡ q✉❛❧ é ❛ ❢❛❧s❛ ✭❢ ✮✳ 1. N = Z+ 2. N+ = Z 3. N+ = Z+ 4. N ⊂ Z ✷✳ ❉❛s s❡❣✉✐♥t❡s ♣r♦♣♦s✐çõ❡s q✉❛❧ é ✈❡r❞❛❞❡✐r❛ ✭✈✮ ♦✉ ❢❛❧s❛ ✭❢ ✮✳ 1. N ⊂ Z ⊂ Q ⊂ R 2. I ⊂ R 7. Z+ = N 8. Z ∩ Q+ = N 4. R − Q = I 3. Q ∩ I = ∅ 5. N ⊂ (Q − Z) 6. N ∪ Z = Q ✸✳ ❱❡r✐✜q✉❡ q✉❛✐s ❞❛s s❡❣✉✐♥t❡s ♣r♦♣♦s✐çõ❡s sã♦ ✈❡r❞❛❞❡✐r❛s✿ √ 3 ∈ Q 3. 5, 41 ∈ (Q − Z) 4. 2 √ 6. 0 ∈ /Z 7. −7 ∈ /R 8. 1. 7, 43333... ∈ I 2. 5. 2, 71854 ∈ /I −5∈ /Q − 3 ∈ (R − Q) 5 ✹✳ ❈♦♥str✉❛ ✉♠ ❞✐❛❣r❛♠❛ ❝♦♥t❡♥❞♦ ♦s ❝♦♥❥✉♥t♦s N✱ Z✱ Q ❡ I ❡ s✐t✉❡ ♦s s❡❣✉✐♥t❡s ♥ú♠❡r♦s✿ 1. 6. 11. √ 3 2 π 2 10 − 3 2. √ 7. −5 12. 0 3 −3 3. 0 8. 13. − 0, 60 3 8 9. 2, 573 4. 5 − (− )2 2 5. 8, 43 10. 0, 333 · · · ✺✳ ▼♦str❡ q✉❡✱ s❡ x2 ❂ ✵✱ ❡♥tã♦ x = 0✳ ✻✳ ▼♦str❡ q✉❡✱ s❡ p é ♥ú♠❡r♦ í♠♣❛r✱ ❡♥tã♦ p2 é í♠♣❛r✳ ✼✳ ▼♦str❡ q✉❡✱ s❡ p é ♥ú♠❡r♦ ♣❛r✱ ❡♥tã♦ p2 é ♣❛r✳ ✽✳ ✶✳ ❙❡ a é r❛❝✐♦♥❛❧ ❡ b ✐rr❛❝✐♦♥❛❧✱ a + b ♥❡❝❡ss❛r✐❛♠❡♥t❡ é ✐rr❛❝✐♦♥❛❧❄ ✷✳ ❙❡ a é ✐rr❛❝✐♦♥❛❧ ❡ b ✐rr❛❝✐♦♥❛❧✱ a + b ♥❡❝❡ss❛r✐❛♠❡♥t❡ é ✐rr❛❝✐♦♥❛❧❄ ✸✳ ❙❡ a é r❛❝✐♦♥❛❧ ❡ b ✐rr❛❝✐♦♥❛❧✱ ab ♥❡❝❡ss❛r✐❛♠❡♥t❡ é ✐rr❛❝✐♦♥❛❧❄ ✶✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✹✳ ❊①✐st❡ ♥ú♠❡r♦ r❡❛❧ a t❛❧ q✉❡ a2 s❡❥❛ ✐rr❛❝✐♦♥❛❧✱ ♣♦ré♠ a4 r❛❝✐♦♥❛❧❄ ✺✳ ❊①✐st❡♠ ❞♦✐s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s t❛✐s q✉❡ s✉❛ s♦♠❛ ❡ ♣r♦❞✉t♦ s❡❥❛♠ r❛❝✐♦♥❛✐s❄ ✾✳ ▼♦str❡ q✉❡ √ 2 é ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧✳ ✶✵✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ A ⊆ R ❞✐③✲s❡ ❡stá✈❡❧ ❛❞✐t✐✈❛♠❡♥t❡ s❡✱ ∀ a, b ∈ A t❡♠♦s (a + b) ∈ A❀ ❡ ❡stá✈❡❧ ♠✉❧t✐♣❧✐❝❛t✐✈❛♠❡♥t❡ s❡✱ ∀ a, b ∈ A t❡♠♦s (a · b) ∈ A✳ ❉❛❞♦s ♦s ❝♦♥❥✉♥t♦s A = { 2, 4, 6, 8, · · · } ❡ B = { 1, 3, 5, 7, 9, · · · }✱ ❞❡t❡r♠✐♥❡ s❡ ❡❧❡s sã♦ ❝♦♥❥✉♥t♦s ❡stá✈❡✐s ❛❞✐t✐✈❛ ❡ ♠✉❧t✐♣❧✐❝❛t✐✈❛♠❡♥t❡✳ ✶✳ ❉❛❞♦s ♦s ❝♦♥❥✉♥t♦s✿ N, Z, Q ❡ R ❞❡t❡r♠✐♥❡ q✉❛✐s sã♦ ❡stá✈❡✐s r❡s♣❡✐t♦ ❞❛s ♦♣❡r❛çõ❡s ❞❡✿ ✐✮ ❛❞✐çã♦❀ ✐✐✮ ♠✉❧t✐♣❧✐❝❛çã♦✳ ✷✳ ✶✶✳ ▼♦str❡ q✉❡ 2 ❡ 3 sã♦ ❛s ú♥✐❝❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ x2 − 5x + 6 = 0✳ ✶✷✳ ❚r❛♥s❢♦r♠❡ ❝❛❞❛ ✉♠❛ ❞❛s ❡①♣r❡ssõ❡s ❡♠ ✉♠ ú♥✐❝♦ r❛❞✐❝❛❧✿ ✶✳ q p √ x y z q p √ 3 x3 y3z ✷✳ ✶✸✳ ❉❡t❡r♠✐♥❡ ❛ ❝♦♥❞✐çã♦ ♣❛r❛ q✉❡ s❡❥❛ ♣♦ssí✈❡❧ ❡①♣r❡ss❛r ♦♥❞❡ a, b, x ❡ y s❡❥❛♠ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✳ ✸✳ p √ √ √ a + b ♥❛ ❢♦r♠❛ x + y ✱ ✶✹✳ ❊s❝r❡✈❛ ❛s ❡①♣r❡ssõ❡s ❛❜❛✐①♦ ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ r❛❞✐❝❛✐s✿ ✶✳ p √ 12 + 140 p √ 13 − 160 ✷✳ ✶✺✳ ❙✐♠♣❧✐✜q✉❡ ❛s s❡❣✉✐♥t❡s ❡①♣r❡ssõ❡s✿ ✶✳ 1 1 1 √ +√ −√ 3 3 3 2 4 16 √ √ 2+ 5 1− 5 √ + √ 2− 3 2+ 3 ✷✳ ✶✻✳ ❙❡❥❛♠ a, b, c, d, m, n ❡ p ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s✳ ▼♦str❡ q✉❡ s❡ √ am + √ bn + √ cp = p (a + b + c)(m + n + p) ✳ q p √ 4 x3 y z ✸✳ ✸✳ p √ 9 − 72 √ √ ( 3 9 − 3 3)2 b c a = = ❡♥tã♦ m n p ✶✼✳ ❉❛❞♦s ♦s ♥ú♠❡r♦s a = 710 ❡ b = 68✳ ✶✳ ❉❡t❡r♠✐♥❡ ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b✳ ✷✳ ❉❡t❡r♠✐♥❡ ♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ a ❡ b✳ ✶✽✳ ❍á s❡✐s ❛♥♦s✱ ❛ ✐❞❛❞❡ ❞❡ ❆❧❜❡rt♦ ❡r❛ q✉❛tr♦ ✈❡③❡s ❛ ✐❞❛❞❡ ❞❡ P❡❞r♦✳ ❈❛❧❝✉❧❛r s✉❛s ✐❞❛❞❡s ❛t✉❛✐s s❛❜❡♥❞♦ q✉❡✱ ❞❡♥tr♦ ❞❡ q✉❛tr♦ ❛♥♦s✱ ❆❧❜❡rt♦ só t❡rá ♦ ❞♦❜r♦ ❞❛ ✐❞❛❞❡ ❞❡ P❡❞r♦✳ 1 2 ✶✾✳ ❆ ✐❞❛❞❡ ❞❡ ▼❛r✐❛ é ✭♠❡t❛❞❡✮ ❞❡ ❞❛ ✐❞❛❞❡ ❞❡ ▼❛r✐s❛✳ ❙❡ ▼❛r✐s❛ t❡♠ 24 ❛♥♦s✳ 2 3 q✉❛♥t♦s ❛♥♦s tê♠ ▼❛r✐❛❄ ✶✹ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✵✳ ❆ s♦♠❛ ❞❛s ✐❞❛❞❡s ❞❡ ❞♦ ♠❡✐♦ 18 3 ♣❡ss♦❛s é 97✳ ❆ ♠❛✐♦r t❡♠ 29 ❛♥♦s ♠❛✐s q✉❡ ❛ ♠❡♥♦r✱ ❡ ❛ ❛♥♦s ♠❡♥♦s q✉❡ ❛ ♠❛✐♦r✱ ❈❛❧❝✉❧❛r ❛ ✐❞❛❞❡ ❞❡ ❝❛❞❛ ✉♠❛✳ 100 cm3 ✷✶✳ ◗✉❛♥t♦ ❞❡ á❣✉❛ ❞❡✈❡ s❡r ❛❞✐❝✐♦♥❛❞❛ ❛ 50% ❜ór✐❝♦✱ ♣❛r❛ r❡❞✉③✐r✲❧❛ ❛ ✷✷✳ ❆♦ ❞✐✈✐❞✐r ♦ ♥ú♠❡r♦ t❛r♠♦s ♦ ❞✐✈✐❞❡♥❞♦ ❉ ♣♦r ❉ ❡♠ ❞ 15 ❞❡ 80% ❞❡ ✉♠❛ s♦❧✉çã♦ ❞❡ á❝✐❞♦ ❞❛ s♦❧✉çã♦ ❄ ♦❜t❡♠♦s ❝♦♠♦ q✉♦❝✐❡♥t❡ ✉♥✐❞❛❞❡s ❡ ♦ ❞✐✈✐s♦r ❞ q ❡ ❝♦♠♦ r❡st♦ ❡♠ 5 r✳ ❙❡ ❛✉♠❡♥✲ ✉♥✐❞❛❞❡s✱ ♦ q✉♦❝✐❡♥t❡ ❡ r❡st♦ ♦r✐❣✐♥❛✐s ♣❡r♠❛♥❡❝❡♠ ✐❣✉❛✐s✳ ◗✉❛❧ ❢♦✐ ♦ q✉♦❝✐❡♥t❡❄ ✷✸✳ ❈♦♠♣r❛♠✲s❡ ❝❛❞❡r♥♦s ❞❡ ❢♦r♠❛ ♣r♦❣r❡ss✐✈❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ 14 ❝❛❞❡r♥♦s❀ ♥♦ s❡❣✉♥❞♦ ❞✐❛ ❝❡ss✐✈❛♠❡♥t❡✳ ❉❡♣♦✐s ❞❡ 30 15 ❝❛❞❡r♥♦s❀ ♥♦ t❡r❝❡✐r♦ ❞✐❛ 16 ♥♦ ♣r✐♠❡✐r♦ ❞✐❛ ❝❛❞❡r♥♦s ❡ ❛ss✐♠ s✉✲ ❞✐❛s ❝♦♥s❡❝✉t✐✈♦s ❝♦♠♣r❛♥❞♦✱ q✉❛♥t♦s ❝❛❞❡r♥♦s ❢♦r❛♠ ❝♦♠♣r❛❞♦s ♥♦ t♦t❛❧ ❄ ✷✹✳ ❖ ❞❡♥♦♠✐♥❛❞♦r ❞❡ ✉♠❛ ❢r❛çã♦ ❞❡❝✐♠❛❧ é ❙❡ ♦ ♥✉♠❡r❛❞♦r ❛✉♠❡♥t❛ ❡♠ 5 3 ❛ ♠❡♥♦s q✉❡ ♦ ❞♦❜r♦ ❞♦ ♥✉♠❡r❛❞♦r✳ ❡ ♦ ❞❡♥♦♠✐♥❛❞♦r ❡♠ 14✱ ♦ ✈❛❧♦r ❞❛ ❢r❛çã♦ é 7/15✳ ❉❡t❡r♠✐♥❡ ❛ ❢r❛çã♦✳ ✷✺✳ ❊①♣❡❞✐çã♦✿ ■♥❢♦r♠❡✿ P❧❛♥❡t❛ K ❆♦ ❝❤❡❣❛r ❛♦ ♣❧❛♥❡t❛ ❡♠❜♦r❛ t❛♠❜é♠ t❡♥❤❛♠ K✱ ❛❝❤❛♠♦s s❡r❡s ✈✐✈♦s ❝♦♠♦ ❡♠ ♥♦ss♦ ♣❧❛♥❡t❛✱ 20 ❞❡❞♦s✱ ❡❧❡s tê♠ ✉♠ ♠❡♠❜r♦ ❛ ♠❡♥♦s✱ ❡ ✉♠ ❞❡❞♦ ❛ ♠❛✐s ❡♠ ❝❛❞❛ ♠❡♠❜r♦✳ P❡r❣✉♥t❛✲s❡✿ P♦ssí✈❡❧♠❡♥t❡ q✉❡ t✐♣♦ ❞❡ s❡r❡s ❤❛❜✐t❛♠ ♦ ♣❧❛♥❡t❛ K ❄ ✷✻✳ ❉❡t❡r♠✐♥❡ ❞♦✐s ♥ú♠❡r♦s t❛✐s q✉❡ s✉❛ s♦♠❛✱ ♣r♦❞✉t♦ ❡ q✉♦❝✐❡♥t❡ s❡♠♣r❡ s❡❥❛♠ ✐❣✉❛✐s✳ ✷✼✳ ❯♠❛ ❧❡❜r❡ s❡❣✉✐❞❛ ♣♦r ✉♠ ❣❛❧❣♦ ❧❡✈❛ ✉♠❛ ✈❛♥t❛❣❡♠ ❞❡ s❛❧t♦s ❡♥q✉❛♥t♦ q✉❡ ❛ ❧❡❜r❡ ❞á 6 s❛❧t♦s✱ ♠❛✐s✱ 9 50 s❛❧t♦s✳ ❖ ❣❛❧❣♦ ❞á s❛❧t♦s ❞❛ ❧❡❜r❡ ❡q✉✐✈❛❧❡♠ ❛ 7 5 ❞♦ ❣❛❧❣♦✳ ◗✉❛♥t♦s s❛❧t♦s ❞❛rá ❛ ❧❡❜r❡ ❛♥t❡s ❞❡ s❡r ❛❧❝❛♥ç❛❞❛ ♣❡❧♦ ❣❛❧❣♦ ❄ ✷✽✳ ❯♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s é ❞✐t❛ ✉♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛ ❞❡ s❡❣✉♥❞❛ ♦r✲ ❞❡♠ q✉❛♥❞♦ ❛ s❡q✉ê♥❝✐❛ ❢♦r♠❛❞❛ ♣❡❧❛s ❞✐❢❡r❡♥ç❛s ❡♥tr❡ t❡r♠♦s s✉❝❡ss✐✈♦s ❢♦r ✉♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛✳ ❆ss✐♥❛❧❡ ❛ ❛❧t❡r♥❛t✐✈❛ ♥❛ q✉❛❧ s❡ ❡♥❝♦♥tr❛ ♣❛rt❡ ❞❡ ✉♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✳ ❆✮❂④✵✱ ❉✮ ✺✱ ✶✷✱ ✷✶✱ ✷✸⑥ ❂ ④✼✱ ✸✱ ✷✱ ✵✱ ✲✶⑥ ✷✾✳ ▼♦str❡ q✉❡✱ s❡ p ❇✮❂ ❊✮ ④✻✱ ✽✱ ✶✺✱ ✷✼✱ ✹✹⑥ ❈✮ ❂ ④✲✸✱ ✵✱ ✹✱ ✺✱ ✽⑥ ❂④✷✱ ✹✱ ✽✱ ✷✵✱ ✸✵⑥ é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❡ ai ∈ Z, i = 1, 2, 3, · · · , n✱ (a1 + a2 + a3 + · · · + an )p = ap1 + ap2 + ap3 + · · · + apn + kp ✶✺ ❡♥tã♦✿ ♣❛r❛ ❛❧❣✉♠ k ∈ Z✳ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✸✵✳ ❙❡❥❛♠ a, b ∈ Z t❛✐s q✉❡ (a, b) = 1 s❡♥❞♦ a ❡ b ❞✐❢❡r❡♥t❡s ❞❡ ③❡r♦✳ ▼♦str❡ q✉❡ ❡①✐st❡♠ x, y ∈ Z t❛✐s q✉❡ 1 x y 1 ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♥❛ ❢♦r♠❛ = + ab ab a b ✸✶✳ ▼♦str❡ q✉❡ t♦❞♦ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ é ❞❛ ❢♦r♠❛ 5n ♦✉ 5n ± 1 ♣❛r❛ n ∈ Z✳ ✸✷✳ ❱❡r✐✜❝❛r q✉❡ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❝♦♠♣♦st♦ ♣♦r ❝✐♥❝♦ ❛❧❣❛r✐s♠♦s n = xmdcu é ♠ú❧t✐♣❧♦ ❞❡✿ ✶✳ ✷✳ 2 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ u = 0, 2, 4, 6, 8✳ ■st♦ é✱ s❡ ♦ ❛❧❣❛r✐s♠♦ ❞❛s ✉♥✐❞❛❞❡s ❞❡ n ❢♦r ♠ú❧t✐♣❧♦ ❞❡ 2 3 ✭♦✉ 9✮ s❡✱ ❡ s♦♠❡♥t❡ s❡✱❛ s♦♠❛ x + m + c + d + u ❢♦r ❞✐✈✐sí✈❡❧ ♣♦r 3 ✭♦✉ 9✮✳ ❖♥❞❡ x, m, c, d, u sã♦ ♦s ❛❧❣❛r✐s♠♦s ❞❡ n ✸✳ 4 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ ♥ú♠❡r♦ du ❢♦r ♠ú❧t✐♣❧♦ ❞❡ 4✱ ♦✉ n é ❞❛ ❢♦r♠❛ a = xm200✳ ✹✳ 5 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ u = 0 ♦✉ u = 5✳ ✺✳ 6 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ n ❢♦r ❞✐✈✐sí✈❡❧ ♣♦r 2 ❡ 3✳ ✻✳ ✼✳ ✽✳ 8 ✭♦✉ 125✮ s❡✱ ❡ s♦♠❡♥t❡ s❡✱♦ ♥ú♠❡r♦ cdu ❢♦r ❞✐✈✐sí✈❡❧ ♣♦r 8 ✭♦✉ 125✮✱ ♦✉ n ❞❛ ❢♦r♠❛ n = x000✳ 11 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ (d + m) − (x + c + u) ❢♦r ❞✐✈✐sí✈❡❧ ♣♦r 11✳ 25 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ ♥ú♠❡r♦ du ❢♦r ♠ú❧t✐♣❧♦ ❞❡ 25✱ ♦✉ du = 00✳ ✸✸✳ ❉❡t❡r♠✐♥❡ ✉♠❛ r❡❣r❛ q✉❡ ♣❡r♠✐t❛ s❛❜❡r s❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ n é ♠ú❧t✐♣❧♦ ❞❡ 7✳ ✸✹✳ ❯♠❛ ❛r❛♥❤❛ s❡ ❡♥❝♦♥tr❛ ♥♦ ✈ért✐❝❡ A ❞❡ ✉♠ ❝✉❜♦ só❧✐❞♦ ❝✉❥❛ ❛r❡st❛ é ❞❡ 10cm✱ ❡ t❡♠ ❛ ✐♥t❡♥çã♦ ❞❡ ❝❛♣t✉r❛r ✉♠❛ ♠♦s❝❛ q✉❡ s❡ ❡♥❝♦♥tr❛ ♥♦ ✈ért✐❝❡ ♦♣♦st♦ B ✭✈❡r ❋✐❣✉r❛ ✭✶✳✹✮✮✳ ❆ ❛r❛♥❤❛ ❞❡✈❡ ❝❛♠✐♥❤❛r s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ ❞♦ ❝✉❜♦ só❧✐❞♦ ❡ ❡♥❝♦♥tr❛r ♦ ❝❛♠✐♥❤♦ ♠❛✐s ❝✉rt♦✳ ❊♥❝♦♥tr❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ss❡ ❝❛♠✐♥❤♦✳ ❋✐❣✉r❛ ✶✳✹✿ ✶✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✶✳✸ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❘❡❧❛çã♦ ❞❡ ♦r❞❡♠ ❆①✐♦♠❛ ✶✳✶✳ ❉❡ ❡①✐stê♥❝✐❛✳ ◆♦ ❝♦♥❥✉♥t♦ R✱ ❡①✐st❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡♥♦t❛❞♦ R+ ✱ ❝❤❛♠❛❞♦✱ ✏❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s✑✱ q✉❡ ❝✉♠♣r❡ ♦ s❡❣✉✐♥t❡✿ ✐✮ ❚♦❞♦ ♥ú♠❡r♦ r❡❛❧ a ❝✉♠♣r❡ ✉♠❛ ❡ s♦♠❡♥t❡ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿ a ∈ R+ , ♦✉ −a ∈ R+ , ✐✐✮ ❙❡ a ∈ R+ ❡ b ∈ R+ ✱ ❡♥tã♦ a + b ∈ R+ ❡ a=0 a · b ∈ R+ ✳ ❉❡✜♥✐çã♦ ✶✳✶✵✳ a, b ∈ R✱ (b − a) ∈ R+ ✳ ❙❡❥❛♠ ❞✐③✲s❡ q✉❡ ✏ a é ♠❡♥♦r q✉❡ b✑ ❡ s❡ ❡s❝r❡✈❡ a < b✱ s♦♠❡♥t❡ q✉❛♥❞♦ ❉❡st❛ ❞❡✜♥✐çã♦ t❡♠♦s q✉❡ a ∈ R+ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ (a − 0) ∈ R+ ✱ ❧♦❣♦ 0 < a✳ ❖❜s❡r✈❛çã♦ ✶✳✸✳ ✐✮ ✐✐✮ ❙❡ a < b✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r b > a✱ ❡ s❡ ❧ê ✏ b é ♠❛✐♦r q✉❡ ❉✐③✲s❡ q✉❡ ✏ a é ♠❡♥♦r ♦✉ ✐❣✉❛❧ q✉❡ b✑ ❡ s❡ ❡s❝r❡✈❡ a = b✳ a✑ ✳ a≤b s❡ ❡ s♦♠❡♥t❡ s❡ a<b ♦✉ ✐✐✐✮ R+ = { a ∈ R /. 0 < a} = {a ∈ R /. a > 0}✳ ✐✈✮ a ∈ R+ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ 0 < a✱ t❛♠❜é♠ ♣♦❞❡♠♦s ❡s❝r❡✈❡r a > 0✳ Pr♦♣r✐❡❞❛❞❡ ✶✳✻✳ P❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ ✶✳ a = b ♦✉ a<b ✷✳ a2 ≥ 0, ✸✳ ❙❡ a < b ❡ tr✐❝♦t♦♠✐❛ a>b b<c✱ ❡♥tã♦ a<c ❡♥tã♦ a + c < b + c ❡ ✻✳ ❙❡ a < b ❡ ✼✳ ❙❡ a < b ♦✉ t❡♠♦s✿ ∀ a ∈ R (a2 > 0 s❡ a 6= 0) ✹✳ ❙❡ a < b ✱ ✺✳ ❙❡ a < b a, b, c, d ❡ c < d✱ ❡♥tã♦ ∀c∈R tr❛♥s✐t✐✈❛ ♠♦♥♦t♦♥✐❛ ♥❛ s♦♠❛ a+c<b+d c > 0 ✱ ❡♥tã♦ a.c < b.c c < 0✱ ❡♥tã♦ ♣♦s✐t✐✈✐❞❛❞❡ ♠♦♥♦t♦♥✐❛ ♥♦ ♣r♦❞✉t♦ a.c > b.c✳ ✶✼ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✽✳ ❙❡ a<b ✱ ✾✳ ❙❡ a>0 ✱ ✶✵✳ ❙❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ −a > −b✳ ❡♥tã♦ a−1 > 0 ❡♥tã♦ 0 < a < b✱ ✭❙❡ ab ≥ 0 s❡ ❡ s♦♠❡♥t❡ s❡ ✭a ≥0 ✶✷✳ ab ≤ 0 s❡ ❡ s♦♠❡♥t❡ s❡ ✭a ✶✹✳ ✶✺✳ ✶✻✳ a≥0 b ≥ 0❀ a ≤ b ❡ a2 + b2 = 0 a < 0✱ a−1 > b−1 > 0 ❡♥tã♦ ✶✶✳ ✶✸✳ ❙❡ R ❡ ✭❙❡ b ≥ 0✮ ≥0 a−1 < 0✮ ❡♥tã♦ a<b<0 ♦✉ ✭a b ≤ 0✮ ❡ ≤0 ♦✉ ❡♥tã♦ b ≤ 0✮ ❡ ✭a 0 > a−1 > b−1 ✮ ≤0 ❡ b ≥ 0✮ a2 ≤ b2 ✳ s❡ ❡ s♦♠❡♥t❡ s❡ a = 0 ❡ b = 0✳ √ √ ❙❡ a2 ≤ b ✱ ❡♥tã♦ ✲ b≤a≤ b √ √ a2 ≥ b ✱ ❡♥tã♦ a ≥ b ♦✉ a ≤ − b ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ s❡ ❡ s♦♠❡♥t❡ s❡ ✭✶✮ a, b ∈ R✳ ❊♥tã♦✱ a − b ∈ R✱ ♣❡❧♦ ❆①✐♦♠❛ ✭✶✳✶✮✲✭✐✮✱ t❡♠♦s q✉❡ ✉♠❛ ❡ s♦♠❡♥t❡ a − b ∈ R+ ♦✉ −(a − b) ∈ R+ ♦✉ a − b = 0✳ ❊♥tã♦✱ a − b > 0 ♦✉ b − a > 0 ♦✉ a = b✱ ✐st♦ é✱ a > b ♦✉ b > a ♦✉ a = b✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ a ∈ R✱ ❡♥tã♦ a > 0 ♦✉ a < 0 ♦✉ a = 0✳ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s s❡ ❝✉♠♣r❡✿ ❉❡♠♦♥str❛çã♦✳ ❙❡ a∈R ✭✷✮ ❡♥tã♦ a=0 ♦✉ a 6= 0✳ a = 0 ⇒ a2 = 0 ❙❡ a 6= 0 ✱ ❡♥tã♦ a ∈ R+ ♦✉ −a ∈ R+ ✱ ❧♦❣♦ ✭✶✳✸✮ a2 = a.a ∈ R+ ♦✉ a2 = (−a)(−a) ∈ R+ ⇒ a2 > 0 ❉❡ ✭✶✳✸✮ ❡ ✭✶✳✹✮ s❡❣✉❡ q✉❡ ❉❡♠♦♥str❛çã♦✳  ✭✶✳✹✮ a2 ≥ 0✳  ✭✻✮ a < b ❡ c > 0 ❡♥tã♦ b − a ∈ R+ ❡ ❝♦♠♦ c ∈ R+ ✱ ❧♦❣♦ c(b − a) ∈ R+ ✳ ❆ss✐♠✱ (bc − ac) ∈ R+ ✱ ❧♦❣♦ (bc − ac) > 0✱ ❡♥tã♦ bc > ac ♦✉ ac < bc✳ ❙❡ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ a−1 = 0✳ a > 0 ✭✾✮ ❡♥tã♦ a−1 a−1 = 0 ✱ ❡♥tã♦ ❡①✐st❡ ❊st❡ ú❧t✐♠♦ ❝❛s♦ ❧❡✈❛r✐❛ à ✐❣✉❛❧❞❛❞❡ ❙❡  1=0 ❡ ♣❡❧♦ ❆①✐♦♠❛ a−1 > 0 ♦✉ a−1 < 0 ♦✉ q✉❡ a.a−1 = a.0 = 0 ♦ q✉❡ ✭✶✳✶✮ t❡♠♦s é ✐♠♣♦ssí✈❡❧✱ ♣♦✐s t❡rí❛♠♦s q✉❡ é ✉♠ ❛❜s✉r❞♦✳ a.a−1 < 0✱ ❡♥tã♦ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ 1 < 0✱ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ ❞❛ ♠♦♥♦t♦♥✐❛ ❞♦ ♣r♦❞✉t♦ r❡s✉❧t❛✿ ✶✽ a−1 .a < 0.a 09/02/2021 ✱ ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❆ss✐♠✱ r❡s✉❧t❛ q✉❡ s❡ a > 0✱ a−1 > 0✳ ❡♥tã♦  ❉❡♠♦♥str❛çã♦✳ ✭✶✶✮ ab > 0 ❡♥tã♦ a 6= 0 ❡ b 6= 0✳ t❡♠♦s a > 0✳ ❆ss✐♠ b = a (a.b) > 0✳ −1 ❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ a < 0 ❡♥tã♦ a < 0 ❡ b = a−1 (a.b) < 0✳ P♦rt❛♥t♦✱ s❡ a.b > 0 ❡♥tã♦ ✭a < 0 ❡ b < 0✮ ♦✉ ✭a > 0 ❡ b > 0✮ P❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✶✳✶✮✲(10)✱ s❡ −1 P♦rt❛♥t♦ q✉❛♥❞♦ a>0 −1  ❆s ❞❡♠❛✐s ♣r♦♣r✐❡❞❛❞❡s sã♦ ❡①❡r❝í❝✐♦s ♣❛r❛ ♦ ❧❡✐t♦r✳ ❉❡✜♥✐çã♦ ✶✳✶✶✳ ❯♠❛ ❡q✉❛çã♦ é ✉♠❛ ❡①♣r❡ssã♦ ❛❧❣é❜r✐❝❛ q✉❡ ❝♦♥té♠ ♦ sí♠❜♦❧♦ ❞❛ r❡❧❛çã♦ ❞❡ ✐❣✉❛❧✲ ❞❛❞❡✳ x2 − 5 = x; √ 2x − 5 = x4 − 6x. ◆♦ q✉❡ s❡❣✉❡✱ ❡♥t❡♥❞❡r❡♠♦s q✉❡ ✏ r❡s♦❧✈❡r ✉♠❛ ❡q✉❛çã♦ E(x) = 0✑✱ ♦♥❞❡ E(x) é ✉♠❛ ❡①♣r❡ssã♦ ❛❧❣é❜r✐❝❛✱ s✐❣♥✐✜❝❛ ❞❡t❡r♠✐♥❛r ♥ú♠❡r♦s x = a ∈ R ❞❡ ♠♦❞♦ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ E(a) = 0 s❡❥❛ ✈❡r❞❛❞❡✐r❛✳ P♦r ❡①❡♠♣❧♦✱ ❛♦ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ 4x − 8 = 0 ♦❜t❡♠♦s x = 2✱ ♣♦✐s 4(2) − 8 = 0✳ 2 2 P♦r ♦✉tr♦ ❧❛❞♦ ❛♦ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ x + 9 = 0 ♦❜t❡♠♦s q✉❡ x = −9 ✱ ❛ q✉❛❧ ♥ã♦ t❡♠ 2 s♦❧✉çã♦ ❡♠ R✳ ▲❡♠❜r❡✲s❡ q✉❡ x ≥ 0 ∀ x ∈ R✳ ❙ã♦ ❡①❡♠♣❧♦s ❞❡ ❡q✉❛çõ❡s✿ x + 7 = 3; ❖❜s❡r✈❛çã♦ ✶✳✹✳ a, b ∈ R t❛✐s √ b✳ ❞❡♥♦t❛✲s❡ a = √ P♦r ❡①❡♠♣❧♦ 4=2 ❙❡❥❛♠ q✉❡ b > 0✳ −2✱ ♦✉ ◆♦ q✉❡ s❡❣✉❡ ❡♥t❡♥❞❡r❡♠♦s q✉❛❞r❛❞❛ ♥❡❣❛t✐✈❛✳ ❆ss✐♠✱ ❙❡ b < 0✱ √ ♣♦✐s √ 4=2 b ❙❡ a2 = b 22 = (−2)2 = 4✳ ❡ ✲ √ 4 = −2✳ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✶✳✻✮✲(2) ♥ã♦ ❡①✐st❡ Pr♦♣r✐❡❞❛❞❡ ✶✳✼✳ √ ❝♦♠♦ ❛ r❛✐③ q✉❛❞r❛❞❛ ♣♦s✐t✐✈❛ ❡ ✲ ♥ã♦ ❡①✐st❡ r❛✐③ q✉❛❞r❛❞❛ ❞❡ ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s✳ ❙❡❥❛♠ ❞✐③✲s❡ q✉❡✿ ✏ a é r❛✐③ q✉❛❞r❛❞❛ ❞❡ a∈R t❛❧ q✉❡ a2 = b✳ b b✑ ❡ ❝♦♠♦ ❛ r❛✐③ P♦rt❛♥t♦ ❡♠ R ✺ ❋ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛✳ ✳ a, b, c ∈ R ✱ ♦♥❞❡ ♣❡❧❛ ❡①♣r❡ssã♦✿ a 6= 0✱ ❡♥tã♦ ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦✿ x= −b ± ax2 + bx + c = 0✱ é ❞❛❞❛ √ b2 − 4ac 2a ❉❡♠♦♥str❛çã♦✳ ❉✐✈✐❞✐♥❞♦ ❛ ❡q✉❛çã♦ ✺ ❇❤❛s❦❛r❛ ❆❝❤❛r②❛ b c ax2 +bx+c = 0 ♣♦r a 6= 0 r❡s✉❧t❛ ❛ ❡①♣r❡ssã♦ x2 + ( )x + = 0✳ a a (1114 − 1185)✱ ♥❛s❝✐❞♦ ♥❛ ❮♥❞✐❛✳ ❋♦✐ ❡❧❡ q✉❡♠ ♣r❡❡♥❝❤❡✉ ❛❧❣✉♠❛s ❧❛❝✉♥❛s ♥❛ ♦❜r❛ ❞❡ ❇r❛❤♠❛❣✉♣t❛✱ ❞❛♥❞♦ ✉♠❛ s♦❧✉çã♦ ❣❡r❛❧ ❞❛ ❡q✉❛çã♦ ❞❡ P❡❧❧ ❡ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐✈✐sã♦ ♣♦r ③❡r♦✳ ✶✾ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈♦♠♣❧❡t❛♥❞♦ q✉❛❞r❛❞♦s c b b b x + 2 x + + ( )2 = ( )2 2a a 2a 2a 2 ⇒ ❖❜t❡♥❞♦ ❛ r❛✐③ q✉❛❞r❛❞❛ r❡s✉❧t❛✿ x =  −b ± b x+ 2a 2 =( b2 − 4ac b 2 c ) − = 2a a 4a2 √ b2 − 4ac 2a ❊①❡♠♣❧♦ ✶✳✶✶✳ ❘❡s♦❧✈❡r ❛ s❡❣✉✐♥t❡s ❡q✉❛çõ❡s✿ ❛✮ ❝✮ 3x + 2 = 14 − x ❙♦❧✉çã♦✳ x4 − 13x2 + 12 = 0 x2 − 2x − 3 = 0 ❜✮ x3 − 3x2 + x + 2 = 0 ❞✮ ✭❛✮ 3x + 2 = 14 − x✱ ❡♥tã♦ (3x + 2) + x = (14 − x) + x✱ ❧♦❣♦ (3x + x) + 2 = 14✱ ❡♥tã♦ 14 − 2 ✱ ❧♦❣♦ x = 3 é s♦❧✉çã♦ ❞❛ 4x + 2 = 14✳ P❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✶✳✷✮ ✲ (6) ✈❡♠ q✉❡ x = 4 ❡q✉❛çã♦✳  ❙♦❧✉çã♦✳ ✭❜✮ x2 − 2x − 3 = 0✱ ❡♥tã♦ (x + 1)(x − 3) = 0✱ ♣❡❧❛ x = −1 ♦✉ x = 3✳ Pr♦♣r✐❡❞❛❞❡ ✭✶✳✶✮✲(10) s❡❣✉❡ q✉❡ ❉❡ ♦✉tr♦ ♠♦❞♦✱ ❝♦♠♣❧❡t❛♥❞♦ q✉❛❞r❛❞♦s x2 − 2x − 3 = 0 ❡♥tã♦ x2 − 2x + 1 − 3 = 0 + 1 ✐st♦ é x2 − 2x + 1 = 4✱ ❧♦❣♦ (x − 1)2 = 4✳ ❉❛ ❞❡✜♥✐çã♦ ❞❡ r❛✐③ q✉❛❞r❛❞❛ ① ✲ ✶ ❂ ✷ ♦✉ ① ✲ ✶ ❂ ✲✷ ✳ P♦rt❛♥t♦ x = 3 ♦✉ x = −1 é s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦✳  ❙♦❧✉çã♦✳ ✭❝✮ x4 − 13x2 + 12 = 0 ❡♥tã♦ ✭x2 − 12)(x2 − 1) = 0✱ ❛ss✐♠ t❡♠♦s q✉❡ x2 − 12 = 0 ♦✉ x2 − 1 = 0✳ ❉❡ x2 − 1 = 0 s❡❣✉❡ q✉❡ (x − 1)(x + 1) = 0 ✱ ❡♥tã♦ x = −1 ♦✉ x = 1 é s♦❧✉çã♦✳ √ √ √ √ ❉❡ x2 − 12 = 0 s❡❣✉❡ q✉❡ (x − 12)(x + 12) = 0 ❡ x = − 12 ♦✉ x = 12 é s♦❧✉çã♦✳ √ √ P♦rt❛♥t♦✱ x = −1, x = 1, x = − 12 ♦✉ x = 12 sã♦ s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦✳  ❙♦❧✉çã♦✳✭❞✮ x3 − 3x2 + x + 2 = 0✱ ❡s❝r❡✈❡♥❞♦ ♥❛ ❢♦r♠❛ ❞❡ ❢❛t♦r❡s x3 − 3x2 + x + 2 = (x − 2)(x2 − x − 1) = 0✱ ❡♥tã♦ x − 2 = 0 ♦✉ x2 − x − 1 = 0✱ ❝♦♠♣❧❡t❛♥❞♦ q✉❛❞r❛❞♦s ❛ ❡st❛ ú❧t✐♠❛ 5 1 ✐❣✉❛❧❞❛❞❡ r❡s✉❧t❛✿ (x − )2 = . 2 4 √ 1 2 5 5 1 ❉❡ x − 2 = 0 s❡❣✉❡ q✉❡ x = 2 é s♦❧✉çã♦❀ ❞❡ (x − ) = s❡❣✉❡ q✉❡ x = + ♦✉ 2 4 2 2 √ 5 1 é s♦❧✉çã♦✳ x= − 2 2 √ √ 1 5 5 1 P♦rt❛♥t♦✱ x = 2, x = + ♦✉ x = − é s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦✳ 2 2 2 2 ❊①❡♠♣❧♦ ✶✳✶✷✳ ✷✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❡t❡r♠✐♥❛r ♦ ♠❡♥♦r ♥ú♠❡r♦ ♣♦s✐t✐✈♦ M ❞❡ ♠♦❞♦ q✉❡✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x✱ ❛❝♦♥✲ 2 6x − x ≤ M ✳ t❡ç❛ ❙♦❧✉çã♦✳ ❉❡ 6x − x2 ≤ M ❝♦♠♣❧❡t❛♥❞♦ q✉❛❞r❛❞♦s t❡♠♦s q✉❡ 32 − 32 + 6x − x2 ≤ M ✳ ❆ss✐♠ 9 − (x − 3)2 ≤ M ✳ ◗✉❛♥❞♦ x = 3 t❡r❡♠♦s ♦ ♠❡♥♦r ♥ú♠❡r♦ ♣♦s✐t✐✈♦ M = 9✳ ❖❜s❡r✈❡✱ q✉❛♥❞♦ M > 9 t❛♠❜é♠ ❝✉♠♣r❡ ❛s ❝♦♥❞✐çõ❡s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡✳ ❉❡✜♥✐çã♦ ✶✳✶✷✳ P❛rt❡ ✐♥t❡✐r❛✳ x ❞❡♥♦t❛❞❛ ♣♦r [|x|] é ♦ ♠❛✐♦r [|x|] = max{ m ∈ Z /. m ≤ x } ❆ ♣❛rt❡ ✐♥t❡✐r❛ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ q✉❡ ♥ã♦ ✉❧tr❛♣❛ss❛ x✳ ■st♦ é ♥ú♠❡r♦ ✐♥t❡✐r♦ ❉❡st❛ ❞❡✜♥✐çã♦ r❡s✉❧t❛ q✉❡ ♦ ♥ú♠❡r♦ [|x|] é ú♥✐❝♦✱ ❡ s❡♠♣r❡ [|x|] ≤ x✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ [|x|] é ♦ ♠❛✐♦r ✐♥t❡✐r♦ q✉❡ ❝✉♠♣r❡ ❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ ❡ t❡♠♦s q✉❡ x < [|x|] + 1✳ P♦rt❛♥t♦✱ [|x|] é ♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ q✉❡ ❝✉♠♣r❡ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s✿ [|x|] ≤ x < [|x|] + 1 ♦✉ (x − 1) < [|x|] ≤ x✳ ❊①❡♠♣❧♦ ✶✳✶✸✳ ❉❛s ❞❡s✐❣✉❛❧❞❛❞❡s✿ q✉❡ √ 17 < 6, −2 < − 2 < −1 3 √ [| − 2|] = −2 ❡ [|5|] = 5✳ 3 < π < 4, h 17 i = 5, 3 [|π|] = 3, 5< ❡ 5=5<6 r❡s✉❧t❛ Pr♦♣r✐❡❞❛❞❡ ✶✳✽✳ ❙❡❥❛ x ✉♠ ♥ú♠❡r♦ r❡❛❧✿ ✐✮ [| − x|] ❂  −[|x|] s❡ x ∈ Z. −[|x|] − 1 s❡ x ∈ / Z. ✐✐✮ [|x + y|] = [|x|] ✰ [|y|] ♦✉ [|x + y|] = [|x|] ✰ [|y|] ✰ ✶ ✐✐✐✮ [|x + n|] = [|x|] + n ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ n✳ ✐✈✮ [|n · x|] = n−1 h X k=1 k i x+ n ❉❡♠♦♥str❛çã♦✳ ❊①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❊①❡♠♣❧♦ ✶✳✶✹✳ ❛✮ [| − 5|] = −[|5|] ❝✮ h 5 13 i = k6k + 3 3 ❜✮ ❞✮ h h 1 i 1 i =− − 1 = 0 − 1 = −1 2 2 h 5 13 i h 5 i h 13 i = + +1=1+4+1=6 + 3 3 3 3 − ✷✶ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ h 4 7 i h 4 i h 7 i = + =1+1=2 + 3 5 3 5 h 4 7 i h 20 + 21 i h 41 i ❢✮ + = = =2 3 5 15 15 h 7 i h 7 i h 7 1 i h 7 2 i h 7 3 i h 7 4 i = + + + + = + + + + ❣✮ 5 9 9 9 5 9 5 9 5 9 5 ❡✮ =0+0+1+1+1=3 Pr✐♥❝í♣✐♦ ❞❡ ❆rq✉✐♠❡❞❡s✻ ✳ ❙❡ a > 0 ❡ b > 0 sã♦ ♥ú♠❡r♦s r❡❛✐s✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n t❛❧ q✉❡ a·n > b Pr♦♣r✐❡❞❛❞❡ ✶✳✾✳ ❉❡♠♦♥str❛çã♦✳ 1 b > 0✱ s❡♥❞♦ b > 0 t❡♠♦s q✉❡ > 0✳ a a h b b i ❀ ✐st♦ é ❛ ♣❛rt❡ ✐♥t❡✐r❛ ❞♦ ♥ú♠❡r♦ r❡❛❧ (1 + )✳ ❉❛ ❉❡✜♥✐♠♦s ♦ ♥ú♠❡r♦ n = 1 + a a h b i b = n✳ ❉❡✜♥✐çã♦ ✭✶✳✾✮ t❡♠♦s q✉❡ (1 + ) − 1 < 1 + a a P♦rt❛♥t♦✱ a · n > b✳ ❙❡ a > 0✱ ❡♥tã♦ ❊①❡♠♣❧♦ ✶✳✶✺✳ ❙❡❥❛♠ a, b ∈ R+ ✱ t❛✐s q✉❡ a · b = 1✳ ▼♦str❡ q✉❡ a + b ≥ 2✳ ❉❡♠♦♥str❛çã♦✳ ❉❛ ❤✐♣ót❡s❡ a.b = 1 t❡♠♦s q✉❡ 0 < a ≤ 1 ❡ 1 ≤ b, ❡♥tã♦ 0 ≤ (1 − a) ❡ 0 ≤ (b − 1) ⇒ 0 ≤ (1 − a)(b − 1) = b − 1 − a.b + a = b − 1 − 1 + a✳ P♦rt❛♥t♦✱ a + b ≥ 2✳ ❖❜s❡r✈❛çã♦ ✶✳✺✳ ➱ ✐♠♣♦rt❛♥t❡ ❧❡♠❜r❛r ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞❡ ♥ú♠❡r♦s r❡❛✐s✿ a0 = 1 s♦♠❡♥t❡ s❡ a 6= 0❀ ❝❛s♦ a = 0 ❛ ❡①♣r❡ssã♦ 00 ♥ã♦ ❡①✐st❡✳ ✐✮ a a ∈ R✱ s♦♠❡♥t❡ s❡ b 6= 0❀ ❝❛s♦ b = 0 ❡♥tã♦ ♥ã♦ ❡①✐st❡✳ b 0 √ √ √ ✐✐✐✮ a.b = a. b ❞❡s❞❡ q✉❡ a ❡ b s❡❥❛♠ ♣♦s✐t✐✈♦s✱ s✉♣♦♥❤❛ a = −1 ❡ b = −1✱ ❡♥tã♦ p √ √ 1 = (−1)(−1) = −1 −1 ♥ã♦ ❡①✐st❡ ❡♠ R✱ ♦ ♥ú♠❡r♦ 1 ♥ã♦ ❞❡✈❡♠♦s ❡s❝r❡✈❡r ❝♦♠ ❡❧❡♠❡♥t♦s q✉❡ ♥ã♦ ❡①✐st❡♠ ❡♠ R✳ ✐✐✮ ✐✈✮ ❆ ❡①♣r❡ssã♦ ✰∞ é ❛ ✐❞❡✐❛ ❞❡ ✉♠ ♥ú♠❡r♦ ♣♦s✐t✐✈♦✱ ♦ ♠❛✐♦r ❞❡ t♦❞♦s ♣♦ré♠ ✭✰∞✮ +∞ ❂ ❄ sã♦ ❢♦r♠❛s ✐♥❞❡t❡r♠✐♥❛❞❛s✳ ◆ã♦ s❡ ❞❡✈❡ ♦♣❡r❛r ❝♦♠ ♦s ✲ ✭✰∞✮ ❂ ❄✱ ♦✉ +∞ sí♠❜♦❧♦s +∞, −∞✱ ❝♦♠♦ s❡ ❢♦ss❡♠ ♥ú♠❡r♦s✱ ♣♦✐s ♥ã♦ ♦ sã♦✳ ✻ ❆rq✉✐♠❡❞❡s (287 − 212 a.C.)✱ ❝❤❛♠❛❞♦ ✏♦ ♠❛✐♦r ✐♥t❡❧❡❝t♦ ❞❛ ❛♥t✐❣✉✐❞❛❞❡✑✱ ❢♦✐ ✉♠ ❞♦s ♣r✐♠❡✐r♦s ❢✉♥❞❛❞♦r❡s ❞♦ ♠ét♦❞♦ ❝✐❡♥tí✜❝♦✳ ✷✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❆♣❧✐❝❛çõ❡s ❝♦♠ ♥ú♠❡r♦s r❡❛✐s ❊①❡♠♣❧♦ ✶✳✶✻✳ ❊♠ ❛♠❜❛s ❛s ♠❛r❣❡♥s ❞❡ ✉♠ r✐♦ ❝r❡s❝❡♠ ♣❛❧♠❡✐r❛s✱ ✉♠❛ ❡♠ ❢r❡♥t❡ à ♦✉tr❛✳ ❆ ❛❧t✉r❛ ❞❡ ✉♠❛ é ❞❡ 30 m✱ ❡ ❞❛ ♦✉tr❛ é 20 m✳ ❆ ❞✐stâ♥❝✐❛ ❡♥tr❡ s❡✉s tr♦♥❝♦s é ❞❡ 50 m✳ ◆❛ ♣❛rt❡ ♠❛✐s ❛❧t❛ ❞❡ ❝❛❞❛ ♣❛❧♠❡✐r❛ ❞❡s❝❛♥s❛♠ ♣áss❛r♦s✱ ❞❡ sú❜✐t♦ ❞♦✐s ♣áss❛r♦s ✭✉♠ ❡♠ ❝❛❞❛ ♣❛❧♠❡✐r❛✮ ❛✈✐st❛♠ ✉♠ ♣❡✐①❡ q✉❡ ❛♣❛r❡❝❡ ♥❛ s✉♣❡r❢í❝✐❡ ❞❛ á❣✉❛✱ ❡♥tr❡ ❛s ❞✉❛s ♣❛❧♠❡✐r❛s✳ ❖s ♣áss❛r♦s ✈♦❛rã♦ ❡ ❛❧❝❛♥ç❛r❛♠ ♦ ♣❡✐①❡ ❛♦ ♠❡s♠♦ t❡♠♣♦✳ ❙✉♣♦♥❞♦ ❛ ♠❡s♠❛ ✈❡❧♦❝✐❞❛❞❡❀ ❛ q✉❡ ❞✐stâ♥❝✐❛ ❞♦ tr♦♥❝♦ ❞❛ ♣❛❧♠❡✐r❛ ♠❡♥♦r ❛♣❛r❡❝❡✉ ♦ ♣❡✐①❡❄ ✉ ❅ ❅ ❅ ❙♦❧✉çã♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ♦ ♣❡✐①❡ ❛♣❛r❡❝❡✉ ❛ x ✉♠❛ ❞✐stâ♥❝✐❛ ❞❡ ♠❡tr♦s ❞♦ ♣é ❞❛ ♣❛❧✲ ♠❡✐r❛ ♠❡♥♦r ❋✐❣✉r❛ ✭✶✳✺✮✱ ❡♥tã♦ ♣❡❧♦ t❡♦✲ r❡♠❛ ❞❡ P✐tá❣♦r❛s✿ √ 202 + x2 = ❅ ❅ ❅ 30 m p ✉ ❅ ❅ (50 − x) 302 + (50 − x)2 20 m ❅ x 50 m 202 + x2 = 302 + (50 − x)2 ❋✐❣✉r❛ ✶✳✺✿ x2 − (50 − x)2 = 302 − 202 ⇒ 2x − 50 = 10 ⇒ x = 30 P♦rt❛♥t♦✱ ♦ ♣❡✐①❡ ❛♣❛r❡❝❡✉ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ❞❡ 30 m ❞❛ ♣❛❧♠❡✐r❛ ♠❡♥♦r✳ ❊①❡♠♣❧♦ ✶✳✶✼✳ ▼♦str❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❙♦❧✉çã♦✳ ❙✉♣♦♥❤❛♠♦s y= 99 1 1 3 5 < ✳ x = . . ··· 2 4 6 100 10 100 2 4 6 · · ··· 3 5 7 101 2 1 < , 2 3 r❡s✉❧t❛ q✉❡ x<y ❧♦❣♦✱ ✱ ❞❡s❞❡ q✉❡✿ 3 4 < , 4 5 5 6 99 100 < , ··· , < 6 7 100 101 1 2 3 4 56 99 100 x2 < xy = . . . · ··· · 2 3 4 5 67 100 101 ✳ ❊①tr❛✐♥❞♦ ❛ r❛✐③ q✉❛❞r❛❞❛ ❞❡ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❡st❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♦❜t❡♠♦s x< √ 1 1 < 10 101 ❊①❡♠♣❧♦ ✶✳✶✽✳ ◗✉❡r❡♠♦s ❝♦♥str✉✐r ✉♠❛ ❝❛✐①❛ ❞❡ ♣❛♣❡❧ã♦ ❞❡ ❞❡ ❧❛r❣✉r❛ 10 cm ♠❡♥♦s q✉❡ s❡✉ ❝♦♠♣r✐♠❡♥t♦✳ ✷✸ 10 cm ❛❧t✉r❛✱ s❡♥❞♦ ❛ ❜❛s❡ ✉♠ r❡tâ♥❣✉❧♦ ❙❡ ♦ ✈♦❧✉♠❡ ❞❛ ❝❛✐①❛ ❞❡✈❡ s❡r ❞❡ 6000 cm3 ✱ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R q✉❛✐s ❛s ❞✐♠❡♥sõ❡s ❞❛ ❝❛✐①❛ q✉❡ s✉♣♦rt❛ ♠❛✐♦r ✈♦❧✉♠❡❄ ❙♦❧✉çã♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ s❡❥❛ xcm✳ ❊♥tã♦ s❡❣✉♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛ t❡♠♦s ✉♠❛ ❝❛✐①❛ ❝♦♠♦ ♥❛ ❋✐❣✉r❛ 10 ♣ ♣ ♣ ♣ ♣ ♣ ✭✶✳✻✮✳ 10x(x−10) = 6000 ⇒ x(x−10) = 2 600 ⇒ x − 10x − 600 = 0✳ ▲♦❣♦ P❡❧❛ x= Pr♦♣r✐❡❞❛❞❡ 10 ± P♦rt❛♥t♦ 30 cm x − 10 x ✭✶✳✼✮✿ p 102 − 4(−600) = 5 ± 25 2 x = 30✱ ♣ · · · · · · · · · · · · · ···· · ❋✐❣✉r❛ ✶✳✻✿ ❡ ❛s ❞✐♠❡♥sõ❡s ❞❛ ❝❛✐①❛ sã♦✿ ❛❧t✉r❛ ❡ ❧❛r❣✉r❛ ❞❛ ❜❛s❡ 10 cm✱ ❡ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❜❛s❡ 20 cm✳  ❊①❡♠♣❧♦ ✶✳✶✾✳ ❉❡t❡r♠✐♥❡ ❛ ♣❛rt❡ ✐♥t❡✐r❛ ❞♦ ♥ú♠❡r♦✿ 1 1 1 1 x=1+ √ + √ + √ + √ ✳ 2 3 4 5 ❙♦❧✉çã♦✳ ❖❜s❡r✈❡✱ ❝❛❧❝✉❧❛♥❞♦ ❛s r❛í③❡s ♣♦r ❢❛❧t❛ ❡ ♣♦r ❡①❝❡ss♦ ❡♠ ♠❡♥♦s ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡s✿ 1 ≤ 1 ≤ 1, 1 0.7 < √ < 0.8, 2 1 0.5 < √ < 0.6, 3 ❙♦♠❛♥❞♦ ❡st❛s ❞❡s✐❣✉❛❧❞❛❞❡s✱ ❡♥❝♦♥tr❛♠♦s q✉❡ 0.8 + 0.6 + 0.5 + 0.5 ▲♦❣♦ ✐st♦ é 0, 1 ♦❜t❡♠♦s ❛s 1 1 0.5 ≤ √ ≤ 0.5 0.4 < √ < 0.5 4 5 1 + 0.7 + 0.5 + 0.5 + 0.4 < x < 1 + 3, 1 < x < 3, 4✳ [|x|] = 3✳ ❊①❡♠♣❧♦ ✶✳✷✵✳ ❉❡❝♦♠♣♦r ♦ ♥ú♠❡r♦ 60 ❡♠ ❞✉❛s ♣❛rt❡s ❞❡ ♠♦❞♦ q✉❡ ♦ ♣r♦❞✉t♦ ❞❡ ❛♠❜❛s ❛s ♣❛rt❡s s❡❥❛ ♦ ♠❛✐♦r ♣♦ssí✈❡❧✳ ❙♦❧✉çã♦✳ ❈♦♥s✐❞❡r❡♠♦s ♦s ♥ú♠❡r♦s ❙❡✉ ♣r♦❞✉t♦ é✿ P❛r❛ q✉❡ ♦ ♣r♦❞✉t♦ sã♦✿ 30 ❡ 60 − 30✳ x ❡ 60 − x✱ ♦❜s❡r✈❡ q✉❡ ❛ ❛❞✐çã♦ ❞❡ ❡ss❡s ♥ú♠❡r♦s é 30 ❡ 2 2 2 60✳ P = x(60−x) = 60x−x = 30 −30 +2(30)x−x = 30 −(30−x)2 ✳ s❡❥❛ ♦ ♠❛✐♦r ♣♦ssí✈❡❧ t❡♠ q✉❡ ❛❝♦♥t❡❝❡r q✉❡ x = 30✳ ❧♦❣♦ ♦s ♥ú♠❡r♦s P♦rt❛♥t♦✱ ♦s ♥ú♠❡r♦s sã♦ 2 2 30✳ ❊①❡♠♣❧♦ ✶✳✷✶✳ ❙❛❜❡✲s❡ q✉❡ ❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛ ❞❡ n ♥ú♠❡r♦s✱ é s❡♠♣r❡ ♠❡♥♦r ♦✉ ✐❣✉❛❧ à s✉❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛✳ ✷✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❡ t♦❞♦s ♦s ♣❛r❛❧❡❧❡♣í♣❡❞♦s ❝♦♠ s♦♠❛ ✜①❛ ❞❡ s✉❛s três ❛r❡st❛s r❡❝✐♣r♦❝❛♠❡♥t❡ ♣❡r♣❡♥✲ ❞✐❝✉❧❛r❡s✱ ❞❡t❡r♠✐♥❡ ♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ✈♦❧✉♠❡ ♠á①✐♠♦✳ ❙♦❧✉çã♦✳ ❙❡❥❛ m = a + b + c ❛ s♦♠❛ ❞❛s ❛r❡st❛s ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦✳ ▲♦❣♦ s❡✉ ✈♦❧✉♠❡ é V = abc✳ ❆♣❧✐❝❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛ s❡❣✉❡ √ 3 V = √ 3 abc ≤ m a+b+c = 3 3 ❖ ✈♦❧✉♠❡ s❡rá ♠á①✐♠♦ s♦♠❡♥t❡ q✉❛♥❞♦ V = c= m ✳ 3 ⇒ V ≤ m3 27 m3 ❡ ✐st♦ ❛❝♦♥t❡❝❡ s♦♠❡♥t❡ s❡ a = b = 27 P♦rt❛♥t♦ ♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ é ♦ ❝✉❜♦✳  ❊①❡♠♣❧♦ ✶✳✷✷✳ ▼♦str❡ q✉❡✱ s❡ ai > 0 i = 1, 2, 3, · · · , n ❡♥tã♦✿ n · a1 a2 a3 . · · · an−1 an ≤ an1 + an2 + an3 + · · · + ann−1 + ann ❙♦❧✉çã♦✳ P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛ t❡♠♦s q✉❡✿ a1 a2 a3 · · · an = p n an1 an2 an3 · · · ann−1 ann ≤ an1 + an2 + an3 + · · · + ann−1 + ann n ❧♦❣♦ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r n s❡❣✉❡✿ n · a1 a2 a3 . · · · an−1 an ≤ an1 + an2 + an3 + · · · + ann−1 + ann ❉❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡❞✉③✐♠♦s✿ 2a1 a2 ≤ a21 + a22 , 3a1 a2 a3 ≤ a31 + a32 + a33 , 4a1 a2 a3 a4 ≤ a41 + a42 + a43 + a44 ❊①❡♠♣❧♦ ✶✳✷✸✳ ❯♠❛ ✜❧❛ ❞❡ ❝❛❞❡✐r❛s ♥♦ ❝✐♥❡♠❛ t❡♠ 10 ♣♦❧tr♦♥❛s✳ ❉❡ q✉❛♥t♦s ♠♦❞♦s 3 ❝❛s❛✐s ♣♦❞❡♠ s❡ s❡♥t❛r ♥❡ss❛s ♣♦❧tr♦♥❛s ❞❡ ♠♦❞♦ q✉❡ ♥❡♥❤✉♠ ♠❛r✐❞♦ s❡ s❡♥t❡ s❡♣❛r❛❞♦ ❞❡ s✉❛ ♠✉❧❤❡r❄ ❙♦❧✉çã♦✳ ▲❡♠❜r❛♥❞♦ q✉❡ ♦ ❢❛t♦r✐❛❧ ❞❡ ✉♠ ♥ú♠❡r♦ n ∈ N ❝♦♠ n ≥ 2 é ❞❡✜♥✐❞♦ ♣♦r n! = n(n − 1)(n − 2) . . . (3)(2)(1), ✷✺ 0! = 1 ❡ 1! = 1 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊♥tã♦✱ ❡s❝♦❧❤✐❞❛ ❛ ♦r❞❡♠ ❞❡ ❝❛❞❛ ❝❛s❛❧✱ ♦ q✉❡ ♣♦❞❡ s❡r ❢❡✐t♦ ❞❡ 23 ♠♦❞♦s t❡♠♦s q✉❡ ❛rr✉♠❛r ❡♠ ✜❧❛ 4 ❡s♣❛ç♦s ✈❛③✐♦s ❡ 3 ❝❛s❛✐s✱ ♦ q✉❡ ♣♦❞❡ s❡r ❢❡✐t♦ ❞❡ 7! ♠♦❞♦s ✭❡s❝♦❧❤❛ (3!)(4!) ❞♦s ❡s♣❛ç♦s ✈❛③✐♦s✮ ✈❡③❡s 3! ✭❝♦❧♦❝❛çã♦ ❞♦s 3 ❝❛s❛✐s ♥♦s 3 ❧✉❣❛r❡s r❡st❛♥t❡s✮✳ ❆ r❡s♣♦st❛ é 23 × 7! × 3! = 1.680✳ (3!)(4!) ❊①❡♠♣❧♦ ✶✳✷✹✳ ❯♠ ♠á❣✐❝♦ s❡ ❛♣r❡s❡♥t❛ ✉s❛♥❞♦ ✉♠ ♣❛❧❡tó ❝✐♥t✐❧❛♥t❡ ❡ ✉♠❛ ❝❛❧ç❛ ❝♦❧♦r✐❞❛ ❡ ♥ã♦ r❡♣❡t❡ ❡♠ s✉❛s ❛♣r❡s❡♥t❛çõ❡s ♦ ♠❡s♠♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❛❧ç❛ ❡ ♣❛❧❡tó✳ P❛r❛ ♣♦❞❡r s❡ ❛♣r❡s❡♥t❛r ❡♠ 500 ❡s♣❡tá❝✉❧♦s✱ q✉❛❧ ♦ ♠❡♥♦r ♥ú♠❡r♦ ❞❡ ♣❡ç❛s ❞❡ r♦✉♣❛ q✉❡ ♣♦❞❡ t❡r s❡✉ ❣✉❛r❞❛✲r♦✉♣❛❄ ❙♦❧✉çã♦✳ ❙❡❥❛ c ♦ ♥ú♠❡r♦ ❞❡ ❝❛❧ç❛s ❡ p ♦ ♥ú♠❡r♦ ❞❡ ♣❛❧❡tós✳ P❡❧♦ ♣r✐♥❝í♣✐♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ❝♦♥t❛❣❡♠ c · p = 500✳ ❖ ♠❡♥♦r ♥ú♠❡r♦ ❞❡ ❡ ♣❡ç❛s é c+p P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❛s ♠é❞✐❛s s❡❣✉❡ √ c·p≤ c+p 2 ⇒ √ 2 c·p≤c+p √ ❈♦♠♦ c · p = 500 ❡♥tã♦ 2 500 ≤ c + p✱ ❧♦❣♦ 44, 72 ≤ c + p P♦rt❛♥t♦ ♦ ♠❡♥♦r ♥ú♠❡r♦ ❞❡ ♣❡ç❛s s❡rá 45✳ ✷✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✶✲✷ 0<a<1 ✶✳ ▼♦str❡ q✉❡✱ s❡ ✷✳ ▼♦str❡ q✉❡✱ a>b≥0 ❡♥tã♦ a, b > 0 ✸✳ ▼♦str❡ q✉❡✱ s❡ a ✹✳ ▼♦str❡ q✉❡✱ s❡ ❡ b a>0 a 2 > b2 ❡ a ✾✳ ▼♦str❡ q✉❡✿ √ ab ≥ ✶✵✳ ▼♦str❡ q✉❡✱ q✉❛♥❞♦ ✶✶✳ ▼♦str❡ q✉❡✿ a3 + ✶✷✳ ▼♦str❡ q✉❡✱ s❡ ♦♥❞❡ ❡♥tã♦ a, b ∈ R✳ a > b✳ 2ab a+b ♠♦str❡ q✉❡ b>a ❡♥tã♦ a−1 > b−1 ✳ 2ab ≤ a2 + b2 ✳ 1 (a + ) ≥ 2✳ a a+b+c = 1 b)(1 − c) ≥ 8abc✳ 0 < a < b✱ b✱ ❡ ❡♥tã♦ ✼✳ ▼♦str❡ q✉❡✱ s❡ ✽✳ ▼♦str❡ q✉❡✿ ❙❡ a2 > b2 sã♦ ♣♦s✐t✐✈♦s ✭♦✉ ♥❡❣❛t✐✈♦s✮ ❡ ✺✳ ❉❛❞♦s ♦s ♥ú♠❡r♦s r❡❛✐s ✻✳ ▼♦str❡ q✉❡✱ s❡ a2 < a✳ ❡♥tã♦ a > 0, b > 0, c > 0✱ ♦♥❞❡✱ ❡♥tã♦ a< a, b, c, d ∈ R✱ a, b, c > 0 ❡♥tã♦ a+b <b 2 ✳ a4 + b4 + c4 + d4 ≥ 4abcd✳ a > 1✳ s❡ bc ac ab + + > a + b + c✳ a b c ❡♥tã♦ ✶✸✳ ❉❡t❡r♠✐♥❛r ♦ ♠❡♥♦r ♥ú♠❡r♦ ab ≤ a, b > 0✳ q✉❛♥❞♦ 1 1 > a2 + 2 3 a a √ (1 − a)(1 − ❡♥tã♦ t❡♠♦s q✉❡✱ M ❞❡ ♠♦❞♦ q✉❡✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x✱ t❡♥❤❛✲s❡ M ❞❡ ♠♦❞♦ q✉❡✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x✱ t❡♥❤❛✲s❡ 2x − x2 ≤ M ✳ ✶✹✳ ❉❡t❡r♠✐♥❛r ♦ ♠❛✐♦r ♥ú♠❡r♦ 2 M ≤ x − 16x✳ ✶✺✳ ❙❡❥❛♠ a ❡ b ♣♦s✐t✐✈♦s✱ ♠♦str❡ q✉❡ ✶✻✳ ❉❡♠♦♥str❛r q✉❡✱ s❡ ✶✼✳ ▼♦str❡ q✉❡✱ s❡ a ❡ b sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ❡♥tã♦ x3 + y 3 + z 3 = 81, ✶✽✳ ▼♦str❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡✿ a b 1 1 + 2 ≥ + ✳ 2 b a a b x > 0, y > 0, z > 0✱ ❡♥tã♦ a3 + b3 ≥ 2  a+b 2 3 xyz ≤ 27✳ x2 + 3 √ ≥ 2✳ x2 + 2 ✷✼ 09/02/2021 ✳ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✶✾✳ ❙❡❥❛♠ 0 < a < b✱ ❞❡t❡r♠✐♥❡ ❛ s♦❧✉çã♦ ❞❛ ✐♥❡q✉❛çã♦ ✷✵✳ ▼♦str❡ q✉❡ s❡ ab ≥ 0✱ ❡♥tã♦ ab ≥ min .{a2 , b2 }✳ 1 1 1 1 + < + ✳ x a+b−x a b 1 8 ✷✶✳ ▼♦str❡ q✉❡✱ s❡ a + b = 1✱ ❡♥tã♦✿ a4 + b4 ≥ ✳ ✷✷✳ ❉❡t❡r♠✐♥❡ t♦❞♦s ♦s ✈❛❧♦r❡s r❡❛✐s ❞❡ r ♣❛r❛ ♦s q✉❛✐s ♦ ♣♦❧✐♥ô♠✐♦✿ (r2 − 1)x2 + 2(r − 1)x + 1✱ s❡❥❛ ♣♦s✐t✐✈♦ ♣❛r❛ t♦❞♦ x ∈ R✳ ✷✸✳ ❉❛❞♦s três ♥ú♠❡r♦s ♣♦s✐t✐✈♦s✱ s❛❜❡✲s❡ q✉❡ s❡✉ ♣r♦❞✉t♦ é 1 ❡ ❛ s♦♠❛ ❞❡ss❡s três ♥ú♠❡r♦s é ♠❛✐♦r q✉❡ ❛ s♦♠❛ ❞♦s s❡✉s ✐♥✈❡rs♦s✳ ▼♦str❡ q✉❡ ✉♠ ❞♦s ♥ú♠❡r♦s é ♠❛✐♦r q✉❡ 1✱ ❡♥q✉❛♥t♦ ♦s ♦✉tr♦s ❞♦✐s sã♦ ♠❡♥♦r❡s q✉❡ 1✳ ✷✹✳ ❖s ❧❛❞♦s a, b ❡ c ❞❡ ✉♠ tr✐â♥❣✉❧♦ s❛t✐s❢❛③❡♠ ❛ r❡❧❛çã♦ a2 + b2 + c2 ≥ ab + ac + bc✳ ❊ss❡ tr✐â♥❣✉❧♦ é ❡q✉✐❧át❡r♦❄  ✷✺✳ ▼♦str❡ q✉❡✱ s❡ a, b ∈ R sã♦ ♥ú♠❡r♦s t❛✐s q✉❡ a+b = 1✱ ❡♥tã♦ a+ 1 2  1 2 25 + b+ ≥ a b 2 ✷✻✳ ❆ r❡❝❡✐t❛ ❞❛ ✈❡♥❞❛ ❞❡ q ✉♥✐❞❛❞❡s ❞❡ ✉♠ ♣r♦❞✉t♦ é R = 240q ❡ ♦ ❝✉st♦ ❞❡ ♣r♦❞✉çã♦ ❞❡ q ✉♥✐❞❛❞❡s é C = 190q + 1500✳ P❛r❛ q✉❡ ❤❛❥❛ ❧✉❝r♦✱ ❛ r❡❝❡✐t❛ ❞❡ ✈❡♥❞❛s ❤á ❞❡ s❡r ♠❛✐♦r q✉❡ ♦ ❝✉st♦✳ P❛r❛ q✉❡ ✈❛❧♦r❡s ❞❡ q ❡st❡ ♣r♦❞✉t♦ ❞❛rá ❧✉❝r♦❄ ✷✼✳ ❆❧é♠ ❞♦ ❝✉st♦ ❛❞♠✐♥✐str❛t✐✈♦ ✜①♦✱ ✭❞✐ár✐♦✮ ❞❡ ❘$350, 00 ♦ ❝✉st♦ ❞❡ ♣r♦❞✉çã♦ ❞❡ q ✉♥✐❞❛❞❡s ❞❡ ❝❡rt♦ ♣r♦❞✉t♦ é ❞❡ ❘$5, 50 ♣♦r ✉♥✐❞❛❞❡✳ ❉✉r❛♥t❡ ♦ ♠ês ❞❡ ♠❛rç♦✱ ♦ ❝✉st♦ t♦t❛❧ ❞❛ ♣r♦❞✉çã♦ ✈❛r✐♦✉ ❡♥tr❡ ♦ ♠á①✐♠♦ ❞❡ ❘$3.210 ❡ ♦ ♠í♥✐♠♦ ❞❡ ❘$1.604 ♣♦r ❞✐❛✳ ❉❡t❡r♠✐♥❡ ♦s ♥í✈❡✐s ❞❡ ♣r♦❞✉çã♦ ♠á①✐♠♦ ❡ ♠í♥✐♠♦ ♣♦r ♠ês✳ ✷✽✳ ❊st❛❜❡❧❡ç❛ ♣❛r❛ q✉❡ ✈❛❧♦r❡s r❡❛✐s ❞❡ x ❛ ár❡❛ ❞❡ ✉♠ ❝ír❝✉❧♦ ❞❡ r❛✐♦ x✿ ❛✮ é ♠❛✐♦r q✉❡ 400π cm2 ❜✮ ♥ã♦ é s✉♣❡r✐♦r ❛ 400π cm2 ✳ ✷✾✳ ❯♠❛ ♣✐s❝✐♥❛ ✐♥❢❛♥t✐❧ ❞❡✈❡ t❡r 1 m ❞❡ ❛❧t✉r❛ ❡ ♦ ❢♦r♠❛t♦ ❞❡ ✉♠ ❜❧♦❝♦ r❡t❛♥❣✉❧❛r✳ ❖ s❡✉ ❝♦♠♣r✐♠❡♥t♦ ♣r❡❝✐s❛ s✉♣❡r❛r à ❧❛r❣✉r❛ ❡♠ 0, 2 m✳ ❈♦♠ q✉❛♥t♦ ❞❡ ❧❛r❣✉r❛ ❡ss❛ ♣✐s❝✐♥❛ ❝♦♠♣♦rt❛rá ♠❛✐s ❞❡ 2.000.000 litros❄ ✭▲❡♠❜r❡t❡✿ 1 m3 = 1.000 litros✮✳ ✸✵✳ ❙❛❜❡✲s❡ q✉❡ s♦❜r❡ ❝❡rt❛s ❝♦♥❞✐çõ❡s ♦ ♥ú♠❡r♦ ❞❡ ❜❛❝tér✐❛s q✉❡ ❝♦♥té♠ ♦ ❧❡✐t❡ s❡ ❞✉♣❧✐❝❛ ❛ ❝❛❞❛ 3 ❤♦r❛s✳ ❈❛❧❝✉❧❛r ♦ ♥ú♠❡r♦ ♣❡❧♦ q✉❛❧ é ♥❡❝❡ssár✐♦ ♠✉❧t✐♣❧✐❝❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❜❛❝tér✐❛s ❞♦ ✐♥✐❝✐♦✱ ♣❛r❛ ♦❜t❡r ♦ ♥ú♠❡r♦ ❞❡ ❜❛❝tér✐❛s ❛♦ ✜♥❛❧ ❞❡ 1 ❞✐❛✳ ✸✶✳ ❖s ❛❧✉♥♦s ❞❛ ❯❋❚✱ ♦r❣❛♥✐③❛r❛♠ ✉♠❛ r✐❢❛ s♦♠❡♥t❡ ♣❛r❛ ❛❧✉♥♦s✳ ❉♦s q✉❛✐s 45 ❝♦♠✲ ♣r❛r❛♠ 2 ♥ú♠❡r♦s✱ ❡ ♦ t♦t❛❧ ❞❡ ❛❧✉♥♦s q✉❡ ❝♦♠♣r❛r❛♠ ✉♠ ♥ú♠❡r♦ ❡r❛ ✷✵% ❞♦ ♥ú♠❡r♦ ❞♦s r✐❢❛s ✈❡♥❞✐❞❛s✱ 80 ♥ã♦ ❝♦♠♣r❛r❛♠ ♥ú♠❡r♦ ♥❡♥❤✉♠ ❡ ♦✉tr♦s ❝♦♠♣r❛r❛♠ 3 ♥ú♠❡r♦s✳ ❙❡ ♦ t♦t❛❧ ❞❡ r✐❢❛s ✈❡♥❞✐❞❛s ❡①❝❡❞❡✉ ❡♠ 33 ❛♦ ♥ú♠❡r♦ ❞❡ ❛❧✉♥♦s✱ ❞✐❣❛ q✉❛♥t♦s ❛❧✉♥♦s ❝♦♠♣r❛r❛♠ s♦♠❡♥t❡ ✉♠ ♥ú♠❡r♦ ❞❛ r✐❢❛✳ ✷✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✶✳✹ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❡s✐❣✉❛❧❞❛❞❡s ❖s ♥ú♠❡r♦s r❡❛✐s ♣♦❞❡♠ s❡r r❡❧❛❝✐♦♥❛❞♦s ❜✐✉♥í✈♦❝❛♠❡♥t❡ ❝♦♠ ♦s ♣♦♥t♦s ❞❡ ✉♠❛ r❡t❛ ❈♦♠ ❡st❛ ✐❞❡♥t✐✜❝❛çã♦✱ ❞❛❞♦s ♦s ♥ú♠❡r♦s ♥❛ r❡t❛ x, y ∈ R ●r❛✜❝❛♠❡♥t❡✳ ✛ r ✲ y−x x x < y ✱ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ (y − x) ✉♥✐❞❛❞❡s✳ ❞❡ ♠♦❞♦ q✉❡ ▲✱ ♦ ♣♦♥t♦ x ❡st❛ à ❡sq✉❡r❞❛ ❞❡ y ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ✛ ▲✳ ▲ r ✲ y ❉❡✜♥✐çã♦ ✶✳✶✸✳ ❆ ❡s❝r✐t❛ ❞❡ ✉♠❛ ♣r♦♣♦s✐çã♦ ♠❛t❡♠át✐❝❛ q✉❡ ❝♦♥tê♠ r❡❧❛çõ❡s ❞♦ t✐♣♦ ♦✉ ✶✳✹✳✶ ≥ <, >, é ❝❤❛♠❛❞❛ ✏❞❡s✐❣✉❛❧❞❛❞❡✑ ≤ ■♥❡q✉❛çã♦ ❯♠❛ ✐♥❡q✉❛çã♦ é ✉♠❛ ❡①♣r❡ssã♦ ❛❧❣é❜r✐❝❛ q✉❡ ❝♦♥té♠ ❛s r❡❧❛çõ❡s ❙ã♦ ❡①❡♠♣❧♦s ❞❡ ✐♥❡q✉❛çõ❡s✿ <, >, ≤ ♦✉ ≥✳ 3x − 4 < 2 + x ■♥❡q✉❛çã♦ ❞❡ ♣r✐♠❡✐r♦ ❣r❛✉ 3x2 − 4x − 5 ≤ 0 ■♥❡q✉❛çã♦ ❞❡ s❡❣✉♥❞♦ ❣r❛✉ x2 − 5x + 4 ≤2 x2 − 4 ■♥❡q✉❛çã♦ r❛❝✐♦♥❛❧ 3x − 4 < 2 + x ≤ 3x2 − 4x ■♥❡q✉❛çã♦ ♠✐st❛ ax − bx ≤ a − b ■♥❡q✉❛çã♦ ❡①♣♦♥❡♥❝✐❛❧ sen2 x − cos2 x ≥ 1 ■♥❡q✉❛çã♦ tr✐❣♦♥♦♠étr✐❝❛ ❘❡s♦❧✈❡r ✉♠❛ ✐♥❡q✉❛çã♦ s✐❣♥✐✜❝❛ ❞❡t❡r♠✐♥❛r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❛❧♦r❡s q✉❡ ❛ ✈❛r✐á✈❡❧ ✭✐♥❝ó❣♥✐t❛✮ t❡♠ q✉❡ ❛ss✉♠✐r✱ ❞❡ ♠♦❞♦ q✉❡✱ ❛♦ s✉❜st✐t✉✐r ♥❛ ✐♥❡q✉❛çã♦ ❡♠ ❡st✉❞♦✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ s❡❥❛ ✈❡r❞❛❞❡✐r❛✳ ❖ ❝♦♥❥✉♥t♦ ❡♠ r❡❢❡rê♥❝✐❛ é ❝❤❛♠❛❞♦ ✏ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ ✑✳ ❖❜s❡r✈❛çã♦ ✶✳✻✳ x<y ❙❡ t✐✈❡r♠♦s ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❡ y<z ❞❡t♦♥❛✲s❡ x < y < z✳ ❉❡ ✐❣✉❛❧ ♠♦❞♦✿ ❛✮ x < y ≤ z x<y s✐❣♥✐✜❝❛ ❡ y ≤ z✳ ❜✮ x ≥ y ≥ z s✐❣♥✐✜❝❛ x≥y ❡ y ≥ z✳ ❝✮ x ≥ y > z s✐❣♥✐✜❝❛ x≥y ❡ y > z✳ ❞✮ x ≥ y ≤ z ♥ã♦ t❡♠ s✐❣♥✐✜❝❛❞♦✱ é ♠❡❧❤♦r ❡s❝r❡✈❡r r❡❧❛çã♦ ❡♥tr❡ x ❡ z✳ ✷✾ y ≤ z ❡ y ≤ x✱ ♥ã♦ ❤❛✈❡♥❞♦ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✶✳✹✳✷ ■♥t❡r✈❛❧♦s ❙❡❥❛♠ a ❡ b ♥ú♠❡r♦s r❡❛✐s t❛✐s q✉❡ R✳ (a, b) = { x ∈ R /. a < x < b} s✉❜❝♦♥❥✉♥t♦s ❞❡ a ≤ b✳ ❙ã♦ ❝❤❛♠❛❞♦s ❞❡ ✐♥t❡r✈❛❧♦s ♦s s❡❣✉✐♥t❡s ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ ❞❡ ❡①tr❡♠♦s a ❞❡ ♥ú♠❡r♦s r❡❛✐s ❝♦♠♣r❡❡♥❞✐❞♦s ❡str✐t❛♠❡♥t❡ ❡♥tr❡ a ❡ b✱ ✐st♦ é✱ ♦ ❝♦♥❥✉♥t♦ b✳ ❡ [a, b] = { x ∈ R /. a ≤ x ≤ b} ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ ❞❡ ❡①tr❡♠♦s (a, b] = { x ∈ R /. a < x ≤ b} ✐♥t❡r✈❛❧♦ s❡♠✐✲❛❜❡rt♦ ♣❡❧❛ ❡sq✉❡r❞❛ ❞❡ ❡①tr❡♠♦s ❞❡ ♥ú♠❡r♦s r❡❛✐s ❝♦♠♣r❡❡♥❞✐❞♦s ❡♥tr❡ a ❡ b ✭✐♥❝❧✉✐♥❞♦ ♦s ♣♦♥t♦s ❡ b é✱ ♦ ❝♦♥❥✉♥t♦ a ❡ b b✮✳ [a, b) = { x ∈ R /. a ≤ x < b} ✐♥t❡r✈❛❧♦ s❡♠✐✲❛❜❡rt♦ ♣❡❧❛ ❞✐r❡✐t❛ ❞❡ ❡①tr❡♠♦s a ❡ b✱ ✐st♦ é ♦ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s ❝♦♠♣r❡❡♥❞✐❞♦s ❡♥tr❡ a ❡ b ✭✐♥❝❧✉✐♥❞♦ ♦ ♣♦♥t♦ a✮✳ ▲♦❣♦✱ ✉♠ s✉❜❝♦♥❥✉♥t♦ I ❞❡ R é ✉♠ ✐♥t❡r✈❛❧♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ s❡❣✉✐♥t❡ ❝✉♠♣r❡ ❛ ✐st♦ é✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s ❝♦♠♣r❡❡♥❞✐❞♦s ❡♥tr❡ a a ❡ b✱ ✐st♦ a ❡ b✮✳ ✭✐♥❝❧✉✐♥❞♦ ♦ ♣♦♥t♦ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ Pr♦♣r✐❡❞❛❞❡ ✶✳✶✵✳ ✳ I ✉♠ ✐♥t❡r✈❛❧♦ ❡♠ R✱ x < z < y ✱ t❡♠♦s z ∈ I ✳ ❙❡❥❛ ❝♦♠ ♣❛r❛ q✉❛✐sq✉❡r x, y ∈ I ❝♦♠ x<y ❡ ♣❛r❛ q✉❛❧q✉❡r z∈R ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ✶✳✹✳✸ ❆ r❡t❛ ❛♠♣❧✐❛❞❛✳ ■♥t❡r✈❛❧♦s ✐♥✜♥✐t♦s ❘❡t❛ ❛♠♣❧✐❛❞❛ é ♦ ❝♦♥❥✉♥t♦ ♥✉♠ér✐❝♦ ❡ +∞ R = R ∪ {−∞, +∞}✱ ♦♥❞❡ −∞ ✭♠❡♥♦s ✐♥✜♥✐t♦✮ ✭♠❛✐s ✐♥✜♥✐t♦✮ sã♦ sí♠❜♦❧♦s q✉❡ s❡ ❝♦♠♣♦rt❛♠ s❡❣✉♥❞♦ ❛s s❡❣✉✐♥t❡s ❝♦♥✈❡♥çõ❡s✳ ✶✳ −∞ < x < +∞ ∀ x ∈ R ✷✳ ✸✳ x.(±∞) = (±∞).x = (±∞) ∀ x ∈ R ✹✳ ✺✳ x.(±∞) = (±∞).x = (∓∞) ∀ x ∈ R x < 0✳ x + (±∞) = (±∞) + x = (±∞✮ (±∞) + (±∞) = (±∞) x > 0✳ ❖s ✐♥t❡r✈❛❧♦s ✐♥✜♥✐t♦s sã♦ ❞❡✜♥✐❞♦s ❝♦♠♦✿ (a, +∞) = { x ∈ R /. a < x } [a, +∞) = { x ∈ R /. a ≤ x } (−∞, b) = { x ∈ R /. x < b } (−∞, b] = { x ∈ R /. x ≤ b } ❖s sí♠❜♦❧♦s ✲∞, +∞ ❡ ∞ s♦♠❡♥t❡ sã♦ ✐❞❡✐❛s ❞❡ ✏ ♥ú♠❡r♦s ✑ ♣♦ré♠ ♥ã♦ s❡ ❝♦♠♣♦rt❛♠ ❝♦♠♦ ♥ú♠❡r♦s✳ ❊①❡♠♣❧♦ ✶✳✷✺✳ ❉❛❞♦s ♦s ✐♥t❡r✈❛❧♦s A = [3, 5], B = (4, 7] ❡ C = [8, 10] ❡♥tã♦✿ a) A ∪ C = [3, 5] ∪ [8, 10] b) A ∪ B = [3, 7] c) A ∩ C = ∅ d) A ∩ B = (4, 5] ✸✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❖❜s❡r✈❛♠♦s q✉❡ ❛ ✉♥✐ã♦ ♦✉ ✐♥t❡rs❡çã♦ ❞❡ ❞♦✐s ✐♥t❡r✈❛❧♦s ♥❡♠ s❡♠♣r❡ é ✉♠ ✐♥t❡r✈❛❧♦✳ ❊①❡♠♣❧♦ ✶✳✷✻✳ ❙❡❥❛ ❙♦❧✉çã♦✳ x ∈ (1, 2] ❉❛ ❤✐♣ót❡s❡ x2 − 2x ∈ (−1, 0]✳ ✱ ♠♦str❡ q✉❡ x ∈ (1, 2] t❡♠♦s q✉❡ 1 < x ≤ 2✱ ❡♥tã♦ 0 < x − 1 ≤ 1✳ ▲♦❣♦ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ♣❛r❛ ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s 2 (x − 1) − 1 ≤ 0✱ P♦rt❛♥t♦✱ ✐st♦ é 2 −1 < x − 2x ≤ 0✳ 0 < (x − 1)2 ≤ 1✱ ❛ss✐♠ −1 < x2 − 2x ∈ (−1, 0]✳ ❊①❡♠♣❧♦ ✶✳✷✼✳ ❙❡ x ∈ (0, 2)✱ ❞❡t❡r♠✐♥❡ ♥ú♠❡r♦s x ∈ (0, 2)✱ ❡♥tã♦ m M ❡ ❞❡ ♠♦❞♦ q✉❡✿ ❙♦❧✉çã♦✳ ❙❡ 0<x<2 ✱ ❧♦❣♦ m< x+2 < M✳ x+5 5 < x + 5 < 7✳ ❉❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ✐♥✈❡rs♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s t❡♠♦s✿ 1 1 1 < < 7 x+5 5 P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❡ x ∈ (0, 2) ✭✶✳✺✮ s❡❣✉❡ q✉❡✿ 2<x+2<4 ✭✶✳✻✮ ❉❡ ✭✶✳✺✮ ❡ ✭✶✳✻✮ t❡♠♦s ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ♠♦♥♦t♦♥✐❛ ♣❛r❛ ♦ ♣r♦❞✉t♦ q✉❡✿ x+2 4 2 < < 7 x+5 5 P♦rt❛♥t♦✱ m= 2 7 ❡ M= 4 5 ✭❡st❡s ♥ã♦ sã♦ ♦s ú♥✐❝♦s ♥ú♠❡r♦s✮✳ ❊①❡♠♣❧♦ ✶✳✷✽✳ ❉❡t❡r♠✐♥❛r ❡♠ t❡r♠♦s ❞❡ ✐♥t❡r✈❛❧♦s ♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ ❞❛ ✐♥❡q✉❛çã♦✿ 3x−4 < 2+x✳ ❙♦❧✉çã♦✳ ❚❡♠♦s q✉❡ 3x − 4 < 2 + x✱ ❡♥tã♦ 2x < 6 P♦rt❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ é ♦ ✐♥t❡r✈❛❧♦ ❀ ❧♦❣♦ x < 3✳ (−∞, 3)✳ ❊①❡♠♣❧♦ ✶✳✷✾✳ ❘❡s♦❧✈❡r ❛ ✐♥❡q✉❛çã♦ ❙♦❧✉çã♦✳ x2 − 4 < x + 2 ✳ ✸✶ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 1o ▼ét♦❞♦✳ x2 − 4 < x + 2 ⇒ x2 − x − 6 < 0 ⇒ (x + 2)(x − 3) < 0 ⇒ {x + 2 > 0 ❡ x − 3 < 0} ♦✉ {x + 2 < 0 ❡ x − 3 > 0} ⇒ {x > −2 ❡ x < 3} ♦✉ {x < −2 ❡ x > 3} ⇒ x ∈ (−2, 3) ♦✉ x ∈ ∅ ⇒ x ∈ (−2, 3) P♦rt❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ ❞❛ ✐♥❡q✉❛çã♦ é (−2, 3) 2o ▼ét♦❞♦✳❈♦♠♣❧❡t❛♥❞♦ q✉❛❞r❛❞♦s✳ 1 1 x2 − 4 < x + 2 ⇒ x2 − x < 6 ⇒ x2 − x + < 6 + 4 4 5 1 5 1 2 25 ⇒ − < x− < ⇒ −2 < x < 3 ⇒ (x − ) < 2 4 2 2 2 P♦rt❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ ❞❛ ✐♥❡q✉❛çã♦ é (−2, 3) ⇒ x ∈ (−2, 3) 3o ▼ét♦❞♦✳ ▼ét♦❞♦ ❞♦s ♣♦♥t♦s ❝rít✐❝♦s✳ x2 − 4 < x + 2 ⇒ x2 − x − 6 < 0 ⇒ (x + 2)(x − 3) < 0. ❖s ✈❛❧♦r❡s ❞❡ x ♣❛r❛ ♦s q✉❛✐s ✈❡r✐✜❝❛✲s❡ ❛ ✐❣✉❛❧❞❛❞❡ (x + 2)(x − 3) = 0✱ sã♦ x = −2 ❡ x = 3✳ + + + + +− + r− − − − − − − r+ + + + + + ✲ + ✛ −∞ −2 3 +∞ ◆♦ ❞✐❛❣r❛♠❛✱ ♦❜s❡r✈❛♠♦s q✉❡ (x + 2)(x − 3) < 0 s❡ x ∈ (−2, 3) P♦rt❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ ❞❛ ✐♥❡q✉❛çã♦ é (−2, 3)✳ ❖❜s❡r✈❛çã♦ ✶✳✼✳ ✶✮ P❛r❛ ❞❡t❡r♠✐♥❛r ♦ s✐♥❛❧ ❞♦ ❢❛t♦r ❙❡✱ ♦ s✐♥❛❧ ❞❡ ❞❡ a✳ (x − a) ❙❡✱ ♦ s✐♥❛❧ ❞❡ ❡sq✉❡r❞❛ ❞❡ ✷✮ a✳ ❝♦♥s✐❞❡r❡✿ é ♣♦s✐t✐✈♦✱ ❡♥tã♦ (x − a) (x − a) > 0 é ♥❡❣❛t✐✈♦✱ ❡♥tã♦ ❡ (x − a) < 0 x > a✱ ❡ ❧♦❣♦ x x < a✱ E1 (x) ❞❛ ❢♦r♠❛ E1 (x) > 0 ♦✉ E1 (x) ≥ 0 P❛r❛ ❞❡t❡r♠✐♥❛r ♦ s✐♥❛❧ ❞❡ ✉♠ ♣r♦❞✉t♦✱ ❝♦♥s✐❞❡r❡✿ (−)(+) = − ❡ (−)(−) = +✳ ♦✉ ❡st❛ à ❞✐r❡✐t❛ ❧♦❣♦ x ❡st❛ à E(x) < 0 E1 (x) ≤ 0✳ ❖ ♠ét♦❞♦ ❞♦s ♣♦♥t♦s ❝rít✐❝♦s ❝♦♥s✐st❡ ❡♠ tr❛♥s❢♦r♠❛r ❛ ✐♥❡q✉❛çã♦ ❞❛❞❛ ♦✉tr❛ ❡q✉✐✈❛❧❡♥t❡ ✸✮ x − a✱ (+)(+) = +, ❡♠ (+)(−) = − ✱ ▲♦❣♦✱ ❞❡✈❡♠♦s ❞❡t❡r♠✐♥❛r ♦s ♣♦♥t♦s ❝rít✐❝♦s ❞❡ E1 (x)❀ ✐st♦ é✱ ♦s ✈❛❧♦r❡s ❞♦ ♥✉♠❡r❛❞♦r ❡ ❞❡♥♦♠✐♥❛❞♦r ❞❡ E1 (x) ♦s q✉❛✐s s❡❥❛♠ ✐❣✉❛✐s ❛ ③❡r♦✱ ♣❛r❛ ❛ss✐♠ ❞❡t❡r♠✐♥❛r ♥❛ r❡t❛ r❡❛❧ R ♦s ✐♥t❡r✈❛❧♦s r❡s♣❡❝t✐✈♦s✳ P♦r ú❧t✐♠♦✱ t❡♠♦s q✉❡ ❞❡t❡r♠✐♥❛r ♦ s✐♥❛❧ ❞❡ E1 (x) ❡♠ ❝❛❞❛ ✉♠ ❞♦s ✐♥t❡r✈❛❧♦s q✉❡ ❝✉♠♣r❡♠ ❛ ✐♥❡q✉❛çã♦✳ ✸✷ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❖ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s s✐♥❛✐s ❡♠ ✉♠❛ ✐♥❡q✉❛çã♦ ♣r♦✈é♠ ❞♦ ❣rá✜❝♦ ❞❡ ❢✉♥çõ❡s ♣♦❧✐♥♦✲ ♠✐❛✐s ♥✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s✳ ❊st❡ tó♣✐❝♦ s❡rá tr❛t❛❞♦ ♣♦st❡r✐♦r♠❡♥t❡✳ ❊①❡♠♣❧♦ ✶✳✸✵✳ ❉❡t❡r♠✐♥❡ ♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ ❞❛ ✐♥❡q✉❛çã♦ ❙♦❧✉çã♦✳ x2 − 9 ≥ 0✳ 25 − x2 x2 − 9 (x − 3)(x + 3) (x − 3)(x + 3) = ≥ 0 s❡✱ ❡ s♦♠❡♥t❡ s❡ ≤ 0✱ sã♦ 2 25 − x (5 − x)(5 + x) (x − 5)(x + 5) ♣♦♥t♦s ❝rít✐❝♦s✿ { −5, −3, 3, 5 }✳ ❚❡♠♦s q✉❡ + ✛ + + + r− − − − + r + + + + +− r − −−+ r + + + + + + ✲ −∞ −5 −3 3 5 +∞ ▲♦❣♦✱ ♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ é ♦ ✐♥t❡r✈❛❧♦ s❡♠✐✲❛❜❡rt♦ (−5, −3] ∪ [3, 5) ❆s ✐♥❡q✉❛çõ❡s ❞♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦ ❞❡✈❡♠ s❡r ❡st✉❞❛❞❛s ❝♦♠ ♠✉✐t❛ ❛t❡♥çã♦✱ ✉♠❛ ✈❡③ q✉❡ sã♦ ❢r❡q✉❡♥t❡s ♦s ❡q✉í✈♦❝♦s ♥❛s s♦❧✉çõ❡s ♣♦r ♣❛rt❡ ❞♦s ❡st✉❞❛♥t❡s ♥❛ ❢❛s❡ ✐♥✐❝✐❛❧ ❞♦ ❡st✉❞♦ ❞♦ ❝á❧❝✉❧♦✳ ❊①❡♠♣❧♦ ✶✳✸✶✳ ❘❡s♦❧✈❡r ❛s s❡❣✉✐♥t❡s ✐♥❡q✉❛çõ❡s✿ ❛✮ x2 < 16 ❜✮ x2 < −9 ❝✮ x3 < 27 ❙♦❧✉çã♦✳ ❞✮ (x + 1)4 < (x + 1)2 ✭❛✮ ❉❛ ✐♥❡q✉❛çã♦ E(x) : x2 < 16 t❡♠♦s ❛ ✐♥❡q✉❛çã♦ E1 (x) : x2 − 16 < 0✱ ❡♥tã♦ ♥❛ ❢♦r♠❛ ❞❡ ❢❛t♦r❡s r❡s✉❧t❛ (x − 4)(x + 4) < 0. + + + + +− + r− − − − − − − r+ + + + + + ✲ + ✛ −∞ −4 4 +∞ ❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ q✉❛❞r♦✿ ■♥t❡r✈❛❧♦s (−∞, −4) ❙✐♥❛❧ ❞❡ E1 (x) ❈♦♥❥✉♥t♦ s♦❧✉çã♦ ❞❡ E1 (x) + (−4, 4) − (4, +∞) + (−4, 4) P♦rt❛♥t♦✱ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ ❞❛ ✐♥❡q✉❛çã♦ é (−4, 4)✳ ❙♦❧✉çã♦✳  ✭❜✮ ✸✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❉❛ ✐♥❡q✉❛çã♦ x2 < −9✱ t❡♠♦s x2 + 9 < 0✱ ✐st♦ é ❛❜s✉r❞♦✱ ❛ s♦♠❛ ❞❡ ♥ú♠❡r♦s ♣♦s✐t✐✈♦s s❡♠♣r❡ é ♣♦s✐t✐✈♦❀ ❧♦❣♦ ♥ã♦ ❡①✐st❡♠ ♥ú♠❡r♦s r❡❛✐s q✉❡ ❝✉♠♣r❛♠ ❛ ✐♥❡q✉❛çã♦✳  P♦rt❛♥t♦ ❛ s♦❧✉çã♦ é ♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦✳ ❙♦❧✉çã♦✳ ✭❝✮ E2 (x) : x3 < 27✳ ❈♦♥s✐❞❡r❡ ❛ ✐♥❡q✉❛çã♦ x3 − 33 < 0✱ ✐st♦ é (x − 3)(x2 + 3x + 9) < 0✳ ❖❜s❡r✈❡ q✉❡ x2 + 3x + 9 = 9 9 3 27 3 > 0 ∀ x ∈ R✱ ❡♥tã♦ x2 + 3x + 9 > 0 ∀ x ∈ R✳ x2 + 2 · x + + 9 − = (x + )2 + 2 4 4 2 4 2 ▲♦❣♦✱ ♥❛ ✐♥❡q✉❛çã♦ (x − 3)(x + 3x + 9) < 0 s❡❣✉❡ q✉❡ x − 3 < 0❀ ✐st♦ é x < 3✳ ❚❡♠♦s P♦rt❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ é ♦ ✐♥t❡r✈❛❧♦ ❙♦❧✉çã♦✳ (−∞, 3)  ✭❞✮ E(x) : (x + 1)4 < (x + 1)2 . ❚❡♠♦s ❛q✉✐ ❛ ✐♥❡q✉❛çã♦ (x + 1)4 < (x + 1)2 (x + 1)2 (x2 + 2x) < 0 ⇔ (x + 1)4 − (x + 1)2 < 0 x(x + 1)2 (x + 2) < 0 ⇔ (x + 1)2 ≥ 0 ♣❛r❛ t♦❞♦ ✐♥❡q✉❛çã♦ E1 (x) : x(x + 2) < 0✳ ❙❡♥❞♦ ❙❡✉s ♣♦♥t♦s ❝rít✐❝♦s sã♦ −2 ❡ (x + 1)2 .[(x + 1)2 − 1] < 0 ⇔ ♥ú♠❡r♦ r❡❛❧✱ ❛ ✐♥❡q✉❛çã♦ E(x) tr❛♥s❢♦r♠♦✉✲s❡ ♥❛ 0✳ + + + +− + r− − − + + r + + + + + + + + + ✲ −∞ −2 0 +∞ + ✛ ❖❜s❡r✈❡ ♦ s❡❣✉✐♥t❡ q✉❛❞r♦✿ ■♥t❡r✈❛❧♦s ❙✐♥❛❧ ❞❡ (−∞, −2) E1 (x) ❈♦♥❥✉♥t♦ s♦❧✉çã♦ ❞❡ E1 (x) + (−2, 0) − {−1} (0, +∞) − (−2, 0) + Pr♦♣r✐❡❞❛❞❡ ✶✳✶✶✳ P❛r❛ t♦❞♦ x∈R ❡ a>0 ❉❡♠♦♥str❛çã♦✳ ❉✐✈✐❞✐♥❞♦ ♥❛ ✐♥❡q✉❛çã♦ t❡♠♦s✿ ax2 + bx + c ≥ 0 ax2 + bx + c ≥ 0 ♣♦r s❡✱ ❡ s♦♠❡♥t❡ s❡ a > 0 r❡s✉❧t❛✿ b2 ≤ 4ac✳ x2 + b c x+ ≥ 0✳ a a ❈♦♠♣❧❡t❛♥❞♦ q✉❛❞r❛❞♦s c b b b x + 2 x + + ( )2 ≥ ( )2 2a a 2a 2a 2 ✸✹ ⇒  b x+ 2a 2 ≥ b2 − 4ac 4a2 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❆ ❞❡s✐❣✉❛❧❞❛❞❡ ✈❛❧❡ ♣❛r❛ t♦❞♦ 02 ≥ b2 − 4ac 4a2 x ∈ R✱ ⇒ ❡♠ ♣❛rt✐❝✉❧❛r ♣❛r❛ 0 ≥ b2 − 4ac ⇒ x=− b ✱ 2a ❛ss✐♠ b2 ≤ 4ac ❆ ❞❡♠♦♥str❛çã♦ ❞❛ r❡❝í♣r♦❝❛ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❊①❡♠♣❧♦ ✶✳✸✷✳ ❘❡s♦❧✈❡r ❛s ✐♥❡q✉❛çõ❡s✿ ❙♦❧✉çã♦✳ ❛✮ 8x − x2 − 20 ≤ 0 ❝✮ x6 − 1 ≤ 0 ✭❛✮ ❜✮x2 + x + 9 > 0 ❞✮xp − 1 > 0✱ ♦♥❞❡ p é ♣r✐♠♦ 0 ≤ x2 − 8x + 20✱ ❝♦♠♦ (−8)2 ≤ 4(1)(20)✱ s❡❣✉❡ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✶✳✺✱ ❛ s♦❧✉çã♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♥ú♠❡r♦s r❡❛✐s✳  ❚❡♠♦s é ♦ ❙♦❧✉çã♦✳ ✭❜✮ ❉❛ ✐♥❡q✉❛çã♦ x2 + x + 9 > 0 ✱ s❡❣✉❡ q✉❡ (1)2 ≤ 4(1)(9)✱ ❡♥tã♦✱ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ❛ s♦❧✉çã♦ é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♥ú♠❡r♦s r❡❛✐s✳ ❙♦❧✉çã♦✳ ✭✶✳✺✮✱  ✭❝✮ x6 − 1 ≤ 0 ♣♦❞❡♠♦s ❡s❝r❡✈❡r s♦❜ ❛ ❢♦r♠❛ (x2 )3 − 13 ≤ 0 ❡♥tã♦✱ ❞❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❝✉❜♦s t❡♠♦s ✭x2 − 12 )[(x2 )2 + x2 + 1] ≤ 0 ✐st♦ é (x + 1)(x − 1)(x4 + x2 + 1) ≤ 0❀ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✶✳✶✶✮ s❡❣✉❡ q✉❡ ✭x4 + x2 + 1) ≥ 0✱ ❧♦❣♦ ❛ ✐♥❡q✉❛çã♦ ♦r✐❣✐♥❛❧ s❡ r❡❞✉③ ❛ ❝❛❧❝✉❧❛r (x + 1)(x − 1) ≤ 0 q✉❡ t❡♠ ❝♦♠♦ s♦❧✉çã♦ ♦ ✐♥t❡r✈❛❧♦ [−1, 1]✳ P♦rt❛♥t♦ ♦ ❝♦♥❥✉♥t♦ ❛ s♦❧✉çã♦ ❞❡ x6 − 1 ≤ 0 é ♦ ✐♥t❡r✈❛❧♦ [−1, 1]✳  ❆ ✐♥❡q✉❛çã♦ ❙♦❧✉çã♦✳ ✭❞✮ xp − 1 > 0 ♦♥❞❡ p é ♣r✐♠♦✱ ♣♦❞❡r♠♦s ❡s❝r❡✈❡r ♥❛ ❢♦r♠❛ ❞❡ ❢❛t♦r❡s ❝♦♠♦ (x−1)(xp−1 +xp−2 +xp−3 + · · · +x2 +x+1) > 0✱ ♦ ❢❛t♦r ✭xp−1 +xp−2 +xp−3 + · · · +x2 +x+1 s❡♠♣r❡ é ♣♦s✐t✐✈♦ ∀x ∈ R ♣♦✐s é ✉♠ ♣♦❧✐♥ô♠✐♦ ✐rr❡❞✉tí✈❡❧ ❞❡ ❣r❛✉ ♣❛r ✭t♦❞❛s s✉❛s r❛í③❡s ❆ ✐♥❡q✉❛çã♦ sã♦ ♥ú♠❡r♦s ♥ã♦ r❡❛✐s✮✳ ❊♥tã♦✱ r❡s♦❧✈❡r ♥♦ss❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♦r✐❣✐♥❛❧ r❡❞✉③✲s❡ ❛ r❡s♦❧✈❡r é x ∈ (1, +∞✮ P♦rt❛♥t♦✱ ❛ s♦❧✉çã♦ ❞❡ xp − 1 > 0 ✱ ♦♥❞❡ p (x−1) > 0✱ ❝✉❥❛ s♦❧✉çã♦ é ♣r✐♠♦ é ♦ ❝♦♥❥✉♥t♦ (1, +∞)✳ ❊①❡♠♣❧♦ ✶✳✸✸✳ ❘❡s♦❧✈❡r ❡♠ R ♦ s❡❣✉✐♥t❡✿ ❛✮ x2 + 6x + 10 = 0 ❝✮ x2 + 6x + 10 < 0 ❙♦❧✉çã♦✳ ❡✱ ✭❛✮ ❜✮ x2 + 6x + 10 ≥ 0 ❞✮ x2 + 10 ≥ 0 √ −6 ± −4 ❈♦♠♦ r❡s✉❧t❛❞♦ ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭✶✳✼✮ ✭❢ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛✮✱ s❡❣✉❡ q✉❡ x = 2 ❝♦♠♦ ♥ã♦ é ♥ú♠❡r♦ r❡❛❧✱ ❡♥tã♦ ♦ ♣r♦❜❧❡♠❛ ♥ã♦ t❡♠ s♦❧✉çã♦ ❡♠ R❀ ✐st♦ é x ∈ / R✳ ❙♦❧✉çã♦✳ ✭❜✮ ✸✺ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ P❡❧❛ ✐st♦ é Pr♦♣r✐❡❞❛❞❡ ∀x∈R 62 ≤ 4(10)✱ x2 + 6x + 10 ≥ 0 ✭✶✳✶✶✮ t❡♠♦s q✉❡ t❡♠♦s q✉❡ ❙♦❧✉çã♦✳ ✭❝✮ Pr♦♣r✐❡❞❛❞❡ ❧♦❣♦ ♦ ♣r♦❜❧❡♠❛ t❡♠ s♦❧✉çã♦ ❡♠ 62 ≤ 4(10)✱ 0 ∀ x ∈ R✱ ❛ss✐♠✱ ♥✉♥❝❛ ♣♦❞❡rá ♦❝♦rr❡r q✉❡ x2 + 6x + 10 < 0✳ ▲♦❣♦✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♠ ❡st✉❞♦ ♥ã♦ t❡♠ s♦❧✉çã♦ ❡♠ R✳ ❈♦♠♦ r❡s✉❧t❛❞♦ ❞❛ ✭✶✳✶✶✮ t❡♠♦s q✉❡ ❧♦❣♦ R❀ x2 + 6x + 10 ≥  ❙♦❧✉çã♦✳ ✭❞✮ x2 + 10 ≥ 0 é ✐♠❡❞✐❛t❛✱ ♥ã♦ ♣r❡❝✐s❛ ❞❛ Pr♦♣r✐❡❞❛❞❡ ✶✳✼✱ ♣♦✐s ∀ x ∈ R, x2 ≥ 0✱ ❡♥tã♦ x2 + 10 ≥ 10 ≥ 0✱ ✐st♦ é ∀ x ∈ R, x2 + 10 ≥ 0✳ P♦rt❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ ❞❛ ✐♥❡q✉❛çã♦ x2 + 10 ≥ 0 sã♦ t♦❞♦s ♦s ♥ú♠❡r♦s r❡❛✐s✳ ❆ s♦❧✉çã♦ ❞❡ ❆♣❧✐❝❛çõ❡s ❞❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❊①❡♠♣❧♦ ✶✳✸✹✳ ❯♠ t❡rr❡♥♦ ❞❡✈❡ s❡r ❧♦t❛❞♦✳ ❖s ❧♦t❡s✱ t♦❞♦s r❡t❛♥❣✉❧❛r❡s✱ ❞❡✈❡♠ t❡r ár❡❛ s✉♣❡r✐♦r ♦✉ ✐❣✉❛❧ q✉❡ 1.500 m2 ✱ ❡ ❛ ❧❛r❣✉r❛ ❞❡ ❝❛❞❛ ✉♠ ❞❡✈❡ t❡r 20 m ❛ ♠❡♥♦s q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦✳ ❉❡t❡r♠✐♥❡ ❛s ❞✐♠❡♥sõ❡s ❞♦ ♠❡♥♦r ❞♦s ❧♦t❡s q✉❡ ❝✉♠♣r❡♠ t❛✐s ❝♦♥❞✐çõ❡s✳ ❙♦❧✉çã♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝❛❞❛ ❧♦t❡ s❡❥❛ x ♠❡tr♦s✱ ❡♥tã♦ ❛ ❧❛r❣✉r❛ ♠❡❞❡ 2 (x − 20) ♠❡tr♦s❀ ❧♦❣♦ s✉❛ ár❡❛ ♠❡❞❡ x(x − 20)m ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠ q✉❡ s❡r s✉♣❡r✐♦r ♦✉ ✐❣✉❛❧ ❛ 1.500m2 ❀ ❛ss✐♠ x(x−20) ≥ 1.500 ♦♥❞❡ x2 −20x−1.500 ≥ 0✱ ✐st♦ é (x−50)(x+30) ≥ 0 ⇒ x ≥ 50 ♦✉ x ≤ −30✳ ❉❡s❝♦♥s✐❞❡r❛♥❞♦ x ≤ −30✱ t❡♠♦s q✉❡ ❛s ♠❡❞✐❞❛s ❞♦ ♠❡♥♦r ❞♦s ❧♦t❡s é✿ ❝♦♠♣r✐♠❡♥t♦ 50 m ❡ ❧❛r❣✉r❛ 30 m✳  ❊①❡♠♣❧♦ ✶✳✸✺✳ ❯♠❛ ❣❛❧❡r✐❛ ✈❛✐ ♦r❣❛♥✐③❛r ✉♠❛ ❡①♣♦s✐çã♦ ❡ ❢❡③ ❞✉❛s ❡①✐❣ê♥❝✐❛s✿ ✐✮ ❛ ár❡❛ ❞❡ ❝❛❞❛ q✉❛❞r♦ ❞❡✈❡ s❡r ♥♦ ♠í♥✐♠♦ ❞❡ 2.800 cm2 ❀ ✐✐✮ ♦s q✉❛❞r♦s ❞❡✈❡♠ s❡r r❡t❛♥❣✉❧❛r❡s ❡ ❛ ❛❧t✉r❛ ❞❡✈❡ t❡r 30 cm ❛ ♠❛✐s q✉❡ ❛ ❧❛r❣✉r❛✳ ❉❡♥tr♦ ❞❡ss❛s ❡s♣❡❝✐✜❝❛çõ❡s✱ ❡♠ q✉❡ ✐♥t❡r✈❛❧♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s ❞❡✈❡♠ s❡ s✐t✉❛r ❛s ❧❛r❣✉r❛s ❞♦s q✉❛❞r♦s❄ ❙♦❧✉çã♦✳ ❉❛ s❡❣✉♥❞❛ ❝♦♥❞✐çã♦✱ s✉♣♦♥❤❛ ❛ ❧❛r❣✉r❛ ❞♦ q✉❛❞r♦ s❡❥❛ x cm✱ ❡♥tã♦ s✉❛ ❛❧t✉r❛ ♠❡❞❡ (30 + x)cm ❡ s✉❛ ár❡❛ ♠❡❞❡ (30 + x)xcm2 ❀ ♣❡❧❛ ♣r✐♠❡✐r❛ ❝♦♥❞✐çã♦ 2.800 ≤ (30 + x)x ✱ ♦♥❞❡ 0 ≤ x2 + 30x − 2.800 ⇒ 0 ≤ (x + 70)(x − 40) ⇒ (x ≤ −70 ♦✉ x ≥ 40)✳ ❉❡s❝♦♥s✐❞❡r❛♠♦s x ≤ −70✳ P♦rt❛♥t♦✱ ❛s ♠❡❞✐❞❛s ❞♦ q✉❛❞r♦ sã♦✿ ❧❛r❣✉r❛ 40 cm ❡ ❛❧t✉r❛ 70 cm✳ ✸✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✶✲✸ ✶✳ ❊①♣r❡ss❡ ❝❛❞❛ ✉♠ ❞♦s ✐♥t❡r✈❛❧♦s ❛❜❛✐①♦ ✉s❛♥❞♦ ♦✉tr❛ ♥♦t❛çã♦ ❛❞❡q✉❛❞❛ ✭❞✉♣❧❛s ❞❡s✐❣✉❛❧❞❛❞❡s ♣♦r ❡①❡♠♣❧♦✮ 1. (1, 14) 2. (4, 7) 3. [−π, π] 5. [−10, −2] 6. (0, 4) 7. [−3π, π) 5 4. [− , 8] 3 8. (−16, 16] A = {x ∈ N x é ✐♠♣❛r }, B = {x ∈ Z/. −3 ≤ x < 4} C = { x ∈ N /. x < 6 }✳ Pr♦✈❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ D✱ t❛❧ q✉❡ D = (A ∩ B) − C ✱ é ✷✳ ❙ã♦ ❞❛❞♦s ♦s ❝♦♥❥✉♥t♦s ❡ ✈❛③✐♦✳ ✸✳ ❘❡s♦❧✈❡r ❛s s❡❣✉✐♥t❡s ❡q✉❛çõ❡s✿ 1. 3x + 2 = 4 − x 4. x3 − 3x2 + x + 2 = 0 2. x2 − 2x − 3 = 0 5. 5x2 − 3x − 4 = 0 ✹✳ ❉❡t❡r♠✐♥❡ ♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ ❡♠ R 3. x4 − 13x2 + 36 = 0 6. x4 − x2 + 20 = 0 ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❞❡s✐❣✉❛❧❞❛❞❡s✿ 1. x2 ≥ 1 2. x3 ≥ x2 3. 2x − 7 < 5 − x 4. 2(x − 4) + 3x < 5x − 7 5. 3 − x < 5 + 3x 6. 2 > −3 − 3x ≥ −1 √ x2 + x − 2 < 4 9. 8. x2 − 4 < x + 2 7. 4x − 3(x + 5) < x − 18 3x + 8 10. 2x − 6 < 5 2x − 74 13. x2 + 3x + 8 < x−7 ✶✻✳ 11. 14. 2x + 6 x x2 + 4x + 10 − < 5 12. >0 3 4 x2 − x − 12 x+4 x < 15. (x + 1)4 ≤ (x + 1)2 x−2 x+1 7(3 − 2x) + 2(2x − 15) < 6(3x − 5) + 3(3x − 1) ✶✼✳ ✺✳ ❘❡s♦❧✈❡r ❛s s❡❣✉✐♥t❡s ✐♥❡q✉❛çõ❡s✿ 3 > 3x − 16 2x − 3 1 1. (x − )(3x + 5) > 0 2 2. (x − 2)(x + 2) ≤ 0 3. x(x + 1) ≤ 0 4. (x − 1)(x + 1) ≤ 0 5. 6. 7. x < x2 − 12 < 4x x−1 ≥0 x 8. 3 − x < 5 + 3x 10. (x − 5)2 < (2x − 3)2 11. x2 + 3x > −2 ✸✼ x+1 <0 x−1 √ 9. x2 + x − 2 ≥ 4 12. 3x − 4 < 2 + x 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ 13. (x − 1)3 (x2 − 4)(x − 5) > 0 14. 2 ≤ 5 − 3x < 11 3 1−x 16. 5x − 4(x + 5) < x − 24 17. 3x − 5 < + 4 3 5 4 3 2 19. x − 2x − 15x > 0 20. 2x − x − 10 > 0 3 9 22. x2 + 8x − 65 < x − 18 23. x2 + x + <0 5 100 25. 3(x + 4) + 4x < 7x + 2 26. 3x2 − 7x + 6 < 0 28. (x5 − 1)(x + 1) ≥ 0 ✸✶✳ 29. x2 + 20x + 100 > 0 (x3 − 5x2 + 7x − 3)(2 − x) ≥ 0 ✸✷✳ R 15. x2 − 3x + 2 > 0 18. 3 − x < 5 + 3x 21. x2 − 3x + 2 > 0 24. x2 − 2x − 5 > 0 27. x2 − 2x − 8 < 0 30. 3x − 4 < x + 6 (x2 − 3)3 (x2 − 7)(x2 − 2x − 3) > 0 ✻✳ ❉❡t❡r♠✐♥❡ ♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ ❞❛s s❡❣✉✐♥t❡s ✐♥❡q✉❛çõ❡s✿ ✶✳ ✷✳ ✸✳ ✹✳ 3x 5 x + < s❡ a > b > 0 2 −b a+b a−b x x x + >1+ s❡ c > b > a > 0 a b c 5x 2x +4> + 2x s❡ b > a > 0 3a 6b 11(2x − 3) − 3(4x − 5) > 5(4x − 5) a2 ✼✳ ❘❡s♦❧✈❡r ❛s s❡❣✉✐♥t❡s ✐♥❡q✉❛çõ❡s r❛❝✐♦♥❛✐s✿ 1. x−1 2x x + < x−1 x x+1 4. (2x + 1)101 (x − 3)99 ≥ 0 7. 2. 5. (1 − x − x2 )(2 − x − x2 ) ≥ 0 8. (3 − x)(2 − x) 2 <0 3. 2x + 3 2x − 3 1 < 6. x+2 3 x5 − 1 x5 − 2 < 9. x4 + 1 x4 + 2 3x + 5 ≤3 2x + 1 3x2 + 12 >3 x2 + 4x − 5 x+4 x > x−7 x+1 ✽✳ ▼♦str❡ q✉❡ s❡ x ❡ y ♥ã♦ sã♦ ❛♠❜♦s ✐❣✉❛✐s ❛ ③❡r♦✱ ❡♥tã♦ 4x2 + 6xy + 4y 2 > 0 3x2 + 5xy + 3y 2 > 0✳ ❡ ✾✳ ❉❡t❡r♠✐♥❛r ♣❛r❛ q✉❛✐s ✈❛❧♦r❡s ❞❡ x ∈ R ✈❡r✐✜❝❛ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ 3(x − a)a2 < x3 − a3 < 3(x − a)x2 1 1 1 1 + 2 + 3 + · · · + n ✱ s❡ n → ∞ 3 3 3 3 √ √ 2 − 4c −b + b b2 − 4c −b − ✶✶✳ ❙✉♣♦♥❤❛ b2 −4c ≥ 0✳ ▼♦str❡ q✉❡ ♦s ♥ú♠❡r♦s ❡ 2 2 ❛♠❜♦s ❝✉♠♣r❡♠ ❛ ❡q✉❛çã♦✿ x2 + bx + c = 0✳ ✶✵✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❡✿ S = 1 + ✸✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ b2 − 4c < 0✳ ▼♦str❡ x2 + bx + c = 0✳ ✶✷✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❡q✉❛çã♦✿ a, b, c ❡ d √ a2 + b 2 . c 2 + d 2 ✳ ✶✸✳ ❙✉♣♦♥❤❛ √ ✶✹✳ ▼♦str❡ q✉❡✿ ✶✺✳ ▼♦str❡ q✉❡✿ √ R ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ q✉❡ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ♥ú♠❡r♦ r❡❛❧ q✉❡ ❝✉♠♣r❡ ♥ú♠❡r♦s r❡❛✐s✳ ▼♦str❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❙❝❤✇❛rt③✿ ac + bd ≤ x2 − 2x − 15 ≥ x + 1 ∀ x ∈ (−∞, −3]✳ 1 ≤ x2 + x + 2 ≤ 8 ∀ x ∈ [−1, 2] − {1}. 4 ✶✻✳ ❖s ♥ú♠❡r♦s ♣♦s✐t✐✈♦s a1 , a 2 , a 3 , · · · , a n ♥ã♦ sã♦ ✐❣✉❛✐s ❛ ③❡r♦ ❡ ❢♦r♠❛♠ ✉♠❛ ♣r♦✲ ❣r❡ssã♦ ❛r✐t♠ét✐❝❛✳ ▼♦str❡ q✉❡✿ √ 1 1 1 1 n−1 √ +√ √ +√ √ + ··· + √ √ =√ √ a1 + a2 a2 + a3 a3 + a4 an−1 + an a1 + an ✶✼✳ ❉❡t❡r♠✐♥❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ♣♦s✐t✐✈♦s ❡ í♠♣❛r❡s✱ ❢♦r♠❛❞♦s ♣♦r três ❛❧❣❛r✐s♠♦s ❞✐st✐♥t♦s✱ ❡s❝♦❧❤✐❞♦s ❞❡♥tr❡ ♦s ❛❧❣❛r✐s♠♦s 0, 1, 2, 3, 4, 5, 6, 7, 8 ✶✽✳ ❈❛❧❝✉❧❡ ❛ s♦♠❛ ❞❡ t♦❞❛s ❛s ❢r❛çõ❡s ✐rr❡❞✉tí✈❡✐s✱ ❞❛ ❢♦r♠❛ ✐♥t❡r✈❛❧♦ [4, 7]✳ p ✱ 72 ❡ 9✳ q✉❡ ♣❡rt❡♥ç❛♠ ❛♦ ✶✾✳ ❉❡♥tr❡ ♦s ♣❛r❛❧❡❧❡♣í♣❡❞♦s ❝♦♠ s♦♠❛ ✜①❛ ❞❡ s✉❛s três ❛r❡st❛s s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♣❡r✲ ♣❡♥❞✐❝✉❧❛r❡s✱ ❛❝❤❛r ♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ✈♦❧✉♠❡ ♠á①✐♠♦✳ C ✈✐s✐t❛♠ ♦ ❛ç✉❞❡ ✏ P❡✐①❡ ♥❛ ❝❤❛♣❛ ✑ ❡ ♣❡s❝❛♠ ♠❛✐s ❞❡ 8 ♣❡✐①❡s❀ B ♣❡♥s❛ ♣❡s❝❛r ♠❛✐s 4 ❝♦♠ ♦ q✉❡ t❡r✐❛ ♠❛✐s ♣❡✐①❡s q✉❡ A ❡ C ♣♦ré♠ B t❡♠ ♠❡♥♦s ♣❡✐①❡s q✉❡ C ❡ ♦ q✉❡ t❡♠ C ♥ã♦ ❝❤❡❣❛♠ ❛ 5✳ ◗✉❛♥t♦s ♣❡✐①❡s tê♠ ❝❛❞❛ ✉♠ ❞❡❧❡s❄ ✷✵✳ ❚rês ♣❡ss♦❛s A, B ❡ ✷✶✳ P❛r❛ ✉♠❛ ❢❡st❛ ♥♦ ◆❛t❛❧✱ ✉♠❛ ❝r❡❝❤❡ ♥❡❝❡ss✐t❛✈❛ ❞❡ ❞♦❛çã♦ ❞❡ ❘$ ✸✼✵✱✵✵✳ ❜♦♥❡❝❛s ❡ ❝❛rr✐♥❤♦s ❥✉♥t♦s✳ 60 ❜r✐♥q✉❡❞♦s✳ ❘❡❝❡❜❡✉ ✉♠❛ ❊s♣❡r❛✈❛✲s❡ ❝♦♠♣r❛r ❝❛rr✐♥❤♦s ❛ ❘$2, 00 ❝❛❞❛✱ ❜♦♥❡❝❛s ❛ ❘$3, 00 ❡ ❜♦❧❛s ❛ ❘$3, 50✳ ❝❛rr✐♥❤♦s ❡ 120 ❙❡ ♦ ♥ú♠❡r♦ ❞❡ ❜♦❧❛s ❞❡✈❡r✐❛ s❡r ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ▼♦str❡ q✉❡ ❛ s♦❧✉çã♦ s❡r✐❛ ❝♦♠♣r❛r✿ 40 ❜♦♥❡❝❛s✱ 20 ❜♦❧❛s✳ ✷✷✳ ❊♠ ✉♠❛ ❢❛③❡♥❞❛✱ ❡①✐st✐❛ ✉♠ ♥ú♠❡r♦ ❞❡ ❝❛❜❡ç❛s ❞❡ ❣❛❞♦s✳ ❉❡♣♦✐s ❞❡ ❞✉♣❧✐❝❛r ❡ss❡ ♥ú♠❡r♦✱ ❢♦✐ r♦✉❜❛❞♦ 1 ❝❛❜❡ç❛✱ s♦❜r❛♥❞♦ ♠❛✐s ❞❡ 54✳ ▼❡s❡s ❞❡♣♦✐s ♦❜s❡r✈♦✉✲s❡ q✉❡ tr✐♣❧✐❝♦✉ ♦ ♥ú♠❡r♦ ❞❡ ❝❛❜❡ç❛s ❞❡ ❣❛❞♦ q✉❡ ❡①✐st✐❛ ♥♦ ✐♥í❝✐♦ ❡ ❢♦r❛♠ r♦✉❜❛❞❛s r❡st❛♥❞♦ ♠❡♥♦s ❞❡ 80✳ 5 ◗✉❛♥t❛s ❝❛❜❡ç❛s ❞❡ ❣❛❞♦ ❡①✐st✐❛♠ ♥♦ ✐♥í❝✐♦❄ ✷✸✳ ❆ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞❛s ✐❞❛❞❡s ❞❡ ✉♠ ❣r✉♣♦ ❞❡ ♠é❞✐❝♦s ❡ ❛❞✈♦❣❛❞♦s é ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞❛s ✐❞❛❞❡s ❞♦s ♠é❞✐❝♦s é 35 40 ❛♥♦s ❡ ❛ ❞♦s ❛❞✈♦❣❛❞♦s é ❛♥♦s✳ ❆ 50 ❛♥♦s✳ P♦❞❡✲s❡✱ ❡♥tã♦✱ ❛✜r♠❛r q✉❡✿ ✸✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✷✹✳ ❯♠❛ ♣❡ss♦❛ ❝♦♠♣r❛ ✉♠ ❛♣❛rt❛♠❡♥t♦ ♣♦r ❘$10.000, 00✱ ❡♠ s❡❣✉✐❞❛ ♦ ❛❧✉❣❛✳ ❉❡✐✲ 1 ①❛♥❞♦ 12 % ❞❛ r❡♥❞❛ ❛♥✉❛❧ ♣❛r❛ r❡♣❛r❛çõ❡s ❡ ♠❛♥✉t❡♥çã♦✱ ♣❛❣❛♥❞♦ ❘$325, 00 ❞❡ 2 1 ■P❚❯ ❡ 5 % ❞❡s❝♦♥t❛♥❞♦ ♣♦r ❝♦♥t❛ ❞❡ ✐♥✈❡st✐♠❡♥t♦✳ ◗✉❛❧ é ❛ r❡♥❞❛ ♠❡♥s❛❧❄ 2 ✷✺✳ ❆ s♦♠❛ ❞❛s ✐❞❛❞❡s ❞❡ três ♣❡ss♦❛s é 96✳ ❆ ♠❛✐♦r t❡♠ 32 ❛♥♦s ♠❛✐s q✉❡ ❛ ♠❡♥♦r ❡ ❛ ❞♦ ♠❡✐♦ 16 ❛♥♦s ♠❡♥♦s q✉❡ ❛ ♠❛✐♦r✳ ❈❛❧❝✉❧❛r ❛ ✐❞❛❞❡ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ♣❡ss♦❛s✳ ✷✻✳ ❊✉ t❡♥❤♦ ❛ ✐❞❛❞❡ q✉❡ ✈♦❝ê t✐♥❤❛✱ q✉❛♥❞♦ ❡✉ t✐♥❤❛ ❛ ♠❡t❛❞❡ ❞❛ ✐❞❛❞❡ q✉❡ ✈♦❝ê t❡♠✳ ❆ s♦♠❛ ❞❡ ♥♦ss❛s ✐❞❛❞❡s ❤♦❥❡ é ✐❣✉❛❧ ❛ 35 ❛♥♦s✳ ◗✉❛✐s sã♦ ❛s ✐❞❛❞❡s ❤♦❥❡❄ ✷✼✳ ▼♦str❡ q✉❡✱ ♣❛r❛ ♥ú♠❡r♦s r❡❛✐s x ❡ y ✱ ❡ n ∈ N n ≥ 2 sã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s✿ ✶✳ xn − y n = (x − y)(xn−1 + xn−2 y + xn−3 y 2 + · · · + x2 y n−3 + xy n−2 + y n−1 ) xn + y n = (x + y)(xn−1 − xn−2 y + xn−3 y 2 − · · · + (−1)n−3 x2 y n−3 − xy n−2 + y n−1 ) s♦♠❡♥t❡ q✉❛♥❞♦ n í♠♣❛r✳ ✷✳ ✷✽✳ ▼♦str❡ q✉❡✱ s❡ p é ♥ú♠❡r♦ ♣r✐♠♦✱ ❡ a ∈ N✱ ❡♥tã♦ ap − a é ♠ú❧t✐♣❧♦ ❞❡ p✳ ✷✾✳ Pr♦✈❡ q✉❡✿ (1 − x)[(1 + x)(1 + x2 )(1 + x4 ) · · · (1 + x2 )] = 1 − x2 ✐♥t❡✐r♦ x✱ ❡ t♦❞♦ n ≥ 0✳ n ✸✵✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❡ E = x3 + 3x + 2✱ q✉❛♥❞♦ x = p √ 3 (n+1) ♣❛r❛ q✉❛❧q✉❡r 1 2−1− p ✳ √ 3 2−1 ✸✶✳ ❈♦♥str✉✐r ♥ú♠❡r♦s 49, 4.489, 444.889, 44.448.889, . . . ❡t❝ ♦❜t❡♥❞♦ ❝❛❞❛ ✉♠ ❞❡❧❡s ✐♥s❡r✐♥❞♦ ♦ ♥ú♠❡r♦ 48 ♥♦ ♠❡✐♦ ❞♦ ♥ú♠❡r♦ ❛♥t❡r✐♦r✳ ❱❡r✐✜❝❛r q✉❡ ❡st❡s ♥ú♠❡r♦s sã♦ q✉❛❞r❛❞♦s ♣❡r❢❡✐t♦s ❡ ❡♥❝♦♥tr❛r ❛ r❛✐③ q✉❛❞r❛❞❛ ❞♦ ♥ú♠❡r♦ q✉❡ ❝♦♥s✐st❡ ❞❡ 2n ❛❧❣❛r✐s♠♦s✳ ✸✷✳ ❉❛❞❛ ❛ ❡q✉❛çã♦ ❞❡ r❛í③❡s x1 ❡ x2 ✿ (m2 − 5m + 6)x2 + (4 − m2 )x + 20 = 0✳ ❉❡t❡r♠✐♥❡ ♦s ✈❛❧♦r❡s ❞♦ ♣❛râ♠❡tr♦ m t❛❧ q✉❡ x1 < 1 < x2 ✳ ✹✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✶✳✺ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❱❛❧♦r ❛❜s♦❧✉t♦ ❉❡✜♥✐çã♦ ✶✳✶✹✳ ❖ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ x é ❞❡♥♦t❛❞♦ ♣♦r | x |✱ s❡ ❢♦r ♣♦s✐t✐✈♦ ♦✉ ✐❣✉❛❧ ❛ ③❡r♦✱ ❡ é ✐❣✉❛❧ ❛ s❡✉ ♦♣♦st♦ ❛❞✐t✐✈♦ ■st♦ é✿ | x |= P♦r ❡①❡♠♣❧♦✱ | 3 |= 3, | 0 |= 0,  x é ♦ ♣ró♣r✐♦ ♥ú♠❡r♦ −x x s❡ ❢♦r ♥❡❣❛t✐✈♦✳ s❡ x ≥ 0, −x s❡ x < 0. | −4 |= −(−4) = 4 Pr♦♣r✐❡❞❛❞❡ ✶✳✶✷✳ ✶✳ | a |≥ 0, ∀ a ∈ R ❡ | a |= 0 s❡ a = 0✳ ✷✳ | a |2 = a2 ✸✳ | −a |=| a | ✹✳ | ab |=| a | . | b | ✺✳ | a + b |≤| a | + | b | ❉❡♠♦♥str❛çã♦✳ ✳ ✳ ✳ ❉❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ✭✷✮ ❙✉♣♦♥❤❛ a ≥ 0✱ ❡♥tã♦ | a |= a ✱ ❧♦❣♦ | a |2 = a.a = a2 ✳ ❙✉♣♦♥❤❛ a < 0✱ ❡♥tã♦ | a |= −a✱ ❧♦❣♦ | a |2 = (−a)(−a) = a2 ✳ ✭✺✮ ❉♦ ❢❛t♦ s❡r ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ s❡♠♣r❡ ♣♦s✐t✐✈♦✱ s❡❣✉❡ q✉❡✿ ❉❡♠♦♥str❛çã♦✳ ✭✶✳✼✮ ab ≤| a | . | b | P❡❧❛ 2a ♣❛rt❡ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ ❡ ❞❡ ✭✶✳✼✮ t❡♠♦s q✉❡ | a+b |2 = (a+b)2 = a2 +2ab+b2 =| a |2 +2ab+ | b |2 ≤| a |2 +2 | ab | + | b |2 = (| a | + | b |)2 ✱ ✐st♦ é | a + b |2 ≤ (| a | + | b |)2 s❡♥❞♦ t♦❞♦s ❡st❡ ú❧t✐♠♦s ♥ú♠❡r♦s ♣♦s✐t✐✈♦s ❝♦♥❝❧✉í♠♦s q✉❡ | a + b |≤| a | + | b |✳ ❖❜s❡r✈❛çã♦ ✶✳✽✳ ✐✮ ✐✐✮ ❆ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ♥ú♠❡r♦s r❡❛✐s ●❡♦♠❡tr✐❝❛♠❡♥t❡✱ |a| a ❡ b ❞❛ r❡t❛ ♥✉♠ér✐❝❛ ❞❡♥♦t❛♠♦s ♣♦r é ❛ ❞✐stâ♥❝✐❛ ♥❛ r❡t❛ ♥✉♠ér✐❝❛ ❞♦ ♥ú♠❡r♦ a | b − a |✳ ❛té ♦ ♣♦♥t♦ ③❡r♦✳ ●r❛✜❝❛♠❡♥t❡✳ ✹✶ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ r r | b − a |=| a − b | ✛ a ✲ ✛ b r 0 |a| r ✲ a Pr♦♣r✐❡❞❛❞❡ ✶✳✶✸✳ ✐✮ ✐✐✮ ✐✐✐✮ ✐✈✮ b>0 ❙❡ | a | = | b |✱ √ | a2 | b |= a✱ ❡ ❡♥tã♦ |a| ❂ a |a| |= b |b| ❉❡♠♦♥str❛çã♦✳ ❡♥tã♦ a=b ♦♥❞❡ √ ♦✉ a2 a=b a = −b✳ ♦✉ a = −b✳ é ❛ r❛✐③ q✉❛❞r❛❞❛ ♣♦s✐t✐✈❛ ❞❡ a2 ✳ b 6= 0 s❡ ✭✐✐✮ ❉❛ ❤✐♣ót❡s❡ | a |=| b | ❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞♦ ♥ú♠❡r♦ b✱ s❡❣✉❡ q✉❡ | a |= b ♦✉ | a |= −b✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ✈❛❧♦r ❛❜s♦❧✉t♦ ♣❛r❛ ♦ ♥ú♠❡r♦ a s❡❣✉❡ ❞❡ | a |= b q✉❡✱ a = b ♦✉ −a = b❀ ❡ ❞❡ | a |= −b s❡❣✉❡ q✉❡ a = −b ♦✉ −a = −b✳ P♦rt❛♥t♦ a = b ♦✉ a = −b✳ Pr♦♣r✐❡❞❛❞❡ ✶✳✶✹✳ ✐✮ ✐✐✮ | x |< b s❡✱ ❡ s♦♠❡♥t❡ s❡ | x |≤ b s❡✱ ❡ s♦♠❡♥t❡ s❡ ✐✐✐✮ ❙❡ b≥ ✐✈✮ ❙❡ b ≥ 0, | x | ≥ b ✈✮ ✵✱ | x |> b −b < x < b✳ −b ≤ x ≤ b✳ s❡✱ ❡ s♦♠❡♥t❡ s❡ x>b s❡✱ ❡ s♦♠❡♥t❡ s❡ ♦✉ x ≥ b x < −b✳ ♦✉ x ≤ −b || a | − | a ||≤| a − b |≤| a | + | b | ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❊①❡♠♣❧♦ ✶✳✸✻✳ ❘❡s♦❧✈❡r ❛s s❡❣✉✐♥t❡s ❡q✉❛çõ❡s✿ ❛✮ | 2x − 4 | ❂✻ ❜✮ || 5 − 2x | −4 | ❞✮ | x2 − 4 | ❂ | 2x | ❡✮| x − 1 | ❂✽ ❝✮ 3x + 1 ❂✹ x−1 ✰ ✹| x − 3 | ❂ ✷| x + 2 | ✹✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✭❛✮ ❉❛ ❞❡✜♥✐çã♦✱ ❞❡ | 2x − 4 |= 6 s❡❣✉❡✲s❡ q✉❡ 2x − 4 = 6 ♦✉ −(2x − 4) = 6✱ ❡♥tã♦ ❙♦❧✉çã♦✳ x= 6−4 6+4 ♦✉ x = ✳ P♦rt❛♥t♦✱ x = 5 ♦✉ x = −1✳ 2 −2  ✭❜✮ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ✈❛❧♦r ❛❜s♦❧✉t♦✱ s❡❣✉❡ q✉❡ | 5 − 2x | −4 = 8 ♦✉ | 5 − 2x | −4 = −8✱ ❡♥tã♦ 5 − 2x = 12 ♦✉ 5 − 2x = −12 ♦✉ | 5 − 2x |= −4✱ s❡♥❞♦ ❡st❛ ú❧t✐♠❛ ✉♠ ❛❜s✉r❞♦✳ ❙♦❧✉çã♦✳ 7 2 ▲♦❣♦✱ ❞❡ 5 − 2x = 12 ♦❜t❡♠♦s x = − ✱ ❡ ❞❡ 5 − 2x = −12 ♦❜t❡♠♦s x = 7 2 P♦rt❛♥t♦ x = − ♦✉ x = 17 é s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛✳ 2 17 ✳ 2  ✭❡✮ ❉❛ ❡q✉❛çã♦ | x − 1 | ✰ ✹| x − 3 | ❂ ✷| x + 2 | t❡♠♦s ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛✿ ❙♦❧✉çã♦✳ ✛ −∞ r −2 r r 1 3 ✲ +∞ ❙❡ x < −2 ❡♥tã♦✱ | x + 2 |= −(x + 2), | x − 1 |= −(x − 1) ❡ | x − 3 |= −(x − 3)✱ ❧♦❣♦ 17 17 ❡✱ ❝♦♠♦ x = ❛ ❡q✉❛çã♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ −(x − 1) − 4(x − 3) = −2(x + 2) ♦♥❞❡ x = 3 3 ♥ã♦ ♣❡rt❡♥❝❡ ❛♦ ✐♥t❡r✈❛❧♦ ❞❛ ❝♦♥❞✐çã♦✱ s❡❣✉❡ q✉❡ x ∈ / R✳ ❙❡ −2 ≤ x < 1 ❡♥tã♦ | x + 2 |= x + 2, | x − 1 |= −(x − 1) ❡ | x − 3 |= −(x − 3)✱ ❧♦❣♦ 9 ❛ ❡q✉❛çã♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ −(x − 1) − 4(x − 3) = 2(x + 2) ♦♥❞❡ x = ❡✱ ♣❡❧❛ ❝♦♥❞✐çã♦ 7 x∈ / R✳ ❙❡ 1 ≤ x < 3 ❡♥tã♦ | x + 2 |= x + 2, | x − 1 |= x − 1 ❡ | x − 3 |= −(x − 3)✱ ❧♦❣♦ ❛ 7 5 ❙❡ x ≥ 3✱ ❡♥tã♦ | x + 2 |= x + 2, | x − 1 |= x − 1 ❡ | x − 3 |= x − 3✱ ❧♦❣♦ ❛ ❡q✉❛çã♦ é 17 ❡q✉✐✈❛❧❡♥t❡ ❛ x − 1 + 4(x − 3) = 2(x + 2) ♦♥❞❡ x = ✳ 3 17 7 sã♦ s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦✳ P♦rt❛♥t♦✱ x = ✱ ❡ x = 5 3 ❡q✉❛çã♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ x − 1 − 4(x − 3) = 2(x + 2) ♦♥❞❡ x = ✳ ❊①❡♠♣❧♦ ✶✳✸✼✳ A = { x ∈ R /. | 12x − 4 |< 10 }, B = { x ∈ R/. | 3x − 1 |≥ 1 } ❡ C = { x ∈ R /. | x2 − 4 |< 2 }✳ ❊①♣r❡ss❛r ♥❛ ❢♦r♠❛ ❞❡ ✐♥t❡r✈❛❧♦s ♦ ❝♦♥❥✉♥t♦ (A ∪ B) ∩ C ✳ ❉❛❞♦s✿ ❙♦❧✉çã♦✳ P❛r❛ ♦ ❝♦♥❥✉♥t♦ A t❡♠♦s q✉❡ | 12x − 4 |< 10✱ ❡♥tã♦ −10 < 12x − 4 < 10 ❧♦❣♦ ✲ 1 14 1 14 <x< ❀ ✐st♦ é A = (− , ✮✳ 2 12 2 12 P❛r❛ ♦ ❝♦♥❥✉♥t♦ B t❡♠♦s q✉❡ | 3x − 1 |≥ 1 ✐♠♣❧✐❝❛ 3x − 1 ≥ 1 ♦✉ 3x − 1 ≤ −1✱ ❧♦❣♦ 2 2 x ≥ ♦✉ x ≤ 0✱ ✐st♦ é B = (−∞, 0] ∪ [ , +∞)✳ 3 3 ✹✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ √ √ −2 < x2 − 4 < 2✱ ❡♥tã♦ 2 < x2 < 6✱ ❧♦❣♦ − 6 < x < − 2 √ √ √ S√ √ √ 2 < x < 6❀ ❛ss✐♠ C = (− 6, − 2) ( 2, 6) √ S√ √ √ 6) é s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛✳ P♦rt❛♥t♦✱ (A ∪ B) ∩ C = (− 6, − 2) ( 2, P❛r❛ ♦ ❝♦♥❥✉♥t♦ ♦✉ C t❡♠♦s ❊①❡♠♣❧♦ ✶✳✸✽✳ ❘❡s♦❧✈❡r ❙♦❧✉çã♦✳✳ | x2 − 4 | + | 2x − 5 |< 1✳ ❚❡♠♦s q✉❡ ▲♦❣♦✿ | x2 − 4 |= x2 − 4 s❡ 2≤x ♦✉ s❡ x ≤ −2 ❡ | 2x − 5 |= 2x − 5 s❡ 5 ≤ x✳ 2 ≤ −2 ✈❡♠ q✉❡ | x2 −4 |= x2 −4 ❡ | 2x−5 |= −(2x−5) ⇒ (x2 −4)−(2x−5) < 1 x2 − 2x < 0 ✐st♦ é (x − 0)(x − 2) < 0✱ ❡ ♣❡❧❛ ❝♦♥❞✐çã♦ x ∈ / R✳ ❙❡ x ♦♥❞❡ −2 < x < 2 t❡♠♦s q✉❡ | x2 −4 |= −(x2 −4) ❡ | 2x−5 |= −(2x−5) ❡♥tã♦ ❛ ✐♥❡q✉❛çã♦ é ❡q✉✐✈❛❧❡♥t❡ à −(x2 − 4) − (2x − 5) < 1 ♦♥❞❡ 0 < x2 + 2x − 8 ✐st♦ é 0 < (x + 4)(x − 2) ❡ ❞❛ ❝♦♥❞✐çã♦ x ∈ / R✳ 5 ❙❡ 2 ≤ x < t❡♠♦s q✉❡ | x2 − 4 |= x2 − 4 ❡ | 2x − 5 |= −(2x − 5) ❡♥tã♦ ✭x2 − 4) − 2 (2x − 5) < 1 ♦♥❞❡ (x − 0)(x − 2) < 0 ❡ ♣❡❧❛ ❝♦♥❞✐çã♦ x ∈ / R✳ 5 ≤ x t❡♠♦s q✉❡ | x2 −4 |= x2 −4 ❡ | 2x−5 |= (2x−5) ❡♥tã♦ ✭x2 −4)+(2x−5) < 1 ❙❡ 2 √ √ ♦♥❞❡ x2 + 2x − 10 < 0 ✱ ✐st♦ é (x − 11 + 1)(x + 11 + 1) < 0✱ ♣❡❧❛ ❝♦♥❞✐çã♦ x ∈ / R✳ ❙❡ P♦rt❛♥t♦✱ ♥ã♦ ❡①✐st❡ s♦❧✉çã♦ ❡♠ R✳ ❊①❡♠♣❧♦ ✶✳✸✾✳ ❘❡s♦❧✈❡r ❙♦❧✉çã♦✳ ❉♦ ❢❛t♦ q✉❛❞r❛❞♦s (x − 1)2 − | x − 1 | +8 > 0✳ (x − 1)2 =| x − 1 |2 ✱ s❡❣✉❡ q✉❡ | x − 1 |2 − | x − 1 | +8 > 0✱ ❧♦❣♦ ❝♦♠♣❧❡t❛♥❞♦ 1 1 1 31 1 > 0✳ | x − 1 |2 −2( ) | x − 1 | + − + 8 > 0✱ ❛ss✐♠ (| x − 1 | − )2 + 2 4 4 2 4 ❖❜s❡r✈❡ q✉❡ ❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✈❛❧❡ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧✳ P♦rt❛♥t♦✱ x∈R é ❛ s♦❧✉çã♦✳ ❖❜s❡r✈❛çã♦ ✶✳✾✳ ❛✮ P♦r ❡①❡♠♣❧♦ ❜✮ ❡ b ❞❡♥♦t❛♠♦s max{−1, 4} = 4 ❡ min{6, −3} = −3✳ ❖ ♠á①✐♠♦ ❞❡ ❞♦✐s ♥ú♠❡r♦s ❙❡ a < x < b✱ ❡♥tã♦ P♦r ❡①❡♠♣❧♦✱ s❡ a max{a, b} ❡ ♦ ♠í♥✐♠♦ ❞❡ min{a, b}✳ | x |< max{| a |, | b | }✳ 2 < x < 6✱ ❡♥tã♦ | x |< 6 ✹✹ ❡ s❡ −12 < x < 6✱ ❡♥tã♦ | x |< 12✳ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✶✲✹ ✶✳ ❘❡s♦❧✈❡r ❛s s❡❣✉✐♥t❡s ❡q✉❛çõ❡s✿ ✶✳ ✹✳ ✼✳ ✶✵✳ ✶✸✳ ✶✻✳ ✷✳ || 5 − 2x | −4 |= 8 ✸✳ | x2 − 4 |=| 2x | ✺✳ | x2 − 4 |= 3x + 4 ✻✳ | x2 + 4 |=| 2x | | 4x + 3 |= 7 x2 − 2 | x |= 3 ✽✳ | x2 + 2 |= 2x + 1 | x − 4 |=| x − 2 | ✾✳ | 2x + 2 |= 6x − 18 | x2 − x − 6 |= x + 2 | 2x − 5 |= 3 ✶✹✳ | 2x − 4 |= 6 3x + 1 =4 x−1 ✶✶✳ | x − 2 |=| 3 − 2x | 2|x+2| − | 2x+1 − 1 |= 2x+1 + 1 ✶✼✳ ✶✷✳ 2 | x − 1 | −x2 + 2x + 7 = 0 ✶✺✳ | x − 1 | + 4 | x − 3 |= 2 | x + 2 | ✷✳ ❘❡♣r❡s❡♥t❡ ❝❛❞❛ ✉♠ ❞♦s ❝♦♥❥✉♥t♦s s❡❣✉✐♥t❡s ❛tr❛✈és ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡s ❡♥✈♦❧✈❡♥❞♦ ✈❛❧♦r❡s ❛❜s♦❧✉t♦s✳ ✶✳ ✸✳ A = { x ∈ R /. x < −4 ♦✉ x > 4 } C = { x ∈ R /. x > −9 ♦✉ x < 9 } ✷✳ ✹✳ B = { x ∈ R /. x ≤ −6 ♦✉ x ≥ 4 } D = { x ∈ R /. x ≥ −9 ♦✉ x ≤ 7 } ✸✳ ❘❡♣r❡s❡♥t❡ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ♦s s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s✱ ♣❛r❛ ❧♦❣♦ ❡♠ s❡❣✉✐❞❛ ❡①♣r❡ssá✲ ❧♦s ♥❛ ❢♦r♠❛ ❞❡ ✐♥t❡r✈❛❧♦s✳ ✶✳ ✸✳ ✺✳ ✼✳ A = { x ∈ R /. 8 < x < 13 } ✷✳ C = { x ∈ R /. − 13 ≤ x < 15 } ✹✳ E = { x ∈ R /. | 9 − x |< 7 } G = { x ∈ R /. x > −9 ✻✳ x<9} ♦✉ ✽✳ B = { x ∈ R /. − 14 ≤ x < 5 } D = { x ∈ R /. | x |< 6 } F = { x ∈ R /. | x + 5 |≥ 8 } H = { x ∈ R /. | 9 − x |<| x + 5 | } ✹✳ ❘❡s♦❧✈❡r ❛s s❡❣✉✐♥t❡s ✐♥❡q✉❛çõ❡s✿ 1. | x + 4 | − | 5 − 2x |> 4 2. | x2 − 4 | + | 2x − 5 |< 6 3. | 3− | 2x + 3 ||< 2 4. | 3x − 2 |≤| 4x − 4 | + | 7x − 6 | ✺✳ ❊♥❝♦♥tr❛r ♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ ❡♠ R✳ 1. | 2x + 3 | +4 = 5x 2. | x2 − 4 |= −2x + 4 3. 4. | 5x − 3 |=| 3x + 5 | 5. | 2x + 6 |=| 4 − 5x | 6. 7. 2 1 ≤ 6 − 3x |x+3| ✻✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❡ E✱ s❡✿ 8. | x | −2 <| x − 1 | E= 9. | 4x + 1 | − | x − 1 | x ✹✺ | 3x − 1 |= 2x + 5 6 − 5x 1 ≤ 3+x 2 | x − 3 | +2 | x |< 5 ∀ x ∈ (0, 1)✳ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✼✳ ❙❡❥❛♠ a ❡ b ♥ú♠❡r♦s r❡❛✐s✱ ♠♦str❡ q✉❡✿ max{ a, b} = a + b+ | b − a | 2 min{ a, b} = ✽✳ ❙✉♣♦♥❤❛ ε > 0 ❡ ♠♦str❡ ♦ s❡❣✉✐♥t❡✿ ε , 1} ❡ | y − y0 |< ✶✳ ❙❡ | x − x0 |< min{ 2(| y0 | +1) ✷✳ ❙❡ | y0 |6= 0 ❡ | y − y0 |< min{ | y0 | ε | y0 | 2 , } 2 2 a + b− | b − a | 2 ε 2(| x0 | +1) ⇒ ⇒ | xy − x0 y0 |< ε y 6= 0 ❡ 1 1 − < ε✳ y y0 ✾✳ ▼♦str❡ q✉❡✱ s❡ ♦s ♥ú♠❡r♦s a1 , a2 , a3 , · · · , an ♥ã♦ sã♦ ✐❣✉❛✐s ❛ ③❡r♦ ❡ ❢♦r♠❛♠ ✉♠❛ 1 1 1 1 n−1 + + + ··· + = . ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛✱ ❡♥tã♦✿ a1 .a2 a2 .a3 a3 .a4 an−1 .an a1 .an ✶✵✳ P❛r❛ t❡st❛r s❡ ✉♠❛ ♠♦❡❞❛ é ❡q✉✐❧✐❜r❛❞❛✱ ✉♠ ♣❡sq✉✐s❛❞♦r ❧❛♥ç❛ 100 ✈❡③❡s ❡ ❛♥♦t❛ ♦ ♥ú♠❡r♦ x ❞❡ ❝❛r❛✳ ❆ t❡♦r✐❛ ❡st❛tíst✐❝❛ ❛✜r♠❛ q✉❡ ❛ ♠♦❡❞❛ ❞❡✈❡ s❡r ❝♦♥s✐❞❡r❛❞❛ ♥ã♦ ❡q✉✐❧✐❜r❛❞❛ s❡ x − 50 ≥ 1, 645✳ P❛r❛ q✉❡ ✈❛❧♦r❡s ❞❡ x ❛ ♠♦❡❞❛ s❡rá ❡q✉✐❧✐❜r❛❞❛ ❄ 5 ✶✶✳ ❆ ♣r♦❞✉çã♦ ❞✐ár✐❛ ❡st✐♠❛❞❛ ① ❞❡ ✉♠❛ r❡✜♥❛r✐❛ é ❞❛❞❛ ♣♦r | x − 300.000 |≤ 275.000✱ ♦♥❞❡ x é ♠❡❞✐❞❛ ❡♠ ❜❛rr✐s ❞❡ ♣❡tró❧❡♦✳ ❉❡t❡r♠✐♥❡ ♦s ♥í✈❡✐s ♠á①✐♠♦ ❡ ♠í♥✐♠♦ ❞❡ ♣r♦❞✉çã♦✳ ✶✷✳ ❆s ❛❧t✉r❛s h ❞❡ ❞♦s t❡rç♦s ❞❡ ❛❧✉♥♦s ❞❛ ▲✐❝❡♥❝✐❛t✉r❛ ❡♠ ▼❛t❡♠át✐❝❛✱ ✈❡r✐✜❝❛♠ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ h − 1, 76 ≤ 1✱ ♦♥❞❡ h é ♠❡❞✐❞♦ ❡♠ ♠❡tr♦s✳ ❉❡t❡r♠✐♥❡ ♦ ✐♥t❡r✈❛❧♦ 0, 22 ❞❛ r❡t❛ r❡❛❧ q✉❡ ❡ss❛s ❛❧t✉r❛s s❡ s✐t✉❛♠✳ ✶✸✳ ❯♠ t❡rr❡♥♦ ❞❡✈❡ s❡r ❧♦t❛❞♦✳ ❖s ❧♦t❡s✱ t♦❞♦s r❡t❛♥❣✉❧❛r❡s✱ ❞❡✈❡♠ t❡r ár❡❛ s✉♣❡r✐♦r ♦✉ ✐❣✉❛❧ ❛ 400 m2 ✱ ❡ ❛ ❧❛r❣✉r❛ ❞❡ ❝❛❞❛ ✉♠ ❞❡✈❡ t❡r ✸✵♠ ❛ ♠❡♥♦s q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦✳ ❉❡t❡r♠✐♥❡ ❛s ❞✐♠❡♥sõ❡s ❞♦ ♠❡♥♦r ❞♦s ❧♦t❡s q✉❡ ❝✉♠♣r❡♠ t❛✐s ❝♦♥❞✐çõ❡s✳ ✶✹✳ ❯♠❛ ❣❛❧❡r✐❛ ✈❛✐ ♦r❣❛♥✐③❛r ✉♠❛ ❡①♣♦s✐çã♦ ❡ ❢❡③ ❛s s❡❣✉✐♥t❡s ❡①✐❣ê♥❝✐❛s✿ ✐✮ ❛ ár❡❛ ❞❡ ❝❛❞❛ q✉❛❞r♦ ❞❡✈❡ s❡r ♥♦ ♠í♥✐♠♦ ❞❡ 3.200 cm2 ❀ ✐✐✮ ♦s q✉❛❞r♦s ❞❡✈❡♠ s❡r r❡t❛♥❣✉❧❛r❡s ❡ ❛ ❛❧t✉r❛ ❞❡✈❡ t❡r 40 cm ❛ ♠❛✐s q✉❡ ❛ ❧❛r❣✉r❛✳ ❉❡♥tr♦ ❞❡ss❛s ❡s♣❡❝✐✜❝❛çõ❡s✱ ❡♠ q✉❡ ✐♥t❡r✈❛❧♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s ❞❡✈❡♠ s❡ s✐t✉❛r ❛s ❧❛r❣✉r❛s ❞♦s q✉❛❞r♦s❄ ✶✺✳ ❯♠❛ ❡♠♣r❡s❛ ❞❡ ✉t✐❧✐❞❛❞❡ ♣ú❜❧✐❝❛ t❡♠ ✉♠❛ ❢r♦t❛ ❞❡ ❛✈✐õ❡s✳ ❊st✐♠❛✲s❡ q✉❡ ♦ ❝✉st♦ ♦♣❡r❛❝✐♦♥❛❧ ❞❡ ❝❛❞❛ ❛✈✐ã♦ s❡❥❛ ❞❡ C = 0, 2k + 20 ♣♦r ❛♥♦✱ ♦♥❞❡ C é ♠❡❞✐❞♦ ❡♠ ♠✐❧❤õ❡s ❞❡ r❡❛✐s ❡ k ❡♠ q✉✐❧ô♠❡tr♦s ❞❡ ✈ô♦❀ s❡ ❛ ❡♠♣r❡s❛ q✉❡r q✉❡ ♦ ❝✉st♦ ♦♣❡r❛❝✐♦♥❛❧ ❞❡ ❝❛❞❛ ❛✈✐ã♦ s❡❥❛ ♠❡♥♦r q✉❡ 100 ♠✐❧❤õ❡s ❞❡ r❡❛✐s✱ ❡♥tã♦ k t❡♠ s❡r ♠❡♥♦r ❛ q✉❡ ✈❛❧♦r❄ ✹✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✶✳✻ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❆①✐♦♠❛ ❞♦ s✉♣r❡♠♦ ❉❡✜♥✐çã♦ ✶✳✶✺✳ ❙❡❥❛ A ✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s R✳ ✐✮ A ❉✐③❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ k1 ∈ R ✐✐✮ t❛❧ q✉❡✿ a ≤ k1 ✱ ✐✐✐✮ t❛❧ q✉❡✿ A k2 ≤ a ❉✐③❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ a ∈ A✳ ♣❛r❛ t♦❞♦ ❉✐③❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ k2 ∈ R é ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡✱ s❡ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ é ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡✱ s❡ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ✱ ♣❛r❛ t♦❞♦ A a ∈ A✳ é ❧✐♠✐t❛❞♦✱ s❡ ❢♦r ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r ❡ ✐♥❢❡r✐♦r♠❡♥t❡✳ ❊①❡♠♣❧♦ ✶✳✹✵✳ ❛✮ ❖s ❝♦♥❥✉♥t♦s N, A = (0, +∞) ❡ ✐♥❢❡r✐♦r♠❡♥t❡❀ ✉♠ ❧✐♠✐t❡ ✐♥❢❡r✐♦r é ❜✮ ❖s ❝♦♥❥✉♥t♦s A = (−∞, 3] ❡ ❧✐♠✐t❛❞♦s s✉♣❡r✐♦r♠❡♥t❡❀ ✉♠ ❝✮ ❖ ❝♦♥❥✉♥t♦ A={ 1 B = { /. n ∈ N } n k1 = −5✳ sã♦ ❝♦♥❥✉♥t♦s ❧✐♠✐t❛❞♦s B = { x ∈ R/. 5 − (x − 1)2 > 0 } ❧✐♠✐t❡ s✉♣❡r✐♦r é k2 = 5✳ 1 /. n ∈ N } n sã♦ ❝♦♥❥✉♥t♦s é ❧✐♠✐t❛❞♦✳ ❉❡✜♥✐çã♦ ✶✳✶✻✳ ❙✉♣r❡♠♦✳ ❮♥✜♠♦✳ ❙❡❥❛ A ⊂ R ❡ A 6= ∅✳ ✐✮ s ❖ ♥ú♠❡r♦ r❡❛❧ 1o ❖ ♥ú♠❡r♦ 2o ❙❡ ✐✐✮ b∈A ❡ s é ❧✐♠✐t❡ s✉♣❡r✐♦r ❞❡ b<s ❖ ♥ú♠❡r♦ r❡❛❧ 1o ❖ ♥ú♠❡r♦ 2o ❙❡ b∈A ❡ é ❝❤❛♠❛❞♦ s✉♣r❡♠♦ ❞❡ r r ❡♥tã♦ ❡①✐st❡ A❀ x∈A r<b ❡♥tã♦ ❡①✐st❡ A❀ a ≤ s✱ t❛❧ q✉❡ A q✉❛♥❞♦✿ a ∈ A✳ b < x ≤ s✳ r ≤ a✱ t❛❧ q✉❡ s = sup A ♣❛r❛ t♦❞♦ ❡ ❞❡♥♦t❛♠♦s ✐st♦ é x∈A ❡ ❞❡♥♦t❛♠♦s ✐st♦ é é ❝❤❛♠❛❞♦ í♥✜♠♦ ❞❡ é ❧✐♠✐t❡ ✐♥❢❡r✐♦r ❞❡ A r = inf A ♣❛r❛ t♦❞♦ q✉❛♥❞♦ ✿ a ∈ A✳ r ≤ x < b✳ ❆ss✐♠✱ s❡❣✉❡ q✉❡ ♦ ♠❡♥♦r ❞♦s ❧✐♠✐t❡s s✉♣❡r✐♦r❡s é ❝❤❛♠❛❞♦ ❞❡ ✏s✉♣r❡♠♦✑ ❡✱ ♦ ♠❛✐♦r ❞♦s ❧✐♠✐t❡s ✐♥❢❡r✐♦r❡s é ❝❤❛♠❛❞♦ ✏í♥✜♠♦✑✳ ❖ í♥✜♠♦ ♦✉ s✉♣r❡♠♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦✱ ♣♦❞❡ ♦✉ ♥ã♦ ♣❡rt❡♥❝❡r ❛♦ ♣ró♣r✐♦ ❝♦♥❥✉♥t♦✳ P♦r ❡①❡♠♣❧♦ ♦ í♥✜♠♦ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ ❝♦♥❥✉♥t♦✳ A={ ✹✼ 1 /. n ∈ N } n é ♦ ③❡r♦ ❡ ♥ã♦ ♣❡rt❡♥❝❡ ❛♦ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❉❡✜♥✐çã♦ ✶✳✶✼✳ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ▼á①✐♠♦✳ ▼í♥✐♠♦✳ ❙❡ ♦ s✉♣r❡♠♦ ❡ í♥✜♠♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ A sã♦ ❝❤❛♠❛❞♦s ❞❡ ✏♠á①✐♠♦✑ ❡ ✏♠í♥✐♠♦✑ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❞❡ min A ❡ inf .(A) = 0 inf(C) = 0 B={ A = (0, 9] ❙❡❥❛♠ ♦s ❝♦♥❥✉♥t♦s✿ ❆①✐♦♠❛ ✶✳✷✳ A ❡ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡✮✳ ❊①❡♠♣❧♦ ✶✳✹✶✳ ❊♥tã♦✿ A✱ ❡♥tã♦ ❞❡♥♦t❛♠♦s max A ♣❡rt❡♥❝❡♠ ❛♦ ♠❡s♠♦ ❝♦♥❥✉♥t♦ sup .(A) = 9 = max(A); ❡ ❡ sup(C) 1 /. n ∈ N } C = N✳ n inf(B) = 0 ❡ sup(B) = 1 = max(B) ♥ã♦ ❡①✐st❡✳ ❆①✐♦♠❛ ❞♦ ❙✉♣r❡♠♦✳ ❚♦❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s ♥ã♦ ✈❛③✐♦ ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡✱ t❡♠ s✉♣r❡♠♦ Pr♦♣r✐❡❞❛❞❡ ✶✳✶✺✳ ❙❡ ♦ ❝♦♥❥✉♥t♦ ♣♦ss✉✐ í♥✜♠♦✳ A ⊂ R é ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡ s❡♥❞♦ A 6= ∅ ✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ A ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ ❙❡ c B = { −x ∈ R /. x ∈ A }, B 6= ∅ é ❧✐♠✐t❡ ✐♥❢❡r✐♦r ❞❡ é ❧✐♠✐t❡ s✉♣❡r✐♦r ❞❡ ❛ss✐♠ B A✱ ❡♥tã♦ c ≤ a ∀ a ∈ A❀ −a ≤ −c ∀ a ∈ A ❡♥tã♦ −c B ♣♦ss✉✐ s✉♣r❡♠♦ s = sup(B)❀ ❧♦❣♦ ❡ ♣❡❧♦ ❛①✐♦♠❛ ❞♦ s✉♣r❡♠♦ ❡♥tã♦ −s = inf(A)✳ Pr♦♣r✐❡❞❛❞❡ ✶✳✶✻✳ Pr✐♥❝í♣✐♦ ❞❛ ❜♦❛ ♦r❞❡♠✳ ❚♦❞♦ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞❡ N ♣♦ss✉✐ ♠í♥✐♠♦✳ ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ❧❡✐t♦r✳ ✶✳✼ ■♥❞✉çã♦ ♠❛t❡♠át✐❝❛ ❉❡✜♥✐çã♦ ✶✳✶✽✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ M ❞❡ ♥ú♠❡r♦s r❡❛✐s ❞✐③✲s❡ q✉❡ é ✏ ❝♦♥❥✉♥t♦ ✐♥❞✉t✐✈♦✑✱ s❡ ❝✉♠♣r❡ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ ✐✮ ✐✐✮ 0 ∈ M✳ ∀x∈M ❡♥tã♦ (x + 1) ∈ M ❊①❡♠♣❧♦ ✶✳✹✷✳ ✹✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R • ❖ ❝♦♥❥✉♥t♦ R ❞❡ ♥ú♠❡r♦s r❡❛✐s é ✐♥❞✉t✐✈♦✱ ♣♦✐s 0 é ✉♠ ♥ú♠❡r♦ r❡❛❧ ❡ x + 1 t❛♠❜é♠ é r❡❛❧ ♣❛r❛ t♦❞♦ x r❡❛❧✳ • ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s é ✐♥❞✉t✐✈♦✳ • ❖ ❝♦♥❥✉♥t♦ { 0, 3 5 1 , 1, , 2, , · · · } é ✐♥❞✉t✐✈♦ 2 2 2 ❊①❡♠♣❧♦ ✶✳✹✸✳ ❙❡❣✉♥❞♦ ♥♦ss❛ ❞❡✜♥✐çã♦✱ ♦s s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s ♥ã♦ sã♦ ✐♥❞✉t✐✈♦s✿ • { 1, 2, 3, 4, 5, · · · } • { 0, 1, 2, 3, 4, 5 } • { 0, 2, 4, 6, · · · } ❊♠ ♠❛t❡♠át✐❝❛✱ ♠✉✐t❛s ❞❡✜♥✐çõ❡s ❡ ♣r♦♣♦s✐çõ❡s s❡ r❡❛❧✐③❛♠ ✉t✐❧✐③❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛✳ ❆ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❛♣ós ✈❡r✐✜❝❛çã♦ ❞❡ q✉❡ ❛ ♣r♦♣r✐❡❞❛❞❡ é ✈á❧✐❞❛ ❡♠ ❛❧❣✉♥s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s✱ ♣♦❞❡ ❝♦♥❞✉③✐r ❛ sér✐♦s ❡♥❣❛♥♦s ❝♦♠♦ ♠♦str❛ ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✿ ❊①❡♠♣❧♦ ✶✳✹✹✳ ❈♦♥s✐❞❡r❡ ❛ r❡❧❛çã♦ f (n) = 22 + 1 ❞❡✜♥✐❞❛ ♣❛r❛ t♦❞♦ ♥ ∈ N✳ ❚❡♠♦s q✉❡✱ q✉❛♥❞♦✿ n n = 0 ❡♥tã♦ f (0) = 22 + 1 = 3 0 n = 1 ❡♥tã♦ f (1) = 22 + 1 = 5 1 n = 2 ❡♥tã♦ f (2) = 22 + 1 = 17 2 n = 3 ❡♥tã♦ f (3) = 22 + 1 = 257 3 n = 4 ❡♥tã♦ f (4) = 22 + 1 = 65.537 4 ❖❜s❡r✈❡ q✉❡ t♦❞♦s ❛q✉❡❧❡s ♥ú♠❡r♦s ❡♥❝♦♥tr❛❞♦s sã♦ ♥ú♠❡r♦s ♣r✐♠♦s❀ P✳ ❋❡r♠❛t ✭1601− 1665✮ ❛❝r❡❞✐t♦✉ q✉❡ ❛ ❢ór♠✉❧❛ f (n) r❡♣r❡s❡♥t❛r✐❛ ♥ú♠❡r♦s ♣r✐♠♦s✱ q✉❛❧q✉❡r q✉❡ ❢♦ss❡ ♦ ✈❛❧♦r ♣❛r❛ n ∈ N✱ ♣♦✐s ❡st❛ ✐♥❞✉çã♦ ❡r❛ ❢❛❧s❛ ❊✉❧❡r✼ ♠♦str♦✉ q✉❡ ♣❛r❛ n = 5 r❡s✉❧t❛ f (5) = 4.294.967.297 = 641 × 6.700.417 ❧♦❣♦✱ ❛ ❛✜r♠❛çã♦ ❞❡ P✳ ❋❡r♠❛t ❢♦✐ ♣r❡❝✐♣✐t❛❞❛✳ ❊①❡♠♣❧♦ ✶✳✹✺✳ ❈♦♥s✐❞❡r❡♠♦s ❛ r❡❧❛çã♦ f (n) = n2 + n + 41 ❞❡✜♥✐❞❛ ♣❛r❛ t♦❞♦ n ∈ N ✼ ▲❡♦♥❛r❞ ❊✉❧❡r (1707 − 1783) ❡st✉❞♦✉ ❝♦♠ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐✱ ❛✐♥❞❛ ♣❛✐ ❞❡ tr❡③❡ ✜❧❤♦s ❡ ✜❝❛♥❞♦ ❝♦♠♣❧❡t❛♠❡♥t❡ ❝❡❣♦✱ ❡s❝r❡✈❡✉ ♠❛✐s ❞❡ ♦✐t♦❝❡♥t♦s tr❛❜❛❧❤♦s ❡ ❧✐✈r♦s ❡♠ t♦❞♦s ♦s r❛♠♦s ❞❛ ♠❛t❡♠át✐❝❛✳ ✹✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R 40, f (n) é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ n = 1, f (1) = 43; s❡ n = 2, f (2) = 47; s❡ n = 3, f (3) = 53; . . . ; s❡ n = 39, f (39) = 1.601✳ 2 P♦ré♠ s❡ n = 40 t❡♠♦s f (40) = 40 + 40 + 41 = (41)(41) ♥ã♦ é ♣r✐♠♦✱ ♠♦str❛♥❞♦ q✉❡ ❛ 2 s❡♥t❡♥ç❛ é ❢❛❧s❛✳ ❊♠ 1.772 ❊✉❧❡r ♠♦str♦✉ q✉❡ f (n) = n + n + 41 ❛ss✉♠❡ ✈❛❧♦r❡s ♣r✐♠♦s ♣❛r❛ n = 0, 1, 2, 3, . . . , 39✳ ❖❜s❡r✈❡ q✉❡✱ ♣❛r❛ ✈❛❧♦r❡s ♠❡♥♦r❡s q✉❡ f (n−1) = f (−n) ♠♦str♦✉ q✉❡ n2 +n+41 ❛ss✉♠❡ ✈❛❧♦r❡s ♣r✐♠♦s ♣❛r❛ 80 ♥ú♠❡r♦s ✐♥t❡✐r♦s ❝♦♥s❡❝✉t✐✈♦s✱ s❡♥❞♦ ❡st❡s ✐♥t❡✐r♦s✿ n = −40, −39, −38, . . . 0, 1, 2, 3, . . . 38, 39❀ s✉❜st✐t✉✐♥❞♦ ❛ ✈❛r✐á✈❡❧ n ♣♦r n − 40 t❡♠♦s f (n − 40) = g(n) = n2 − 79n + 1.601❀ ❧♦❣♦ g(n) = n2 − 79n + 1.601 ❛ss✉♠❡ ✈❛❧♦r❡s ♣r✐♠♦s ♣❛r❛ t♦❞♦s ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❞❡ 0 ❛té 79✳ ❊✉❧❡r ♦❜s❡r✈❛♥❞♦ q✉❡ ❊①❡♠♣❧♦ ✶✳✹✻✳ ❆ s❡♥t❡♥ç❛✿ ✏ 2n ♣❛r❛ + 2 é ❛ s♦♠❛ ❞❡ ❞♦✐s ♥ú♠❡r♦s ♣r✐♠♦s✑ é ✉♠❛ s❡♥t❡♥ç❛ ✈❡r❞❛❞❡✐r❛ n = 1, n = 2, n = 3, n = 4, . . . ❡✱ ❝♦♠♦ ♥♦s ❡①❡♠♣❧♦s ❛♥t❡r✐♦r❡s ❛♣ós ♠✉✐t❛s t❡♥t❛t✐✈❛s✱ ♥ã♦ ❛❝❤❛♠♦s ♥❡♥❤✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ q✉❡ ❛ t♦r♥❡ ❢❛❧s❛✳ ◆✐♥❣✉é♠ ❛té ❤♦❥❡✱ ❛❝❤♦✉ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ q✉❡ t♦r♥❛ss❡ ❛ s❡♥t❡♥ç❛ ❢❛❧s❛ ❡ ♥✐♥❣✉é♠✱ ❛té ❤♦❥❡✱ s❛❜❡ ❞❡♠♦♥str❛r q✉❡ ❛ s❡♥t❡♥ç❛ é s❡♠♣r❡ ✈❡r❞❛❞❡✐r❛✳ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❝♦♥❥❡t✉r❛ ❞❡ ●♦❧❞❜❛❝❤✱ ❢♦✐ ❢❡✐t❛ ❡♠ 1742✱ ❊st❛ ❢❛♠♦s❛ s❡♥t❡♥ç❛ ❡♠ ✉♠❛ ❝❛rt❛ ❞✐r✐❣✐❞❛ ❛ ❊✉❧❡r ❞✐③✿ ✏ ❚♦❞♦ ✐♥t❡✐r♦ ♣❛r✱ ♠❛✐♦r ❞♦ q✉❡ 2✱ é ❛ s♦♠❛ ❞❡ ❞♦✐s ♥ú♠❡r♦s ♣r✐♠♦s✳✑ ◆ã♦ s❛❜❡♠♦s ❛té ❤♦❥❡ s❡ ❡st❛ s❡♥t❡♥ç❛ é ✈❡r❞❛❞❡✐r❛ ♦✉ ❢❛❧s❛✳ ❊♠ r❡s✉♠♦✱ ❞❛❞❛ ✉♠❛ ❛✜r♠❛çã♦ s♦❜r❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ s❡ ❡♥❝♦♥tr❛♠♦s ✉♠ ❝♦♥tr❛✲ ❡①❡♠♣❧♦✱ s❛❜❡♠♦s q✉❡ ❛ ❛✜r♠❛çã♦ ♥ã♦ é s❡♠♣r❡ ✈❡r❞❛❞❡✐r❛✳ ❊ s❡ ♥ã♦ ❛❝❤❛♠♦s ✉♠ ❝♦♥tr❛✲❡①❡♠♣❧♦❄ ◆❡st❛ ❝❛s♦✱ s✉s♣❡✐t❛♥❞♦ q✉❡ ❛ ❛✜r♠❛çã♦ s❡❥❛ ✈❡r❞❛❞❡✐r❛ s❡♠♣r❡✱ ✉♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡ é t❡♥t❛r ❞❡♠♦♥str❛✲❧❛ r❡❝♦rr❡♥❞♦ ❛♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦❀ é ♥❡❝❡ssár✐♦ ♣♦rt❛♥t♦✱ ❞✐s♣♦r ❞❡ ✉♠ ♠ét♦❞♦ ❝♦♠ ❜❛s❡ ❧ó❣✐❝❛ q✉❡ ♣❡r♠✐t❛ ❞❡❝✐❞✐r s♦❜r❡ ❛ ✈❛❧✐❞❛❞❡ ♦✉ ♥ã♦ ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ✐♥❞✉çã♦✱ ✐st♦ ❡st❛ ❣❛r❛♥t✐❞♦ ❝♦♠ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ Pr♦♣r✐❡❞❛❞❡ ✶✳✶✼✳ Pr✐♠❡✐r♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛✳ ❙❡ P (n) é ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❡♥✉♥❝✐❛❞❛ ❡♠ t❡r♠♦s ❞❡ 1o P (1) é ✈❡r❞❛❞❡✐r♦ 2o P (h) é ✈❡r❞❛❞❡✐r♦ ♣❛r❛ ❊♥tã♦ P (n) h > 1✱ ✐♠♣❧✐❝❛ é ✈❡r❞❛❞❡✐r♦✱ ♣❛r❛ t♦❞♦ P (h + 1) n✱ ♣❛r❛ n∈N t❛❧ q✉❡✿ é ✈❡r❞❛❞❡✐r♦✳ n∈N✳ ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ✺✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✶✳✹✼✳ ▼♦str❡ q✉❡ 1 + 2 + 3 + 4 + . . . + n = ❙♦❧✉çã♦✳ n(n + 1) ✳ 2 ◆❡st❡ ❡①❡♠♣❧♦ ♦❜s❡r✈❡ q✉❡ P (n) : 1 + 2 + 3 + 4 + . . . + n = P❛r❛ n = 1, P (1) : 1 = 1(1 + 1) é ✈❡r❞❛❞❡✐r❛✳ 2 n(n + 1) ✳ 2 h(h + 1) s❡❥❛ ✈❡r❞❛❞❡✐r❛✳ 2 (h + 1)[(h + 1) + 1] é ▼♦str❛r❡✐ q✉❡ P (h + 1) : 1 + 2 + 3 + 4 + . . . + h + (h + 1) = 2 ❙✉♣♦♥❤❛♠♦s q✉❡ P (h) : 1 + 2 + 3 + 4 + . . . + h = ✈❡r❞❛❞❡✐r♦✳ ❈♦♠ ❡❢❡✐t♦✱ t❡♠♦s q✉❡✿ h(h + 1) + (h + 1) = 2 (h + 1)(h + 2) (h + 1)[(h + 1) + 1 ] h = ✳ = (h + 1)( + 1) = 2 2 2 1 + 2 + 3 + 4 + . . . + h + (h + 1) = ▲♦❣♦✱ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛ ❝✉♠♣r❡✿ 1 + 2 + 3 + 4 + ... + n = n(n + 1) 2 ∀n∈N ❊①❡♠♣❧♦ ✶✳✹✽✳ ❉❡s❡❥❛✲s❡ ❝♦♥str✉✐r ✉♠❛ ♣❛r❡❞❡ ❞❡❝♦r❛t✐✈❛ ❝♦♠ t✐❥♦❧♦s ❞❡ ✈✐❞r♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❛ ♣r✐♠❡✐r❛ ✜❧❛ ✭❜❛s❡✮ ❞❡✈❡rá t❡r 100 t✐❥♦❧♦s✱ ❛ s❡❣✉♥❞❛ ✜❧❛✱ 99 t✐❥♦❧♦s✱ ❛ t❡r❝❡✐r❛✱ 98 t✐❥♦❧♦s ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡ ❛té ❛ ú❧t✐♠❛ ✜❧❛ q✉❡ ❞❡✈❡rá t❡r ❛♣❡♥❛s 1 t✐❥♦❧♦✳ ❉❡t❡r♠✐♥❡ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ t✐❥♦❧♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❝♦♥str✉✐r ❞❡st❛ ♣❛r❡❞❡✳ s❡rá ✐❣✉❛❧ ❛✿ ❙♦❧✉çã♦✳ ❖❜s❡r✈❡ q✉❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♥ú♠❡r♦ ❞❡ t✐❥♦❧♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❝❛❞❛ ✜❧❛ é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❞❡❝r❡s❝❡♥t❡ ❛ ♣❛rt✐r ❞❡ 100 ❧♦❣♦✱ t❡♠♦s ❛♣❧✐❝❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞♦ ❊①❡♠♣❧♦ ✭✶✳✹✼✮✱ q✉❡ ♦ t♦t❛❧ ❞❡ t✐❥♦❧♦s é✿ 100 + 99 + · · · + 3 + 2 + 1 = P♦rt❛♥t♦✱ sã♦ ♥❡❝❡ssár✐♦s 5.050 t✐❥♦❧♦s✳ 100(100 + 1) = 5.050✳ 2 ❊①❡♠♣❧♦ ✶✳✹✾✳ ▼♦str❡ q✉❡✱ ♣❛r❛ t♦❞♦ n ∈ N ❛ ❡①♣r❡ssã♦ n3 − n é ❞✐✈✐sí✈❡❧ ♣♦r s❡✐s✳ ❙♦❧✉çã♦✳ ❚❡♠♦s q✉❡ P (n) : n3 − n P (1) : 13 − 1 = 0 é ❞✐✈✐sí✈❡❧ ♣♦r 6✳ ❙✉♣♦♥❤❛ q✉❡ P (h) : h3 − h s❡❥❛ ❞✐✈✐sí✈❡❧ ♣♦r 6 s❡♥❞♦ h ∈ N✳ P❛r❛ n = h + 1 t❡♠♦s P (h + 1) : (h + 1)3 − (h + 1) = (h + 1)[(h + 1)2 − 1] = h3 − h + 3h(h + 1) ✺✶ ✭✶✳✽✮ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❖❜s❡r✈❡ q✉❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ 3h(h + 1) é ❞✐✈✐sí✈❡❧ ♣♦r 6✳ h = 1 t❡♠♦s q✉❡ 3(1)(2) é ❞✐✈✐sí✈❡❧ ♣♦r 6✳ h ∈ N✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ ♣♦r 6✱ ♣❛r❛ t♦❞♦ ❙✉♣♦♥❤❛ 3h(h + 1) é ❞✐✈✐sí✈❡❧ h + 1 s❡❣✉❡ q✉❡ 3(h + 1)(h + 2) = 3h(h + 1) + 6 s❡♥❞♦ ❞✐✈✐sí✈❡❧ ♣♦r 6✳ ❊♥tã♦ ❤✐♣ót❡s❡ ❛✉①✐❧✐❛r ♣❛r❛ P (n) ❝♦♥❝❧✉í♠♦s q✉❡ ♣❛r❛ t♦❞❛ n ∈ N ❛ ❡①♣r❡ssã♦ ▲♦❣♦ ♣❛r❛ ❡♠ ✭✶✳✽✮ ❞❛ n3 − n R é ❞✐✈✐sí✈❡❧ ♣♦r s❡✐s✳ ❊①❡♠♣❧♦ ✶✳✺✵✳ (1 + x)n ≥ −1 (1 + x)n ≥ 1 + nx✳ ▼♦str❡ q✉❡✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ t❡♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❙♦❧✉çã♦✳ ❙❡❥❛ S 1o 1 ∈ S 2o ❙❡ ❡ ♣❛r❛ q✉❛❧q✉❡r ♥❛t✉r❛❧ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣❛r❛ ♦s q✉❛✐s ♣♦✐s✱ n∈N ❡♥tã♦ (1 + x)n ≥ 1 + nx✳ (1 + x)1 ≥ 1 + (1)x✳ h ∈ S ✱ t❡♠♦s q✉❡ (1 + x)h ≥ 1 + hx✱ ❡♥tã♦ (1 + x)h+1 = (1 + x)(1 + x)h ≥ (1 + x)(1 + hx) ≥ 1 + x + hx + hx2 ≥ 1 + (h + 1)x✳ ▲♦❣♦✱ s❡ h∈S ❡♥tã♦ (h + 1) ∈ S ✳ S = N✳ ❆♣❧✐❝❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛ t❡♠♦s q✉❡ Pr♦♣r✐❡❞❛❞❡ ✶✳✶✽✳ ❙❡❣✉♥❞♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛✳ ❙❡ P (n) é ✉♠❛ ♣r♦♣♦s✐çã♦ ❡♥✉♥❝✐❛❞❛ ♣❛r❛ 1o P (no ) o 2 P (h) é ✈❡r❞❛❞❡✐r♦✳ h > no ✱ n ≥ no ✳ é ✈❡r❞❛❞❡✐r♦ ♣❛r❛ ∀ n ∈ N✱ ✈❡r❞❛❞❡✐r♦ t❛❧ q✉❡ n∈N t❛❧ q✉❡✿ P (h + 1) é ✈❡r❞❛❞❡✐r♦✳ 1 3 (n + 2n) 3 é ✉♠ ✐♥t❡✐r♦✳ ✐♠♣❧✐❝❛ ❊♥tã♦ P (n) é ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❊①❡♠♣❧♦ ✶✳✺✶✳ ▼♦str❡ q✉❡ s❡ n é q✉❛❧q✉❡r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ ❙♦❧✉çã♦✳ ❙❡❥❛ S ♦ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s t❛✐s q✉❡ 1 3 (n + 2n) 3 é ✉♠ ✐♥t❡✐r♦✳ 1 3 (1 + 2(1)) = 1✳ 3 1 3 ❙✉♣♦♥❤❛ q✉❡ h ∈ S ❀ ✐st♦ é (h + 2h) é ✉♠ ✐♥t❡✐r♦✳ 3 1 1 3 1 3 3 2 2 ❊♥tã♦✱ [(h+1) +2(h+1)] = [(h +3h +3h+1)+(2h+2)] = (h +2h)+(h +h+1) 3 3 3 ❖ ♥ú♠❡r♦ 1∈S ♣♦✐s é ✉♠ ✐♥t❡✐r♦✳ ❆ss✐♠ h∈S ✐♠♣❧✐❝❛ (h + 1) ∈ S ✳ ▲♦❣♦ S=N ✺✷ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦✳  09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✶✳✺✷✳ ❙❡❥❛♠ a, b ∈ R+ t❛✐s q✉❡ s❡♠♣r❡ é ✈❡r❞❛❞❡✐r❛✳ a 6= b✳ ▼♦str❡ q✉❡ 2n−1 (an + bn ) > (a + b)n , ∀ n ∈ N+ ✱ ❉❡♠♦♥str❛çã♦✳ P❛r❛ n=2 ❛ ❞❡s✐❣✉❛❧❞❛❞❡ é ❞❛ ❢♦r♠❛✿ 2(a2 + b2 ) > (a + b)2 ✭✶✳✾✮ a 6= b✱ t❡♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ a2 +b2 > 2ab✱ s♦♠❛♥❞♦ a2 +b2 ♦❜t❡♠♦s 2(a2 +b2 ) > 2ab + a2 + b2 = (a + b)2 ✐st♦ ✐♠♣❧✐❝❛ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✾✮✱ ♣♦rt❛♥t♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ é ✈á❧✐❞❛ ♣❛r❛ n = 2✳ ❈♦♠♦ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ s❡❥❛ ✈á❧✐❞❛ ♣❛r❛ n = h✱ ✐st♦ é (a + b)h < 2h−1 (ah + bh ) ▼♦str❛r❡♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣❛r❛ n = h + 1✱ ✭✶✳✶✵✮ ✐st♦ é ❛ ♠♦str❛r q✉❡ (a + b)h+1 < 2h (ah+1 + bh+1 ) ❚❡♠♦s ❞❡ ✭✶✳✶✵✮ q✉❡ ✭✶✳✶✶✮ (a + b)h+1 = (a + b)h (a + b) < 2h−1 (ah + bh )(a + b)✱ (a + b)h+1 < 2h−1 [ah+1 + bh+1 + abh + bah ] ❈♦♠♦ ✐st♦ é✿ ✭✶✳✶✷✮ a 6= b✱ s✉♣♦♥❤❛♠♦s a > b✱ ❝♦♠♦ a, b ∈ R ❡♥tã♦ ah > bh ✱ ❧♦❣♦ (ah − bh )(a − b) > 0 s❡♠♣r❡ é ✈❡r❞❛❞❡✐r❛✳ P❛r❛ ♦ ❝❛s♦ a < b✱ ❡♥tã♦ ah < bh ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s✱ ❧♦❣♦ (ah − bh )(a − b) > 0 é ♦ ♣r♦❞✉t♦ ❞❡ (ah − bh )(a − b) > 0 ✐st♦ ✐♠♣❧✐❝❛ q✉❡ ah+1 + bh+1 − abh − bah > 0 ⇒ abh + bah < ah+1 + bh+1 ❊♠ ✭✶✳✶✷✮ t❡♠♦s (a + b)h+1 < 2h−1 [ah+1 + bh+1 + abh + bah < 2h (ah+1 + bh+1 ) P♦rt❛♥t♦✱ s❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✶✵✮ t❛♠❜é♠ ✈❛❧❡ ♣❛r❛ n = h + 1✱ ❧♦❣♦ ✈❛❧❡ ♣❛r❛ t♦❞♦ n ∈ N✳ ✺✸ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✶✳✽ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R Pr♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❊st✉❞❛r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ♣❛r❛ ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❡st❡ ❝♦♥❥✉♥t♦ Z ♣♦❞❡♠♦s ❡st✉❞❛r ❝♦♠♦ ✉♠❛ ❡①t❡♥sã♦ ❞♦ ❝♦♥❥✉♥t♦ N ✶✳✽✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ❉❡✜♥✐çã♦ ✶✳✶✾✳ d, n ∈ Z✱ ❛❧❣✉♠ c ∈ Z✳ ❙❡❥❛♠ ♦s ♥ú♠❡r♦s n=c·d ♣❛r❛ ❞✐③✲s❡ q✉❡ d é ❞✐✈✐s♦r ❞❡ n ❡ ❡s❝r❡✈❡♠♦s d|n q✉❛♥❞♦ ❖❜s❡r✈❡ q✉❡ ❛ ♥♦t❛çã♦ d | b ♥ã♦ r❡♣r❡s❡♥t❛ ♥❡♥❤✉♠❛ ♦♣❡r❛çã♦ ❡♠ Z✱ ♥❡♠ r❡♣r❡s❡♥t❛ ✉♠❛ ❢r❛çã♦✳ ❚r❛t❛✲s❡ ❞❡ ✉♠❛ s❡♥t❡♥ç❛ q✉❡ ❞✐③ s❡r ✈❡r❞❛❞❡ q✉❡ ❡①✐st❡ c ∈ Z t❛❧ q✉❡ n = cd✳ ❆ ♥❡❣❛çã♦ ❞❡ss❛ s❡♥t❡♥ç❛ é r❡♣r❡s❡♥t❛❞❛ ♣♦r ❛ d ∤ n✱ s✐❣♥✐✜❝❛♥❞♦ q✉❡ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ c ∈ Z t❛❧ q✉❡ n = cd✳ P♦rt❛♥t♦✱ t❡♠♦s q✉❡ 0 ∤ n✱ s❡ n 6= 0 ❆ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡st❛❜❡❧❡❝❡ ✉♠❛ r❡❧❛çã♦ ❜✐♥ár✐❛ ❡♥tr❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s ❝♦♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✳ Pr♦♣r✐❡❞❛❞❡ ✶✳✶✾✳ ❙❡❥❛♠ a, b, d, , n , m ∈ Z ✶✳ n|n ✷✳ d|n ❡ n|m ⇒ d|m ✸✳ d|n ❡ d|m ⇒ d | (an + bm) ♣❛r❛ ❛❧❣✉♠ a, b ∈ Z ✹✳ d|n ⇒ ✺✳ ad | an ❡ a 6= 0 ✻✳ 1|n 1 é ❞✐✈✐s♦r ❞❡ t♦❞♦s ♦s ✐♥t❡✐r♦s ✼✳ n|0 ❝❛❞❛ ✐♥t❡✐r♦ é ❞✐✈✐s♦r ❞♦ ③❡r♦ ✽✳ 0|n ✾✳ d | n ❡ n 6= 0 ⇒ | d |≤| n | ✶✵✳ d|n ❡ n|d ⇒ | d |=| n | ✶✶✳ d | n ❡ d 6= 0 r❡✢❡①✐✈❛ ⇒ tr❛♥s✐t✐✈❛ ❧✐♥❡❛r ♠✉❧t✐♣❧✐❝❛çã♦ ad | an ⇒ s✐♠♣❧✐✜❝❛çã♦ d|n ③❡r♦ é ❞✐✈✐s♦r s♦♠❡♥t❡ ❞♦ ③❡r♦ n=0 ⇒ ❝♦♠♣❛r❛çã♦ (n | d) | n ❉✐③❡r q✉❡ ✉♠ ♥ú♠❡r♦ a é ❞✐✈✐s♦r ❞❡ ♦✉tr♦ b✱ ♥ã♦ s✐❣♥✐✜❝❛ ❞✐③❡r q✉❡ a ❞✐✈✐❞❡ b✱ ♦❜s❡r✈❡ ❛ ♣❛rt❡ 8. ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡✱ ❛q✉✐ ③❡r♦ é ❞✐✈✐s♦r ❞♦ ③❡r♦✳ ✺✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✶✳✽✳✷ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ▼á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✳ ▼í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❉❡✜♥✐çã♦ ✶✳✷✵✳ ❙❡❥❛♠ ❉✐✈✐s♦r ❝♦♠✉♠✳ a, b, d ∈ Z✱ s❡ ♦ ♥ú♠❡r♦ dé a ❡ b✳ ❝❤❛♠❛❞♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ ✉♠ ❞✐✈✐s♦r ❞♦s ♥ú♠❡r♦s a ❡ b✱ ♦ ♥ú♠❡r♦ d é Pr♦♣r✐❡❞❛❞❡ ✶✳✷✵✳ ❉❛❞♦s ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❛❧❣✉♠ x, y ∈ Z a ❡ b✱ ❡①✐st❡ ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❛ ❢♦r♠❛ ❡✱ t♦❞♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ ❡ a b ❞✐✈✐❞❡ ❡st❡ d = ax + by ♣❛r❛ d✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳  Pr♦♣r✐❡❞❛❞❡ ✶✳✷✶✳ a, b ∈ Z✱ ❉❛❞♦s ❛✮ d≥0 ❜✮ d|a ❝✮ d∈Z ❝♦♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ d d|b ❡ c|a ❙❡ ❡①✐st❡ ✉♠ ❡ s♦♠❡♥t❡ ✉♠ ❡ d c|b ⇒ c|d ♥ã♦ é ♥❡❣❛t✐✈♦ ❡ b ❝❛❞❛ ❞✐✈✐s♦r ❝♦♠✉♠ é ❞✐✈✐s♦r ❞❡ d é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❉❡♠♦♥str❛çã♦✳ P❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✶✳✷✵✮ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❧♦❣♦ −d d′ ❝✉♠♣r❡ ✭❜✮ ❡ ✭❝✮ ❡♥tã♦ d | d′ ▲♦❣♦ ❡①✐st❡ s♦♠❡♥t❡ ✉♠ ❉❡✜♥✐çã♦ ✶✳✷✶✳ ❖ ♥ú♠❡r♦ a ❡ b d ✭❜✮ ❡ ✭❝✮✱ a | bc ❡ ❉❡♠♦♥str❛çã♦✳ d≥0 q✉❡ ❝✉♠♣r❡ ❡ d′ | d✱ ♣♦rt❛♥t♦ ✭❜✮ ❡ ✭❝✮✳ | d |=| d′ |✳ ▼á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✳ ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭✶✳✷✶✮ é ❝❤❛♠❛❞♦ ❞❡ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ✭♠✳❞✳❝✳✮ ❡ ❞❡♥♦t❛✲s❡ Pr♦♣r✐❡❞❛❞❡ ✶✳✷✷✳ ❙❡ q✉❡ ❝✉♠♣r❡ ❛s ❝♦♥❞✐çõ❡s t❛♠❜é♠ ❝✉♠♣r❡✱ ❧♦❣♦ ❡st❛ ♣r♦✈❛❞♦ ❛ ❡①✐stê♥❝✐❛✳ P♦ré♠✱ s❡ ❞❡ d mdc{ a, b }✳ ▲❡♠❛ ❞❡ ❊✉❝❧✐❞❡s✽ ✳ mdc{ a, b } = 1 ❡♥tã♦ a | c✳ mdc{ a, b } = 1 ♣♦❞❡♠♦s ❡s❝r❡✈❡r 1 = ax + by ♣❛r❛ ❛❧❣✉♠ x, y ∈ Z✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ c = cax + cby ✱ ❈♦♠♦ a | acx ❡ a | bc✱ ❡♥tã♦ c = cax + zay ❧♦❣♦ a | c✳ ❉❡s❞❡ q✉❡ ✽ ❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛ ✏ 300 a.C.✑ ❢♦✐ ♣r♦❢❡ss♦r✱ ♠❛t❡♠át✐❝♦✱ ♣❧❛tó♥✐❝♦ ❡ ❡s❝r✐t♦r ♣♦ss✐✈❡❧♠❡♥t❡ ❣r❡❣♦✱ ♠✉✐t❛s ✈❡③❡s r❡❢❡r✐❞♦ ❝♦♠♦ ♦ ✏P❛✐ ❞❛ ●❡♦♠❡tr✐❛✑✳ ❆❧é♠ ❞❡ s✉❛ ♣r✐♥❝✐♣❛❧ ♦❜r❛ ✏❖s ❊❧❡♠❡♥t♦s✑✱ ❊✉❝❧✐❞❡s t❛♠❜é♠ ❡s❝r❡✈❡✉ s♦❜r❡ s❡çõ❡s ❝ô♥✐❝❛s✱ ❣❡♦♠❡tr✐❛ ❡s❢ér✐❝❛✱ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✳ ✺✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❡✜♥✐çã♦ ✶✳✷✷✳ ▼í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠✳ ❙❡❥❛♠ a, b ∈ Z ♥ã♦ ♥✉❧♦s✱ ♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ a ❡ b✱ ❞❡♥♦t❛❞♦ mmc{a, b} é ❞❡✜♥✐❞♦ ♣♦r mmc{a, b} = ✶✳✽✳✸ a·b mdc{ a, b } ◆ú♠❡r♦s ♣r✐♠♦s ❉❡✜♥✐çã♦ ✶✳✷✸✳ ❉✐③✲s❡ q✉❡ ♦ ✐♥t❡✐r♦ n é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✱ s❡ n > 1 ❡ ♦s ú♥✐❝♦s ❞✐✈✐s♦r❡s ♣♦s✐t✐✈♦s ❞❡ n sã♦ 1 ❡ ♦ ♣ró♣r✐♦ n✳ ❙❡ n ♥ã♦ é ♥ú♠❡r♦ ♣r✐♠♦ ❡♥tã♦ é ❝❤❛♠❛❞♦ ❞❡ ♥ú♠❡r♦ ❝♦♠♣♦st♦✳ ❊①❡♠♣❧♦ ✶✳✺✸✳ ❙ã♦ ♥ú♠❡r♦s ♣r✐♠♦s✿ 2, 3, 7, 11, 13, 17, 19 ❙ã♦ ♥ú♠❡r♦s ❝♦♠♣♦st♦s✿ 4, 6, 8, 10, 16, 24 ❖ ♥ú♠❡r♦ 1 ♥ã♦ é ♣r✐♠♦❀ ♦❜s❡r✈❡ q✉❡ ♥ã♦ ❝✉♠♣r❡ ❛ ❞❡✜♥✐çã♦✳ Pr♦♣r✐❡❞❛❞❡ ✶✳✷✸✳ ❚♦❞♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ n > 1 é ♥ú♠❡r♦ ♣r✐♠♦ ♦✉ ♣r♦❞✉t♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ Pr♦♣r✐❡❞❛❞❡ ✶✳✷✹✳ ❊✉❝❧✐❞❡s✳ ❊①✐st❡ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ Pr♦♣r✐❡❞❛❞❡ ✶✳✷✺✳ ❚❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ❛r✐t♠ét✐❝❛✳ ❚♦❞♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ n > 1 ♣♦❞❡♠♦s ❡①♣r❡ss❛r ❝♦♠♦ ♣r♦❞✉t♦ ❞❡ ❢❛t♦r❡s ♣r✐♠♦s ❞❡ ♠♦❞♦ ú♥✐❝♦✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❊①❡♠♣❧♦ ✶✳✺✹✳ ▼♦str❡ q✉❡ 13|270 + 370 ✳ ❉❡♠♦♥str❛çã♦✳ ❉❡♥♦t❡♠♦s ♣❛r❛ ❡st❡ ❡①❡♠♣❧♦ m(13) ❝♦♠♦ ❛❧❣✉♠ ♠ú❧t✐♣❧♦ ❞❡ 13✱ ✐st♦ é m(13) = 13α, α ∈ Z✳ ✺✻ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❚❡♠✲s❡✿ 24 = 13 + 3, 25 = m(13) + 6, 26 = m(13) + 4 = m(13) − 1✳ ▲♦❣♦✱ 270 = 24 × (26 )11 = 24 [m(13) − 1]11 = m(13) − 24 = m(13) − 3 P♦r ♦✉tr♦ ❧❛❞♦✱ 32 = 13 − 4, 33 = m(13) + 1✳ ❡♥tã♦ 370 = 3 · (33 )23 = 3 · (m(13) + 1)23 = 3(m(13) + 123 ) = m(13) + 3 ❆ss✐♠✱ 270 + 370 = [m(13) − 3] + m(13) + 3 = m(13)✳ P♦rt❛♥t♦✱ 13|270 + 370 ✳ ❊①❡♠♣❧♦ ✶✳✺✺✳ ▼♦str❡ q✉❡ ❡①✐st❡♠ ✐♥✜♥✐t♦s ✈❛❧♦r❡s ❞❡ n ∈ Z ♣❛r❛ ♦s q✉❛✐s 7 ❡ 11 sã♦ ❞✐✈✐s♦r❡s ❞❡ 8n + 5✳ ❉❡♠♦♥str❛çã♦✳ 2 ❙❡ 7 ❡ 11 sã♦ ❞✐✈✐s♦r❡s ❞❡ 8n2 + 5✱ s❡❣✉❡ q✉❡ 77 t❛♠❜é♠ é ✉♠ ❞✐✈✐s♦r ❞❡ 8n2 + 5✱ ♣♦✐s ♦ m.d.c{ 7, 11 } = 1✳ ❙❡ 77 é ✉♠ ❞✐✈✐s♦r ❞❡ 8n2 + 5 ❡♥tã♦ ❡①✐st❡ β ∈ Z t❛❧ q✉❡ 8n2 + 5 = 77β ⇒ 8n2 + 5 − 77 = 77(β − 1) ⇒ 8(n2 − 9) = 77(β − 1) ❈♦♠♦ 8 ∤ 77✱ s❡❣✉❡ q✉❡ 8|(β − 1) ❡ 77|(n2 − 9)✳ ❆ss✐♠✱ ♣❛r❛ t♦❞♦ α ∈ N t❡♠♦s β − 1 = 8α ❡ n2 − 9 = 77α, α ∈ N✱ ❧♦❣♦ β = 1 + 8α✳ P♦rt❛♥t♦✱ 8n2 + 5 = 77(1 + 8α) ♣❛r❛ t♦❞♦ α ∈ Z✱ ❛ss✐♠ ❡①✐st❡♠ ✐♥✜♥✐t♦s ✈❛❧♦r❡s ❞❡ n ∈ Z ♣❛r❛ ♦s q✉❛✐s 8n2 + 5 é ❞✐✈✐sí✈❡❧ ♣♦r 7 ❡ ♣♦r 11✳ ❊①❡♠♣❧♦ ✶✳✺✻✳ ❙❡❥❛ n ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ ▼♦str❡ q✉❡ ✉♠✱ ❡ ❛♣❡♥❛s ✉♠✱ ♥ú♠❡r♦ ❞❡ ❝❛❞❛ t❡r♥❛ ❛❜❛✐①♦ é ❞✐✈✐sí✈❡❧ ♣♦r 3✳ ❛✮ n, n + 2, n + 4 ❜✮ n, n + 10, n + 23 ❝✮ n, n + 1, 2n + 1✳ ❉❡♠♦♥str❛çã♦✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ♠❡❞✐❛♥t❡ ♦ ❝♦♥❥✉♥t♦ A = { 3k, 3k + 1, 3k + 2, k ∈ N }✳ ❙❡ n = 3k ✱ ❡♥tã♦ ♣❛r❛ t♦❞♦s ♦s 4 ❡①❡r❝í❝✐♦s ✉♠✱ ❡ ❛♣❡♥❛s ✉♠✱ ♥ú♠❡r♦ ❞❡ ❝❛❞❛ t❡r♥❛ é ❞✐✈✐sí✈❡❧ ♣♦r 3 ❛✮ P❛r❛ ♦ ❝♦♥❥✉♥t♦ n, n + 2, n + 4 ❙❡ n = 3k+1 ❡♥tã♦ ❛ t❡r♥❛ ❞❛❞❛ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♥❛ ❢♦r♠❛ 3k+1, 3k+3, 3k+5 ❧♦❣♦ ✉♠✱ ❡ ❛♣❡♥❛s ✉♠✱ ♥ú♠❡r♦ ❞❛ t❡r♥❛ é ❞✐✈✐sí✈❡❧ ♣♦r 3✱ ♦ ♥ú♠❡r♦ 3k + 3✳ • ✺✼ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ • n = 3k+2 ❡♥tã♦ ❛ t❡r♥❛ ❞❛❞❛ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♥❛ ❢♦r♠❛ 3k+2, 3k+4, 3k+6 ✉♠✱ ❡ ❛♣❡♥❛s ✉♠✱ ♥ú♠❡r♦ ❞❛ t❡r♥❛ é ❞✐✈✐sí✈❡❧ ♣♦r 3✱ ♦ ♥ú♠❡r♦ 3k + 6✳ ❙❡ ❧♦❣♦ ❈♦♠ q✉❛❧q✉❡r ❞❛s três ❤✐♣ót❡s❡s ♥❛ t❡r♥❛ ❞❛ é ❞✐✈✐sí✈❡❧ ♣♦r ❜✮ P❛r❛ ♦ ❝♦♥❥✉♥t♦ • ✉♠✱ ❡ ❛♣❡♥❛s ✉♠✱ ♥ú♠❡r♦ 3✳ n, n + 10, n + 23 n = 3k+1 ❡♥tã♦ ❛ t❡r♥❛ ❞❛❞❛ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♥❛ ❢♦r♠❛ 3k+1, 3k+11, 3k+24 ✉♠✱ ❡ ❛♣❡♥❛s ✉♠✱ ♥ú♠❡r♦ ❞❛ t❡r♥❛ é ❞✐✈✐sí✈❡❧ ♣♦r 3✱ ♦ ♥ú♠❡r♦ 3k + 24✳ ❙❡ ❧♦❣♦ • n, n + 2, n + 4 n = 3k+2 ❡♥tã♦ ❛ t❡r♥❛ ❞❛❞❛ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♥❛ ❢♦r♠❛ 3k+2, 3k+12, 3k+25 ✉♠✱ ❡ ❛♣❡♥❛s ✉♠✱ ♥ú♠❡r♦ ❞❛ t❡r♥❛ é ❞✐✈✐sí✈❡❧ ♣♦r 3✱ ♦ ♥ú♠❡r♦ 3k + 12✳ ❙❡ ❧♦❣♦ ❈♦♠ q✉❛❧q✉❡r ❞❛s três ❤✐♣ót❡s❡s ♥❛ t❡r♥❛ ✉♠✱ ❡ ❛♣❡♥❛s ✉♠✱ ♥ú♠❡r♦ ❞❛ é ❞✐✈✐sí✈❡❧ 3✳ ♣♦r ❝✮ P❛r❛ ♦ ❝♦♥❥✉♥t♦ • n = 3k+1 ❡♥tã♦ ❛ t❡r♥❛ ❞❛❞❛ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♥❛ ❢♦r♠❛ 3k+1, 3k+2, 6k+3 ✉♠✱ ❡ ❛♣❡♥❛s ✉♠✱ ♥ú♠❡r♦ ❞❛ t❡r♥❛ é ❞✐✈✐sí✈❡❧ ♣♦r 3✱ ♦ ♥ú♠❡r♦ 6k + 3✳ ❙❡ ❧♦❣♦ • n, n + 1, 2n + 1 n = 3k+2 ❡♥tã♦ ❛ t❡r♥❛ ❞❛❞❛ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♥❛ ❢♦r♠❛ 3k+2, 3k+3, 6k+5 ✉♠✱ ❡ ❛♣❡♥❛s ✉♠✱ ♥ú♠❡r♦ ❞❛ t❡r♥❛ é ❞✐✈✐sí✈❡❧ ♣♦r 3✱ ♦ ♥ú♠❡r♦ 3k + 3✳ ❙❡ ❧♦❣♦ ❈♦♠ q✉❛❧q✉❡r ❞❛s três ❤✐♣ót❡s❡s ♥❛ t❡r♥❛ ✉♠✱ ❡ ❛♣❡♥❛s ✉♠✱ ♥ú♠❡r♦ ❞❛ é ❞✐✈✐sí✈❡❧ ♣♦r 3✳ ✺✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✶✲✺ ✶✳ ❈❛s♦ ❡①✐st❛♠✱ ❞❡t❡r♠✐♥❡ ♦ s✉♣r❡♠♦✱ ♦ í♥✜♠♦✱ ♦ ♠á①✐♠♦ ❡ ♦ ♠í♥✐♠♦ ♣❛r❛ ❝❛❞❛ ✉♠ ❞♦s s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s✿ ✶✳ B = { x ∈ Q/. | x2 − 4 |< 16 } ✷✳ A = { x ∈ Z /. | x2 − 9 | +3 | x − 4 |< 16 } ✸✳ C = { x ∈ N /. | x2 − x + 1 |< 3 } ✹✳ D = { x ∈ I /. | 5x − 10 | + | x |≥ 1 } ✺✳ F = {x ∈ R /. | x2 − 9 |≥ 16 − x } ✻✳ E = {x ∈ Z/. | x2 − 16 | + | x − 4 |> 1 } ✼✳ H = {x ∈ R/. | x2 − 9 |< 16 − x } ✽✳ G = { x ∈ R /. | 9 − x2 | − | x − 4 |< 1 } ✷✳ ▼♦str❡ q✉❡ 1 é ♦ s✉♣r❡♠♦ ❞♦ ❝♦♥❥✉♥t♦ E = { x/. x = 2n − 1 , 2n n ∈ N }✳ ✸✳ ▼♦str❡ q✉❡✱ s❡ ♦ ♣r♦❞✉t♦ ❞❡ ♥ ♥ú♠❡r♦s ♣♦s✐t✐✈♦s é ✐❣✉❛❧ ❛ 1 ✭✉♠✮✱ ❛ s♦♠❛ ❞♦s ♠❡s♠♦s ♥ã♦ é ♠❡♥♦r q✉❡ n✳ ✹✳ ▼♦str❡ q✉❡✱ s❡ x1 ✱ x2 ✱ x3 ✱ x4 , · · · , xn sã♦ ♥ú♠❡r♦s ♣♦s✐t✐✈♦s✱ t❡♠♦s✿ xn−1 xn x1 x2 x3 x4 + + + + ··· + + ≥n x2 x3 x4 x5 xn x1 ✺✳ ❯t✐❧✐③❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛✱ ♠♦str❡ ❝❛❞❛ ✉♠ ❞♦s s❡❣✉✐♥t❡s ❡♥✉♥✲ ❝✐❛❞♦s✱ ♦♥❞❡ n ∈ N✿ ✶✳ ✷✳ ✸✳ ✹✳ ✺✳ ✻✳ n(n + 1)(2n + 1) . 6 n2 (n + 1)2 3 3 3 3 . 1 + 2 + 3 + ··· + n = 4 n(3n − 1) 1 + 4 + 7 + · · · + (3n − 2) = . 2 n(4n2 − 1) 2 2 2 2 . 1 + 3 + 5 + · · · + (2n − 1) = 3 n(1 + 3n) 2 + 5 + 8 + · · · + (3n − 1) = , n≥1 2 20 + 21 + 22 + · · · + 2n−1 = 2n − 1, n > 1 12 + 2 2 + 3 2 + · · · + n2 = ✺✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✼✳ ✽✳ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R n(n + 1)(n + 2) . 3 1 1 1 1 n + + + ··· + = . 1×3 3×5 5×7 (2n − 1)(2n + 1) 2n + 1 1 × 2 + 2 × 3 + 3 × 4 + · · · + n(n + 1) = ✻✳ ❯t✐❧✐③❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛✱ ✈❡r✐✜q✉❡ ❛ ✈❛❧✐❞❛❞❡ ❞❡ ❝❛❞❛ ✉♠ ❞♦s s❡❣✉✐♥t❡s ❡♥✉♥❝✐❛❞♦s✿ é ❞✐✈✐sí✈❡❧ ♣♦r 2, ✶✳ (n2 + n) ✷✳ (n3 + 2n) ✸✳ n(n + 1)(n + 2) ✹✳ (32n − 1) ✺✳ (10n − 1) ✻✳ 2 n ≥ n2 ; ✼✳ 3n ≥ (1 + 2n); ✽✳ 8 é ❞✐✈✐sí✈❡❧ ♣♦r 3, ∀ n ∈ N✳ ∀ n ∈ N✳ é ❞✐✈✐sí✈❡❧ ♣♦r 6. ∀ n ∈ N, é ❞✐✈✐sí✈❡❧ ♣♦r 8, é ❞✐✈✐sí✈❡❧ ♣♦r 9, ∀ n ∈ N, ∀ n ∈ N✳ n 6= 0✳ ∀ n ∈ N✳ n≥4 ∀ n ∈ N✳ é ✉♠ ❞✐✈✐s♦r ❞❡ 52n + 7 ∀ n ∈ N, n≥1 ✼✳ ❉❡t❡r♠✐♥❡ ❛ ✈❛❧✐❞❛❞❡ ❞❛s s❡❣✉✐♥t❡s ♣r♦♣♦s✐çõ❡s❀ ❥✉st✐✜q✉❡ s✉❛ r❡s♣♦st❛✳ ✶✳ ❙❡ x, y ∈ R ✱ ❝♦♠ 0 < x < y ✱ ❡♥tã♦ xn < y n ∀ n ∈ N, ✷✳ 4n − 1 ✸✳ (8n − 5n ) ✹✳ (10n+1 + 10n + 1) ✺✳ ✻✳ é ❞✐✈✐sí✈❡❧ ♣♦r 3, ∀ n ∈ N✳ é ❞✐✈✐sí✈❡❧ ♣♦r 3, n 6= 0✳ ∀ n ∈ N✳ é ❞✐✈✐sí✈❡❧ ♣♦r 3, 4n > n4 ; ∀ n ∈ N, n ≥ 5✳ 22n+1 + 32n+1 é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✳ 5 ∀ n ∈ N✳ ✽✳ ▼♦str❡ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s ♥ú♠❡r♦s ♣♦s✐t✐✈♦s ❞✐❢❡r❡♥t❡s a ❡ b é ✈á❧✐❞❛ √ a + bn ✳ n+1 n n P P ✾✳ ▼♦str❡ ❛ s❡❣✉✐♥t❡ ✐❣✉❛❧❞❛❞❡✿ (b + ai ) = nb + ai ❛ ❞❡s✐❣✉❛❧❞❛❞❡✿ n+1 abn < i=1 i=1 ✶✵✳ ❙❡ n ∈ N✱ ♦ ❢❛t♦r✐❛❧ ❞♦ ♥ú♠❡r♦ n é ❞❡♥♦t❛❞♦ n✦✱ ❡ ❞❡✜♥✐❞♦ ❞♦ ♠♦❞♦ s❡❣✉✐♥t❡✿ 0✦ ❂ 1, 1✦ ❂ 1 ❡ q✉❛♥❞♦ n > 1 ❞❡✜♥❡✲s❡ n✦ ❂ 1 × 2 × 3 × 4 × 5 × · · · (n − 1) × n ♦✉ n✦ ❂ n(n − 1)(n − 2)(n − 3) · · · 4 × 3 × 2 × 1✳ ▼♦str❡ q✉❡✿ ✶✳ 2n−1 ≤ n✦ ∀ n ∈ N✳ ✶✶✳ ▼♦str❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡✿ n✦ < ✷✳  n+1 2 n 2n < n✦ < nn ♣❛r❛ ∀ n ∈ N n ≥ 4✳ ♣❛r❛ n ♥❛t✉r❛❧✱ ❝♦♠ n ≥ 2✳ ✶✷✳ ▼♦str❡ q✉❡✱ s❡ | x |< 1✱ ♣❛r❛ q✉❛❧q✉❡r ✐♥t❡✐r♦ n ≥ 2✱ ❡♥tã♦ é ✈á❧✐❞❛ ❛ ❞❡s✐❣✉❛❧❞❛❞❡✿ (1 − x)n + (1 + x)n < 2n ✳ ✻✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ▼✐s❝❡❧â♥❡❛ ✶✲✶ a, b ❡ c r❛í③❡s 1 1 69 1 + 2+ 2 = ✳ 2 a b c 4 ✶✳ ❙❡❥❛♠ ❞❛ ❡q✉❛çã♦ x3 − 3x2 + 9x − 2 = 0✳ ▼♦str❡ q✉❡ ♦ ✈❛❧♦r ❞❡ ✷✳ ❉❡t❡r♠✐♥❡ ❛ s♦♠❛✿ S = 1 + 2x + 3x2 + 4x3 + · · · + (n + 1)xn ✳ ✸✳ ❉❡t❡r♠✐♥❡ ❛ s♦♠❛✿ 1 + 11 + 111 + 1111 + · · · + 111111111 · · · 1 ✱ s❡ ♦ ú❧t✐♠♦ s♦♠❛♥❞♦ é ✉♠ ♥ú♠❡r♦ ❞❡ n ❛❧❣❛r✐s♠♦s✳ ✹✳ ❉❡t❡r♠✐♥❡ ❛ s♦♠❛✿ S = nx + (n − 1)x2 + (n − 2)x3 + · · · + 2xn−1 + xn ✳ ✺✳ ❉❡t❡r♠✐♥❡ ❛ s♦♠❛✿ S= 1 5 7 2n − 1 3 + 2 + 3 + 4 + ··· + 2 2 2 2 2n ✻✳ ▼♦str❡ q✉❡ ❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛ ❞❡ ♥ ♥ú♠❡r♦s ♣♦s✐t✐✈♦s ♥ã♦ ✉❧tr❛♣❛ss❛ ❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞❡st❡s ♠❡s♠♦s ♥ ♥ú♠❡r♦s✳ m > 1, m ∈ N sã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s ❞❡s✐❣✉❛❧❞❛❞❡s✿ 1 1 1 1 1 + + + ··· + > m+1 m+2 m+3 2m 2 1 1 1 1 + + + ··· + >1 m+1 m+2 m+3 m + (2m + 1) ✼✳ ▼♦str❡ q✉❡✱ s❡ ✶✳ ✷✳ ✽✳ Pr♦✈❡ q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n é ✈á❧✐❞♦ ♦ s❡❣✉✐♥t❡✿ 1 1 1 1 1 n−1 + 2 + 2 + 2 + ··· + 2 < 2 2 3 4 5 n n ✾✳ ▼♦str❡ ♣♦r ✐♥❞✉çã♦ s♦❜r❡ ✶✳ ❙❡ x = p+ √ q✱ ♦♥❞❡ p n✱ ❡ q q✉❡✿ sã♦ r❛❝✐♦♥❛✐s✱ ❡ n∈N ❡♥tã♦ √ xn = a + b q s❡♥❞♦ a ❡ b ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✳ ✷✳ ▼♦str❡ q✉❡ : (p − √ √ q)n = a − b q ✳ a, b, c ❢♦r♠❛♠ ✉♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛❀ 1 1 1 √ , √ √ t❛♠❜é♠ ❢♦r♠❛♠ ✉♠❛ ♣r♦❣r❡ssã♦ √ √ , √ c+ a b+ a c+ b ✶✵✳ ▼♦str❡ q✉❡✱ s❡ ♦s ♥ú♠❡r♦s ♣♦s✐t✐✈♦s ❡♥tã♦ ♦s ♥ú♠❡r♦s ❛r✐t♠ét✐❝❛✳ ✶✶✳ ❖ sí♠❜♦❧♦ n P ai é ✉s❛❞♦ ♣❛r❛ r❡♣r❡s❡♥t❛r ❛ s♦♠❛ ❞❡ t♦❞♦s ♦s i=1 ✐♥t❡✐r♦ n P i ❞❡s❞❡ 1 ❛té 1 n = ✳ n+1 i=1 i(i + 1) n❀ ✐st♦ é n P i=1 ai ❂ ai ♣❛r❛ ✈❛❧♦r❡s ❞♦ a1 + a2 + a3 + · · · + an−1 + an ✳ ✻✶ ▼♦str❡ q✉❡✿ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✶✷✳ ❈❛❧❝✉❧❛r ❛ s♦♠❛ S = n P i=1 ✶✸✳ ▼♦str❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ R ai s❡♥❞♦ ai = k ✉♠❛ ❝♦♥st❛♥t❡✳ 1 x2 ≤ é ✈❡r❞❛❞❡✐r❛ ∀ x ∈ R✳ 1 + x4 2 ✶✹✳ ❯s❛♥❞♦ ♦ ❢❛t♦ q✉❡ x2 + xy + y 2 ≥ 0✱ ♠♦str❡ q✉❡ ❛ s✉♣♦s✐çã♦ x2 + xy + y 2 < 0 ❧❡✈❛ ❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ✶✺✳ ❯♠❛ ♣✐râ♠✐❞❡ ❤❡①❛❣♦♥❛❧ r❡❣✉❧❛r✱ ❝♦♠ ❛ ❛r❡st❛ ❞❛ ❜❛s❡ 9 cm ❡ ❛r❡st❛ ❧❛t❡r❛❧ 15 cm✱ ❢♦✐ s❡❝❝✐♦♥❛❞❛ ♣♦r ❞♦✐s ♣❧❛♥♦s ♣❛r❛❧❡❧♦s à s✉❛ ❜❛s❡ q✉❡ ❞✐✈✐❞✐r❛♠ s✉❛ ❛❧t✉r❛ ❡♠ três ♣❛rt❡s ✐❣✉❛✐s✳ ▼♦str❡ q✉❡ ❛ ♣❛rt❡ ❞❛ ♣✐râ♠✐❞❡✱ ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡ ❡ss❡s ♣❧❛♥♦s✱ √ t❡♠ ✈♦❧✉♠❡✱ ✶✷✻ 3 cm3 ✳ ✶✻✳ Pr♦✈❡✱ ♣♦r ✐♥❞✉çã♦✱ q✉❡ s❡q✉ê♥❝✐❛ 1, √ 2, √ 3 3, √ 4  n + 1 n ≤ n ♣❛r❛ t♦❞♦ n ≥ 3 ❡ ❝♦♥❝❧✉❛ ❞❛í q✉❡ ❛ n 4, . . . é ❞❡❝r❡s❝❡♥t❡ ❛ ♣❛rt✐r ❞♦ t❡r❝❡✐r♦ t❡r♠♦✳ ✶✼✳ ❯♠❛ ✐♥❞ústr✐❛ ❞❡ ❝♦s♠ét✐❝♦s ❞❡s❡❥❛ ❡♠❜❛❧❛r s❛❜♦♥❡t❡s ❡s❢ér✐❝♦s ❞❡ r❛✐♦ 3 cm✳ ❆ ❡♠❜❛❧❛❣❡♠ ❞❡✈❡rá t❡r ❢♦r♠❛t♦ ❝✐❧í♥❞r✐❝♦ ❞❡ ❢♦r♠❛ ❛ ❛❝♦♥❞✐❝✐♦♥❛r 3 s❛❜♦♥❡t❡s✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✭✶✳✼✮ ✭✈✐st❛ s✉♣❡r✐♦r ❞❛ ❡♠❜❛❧❛❣❡♠ ❛❜❡rt❛✮✳ ❋✐❣✉r❛ ✶✳✼✿ ▼♦str❡ q✉❡ ❛ ♠❡❞✐❞❛ ❞♦ r❛✐♦ ❡ ❛ ❛❧t✉r❛ ❞❛ ❡♠❜❛❧❛❣❡♠✱ ❡♠ cm✱ ❞❡✈❡rã♦ s❡r ❞❡✱ √ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✿ 6, 92 ❡ 6 r❡s♣❡❝t✐✈❛♠❡♥t❡✭✳ ❙✉❣❡stã♦✿ 3 = 1, 73✮✳ ✶✽✳ ❱❡r✐✜q✉❡✱ q✉❡ ♦ ♠á①✐♠♦ ♥ú♠❡r♦ ❞❡ ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ ❞❡ n ❧❛❞♦s é✿ Nd = n(n − 3) 2 ∀ n ∈ N, n > 3✳ ✶✾✳ ▼♦str❡ q✉❡ s❡ ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ p ♥ã♦ ❞✐✈✐❞❡ a✱ ❡♥tã♦ mmc{ p, a } = 1✳ ✷✵✳ Pr♦✈❡ q✉❡ s❡ m é ✉♠ ✐♥t❡✐r♦ ♥ã♦ ♥❡❣❛t✐✈♦✱ ❡♥tã♦ 1m + 2m + 3m + · · · (n − 1)m + nm ≤ nm+1 , ✻✷ n≥1 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✷✶✳ ▼♦str❡ ♣♦r ✐♥❞✉çã♦ q✉❡ ♣❛r❛ q✉❛❧q✉❡r ✐♥t❡✐r♦ k > 1 ❡ n ∈ N✿ ✶✳ nk+1 ≥ 1 + 2k + 3k + · · · + (n − 2)k + (n − 1)k (k + 1) k−1 ✷✳ 1 1 1 n k − k1 + 3− k + · · · + (n − 1)− k + n− k 1 ≥ 1+2 1− k ✷✷✳ ▼♦str❡ ♣♦r ✐♥❞✉çã♦ ♦ s❡❣✉✐♥t❡✿ ✶✳ ❆ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤② ✿ n X ai bi i=1 !2 ≤ n X a2i i=1 ! · n X i=1 b2i ! 2(n+1) ✷✳ 2 4 (1 + q)(1 + q )(1 + q ) · · · (1 + q 2(n−1) 1−q )(1 + q ) = 1−q 2n ! n n✦ ✷✸✳ ❉❡✜♥❡✲s❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❜✐♥♦♠✐❛❧ = s❡ 0 ≤ m ≤ n✳ ▼♦str❡ q✉❡✿ m✦(m − n)✦ m ! ! ! n+1 n n ✶✳ = + s❡ 1 ≤ m ≤ n✳ m m−1 m ! n P n n . ✷✳ (a + b) = an−j bj ∀ a, b ∈ R✳ j j=0 ✷✹✳ ❉❡s❝♦❜r❛ ♦ ❡rr♦ ♥♦ s❡❣✉✐♥t❡ r❛❝✐♦❝í♥✐♦ ♣♦r ✐♥❞✉çã♦✿ ❙❡❥❛ P (n)✿ ❙❡ a ❡ b sã♦ ✐♥t❡✐r♦s ♥ã♦ ♥❡❣❛t✐✈♦s t❛✐s q✉❡ a + b ≤ n ⇒ a = b✳ ❖❜s❡r✈❡ q✉❡ P (0) é ✈❡r❞❛❞❡✐r❛✳ ❙❡❥❛♠ a ❡ b ✐♥t❡✐r♦s t❛✐s q✉❡ a + b ≤ h + 1✱ ❞❡✜♥❛ c = a − 1 ❡ d = b − 1✱ ❡♥tã♦ c + d = a + b − 2 ≤ h + 1 − 2 ≤ h✳ ❆ ✈❡r❞❛❞❡ ❞❡ P (h) ✐♠♣❧✐❝❛ q✉❡ a = b❀ ✐st♦ é P (h + 1) é ✈❡r❞❛❞❡✐r❛✳ P♦rt❛♥t♦ P (n) é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ n ≥ 0, n ∈ N✳   2  3 n  1 1 1 (n + 1)n 1 . 1+ ··· 1 + = ✷✺✳ ▼♦str❡ q✉❡✿ 1 + . 1 + 1 2 3 n n✦  ∀ n ∈ N+ ✳ ✷✻✳ ❙❡ a, b ❡ n sã♦ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✱ ♠♦str❡ ♦ s❡❣✉✐♥t❡✿ ! b n !2 + ✶✳ a 0 ✷✳ n 0 ! ! ! ! ! ! ! a b a b a b + + ··· + + = 1 n−1 n−1 1 n 0 ! !2 !2 !2 !2 2n n n n n = + + ··· + + n n n−1 2 1 ✻✸ a+b n ! 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✷✼✳ ❙❡❥❛ r 6= 1✳ ✶✳ ❉❡❞✉③✐r q✉❡✱ a + ar + ar + ar + ar + · · · + ar ✷✳ ▼♦str❡ ♣♦r ✐♥❞✉çã♦ s♦❜r❡ n ∈ N, 2 2 3 3 4 n−1 n ≥ 1 q✉❡✿ 4 a + ar + ar + ar + ar + · · · + ar n−1  1 − rn =a 1−r  1 − rn =a 1−r R   ✷✽✳ ▼♦str❡ q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r x > 0 ❡ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ n✲♣❛r✱ ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡ é ✈❡r❞❛❞❡✐r❛✿ xn + xn−2 + xn−4 + · · · + 1 xn−4 + 1 xn−2 + 1 ≥n+1 xn ✷✾✳ ▼♦str❡ q✉❡ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❝♦♠♦ ♦ ♣r♦❞✉t♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✳ ✸✵✳ ❆ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ❞❡✜♥❡✲s❡ ❝♦♠♦ s❡❣✉❡✿ a1 = 1, ♣❛r❛ n ≥ 3✳ ▼♦str❡ ♣♦r ✐♥❞✉çã♦ q✉❡✿ an =  a2 = 1, an = an−1 +an−2 √ n 1+ 5 2  √ n − 1−2 5 √ 5 ✸✶✳ ◆❛ ✜❣✉r❛ ❛♦ ❧❛❞♦✱ ♦ tr✐â♥❣✉❧♦ ABC é ❡q✉✐❧át❡r♦✱ M é ♣♦♥t♦ ♠é❞✐♦ ❞♦ ❧❛❞♦ AB ✱ ♦ s❡❣♠❡♥t♦ M N é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❧❛❞♦ BC ❡ ♦ s❡❣♠❡♥t♦ N P é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❧❛❞♦ AC ✳ ❙❛❜❡♥❞♦ q✉❡ ♦ ❧❛❞♦ AP = 12✱ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ABC M✡ ✡ ✡ A ✡ ✡ ❏❏ ❏ ❏ P ✟❏ ✡ ✟✟ ❏ ❏ ✡ ✟✟ B N C ✸✷✳ ▼♦str❡ q✉❡✱ s❡ a1 , a2 , a3 , · · · , an sã♦ ♥ú♠❡r♦s r❡❛✐s t❛✐s q✉❡ | a1 |≤ 1 ❡ | an −an−1 |≤ 1✱ ❡♥tã♦ | an |≤ 1✳ ✸✸✳ ▼♦str❡ q✉❡✱ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n ❡ ♣❛r❛ p > 0 ♥ú♠❡r♦ r❡❛❧ ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡ é ✈á❧✐❞❛✿(1 + p)n ≥ 1 + np + ✸✹✳ ▼♦str❡ q✉❡✿ | n P i=1 ai |≤ n P i=1 n(n − 1) 2 p. 2 | ai | ✻✹ 09/02/2021 ❈❛♣ít✉❧♦ ✷ ❋❯◆➬Õ❊❙ ▲❡♦♥❤❛r❞ ❊✉❧❡r ♥❛s❝❡✉ ❡♠ ❇❛s✐❧❡✐❛✱ ♥❛ ❙✉íç❛✱ ❡♠ 15 ❞❡ ❛❜r✐❧ ❞❡ 1707✱ ❡ ♠♦rr❡✉ ❡♠ 18 ❞❡ s❡t❡♠❜r♦ ❞❡ 1783✱ ❡♠ ❙ã♦ P❡t❡rs❜✉r❣♦✱ ❘úss✐❛✳ ❋♦✐ ♦ ♠❛t❡♠át✐❝♦ ♠❛✐s ♣r♦❞✉t✐✈♦ ❞♦ sé❝✉❧♦ XV II ✲ ❤á q✉❡♠ ♦ ❝♦♥s✐❞❡r❡ ♦ ♠❛t❡♠át✐❝♦ ♠❛✐s ♣r♦❞✉t✐✈♦ ❞❡ t♦❞♦s ♦s t❡♠♣♦s✳ ❊✉❧❡r ❡st✉❞♦✉ ▼❛t❡♠át✐❝❛ ❝♦♠ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐✳ ◗✉❛♥❞♦✱ ❡♠ 1725✱ ◆✐❦♦❧❛✉s✱ ✜❧❤♦ ❞❡ ❏♦❤❛♥✱ ✈✐❛❥♦✉ ♣❛r❛ ❙ã♦ P❡t❡rs❜✉r❣♦✱ ❝♦♥✈✐❞♦✉ ♦ ❥♦✈❡♠ ❊✉❧❡r ♣❛r❛ s❡❣✉✐✲❧♦ ❡ s❡ ✐♥s❝r❡✈❡r ♥❛ ❆❝❛❞❡♠✐❛✱ ❛té 1741✳ ❊♠ 1726✱ ❊✉❧❡r ❥á t✐♥❤❛ ✉♠ ♣❡q✉❡♥♦ ❛rt✐❣♦ ♣✉❜❧✐❝❛çã♦ ❡♠ 1727✱ ♣✉❜❧✐❝♦✉ ♦✉tr♦ ❛rt✐❣♦ s♦❜r❡ tr❛❥❡tór✐❛s r❡❝í♣r♦❝❛s✳ ❊st❡ ❛rt✐❣♦ ❣❛♥❤♦✉ ♦ s❡❣✉♥❞♦ ❧✉❣❛r ♥♦ ●r❛♥❞❡ Pr❡♠✐♦ ❞❛ ❆❝❛❞❡♠✐❛ ▲❡♦♥❤❛r❞ ❊✉❧❡r ❞❡ P❛r✐s✱ ♦ q✉❡ ❢♦✐ ✉♠ ❣r❛♥❞❡ ❢❡✐t♦ ♣❛r❛ ♦ ❥♦✈❡♠ ❧✐❝❡♥❝✐❛❞♦✳ ❉❡ 1741 ❛té 1766✱ ❊✉❧❡r ❡st❡✈❡ ♥❛ ❆❧❡♠❛♥❤❛✱ ♥❛ ❆❝❛❞❡♠✐❛ ❞❡ ❇❡r❧✐♠✱ s♦❜ ❛ ♣r♦t❡çã♦ ❞❡ ❋r❡❞❡r✐❝♦✲♦✲●r❛♥❞❡❀ ❞❡ 1.766 ❛ 1783 ✈♦❧t♦✉ ❛ ❙ã♦ P❡t❡rs❜✉r❣♦✱ ❛❣♦r❛ s♦❜ ❛ ♣r♦t❡çã♦ ❞❛ ✐♠♣❡r❛tr✐③ ❈❛t❛r✐♥❛✳ ❆ ✈✐❞❛ ❞❡st❡ ♠❛t❡♠át✐❝♦ ❢♦✐ q✉❛s❡ ❡①❝❧✉s✐✈❛♠❡♥t❡ ❞❡❞✐❝❛❞❛ ❛♦ tr❛❜❛❧❤♦ ♥♦s ❞✐❢❡r❡♥t❡s ❝❛♠✲ ♣♦s ❞❛ ▼❛t❡♠át✐❝❛✳ ❊♠❜♦r❛ t✐✈❡ss❡ ♣❡r❞✐❞♦ ✉♠ ♦❧❤♦✱ ❡♠ 1735 ❡ ♦ ♦✉tr♦ ❡♠ 1766✱ ♥❛❞❛ ♣♦❞✐❛ ✐♥t❡rr♦♠♣❡r ❛ s✉❛ ❡♥♦r♠❡ ♣r♦❞✉t✐✈✐❞❛❞❡✳ ❊✉❧❡r✱ ❝❡❣♦✱ ❛❥✉❞❛❞♦ ♣♦r ✉♠❛ ♠❡♠ór✐❛ ❢❡♥♦♠❡♥❛❧✱ ❝♦♥t✐♥✉♦✉ ❛ ❞✐t❛r ❛s s✉❛s ❞❡s❝♦❜❡rt❛s✳ ❉✉r❛♥t❡ ❛ s✉❛ ✈✐❞❛ ❡s❝r❡✈❡✉ 560 ❧✐✈r♦s ❡ ❛rt✐❣♦s❀ ❡♠ ✈♦❞❛ ❞❡✐①♦✉ ♠✉✐t♦s ♠❛♥✉s❝r✐t♦s✱ q✉❡ ❢♦r❛♠ ♣✉❜❧✐❝❛❞♦s ♣❡❧❛ ❆❝❛❞❡♠✐❛ ❞❡ ❙ã♦ P❡t❡rs❜✉r❣♦ ❞✉r❛♥t❡ ♦s q✉❛r❡♥t❛ ❡ s❡t❡ ❛♥♦s s❡❣✉✐♥t❡s à s✉❛ ♠♦rt❡✳ ✷✳✶ ■♥tr♦❞✉çã♦ ❆ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❞❛ ♠❛t❡♠át✐❝❛✱ ❡♥q✉❛♥t♦ ✐♥str✉♠❡♥t♦ ❞❡ ❡st✉❞♦ ❞♦s ❢❡♥ô♠❡♥♦s r❡❛✐s✱ ❞❡♣❡♥❞❡ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❞❛ s✉❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ r❡♣r❡s❡♥t❛r ❡ss❡s ❢❡♥ô♠❡♥♦s✱ ✐st♦ é✱ ❞❛ ❝♦♥❝❡♣çã♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ q✉❡ s✐♥t❡t✐③❡ ❡ r❡❧❛❝✐♦♥❡ ❛s ♣r✐♥❝✐♣❛✐s ❝❛r❛❝t❡ríst✐❝❛s ❞♦ ❢❡♥ô♠❡♥♦ ❛ ❡st✉❞❛r✳ ◆❡ss❡s ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s✱ t❛✐s r❡❧❛çõ❡s sã♦ ❤♦❥❡ r❡♣r❡s❡♥t❛❞❛s ♣♦r ❢✉♥çõ❡s✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦ ❤♦❥❡ ♥♦s ♣♦❞❡ ♣❛r❡❝❡r s✐♠♣❧❡s✱♠❛s✱ é ♦ r❡s✉❧t❛❞♦ ❞❡ ✉♠❛ ❧❡♥t❛ ❡ ❧♦♥❣❛ ❡✈♦❧✉çã♦ ❤✐stór✐❝❛ ✐♥✐❝✐❛❞❛ ♥❛ ❛♥t✐❣✉✐❞❛❞❡✱ q✉❛♥❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ✻✺ ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ♦s ♠❛t❡♠át✐❝♦s ❞❛ ❇❛❜✐❧ô♥✐❛ ✉t✐❧✐③❛r❛♠ t❛❜❡❧❛s ❞❡ q✉❛❞r❛❞♦s ❡ ❞❡ r❛í③❡s q✉❛❞r❛❞❛s ❡ ❝ú❜✐❝❛s✱ ♦✉ q✉❛♥❞♦ ♦s P✐t❛❣ór✐❝♦s t❡♥t❛r❛♠ r❡❧❛❝✐♦♥❛r ❛ ❛❧t✉r❛ ❞♦ s♦♠ ❡♠✐t✐❞♦ ♣♦r ❝♦r❞❛s s✉❜♠❡t✐❞❛s à ♠❡s♠❛ t❡♥sã♦ ❝♦♠ ♦ s❡✉ ❝♦♠♣r✐♠❡♥t♦✳ ◆❛ é♣♦❝❛ ❛♥t✐❣❛✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦ ♥ã♦ ❡st❛✈❛ ❝❧❛r❛♠❡♥t❡ ❞❡✜♥✐❞♦✳ ❆s r❡❧❛çõ❡s ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s s✉r❣✐❛♠ ❞❡ ❢♦r♠❛ ✐♠♣❧í❝✐t❛ ❡ ❡r❛♠ ❞❡s❝r✐t❛s ✈❡r❜❛❧♠❡♥t❡ ♦✉ ♣♦r ✉♠ ❣rá✜❝♦✳ ❙ó ♥♦ sé❝✉❧♦ XV II ✱ q✉❛♥❞♦ ❉❡s❝❛rt❡s✶ ❡ P✐❡rr❡ ❋❡r♠❛t ✐♥tr♦❞✉③❡♠ ❛s ❝♦♦r❞❡♥❛✲ ❞❛s ❝❛rt❡s✐❛♥❛s✱ é q✉❡ s❡ t♦r♥❛ ♣♦ssí✈❡❧ tr❛♥s❢♦r♠❛r ♣r♦❜❧❡♠❛s ❣❡♦♠étr✐❝♦s ❡♠ ♣r♦❜❧❡♠❛s ❛❧❣é❜r✐❝♦s ❡ ❡st✉❞❛r ❛♥❛❧✐t✐❝❛♠❡♥t❡ ❛s ❢✉♥çõ❡s✳ ❆ ♠❛t❡♠át✐❝❛ r❡❝❡❜❡ ❛ss✐♠ ✉♠ ❣r❛♥❞❡ ✐♠♣✉❧s♦✱ ♥♦t❛❞❛♠❡♥t❡ ♣❡❧❛ s✉❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❛ ♦✉tr❛s ❝✐ê♥❝✐❛s✳ ❆ ♣❛rt✐r ❞❡ ♦❜s❡r✈❛çõ❡s ♦✉ ❡①♣❡r✐ê♥❝✐❛s r❡❛❧✐③❛❞❛s✱ ♦s ❝✐❡♥t✐st❛s ♣❛ss❛r❛♠ ❛ ❞❡t❡r♠✐♥❛r ❛ ❢ór♠✉❧❛ ♦✉ ❢✉♥çã♦ q✉❡ r❡❧❛❝✐♦♥❛ ❛s ✈❛r✐á✈❡✐s ❡♠ ❡st✉❞♦✳ ❆ ♣❛rt✐r ❞❛q✉✐ t♦❞♦ ♦ ❡st✉❞♦ s❡ ❞❡s❡♥✈♦❧✈❡ ❡♠ t♦r♥♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ t❛✐s ❢✉♥çõ❡s✳ ➱ ♣♦r ✐ss♦ q✉❡ ✉♠ ❞♦s ❝♦♥❝❡✐t♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❛ ♠❛t❡♠át✐❝❛ é ♦ ❞❡ ❢✉♥çã♦✳ ❊♠ q✉❛s❡ t♦❞❛s ❛s ♣❛rt❡s ❞❛ ❝✐ê♥❝✐❛ ♦ ❡st✉❞♦ ❞❡ ❢✉♥çõ❡s é ❛ ♣❛rt❡ ❝❡♥tr❛❧ ❞❛ t❡♦r✐❛✳ ✷✳✷ ❘❡❧❛çõ❡s ❉❛❞♦s ♦s ❝♦♥❥✉♥t♦s A = { 1, 2, 3, 4 } ❡ B = { a, b, c, d }✳ P♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ✉♠❛ r❡❧❛çã♦ ✭❝♦rr❡s♣♦♥❞ê♥✲ A ✲ B ❝✐❛✮ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s A ❡ B ❞❡ ♠♦❞♦ q✉❡✱ ❛ ❝❛❞❛ ♥ú✲ ✬ ✬ ✩ ♠❡r♦ ❡♠ ♦r❞❡♠ ❝r❡s❝❡♥t❡ ❞♦ ❝♦♥❥✉♥t♦ A ❝♦rr❡s♣♦♥❞❛ 1 ✲ a ✉♠❛ ❧❡tr❛ ♥❛ ♦r❞❡♠ ❛❧❢❛❜ét✐❝❛ ❞♦ ❝♦♥❥✉♥t♦ B ✳ ✲ b 2 ❖✉tr♦ ♠♦❞♦ ❞❡ ❛♣r❡s❡♥t❛r ♦ ❡sq✉❡♠❛ ❞❛ ❋✐❣✉r❛ ✲ c 3 ✭✷✳✶✮ s❡r✐❛ ✉t✐❧✐③❛♥❞♦ ❛ ❢♦r♠❛ ❞❡ ♣❛r ♦r❞❡♥❛❞♦✱ ✐st♦ é✿ ✲ d 4 (1, a), (2, b), (3, c) ❡ (4, d)✳ ✫ ✪ ✫ ❖❜s❡r✈❛♠♦s q✉❡ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡st❛❜❡❧❡❝✐❞❛ ❞❡t❡r♠✐♥❛ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ♣r♦❞✉t♦ ❝❛r✲ ❋✐❣✉r❛ ✷✳✶✿ t❡s✐❛♥♦ A × B ✳ ❊st❡ ❝♦♥❥✉♥t♦ é ❞❡♥♦t❛❞♦ ❝♦♠♦✿ {(1, a), (2, b), (3, c), (4, d)}✳ ✩ ✪ ❉❡✜♥✐çã♦ ✷✳✶✳ S é ✉♠❛ r❡❧❛çã♦ ❞❡ A ❡♠ B ✱ A × B ❀ ✐st♦ é✱ S ⊆ A × B ✳ ❉✐③❡♠♦s q✉❡ ❝❛rt❡s✐❛♥♦ s❡ S é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞♦ ♣r♦❞✉t♦ ❆ss✐♠✱ ✉♠❛ r❡❧❛çã♦ é ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡①✐st❡♥t❡ ❡♥tr❡ ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦s✳ ✶ ❘❡♥❡ ❉❡s❝❛rt❡s ✭1596 − 1650✮✱ ❝r✐❛❞♦r ❞❛ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛✱ ❢♦✐ ✉♠ ❣❡♥t✐❧ ❤♦♠❡♠✱ ♠✐❧✐t❛r✱ ♠❛t❡✲ ♠át✐❝♦ ❡ ✉♠ ❞♦s ♠❛✐♦r❡s ✜❧ós♦❢♦s ❞❡ t♦❞♦s ♦s t❡♠♣♦s ✻✻ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❆ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ♦s ❞♦✐s ❝♦♥❥✉♥t♦s é ❞❛❞❛ ❡♠ t❡r♠♦s ❞❡ ♣❛r❡s ♦r❞❡♥❛❞♦s✱ ♦♥❞❡ ♦ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦ ❞♦ ♣❛r ♣r♦❝❡❞❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛ A ❡ ♦ s❡❣✉♥❞♦ ❡❧❡♠❡♥t♦ ❞♦ ♣❛r ♣r♦❝❡❞❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❤❡❣❛❞❛ B ✳ ❖❜s❡r✈❛çã♦ ✷✳✶✳ ✶✮ ❙❡ ✷✮ ❙❡ x ∈ A ❡ y ∈ B ❡ ❝✉♠♣r❡ (x, y) ∈ S ✱ ❡♥tã♦ ❞✐③❡♠♦s y ♠❡❞✐❛♥t❡ S ❡ ❞❡♥♦t❛♠♦s ❝♦♠ ♦ sí♠❜♦❧♦ xSy ✳ S é ✉♠❛ r❡❧❛çã♦ ❞❡ ♦ ❝♦♥❥✉♥t♦ ✸✮ B A ❡♠ B✱ ♦ ❝♦♥❥✉♥t♦ ❈♦♠♦ ♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ ❚❡♠♦s q✉❡ S x ❡stá ❡♠ r❡❧❛çã♦ ❝♦♠ é ❝❤❛♠❛❞♦ ❞❡ ✏ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛✑ ❡ é ❝❤❛♠❛❞♦ ❞❡ ✏❝♦♥❥✉♥t♦ ❞❡ ❝❤❡❣❛❞❛✑✳ ∅ ⊆ A × B✱ ❡♥tã♦ ❞❡ ✏ r❡❧❛çã♦ ♥✉❧❛ ♦✉ ✈❛③✐❛✑✳ ✹✮ A q✉❡ é ✉♠❛ r❡❧❛çã♦ ❞❡ A ❡♠ B✱ ∅ é ✉♠❛ r❡❧❛çã♦ ❞❡ s❡ ❡ s♦♠❡♥t❡ s❡ A ❡♠ B ❡ é ❝❤❛♠❛❞❛ S ⊆ A × B✳ ❖s ❝♦♥❥✉♥t♦s ❞❡ ♣❛rt✐❞❛ ❡ ❞❡ ❝❤❡❣❛❞❛ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ tê♠ ✉♠❛ ❡str✉t✉r❛✳ ◆♦ ❡♥t❛♥t♦✱ s❡❣✉♥❞♦ ♦ t✐♣♦ ❞❡ ❡str✉t✉r❛ ✐♠♣♦st❛ ❛ ❡ss❡s ❝♦♥❥✉♥t♦s✱ ❡ ♦ t✐♣♦ ❞❡ r❡str✐çã♦ q✉❡ s❡ ✐♠♣õ❡ à ♣ró♣r✐❛ r❡❧❛çã♦✱ ♦❜t❡♠♦s ❛❧❣✉♥s t✐♣♦s ❡s♣❡❝✐❛✐s ❞❡ r❡❧❛çõ❡s✱ ❝❛❞❛ ✉♠❛ ❞❡❧❛s ❝♦♠ ✉♠ ♥♦♠❡ ❡s♣❡❝í✜❝♦ ❊①❡♠♣❧♦ ✷✳✶✳ ❙❡❥❛♠ ♦s ❝♦♥❥✉♥t♦s ❝♦♥❥✉♥t♦s A ❡ B A = { ❛❧✉♥♦s ❞♦ 1o ❛♥♦ ❞❡ ❈á❧❝✉❧♦ ♣♦❞❡♠♦s ❢♦r♠❛r ❛❧❣✉♠❛s r❡❧❛çõ❡s ❝♦♠♦✿ I} B = N✱ ❡ S1 = { (x, y) ∈ A × B /. x tê♠ y ❛♥♦s } S2 = { (x, y) ∈ A × B /. x tê♠ y r❡❛✐s } ❡♥tã♦ ❝♦♠ ♦s ❊①❡♠♣❧♦ ✷✳✷✳ ❙❡❥❛♠ ♦s ❝♦♥❥✉♥t♦s✿ A = {3, 4, 5, 6}✱ B = {1, 2, 3, 4} S = {(x, y) ∈ A × B /. ❆ss✐♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿ ✷✳✷✳✶ ❡ ❛ r❡❧❛çã♦✿ x = y + 2} S = { (3, 1), (4, 2), (5, 3), (6, 4) }✳ ❉♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ❞❡ ✉♠❛ r❡❧❛çã♦ ❙❡❥❛ S ✉♠❛ r❡❧❛çã♦ ♥ã♦ ✈❛③✐❛ ❞❡ A ❡♠ B ✱ ✐st♦ é✿ S = { (x, y) ∈ A × B /. x ❡st❛ ❡♠ r❡❧❛çã♦ ❝♦♠ y } ✻✼ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❡✜♥✐çã♦ ✷✳✷✳ ❉♦♠í♥✐♦ ❞❡ ✉♠❛ r❡❧❛çã♦✳ ❖ ❞♦♠í♥✐♦ ❞❛ r❡❧❛çã♦ S ❞❡ A ❡♠ B é ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s x ∈ A ♣❛r❛ ♦s q✉❛✐s ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ y ∈ B t❛❧ q✉❡ (x, y) ∈ S ✳ ■st♦ é ♦ ❞♦♠í♥✐♦ ❞❡ S é ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ A ❢♦r♠❛❞♦ ♣❡❧❛s ♣r✐♠❡✐r❛s ❝♦♠♣♦♥❡♥t❡s ❞♦s ♣❛r❡s ♦r❞❡♥❛❞♦s q✉❡ ♣❡rt❡♥❝❡♠ ❛ r❡❧❛çã♦✳ ❆ ♥♦t❛çã♦ ♣❛r❛ ✐♥❞✐❝❛r ♦ ❞♦♠í♥✐♦ ❞❛ r❡❧❛çã♦ S é D(S) ❛ss✐♠ ❞❡✜♥✐❞♦✿ D(S) = { x ∈ A /. y ∈ B; (x, y) ∈ S } ❉❡✜♥✐çã♦ ✷✳✸✳ ■♠❛❣❡♠ ❞❡ ✉♠❛ r❡❧❛çã♦✳ ❆ ✐♠❛❣❡♠ ♦✉ ❝♦♥tr❛❞♦♠í♥✐♦ ❞❛ r❡❧❛çã♦ S ❞❡ A ❡♠ B é ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s y ∈ B ♣❛r❛ ♦s q✉❛✐s ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ x ∈ A t❛❧ q✉❡ (x, y) ∈ S ✳ ■st♦ é✱ ❛ ✐♠❛❣❡♠ ❞❡ S é ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡ B ❢♦r♠❛❞♦ ♣❡❧❛s s❡❣✉♥❞❛s ❝♦♠♣♦♥❡♥t❡s ❞♦s ♣❛r❡s ♦r❞❡♥❛❞♦s q✉❡ ♣❡rt❡♥❝❡♠ ❛ r❡❧❛çã♦✳ ❆ ♥♦t❛çã♦ ♣❛r❛ ✐♥❞✐❝❛r ❛ ✐♠❛❣❡♠ ❞❛ r❡❧❛çã♦ S é Im(S) = { y ∈ B /. x ∈ A; (x, y) ∈ S } ❊①❡♠♣❧♦ ✷✳✸✳ ❖ ❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ❞❛ r❡❧❛çã♦ ❞♦ ❊①❡♠♣❧♦ ✭✷✳✷✮ sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡✿ D(S) = {3, 4, 5, 6} ✷✳✷✳✷ ❘❡❧❛çõ❡s ❞❡ Im(S) = {1, 2, 3, 4} R ❡♠ R ◆♦ q✉❡ s❡❣✉❡✱ ✉t✐❧✐③❛r❡♠♦s r❡❧❛çõ❡s ❞❡ A ❡♠ B ♦♥❞❡ A ❡ B sã♦ s✉❜❝♦♥❥✉♥t♦s ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s R✳ ❊①❡♠♣❧♦ ✷✳✹✳ ❙❡❥❛ S ✉♠❛ r❡❧❛çã♦ ❞❡✜♥✐❞❛ ♣♦r✿ S = {(x, y) ∈ N+ × N+ /. x2 + y 2 ≤ 9} ▲♦❣♦✱ ♥♦ss❛ r❡❧❛çã♦ é✿ S = {(1, 1), (1, 2), (2, 1), (2, 2)}✳ ❯♠ ❞✐❛❣r❛♠❛ ❞❛ r❡❧❛çã♦ S ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✷✳✷✮✳ ❖❜s❡r✈❡✱ s♦♠❡♥t❡ sã♦ q✉❛tr♦ ♣♦♥t♦s ❞♦ ♣❧❛♥♦✳ ❊①❡♠♣❧♦ ✷✳✺✳ ❙❡❥❛ T ❛ r❡❧❛çã♦ ❡♠ R ❞❡✜♥✐❞❛ ❝♦♠♦ s❡❣✉❡✿ T = {(x, y) ∈ R × R /. x2 + y 2 ≤ 9} ❯♠ ❞✐❛❣r❛♠❛ ❞❛ r❡❧❛çã♦ ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✷✳✸✮✳ ❖❜s❡r✈❡✱ é ✐♠♣♦ssí✈❡❧ ❞❡s❡♥❤❛r ✉♠ ❛ ✉♠ ♦s ✐♥✜♥✐t♦s ❡❧❡♠❡♥t♦s ❞❛ r❡❧❛çã♦ T ❀ ✐st♦ ❛❝♦♥t❡❝❡ ♣❡❧♦ ❢❛t♦ ❛ r❡❧❛çã♦ T ❡st❛r ❞❡✜♥✐❞❛ ❝♦♠ s✉❜❝♦♥❥✉♥t♦s ❞❡ ✐♥✜♥✐t♦s ♥ú♠❡r♦s r❡❛✐s R✳ ✻✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R y ✻ 3 r 2 r 1 ✛ ❄0 −1 r r 1 2 x✲ 3 4 ··· ❋✐❣✉r❛ ✷✳✷✿ ❋✐❣✉r❛ ✷✳✸✿ ❊①✐st❡♠ ♦✉tr♦s t✐♣♦s ❞❡ r❡❧❛çõ❡s✱ ❝♦♠♦ ♠♦str❛ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦ ❬✻❪✳ ❉❡✜♥✐çã♦ ✷✳✹✳ ❙❡❥❛♠ k ♥ú♠❡r♦ r❡❛❧ ❝♦♥st❛♥t❡ ♥ã♦ ♥✉❧♦✱ ❡ n ∈ N✳ ✐✮ ❉✐③❡♠♦s q✉❡ y é ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛ x✱ s❡ y = kx❀ ❡ ❞✐③❡♠♦s q✉❡ y é 1 x ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛ x✱ s❡ y = k( )✳ ✐✐✮ ❉✐③❡♠♦s q✉❡ y é ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ á n✲és✐♠❛ ♣♦tê♥❝✐❛ ❞❡ x✱ s❡ y = k.xn ❀ ❡ ❞✐③❡♠♦s q✉❡ y é ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ á n✲és✐♠❛ ♣♦tê♥❝✐❛ ❞❡ x✱ s❡ 1 y = k( n )✳ x ✐✐✐✮ ❉✐③❡♠♦s q✉❡ z é ❝♦♥❥✉♥t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛ x ❡ y s❡ z = kxy ❊①❡♠♣❧♦ ✷✳✻✳ ❉❡ ✉♠ ❣r✉♣♦ ❞❡ 100 ❛❧✉♥♦s✱ ❛ r❛③ã♦ s❡❣✉♥❞♦ ❛ q✉❛❧ ✉♠ ❜♦❛t♦ s❡ ❡s♣❛❧❤❛ é ❝♦♥❥✉♥✲ t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❛❧✉♥♦s q✉❡ ♦✉✈✐r❛♠ ♦ ❜♦❛t♦ ❡ ❛♦ ♥ú♠❡r♦ ❞❡ ❛❧✉♥♦s q✉❡ ♥ã♦ ♦✉✈✐r❛♠ ♦ ❜♦❛t♦✳ ♠✐♥✉t♦✱ q✉❛♥❞♦ 30 ❛✮ ❙❡ ♦ ❜♦❛t♦ ❡stá s❡ ❡s♣❛❧❤❛♥❞♦ ❛ ✉♠❛ r❛③ã♦ ❞❡ 5 ❛❧✉♥♦s ♣♦r ♦ ♦✉✈✐r❛♠✳ ❊①♣r❡ss❡ ❛ t❛①❛ s❡❣✉♥❞♦ ♦ q✉❛❧ ♦ ❜♦❛t♦ s❡ ❡stá ❡s♣❛❧❤❛♥❞♦ ❝♦♠♦ ❢✉♥çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❛❧✉♥♦s q✉❡ ♦ ♦✉✈✐r❛♠✳ q✉❛♥❞♦ 90 ❙♦❧✉çã♦✳ ❜✮ ◗✉ã♦ rá♣✐❞♦ ♦ ❜♦❛t♦ s❡ ❡s♣❛❧❤♦✉ ❛❧✉♥♦s ♦ ♦✉✈✐r❛♠❄ ❛✮ ❙✉♣♦♥❤❛♠♦s f (x) s❡❥❛ ❛ t❛①❛ ♣❡❧♦ q✉❛❧ ♦ ❜♦❛t♦ s❡ ❡st❛ ❡s♣❛❧❤❛♥❞♦✱ q✉❛♥❞♦ x ❛❧✉♥♦s ♦ ♦✉✈✐r❛♠ ✭❧♦❣♦ ♥ã♦ ♦✉✈✐r❛♠ 100 − x)❀ ❡♥tã♦ f (x) = kx(100 − x)✳ ◗✉❛♥❞♦ x = 30✱ t❡♠♦s f (30) = 5 ⇒ 5 = k(30)(100 − 30) ⇒ 5 = 2100k ⇒ 1 5 = ✳ 2100 420 x(100 − x) ▲♦❣♦ f (x) = ✳ 420 k= ❙♦❧✉çã♦✳  ❜✮ ✻✾ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ◗✉❛♥❞♦ x = 90✱ t❡♠♦s 1 900 [90(100 − 90)] = = 2, 142✱ 420 420 ♦✉✈✐r❛♠ é 2, 142 ♦✉✈✐♥t❡s ♣♦r ♠✐♥✉t♦✳ f (90) = ❝r❡s❝✐♠❡♥t♦ q✉❛♥❞♦ ✾✵ ❛❧✉♥♦s ♦ ❛ t❛①❛ ❞❡ ❊①❡♠♣❧♦ ✷✳✼✳ ❖ ♣❡s♦ ❛♣r♦①✐♠❛❞♦ ❞❛ ❜❛♥❤❛ ❡♠ ✉♠ ♣♦r❝♦ é ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛ s❡✉ ♣❡s♦ ❝♦r♣♦r❛❧✳ ❛✮ ❊①♣r❡ss❡ ♦ ♥ú♠❡r♦ ❞❡ q✉✐❧♦s ❞♦ ♣❡s♦ ❛♣r♦①✐♠❛❞♦ ❞❛ ❜❛♥❤❛ ❞❡ ✉♠ ♣♦r❝♦ ❝♦♠♦ ❢✉♥çã♦ ❞❡ s❡✉ ♣❡s♦ ❝♦r♣♦r❛❧ s❛❜❡♥❞♦ q✉❡ ✉♠ ♣♦r❝♦ ❝♦♠ 98 kg t❡♠ ✉♠ ♣❡s♦ ❛♣r♦①✐♠❛❞♦ ❞❡ 32 kg ❞❡ ❜❛♥❤❛✳ ❜✮ ❆❝❤❡ ♦ ♣❡s♦ ❞❛ ❜❛♥❤❛ ❞❡ ✉♠ ♣♦r❝♦ ❝✉❥♦ ♣❡s♦ ❝♦r♣♦r❛❧ s❡❥❛ 72 kg ✳ ❙♦❧✉çã♦✳ ❙❡❥❛ ✭❛✮ y = f (x) ♦ ♣❡s♦ ❛♣r♦①✐♠❛❞♦ ❞❡ ❜❛♥❤❛ ❞❡ ✉♠ ♣♦r❝♦ ❝✉❥♦ ♣❡s♦ ❝♦r♣♦r❛❧ é x kg ✱ s❡♥❞♦ ♦ ♣❡s♦ ❞❛ ❜❛♥❤❛ ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛ s❡✉ ♣❡s♦ ❝♦r♣♦r❛❧✱ t❡♠♦s q✉❡ ❡①✐st❡ k t❛❧ q✉❡ f (x) = kx❀ q✉❛♥❞♦ x = 98 t❡♠♦s f (98) = 32✱ ❧♦❣♦ 32 = k.(98) 32 ✳ ♦♥❞❡ k = 98 32 P♦rt❛♥t♦ f (x) = x✳  98 ✉♠❛ ❝♦♥st❛♥t❡ ❙♦❧✉çã♦✳ ✭❜✮ P♦r ♦✉tr♦ ❧❛❞♦✱ q✉❛♥❞♦ x = 72 t❡♠♦s f (72) = 72( ▲♦❣♦ ♦ ♣❡s♦ ❞❛ ❜❛♥❤❛ é ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 32 1152 ) = = 23, 51✳ 98 49 23, 51 kg ✳ ❊①❡♠♣❧♦ ✷✳✽✳ ❯♠❛ t♦r♥❡✐r❛ ❡♥❝❤❡ ✉♠ t❛♥q✉❡ ❡♠ 12 ♠✐♥✉t♦s✱ ❡♥q✉❛♥t♦ ✉♠❛ s❡❣✉♥❞❛ t♦r♥❡✐r❛ ❣❛st❛ 18 ♠✐♥✉t♦s ♣❛r❛ ❡♥❝❤❡r ♦ ♠❡s♠♦ t❛♥q✉❡✳ ❈♦♠ ♦ t❛♥q✉❡ ✐♥✐❝✐❛❧♠❡♥t❡ ✈❛③✐♦✱ ❛❜r❡✲s❡ ❛ ♣r✐♠❡✐r❛ t♦r♥❡✐r❛ ❞✉r❛♥t❡ x ♠✐♥✉t♦s❀ ❛♦ ✜♠ ❞❡ss❡ t❡♠♣♦✱ ❢❡❝❤❛✲s❡ ❡ss❛ t♦r♥❡✐r❛ ❡ ❛❜r❡✲s❡ ❛ s❡❣✉♥❞❛✱ ❛ q✉❛❧ t❡r♠✐♥❛ ❞❡ ❡♥❝❤❡r ♦ t❛♥q✉❡ ❡♠ (x + 3) ♠✐♥✉t♦s✳ ❈❛❧❝✉❧❛r ♦ t❡♠♣♦ ❣❛st♦ ♣❛r❛ ❡♥❝❤❡r ♦ t❛♥q✉❡✳ ❙♦❧✉çã♦✳ ❙❡❥❛ V ♦ ✈♦❧✉♠❡ ❞♦ t❛♥q✉❡✱ ❞♦ ❡♥✉♥❝✐❛❞♦✱ ❝♦♥❝❧✉✐✲s❡ q✉❡✱ ❡♠ ❞❡ ❝❛❞❛ t♦r♥❡✐r❛ s❡rá V 12 ❡ V 18 ❞♦ ✈♦❧✉♠❡ t♦t❛❧ ❞♦ t❛♥q✉❡✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P♦❞❡♠♦s ❡♥tã♦ ❡s❝r❡✈❡r ❛ r❡❧❛çã♦✿ V V ·x+ · (x + 3) = V ❀ 12 18 x x+3 + =1 12 18 ❛ss✐♠ 1 ♠✐♥✉t♦ ❛ ❝♦♥tr✐❜✉✐çã♦ ⇒ ❡♥tã♦ 3x + 2x + 6 = 36 x = 6✳ ✼✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ x=6 ▲♦❣♦✱ ♦ t❡♠♣♦ ♣❛r❛ ❛ ♣r✐♠❡✐r❛ t♦r♥❡✐r❛ é 9 R ❡ ♦ t❡♠♣♦ ♣❛r❛ ❛ s❡❣✉♥❞❛ t♦r♥❡✐r❛ é ♠✐♥✉t♦s✳ 15 ❈♦♥❝❧✉✐✲s❡✱ q✉❡ ♦ t❡♠♣♦ t♦t❛❧ ❣❛st♦✱ s❡rá ✐❣✉❛❧ ❛ ♠✐♥✉t♦s✳ ❊①❡♠♣❧♦ ✷✳✾✳ 1 65 ❯♠ ❛❝✐❞❡♥t❡ ❢♦✐ ♣r❡s❡♥❝✐❛❞♦ ♣♦r q✉❡ s♦✉❜❡ ❞♦ ❛❝♦♥t❡❝✐♠❡♥t♦ ❛♣ós x ❞❛ ♣♦♣✉❧❛çã♦ ❞❡ P❛tó♣♦❧✐s✳ ❖ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s ❤♦r❛s✱ é ❞❛❞♦ ♣♦r✿ f (x) = t♦t❛❧ ❞❛ ♣♦♣✉❧❛çã♦✳ ❙❛❜❡♥❞♦ q✉❡ 1 9 ❞❛ ♣♦♣✉❧❛çã♦ s♦✉❜❡ ❞♦ ❛❝✐❞❡♥t❡ ❛♣ós tr❛♥s❝♦rr✐❞♦ ❛té q✉❡ ❙♦❧✉çã♦✳ 1 5 B✳ ❋❛③❡♥❞♦ x=0 ❡ f (0) = 1 · B✱ 65 ✈❡♠✿ ❚❛♠❜é♠ ♣❡❧♦ ❡♥✉♥❝✐❛❞♦ ❞♦ ♣r♦❜❧❡♠❛✱ f (3) = ❉❛í ✈❡♠✿ ❤♦r❛s✳ ♦♥❞❡ B é ♦ ❉❡t❡r♠✐♥❡ ♦ t❡♠♣♦ ❞❛ ♣♦♣✉❧❛çã♦ s♦✉❜❡ss❡ ❞❛ ♥♦tí❝✐❛✳ P❡❧♦ ❡♥✉♥❝✐❛❞♦ ❞♦ ♣r♦❜❧❡♠❛✱ ♥♦ t❡♠♣♦ ♣♦♣✉❧❛çã♦ 3 B ✱ 1 + Ca−kx 9 = 1 + 64a−3k ✱ 1 ·B 9 ❧♦❣♦ ⇒ x = 0✱ 1 B ❞❡ ·B = 65 1 + Ca−0 q✉❛♥❞♦ x = 3 t❡♠♦s✱ ⇒ ak = 2 s✱ ⇒ C = 64✳ k = loga 2 é ✈á❧✐❞❛ ❛ ✐❣✉❛❧❞❛❞❡ ❢✉♥çã♦ ❞❛❞❛ ♥♦ ❡♥✉♥❝✐❛❞♦ ♣♦❞❡rá s❡r ❡s❝r✐t❛ ❝♦♠♦✿ 1 5 ❞❛ ❞❡ ♦♥❞❡ ❙❛❜❡✲s❡ q✉❡✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ ◗✉❛❧ ♦ t❡♠♣♦ tr❛♥s❝♦rr✐❞♦ ❛té q✉❡ ♦♥❞❡ 1 65 1 B ·B = 9 1 + 64a−3k 1 = a−3k ✱ 8 2−3 = (ak )−3 ♦ ❛❝✐❞❡♥t❡ ❢♦✐ ♣r❡s❡♥❝✐❛❞♦ ♣♦r B f (x) = 1 + 2−x · 64 s = aloga s ✱ ❧♦❣♦ ❛ ❞❛ ♣♦♣✉❧❛çã♦ s♦✉❜❡ss❡ ❞❛ ♥♦tí❝✐❛ ❞♦ ❛❝✐❞❡♥t❡❄ 1 f (x) = · B ❡ ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r r❡s♣❡❝t✐✈♦ ❞❡ x✳ 5 B B = ⇒ 4 = 2−x · 64 ⇒ x = 4 ❚❡r❡♠♦s ❡♥tã♦✿ −x 5 1 + 2 · 64 1 P♦rt❛♥t♦✱ ♦ t❡♠♣♦ tr❛♥s❝♦rr✐❞♦ ❛té q✉❡ ❞❛ ♣♦♣✉❧❛çã♦ s♦✉❜❡ss❡ ❞❛ ♥♦tí❝✐❛ ❞♦ ❛❝✐❞❡♥t❡ 5 x = 4 ❤♦r❛s✳ ❖r❛✱ ❜❛st❛ ❢❛③❡r ❢♦✐ ❊①❡♠♣❧♦ ✷✳✶✵✳ ❉❛❞❛s ❛s r❡❧❛çõ❡s✿ f (x) = x + 1; g(x) = x − 2❀ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦✿ | f (x) + g(x) |=| f (x) | + | g(x) | ❙♦❧✉çã♦✳ ✼✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R | (x + 1) + (x − 2) |=| (x + 1) | + | (x − 2) | ⇔ | 2x − 1 |=| x + 1 | + | x − 2 |✳ ❙❡ x < −1✱ ❡♥tã♦ −(2x − 1) = −(x + 1) − (x − 2) ⇒ −2x + 1 = −2x + 1✱ ❧♦❣♦ x ∈ (−∞, −1)✳ 1 ✱ ❡♥tã♦ −(2x − 1) = (x + 1) − (x − 2) ⇒ −2x + 1 = 3✱ ❧♦❣♦ x = −1✳ ❙❡ −1 ≤ x < 2 1 ❙❡ ≤ x < 2✱ ❡♥tã♦ (2x − 1) = x + 1 − (x − 2) ⇒ 2x − 1 = 3✱ ❧♦❣♦ x = 2 2 ❚❡♠♦s✿ ✭❛❜s✉r❞♦✦✮✳ ❙❡ 2 ≤ x✱ P♦rt❛♥t♦✱ 2x − 1 = x + 1 + x − 2 x ∈ (−∞, −1) ∪ [2, +∞)✳ ⇒ ❡♥tã♦ −1 = −1✱ ❧♦❣♦ x ≥ 2.✳ ❊①❡♠♣❧♦ ✷✳✶✶✳ ❉❡t❡r♠✐♥❡ ♦s ✈❛❧♦r❡s ❞❡ a ❡ b ♥❛ r❡❧❛çã♦ f (x) = ax2 + bx + 5✱ ♣❛r❛ ♦s q✉❛✐s s❡❥❛ ✈á❧✐❞❛ ❛ ✐❞❡♥t✐❞❛❞❡ f (x + 1) − f (x) = 8x + 3✳ ❙♦❧✉çã♦✳ ❚❡♠♦s f (x + 1) − f (x) = a(x + 1)2 + b(x + 1) + 5 − (ax2 + bx + 5) = 8x + 3 ⇒ P♦rt❛♥t♦✱ x(2a) + a + b = 8x + 3 a=4 ❡ ⇒ 2a = 8 ⇒ a+b=3 ♦✉ b = −1✳ ❊①❡♠♣❧♦ ✷✳✶✷✳ ❉❡t❡r♠✐♥❛r ♦ ❞♦♠í♥✐♦ ❞❡ ❞❡✜♥✐çã♦ ❞❛ s❡❣✉✐♥t❡s r❡❧❛çã♦✿ R(x) = √ 4 x2 − 4x + 12 + √ 4 3x2 −x − 20 + x2 ❙♦❧✉çã♦✳ ❖ ♥ú♠❡r♦s r❡❛✐s ❞♦ ❞♦♠í♥✐♦ ❞❛ r❡❧❛çã♦ x2 − 4x + 12 ≥ 0 ⇔ x∈R ⇔ ❡ ❝✉♠♣r❡✿ − x − 20 + x2 > 0 ❡ (x − 2)2 + 8 ≥ 0 ⇔ R ❡ ⇔ (x − 5)(x + 4) > 0 (−∞, −4) ∪ (5, +∞) ⇔ ⇔ D(R) = (−∞, −4)(5, +∞) P♦rt❛♥t♦✱ ♦ ❞♦♠í♥✐♦ ❞❛ r❡❧❛çã♦ R(x) é D(f ) = (−∞, −4) ∪ (5, +∞). ✼✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✷✲✶ ✶✳ ❙❡❥❛♠ ♦s ❝♦♥❥✉♥t♦s A = {0, 1, 2} ❡ B = {3, 2, 1}✱ ❡s❝r❡✈❡r ❡♠ ❢♦r♠❛ ❞❡ ❝♦♥❥✉♥t♦s ❛ r❡❧❛çã♦ ❞❡ A ❡♠ B ❞❡✜♥✐❞❛ ♣♦r x = y ♣❛r❛ x ∈ A ❡ y ∈ B ✳ ✷✳ ❙❡❥❛♠ ❛s r❡❧❛çõ❡s✿ f (x) = x ❡ g(x) = x − 2✳ P❛r❛ q✉❛✐s ✈❛❧♦r❡s ❞❡ x✱ é ✈á❧✐❞❛ ❛ r❡❧❛çã♦✿ | f (x) − g(x) |>| f (x) | − | g(x) |❄ ✸✳ ❙✉♣♦♥❤❛ ♦s ❝♦♥❥✉♥t♦s A = {3, 5, 8, 9} ❡ B = {1, 3, 5, 7} ✱ ❡s❝r❡✈❡r ❡♠ ❢♦r♠❛ ❞❡ ❝♦♥❥✉♥t♦s ❛ r❡❧❛çã♦ ❞❡ A ❡♠ B ❞❡✜♥✐❞❛ ♣♦r✿ ✶✳ x < y; x ∈ A ❡ y ∈ B ✷✳ x ≥ y; x ∈ A ❡ y ∈ B ✸✳ x = y; x ∈ A ❡ y ∈ B ✹✳ y + x = 4; x ∈ A ❡ y ∈ B ✺✳ x é ❞✐✈✐sí✈❡❧ ♣♦r y; x ∈A ❡ y ∈B ✹✳ P❛r❛ ♦ ❡①❡r❝í❝✐♦ ❛♥t❡r✐♦r✱ ❞❡t❡r♠✐♥❡ ♦ ❞♦♠í♥✐♦✱ ✐♠❛❣❡♠ ❞❡ ❝❛❞❛ r❡❧❛çã♦✳ ✺✳ ❈♦♥str✉✐r ✉♠ ❞❡s❡♥❤♦✱ ❛❝❤❛r ♦ ❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s ❞❡✜♥✐❞❛s ❡♠ R✳ ✶✳ S = {(x, y) ∈ R2 /.x − 5y = 0} ✷✳ ✸✳ S = {(x, y) ∈ R2 /. y < 2x }. ✹✳ ✺✳ S = {(x, y) ∈ R2 /. (x − 2)(y + 3) = 0} ✻✳ ✼✳ ✽✳ ✾✳ ✶✵✳ S = {(x, y) ∈ R2 /. x = 3 ❡ S = {(x, y) ∈ R2 /. x = 3y }. 1 S = {(x, y) ∈ R2 /. y = }. x − 2 < y < 2}. S = {(x, y) ∈ R2 /. y = 2x ❡ x ∈ [−2, 1]}. 9 − x2 2 }. S = {(x, y) ∈ R /. y = 2 x −4 S = {(x, y) ∈ R2 /. x = 3 ❡ y > 0}. 3x2 − 8x + 4 S = {(x, y) ∈ R2 /. y = }✳ x2 ✻✳ P❛r❛ ❛s r❡❧❛çõ❡s ❞♦ ❡①❡r❝í❝✐♦ ❛♥t❡r✐♦r✱ ❛❝❤❛r ❛s r❡❧❛çõ❡s ✐♥✈❡rs❛s✱ ✐♥❞✐❝❛r s❡✉ ❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ❡ ❞❡s❡♥❤❛r✲❧❛✳ ✼✳ ❉❡s❡♥❤❛r✱ ❧♦❣♦ ❞❡t❡r♠✐♥❡ ♦ ❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ❞❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s✿ ✶✳ S = { (x, y) ∈ R2 /. 1 ≤ x + y ≤ 2 } ✷✳ S = { (x, y) ∈ R2 /. | x | + | y |= 5 } ✸✳ S = { (x, y) ∈ R2 /. | x | + | y |≤ 8 } ✹✳ S = { (x, y) ∈ R2 /. y ≤ 2x ❡ x2 + y 2 ≤ 1 } ✼✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✽✳ ❙❡♥❞♦ y =| x − 5 | + | 3x − 21 | + | 12 − 3x | ✱ s❡ 4 < x < 5✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ❛ r❡❧❛çã♦ é ❡q✉✐✈❛❧❡♥t❡ ❛✿ ✾✳ ❙❡❥❛ A = { 4, 5, 6 } ❞❡✜♥❡✲s❡ ❛ r❡❧❛çã♦ ❡♠ A × A ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦ (a, b)S(c, d) s❡✱ ❡ s♦♠❡♥t❡ s❡ a + d = b + c✳ ❆❝❤❛r ♦s ❡❧❡♠❡♥t♦s ❞❛ r❡❧❛çã♦ S ❡ ❞❡t❡r♠✐♥❡ s❡✉ ❞♦♠í♥✐♦✳ ✶✵✳ ❙❡❥❛ A = { 1, 2, 3 } ❞❡✜♥❡✲s❡ ❛ r❡❧❛çã♦ ❡♠ A × A ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦ (a, b)T (c, d) s❡✱ ❡ s♦♠❡♥t❡ s❡ a − d = b − c✳ ❆❝❤❛r ♦s ❡❧❡♠❡♥t♦s ❞❛ r❡❧❛çã♦ T ❡ ❞❡t❡r♠✐♥❡ s❡✉ ❞♦♠í♥✐♦✳ ✶✶✳ ❆ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ❝♦♥✈❡①♦ ♣❧❛♥♦ ❡stá ❡♠ r❡❧❛çã♦ ❝♦♠ ♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s✳ ❊①♣r❡ss❛r ❛♥❛❧✐t✐❝❛♠❡♥t❡ ❡st❛ r❡❧❛çã♦✳ ◗✉❛✐s ✈❛❧♦r❡s ♣♦❞❡ ❛ss✉♠✐r ❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡❄ ✶✷✳ ❊s❝r❡✈❡r ❛ r❡❧❛çã♦ q✉❡ ❡①♣r❡ss❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❡♥tr❡ ♦ r❛✐♦ r ❞❡ ✉♠ ❝✐❧✐♥❞r♦ ❡ s✉❛ ❛❧t✉r❛ h s❡♥❞♦ ♦ ✈♦❧✉♠❡ V = 1✳ ✶✸✳ ❉❡t❡r♠✐♥❡ ♦s ✈❛❧♦r❡s ❞❡ a ❡ b ♥❛ r❡❧❛çã♦ y = S(x) ♦♥❞❡ S(x) = ax2 + bx + 5 ♣❛r❛ ♦s q✉❛✐s é ✈á❧✐❞❛ ❛ ✐❣✉❛❧❞❛❞❡ S(x + 1) − S(x) = 8x + 3✳ 1 ❝♦♠ x 6= 0 ❡ x 6= −1✱ ❡♥tã♦ ♦ ✈❛❧♦r ❞❡ S = f (1) + f (2) + f (3) + x(x + 1) · · · + f (100) é✿ ✶✹✳ ❙❡ f (x) = ✶✺✳ ❖ ❣rá✜❝♦ ❞❛ r❡❧❛çã♦ f ❞❡ R ❡♠ R✱ ❞❛❞❛ ♣♦r f (x) =| 1 − x | −2✱ ✐♥t❡r❝❡♣t❛ ♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ♥♦s ♣♦♥t♦s (a, b) ❡ (c, d)✱ ❝♦♠ a < c✳ ◆❡st❛s ❝♦♥❞✐çõ❡s ♦ ❞❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❡ E = d + c − b − a✳ ✶✻✳ ❆ ✈❛r✐á✈❡❧ x é ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛ y ❀ y é ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛ z ❀ z é ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛ u✱ q✉❡ ♣♦r s✉❛ ✈❡③ é ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛ v ✳ ◗✉❡ ❞❡♣❡♥❞ê♥❝✐❛ ❡①✐st❡ ❡♥tr❡ x ❡ v ❄ ✶✼✳ ❆ ❢♦❧❤❛ ❞❡ ♣❛❣❛♠❡♥t♦ ✭F.P.✮ ♠❡♥s❛❧ ❞❡ ✉♠❛ ❡♠♣r❡s❛ é ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ tr❛❜❛❧❤❛❞♦r❡s ✭T ✮✱ s❛❜❡♥❞♦ q✉❡ 20 ❞♦s tr❛❜❛❧❤❛❞♦r❡s t❡♠ ✉♠❛ ❢♦❧❤❛ ❞❡ ♣❛❣❛♠❡♥t♦ ❞❡ ❘$3000, 00✳ ❛✮ ❊①♣r❡ss❡ ♦ ✈❛❧♦r ❞❛ ❢♦❧❤❛ ❞❡ ♣❛❣❛♠❡♥t♦ ♠❡♥s❛❧ ❝♦♠♦ ❢✉♥çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ tr❛❜❛❧❤❛❞♦r❡s❀ ❜✮ q✉❛❧ ❛ ❢♦❧❤❛ ❞❡ ♣❛❣❛♠❡♥t♦ ♣❛r❛ 18 tr❛❜❛❧❤❛❞♦r❡s❄ ✶✽✳ ❙❡❥❛ a ∈ R ✉♠ ♥ú♠❡r♦ ✜①♦✱ ❡ f (x) = ax ✉♠❛ r❡❧❛çã♦ ❡♠ R ✶✳ ✷✳ ▼♦str❡ q✉❡✱ ♣❛r❛ ∀ x ∈ R é ✈á❧✐❞❛ ❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✿ f (−x) · f (x) = 1✳ ▼♦str❡ q✉❡ f (x) · f (y) = f (x + y) ✼✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✳✸ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❋✉♥çõ❡s ❖ ❝♦♥❝❡✐t♦ ❜ás✐❝♦ ❞❡ ❢✉♥çã♦ é ♦ s❡❣✉✐♥t❡✿ ✏t♦❞❛ ✈❡③ q✉❡ t❡♠♦s ❞♦✐s ❝♦♥❥✉♥t♦s ❡ ❛❧❣✉♠ t✐♣♦ ❞❡ ❛ss♦❝✐❛çã♦ ❡♥tr❡ ❡❧❡s q✉❡ ❢❛ç❛ ❝♦rr❡s♣♦♥❞❡r ❛ t♦❞♦ ❡❧❡♠❡♥t♦ ❞♦ ♣r✐♠❡✐r♦ ❝♦♥❥✉♥t♦ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞♦ s❡❣✉♥❞♦✱ ♦❝♦rr❡ ✉♠❛ ❢✉♥çã♦ ✑ ❉❡ ♦✉tr♦ ♠♦❞♦✱ ❞❛❞♦s ♦s ❝♦♥❥✉♥t♦s A ❡ B✱ ❡①✐st❡♠ ❞✐✈❡rs❛s r❡❧❛çõ❡s ❞❡ A ❡♠ B✱ ❡♥tr❡ ❡st❛s tê♠ ♣❛rt✐❝✉❧❛r ✐♠♣♦rtâ♥❝✐❛ ❛q✉❡❧❛s q✉❡ ❝✉♠♣r❡♠ ♦ s❡❣✉✐♥t❡✿ ❉❡✜♥✐çã♦ ✷✳✺✳ f ❞❡ A ❡♠ B ❞❡♥♦t❛❞♦ f : A −→ B ✱ é x ∈ A✱ ❝♦rr❡s♣♦♥❞❡ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ y ∈ B ✳ ❯♠❛ r❡❧❛çã♦ ❡❧❡♠❡♥t♦ ✉♠❛ ✏ ❢✉♥çã♦✑ s❡✱ ❛ t♦❞♦ ❆ ❞❡✜♥✐çã♦ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦✿ ✏ ❝♦♥❝❡✐t♦ ✐♥t✉✐t✐✈♦ ❞❡ ❢✉♥çã♦ ✑✳ ❙❡ f (a) = b ❡ s❡ ❧ê ✏ f ❞❡ a✑ ♦✉ ✏ f ❛♣❧✐❝❛❞♦ ❡♠ a✑✳ (a, b) ∈ f ✱ ❡s❝r❡✈❡✲s❡ ❖❜s❡r✈❡✱ ♣♦r ❡①❡♠♣❧♦✱ ♦s ❞✐❛❣r❛♠❛s ❞❛s r❡❧❛çõ❡s ❞❛s ❋✐❣✉r❛s ✭✷✳✹✮ ❡ ✭✷✳✺✮ A ✬ 1 ✲ B ✩ ✬ 2 ✒ ✲ 3 ✲ 4 ✫ 4 ✬ 1 ✲ B ✩ ✬ ✲ a 1 2 ✲ b 2 3 ✲ c ✲ 3 ✪ ✫ ❋✐❣✉r❛ ✷✳✹✿ A ✩ 4 ✪ ✫ ❆ r❡❧❛çã♦ ❞❛ ❋✐❣✉r❛ ✭✷✳✹✮ ♥ã♦ é ✉♠❛ ❢✉♥çã♦✱ ♣♦✐s ❡①✐st❡ ♦ ❡❧❡♠❡♥t♦ A ❛ss♦❝✐❛❞♦ ❛ ♠❛✐s ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ B✳ e ✲ d ✪ ✫ ❋✐❣✉r❛ ✷✳✺✿ 4 ✩ ✪ ♥♦ ❝♦♥❥✉♥t♦ Pr❡st❡ ♠✉✐t❛ ❛t❡♥çã♦ ♥♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦✿ ❆ r❡❧❛çã♦ ❞❛ ❋✐❣✉r❛ ✭✷✳✺✮ é ✉♠❛ ❢✉♥çã♦✱ ♣♦✐s t♦❞♦ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ❛ss♦❝✐❛❞♦ ❛ s♦♠❡♥t❡ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ A✱ ❡stá B✳ ✷✳✸✳✶ ●rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❉❡✜♥✐çã♦ ✷✳✻✳ ●rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦✳ ❉❡♥♦♠✐♥❛✲s❡ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❛♦ ❝♦♥❥✉♥t♦✿ Gf = { (x, y) /. x ∈ D(f ) ✼✺ ❡ y = f (x) ∈ Im(f ) } 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✷✳✸✳✷ ❉❡✜♥✐çã♦ ❢♦r♠❛❧ ❞❡ ❢✉♥çã♦ ❉❡✜♥✐çã♦ ✷✳✼✳ ❯♠❛ ❢✉♥çã♦ f ❞❡✜♥✐❞❛ ❡♠ A ❝♦♠ ✈❛❧♦r❡s ❡♠ B ❡ ❞♦♠í♥✐♦ D(f ) ⊆ A✱ é ✉♠ s✉❜❝♦♥❥✉♥t♦ Gf ⊆ A × B q✉❡ ❝✉♠♣r❡ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿ ✐✮ ✐✐✮ ∀ x ∈ D(f ), ❙❡ (x, y) ∈ Gf ∃ y∈B ❡ t❛❧ q✉❡ (x, z) ∈ Gf ✱ (x, y) ∈ Gf ✳ ❡♥tã♦ y = z✳ ✐✮ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ❛ t♦❞♦ ❡❧❡♠❡♥t♦ x ∈ D(f ) ❝♦rr❡s♣♦♥❞❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❡❧❡♠❡♥t♦ y ∈ B t❛❧ q✉❡ (x, y) ∈ Gf ❀ ❡ ❞❡ ✐✐✮ ♦ ❡❧❡♠❡♥t♦ y ❛ss♦❝✐❛❞♦ ❛♦ ❡❧❡♠❡♥t♦ x é ❉❛ ♣❛rt❡ ú♥✐❝♦✳ ✷✳✸✳✸ ❉♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ❞❡ ✉♠❛ ❢✉♥çã♦ ❉❛ ❞❡✜♥✐çã♦ ❞❡ ❢✉♥çã♦✱ ✏ t♦❞❛ ❢✉♥çã♦ é ✉♠❛ r❡❧❛çã♦✱ ♠❛s ♥❡♠ t♦❞❛ r❡❧❛çã♦ é ✉♠❛ ❢✉♥çã♦✑✱ ♦ ❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ❞❡ ✉♠❛ ❢✉♥çã♦ sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ ♦ ❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ❞❛ r❡❧❛çã♦ q✉❡ ❡❧❛ r❡♣r❡s❡♥t❛✳ f : A −→ B é s❡♠♣r❡ ♦ ♣ró♣r✐♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛✱ ♦✉ s❡❥❛✱ D(f ) = A✳ ❙❡ ✉♠ ❡❧❡♠❡♥t♦ x ∈ A ❡st✐✈❡r ❛ss♦❝✐❛❞♦ ❛ ✉♠ ❡❧❡♠❡♥t♦ y ∈ B ✱ ❞✐③❡♠♦s q✉❡ y é ❛ ✐♠❛❣❡♠ ❞❡ x ✭✐♥❞✐❝❛ q✉❡ y = f (x) ❡ ❧ê✲s❡ ✏ y é ✐❣✉❛❧ ❛ f ❞❡ x✑✮✳ ❖ ❞♦♠í♥✐♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❈♦♠ ❜❛s❡ ♥♦s ❞✐❛❣r❛♠❛s ❞❛s ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ✉♠❛ r❡❧❛çã♦ 1o ❋✐❣✉r❛s f ✭✷✳✹✮ ✲✭✷✳✺✮ ❛❝✐♠❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❡①✐st❡♠ ❞✉❛s s❡❥❛ ✉♠❛ ❢✉♥çã♦✿ ❖ ❞♦♠í♥✐♦ ❞❡✈❡ s❡♠♣r❡ ❝♦✐♥❝✐❞✐r ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛✱ ♦✉ s❡❥❛✱ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ A é ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ❞❡ ✢❡❝❤❛✳ ❙❡ t✐✈❡r♠♦s ✉♠ ❡❧❡♠❡♥t♦ ❞❡ A ❞♦ q✉❛❧ ♥ã♦ ♣❛rt❛ ✢❡❝❤❛✱ ❛ r❡❧❛çã♦ ♥ã♦ é ❢✉♥çã♦✳ 2o ❉❡ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡ A ❞❡✈❡ ♣❛rt✐r ✉♠❛ ú♥✐❝❛ ✢❡❝❤❛✳ ❙❡ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ A ♣❛rt✐r ♠❛✐s ❞❡ ✉♠❛ ✢❡❝❤❛✱ ❛ r❡❧❛çã♦ ♥ã♦ é ❢✉♥çã♦✳ ❖❜s❡r✈❛çã♦ ✷✳✷✳ • ❈♦♠♦ x ❡ y tê♠ s❡✉s ✈❛❧♦r❡s ✈❛r✐❛♥❞♦ ♥♦s ❝♦♥❥✉♥t♦s A ❡ B✱ r❡❝❡❜❡♠ ♦ ♥♦♠❡ ❞❡ ✈❛r✐á✈❡✐s✳ x é ❝❤❛♠❛❞❛ ✏ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ ✑ ❡ ❛ ✈❛r✐á✈❡❧ y ✱ ✏ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ ✑✱ ♦❜t❡r ♦ ✈❛❧♦r ❞❡ y ❞❡♣❡♥❞❡♠♦s ❞❡ ✉♠ ✈❛❧♦r ❞❡ x✳ • ❆ ✈❛r✐á✈❡❧ • ❯♠❛ ❢✉♥çã♦ ♣♦✐s ♣❛r❛ f ✜❝❛ ❞❡✜♥✐❞❛ q✉❛♥❞♦ sã♦ ❞❛❞♦s s❡✉ ❞♦♠í♥✐♦ ✭❝♦♥❥✉♥t♦ tr❛❞♦♠í♥✐♦ ✭❝♦♥❥✉♥t♦ B✮ ❡ ❛ ❧❡✐ ❞❡ ❛ss♦❝✐❛çã♦ ✼✻ A✮✱ s❡✉ ❝♦♥✲ y = f (x)✳ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✳✸✳✹ ❖❜t❡♥çã♦ ❞♦ ❞♦♠í♥✐♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❖ ❞♦♠í♥✐♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❡♠ R é ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡ R ♥♦ q✉❛❧ ♦ ♥ú♠❡r♦ y = f (x) ∈ R✳ ❚❡♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❢✉♥çõ❡s✿ ✶✮ ❙❡❥❛ f (x) = √ 3x − 6 R q✉❛♥❞♦ 3x − 6 ≥ 0 ✱ ❡♥tã♦ ♦ ❞♦♠í♥✐♦ ❞❡ ❞❡✜♥✐çã♦ ♣❛r❛ ❛ ❢✉♥çã♦ é✿ D(f ) = { x ∈ R /. x ≥ 2 }✳ √ x−2 ✷✮ ◗✉❛♥❞♦ f (x) = √ 3−x √ x − 2 só é ♣♦ssí✈❡❧ ♣❛r❛ x ≥ 2 ❡✱ ♦ ❞❡♥♦♠✐♥❛❞♦r é ♣♦ssí✈❡❧ ♣❛r❛ x < 3 ❡♥tã♦ ❈♦♠♦ ♣❛r❛ ❛ ❢✉♥çã♦ f ❡st❛r ❜❡♠ ❞❡✜♥✐❞❛✱ D(f ) = { x ∈ R /. 2 ≤ x < 3 }✳ ❉♦ ❢❛t♦ s❡r ♣♦ssí✈❡❧ ❡♠ ✸✮ ❈♦♥s✐❞❡r❡♠♦s ❛ ❢✉♥çã♦ g(x) = 7 ✳ x−1 x − 1 ♥ã♦ D(g) = { x ∈ R /. x 6= 1 }✳ ❈♦♠♦ ♦ ❞❡♥♦♠✐♥❛❞♦r ♣♦❞❡ s❡r ♥✉❧♦ ✭♥ã♦ ❡①✐st❡ ❞✐✈✐sã♦ ♣♦r ③❡r♦✮✱ ❡♥tã♦✿ ❊①❡♠♣❧♦ ✷✳✶✸✳ ❙❡❥❛ f : N −→ N ✭✐st♦ s✐❣♥✐✜❝❛ q✉❡ ♦ ❞♦♠í♥✐♦ ❡ ♦ ❝♦♥tr❛❞♦♠í♥✐♦ sã♦ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✮ ❞❡✜♥✐❞❛ ♣♦r y = x + 2✳ ❊♥tã♦ t❡♠♦s q✉❡✿ ❉❡ ♠♦❞♦ ❣❡r❛❧✱ ❛ ✐♠❛❣❡♠ ❞❡ x ❛tr❛✈és ❞❡ f é x + 2✱ ♦✉ s❡❥❛✿ f (x) = x + 2✳ • ❆ ✐♠❛❣❡♠ ❞❡ 1 ❛tr❛✈és ❞❡ f é 3✱ ♦✉ s❡❥❛✱ f (1) = 1 + 2 = 3✳ • ❆ ✐♠❛❣❡♠ ❞❡ 2 ❛tr❛✈és ❞❡ f é 4✱ ♦✉ s❡❥❛✱ f (2) = 2 + 2 = 4✳  f : A −→ B ✱ ♦s ❡❧❡♠❡♥t♦s ❞❡ B q✉❡ sã♦ ✐♠❛❣❡♥s ❞♦s ❡❧❡♠❡♥t♦s ❞❡ f ❡ ❢♦r♠❛♠ ♦ ✏ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠ ❞❡ f ✑ ♦✉ ✏ ❝♦♥tr❛❞♦♠í♥✐♦ ❞❡ ▲❡♠❜r❡✱ ❡♠ ✉♠❛ ❢✉♥çã♦ ❞❡ A ❛tr❛✈és ❞❛ r❡❧❛çã♦ f ✑✳ ❊①❡♠♣❧♦ ✷✳✶✹✳ ❙❡❥❛♠ A = { 1, 3, 4, 5 } ❡ B = { 2, 4, 5, 7 } ❡ f = { (1, 2), (3, 4), (4, 5), (5, 7) }✳ f ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ❚❡♠♦s q✉❡✿ f (1) = 2, f (3) = 4, f (4) = 5, f (5) = 7✳ Im(f ) = B ❡ D(f ) = A Gf = { (1, 2), (3, 4), (4, 5), (5, 7) } ❖ ❞✐❛❣r❛♠❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❞❛ ❢✉♥çã♦ ✭✷✳✻✮✳  ❊①❡♠♣❧♦ ✷✳✶✺✳ ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ f : A −→ B r❡♣r❡s❡♥t❛❞❛ ♥♦ ❞✐❛❣r❛♠❛ ❞❛ ❋✐❣✉r❛ ✭✷✳✼✮✱ ❞❡t❡r♠✐♥❡✿ ❛✮ ♦ ❞♦♠í♥✐♦ D(f )❀ ❜✮ f (1), f (−3), f (3) ❡ f (2)❀ ❝✮ ♦ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠ Im(f )❀ ❞✮ ❛ ❧❡✐ ❛ss♦❝✐❛t✐✈❛✳ ❙♦❧✉çã♦✳ ✼✼ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ A ✬ 1 ✲ B ✩ ✬ ✲ 2 4 −3 4 ✲ 5 2 ✲ 7 ✪ ✫ ❋✐❣✉r❛ ✷✳✻✿ ✪ D(f ) = A✳ f (2) = 4✳ ❖ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠ é ❢♦r♠❛❞♦ ♣♦r t♦❞❛s ❛s ✐♠❛❣❡♥s ❞♦s ❡❧❡♠❡♥t♦s ❞♦ ❞♦♠í♥✐♦✱ ♣♦r✲ t❛♥t♦✿ ❞✮ ✩ 9 ❋✐❣✉r❛ ✷✳✼✿ ❖ ❞♦♠í♥✐♦ é ✐❣✉❛❧ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛✱ ♦✉ s❡❥❛✱ ❡ ✲ ✚ ❃ ✚ ✚ ✲ 4 ✚ ✚ 3 ✚ 7 ✫ ✪ ✫ ✪ ❜✮ f (1) = 1, f (−3) = 9, f (3) = 9 ❝✮ 1 ✲ ✫ ❛✮ ✬ 3 5 ✲ B ✩ ✬ ✲ 1 A ✩ ❈♦♠♦ Im(f ) = { 1, 4, 9 }✳ 12 = 1, (−3)2 = 9, 32 = 9 ❡ 2 2 = 4✱ t❡♠♦s y = x2 ✳  ✷✳✸✳✺ ❈♦♥str✉çã♦ ❞♦ ❣rá✜❝♦ ❝❛rt❡s✐❛♥♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❯♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ❝♦♥s✐st❡ ❡♠ ✉♠ ♣❛r ❞❡ r❡t❛s ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ ❛s q✉❛✐s s❡ y ✐♥t❡r❝❡♣t❛♠ ❢♦r♠❛♥❞♦ â♥❣✉❧♦ r❡t♦✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✭✷✳✽✮❀ ❛ r❡t❛ ❤♦r✐③♦♥t❛❧ é ❝❤❛♠❛❞❛ ✏❡✐①♦✲x✑ ♦✉ ✏ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ✑ ❡ ❛ r❡t❛ ✈❡rt✐❝❛❧ é ❝❤❛♠❛❞❛ ❞❡ ✏❡✐①♦✲y ✑ ♦✉ ✏ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s ✑✳ y = f (x)✱ ❜❛st❛ ❛tr✐❜✉✐r ✈❛❧♦r❡s ❞♦ ❞♦♠í♥✐♦ à ✈❛r✐á✈❡❧ x P❛r❛ ❝♦♥str✉✐r ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ 2 1 −x ✛ −3 x ✲ −2 −1 f (x)✳ y = y = 2x−1✳ Pr✐♠❡✐r♦ ♦❜s❡r✈❡ 1 2 3 −2 −y ❄ P♦r ❡①❡♠♣❧♦✱ s❡ ❞❡s❡❥❛♠♦s ❝♦♥str✉✐r ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r 0 −1 ❡✱ ✉s❛♥❞♦ ❛ s❡♥t❡♥ç❛ ♠❛t❡♠át✐❝❛ q✉❡ ❞❡✜♥❡ ❛ ❢✉♥✲ çã♦✱ ❝❛❧❝✉❧❛r ♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ✈❛❧♦r❡s ♣❛r❛ ✻3 ❋✐❣✉r❛ ✷✳✽✿ P❧❛♥♦ ❝❛rt❡s✐❛♥♦ q✉❡ ♦ ❞♦♠í♥✐♦ sã♦ t♦❞♦s ♦s ♥ú♠❡r♦s r❡❛✐s✱ ❧♦❣♦✱ ♣♦✲ ❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ss✐♠ ❝❛❧❝✉❧❛♠♦s x = 2, x = 4, x = 6, x = 8✱ ❡ ♦s r❡s♣❡❝t✐✈♦s ✈❛❧♦r❡s ♣❛r❛ y ✱ ❝♦♠♦ ✐♥❞✐❝❛ ❛ t❛❜❡❧❛✿ ■❞❡♥t✐✜❝❛♠♦s ♦s ♣♦♥t♦s ❡♥❝♦♥tr❛❞♦s ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ❖ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ é ✉♠❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s q✉❛tr♦ ♣♦♥t♦s ❡♥❝♦♥tr❛❞♦s✳ ✭✷✳✾✮✳ ❇❛st❛ tr❛ç❛r ❛ r❡t❛✱ ❡ ♦ ❣rá✜❝♦ ❡st❛rá ❝♦♥str✉í❞♦✳ P❛r❛ ❞❡s❡♥❤❛r ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ r❡t❛ sã♦ ♥❡❝❡ssár✐♦s ❛♣❡♥❛s ❞♦✐s ♣♦♥t♦s✳ ◆♦ ❡①❡♠♣❧♦ ✼✽ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✻y x 2 y 4 3 6 7 11 8 15 10 11 19 ✂ ✛ 21 ✂ ✂ ✂✂ ✂0, 5 −x ✂ ✂ ✂ ✂ ✂ −1 ✲ x −y ❄ ❋✐❣✉r❛ ✷✳✾✿ ❛❝✐♠❛✱ ❡s❝♦❧❤❡♠♦s 6 ♣♦♥t♦s✳ ❊♠ ✈❡r❞❛❞❡ é s✉✜❝✐❡♥t❡ ❡s❝♦❧❤❡r ❞♦✐s ❡❧❡♠❡♥t♦s ❞♦ ❞♦♠í♥✐♦✱ ❡♥❝♦♥tr❛r s✉❛s ✐♠❛❣❡♥s ❡✱ ❧♦❣♦ ❛♣ós✱ tr❛ç❛r ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r ❡ss❡s ❞♦✐s ♣♦♥t♦s✳ ❉❡✜♥✐çã♦ f : A −→ B t❡♠ ❝♦♠♦ ❞♦♠í♥✐♦ D(f ) = A❀ ♣♦ré♠✱ q✉❛♥❞♦ ❞✐③❡♠♦s q✉❡ t❡♠♦s ✉♠❛ ❢✉♥çã♦ ❞❡ A ❡♠ B ❡ ❛❝❤❛♠♦s s❡✉ ❞♦♠í♥✐♦ D(f ) ⊆ A✱ ♦❜s❡r✈❡✱ t❡♠♦s ✉♠❛ r❡❧❛çã♦ ❞❡ A ❡♠ B ✱ ❡ ❛♦ ❝❛❧❝✉❧❛r s❡✉ ❞♦♠í♥✐♦ D(f )✱ ❛ tr❛♥s❢♦r♠❛♠♦s ❡♠ ✉♠❛ ❢✉♥çã♦ ✭s❡♠♣r❡ q✉❡ ❢♦r ♣♦ssí✈❡❧✮ ❞❡ D(f ) ❡♠ B ❀ ✐st♦ ♦❝♦rr❡ ❝♦♠ ❢r❡q✉ê♥❝✐❛ q✉❛♥❞♦ t❡♠♦s ✉♠❛ r❡❧❛çã♦ ❞❡ R ❡♠ R ❡ ❢❛❧❛♠♦s ❞❡ ✏ ❢✉♥çã♦ ❞❡ R ❡♠ R✑✳ ❙❡❣✉♥❞♦ ❛ ✭✷✳✺✮✱ t♦❞❛ ❢✉♥çã♦ ❊①❡♠♣❧♦ ✷✳✶✻✳ ❙❡❥❛ f : R −→ R ❞❡✜♥✐❞❛ ♣♦r✿ f (x) = ❞❡t❡r♠✐♥❡✿ ❛✮ f (0, 12) 1, −1, √ 1 2 ❝✮ f ( 2) ❜✮ f ( ) ❙♦❧✉çã♦✳ ( 12 )=1 a) f (0.12) = f ( 100 √ c) f ( 2) = −1 s❡✱ s❡✱ x∈Q x∈I ❞✮ f (0, 333333...) 1 b) f ( ) = 1 2 3 d) f (0, 333333...) = f ( ) = 1 9 ❊①❡♠♣❧♦ ✷✳✶✼✳ ❉❛❞❛ ❛ ❢✉♥çã♦ f : R −→ R ✭♦✉ s❡❥❛✱ ♦ ❞♦♠í♥✐♦ ❡ ♦ ❝♦♥tr❛❞♦♠í♥✐♦ sã♦ ♦s ♥ú♠❡r♦s r❡❛✐s✮ ❞❡✜♥✐❞❛ ♣♦r f (x) = x2 − 5x + 6✱ ❝❛❧❝✉❧❡✿ ❛✮ f (2), f (3) ❡ f (0)❀ ❜✮ ♦ ✈❛❧♦r ❞❡ x ❝✉❥❛ ✐♠❛❣❡♠ s❡❥❛ 2✳ ❙♦❧✉çã♦✳ ❛✮ f (2) = 22 − 5(2) + 6 = 0; ❙♦❧✉çã♦✳ f (3) = 32 − 5(3) + 6 = 0 ❡ f (0) = 6  ❜✮ ✼✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞❡ s❡❥❛✱ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R x 2 ❝✉❥❛ ✐♠❛❣❡♠ ✈❛❧❡ ❡q✉✐✈❛❧❡ ❛ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ f (x) = 2✱ ♦✉ 2 x − 5x + 6 = 2✳ ❯t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛✱ ❡♥❝♦♥tr❛♠♦s ❛s r❛í③❡s P♦rt❛♥t♦✱ ♦s ✈❛❧♦r❡s ❞❡ x ❝✉❥❛ ✐♠❛❣❡♠ é 2 sã♦ x=1 ❡ 1 4✳ ❡ x = 4✳ ❊①❡♠♣❧♦ ✷✳✶✽✳ ❙❡❥❛ ❛ ❢✉♥çã♦ f : R −→ R ❞❡✜♥✐❞❛ ♣♦r✿ f (x) = x2 − 3x + 2✳ a) f (−3) b) f (x2 ) e) f (a2 ) c) f (y − z) f ) f (x + h) g) f (f (x)) ❉❡t❡r♠✐♥❡✿ d) f (2x − 3) − f (x + 3) h) f (x2 − 3x + 2) ❙♦❧✉çã♦✳ ❛✮ f (−3) = (−3)2 − 3(−3) + 2 = 20 ❜✮ f (x2 ) = [x2 ]2 − 3[x2 ] + 2 = x4 − 3x2 + 2 ❝✮ f (y − z) = (y − z)2 − 3(y − z) + 2 = y 2 + z 2 − 2yz − 3y + 3z + 2 ❞✮ f (2x − 3) − f (x + 3) = [2x − 3]2 − 3[2x − 3] + 2 − [x + 3]2 − 3[x + 3] + 2 = = [4x2 − 18x + 20] − [x2 + 3x + 2] = 3x2 − 21x + 18 ❡✮ f (a2 ) = [a2 ]2 − 3[a2 ] + 2 = a4 − 3a2 + 2 ❢✮ f (x + h) = (x + h)2 − 3(x + h) + 2 = x2 + h2 + 2hx − 3x − 3h + 2 ❣✮ f (f (x)) = [f (x)]2 − 3[f (x)] + 2 ❤✮ f (x2 − 3x + 2) = [x2 − 3x + 2]2 − 3[x2 − 3x + 2] + 2 = x4 − 6x3 + 10x2 − 3x ✷✳✸✳✻  ❋✉♥çã♦✿ ■♥❥❡t✐✈❛✳ ❙♦❜r❡❥❡t✐✈❛✳ ❇✐❥❡t✐✈❛ ❉❡✜♥✐çã♦ ✷✳✽✳ ❋✉♥çã♦ ✐♥❥❡t✐✈❛✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ f : A ⊆ R −→ B D(f ) ⊆ A ❝♦rr❡s♣♦♥❞❡♠ ✐♠❛❣❡♥s ❞✐st✐♥t❛s❀ ❝♦♠ x1 6= x2 ❝♦rr❡s♣♦♥❞❡ f (x1 ) 6= f (x2 )✳ ❞♦ é ✐♥❥❡t✐✈❛✱ s❡ ❛ ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s ✐st♦ é ♣❛r❛ q✉❛❧q✉❡r x1 , x2 ∈ D(f ) ❊st❛ ❉❡✜♥✐çã♦ ✭✷✳✽✮ é ❡q✉✐✈❛❧❡♥t❡ ❛✿ f : A ⊆ R −→ B é ✏✐♥❥❡t✐✈❛✑ f (x1 ) = f (x2 ) t❡♠♦s q✉❡ x1 = x2 ✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ x1 , x2 ∈ D(f ) ❝♦♠ ✽✵ s❡✱ ♣❛r❛ q✉❛❧q✉❡r 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R P♦r ❡①❡♠♣❧♦✱ ❛ ❢✉♥çã♦ f : R −→ R ❞❡✜♥✐❞❛ ♣♦r f (x) = 3x é ✐♥❥❡t✐✈❛ ♣♦✐s s❡ x1 6= x2 ❡♥tã♦ 3x1 6= 3x2 ✱ ♣♦rt❛♥t♦ f (x1 ) 6= f (x2 )✳ ❉❡✜♥✐çã♦ ✷✳✾✳ ❋✉♥çã♦ s♦❜r❡❥❡t✐✈❛✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ f : A ⊆ R −→ B é s♦❜r❡❥❡t✐✈❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ s❡✉ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠ ❢♦r ✐❣✉❛❧ ❛♦ ❝♦♥tr❛❞♦♠í♥✐♦✳ ■st♦ é✱ ♣❛r❛ t♦❞♦ y ∈ B ✱ ❡①✐st❡ x ∈ A t❛❧ q✉❡ f (x) = y ❀ ❧♦❣♦✱ ❛ ❢✉♥çã♦ f : A ⊆ R −→ B é s♦❜r❡❥❡t✐✈❛ s❡ Im(f ) = B ✳ ❉❡✜♥✐çã♦ ✷✳✶✵✳ ❋✉♥çã♦ ❜✐❥❡t✐✈❛✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ f : A ⊆ R −→ R é ❜✐❥❡t✐✈❛ ❡♥tr❡ A ❡ R q✉❛♥❞♦ ❡❧❛ é s♦❜r❡❥❡t✐✈❛ ❡ ✐♥❥❡t✐✈❛ ✭❛♠❜❛s ❛s ❝♦♥❞✐çõ❡s✮✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❢✉♥çã♦ f : R −→ R ❞❡✜♥✐❞❛ ♣♦r y = 3x é ✐♥❥❡t✐✈❛✱ ❝♦♠♦ ✈✐♠♦s ♥♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✳ ❊❧❛ t❛♠❜é♠ é s♦❜r❡❥❡t✐✈❛✱ ♣♦✐s Im(f ) = B = R✳ ▲♦❣♦✱ ❛ ❢✉♥çã♦ f é ❜✐❥❡t✐✈❛✳ ❆ ❢✉♥çã♦ g : N −→ N ❞❡✜♥✐❞❛ ♣♦r y = x + 5 ♥ã♦ é s♦❜r❡❥❡t✐✈❛✳ P♦✐s Im(g) = { 5, 6, 7, 8, · · · } ❡ ♦ ❝♦♥tr❛❞♦♠í♥✐♦ é N✱ ❡st❛ ❢✉♥çã♦ é ✐♥❥❡t✐✈❛✱ ♣♦✐s ✈❛❧♦r❡s ❞✐❢❡r❡♥t❡s ❞❡ x tê♠ ✐♠❛❣❡♥s ❞✐st✐♥t❛s✳ ❊♥tã♦ ❡ss❛ ❢✉♥çã♦ ♥ã♦ é ❜✐❥❡t✐✈❛✳ ❖❜s❡r✈❛çã♦ ✷✳✸✳ • ➱ s✐♥ô♥✐♠♦ ❞❡ ❢✉♥çã♦ ✐♥❥❡t✐✈❛✿ ❋✉♥çã♦ ✐♥❥❡t✐✈❛✳ ❋✉♥çã♦ ✉♥í✈♦❝❛ • ➱ s✐♥ô♥✐♠♦ ❞❡ ❢✉♥çã♦ s♦❜r❡❥❡t✐✈❛✿ ❋✉♥çã♦ s♦❜r❡❥❡t♦r❛✳ • ➱ s✐♥ô♥✐♠♦ ❞❡ ❢✉♥çã♦ ❜✐❥❡t✐✈❛✿ ❋✉♥çã♦ ❜✐✉♥í✈♦❝❛✳ ❈♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈✐❝❛✳ ❇✐✲ ❥❡çã♦✳ ❋✉♥çã♦ ✉♠✲❛✲✉♠✳ ❊①❡♠♣❧♦ ✷✳✶✾✳ ❈♦♥s✐❞❡r❡ ♦s ❝♦♥❥✉♥t♦s A = { 5, 6, 7, 8} ❡ B = { 1, 2, 3, 4, 9 } ❞❡✜♥✐❞❛ ♣❡❧❛ ❡q✉❛çã♦ y = x − 4✳ P❛r❛ ❝❛❞❛ a ∈ A ✜❝❛ ❛ss♦❝✐❛❞♦ ✉♠ ú♥✐❝♦ y ∈ B ✳ ❈♦♥s✐❞❡r❛♥❞♦ y = f (x) = x − 4 t❡♠♦s f (5) = 1, f (6) = 2, f (7) = 3 ❡ f (8) = 4✳ ❊st❛ ❢✉♥çã♦ é ✐♥❥❡t✐✈❛✱ ♥ã♦ é s♦❜r❡❥❡t✐✈❛ ✭♣❛r❛ ♦ ❡❧❡♠❡♥t♦ 9 ∈ B ✱ ♥ã♦ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ❡♠ A✮✱ ❧♦❣♦ ♥ã♦ é ❜✐❥❡t✐✈❛✳ ❊①❡♠♣❧♦ ✷✳✷✵✳ ❛✮ ❙❡❥❛♠ A = { 1, 3, 9, 10 } ❡ B = { 2, 3, 4, 5 } ❡ f : A → B ❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r f (1) = 2, f (9) = 3, f (3) = 4 ❡ f (10) = 5 é ❢✉♥çã♦ ❜✐❥❡t✐✈❛✳ ✽✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❜✮ ❆ ❢✉♥çã♦ h = { (x, y) ∈ R2 /. y = x2 + 1; −3 < x ≤ 3 } ♥ã♦ é ✐♥❥❡t✐✈❛✳ ✷✳✸✳✼ ❋✉♥çã♦ r❡❛❧ ❞❡ ✈❛r✐á✈❡❧ r❡❛❧ ❉❡✜♥✐çã♦ ✷✳✶✶✳ ❙❡❥❛♠ A ❡ B s✉❜❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦s ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ ✉♠❛ ❢✉♥çã♦ f : A −→ B é ❞❡♥♦♠✐♥❛❞❛ ❢✉♥çã♦ r❡❛❧ ❞❡ ✈❛r✐á✈❡❧ r❡❛❧ ♦✉ ❢✉♥çã♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧ ❛ ✈❛❧♦r❡s r❡❛✐s✳ ❉❛q✉✐ ♣♦r ❞✐❛♥t❡✱ t♦❞❛s ❛s ❢✉♥çõ❡s ❡st✉❞❛❞❛s s❡rã♦ r❡❛✐s ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧✳ ❊①❡♠♣❧♦ ✷✳✷✶✳ ❙❡❥❛ f = { (x, y) ∈ A × B /. y = 2x + 1 } ♦♥❞❡ A = R ❡ B = N✱ ❡♥tã♦ t❡♠♦s✿ 1 3 n−1 1 , n) } f = { (− , 0), (0, 1), ( , 2), (1, 3), ( , 4), · · · , ( 2 2 2 2 n−1 n ∈ N} ⊆ A ❡ ❛ ✐♠❛❣❡♠ Im(f ) = B ✳ 2 ❆ ❋✐❣✉r❛ ✭✷✳✶✵✮ ♠♦str❛ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f ✱ sã♦ ♣♦♥t♦s ✐s♦❧❛❞♦s✳ ♥❡st❡ ❝❛s♦ ♦ ❞♦♠í♥✐♦ D(f ) = {x ∈ R/. x = ✻y 4 · · · · · · · · · · · · · · · · ✳q ✳ ✁ ✳✳ 3 y ✻ 2 · · ·· · ·✳q −x ✛ q 1q − 12 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 0 1 2 −x✛ x ✲ 1 ❄ 3 2 ··· 1 −2 −1 n−1 2 ✁ 0 1 2 ✁ ✁ ✁ 3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 4 x ✲ 5 −1 ❄ ❋✐❣✉r❛ ✷✳✶✵✿ ❋✐❣✉r❛ ✷✳✶✶✿ ❊①❡♠♣❧♦ ✷✳✷✷✳ ❙❡❥❛ g : A −→ B ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r✿    0, 5, s❡✱ 0 ≤ x < 2 g(x) = 0, 5 + x, s❡✱ 2 ≤ x ≤ 4   −1, s❡✱ x < 0, ♦✉ x > 4 ♦♥❞❡ A ❡ B sã♦ s✉❜❝♦♥❥✉♥t♦s ❞❡ R✳ ✽✷ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❚❡♠♦s D(f ) = A = R ❡ Im(g) = { −1 } ∪ [1, 4]✳ ❖ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ g(x) ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✷✳✶✶✮✳ ❊①❡♠♣❧♦ ✷✳✷✸✳ ❙❡❥❛ h(x) = x3 ✱ ❞❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❛ ❡①♣r❡ssã♦✿ ❙♦❧✉çã♦✳ h(b) − h(a) s❡♥❞♦ (a − b) 6= 0✳ b−a ❉❡t❡r♠✐♥❛♠♦s ♦s ✈❛❧♦r❡s ❞❛ ❢✉♥çã♦ ❞❛❞❛ ♣❛r❛ x = b ❡ x = a❀ ✐st♦ é h(b) = b3 ❡ h(a) = a3 ✳ ❆ss✐♠✱ b3 − a3 (b − a)(a2 + ab + b2 ) h(b) − h(a) = = = a2 + ab + b2 b−a b−a b−a ♦ ú❧t✐♠♦ ❛❝♦♥t❡❝❡ ♣❡❧♦ ❢❛t♦ a 6= b✳ ❖❜s❡r✈❛çã♦ ✷✳✹✳ ◆♦ q✉❡ s❡❣✉❡✱ ❛ ❢✉♥çã♦ t❡rá ❝♦♠♦ r❡❣r❛ ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ x 7−→ f (x)✱ s❡♠ ❡①♣❧✐❝✐t❛r s❡✉ ❞♦♠í♥✐♦ D(f ) ❡ ✐♠❛❣❡♠ Im(f )✳ ❋✐❝❛ ❡st❛❜❡❧❡❝✐❞♦ q✉❡ ♦ ❞♦♠í♥✐♦ é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s R✱ ♣❛r❛ ♦ q✉❛❧ f (x) é ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ❖ ❣rá✜❝♦ ❞❛s ❢✉♥çõ❡s s❡rá ❢❡✐t♦ ♥✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s✳ ❊①❡♠♣❧♦ ✷✳✷✹✳ ❉❡t❡r♠✐♥❡ ♦ ❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ❞❛ ❢✉♥çã♦ f (x) = x2 − 6x + 5✳ ❙♦❧✉çã♦✳ ❖❜s❡r✈❡✱ f (x) = x2 − 6x + 5 = (x − 3)2 − 4✱ s❡♥❞♦ (x − 3)2 s❡♠♣r❡ ♣♦s✐t✐✈♦✱ ❡♥tã♦ ∀ x ∈ R, f (x) ≥ −4✳ ▲♦❣♦ D(f ) = R ❡ Im(f ) = [−4, +∞)✳ ❊①❡♠♣❧♦ ✷✳✷✺✳ ❛✮ P❛r❛ q✉❛✐s ❢✉♥çõ❡s f (x) ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ g(x) t❛❧ q✉❡ f (x) = [g(x)]2 ❄ ❜✮ P❛r❛ q✉❡ ❢✉♥çã♦ f (x) ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ g(x) t❛❧ q✉❡ f (x) = ❝✮ P❛r❛ q✉❛✐s ❢✉♥çõ❡s b(x) ❡ c(x) ♣♦❞❡♠♦s ❛❝❤❛r ✉♠❛ ❢✉♥çã♦ f (x) t❛❧ q✉❡✿ 1 ❄ g(x) [f (x)]2 + b(x)[f (x)] + c(x) = 0 ♣❛r❛ t♦❞♦s ♦s ♥ú♠❡r♦s r❡❛✐s x ❄ ❞✮ ◗✉❡ ❝♦♥❞✐çõ❡s s❛t✐s❢❛③❡♠ ❛s ❢✉♥çõ❡s a(x) ❡ b(x) s❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ f (x) t❛❧ q✉❡ a(x)f (x) + b(x) = 0 ♣❛r❛ t♦❞♦s ♦s ♥ú♠❡r♦s r❡❛✐s x ❄ ✽✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❙♦❧✉çã♦✳ ❛✮ ❈♦♠♦ f (x) = [g(x)]2 ≥ 0✱ ❡♥tã♦ ✐st♦ é ♣♦ssí✈❡❧ s♦♠❡♥t❡ ♣❛r❛ ❛s ❢✉♥çõ❡s f (x) ≥ 0, ∀ x ∈ R✳ ❜✮ ❈♦♥s✐❞❡r❛♥❞♦ q✉❡ ❡st❛♠♦s tr❛❜❛❧❤❛♥❞♦ ❝♦♠ ❢✉♥çõ❡s ❞❡ R ❡♠ R✱ ♣♦❞❡♠♦s ✐♥t✉✐t✐✈❛✲ ♠❡♥t❡ ❡♥t❡♥❞❡r f (x) ❝♦♠♦ s❡♥❞♦ ✉♠ ♥ú♠❡r♦ r❡❛❧❀ ❛ss✐♠ g(x) ❡①✐st❡ s♦♠❡♥t❡ q✉❛♥❞♦ g(x) = ❝✮ 1 ❡①✐st❛✱ ✐st♦ s♦♠❡♥t❡ é ♣♦ssí✈❡❧ s❡ f (x) 6= 0, f (x) ❉❡ [f (x)]2 + b(x)[f (x)] + c(x) = 0✱ ♣❡❧❛ ❢ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛ s❡❣✉❡✿ f (x) = −b(x) ± p ❧♦❣♦ ❡①✐st❡ f (x) q✉❛♥❞♦ [b(x)]2 ≥ 4 · c(x), ❞✮ ∀x∈R [b(x)]2 − 4 · c(x) 2 ∀ x ∈ R✳ −b(x) , a(x) ❝♦♠ ❡st❛ ❝♦♥❞✐çã♦✳ ◗✉❛♥❞♦ ❛ ❢✉♥çã♦ b(x) = 0, ∀ x ∈ R ❡♥tã♦ a(x) = 0✳ P❛r❛ ♦ ❝❛s♦ a(x) 6= 0, ∀ x ∈ R✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❢✉♥çã♦ f (x) = ∀x ∈ R ❖❜s❡r✈❡✱ s❡ a(x) = 0 ♣❛r❛ ❛❧❣✉♠ x ∈ R✱ ❡♥tã♦ ♣♦❞❡♠♦s ❡❧❡❣❡r ❛r❜✐tr❛r✐❛♠❡♥t❡ f (x) ❞❡ ♠♦❞♦ q✉❡ ❡①✐st❡♠ ✐♥✜♥✐t❛s ❢✉♥çõ❡s q✉❡ ❝✉♠♣r❡♠ ❡st❛ ❝♦♥❞✐çã♦✳ ❊①❡♠♣❧♦ ✷✳✷✻✳ ❯♠ ❡st✉❞♦ s♦❜r❡ ❛ ❡✜❝✐ê♥❝✐❛ ❞❡ ♦♣❡rár✐♦s ❞♦ t✉r♥♦ ❞❛ ♠❛♥❤❛ ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❢á❜r✐❝❛✱ ✐♥❞✐❝❛ q✉❡ ✉♠ ♦♣❡rár✐♦ ♠é❞✐♦ q✉❡ ❝❤❡❣❛ ❛♦ tr❛❜❛❧❤♦ ❛s 8 ❤♦r❛s ❞❛ ♠❛♥❤❛✱ ♠♦♥t❛ x ❤♦r❛s ❛♣ós ❞❡ ✐♥✐❝✐❛❞♦ s❡✉ tr❛❜❛❧❤♦ f (x) = −x3 + 6x2 + 15x rá❞✐♦s tr❛♥s✐st♦r✐③❛❞♦s✳ ❛✮ ◗✉❛♥t♦s rá❞✐♦s ♦ ♦♣❡rár✐♦ t❡rá ♠♦♥t❛❞♦ ❛té ❛s 10 h ❞❛ ♠❛♥❤❛❄ ❜✮ ◗✉❛♥t♦s rá❞✐♦s ♦ ♦♣❡rár✐♦ t❡rá ♠♦♥t❛❞♦ ❡♥tr❡ ❛s 9 ❡ 10 ❤♦r❛s ❞❛ ♠❛♥❤❛❄ ❙♦❧✉çã♦✳ ❛✮ ❚❡♠♦s q✉❡✱ ❞❛s 08 : 00 ❛té ❛s 10 : 00 ♦ ♦♣❡rár✐♦ tr❛❜❛❧❤♦✉ x = 2 ❤♦r❛s✱ ❧♦❣♦ ❡❧❡ ♠♦♥t♦✉ f (2) = −23 + 6(22 ) + 15(2)✱ ❡♥tã♦ f (2) = 46. P♦rt❛♥t♦✱ ❡❧❡ ♠♦♥t♦✉ 46 ❛♣❛r❡❧❤♦s✳ ❜✮ ❊♥tr❡ ❛s 08 : 00 ❡ 09 : 00 ❞❛ ♠❛♥❤❛ ❡❧❡ ♠♦♥t♦✉ f (1) = −13 + 6(12 ) + 15(1) = 20 ❛♣❛r❡❧❤♦s❀ ❧♦❣♦ ❡♥tr❡ ❛s 09 : 00 ❡ 10 : 00 ❡❧❡ ♠♦♥t♦✉ 46 − 20 ❛♣❛r❡❧❤♦s✱ ✐st♦ é 26✳ ❊①❡♠♣❧♦ ✷✳✷✼✳ ❉❡✈❡♠♦s ❝♦♥str✉✐r ✉♠❛ ❝❛✐①❛ ❛❜❡rt❛ s❡♠ t❛♠♣❛ ❝♦♠ ✉♠ ♣❡❞❛ç♦ r❡t❛♥❣✉❧❛r ❞❡ ❝❛rt♦❧✐♥❛ ❞❡ 60 × 86 cm ❝♦rt❛♥❞♦✲s❡ ✉♠❛ ár❡❛ ❞❡ x cm2 ❡♠ ❝❛❞❛ ❝❛♥t♦ ❡ ❞♦❜r❛♥❞♦✲s❡ ♦s ❧❛❞♦s ❝♦♠♦ ✐♥❞✐❝❛ ❛ ❋✐❣✉r❛ ✭✷✳✾✮✳ ❊①♣r❡ss❡ ♦ ✈♦❧✉♠❡ ❞❛ ❝❛✐①❛ ❡♠ ❢✉♥çã♦ ❞❡ x✳ ❙♦❧✉çã♦✳ ✽✹ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✛ ✲ 86 cm ✳✳x · · ·✳ x x✳✳✳ ✻ ··· x 60 − 2x 60 cm x · · ·✳✳x x x x✳✳✳· · · ❄ ✳ 86 − 2x ❋✐❣✉r❛ ✷✳✶✷✿ ❆s ❞✐♠❡♥sõ❡s ❞❛ ❝❛✐①❛ sã♦✿ ❛❧t✉r❛ (86 − 2x) ❝♦♠♦ ♦❜s❡r✈❛♠♦s ♥❛ ▲♦❣♦ ♦ ✈♦❧✉♠❡ é ❋✐❣✉r❛ x ❝♠✱ ❡ ❛ ❜❛s❡ é ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s ✭✷✳✶✷✮✳ V = x(60 − 2x)(86 − 2x)❀ ✐st♦ é (60 − 2x) ❡ V = 4x(30 − x)(43 − x) ❊①❡♠♣❧♦ ✷✳✷✽✳ 900x ❙✉♣õ❡✲s❡ q✉❡ f (x) = s❡❥❛ ♦ ♥ú♠❡r♦ ♥❡❝❡ssár✐♦ ❞❡ ❤♦♠❡♥s ✲ ❤♦r❛ ♣❛r❛ ❞✐str✐✲ 400 − x ❜✉✐r ♣❛♥✢❡t♦s ❡♥tr❡ x ♣♦r ❝❡♥t♦ ❞❡ ♠♦r❛❞♦r❡s ❞❡ ✉♠❛ ❝✐❞❛❞❡✳ ❛✮ ❉❡t❡r♠✐♥❡ ♦ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦✳ ❜✮ P❛r❛ q✉❛❧ ✈❛❧♦r ❞❡ x ♦ ♣r♦❜❧❡♠❛ t❡♠ ✐♥t❡r♣r❡t❛çã♦ ♣rát✐❝❛❄ ❝✮ ◗✉❛♥t♦s ❤♦♠❡♥s✲❤♦r❛ sã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ❞✐str✐❜✉✐r ♣❛♥✢❡t♦s ❡♥tr❡ ♦s ♣r✐♠❡✐r♦s 50% ❞❡ ♠♦r❛✲ ❞♦r❡s❄ ❞✮ ◗✉❛♥t♦s ❤♦♠❡♥s✲❤♦r❛ sã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ❞✐str✐❜✉✐r ♣❛♥✢❡t♦s à ❝♦♠✉♥✐❞❛❞❡ ✐♥t❡✐r❛✳ ❡✮ ◗✉❡ ♣♦r❝❡♥t❛❣❡♠ ❞❡ ♠♦r❛❞♦r❡s ❞❛ ❝✐❞❛❞❡ r❡❝❡❜❡✉ ♣❛♥✢❡t♦s✱ q✉❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❤♦♠❡♥s✲❤♦r❛ ❢♦✐ ❞❡ 100 ❄ ❙♦❧✉çã♦✳ 900 ❝♦♠♦ 400 − x ❡①❝❡t♦ x = 400✳ ❛✮ ❖❜s❡r✈❛♥❞♦ ❛ ❢✉♥çã♦ f (x) = t♦❞♦s ♦s ♥ú♠❡r♦s r❡❛✐s ✉♠❛ r❡❧❛çã♦ ❞❡ R ❡♠ R✱ s❡✉ ❞♦♠í♥✐♦ sã♦ ❜✮ ❙❡♥❞♦ x ✉♠❛ ✈❛r✐á✈❡❧ q✉❡ r❡♣r❡s❡♥t❛ ♣♦r❝❡♥t❛❣❡♠✱ ♦ ♣r♦❜❧❡♠❛ t❡♠ ❛♣❧✐❝❛çã♦ ♣rát✐❝❛ q✉❛♥❞♦ 0 ≤ x ≤ 100✳ ❝✮ ◗✉❛♥❞♦ x = 50✱ ❡♥tã♦ f (50) = ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 129 ❤♦♠❡♥s✳ 900 (900)(50) = = 128, 59 400 − 50 7 ❤♦♠❡♥s✲❤♦r❛❀ ✐st♦ é ❞✮ ❆ ❝♦♠✉♥✐❞❛❞❡ ✐♥t❡✐r❛ r❡♣r❡s❡♥t❛ ♦ 100%❀ ❧♦❣♦ x = 100✱ ❡ f (100) = ❙ã♦ ♥❡❝❡ssár✐♦s 300 ❤♦♠❡♥s✳ ❡✮ P❛r❛ ❝❛❧❝✉❧❛r x q✉❛♥❞♦ f (x) = 100 t❡♠♦s✱ 100 = x = 40✳ ❘❡❝❡❜❡✉ ♦ 40% ❞❛ ♣♦♣✉❧❛çã♦✳ ✽✺ 900x 400 − x ⇒ (900)(100) = 300✳ 400 − 100 (400 − x) = 9x ⇒ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✷✳✷✾✳ A✱ ❈❡rt♦ ❇❛♥❝♦ ❖✉tr♦ ❇❛♥❝♦ B $30.00 ♣♦r t❛❧ã♦ ❞❡ ❝❤❡q✉❡s ❡ ❘$5.00 ♣❛r❛ ❝❛❞❛ ❝❤❡q✉❡ ✉s❛❞♦✳ ❘$10.00 ♣♦r t❛❧ã♦ ❞❡ ❝❤❡q✉❡ ❡ ❘$9.00 ♣❛r❛ ❝❤❡q✉❡ ✉s❛❞♦✳ ❈❛❧❝✉❧❛r ❝♦❜r❛ ❘ ❝♦❜r❛ ♦✉ ❝r✐tér✐♦ ♣❛r❛ ❞❡❝✐❞✐r ❡♠ q✉❡ ❇❛♥❝♦ ✈♦❝ê ❛❜r✐rá s✉❛ ❝♦♥t❛✳ ❙♦❧✉çã♦✳ ❙✉♣♦♥❤❛ s❡❥❛♠ ✉s❛❞❛s x ❢♦❧❤❛s ❞❡ ❝❤❡q✉❡✱ ❡♥tã♦ t❡♠♦s✿ ●❛st♦s ♥♦ ❇❛♥❝♦ A : R$30.00 + (R$5.00)x. ●❛st♦s ♥♦ ❇❛♥❝♦ B : R$10.00 + (R$9.00)x✳ ❋❛③❡♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ R$30.00 + (R$5.00)x < R$10.00 + (R$9.00)x t❡♠♦s 5 < x ✐st♦ s✐❣♥✐✜❝❛ q✉❡ s❡ ✉s❛♠♦s ♠❛✐s ❞❡ 5 ❢♦❧❤❛s é ♠❡❧❤♦r ♦s s❡r✈✐ç♦s ❞♦ ❇❛♥❝♦ A❀ s❡ ✉s❛♠♦s x = 5 ❢♦❧❤❛s ♥ã♦ ❢❛③ ❞✐❢❡r❡♥ç❛ ❡ s❡ ✉s❛♠♦s ♠❡♥♦s ❞❡ 5 ❢♦❧❤❛s é ♠❡❧❤♦r ♦ ❇❛♥❝♦ B ✳ ❊①❡♠♣❧♦ ✷✳✸✵✳ f (x) = kx + b ❡ ♦s ♥ú♠❡r♦s a1 , a2 ❡ a3 ❝♦♥st✐t✉❡♠ ✉♠❛ ♣r♦❣r❡ssã♦ ♥ú♠❡r♦s f (a1 ), f (a2 ) ❡ f (a3 ) t❛♠❜é♠ ❝♦♥st✐t✉❡♠ ✉♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠é✲ ▼♦str❡ q✉❡✱ s❡ ❛r✐t♠ét✐❝❛✱ ♦s t✐❝❛✳ ❙♦❧✉çã♦✳ ❙✉♣♦♥❤❛♠♦s a1 = a − r, a2 = a, ❡ a3 = a + r ❡♥tã♦ f (a1 ) = f (a − r), f (a2 ) = f (r), ❡ f (a3 ) = f (a + r)✱ ❧♦❣♦✿ f (a1 ) = k(a − r) + b = (ka + b) − kr❀ f (a2 ) = kr + b = (kr + b)❀ f (a3 ) = k(a + r) + b = (ka + b) + kr✳ P♦rt❛♥t♦ ♦s ♥ú♠❡r♦s f (a1 ), f (a2 ), ❡ f (a3 ) ❝♦♥st✐t✉❡♠ ✉♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛ ❞❡ r❛③ã♦ kr✳ ❊①❡♠♣❧♦ ✷✳✸✶✳ 24π ❝❡♥tí♠❡tr♦s ❝ú❜✐❝♦s✳ ❖ ♠❡t❛❧ ✉t✐❧✐③❛❞♦ ♣❛r❛ 2 ❛ t❛♠♣❛ ❡ ♣❛r❛ ❛ ❜❛s❡ ❝✉st❛ ❘$3, 00 ♣♦r cm ❡ ♦ ♠❛t❡r✐❛❧ ❡♠♣r❡❣❛❞♦ ♥❛ ♣❛rt❡ ❧❛t❡r❛❧ ❝✉st❛ 2 ❘$2, 00 ♣♦r cm ✳ ❈❛❧❝✉❧❛r ♦ ❝✉st♦ ❞❡ ♣r♦❞✉çã♦ ❞❛ ❧❛t❛ ❡♠ ❢✉♥çã♦ ❞❡ s❡✉ r❛✐♦✳ ❖ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❧❛t❛ ❝✐❧í♥❞r✐❝❛ é ❞❡ ❙♦❧✉çã♦✳ ❙✉♣♦♥❤❛ ♦ r❛✐♦ r ❞❛ ❜❛s❡ ❡ h ❛ ❛❧t✉r❛✱ ❧♦❣♦ s❡✉ ✈♦❧✉♠❡ é ❞❛❞♦ ♣♦r πr2 h ❡ ❞❛ ❝♦♥❞✐çã♦ 24 ❞♦ ♣r♦❜❧❡♠❛ r❡s✉❧t❛ 24π = πr2 h ♦♥❞❡ h = 2 ✳ r ❆ ár❡❛ t♦t❛❧ ❞♦ ❝✐❧✐♥❞r♦ é ❞❛❞❛ ♣❡❧❛ ❡①♣r❡ssã♦✿ ár❡❛ t♦t❛❧ ❂ ✷✭ár❡❛ ❞❛ ❜❛s❡✮ ✰ ✭ár❡❛ ❧❛t❡r❛❧✮ ✽✻ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ P♦r ♦✉tr♦ ❧❛❞♦✱ s❛❜❡♠♦s q✉❡✿ ár❡❛ ❞❛ ❜❛s❡ ❂ = 2πrh = 2πr ár❡❛ ❧❛t❡r❛❧ ❙❡❥❛ C(r) πr2 ❡ 48 24 = π 2 r r ♦ ❝✉st♦ ❞❡ ♣r♦❞✉çã♦❀ ❡♥tã♦✿ C(r) = (❘$3, 00).2(ár❡❛ ❞❛ ❜❛s❡) + (❘$2, 00) · (ár❡❛ ❧❛t❡r❛❧) = (❘$6, 00) · (πr2 ) + (❘$2, 00) · ( 48 π) r ✐st♦ é C(r) = 6πr2 + = 96 π r❡❛✐s r ❊①❡♠♣❧♦ ✷✳✸✷✳ ❯♠ ❢❛❜r✐❝❛♥t❡ ❞❡ ♣❛♥❡❧❛s ♣♦❞❡ ♣r♦❞✉③✐r ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♣❛♥❡❧❛ ❛ ✉♠ ❝✉st♦ ❞❡ ❘$10 ♣♦r ✉♥✐❞❛❞❡✳ ❊st❛ ❡st✐♠❛❞♦ q✉❡ s❡ ♦ ♣r❡ç♦ ❞❡ ✈❡♥❞❛ ❢♦r ❞❡ ❞❡ ♣❛♥❡❧❛s ✈❡♥❞✐❞♦s ♣♦r ♠ês s❡rí❛ x✳ ❜✮ ❯t✐❧✐③❡ ❘$35 ❝❛❞❛✳ ❝♦♠♦ ❢✉♥çã♦ ❞❡ ❞❡ ✈❡♥❞❛ ❢♦r (300 − x)✳ ❛✮ x ❝❛❞❛ ♣❛♥❡❧❛✱ ❡♥tã♦ ♦ ♥ú♠❡r♦ ❡①♣r❡ss❡ ♦ ❧✉❝r♦ ♠❡♥s❛❧ ❞♦ ❢❛❜r✐❝❛♥t❡ ♦ r❡s✉❧t❛❞♦ ❞❛ ♣❛rt❡ ❛✮ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ ❧✉❝r♦ s❡ ♦ ♣r❡ç♦ ❙♦❧✉çã♦✳ ❛✮ ❖ ❧✉❝r♦ ♣♦❞❡♠♦s ♦❜t❡r s✉❜tr❛✐♥❞♦ ❞❛ r❡❝❡✐t❛ t♦t❛❧ R(x)✱ ♦ ❝✉st♦ t♦t❛❧ C(x)❀ ✐st♦ é✿ R(x) = x(300−x) ❡ ❝✉st♦ t♦t❛❧ C(x) = 10(300−x)❀ ❧♦❣♦ ♦ ❧✉❝r♦ ♠❡♥s❛❧ L(x) = x(300 − x) − 10(300 − x) = (x − 10)(300 − x)✳ r❡❝❡✐t❛ t♦t❛❧ L(x)✱ é ❜✮ ◗✉❛♥❞♦ x = 35 r❡❛✐s✱ ♦ ❧✉❝r♦ L(35) = 6.625 r❡❛✐s✳ ❊①❡♠♣❧♦ ✷✳✸✸✳ ❊①♣r❡ss❛r ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❢✉♥❝✐♦♥❛❧ ❝♦♠♣r✐♠❡♥t♦ x f (x) ❡♥tr❡ ♦ ❝❛t❡t♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❡ ♦ ❞♦ ♦✉tr♦ ❝❛t❡t♦✱ s❡♥❞♦ ❛ ❤✐♣♦t❡♥✉s❛ ❝♦♥st❛♥t❡ ✐❣✉❛❧ ❛ 5✳ ❙♦❧✉çã♦✳ ❆♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ t❡♠♦s ❞❛ ❋✐❣✉r❛ ✭✷✳✶✸✮✿ 2 2 AC = AB + BC 2 2 2 BC = AC − AB ✱ ✐st♦ √ ❆ss✐♠✱ f (x) = 25 − x2 ✳ ❧♦❣♦✱ é 2 BC = 52 − x2 2 ♦♥❞❡✿ BC = √ 25 − x2 ✳ ❊①❡♠♣❧♦ ✷✳✸✹✳ ❊①♣r❡ss❛r ❛ ár❡❛ ❞❡ ✉♠ tr❛♣é③✐♦ ✐sós❝❡❧❡s ❞❡ ❜❛s❡ ❜❛s❡ a ❡ b ❝♦♠♦ ❢✉♥çã♦ ❞♦ â♥❣✉❧♦ α ❞❛ a✳ ❙♦❧✉çã♦✳✳ ✽✼ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 5✑ ✑ ✑ ✑ ✑ ✑✑ C B α x C ❅ ❅ ❅ ✑ ✑ A R ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ B ❅ ❅ E A ❋✐❣✉r❛ ✷✳✶✸✿ D ❋✐❣✉r❛ ✷✳✶✹✿ P❡❧♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ ❛ ❛❧t✉r❛ ❞♦ tr❛♣é③✐♦ ❞❛ ❋✐❣✉r❛ ✭✷✳✶✹✮ é BE ✱ ❞❛ ❞❡✜♥✐çã♦ ❞❛ t❛♥❣❡♥t❡ ❞❡ ✉♠ â♥❣✉❧♦✱ t❡♠♦s q✉❡✱ tan α = BE ❀ ❞♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛ ✈❡♠ q✉❡ AE a−b ✳ AD = a ❡ BC = b✱ ❧♦❣♦ AE = 2       a+b AD + BC a−b × BE = · tan α ➪r❡❛ ❞♦ tr❛♣é③✐♦ ❂ 2 2 2  2  a − b2 P♦rt❛♥t♦✱ ❛ ár❡❛ ❞♦ tr❛♣é③✐♦ é✿ f (α) = tan α✳ 4 ❊①❡♠♣❧♦ ✷✳✸✺✳ ❊①♣r❡ss❛r ❛ ár❡❛ ❞❡ ✉♠ tr❛♣é③✐♦ ✐sós❝❡❧❡s ❞❡ ❜❛s❡ ❜❛s❡ a ❡ b ❝♦♠♦ ❢✉♥çã♦ ❞♦ â♥❣✉❧♦ α ❞❛ a✳ ❙♦❧✉çã♦✳✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ ❛ ❛❧t✉r❛ ❞♦ tr❛♣é③✐♦ ❞❛ ❋✐❣✉r❛ ✭✷✳✶✹✮ é BE ✱ ❞❛ ❞❡✜♥✐çã♦ ❞❛ t❛♥❣❡♥t❡ ❞❡ ✉♠ â♥❣✉❧♦✱ t❡♠♦s q✉❡✱ tan α = BE ❀ ❞♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛ ✈❡♠ q✉❡ AE a−b ✳ AD = a ❡ BC = b✱ ❧♦❣♦ AE = 2       a−b a+b AD + BC × BE = · tan α ➪r❡❛ ❞♦ tr❛♣é③✐♦ ❂ 2 2 2  2  a − b2 P♦rt❛♥t♦✱ ❛ ár❡❛ ❞♦ tr❛♣é③✐♦ é✿ f (α) = tan α✳ 4 ✽✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✷✲✷ 1 ✐♥t❡r♣r❡t❛r ♦ s❡❣✉✐♥t❡✿ 1+x f (f (x)) ✷✳ f (cx) ✸✳ ✶✳ ❙❡❥❛ f (x) = ✶✳ f (x + y) ✹✳ f (x) + f (y) ✺✳ ❉❡t❡r♠✐♥❡ ♥ú♠❡r♦s c ❞❡ ♠♦❞♦ q✉❡ ❡①✐st❛♠ x t❛✐s q✉❡ f (cx) = f (x)✳ ✻✳ ❉❡t❡r♠✐♥❡ ♥ú♠❡r♦s c✱ t❛✐s q✉❡ f (cx) = f (x) ♣❛r❛ ✈❛❧♦r❡s ❞✐st✐♥t♦s ❞❛ ✈❛r✐á✈❡❧ x✳ ✷✳ ❉❡t❡r♠✐♥❡ ♦ ❞♦♠í♥✐♦ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ ✶✳ ✹✳ p √ √ ✷✳ g(x) = 1 − 1 − x f (x) = 1 − x √ √ f (x) = 1 − x2 + x2 − 1 ✸✳ ✺✳ 1 1 + h(x) = x−1 x−2 √ √ h(x) = 1 − x + x − 2 ✸✳ ❈❛❧❝✉❧❛r f (a) ♣❛r❛ ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ ✶✳ ✸✳ ✺✳ f (x) = x2 + 6x − 2 a = −2 ✷✳ √ 1 5x2 + 11 a=− 3   | x − 2 | , s❡✱ x 6= 2 f (x) = x−2  1, s❡✱ x = 2 f (x) = ✹✳ x+1 a=0 3 − x5 3x2 − 2x − 1 f (x) = 3 a=1 2x − 5x + 1 f (x) = a = −2 ✹✳ ❉❡s❡♥❤❛r ♦ ❣rá✜❝♦ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ 1. g(x) = f (x) + c 2. g(x) = f (x + c) 3. g(x) = c.f (x) 4. g(x) = f (cx) 5. g(x) = f (1/x) 6. g(x) = f (| x |) 7. g(x) = min .{f (x), 0} 8. g(x) = max .{f (x), 0} ✺✳ ❙❡❥❛♠ ♦s ❝♦♥❥✉♥t♦s A = [1, 4], B = [−1, 1] ❡ C = [−3, 1] ❡ ❝♦♥s✐❞❡r❡ ❛s ❢✉♥çõ❡s f : A −→ R, g : B −→ R ❡ h : C −→ R✱ ❛ss✐♠ ❞❡✜♥✐❞❛s✿ ❛ ❝❛❞❛ ♥ú♠❡r♦ x ❝♦rr❡s♣♦♥❞❡ s❡✉ q✉❛❞r❛❞♦ x2 ✳ ◗✉❛✐s ❞❛s ❢✉♥çõ❡s sã♦ ✐♥❥❡t♦r❛s❄ ✻✳ ❆ ❢✉♥çã♦ ❝♦♥st❛♥t❡ f (x) = k ✱ ♣♦❞❡ s❡r ✐♥❥❡t✐✈❛❄ ❊✱ s♦❜r❡❥❡t✐✈❛❄ ✼✳ ❙❛❜❡✲s❡ q✉❡ −2 ❡ 3 sã♦ r❛í③❡s ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✳ ❙❡ ♦ ♣♦♥t♦ (−1, 8) ♣❡rt❡♥❝❡ ❛♦ ❣rá✜❝♦ ❞❡ss❛ ❢✉♥çã♦✱ ❡♥tã♦✿ ✽✳ ◆✉♠ ❝✐r❝✉✐t♦ ❛ t❡♥sã♦ ✈á ❞❡❝r❡s❝❡♥❞♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ✭❝♦♥❢♦r♠❡ ❛ ❧❡✐ ❧✐♥❡❛r✮✳ ❆♦ ✐♥✐❝✐♦ ❞♦ ❡①♣❡r✐♠❡♥t♦ ❛ t❡♥sã♦ ❡r❛ ✐❣✉❛❧ ❛ 12V ❡ ❛♦ ✜♥❛❧ ❞♦ ♠❡s♠♦ ❡①♣❡r✐♠❡♥t♦✱ ✽✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R q✉❡ ❞✉r♦ 8sg ✱ ❛ t❡♥sã♦ ❜❛✐①♦ ❛té 6, 4V ✳ ❊①♣r❡ss❛r ❛ t❡♥sã♦ V ❝♦♠♦ ❢✉♥çã♦ ❞♦ t❡♠♣♦ t✳ ✾✳ ❯♠❛ ❡s❢❡r❛ ❞❡ r❛✐♦ R t❡♠ ✐♥s❝r✐t♦ ✉♠ ❝♦♥❡ r❡t♦✳ ❆❝❤❛r ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❢✉♥❝✐♦♥❛❧ ❡♥tr❡ ❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❧❛t❡r❛❧ S ❞♦ ❝♦♥❡ ❡ s✉❛ ❣❡r❛tr✐③ x✳ ■♥❞✐❝❛r ♦ ❞♦♠í♥✐♦ ❞❡ ❞❡✜♥✐çã♦ ❞❡ ❡st❛ ❢✉♥çã♦✳ ✶✵✳ ❈❡rt❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❣ás ♦❝✉♣♦ ♦ ✈♦❧✉♠❡ ❞❡ 107cm3 à t❡♠♣❡r❛t✉r❛ ❞❡ 20o C ❀ ♣❛r❛ ✉♠❛ t❡♠♣❡r❛t✉r❛ ❞❡ 40o C ♦ ✈♦❧✉♠❡ ❝❤❡❣♦✉ ❛ s❡r ✐❣✉❛❧ ❛ 114cm3 ✿ ❆♣❧✐❝❛♥❞♦ ❛ ❧❡✐ ❞❡ ●❛②✲▲✉ss❛❝ ❢♦r♠❛r ❛ ❡q✉❛çã♦ q✉❡ ❡①♣r❡ss❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❡♥tr❡ ♦ ✈♦❧✉♠❡ V ❞♦ ❣ás ❡ ❛ t❡♠♣❡r❛t✉r❛ T o C ✳ ✶✳ ✷✳ ◗✉❛❧ s❡r✐❛ ♦ ✈♦❧✉♠❡ ❛ 0o C ❄ ✶✶✳ ❖ ❞♦♥♦ ❞❡ ✉♠ r❡st❛✉r❛♥t❡ r❡s♦❧✈❡✉ ♠♦❞✐✜❝❛r ♦ t✐♣♦ ❞❡ ❝♦❜r❛♥ç❛✱ ♠✐st✉r❛♥❞♦ ♦ s✐st❡♠❛ ❛ q✉✐❧♦ ❝♦♠ ♦ ♣r❡ç♦ ✜①♦✳ ❊❧❡ ✐♥st✐t✉♦ ♦ s❡❣✉✐♥t❡ s✐st❡♠❛ ❞❡ ♣r❡ç♦ ♣❛r❛ ❛s r❡❢❡✐çõ❡s✿ ❆té 300g R$3.00 ♣♦r r❡❢❡✐çã♦ ❊♥tr❡ 300g ❡ 1kg R$10.00 ♣♦r q✉✐❧♦ ❆❝✐♠❛ ❞❡ 1kg R$10.00 ♣♦r r❡❢❡✐çã♦ ❘❡♣r❡s❡♥t❛r ❣r❛✜❝❛♠❡♥t❡ ♦ ♣r❡ç♦ ❞❛s r❡❢❡✐çõ❡s ♥❡ss❡ r❡st❛✉r❛♥t❡✳ ✶✷✳ ❆ ♠❡❞✐❞❛ ❞❛ t❡♠♣❡r❛t✉r❛ ❡♠ ❣r❛✉s ❋❛❤r❡♥❤❡✐t é ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r ❞❛ ♠❡❞✐❞❛ ❡♠ ❣r❛✉s ❝❡♥tí❣r❛❞♦s✿ ✶✳ ❊s❝r❡✈❡r ❛ ❡q✉❛çã♦ ❞❡ ❡st❛ ❢✉♥çã♦ ✭❧❡♠❜r❡ q✉❡ 0o C = 32o F ❡ 100o C = 212o F ✮✳ ❯t✐❧✐③❛r ❛ ❢✉♥çã♦ ♦❜t✐❞❛ ♥♦ ✐t❡♠ ❛♥t❡r✐♦r ♣❛r❛ tr❛♥s❢♦r♠❛r 15o C ❛ ❣r❛✉s ❋❛❤r❡✲ ♥❤❡✐t✳ ✷✳ ✶✸✳ ❖ ✈❛❧♦r ❞❛ ❢✉♥çã♦ ❞❡ ❛r❣✉♠❡♥t♦ ✐♥t❡✐r♦ u = f (n) é ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❞✐✈✐s♦r❡s ✐♥t❡✐r♦s ❞♦ ❛r❣✉♠❡♥t♦ n ❞✐st✐♥t♦s ❞❡ 1 ❡ ❞♦ ♠❡s♠♦ n✳ ❋♦r♠❛r ❛ t❛❜❡❧❛ ❞♦s ✈❛❧♦r❡s ❞❡ u ♣❛r❛ 1 ≤ n ≤ 18✳ ✶✹✳ ❯♠❛ ❜♦❧❛ ❢♦✐ ❛❜❛♥❞♦♥❛❞❛ ❞♦ t❡t♦ ❞❡ ✉♠ ❡❞✐❢í❝✐♦✳ ❆ ❛❧t✉r❛ ❞❛ ❜♦❧❛ ❡♠ ♠❡tr♦s ❞❡♣♦✐s ❞❡ t s❡❣✉♥❞♦s é ❞❛❞❛ ♣❡❧❛ ❢✉♥çã♦ H(t) = −16t2 + 256✳ ✶✳ ❊♠ q✉❡ ❛❧t✉r❛ ❡st❛rá ❛ ❜♦❧❛ ❞❡♣♦✐s ❞❡ 2 s❡❣✉♥❞♦s ❄ ✷✳ ◗✉❡ ❞✐stâ♥❝✐❛ t❡rá r❡❝♦rr✐❞♦ ❛ ❜♦❧❛ ♥♦ 3o s❡❣✉♥❞♦ ❄ ✸✳ ◗✉❛❧ é ❛ ❛❧t✉r❛ ❞♦ ❡❞✐❢í❝✐♦ ❄ ✹✳ ❉❡♣♦✐s ❞❡ q✉❛♥t♦s s❡❣✉♥❞♦s ❛ ❜♦❧❛ ❝❤❡❣❛rá ❛♦ s♦❧♦ ❄ ✾✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✳✹ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❋✉♥çõ❡s ❡s♣❡❝✐❛✐s ✷✳✹✳✶ ❋✉♥çã♦ ❛✜♠ f : R −→ R é ❛q✉❡❧❛ ❞❡✜♥✐❞❛ ♣♦r f (x) = ax + b ∀ x ∈ R✱ ♦♥❞❡ a ❡ b sã♦ ❝♦♥st❛♥t❡s r❡❛✐s ♥ã♦ ♥✉❧❛s❀ ♦ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ D(f ) = R ❡ ❛ ✐♠❛❣❡♠ Im(f ) = R❀ s❡✉ ❣rá✜❝♦ é ✉♠❛ r❡t❛ ♦❜❧íq✉❛ ❛♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ✭❡✐①♦✲x✮ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✭✷✳✶✺✮❀ ❡❧❛ b ✐♥t❡r❝❡♣t❛ ♦ ❡✐①♦✲y ♥♦ ♣♦♥t♦ (0, b) ❡ ♦ ❡✐①♦✲x ♥♦ ♣♦♥t♦ (− , 0)✳ a ❋✉♥çã♦ ❛✜♠ ❊①❡♠♣❧♦ ✷✳✸✻✳ ❆ ❢✉♥çã♦ f (x) = 3x + 5 é ✉♠❛ ❢✉♥çã♦ ❛✜♠✱ s❡✉ ❞♦♠í♥✐♦ D(f ) = R ❡ s✉❛ ✐♠❛❣❡♠ ♦ ❝♦♥❥✉♥t♦ Im(f ) = R✳ y✻ y✻ ✒ 4 4 3 −x ✛ · · ·−3 −2 −1 0 2 ✲ 1 −x ✛ x ✲ 1 f (x) = b 2 2 1 3 ✛ f (x) = ax + b · · ·−3 −2 −1 3··· 0 x ✲ 1 2 3··· ✠ −y ❄ −y ❄ ❋✐❣✉r❛ ✷✳✶✻✿ ❋✐❣✉r❛ ✷✳✶✺✿ ❊①❡♠♣❧♦ ✷✳✸✼✳ x2 − 9 é ✉♠❛ ❢♦r♠❛ ❞✐s❢❛rç❛❞❛ ❞❛ ❢✉♥çã♦ ❛✜♠ g(x) = x + 3✳ x−3 ❙❡✉ ❞♦♠í♥✐♦ é D(f ) = R − {3} ❡ Im(f ) = R − {6}✳ ❆ ❢✉♥çã♦ f (x) = ✷✳✹✳✷ ❋✉♥çã♦ ❝♦♥st❛♥t❡ a = 0 ❡♥tã♦ ❛ ❢✉♥çã♦ f : R −→ R é ❝❤❛♠❛❞❛ ✏ ❢✉♥çã♦ f (x) = b ∀ x ∈ R✱ ♦♥❞❡ b é ✉♠ ♥ú♠❡r♦ r❡❛❧ ❝♦♥st❛♥t❡✳ ◗✉❛♥❞♦✱ ♥❛ ❢✉♥çã♦ ❛✜♠✱ t❡♠♦s ❝♦♥st❛♥t❡ ✑ s❡♥❞♦ ❞❡✜♥✐❞❛ ♣♦r ❖ ❞♦♠í♥✐♦ ❛ ❋✐❣✉r❛ D(f ) = R ✭✷✳✶✻✮✳ ❡ Im(f ) = { b } ❖❜s❡r✈❡✱ ❛ ❢✉♥çã♦ ❛ss♦❝✐❛ ❛ t♦❞♦ ❡ ♦ ❣rá✜❝♦ é ✉♠❛ r❡t❛ ❤♦r✐③♦♥t❛❧ ❝♦♠♦ ♠♦str❛ x∈R ✉♠ ♠❡s♠♦ ♥ú♠❡r♦ r❡❛❧ b✳ ❊①❡♠♣❧♦ ✷✳✸✽✳ ❙❡❥❛ y = f (x) ♦♥❞❡ f (x) = 5✱ ❡♥tã♦ y = 5 r❡♣r❡s❡♥t❛ ❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡✱ é ✉♠❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ❛ ❝✐♥❝♦ ✉♥✐❞❛❞❡s ❞❡ ❞✐stâ♥❝✐❛ s✉♣❡r✐♦r♠❡♥t❡✳ ✾✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✳✹✳✸ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ❋✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ ❡♠ R R ◗✉❛♥❞♦✱ ♥❛ ❢✉♥çã♦ ❛✜♠✱ t❡♠♦s a = 1 ❡ b = 0✱ r❡s✉❧t❛ ❛ ❢✉♥çã♦ f : R −→ R ❡✱ é ❝❤❛♠❛❞❛ ✏ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ ✑ ❞❡✜♥✐❞❛ ♣♦r f (x) = x ∀ x ∈ R✳ ❖ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ D(f ) = R ❡ ❛ ✐♠❛❣❡♠ Im(f ) = R❀ ♦ ❣rá✜❝♦ é ✉♠❛ r❡t❛ ♦❜❧íq✉❛✱ q✉❡ ❢❛③ â♥❣✉❧♦ ❞❡ 45o ❝♦♠ ♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s✱ ✐st♦ é✱ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ é ✉♠❛ r❡t❛ q✉❡ ❝♦♥tê♠ ❛s ❜✐ss❡tr✐③❡s ❞♦ 1o ❡ 3o q✉❛❞r❛♥t❡s ❡ q✉❡ ♣❛ss❛ ♣❡❧❛ ♦r✐❣❡♠✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✭✷✳✶✼✮✳ y✻ y✻ 4 3 2 · · ·−3 −2 −1 ✠ ✷✳✹✳✹ 0 3 −x ✛ x ✲ 1 2 f (x) = x 2 f (x) = x 1 −x ✛ 4 ✒ ✏ · · ·−3 −2 ✏ −1✏ 3··· ✏ ✮✏ ✏ ✏✏ ✏✏ 1 0 1 −y ❄ −y ❄ ❋✐❣✉r❛ ✷✳✶✼✿ ❋✐❣✉r❛ ✷✳✶✽✿ 2 ✶ ✏ ✏✏ x ✲ 3··· ❋✉♥çã♦ ❧✐♥❡❛r ❙❡✱ ♥❛ ❢✉♥çã♦ ❛✜♠ ❛ ❝♦♥st❛♥t❡ b = 0✱ t❡♠✲s❡ ❛ ❢✉♥çã♦ f : R −→ R ❞❡✜♥✐❞❛ ♣♦r f (x) = ax ∀ x ∈ R ❡ ❝❤❛♠❛❞❛ ✏ ❢✉♥çã♦ ❧✐♥❡❛r ✑❀ ♦ ❞♦♠í♥✐♦ D(f ) = R ❡ Im(f ) = R✱ s❡✉ ❣rá✜❝♦ é ✉♠❛ r❡t❛ ♦❜❧íq✉❛ q✉❡ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❢❛③ â♥❣✉❧♦ ❞❡ 45o ❣r❛✉s ❝♦♠ ♦ ❡✐①♦✲x✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✭✷✳✶✽✮✳ ➱ ✉♠❛ r❡t❛ q✉❡ ♥ã♦ é ♣❛r❛❧❡❧❛ ❛ ♥❡♥❤✉♠ ❞♦s ❡✐①♦s❀ ♦ ♥ú♠❡r♦ a 6= 0 é ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❡ss❛ r❡t❛✳ y = a é ❝♦♥st❛♥t❡✳ ❊①✲ ❊st❛ ❢✉♥çã♦ q✉❡ ❡st❛❜❡❧❡❝❡ ❡♥tr❡ x ❡ y ✉♠❛ r❡❧❛çã♦ t❛❧ q✉❡ x ♣r❡ss❛♠♦s ❛ r❡❧❛çã♦ ♣♦r y = a · x✱ ♦♥❞❡ ✏ a✑ ❝♦♥st❛♥t❡✱ ❞✐③❡♠♦s q✉❡ ❛ ✈❛r✐❛çã♦ ❞❡ ✁✁y ✑ é ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛ ✈❛r✐❛çã♦ ❞❡ ✏ x✑✳ ❙❡❥❛♠ x1 , x2 ∈ D(f ), ❆❞✐t✐✈✐❞❛❞❡✿ a, b ∈ R✱ ❛ ❢✉♥çã♦ ❧✐♥❡❛r ♣♦ss✉✐ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ f (x1 + x2 ) = f (x1 ) + f (x2 )❀ ❍♦♠♦❣❡♥❡✐❞❛❞❡✿ f (a · x1 ) = a · f (x1 )✳ f (a · x1 + b · x2 ) = a · f (x1 ) + b · f (x2 ) ❆ ❢✉♥çã♦ ❛✜♠ f (x) = ax + b✱ ♦♥❞❡ a ❡ b sã♦ ❝♦♥st❛♥t❡s✱ é ❛ ❡q✉❛çã♦ ❞❡ ✉♠❛ r❡t❛ ♥♦ ♣❧❛♥♦ R2 ❀ s❡✉ ❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ sã♦ t♦❞♦s ♦s ♥ú♠❡r♦s r❡❛✐s s❛❧✈♦ ❛❧❣✉♠❛ r❡str✐çã♦✱ ❡ ♥ã♦ s❛t✐s❢❛③ ❡st❛s ❞✉❛s ú❧t✐♠❛s ♣r♦♣r✐❡❞❛❞❡s✳ ✾✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✳✹✳✺ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊q✉❛çã♦ ❞❡ ✉♠❛ r❡t❛ ❊①✐st❡♠ s✐t✉❛çõ❡s ♥❛s q✉❛✐s ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ❝♦♠ r❡❧❛çã♦ ❛ ♦✉tr❛ é ❝♦♥st❛♥t❡✳ P♦r ❡①❡♠♣❧♦✱ s✉♣♦♥❤❛♠♦s q✉❡ ♣❛r❛ ❢❛❜r✐❝❛r ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣r♦❞✉t♦ t❡♥❤❛♠♦s ❛ ♣❛❣❛r ❘$20, 00✱ ❛❧é♠ ❞❡ ✉♠❛ ❞❡s♣❡s❛ ✜①❛ s❡♠❛♥❛❧ ❞❡ ❘$300, 00✳ s❡ x ✉♥✐❞❛❞❡s ❢♦r❡♠ ♣r♦❞✉③✐❞❛s ♣♦r s❡♠❛♥❛ ❡ ❢❛❜r✐❝❛♥t❡❀ ❡♥tã♦ y ❊♥tã♦ r❡❛✐s ❢♦r ♦ ❝✉st♦ t♦t❛❧ s❡♠❛♥❛❧ ♣❛r❛ ♦ y = 20x + 300✳ ❙♦❧✉çõ❡s ♣❛r❛ ❡st❛ ❡q✉❛çã♦ sã♦ ❞❛❞❛s ♥❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✿ x y = 20x + 300 0 300 10 500 20 700 30 900 40 1500 ❆ r❡❧❛çã♦ ❞❛❞❛ ♥♦ ❡①❡♠♣❧♦ ♣r❡❝❡❞❡♥t❡ r❡♣r❡✲ s❡♥t❛ ❛ ❡q✉❛çã♦ ❞❡ ✉♠❛ r❡t❛❀ ❡♠ ❣❡r❛❧✱ ❞❛❞♦s ❞♦✐s ♣♦♥t♦s P (x1 , y1 ) ❡ Q(x2 , y2 ) ❞❡ ✉♠❛ r❡t❛✱ ♣❛r❛ ❞❡✲ R t❡r♠✐♥❛r s✉❛ ❡q✉❛çã♦ ♥♦ ♣❧❛♥♦ 2 P (x1 , y1 ) ✳q ✳ ✳✳ ✳ ✳✳ (x, y) • ✳ ✳✳ q · · · · · ·✳✳q· · · Q(x2 , y2 ) R(x1 , y2 ) ✻y y1 ♣r♦❝❡❞❡♠♦s ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿ ❈♦♥s✐❞❡r❡ ♦s ♣♦♥t♦s tr✐â♥❣✉❧♦ P RQ P (x1 , y1 ) ❝♦♠♦ ♠♦str❛ ❛ ❡ Q(x2 , y2 ) ❋✐❣✉r❛ ❞♦ ✭✷✳✶✾✮✳ y2 ✛ [ t❛♥❣❡♥t❡ ❞♦ â♥❣✉❧♦ P QR é ❞❛❞❛ ♣♦r✿ y − y 1 2 ❡st❡ ✈❛❧♦r ❞❛ t❛♥❣❡♥t❡ é ❞❡✲ tan(P[ QR) = x1 − x2 ♥♦♠✐♥❛❞♦ ❵❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ q✉❡ ♣❛ss❛ y1 − y2 ♣❡❧♦s ♣♦♥t♦s P Q✑ ❀ ❡ ❞❡♥♦t❛❞❛ ♣♦r✿ m = ✳ x1 − x2 ❙❡ (x, y) é ✉♠ ♣♦♥t♦ q✉❛✐sq✉❡r ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r ❆ ✲ −x 0 x2 x1 x −y ❄ ❵ ♣❛r❛ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ t❡♠♦s q✉❡✿ L✱ q✉❡ L : y − y1 = m(x − x1 )✳ P♦rt❛♥t♦✱ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ ♣❡❧❛ ❢ór♠✉❧❛✿ y1 − y2 y − y1 = x − x1 x1 − x2 ❋✐❣✉r❛ ✷✳✶✾✿ P ❡ ✐st♦ é ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s Q✱ ❞❛s r❡❧❛çõ❡s ❣❡♦♠étr✐❝❛s y − y1 = m(x − x1 )✳ P (x1 , y1 ) ❡ Q(x2 , y2 ) é ❞❛❞❛ ❊①❡♠♣❧♦ ✷✳✸✾✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s P (−2, −5) ❡ Q(4, 3)✳ ❙♦❧✉çã♦✳ ❚❡♠♦s q✉❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r 8 y − 3 = (x − 4)✳ 6 m= ▲♦❣♦ ❛ ❡q✉❛çã♦ ♣❡❞✐❞❛ −5 − 3 8 = ❡ ❝♦♥s✐❞❡r❡ ♦ ♣♦♥t♦ Q(4, 3)❀ ❡♥tã♦ −2 − 4 6 é✿ 4x − 3y − 7 = 0✳ ❖❜s❡r✈❛çã♦ ✷✳✺✳ ❙✉♣♦♥❤❛ t❡♠♦s ❞✉❛s r❡t❛s L1 ❡ L2 ❞❡ ❝♦❡✜❝✐❡♥t❡s ❛♥❣✉❧❛r❡s m1 ❡ m2 ❡♥tã♦✱ ❛s ❞✉❛s r❡t❛s sã♦ ♣❛r❛❧❡❧❛s s❡ m1 = m2 ❀ ❝❛s♦ ♦ ♣r♦❞✉t♦ m1 · m2 = −1 ❡❧❛s sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s✳ ✾✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❆ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ A(a, b) ❡ B(c, d) é ❞❛❞❛ ♣❡❧❛ ❢ór♠✉❧❛ d(A, B) = ❊①❡♠♣❧♦ ✷✳✹✵✳ p (c − a)2 + (d − b)2 ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ P (2, 5) ❡ t❡♠ ❝♦♠♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r m = 3✳ ❙♦❧✉çã♦✳ ❆♣❧✐❝❛♥❞♦ ❞✐r❡t❛♠❡♥t❡ ❛ ❢ór♠✉❧❛ t❡♠♦s q✉❡✿ y − 5 = 3(x − 2)❀ ❧♦❣♦ 3x − y − 1 = 0 é ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ ♣❡❞✐❞❛✳ ❊①❡♠♣❧♦ ✷✳✹✶✳ ❉❛❞❛ ❛ r❡t❛ L1 : y = 5x − 3✱ ❞❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛✿ ❛✮ L2 q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ A(4, 9) ❡ s❡❥❛ ♣❛r❛❧❡❧❛ ❛ L1 ❀ ❜✮ L3 q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ B(−4, 6) ❡ s❡❥❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ L1 ✳ ❙♦❧✉çã♦✳ ✭❛✮ ❚❡♠♦s q✉❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❡ L1 é m1 = 5 ❧♦❣♦✱ t❡♠ q✉❡ s❡r ✐❣✉❛❧ ❛♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ L2 ✱ ❛ss✐♠ m2 = 5 ❡ L2 : y − 9 = 5(x − 4) ✐st♦ é L2 : y = 5x − 11✳ ✭❜✮ ❙❡♥❞♦ m1 = 5 ❡♥tã♦ ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❡ L3 é m3 = − ❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ L3 é y−6= x 26 −1 (x − (−4)) ✐st♦ é L3 : y = − + ✳ 5 5 5 1 5  ❖❜s❡r✈❛çã♦ ✷✳✻✳ ❆ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦s ♣♦♥t♦s P (x1 , y1 ), Q(x2 , y2 ) ❡ R(x3 , y3 ) é ❞❛❞❛ ♣❡❧♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞♦ ❞❡t❡r♠✐♥❛♥t❡✿ AP QR 1 = 2 x1 y1 1 x2 y2 1 ✳ x3 y3 1 ❊①❡♠♣❧♦ ✷✳✹✷✳ ❉❡t❡r♠✐♥❡ s❡ ♦s ♣♦♥t♦s P (2, 3), Q(7, 9) ❡ R(3, 8) ♣❡rt❡♥❝❡♠ ❛ ✉♠❛ ♠❡s♠❛ r❡t❛✳ ❙♦❧✉çã♦✳ ❖s três ♣♦♥t♦s ♣❡rt❡♥❝❡♠ ❛ ✉♠❛ ♠❡s♠❛ r❡t❛✱ s❡❀ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ❢♦r♠❛❞❛ ♣♦r ❡❧❡s é ✐❣✉❛❧ ❛ ③❡r♦✳ AP QR 1 = 2 2 3 1 7 9 1 3 8 1 19 1 = [(18 + 56 + 9) − (27 + 16 + 21)] = 2 2 ▲♦❣♦✱ ♦s três ♣♦♥t♦s ♥ã♦ ♣❡rt❡♥❝❡♠ ❛ ✉♠❛ ♠❡s♠❛ r❡t❛✳ ✾✹ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❊①❡♠♣❧♦ ✷✳✹✸✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❡ ♣❡❧♦s s❡❣✉✐♥t❡s ♣♦♥t♦s✿ ❛✮ A(3, 6) ❡ B(7, 6) ❜✮ M (5, 7) ❡ N (5, 9) ❙♦❧✉çã♦✳ ❛✮ ❖ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r é m = 6−6 = 0✱ ❛ ❡q✉❛çã♦ ♣❡❞✐❞❛ é✿ y − 6 = 0(x − 3) = 0✱ 3−7 ❡♥tã♦ y = 6✳ ➱ ✉♠❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s✳ ❜✮ 9−7 2 2 = ✱ ❛ ❡q✉❛çã♦ ♣❡❞✐❞❛ é✿ y − 9 = (x − 5)✱ ❡♥tã♦ 5−5 0 0 0(y − 9) = 2(x − 5)✱ ❧♦❣♦ 0 = 2(x − 5) ✐st♦ é x = 5✳ ➱ ✉♠❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ ❞❛s ❖ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r é m = ♦r❞❡♥❛❞❛s✳ ❊①❡♠♣❧♦ ✷✳✹✹✳ ❖s ✈ért✐❝❡s ❞❡ ✉♠ tr✐â♥❣✉❧♦ sã♦ ♦s ♣♦♥t♦s A(2, 4), B(3, −1) ❡ C(−5, 3)✳ ❉❡t❡r♠✐♥❡ ❛ ❞✐stâ♥❝✐❛ ❞♦ ♣♦♥t♦ A ❛♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛s ♠❡❞✐❛♥❛s✳ ❙♦❧✉çã♦✳ ❖s ♣♦♥t♦s ♠é❞✐♦s ❞♦s ❧❛❞♦s (−1, 1)✳ AB, AC 5 3 3 7 BC sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡✿ ( , ), (− , ) 2 2 2 2 (−1, 1) ♣❛r❛ A é ❡ ❆ ❡q✉❛çã♦ ❞❛ ♠❡❞✐❛♥❛ ❞♦ ♣♦♥t♦ y−1= 4−1 (x + 1) 2+1 ❆ ❡q✉❛çã♦ ❞❛ ♠❡❞✐❛♥❛ ❞♦ ♣♦♥t♦ y− 3 7 (− , ) 2 2 ♣❛r❛ −1 − 72 3 1 = 3 (x + ) 2 2 3+ 2 ❆ ❡q✉❛çã♦ ❞❛ ♠❡❞✐❛♥❛ ❞♦ ♣♦♥t♦ y− ⇒ 5 3 ( , ) 2 2 3 − 32 5 3 = 5 (x − ) 2 2 −5 − 2 x−y+2=0 B ⇒ ♣❛r❛ ⇒ C é x+y+1=0 é 2x + 10y − 20 = 0 ❘❡s♦❧✈❡♥❞♦ ❡st❛s três ❡q✉❛çõ❡s t❡♠♦s ❛ ✐♥t❡rs❡çã♦ ❞❛s três ♠❡❞✐❛♥❛s é ♦ ♣♦♥t♦ ❆ ❞✐stâ♥❝✐❛ ❞♦ ♣♦♥t♦ (0, 2) ♣❛r❛ ♦ ♣♦♥t♦ P♦rt❛♥t♦ ❛ ❞✐stâ♥❝✐❛ ♣r♦❝✉r❛❞❛ é ✷✳✹✳✻ √ A é 2 2✳ ❡ p √ (2 − 0)2 + (4 − 2)2 = 2 2✳ (0, 2)✳ ❋✉♥çã♦ ♠❛✐♦r ✐♥t❡✐r♦ f : R −→ R ❞❡♥♦t❛❞❛ f (x) = [|x|] ❞❡ ♠♦❞♦ q✉❡ ❛ ❝❛❞❛ ♥ú♠❡r♦ r❡❛❧ ❞♦ ✐♥t❡r✈❛❧♦ n ≤ x < n + 1 ∀ n ∈ Z ❛ss♦❝✐❛ ♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ n ❀ ✐st♦ é [|x|] = n é ♦ ♠❛✐♦r ✐♥t❡✐r♦ q✉❡ ♥ã♦ s✉♣❡r❛ ♦ ♥ú♠❡r♦ x✳ ➱ ❛ ❢✉♥çã♦ ✾✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✷ ❆ ❢✉♥çã♦ ♠❛✐♦r ✐♥t❡✐r♦ R t❛♠❜é♠ é ❝❤❛♠❛❞❛ ❝♦♠♦ ✏ ❢✉♥çã♦ ❝♦❧❝❤❡t❡ ✑✳ ❖ ❣rá✜❝♦ ♠♦str❛✲ D(f ) = R s❡ ♥❛ ❋✐❣✉r❛ ✭✷✳✷✵✮✳ ❆q✉✐✱ ❡ Im(f ) = Z ❊①❡♠♣❧♦ ✷✳✹✺✳ ❖❜s❡r✈❡✱ s❡ x y = [|x|] ✷✳✹✳✼ f (x) = [|x|] x ∈ [−2 − 1) x ∈ [−1, 0) −2 x ∈ [0, 1) −1 x ∈ [1, 2) 0 x ∈ [2, 3) 1 2 f : R −→ R Im(f ) = [0, +∞)✳ ❞❡✜♥✐❞❛ ♣♦r✿ f (x) = √ x✳ ❙❡✉ ❞♦♠í♥✐♦ · · −3 · −2 −1 ❙❡✉ ❣rá✜❝♦ ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✷✳✷✶✮✳ 4 3 2 1 −x ✛ x ✲ · · −3 · −2 −1 0 0 1 2 3· · · −y ❄ ❋✐❣✉r❛ ✷✳✷✵✿ ❋✐❣✉r❛ ✷✳✷✶✿ ❋✉♥çã♦ s✐♥❛❧ ❙❡✉ ❞♦♠í♥✐♦ D(f ) = R ❡ s✉❛ ✐♠❛❣❡♠ ❋✐❣✉r❛ ✭✷✳✷✷✮✳ Im(f ) = { −1, 0, 1 }✱ ❋✉♥çã♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❡ ❆ ❢✉♥çã♦ x✑✳ x ✲ 2 3· · · −y ❄    −1, ➱ ❛ ❢✉♥çã♦ f : R −→ R ❞❡✜♥✐❞❛ ♣♦r✿ f (x) = Sgn(x) = 0,   1, ❖❜s❡r✈❡✱ ❛ ❢✉♥çã♦ f (x) = Sgn(x) é ❢✉♥çã♦ ❝♦♥st❛♥t❡ ∀ x ∈ R✳ ❞❡ ❡ s✉❛ y✻ 4 3 2 1 −x ✛ ✷✳✹✳✾ 3 D(f ) = [0, +∞) y✻ ✷✳✹✳✽ x ∈ [3, 4) ❋✉♥çã♦ r❛✐③ q✉❛❞r❛❞❛ ➱ ❛ ❢✉♥çã♦ ✐♠❛❣❡♠ t❡♠♦s ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✿ f : R −→ R ❙❡✉ ❞♦♠í♥✐♦ é ❞❡✜♥✐❞❛ ♣♦r✿ s❡✱ s❡✱ s❡✱ x<0 x=0 x>0 ♦ ❣rá✜❝♦ ♠♦str❛✲s❡ ♥❛ x f (x) =| x | é ❝❤❛♠❛❞❛ ✏ ❢✉♥çã♦ ✈❛❧♦r ❛❜s♦❧✉t♦ D(f ) = R ❡ s✉❛ ✐♠❛❣❡♠ é Im(f ) = R+ = [0, +∞)✳ ❙❡✉ ❣rá✜❝♦ ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✷✳✷✸✮✳ ✷ ❊st❛ ❢✉♥çã♦ t❛♠❜é♠ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❢✉♥çã♦ ♣✐s♦ ❡ ❞❡♥♦t❛❞❛ ♣♦r t❛♠❡♥t♦ ❞❛ ❢✉♥çã♦ ✐♥t❡✐r♦ ♠❛✐♦r✳ ❚❛♠❜é♠ ❡①✐st❡ ❛ ❢✉♥çã♦ t❡t♦ ✾✻ f (x) = ⌊x⌋ ❝♦♠ ♦ ♠❡s♠♦ ❝♦♠♣♦r✲ g(x) = ⌈x⌉ ♦♥❞❡ k ≤ x < k + 1, k ∈ Z 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R y✻ 4 y✻ 3 2 −x ✛ · · ·−3 −2 −1 ✛ ❅ ■ ❅ ❅ ✲ 1 x ✲ 0 1 −1 −2 2 3··· −x ✛ ✒ 3 ❅ f (x) =| x | 2 ❅ 1 ❅ · · ·−3 −2 −1 ✳ ✳ ✳ x ✲ 0 1 2 3··· −y ❄ −y ❄ ❋✐❣✉r❛ ✷✳✷✸✿ ❋✐❣✉r❛ ✷✳✷✷✿ ✷✳✹✳✶✵ 4 ❋✉♥çã♦ q✉❛❞rát✐❝❛ f : R −→ R ❞❡✜♥✐❞❛ ♣♦r f (x) = ax2 + bx + c✱ ♦♥❞❡ a, b ❡ c sã♦ ❝♦♥st❛♥t❡s 2 r❡❛✐s ❝♦♠ a 6= 0❀ ♦ ❞♦♠í♥✐♦ D(f ) = R ❡ ❛ ✐♠❛❣❡♠ ✈❛r✐❛♠ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ r❡❧❛çã♦ b = 4ac✱ ➱ ❛ ❢✉♥çã♦ s❡✉ ❣rá✜❝♦ é ✉♠❛ ♣❛rá❜♦❧❛ ❡ s❡rá ❡st✉❞❛❞♦ ❡♠ ❞❡t❛❧❤❡s ♣♦st❡r✐♦r♠❡♥t❡✳ ❖ ❣rá✜❝♦ ❞❡ ✉♠❛ ♣❛rá❜♦❧❛ ❛♣r❡s❡♥t❛ ✉♠ ♣♦♥t♦ ♠❛✐s ❛❧t♦ ✭a < 0✮ ♦✉ ✉♠ ♣♦♥t♦ ♠❛✐s ❜❛✐①♦ ✭a > 0✮ r❡s♣❡✐t♦ ❞♦ ❡✐①♦✲x✱ ❡ss❡ ♣♦♥t♦ ❞♦ ❣rá✜❝♦ é ❝❤❛♠❛❞♦ ❞❡ ✈ért✐❝❡✳ 2xa+b = 0✱ b b ❛ss✐♠✱ ♦ ♣♦♥t♦ (− , f (− )) é ♦ ✈ért✐❝❡ ♣r♦❝✉r❛❞♦❀ ♣❛r❛ ♦ ❣rá✜❝♦ ❞❡ f (x) 2a 2a b b b ❝♦♥s✐❞❡r❛r ♦s ♣♦♥t♦s x = − +1 ❡ x = − −1✱ r❡❝♦♠❡♥❞❛✲s❡ ❛❧é♠ ❞♦ ✈❛❧♦r ❞❡ x = − 2a 2a 2a b b ♣❛r❛ ❡st❡s ♣♦♥t♦s ♦❜t❡r❡♠♦s f (− + 1) = f (− − 1)✳ 2a 2a P♦❞❡♠♦s ❞❡st❛❝❛r✱ ♣❛r❛ ❛❝❤❛r ♦ ✈ért✐❝❡ ❞❛ ♣❛rá❜♦❧❛ ♣♦❞❡♠♦s ✉s❛r ❛ r❡❧❛çã♦ b ♦♥❞❡ x = − 2a ✷✳✹✳✶✶ ❋✉♥çã♦ r❛❝✐♦♥❛❧ ✐♥t❡✐r❛ ♦✉ ♣♦❧✐♥ô♠✐❝❛ ❊♠ ♠❛t❡♠át✐❝❛✱ ❢✉♥çõ❡s ♣♦❧✐♥ô♠✐❝❛s ♦✉ ♣♦❧✐♥ô♠✐♦s sã♦ ✉♠❛ ❝❧❛ss❡ ✐♠♣♦rt❛♥t❡ ❞❡ ❢✉♥çõ❡s s✐♠♣❧❡s ❡ ✐♥✜♥✐t❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❉❡✈✐❞♦ à ♥❛t✉r❡③❛ ❞❛ s✉❛ ❡str✉t✉r❛✱ ♦s ♣♦❧✐♥ô♠✐♦s sã♦ ♠✉✐t♦ ❢❛❝❡✐s ❞❡ s❡ ❛✈❛❧✐❛r ❡ ♣♦r ❝♦♥s❡q✉ê♥❝✐❛ sã♦ ✉s❛❞♦s ❡①t❡♥s✐✈❛♠❡♥t❡ f : R −→ R ❞❡✜♥✐❞❛ ♣♦r f (x) = an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0 ✱ ♦♥❞❡ an 6= 0 ❡ an , an−1 , · · · a2 , a1 ❡ a0 sã♦ ❝♦♥st❛♥t❡s r❡❛✐s✱ ❡st❛ ❢✉♥çã♦ t❛♠❜é♠ é ❝❤❛♠❛❞❛ ✏ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞❡ ❣r❛✉ n✑❀ (n ∈ N✮✳ ♥❛ ❛♥á❧✐s❡ ♥✉♠ér✐❝❛✳ ❉✐③❡♠♦s ❢✉♥çã♦ ♣♦❧✐♥ô♠✐❝❛ à ❢✉♥çã♦ n ❝♦♠ n ≥ 2 Im(f ) ❞❡♣❡♥❞❡ ❖ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ♣♦❧✐♥ô♠✐❝❛ ❞❡ ❣r❛✉ n❀ s❡✉ ❞♦♠í♥✐♦ D(f ) = R ❡ s✉❛ ✐♠❛❣❡♠ ❞❡♥♦♠✐♥❛✲s❡ ♣❛rá❜♦❧❛ ❞❡ ♦r❞❡♠ ❞❡ n ❡ ❞❛ ❝♦♥st❛♥t❡ an ✳ ❆ss✐♠✱ ♦ ❣r❛✉ ❞❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥ô♠✐❝❛ é ❡①♣r❡ss♦ ❛tr❛✈és ❞♦ ♠❛✐♦r ❡①♣♦❡♥t❡ ♥❛t✉r❛❧ ❡♥tr❡ ♦s ♠♦♥ô♠✐♦s q✉❡ ♦ ❢♦r♠❛♠✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥ô♠✐❝❛ é ♥✉❧❛ q✉❛♥❞♦ t♦❞♦s ♦s s❡✉s ❝♦❡✜❝✐❡♥t❡s ai ❢♦r❡♠ ✐❣✉❛✐s ❛ ③❡r♦✳ ❉✉❛s ❢✉♥çõ❡s ♣♦❧✐♥ô♠✐❝❛s sã♦ ✐❞ê♥t✐❝❛s q✉❛♥❞♦ ❛ s✉♠❛ ♦✉ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❡❧❛s ❛s tr❛♥s❢♦r♠❛ ❡♠ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥ô♠✐❝❛ ♥✉❧❛✳ ✾✼ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❉❡t❡r♠✐♥❛r ❛s r❛í③❡s ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ♦✉ ✏r❡s♦❧✈❡r ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s✑✱ é ✉♠ ❞♦s ♣r♦✲ ❜❧❡♠❛s ♠❛✐s ❛♥t✐❣♦s ❞❛ ♠❛t❡♠át✐❝❛✳ ❆❧❣✉♥s ♣♦❧✐♥ô♠✐♦s✱ t❛✐s ❝♦♠♦✿ f (x) = x2 + 1 ♥ã♦ ♣♦ss✉❡♠ r❛í③❡s ❞❡♥tr♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳ ❙❡✱ ♥♦ ❡♥t❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❛♥❞✐✲ ❞❛t♦s ♣♦ssí✈❡✐s ❢♦r ❡①♣❛♥❞✐❞♦ ❛♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♠❛❣✐♥ár✐♦s✱ ♦✉ s❡❥❛✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❡♥tã♦ t♦❞♦ ♦ ♣♦❧✐♥ô♠✐♦ ✭♥ã♦✲❝♦♥st❛♥t❡✮ ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ✉♠❛ r❛✐③ ✭t❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ á❧❣❡❜r❛✮✳ ✷✳✹✳✶✷ ❋✉♥çã♦ r❛❝✐♦♥❛❧ ❢r❛❝✐♦♥ár✐❛ ➱ ❛ ❢✉♥çã♦ f : R −→ R f (x) = ❞❡✜♥✐❞❛ ♣♦r✿ P (x) an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0 = Q(x) bm xm + bm−1 xm−1 + · · · + b2 x2 + b1 x + b0 P (x) ❡ Q(x) sã♦ ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ ❣r❛✉s n ❡ m r❡s♣❡❝t✐✈❛♠❡♥t❡ an bm 6= 0✱ ❞♦♠í♥✐♦ D(f ) = { x ∈ R /. Q(x) 6= 0 } ❡ ❛ ✐♠❛❣❡♠ ✈ár✐❛✱ ❞❡♣❡♥❞❡ ❞❡ n, m ❡ an bm ✳ ♦♥❞❡ ❆❧❣✉♠❛s ✈❡③❡s✱ ✉♠❛ ❢✉♥çã♦ é ❞❡✜♥✐❞❛ ♣♦r ✉♠❛ r❡❣r❛ f (x) s❡♠ ❡①♣❧✐❝✐t❛r♠♦s s❡✉ ❞♦♠í♥✐♦ ❡ ❝♦♥tr❛❞♦♠í♥✐♦✳ R ❋✐❝❛ s✉❜❡♥t❡♥❞✐❞♦ q✉❡ ♦ ❝♦♥tr❛❞♦♠í♥✐♦ é ♣❛r❛ ♦ q✉❛❧ f (x) x 7−→ f (x) ♦ ♦✉ s✐♠♣❧❡s♠❡♥t❡✱ ❡ ♦ ❞♦♠í♥✐♦ é ♦ ♠❛✐♦r s✉❜❝♦♥❥✉♥t♦ ❞❡ R é ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ❊①❡♠♣❧♦ ✷✳✹✻✳ ❊s❝r❡✈❡r s♦♠❡♥t❡ ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ❛ ❢✉♥çã♦✿ f (x) = ( 0, x, s❡✱ s❡✱ x≤0 x>0 ❙♦❧✉çã♦✳ x+ | x | x+x = ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ 2 2 x+ | x | ❡♥tã♦ | x |= −x ❛ss✐♠ 0 = x − x = x + (−x) = x+ | x |= = f (x)✳ 2 x+ | x | ✳ P♦rt❛♥t♦✱ f (x) = 2 ◗✉❛♥❞♦ x > 0 t❡♠♦s | x |= x✱ ❧♦❣♦ f (x) = s❡ x≤0 ❊①❡♠♣❧♦ ✷✳✹✼✳ ❛✮ ♣♦❧✐♥ô♠✐❝❛ ❜✮ g ▼♦str❡ q✉❡ s❡ ❡ ✉♠ ♥ú♠❡r♦ f (a) = 0✱ b ❡♥tã♦ é ❡✈✐❞❡♥t❡✮✳ ❝✮ f ❡ q✉❛❧q✉❡r ♥ú♠❡r♦ a ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ f (x) = (x − a)g(x) + b✳ ▼♦str❡ q✉❡ ♣❛r❛ q✉❛❧q✉❡r ❢✉♥çã♦ ♣♦❧✐♥ô♠✐❝❛ ▼♦str❡ q✉❡ s❡ f t❛❧ q✉❡ f (x) = (x − a)g(x) ♣❛r❛ ❛❧❣✉♠❛ ❢✉♥çã♦ n ∈ N✱ ❡♥tã♦ f q✉❡ f (a) = 0✳ é ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥ô♠✐❝❛ ❞❡ ❣r❛✉ r❛í③❡s ❡ ❡①✐st❡♠ ♥♦ ♠á①✐♠♦ n ♥ú♠❡r♦s a ✾✽ t❛✐s g ✭❆ r❡❝í♣r♦❝❛ t❡♠ ♥♦ ♠á①✐♠♦ n 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❞✮ ▼♦str❡ q✉❡ ♣❛r❛ t♦❞♦ n ∈ N ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥ô♠✐❝❛ ❞❡ ❣r❛✉ n ❝♦♠ r❛í③❡s✳ ❙❡ n é ♣❛r ❛❝❤❛r ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥ô♠✐❝❛ ❞❡ ❣r❛✉ n s❡♠ r❛í③❡s✱ ❡ s❡ n é í♠♣❛r ❛❝❤❛r s♦♠❡♥t❡ ❝♦♠ ✉♠❛ r❛✐③✳ ❛✮ ❙♦❧✉çã♦✳ ❙❡ ♦ ❣r❛✉ ❞❡ ♦♥❞❡ g(x) = c f ❡ é 1✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r f (x) = cx+d = c(x−a)+(d+ac) = (x−a)g(x)+b✱ b = d + ac✳ n ∈ N✳ ❙✉♣♦♥❤❛ ♦ r❡s✉❧t❛❞♦ ✈á❧✐❞♦ ♣❛r❛ n = h✳ ❙❡ f é ❞❡ ❣r❛✉ h + 1 t❡♠ ❛ ❢♦r♠❛ f (x) = ah+1 xh+1 + ah xh + · · · + a1 x + a0 ✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❢✉♥çã♦ p(x) = f (x) − ah+1 (xh+1 − a) ❡♥tã♦ ♦ ❣r❛✉ ❞❡ p(x) é n = h ❡ ♣❡❧❛ ❤✐♣ót❡s❡ ✐♥❞✉t✐✈❛ ♣♦❞❡♠♦s ❡s❝r❡✈❡r p(x) = f (x) − ah+1 (xh+1 − a) = (x − a)g(x) + b ⇒ f (x) = (x − a)[p(x) + ah+1 ] + b✱ ❡ P♦r ✐♥❞✉çã♦ s♦❜r❡ t❡♠♦s ❛ ❢♦r♠❛ r❡q✉❡r✐❞❛✳ ❜✮ ❙♦❧✉çã♦✳ P❡❧❛ ♣❛rt❡ ❛✮ ♣♦❞❡♠♦s s✉♣♦r ❛ss✐♠ f (x) = (x − a)g(x)✳ f (x) = (x − a) + b✱ ❡♥tã♦ 0 = f (a) = (a − a)g(a) + b = b✱ ❝✮ ❙♦❧✉çã♦✳ f t❡♠ n r❛í③❡s✱ a1 , a2 , · · · , an ❝♣♠ a1 6= a2 ✱ ❡♥tã♦ ♣❡❧❛ ♣❛rt❡ ❜✮ ♣♦❞❡♠♦s ❡s❝r❡✈❡r f (x) = (x−a1 )g1 (x) ♦♥❞❡ ♦ ❣r❛✉ ❞❡ g1 (x) é n−1✳ P♦ré♠ f (a2 ) = (a2 −a1 )g1 (a2 ) ❞❡ ♠♦❞♦ q✉❡ g1 (a2 ) = 0 ♣❡❧♦ ❢❛t♦ a1 6= a2 ✳ ▲♦❣♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r f (x) = (x−a1 )(x−a2 )g2 (x) ♦♥❞❡ ♦ ❣r❛✉ ❞❡ g2 (x) é n − 2✳ ❙✉♣♦♥❤❛ Pr♦ss❡❣✉✐♥❞♦ ❞❡st❡ ♠♦❞♦ ♣♦❞❡♠♦s ♦❜t❡r ♣❛r❛ ❛❧❣✉♠ c 6= 0✳ ➱ ó❜✈✐♦ q✉❡ ❞✮ ❙♦❧✉çã♦✳ f (a) 6= 0 s❡ f (x) = (x − a1 )(x − a2 )(x − a3 ) · · · (x − an ) · c a 6= a1 , a2 , ·, an ✳ ❧♦❣♦ f ♣♦❞❡ t❡r n r❛í③❡s✳ f (x) = (x − 1)(x − 2)(x − 3) · (x − n)✱ ❡♥tã♦ f t❡♠ n r❛í③❡s✳ ❙❡ n é ♣❛r f (x) = xn + 2 t❡♠ r❛í③❡s ✭❡♠ R✮✱ s❡ n é í♠♣❛r f (x) = xn t❡♠ ❝♦♠♦ ú♥✐❝❛ r❛✐③ x = 0✳ ❙❡ ♥ã♦ ✷✳✹✳✶✸ ❋✉♥çõ❡s ❞❡ ♦❢❡rt❛ ❡ ❞❡♠❛♥❞❛✳ ❊①✐st❡♠ ❝✐r❝✉♥st❛♥❝✐❛s r❡❧❛t✐✈❛s ❛ ✉♠ ❢❛❜r✐❝❛♥t❡✱ ♣❛r❛ ❛s q✉❛✐s ❛s ú♥✐❝❛s ✈❛r✐á✈❡✐s sã♦ ♦ ♣r❡ç♦ ❞❡ ❝✉st♦ ❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♠❡r❝❛❞♦r✐❛ ❞❡♠❛♥❞❛❞❛ ✭✈❡♥❞✐❞❛✮✳ ❊♠ ❣❡r❛❧✱ ♦ ♥ú♠❡r♦ ❞❡ ♠❡r❝❛❞♦r✐❛s ❞❡♠❛♥❞❛❞❛s ♥♦ ♠❡r❝❛❞♦ ♣❡❧♦s ❝♦♥s✉♠✐❞♦r❡s ❞❡✲ ♣❡♥❞❡ ❞♦ ♣r❡ç♦ ❞❛ ♠❡s♠❛✳ ◗✉❛♥❞♦ ♦s ♣r❡ç♦s ❜❛✐①❛♠✱ ❡♠ ❣❡r❛❧✱ ♦s ❝♦♥s✉♠✐❞♦r❡s ♣r♦❝✉r❛♠ ♠❛✐s ❛ ♠❡r❝❛❞♦r✐❛❀ ❝❛s♦ ♦ ♣r❡ç♦ s✉❜❛✱ ♦ ♦♣♦st♦ ❛❝♦♥t❡❝❡✱ ♦s ❝♦♥s✉♠✐❞♦r❡s ✐rã♦ ♣r♦❝✉r❛r ♠❡♥♦s ♠❡r❝❛❞♦r✐❛s✳ ❙❡❥❛ p ♦ ♣r❡ç♦ ❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ❞❛ ♠❡r❝❛❞♦r✐❛✱ ❡ s❡❥❛ ❞❡♠❛♥❞❛❞❛s✱ ✉♠❛ ❡q✉❛çã♦ q✉❡ r❡❧❛❝✐♦♥❛ ❛ q✉❛♥t✐❞❛❞❡ ♣r❡ç♦ ❞❛❞♦ ♣♦r p s❡❣✉✐♥t❡s ❢♦r♠❛s✿ q✱ q ♦ ♥ú♠❡r♦ ❞❛s ♠❡r❝❛❞♦r✐❛s ❞❛ ♠❡r❝❛❞♦r✐❛ ❞❡♠❛♥❞❛❞❛ ❡ ♦ é ❝❤❛♠❛❞❛ ❞❡ ✏❡q✉❛çã♦ ❞❛ ❞❡♠❛♥❞❛✑ ✱ ❡❧❛ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❡♠ ✉♠❛ ❞❛s p = C(q) ♦✉ q = D(p)✳ ✾✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❖s ❡❝♦♥♦♠✐st❛s✱ ❝♦♥tr❛r✐❛♥❞♦ ♦ ❝♦st✉♠❡ ❞♦s ♠❛t❡♠át✐❝♦s✱ r❡♣r❡s❡♥t❛♠ ❛ ✈❛r✐á✈❡❧ ✐♥✲ ❞❡♣❡♥❞❡♥t❡ p ✭♣r❡ç♦✮ ❞❛ ❡q✉❛çã♦ q = D(p) ♥♦ ❡✐①♦ ✈❡rt✐❝❛❧ ❡ ❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ q ✭q✉❛♥t✐❞❛❞❡ ❞❛ ❞❡♠❛♥❞❛✮ ♥♦ ❡✐①♦ ❤♦r✐③♦♥t❛❧✳ ❯♠❛ ❝✉r✈❛ ❞❛ ❞❡♠❛♥❞❛ ✭♣r♦❝✉r❛✮ ❞❡✈❡ t❡r ♦ ❛s♣❡❝t♦ ❞❛ ❝✉r✈❛ ♠♦str❛❞❛ ♥❛ ❋✐❣✉r❛ ✭✷✳✷✹✮❀ ♥✉♠❛ s✐t✉❛çã♦ ♥♦r♠❛❧✱ s❡ ♦ ♣r❡ç♦ ❛✉♠❡♥t❛✱ ❛ q✉❛♥t✐❞❛❞❡ ♦❢❡rt❛❞❛ ❛✉♠❡♥t❛rá✳ ❖ ❣rá✜❝♦ ❞❛ ❡q✉❛çã♦ ❞❡ ♦❢❡rt❛ é s✐♠✐❧❛r ❝♦♠ ♦ ❞❛ ❋✐❣✉r❛ ✭✷✳✷✺✮✳ p ✻ p ✻ ✲ ✲ q q ❈✉r✈❛ ❞❡ ♦❢❡rt❛ ❋✐❣✉r❛ ✷✳✷✺✿ ❈✉r✈❛ ❞❡ ❞❡♠❛♥❞❛ ❋✐❣✉r❛ ✷✳✷✹✿ ❉❡✜♥✐çã♦ ✷✳✶✷✳ • ❆ r❡❧❛çã♦ q = D(p) é ❝❤❛♠❛❞❛ ✏❢✉♥çã♦ ❞❛ ❞❡♠❛♥❞❛✑ ✱ ❡ D(p) é ♦ ♥ú♠❡r♦ ❞❡ ✉♥✐❞❛❞❡s ❞❡ ♠❡r❝❛❞♦r✐❛ q✉❡ s❡rá ❞❡♠❛♥❞❛❞❛s s❡ p ❢♦r ♦ ♣r❡ç♦ ♣♦r ✉♥✐❞❛❞❡✳ • ❆ r❡❧❛çã♦ p = C(q) é ❝❤❛♠❛❞❛ ✏❢✉♥çã♦ ❞♦ ❝✉st♦ t♦t❛❧✑ ✱ ❡ C(q) é ♦ ♣r❡ç♦ ❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ❞❛ ♠❡r❝❛❞♦r✐❛ q✉❛♥❞♦ q ✉♥✐❞❛❞❡s sã♦ ❞❡♠❛♥❞❛❞❛s✳ • ❆ r❡❧❛çã♦ R = R(q) r❡♣r❡s❡♥t❛ ❛ ❢✉♥çã♦ r❡❝❡✐t❛ t♦t❛❧✱ ❣❡r❛❞❛ ♣❡❧❛ ✈❡♥❞❛ ❞❡ q ✉♥✐❞❛❞❡s ❞♦ ♣r♦❞✉t♦✳ • ❆ ❢✉♥çã♦ ❧✉❝r♦ t♦t❛❧ é ❞❡✜♥✐❞♦ ❝♦♠♦ s❡♥❞♦ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛ r❡❝❡✐t❛ t♦t❛❧ ❡ ♦ ❝✉st♦ t♦t❛❧❀ L(q) = R(q)−C(q) ✐st♦ r❡♣r❡s❡♥t❛ ♦ ❧✉❝r♦ ❛♦ ✈❡♥❞❡r q ✉♥✐❞❛❞❡s ❞♦ ♣r♦❞✉t♦✳ ◆♦ q✉❡ s❡❣✉❡ ✉t✐❧✐③❛r❡♠♦s ❛ s❡❣✉✐♥t❡ ♥♦t❛çã♦ ❞❡ ❢✉♥çõ❡s✿ a) C = C(q) ❈✉st♦ t♦t❛❧. b) CM = CM (q) ❈✉st♦ ▼é❞✐♦. c) R = R(q) ❘❡❝❡✐t❛ t♦t❛❧. d) RM = RM (q) ❘❡❝❡✐t❛ ▼é❞✐❛. e) D = D(q) ❉❡♠❛♥❞❛. f ) S = S(p) ❖❢❡rt❛. ❊①❡♠♣❧♦ ✷✳✹✽✳ ✶✵✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R p2 + 2q − 16 = 0✳ ❊♠ s✐t✉❛çõ❡s √ 16 − 2q q✉❛♥❞♦ 16 − 2q ≥ ✈❛r✐á✈❡✐s q ❡ p ♥ã♦ sã♦ ♥❡❣❛t✐✈❛s✱ t❡♠♦s p = ❢✉♥çã♦ ❝✉st♦ t♦t❛❧ ❞♦ ♣r❡ç♦ ♣❛r❛ ❛ ❡q✉❛çã♦ ❞❛ ❞❡♠❛♥❞❛ é p = C(q) = ❈♦♥s✐❞❡r❡♠♦s ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦ ❞❛ ❞❡♠❛♥❞❛✿ ❡❝♦♥ô♠✐❝❛s✱ ❛s 0✳ √ P♦rt❛♥t♦ ❛ 16 − 2q ✳ ❉❛ ❡q✉❛çã♦ ❞❛ ❞❡♠❛♥❞❛ t❡♠♦s 1 q = D(p) = 8 − p2 2 q✉❡ ❡①♣r❡ss❛ q ❝♦♠♦ ❢✉♥çã♦ ❞❡ p✳ ❉❡✜♥✐çã♦ ✷✳✶✸✳ • ❛ r❡❧❛çã♦ • CM = CM (q) ❞❡ ❝❛❞❛ ✉♥✐❞❛❞❡ é ♦❜t✐❞♦ ♠❡❞✐❛♥t❡ ❖ ❝✉st♦ ♠é❞✐♦ ❞❛ ♣r♦❞✉çã♦ C(q) CM (q) = q ❝❤❛♠❛❞❛ ✏❢✉♥çã♦ ❝✉st♦ ♠é❞✐♦ ✑✳ ❆♦ ❞✐✈✐❞✐r ❛ r❡❝❡✐t❛ t♦t❛❧ ♦❜té♠✲s❡ R(q) RM (q) = q R(q) ♣❡❧❛ q✉❛♥t✐❞❛❞❡ q ❞❡ ✉♥✐❞❛❞❡s ♣r♦❞✉③✐❞❛s ❝❤❛♠❛❞❛ ✏ ❢✉♥çã♦ r❡❝❡✐t❛ ♠é❞✐❛ ✑✳ ❊①❡♠♣❧♦ ✷✳✹✾✳ ❉❛❞❛s ❛s ❢✉♥çõ❡s ❞❡ ❝✉st♦ t♦t❛❧✱ ❞❡t❡r♠✐♥❡ ❛ ❢✉♥çã♦ ❞❡ ❝✉st♦ ♠é❞✐♦✿ ❛✮ ❜✮ C(q) = 300 + C(q) = 2q 3 − 12q 2 + 50q + 40 60 q 2 + q 6 ❙♦❧✉çã♦✳ ❛✮ ❜✮ 2q 3 − 12q 2 + 50q + 40 40 = 2q 2 − 12q + 50 + q q q 300 + 60 + 300 q q 6 CM (q) = = + 60 + ✳ q q 6 CM = ❊①❡♠♣❧♦ ✷✳✺✵✳ L ❡♠ r❡❛✐s q✉❡ ♦❜té♠ ❛♦ ❛❧✉❣❛r ✉♠ ♣ré❞✐♦ é ❞❛❞♦ ♣❡❧❛ ❡q✉❛çã♦ L(q) = −2q + 92q ✱ q✉❛❧ é ♥ú♠❡r♦ ❞❡ ❛♥❞❛r❡s q✉❡ t♦r♥❛ ❯♠❛ ✐♠♦❜✐❧✐ár✐❛ ❡st✐♠❛ q✉❡ ♦ ❧✉❝r♦ ♠❡♥s❛❧ ❞❡ q ❛♥❞❛r❡s✱ 2 ♠❛✐s r❡♥t❛❜❧❡ ♦ ❛❧✉❣✉❡❧ ❞♦ ♣ré❞✐♦❄ ❙♦❧✉çã♦✳ L(q) = −2q 2 + 92q = 2(46q − q 2 ) ⇒ L(q) = 2[232 − 232 + 46q − q 2 ] = 2[232 − (23 − q)2 ] q✉❛♥❞♦ q = 23, L(23) = 1058 é ♦ ♠á①✐♠♦ ❛❜s♦❧✉t♦✳ P♦rt❛♥t♦✱ é ♠❛✐s r❡♥tá✈❡❧ ♦ ❛❧✉❣✉❡❧ ❞❡ ✉♠ ♣ré❞✐♦ ❞❡ 23 ❛♥❞❛r❡s✳ ❚❡♠♦s✱ ❊♠ ❣❡r❛❧✱ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❡♠♣r❡s❛s q✉❡ ♣r♦❞✉③❡♠ ✉♠❛ ♠❡s♠❛ ♠❡r❝❛❞♦r✐❛ ❝❤❛♠❛♠♦s ❞❡ ✐♥❞ústr✐❛❀ ♣♦r ❡①❡♠♣❧♦✱ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❡♠♣r❡s❛s ❞❡ ❝♦♥❢❡❝çã♦ ❞❡ ❝❛❧ç❛❞♦s ❞♦ ❇r❛s✐❧✱ ❝❤❛♠❛♠♦s ✐♥❞ústr✐❛ ❞❡ ❝❛❧ç❛❞♦s ❞♦ ❇r❛s✐❧✳ ❖ ♠❡r❝❛❞♦ ♣❛r❛ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♠❡r❝❛❞♦r✐❛ ❝♦♥st❛ ❞❛ ✐♥❞ústr✐❛ ❡ ❞♦s ❝♦♥s✉♠✐❞♦r❡s ✭❡♠ ❣❡r❛❧✮❀ ❛ ❡q✉❛çã♦ ❞❡ ♦❢❡rt❛ ❞♦ ♠❡r❝❛❞♦ é ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛s ❡q✉❛çõ❡s ❞❡ ♦❢❡rt❛ ❞❛s ❡♠♣r❡s❛s ✐♥t❡❣r❛♥t❡s ❞♦ ♠❡r❝❛❞♦❀ ❡ ❛ ❡q✉❛çã♦ ❞❡ ❞❡♠❛♥❞❛ ❞♦ ♠❡r❝❛❞♦ é ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛s ❡q✉❛çõ❡s ❞❡ ❞❡♠❛♥❞❛ ❞❡ t♦❞♦s ♦s ❝♦♥s✉♠✐❞♦r❡s✳ ✶✵✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✷✳✺✶✳ ❯♠❛ ❝♦♠♣❛♥❤✐❛ ❛ér❡❛ t❡♠ ❝♦♠♦ t❛r✐❢❛ ✜①❛ ❘$800 ❡ tr❛♥s♣♦rt❛ 8.000 ♣❛ss❛❣❡✐r♦s ❝❛❞❛ ❞✐❛✳ ❆♦ ❝♦♥s✐❞❡r❛r ✉♠ ❛✉♠❡♥t♦ ♥❛ t❛r✐❢❛✱ ❛ ❝♦♠♣❛♥❤✐❛ ❞❡t❡r♠✐♥❛ q✉❡ ♣❡r❞❡rá 400 ♣❛s✲ s❛❣❡✐r♦s ♣♦r ❝❛❞❛ ❘$50 ❞❡ ❛✉♠❡♥t♦✳ ❙♦❜ ❡st❛s ❝♦♥❞✐çõ❡s❀ q✉❛❧❀ ❞❡✈❡r s❡r ♦ ❛✉♠❡♥t♦ ♣❛r❛ q✉❡ ♦ ✐♥❣r❡ss♦ s❡❥❛ ♠á①✐♠♦❄ ❙♦❧✉çã♦✳ ❙❡❥❛ x ♦ ♥ú♠❡r♦ ❞❡ ❛✉♠❡♥t♦s ❞❡ ❘$50 ♥❛ t❛r✐❢❛✱ ❡♥tã♦ ❛ t❛r✐❢❛ r❡s✉❧t❛♥t❡ é ❘$(800 + 50x) ❡ ♦ ♥ú♠❡r♦ ❞❡ ♣❛ss❛❣❡✐r♦s s❡rá ❞❡ 8.000 − 400x✳ ❆ ❢✉♥çã♦ q✉❡ ❞❡t❡r♠✐♥❛ ♦ ✐♥❣r❡ss♦ t♦t❛❧ é✿ I(x) = (800 + 50x)(8000 − 400x) = 20.000(320+4x−x2 ) ❝♦♠ 0 ≤ x ≤ 20 ⇒ I(x) = 20.000(320+4x−x2 ) = 20.000[324− (4 − 4x + x2 )] = 20.000[324 − (x − 2)2 ]✳ ❖❜s❡r✈❡ q✉❡✱ q✉❛♥❞♦ x = 2 t❡r❡♠♦s ♠á①✐♠♦ ✈❛❧♦r ♣❛r❛ I(x)✳ ▲♦❣♦✱ ♦ ❛✉♠❡♥t♦ t❡♠ q✉❡ s❡r ❞❡ ❘$100 ❡ ♦ ❝✉st♦ ❞❡ ❝❛❞❛ ♣❛ss❛❣❡♠ s❡rá ❞❡ ❘$900✳ ❖❜s❡r✈❛çã♦ ✷✳✼✳ ❖ ❡q✉✐❧í❜r✐♦ ❞❡ ♠❡r❝❛❞♦ ♦❝♦rr❡ q✉❛♥❞♦ ❛ q✉❛♥t✐❞❛❞❡ ❞❛ ♠❡r❝❛❞♦r✐❛ ❞❡♠❛♥❞❛❞❛✱ ❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣r❡ç♦✱ é ✐❣✉❛❧ à q✉❛♥t✐❞❛❞❡ ❞❡ ♠❡r❝❛❞♦r✐❛ ♦❢❡r❡❝✐❞❛ àq✉❡❧❡ ♣r❡ç♦✳ ◗✉❛♥❞♦ ♦❝♦rr❡ ♦ ❡q✉✐❧í❜r✐♦ ❞❡ ♠❡r❝❛❞♦✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♠❡r❝❛❞♦r✐❛ ♣r♦❞✉③✐❞❛ é ❝❤❛✲ ♠❛❞❛ ✏q✉❛♥t✐❞❛❞❡ ❞❡ ❡q✉✐❧í❜r✐♦✑ ❀ ❡✱ ♦ ♣r❡ç♦ ❞❛ ♠❡r❝❛❞♦r✐❛ é ❝❤❛♠❛❞♦ ♣r❡ç♦ ❞❡ ❡q✉✐❧í❜r✐♦✳ ❉❡✜♥✐çã♦ ✷✳✶✹✳ ❉❡✜♥✐♠♦s ♦ ✏♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✑ ❝♦♠♦ ❛q✉❡❧❡ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞♦ ❣rá✜❝♦ ❞❛ ❝✉r✈❛ ❞❛ ♦❢❡rt❛ ❝♦♠ ♦ ❞❛ ❞❡♠❛♥❞❛✳ ❙✉❛s ❝♦♦r❞❡♥❛❞❛s sã♦ ♦ ♣r❡ç♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡q✉✐❧í❜r✐♦✳ ◆❛ ❋✐❣✉r❛ ✭✷✳✷✻✮ ♠♦str❛✲s❡ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦❀ s❡ ♦ ♣r❡ç♦ ❡stá ❛❝✐♠❛ ❞♦ ♣r❡ç♦ ❞❡ ❡q✉✐❧í❜r✐♦✱ ❤á ❡①❝❡ss♦ ❞❡ ♦❢❡rt❛ ❡ ♦ ♣r❡ç♦ t❡♥❞❡ ❛ ❝❛✐r❀ s❡ ♦ ♣r❡ç♦ ❡stá ❛❜❛✐①♦ ❞♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✱ ❤á ❡s❝❛ss❡③ ❞❡ ♦❢❡rt❛ ❡ ♦ ♣r❡ç♦ t❡♥❞❡ ❛ s✉❜✐r✳ ❊♠ ❡❝♦♥♦♠✐❛✱ ♣❛rt✐❝✉❧❛r♠❡♥t❡ ♥♦s ❡st✉❞♦s r❡✲ ❢❡r❡♥t❡s à ❝♦♥t❛❜✐❧✐❞❛❞❡ ❞❡ ❝✉st♦s✱ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐✲ ❧í❜r✐♦ ❡❝♦♥ô♠✐❝♦ é ♦ ♠♦♠❡♥t♦ q✉❛♥❞♦ ❛s r❡❝❡✐t❛s s❡ ✐❣✉❛❧❛♠ ❛♦s ❝✉st♦s ❡ ❞❡s♣❡s❛s✳ ➱✱ ♣♦rt❛♥t♦✱ ♦ ♠♦✲ ♠❡♥t♦ ❡♠ q✉❡ ✉♠ ♣r♦❞✉t♦ ❞❡✐①❛ ❞❡ ❝✉st❛r ❡ ♣❛ss❛ ❛ ❞❛r ❧✉❝r♦✳ ❆ ❡❧❡ ❛❞✐❝✐♦♥❛♠✲s❡ ♦s ❝✉st♦s ✜①♦s ❡ t♦❞♦s ♦s ❝✉st♦s ❞❡ ♦♣♦rt✉♥✐❞❛❞❡✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ♦s r❡❢❡✲ r❡♥t❡s ❛♦ ✉s♦ ❞♦ ❝❛♣✐t❛❧ ♣ró♣r✐♦✱ ❛♦ ♣♦ssí✈❡❧ ❛❧✉❣✉❡❧ ❋✐❣✉r❛ ✷✳✷✻✿ ❞❛s ❡❞✐✜❝❛çõ❡s ✭❝❛s♦ ❛ ❡♠♣r❡s❛ s❡❥❛ ♣r♦♣r✐❡tár✐❛✮✱ ♣❡r❞❛ ❞❡ s❛❧ár✐♦s✱ ❡t❝✳ ✶✵✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✷✲✸ ✶✳ ◗✉❛❧ ♦ ♥ú♠❡r♦ q✉❡ ❡①❝❡❞❡ ❛ s❡✉ q✉❛❞r❛❞♦ ♦ ♠á①✐♠♦ ♣♦ssí✈❡❧❄ ✷✳ ❆ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s é 8✳ 1.) ❉❡t❡r♠✐♥❡ ♦ ♠❡♥♦r ❞❡❧❡s ♣❛r❛ q✉❡ ♦ ♣r♦❞✉t♦ s❡❥❛ ♦ ♠❡♥♦r ♣♦ssí✈❡❧❀ 2.) ◗✉❛❧ é ♦ ♠❡♥♦r ✈❛❧♦r ❞❡ss❡ ♣r♦❞✉t♦ ❄ ✸✳ ❙❡❥❛♠ f ❡ g ❢✉♥çõ❡s ❞❡ R ❡♠ R✱ s❡♥❞♦ R ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ❞❛❞❛s ♣♦r f (x) = 2x − 3 ❡ f (g(x)) = −4x + 1✳ ◆❡st❛s ❝♦♥❞✐çõ❡s✱ ❞❡t❡r♠✐♥❡ g(−1)✳ ✹✳ ❉❡t❡r♠✐♥❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ ❡q✉❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s ✐♥❞✐❝❛❞♦s✿ 1. A(1, −3) ❡ B(0, 1) 4. C(0, 1) ❡ D(0, 5) 7. M (−1, 6) ❡ P (5, 6) 2. M (0, 1) ❡ N (3, 2) 5. B(−1, 2) ❡ C(3, −5) 8. G(3, 6) ❡ H(1, 4) 3. P (−1, 3) ❡ Q(5, −2) 6. S(3, 9) ❡ T (3, 7) 9. P (5, 3) ❡ S(5, 2) ✺✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s ✐♥❞✐❝❛❞♦s❀ ❞❡s❡♥❤❛r ♦ ❣rá✜❝♦✿ 1. A(1, −3) ❡ B(0, 1) 4. D(3, −1) ❡ E(1, 1) 7. F (2, 8) ❡ G(0, 0) 2. M (0, 1) ❡ N (3, 2) 5. A(3, −2) ❡ B(3, 2) 8. Q(7, 1) ❡ S(8, 12) √ 3. P (−1, 3) ❡ Q(5, −2) 6. R(−1, 3) ❡ U (3, −2) 9. S(6, 8) ❡ R(5, 12) √ ✻✳ ▼♦str❛r q✉❡ ♦s ♣♦♥t♦s P1 (3, 3), P2 (−3, −3), P3 (−3 3, 3 3) sã♦ ♦s ✈ért✐❝❡s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦✳ ✼✳ ❙❡ P1 (−4, 2) ❡ P2 (4, 6) sã♦ ♦s ♣♦♥t♦s ❡①tr❡♠♦s ❞♦ s❡❣♠❡♥t♦s r❡t✐❧í♥❡♦ ♦r✐❡♥t❛❞♦ −−→ P1 P2 ✱ ❛❝❤❛r ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣♦♥t♦ P (x, y) q✉❡ ❞✐✈✐❞❡ ❡st❡ s❡❣♠❡♥t♦ ♥❛ r❛③ã♦ P1 P : P P2 = −3✳ ✽✳ ❉❡t❡r♠✐♥❛r ♦ â♥❣✉❧♦ ❛❣✉❞♦ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❝✉❥♦s ✈ért✐❝❡s sã♦ ♣♦♥t♦s A(−2, 1)✱ B(1, 5)✱ C(10, 7) ❡ D(7, 3)✳ ✾✳ ❉❡♠♦♥str❛r ❛♥❛❧✐t✐❝❛♠❡♥t❡ q✉❡ ♦s s❡❣♠❡♥t♦s q✉❡ ✉♥❡♠ ♦s ♣♦♥t♦s ♠é❞✐♦s ❞♦s ❧❛❞♦s s✉❝❡ss✐✈♦s ❞❡ q✉❛❧q✉❡r q✉❛❞r✐❧át❡r♦ ❢♦r♠❛♠ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ ✶✵✳ Pr♦✈❛r ❛♥❛❧✐t✐❝❛♠❡♥t❡ q✉❡✱ s❡ ❛s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ sã♦ ♠✉t✉❛♠❡♥t❡ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ é ✉♠ ❧♦s❛♥❣♦✳ ✶✵✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✶✶✳ ❉❡t❡r♠✐♥❛r ❛ ❡q✉❛çã♦ ❞❛ ❧✐♥❤❛ r❡t❛ q✉❡ ❝♦♥tê♠ ♦ ♣♦♥t♦ (−3, 1) ❡ é ♣❛r❛❧❡❧❛ à r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ❞♦✐s ♣♦♥t♦s (0, −2) ❡ (5, 2)✳ ✶✷✳ ❉❡t❡r♠✐♥❛r ❛ ❡q✉❛çã♦ ❞❛ ♠❡❞✐❛tr✐③ ❞♦ s❡❣♠❡♥t♦ r❡t✐❧í♥❡♦ ❝✉❥♦s ❡①tr❡♠♦s sã♦ ♦s ♣♦♥t♦s (−2, 1) ❡ (3, −5)✳ ✶✸✳ ▼♦str❡ q✉❡ ❞✉❛s r❡t❛s✱ L1 : Ax + By + C = 0 ❡ L2 : A′ x + B ′ y + C ′ = 0 sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s✱ s❡ A.A′ + B.B ′ = 0✳ ✶✹✳ ❆ ❡q✉❛çã♦ ❞❡ ✉♠❛ r❡t❛ L é 5x − 7y + 11 = 0✳ ❛✮ ❊s❝r❡✈❡r ❛ ❡q✉❛çã♦ q✉❡ r❡♣r❡s❡♥t❛ t♦❞❛s ❛s r❡t❛s ♣❛r❛❧❡❧❛s ❛ L✳ ❜✮ ❉❡t❡r♠✐♥❛r ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛ L q✉❡ ♣❛ss❡ ♣♦r P (4, 2)✳ ✶✺✳ ❖ ♣r❡ç♦ ✉♥✐tár✐♦ ❞❡ ❝❡rt♦ ♣r♦❞✉t♦ é 5✱ ❡ ♦ ❝✉st♦ ✜①♦ ❞❡ ♣r♦❞✉çã♦ é 40❀ ❝♦❧♦❝❛❞♦ ♥♦ ♠❡r❝❛❞♦✱ ✈❡r✐✜❝♦✉✲s❡ q✉❡ ❛ ❞❡♠❛♥❞❛ ♣❛r❛ ❡ss❡ ♣r♦❞✉t♦ ❡r❛ ❞❛❞❛ ♣❡❧❛ r❡❧❛çã♦ q p = 15 − ✳ ✭❛✮ ❉❡t❡r♠✐♥❡ ❛s ❢✉♥çõ❡s C ✭❈✉st♦✮ ❡ R ✭❘❡❝❡✐t❛✮ ♣❛r❛ ❡ss❡ ♣r♦❞✉t♦ 5 ❡ ❢❛ç❛ s❡✉s ❣rá✜❝♦s ♥✉♠ ♠❡s♠♦ s✐st❡♠❛ ❞❡ ❡✐①♦s✳ ✭❜✮ ❉❡t❡r♠✐♥❡ ❛ ❢✉♥çã♦ ▲✉❝r♦ ❡ ❢❛ç❛ ♦ s❡✉ ❣rá✜❝♦✳ ❖❜s❡r✈❡ q✉❡ ♦ ❧✉❝r♦ L é ③❡r♦ q✉❛♥❞♦ C = R✳ ✭❝✮ P❛r❛ q✉❡ ✈❛❧♦r❡s ❞❡ q t❡♠♦s L ≥ 0❄ ✭❞✮ ❉❡t❡r♠✐♥❡ ❢✉♥çõ❡s ❞❡ ❘❡❝❡✐t❛ ▼é❞✐❛ ❡ ❈✉st♦ ▼é❞✐♦ ❛ ❢❛ç❛ s❡✉s ❣rá✜❝♦s✳ ✶✻✳ ❚r❛ç❛r ❛ ❝✉r✈❛ ❝✉❥❛ ❡q✉❛çã♦ é✿ x2 + xy 2 − y 2 = 0✳ ✶✼✳ ❯♠❛ ❢á❜r✐❝❛ ❞❡ ❡q✉✐♣❛♠❡♥t♦s ❡❧❡trô♥✐❝♦s ❡st❛ ❝♦❧♦❝❛♥❞♦ ✉♠ ♥♦✈♦ ♣r♦❞✉t♦ ♥♦ ♠❡r✲ ❝❛❞♦✳ ❉✉r❛♥t❡ ♦ ♣r✐♠❡✐r♦ ❛♥♦ ♦ ❝✉st♦ ✜①♦ ♣❛r❛ ✐♥✐❝✐❛r ❛ ♥♦✈❛ ♣r♦❞✉çã♦ é ❞❡ ❘$140.000 ❡ ♦ ❝✉st♦ ✈❛r✐á✈❡❧ ♣❛r❛ ♣r♦❞✉③✐r ❝❛❞❛ ✉♥✐❞❛❞❡ é ❘$25✳ ❉✉r❛♥t❡ ♦ ♣r✐✲ ♠❡✐r♦ ❛♥♦ ♦ ♣r❡ç♦ ❞❡ ✈❡♥❞❛ é ❘$65 ♣♦r ✉♥✐❞❛❞❡✳ ✭❛✮ ❙❡ X ✉♥✐❞❛❞❡s sã♦ ✈❡♥❞✐❞❛s ❞✉r❛♥t❡ ♦ ♣r✐♠❡✐r♦ ❛♥♦✱ ❡①♣r❡ss❡ ♦ ❧✉❝r♦ ❞♦ ♣r✐♠❡✐r♦ ❛♥♦ ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❞❡ X ✳ ✭❜✮ ❊st✐♠❛✲s❡ q✉❡ 23.000 s❡rã♦ ✈❡♥❞✐❞❛s ❞✉r❛♥t❡ ♦ ♣r✐♠❡✐r♦ ❛♥♦✳ ❯s❡ ♦ r❡s✉❧t❛❞♦ ❞❛ ♣❛rt❡ ✭❛✮ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ ❧✉❝r♦ ❞♦ ♣r✐♠❡✐r♦ ❛♥♦✱ s❡ ♦s ❞❛❞♦s ❞❡ ✈❡♥❞❛ ❢♦r❡♠ ❛t✐♥❣✐❞♦s✳ ✭❝✮ ◗✉❛♥t❛s ✉♥✐❞❛❞❡s ♣r❡❝✐s❛♠ s❡r ✈❡♥❞✐❞❛s ❞✉r❛♥t❡ ♦ ♣r✐♠❡✐r♦ ❛♥♦ ♣❛r❛ q✉❡ ❛ ❢á❜r✐❝❛ ♥ã♦ ❣❛♥❤❡ ♥❡♠ ♣❡r❞❛ ❄ ✶✽✳ ❉❛❞❛s q = 4p−5 ❡ q = 150 +29 r❡s♣❡❝t✐✈❛♠❡♥t❡ ❢✉♥çõ❡s ❞❡ ♦❢❡rt❛ ❡ ❞❡♠❛♥❞❛ ♣❛r❛ p + 15 ✉♠ ❝❡rt♦ ♣r♦❞✉t♦✱ ❢❛ç❛ s❡✉s ❣rá✜❝♦s ♥✉♠ ♠❡s♠♦ ❡✐①♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡ ❞❡t❡r♠✐♥❡ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✳ ✶✾✳ ❖ ❝✉st♦ t♦t❛❧ ♣❛r❛ ♣r♦❞✉③✐r q ✉♥✐❞❛❞❡s ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣r♦❞✉t♦ é C(q) = q 2 + 20q + 5 r❡❛✐s✱ ❡ ♦ ♣r❡ç♦ ❞❡ ✈❡♥❞❛ ❞❡ ✉♠❛ ✉♥✐❞❛❞❡ é ❞❡ (30 − q) r❡❛✐s✳ ❛✮ ❆❝❤❛r ❛ ❢✉♥çã♦ ❞❡ ❧✉❝r♦ t♦t❛❧✳ ❜✮ ❆❝❤❛r ❛ ❢✉♥çã♦ ❞❡ r❡❝❡✐t❛ t♦t❛❧❀ ❝✮ ◗✉❛❧ é ♦ ❝✉st♦ ♠é❞✐♦ ♣❛r❛ q = 10❄✳ ❞✮ ❉❡t❡r♠✐♥❡ ❛ ❢✉♥çã♦ ❞❡ ❞❡♠❛♥❞❛✳ ✶✵✹ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✵✳ ❖ ❝✉st♦ ♠❡♥s❛❧ ✜①♦ ❞❡ ✉♠❛ ❢á❜r✐❝❛ q✉❡ ♣r♦❞✉③ ❡sq✉✐s✱ é ❘✩4.200 ❡ ♦ ❝✉st♦ ✈❛r✐á✈❡❧ ❘$55 ♣♦r ♣❛r ❞❡ ❡sq✉✐s✳ ❖ ♣r❡ç♦ ❞❡ ✈❡♥❞❛ é ❘$105 ♣♦r ♣❛r ❞❡ ❡sq✉✐s✳ ✭❛✮ ❙❡ x ♣❛r❡s ❞❡ ❡sq✉✐s sã♦ ✈❡♥❞✐❞♦s ❞✉r❛♥t❡ ✉♠ ♠ês✱ ❡①♣r❡ss❡ ♦ ❧✉❝r♦ ♠❡♥s❛❧ ❝♦♠♦ ❢✉♥çã♦ ❞❡ x✳ ✭❜✮ ❯s❡ ♦ r❡s✉❧t❛❞♦ ❞❛ ♣❛rt❡ ✭❛✮ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ ❧✉❝r♦ ❞❡ ❞❡③❡♠❜r♦ s❡ ♣❛r❡s ❞❡ ❡sq✉✐s ❢♦r❛♠ ✈❡♥❞✐❞♦s ♥❡ss❡ ♠ês✳ ✭❝✮ 600 ◗✉❛♥t♦s ♣❛r❡s ❞❡ ❡sq✉✐s ❞❡✈❡♠ s❡r ✈❡♥❞✐❞♦s ♣❛r❛ q✉❡ ❛ ❢á❜r✐❝❛ ❡♥❝❡rr❡ ✉♠ ♠ês s❡♠ ❧✉❝r♦ ♥❡♠ ♣r❡❥✉í③♦❄ ✷✶✳ ❯♠ ❢❛❜r✐❝❛♥t❡ ❞❡ ❞♦✐s t✐♣♦s ❞❡ r❛çã♦ ♣❛r❛ ❛✈❡s✱ ♣r♦❞✉③ x−3 r❛çã♦ A ❡ y t♦♥❡❧❛❞❛s ❞❛ r❛çã♦ B ♦♥❞❡ y = ✳ x−1 x t♦♥❡❧❛❞❛s ♣♦r ❞✐❛ ❞❛ ❉❡t❡r♠✐♥❡ ❛ ❢✉♥çã♦ r❡❝❡✐t❛ t♦t❛❧✱ s❛❜❡♥❞♦ q✉❡ ♦s ♣r❡ç♦s ✜①♦s ♣♦r t♦♥❡❧❛❞❛ sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ p1 ❡ p2 ✷✷✳ ❆s ❡q✉❛çõ❡s ❞❡ ❞❡♠❛♥❞❛ ❡ ♦❢❡rt❛ ❞♦ ♠❡r❝❛❞♦ sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ qp + 2 = 5p ❡ ♦♥❞❡ p ♦♥❞❡ 3 p2 = p1 ✳ 4 q 2 + p2 − 36 = 0 é ♦ ♣r❡ç♦ ❡♠ r❡❛✐s ❘✩✳ ❚r❛❝❡ ✉♠ ❡s❜♦ç♦ ❞❛s ❝✉r✈❛s ❞❡ ♦❢❡rt❛ ❡ ❞❡♠❛♥❞❛ ♥✉♠ ♠❡s♠♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ❉❡t❡r♠✐♥❡ ❛ q✉❛♥t✐❞❛❞❡ ❡ ♦ ♣r❡ç♦ ❞❡ ❡q✉✐❧í❜r✐♦✳ ✷✸✳ ❖ ♣❡rí♦❞♦ ❞❡ ✉♠ ♣ê♥❞✉❧♦ ✭♦ t❡♠♣♦✱ ♣❛r❛ ✉♠❛ ♦s❝✐❧❛çã♦ ❝♦♠♣❧❡t❛✮ é ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ à r❛✐③ q✉❛❞r❛❞❛ ✭❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ♣ê♥❞✉❧♦✳ ❡ s❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❢♦r 240 cm ♦ ♣❡rí♦❞♦ s❡rá ❞❡ 3 s✳ ✭❛✮ ❊①♣r❡ss❡ ♦ ♥ú♠❡r♦ ❞❡ s❡❣✉♥❞♦s ❞♦ ♣❡rí♦❞♦ ❞❡ ✉♠ ♣ê♥❞✉❧♦ ❝♦♠♦ ❢✉♥çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❝❡♥tí♠❡tr♦s ❞❡ s❡✉ ❝♦♠♣r✐♠❡♥t♦✳ ♣❡rí♦❞♦ ❞❡ ✉♠ ♣ê♥❞✉❧♦ ❞❡ 60 cm Kg ✳ ❆❝❤❡ ♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦✳ ✷✹✳ ❆ ❢✉♥çã♦ ❞❡ ❝✉st♦ t♦t❛❧ ❞❡ ✉♠❛ ❡♠♣r❡s❛ ❞❛❞♦ ❡♠ ✭❜✮ C(x) = 0, 2x2 − 6x + 100 ♦♥❞❡ x é ♠é❞✐♦ ❡ ♦ ✈❛❧♦r ❞❡ x ♣❛r❛ q✉❡ ♦ ❝✉st♦ A&A ❉❡t❡r♠✐♥❡ ❛ ❢✉♥çã♦ ❞❡ ❝✉st♦ é t♦t❛❧ s❡❥❛ ♠í♥✐♠♦✳ ✷✺✳ ❈❛❧❝✉❧❛r ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ ✉♠ ♠♦♥♦♣♦❧✐st❛ s❡ ❛ ❢✉♥çã♦ ❞❡ ❝✉st♦ é 2 0, 5q + 20q + 45 ❡ ♦ ♣r❡ç♦ ❞❡ ✈❡♥❞❛ ❞❡ ❝❛❞❛ ✉♥✐❞❛❞❡ é C(q) = p = 60 − q ✳ q ✉♥✐❞❛❞❡s ❞❡ ✉♠ ❝❡rt♦ ♣r♦❞✉t♦✱ ♦ ❝✉st♦ t♦t❛❧ ❞❡ C(q) = q − 6q + 15q r❡❛✐s✳ ❊♠ q✉❡ ♥í✈❡❧ ❞❡ ♣r♦❞✉çã♦ ♦ ✏❝✉st♦ ✷✻✳ ❆❞♠✐t❛♠♦s q✉❡✱ ❛♦ s❡ ❢❛❜r✐❝❛r❡♠ ❢❛❜r✐❝❛çã♦ é ❞❡ 3 2 ♠é❞✐♦✑ ♣♦r ✉♥✐❞❛❞❡ s❡rá ♦ ♠❡♥♦r❄ ✷✼✳ ❙ã♦ ❞❛❞❛s ❛s ❡q✉❛çõ❡s ❞❡ ♦❢❡rt❛ ❡ ❞❡♠❛♥❞❛ ❞❡ ✉♠ ❝❡rt♦ ♣r♦❞✉t♦✿ 2 q − p + 4 = 0✳ ❉❡t❡r♠✐♥❡ ❛ q✉❛♥t✐❞❛❞❡ ❡ ♦ ♣r❡ç♦ ❞❡ ❡q✉✐❧í❜r✐♦✳ 2q = p − 12 ❡ ✷✽✳ ❉❡t❡r♠✐♥❡ ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❡ ❞❡s❡♥❤❛r ♦ ❣rá✜❝♦ ❞❛s ❝✉r✈❛s✿ ✶✳ R(q) = 100q, ✷✳ R(q) = 10q − 0, 5q 2 , ✸✳ R(q) = 80q, C(q) = 50 + 3q C(q) = 10 + q C(q) = 0, 1q 2 + 5q + 200 ✶✵✺ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✾✳ ❚❡♠♦s ❛s ❡q✉❛çõ❡s ❞❡ ♦❢❡rt❛ ❡ ❞❡♠❛♥❞❛✱ ❞❡t❡r♠✐♥❛r ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡ ❞❡s❡♥❤❛r ♦ ❣rá✜❝♦ ♥✉♠ ♠❡s♠♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ q = 50 + 2p ❡ ❛✮ q = p+1 ❡ q(p + 10) = 500✳ q = 10 − p ❀ ❜✮ q ✉♥✐❞❛❞❡s ❞❡ ✉♠❛ ♠❡r❝❛❞♦r✐❛ é C(q) = 20q + 20.000✱ ❛ ❡q✉❛çã♦ ❞❛ ❞❡♠❛♥❞❛ é p + q = 5.000✱ ♦♥❞❡ q sã♦ ❛s ✉♥✐❞❛❞❡s ❞❡♠❛♥❞❛❞❛s ❛ ❝❛❞❛ s❡♠❛♥❛ ❛♦ ♣r❡ç♦ ✉♥✐tár✐♦ ❞❡ p r❡❛✐s✳ ❉❡t❡r♠✐♥❡ ♦ ❧✉❝r♦ ❛♦ ✈❡♥❞❡r ❛s q ✉♥✐❞❛❞❡s✳ ✸✵✳ ❯♠ ❝♦♠❡r❝✐❛♥t❡ ❡st✐♠❛ q✉❡ ♦ ❝✉st♦ ❞❡ ♣r♦❞✉çã♦ ❞❡ ✸✶✳ ❙✉♣♦♥❤❛ q✉❡ ♦ ❝✉st♦ t♦t❛❧ s❡❥❛ ❞❛❞♦ ♣♦r 10q − 0, 5q 2 ✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❡ q C(q) = 10 + q R(q) = ♣❛r❛ ♦ q✉❛❧ s❡ ♦❜té♠ ✉t✐❧✐❞❛❞❡ ♠á①✐♠❛✳ ✸✷✳ ❆ s❡❣✉✐♥t❡ ✏ ❜❛rr❛ ✑ ❡stá ❢♦r♠❛❞❛ ♣♦r três s❡❣✲ 1; 2; 1 ❝❡♥✲ 2; 3; 1 ✉♥✐❞❛❞❡s ♠❡♥t♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦s ✐❣✉❛✐s ❛ tí♠❡tr♦s✱ ❡ ♦ ♣❡s♦ é ✐❣✉❛❧ ❛ ❡ ❛ r❡❝❡✐t❛ t♦t❛❧ ✛ ✛ A ✳ ✳ ✳ 1 ✲✛ x 2g ✳ ✳ ✳ 2 ✲ M 3g ✲✛ ✳ ✳ ✳ ✳ ✳ ✳ 1 ✲ 1g ✳ ✳ ✳ ✳ ✳ ✳ ❞❡ ♣❡s♦ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖ ♣❡s♦ ❞♦ s❡❣♠❡♥t♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ q✉❡ ✈❛❧♦r❡s ❞❡ x AM é ✐❣✉❛❧ ❛ f (x)✱ q✉❡ é ❢✉♥çã♦ ❞❡ x✳ P❛r❛ ❡stá ❞❡✜♥✐❞❛ ❡st❛ ❢✉♥çã♦❄✳ ❆♣r❡s❡♥t❛r s✉❛ ❢♦r♠❛ ❛♥❛❧ít✐❝❛ ❞❡st❛ ❢✉♥çã♦ ❡ ❝♦♥str✉✐r s✉❡ ❣rá✜❝♦✳ f ❞❡ R ❡♠ R f [f (x)] + f [f (y)] + 2f (x)f (y)✳ ✸✸✳ ❉❛❞❛ ❛ r❡❧❛çã♦ ❞❡ ❞❡✜♥✐❞❛ ♣♦r ✶✵✻ f (x) = x2 ✱ ♠♦str❡ q✉❡ f (x2 + y 2 ) = 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✳✺ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❖♣❡r❛çõ❡s ❝♦♠ ❢✉♥çõ❡s ❉❡✜♥✐çã♦ ✷✳✶✺✳ ❉✐③❡♠♦s q✉❡ ❞✉❛s ❢✉♥çõ❡s f : A −→ R ❡ g : A −→ R sã♦ ✐❣✉❛✐s q✉❛♥❞♦ D(f ) = D(g) ❡ f (x) = g(x) ∀ x ∈ D(f )✳ ❉❡✜♥✐çã♦ ✷✳✶✻✳ ❙❡❥❛♠ f ❡ g ❞✉❛s ❢✉♥çõ❡s r❡❛✐s ❝♦♠ D(f ) = A ❡ D(g) = B s❡ A∩B 6= ∅ ❞❡✜♥✐♠♦s✿ . (f + g)(x) = f (x) + g(x) ❛✮ ❋✉♥çã♦ s♦♠❛ ❞❡ f ❡ g : ❜✮ ❋✉♥çã♦ ❞✐❢❡r❡♥ç❛ ❞❡ f ❡ ❝✮ ❋✉♥çã♦ ♣r♦❞✉t♦ ❞❡ f g: ❡ ❞✮ ❋✉♥çã♦ q✉♦❝✐❡♥t❡ ❞❡ f g: ❡ ❡ D(f + g) = A ∩ B ✳ . (f − g)(x) = f (x) − g(x) ❡ D(f − g) = A ∩ B . (f · g)(x) = f (x)g(x) ❡ D(f · g) = A ∩ B   f . f (x) (x) = s❡♠♣r❡ q✉❡ ♦ ❞♦♠í♥✐♦ g : g g(x) f D( ) = { x ∈ A ∩ B /. g(x) 6= 0 } g ❝✉♠♣r❛✿ . (kf )(x) = kf (x) ❡✮ Pr♦❞✉t♦ ❞❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦r ✉♠❛ ❢✉♥çã♦ ✿ t❛♥t❡ ✳ ◆❡st❛ ❝❛s♦ k é ❝♦♥s✲ D(kf ) = D(f ) . | f | (x) =| f (x) | ❢✮ ❋✉♥çã♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ✿ ❊①❡♠♣❧♦ ✷✳✺✷✳ ♦♥❞❡ ❡ D(| f |) = D(f ) √ √ ❉❛❞❛ ❛s ❢✉♥çõ❡s f (x) = 25 − x2 ❡ g(x) = x2 − 9 ❝♦♠ s❡✉s r❡s♣❡❝t✐✈♦s ❞♦♠í♥✐♦s D(f ) = [−5, 5] ❡ D(g) = (−∞, −3] ∪ [3, +∞)✱ t❡♠♦s✿ √ ❛✮ (f + g)(x) = ❜✮ (f − g)(x) = ❝✮ (f · g)(x) = ❞✮ √ 25 − x2 f (x) = √ , g x2 − 9 √ √ 25 − x2 − 25 − x2 · ❡✮ √ (kf )(x) = k 25 − x2 ❢✮ | f | (x) =| √ √ 25 − x2 + √ x2 − 9 √ x2 − 9 x 2−9 ❡ ❡ D(f + g) = [−5, −3] ∪ [3, 5]✳ ❡ D(f − g) = [−5, −3] ∪ [3, 5]✳ D(f · g) = [−5, −3] ∪ [3, 5]✳ f D( ) = [−5, −3) ∪ (3, 5] g ❡ 25 − x2 |= D(kf ) = [−5, 5] √ 25 − x2 ❡ D(| f |) = [−5, 5] ✶✵✼ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✳✺✳✶ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s ❉❡✜♥✐çã♦ ✷✳✶✼✳ f : A −→ R ❡ g : B −→ R ❞✉❛s ❢✉♥çõ❡s t❛✐s q✉❡ Im(f ) ⊆ B ❀ ❛ ❢✉♥çã♦ . (g ◦ f ) ❞❡✜♥✐❞❛ ♣♦r (g ◦ f )(x) = g(f (x)) ❞❡♥♦♠✐♥❛✲s❡ ✏ ❢✉♥çã♦ ❝♦♠♣♦st❛ ❞❡ g ❡ f ✑ ❙❡❥❛♠ ✭♥❡ss❛ ♦r❞❡♠✮✳ ❖ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ g ◦ f é✿ D(g ◦ f ) = { x ∈ D(f ) /. f (x) ∈ D(g) } ❖ ❡sq✉❡♠❛ ❞❛ ❋✐❣✉r❛ ✭✷✳✷✼✮ ♠♦str❛ ♦ q✉❡ ❛❝♦♥t❡❝❡ ♥❛ ❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s✳ A ✬ x ✫ ✩ f ❩ g◦f ✬ ✲ ✩ ✤✜ B ✲ g ✲ ✲ f (x) ❩ ❩ ❩ ❩ ❩ ✪ ❩ ❩ ✫ ✪ ✚ ✚ ✚ ✩ ✬ ✩ ✲ Im(f ) ✣✢ C ✬ g(f (x)) ❃ ✚ ✚ ✚Im(g) ✚ ✫ ✪ ✚ ✫ (g ◦ f )(x) ✪ ❋✐❣✉r❛ ✷✳✷✼✿ ❊①❡♠♣❧♦ ✷✳✺✸✳ A = { 1, 2, 3, 4, 5 } ❡ s❡❥❛♠ f, g : A −→ A ❞❡✜♥✐❞❛s ♣♦r✿ f (1) = 3, f (2) = 5, f (3) = 3, f (4) = 1, f (5) = 2, g(1) = 4, g(2) = 1, g(3) = 1, g(4) = 2, g(5) = 3 ✳ ❉❡t❡r♠✐♥❡ g ◦ f ❡ f ◦ g ✳ ❙❡❥❛ ❙♦❧✉çã♦✳ (g ◦ f )(1) = g(f (1)) = g(3) = 1 (g ◦ f )(2) = g(f (2)) = g(5) = 3 (g ◦ f )(3) = g(f (3)) = g(3) = 1 (g ◦ f )(4) = g(f (4)) = g(1) = 4 (g ◦ f )(5) = g(f (5)) = g(2) = 1 ❖❜s❡r✈❡✱ ❛s ❢✉♥çõ❡s g ◦ f ❡ f ◦ g ♥ã♦ (f ◦ g)(1) = f (g(1)) = f (4) = 1 (f ◦ g)(2) = f (g(2)) = f (1) = 3 (f ◦ g)(3) = f (g(3)) = f (1) = 3 (f ◦ g)(4) = f (g(4)) = f (2) = 5 (f ◦ g)(5) = f (g(5)) = f (3) = 3 tê♠ ❛ ♠❡s♠❛ ❞❡✜♥✐çã♦✳ ❊①❡♠♣❧♦ ✷✳✺✹✳ ❛✮ ❉❛❞❛s ❛s ❢✉♥çõ❡s f (x) = x2 − 1 ❡ g(x) = 2x✱ ✶✵✽ ❝❛❧❝✉❧❛r f [g(x)] ❡ g[f (x)]✳ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❜✮ ❉❛❞❛s ❛s ❢✉♥çõ❡s f (x) = 5x ❡ f [g(x)] = 3x + 2✱ ❝❛❧❝✉❧❛r g(x)✳ ❝✮ ❉❛❞❛s ❛s ❢✉♥çõ❡s f (x) = x2 + 1 ❡ g(x) = 3x − 4✱ ❞❡t❡r♠✐♥❡ f [g(3)]✳ ❙♦❧✉çã♦✳ g[f (x)] = g(x2 −1) = 2(x2 −1) = 2x2 −2✳ ✭❛✮ f [g(x)] = f (2x) = (2x)2 −1 = 4x2 −1 ✭❜✮ ❈♦♠♦ f (x) = 5x✱ ❡♥tã♦ f [g(x)] = 5 · g(x)✳ P♦ré♠✱ f [g(x)] = 3x + 2❀ ❧♦❣♦ 5 · g(x) = 3x + 2✱ ❡ ❞❛í g(x) = ✭❝✮ g(3) = 3(3) − 4 = 5 (3x + 2) ✳ 5 ❡♥tã♦ f [g(3)] = f (5) = 52 + 1 = 25 + 1 = 26✳ ❊①❡♠♣❧♦ ✷✳✺✺✳ ❙❡❥❛♠ f ❡ g ❞✉❛s ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ♣♦r f (x) = 3x − 2 ❡ g(x) = x2 + 4x✳ ❉❡t❡r♠✐♥❡ ❛s ❢✉♥çõ❡s g ◦ f ❡ f ◦ g ✳ ❙♦❧✉çã♦✳ ❚❡♠♦s ♦s s❡❣✉✐♥t❡s ❞♦♠í♥✐♦s ❡ ✐♠❛❣❡♥s ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ❢✉♥çõ❡s ✿ D(f ) = R, Im(f ) = R, D(g) = R ❡ Im(g) = [−4, +∞)✳ ✐✮ ❉♦ ❢❛t♦ Im(f ) ⊆ D(g) ❡♥tã♦ (g ◦ f )(x) = g(f (x)) = [f (x)]2 + 4f (x) [3x − 2]2 + 4[3x − 2] = 9x2 − 4✳ ⇒ g(f (x)) = P♦rt❛♥t♦✱ (g ◦ f )(x) = 9x2 − 4 ❡ D(g ◦ f ) = R✳ ✐✐✮ ❉♦ ❢❛t♦ Im(g) ⊆ D(f ) ❡♥tã♦ (f ◦ g)(x) = f (g(x)) = 3g(x) − 2 3(x2 + 4x) − 2 = 3x2 + 12x − 2✳ ⇒ f (g(x)) = P♦rt❛♥t♦✱ (f ◦ g)(x) = 3x2 + 12x − 2 ❡ D(f ◦ g) = R✳ ▼✉✐t❛s ✈❡③❡s sã♦ ❞❛❞❛s ❢✉♥çõ❡s f (x) ❡ g(x) s❡♠ ❡s♣❡❝✐✜❝❛r q✉❛✐s sã♦ s❡✉s ❞♦♠í♥✐♦s❀ ♣❛r❛ ♦❜t❡r gof ♦ ❞♦♠í♥✐♦ ❞❡ f ❞❡✈❡ s❡r ❡s❝♦❧❤✐❞♦ ❞❡ ♠♦❞♦ q✉❡ Im(f ) ⊆ D(g)✳ ❊①❡♠♣❧♦ ✷✳✺✻✳ ❙❡❥❛♠ ❛s ❢✉♥çõ❡s h(x) = 10 ❞❡✜♥✐❞❛ ❡♠ [−3, 4] ❡ s(x) = x2 − 8 ❞❡✜♥✐❞❛ ❡♠ [0, 7]✳ ❉❡t❡r♠✐♥❡ (h ◦ s)(x) ❡ (s ◦ h)(x)✳ ❙♦❧✉çã♦✳ ✐✮ ❙♦❧✉çã♦ ❞❡ (h ◦ s)(x) ❚❡♠♦s D(h) = [−3, 4] ❡ D(s) = [0, 7]✳ P♦r ♦✉tr♦ ❧❛❞♦✱ (h ◦ s)(x) = h(s(x)) = 10 ∀ x ∈ [0, 7] ❡ s(x) ∈ [−3, 4]❀ ✐st♦ é✱ ∀ x ∈ [0, 7] ❡ −3 ≤ x2 − 8 ≤ 4 ❡♥tã♦ x ∈ [0, 7] ❡ 5 ≤ x2 ≤ 12✳ √ P♦rt❛♥t♦✱ (hos)(x) = 10 ∀ x ∈ [ 5, √ 12] ✶✵✾ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✐✐✮ ❙♦❧✉çã♦ ❞❡ (s ◦ h)(x)✳ ❖❜s❡r✈❡ q✉❡✱ (s◦h)(x) = s(h(x)) = [h(x)]2 −8 = 102 −8 = 92✱ ♣❛r❛ t♦❞♦ x ∈ [−3, 4] ❡ h(x) ∈ [0, 7]❀ ✐st♦ é ∀ x ∈ [−3, 4] ❡ 0 ≤ 10 ≤ 7 ✭✐st♦ ú❧t✐♠♦ é ❛❜s✉r❞♦✮✳ P♦rt❛♥t♦✱ ♥ã♦ ❡①✐st❡ (soh)(x)✳ ❊①❡♠♣❧♦ ✷✳✺✼✳ ❈♦♥s✐❞❡r❡♠♦s ❛s ❢✉♥çõ❡s h(x) = √ (g ◦ h)(x)✳ x − 15 ❡ g(x) = x2 + 5❀ ❞❡t❡r♠✐♥❡ (h ◦ g)(x) ❡ ❙♦❧✉çã♦✳ ✐✮ ❚❡♠♦s D(h) = [15, +∞) ❡ D(g) = R✳ P♦r ♦✉tr♦ ❧❛❞♦✱ (h ◦ g)(x) = h(g(x)) = p p √ g(x) − 15 = (x2 + 5) − 15 = x2 − 10✳ √ D(h ◦ g) = { x ∈ R /. g(x) ∈ [15, +∞)}✱ ✐st♦ é x ∈ R ❡ 15 ≤ x2 + 5✱ ❡♥tã♦ x ≤ − 10 √ ♦✉ x ≥ 10✳ √ √ √ P♦rt❛♥t♦✱ (h ◦ g)(x) = x2 − 10 ∀ x ∈ (−∞, − 10] ∪ [ 10, +∞)✳ ✐✐✮ √ ❚❡♠♦s (g ◦ h)(x) = g(h(x)) = [h(x)]2 + 5 = [ x − 15]2 + 5 = x − 10✱ ✐st♦ [15, +∞) ❡ h(x) ∈ D(g) = R✱ ❡♥tã♦ ∀ x ∈ [15, +∞) ❡ x ∈ R✳ ∀x ∈ P♦rt❛♥t♦✱ (g ◦ h)(x) = x − 10 ∀ x ∈ [15, +∞)✳ ❊①❡♠♣❧♦ ✷✳✺✽✳ ❈♦♥s✐❞❡r❡ ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ f (x) = ❞❡t❡r♠✐♥❡ ❙♦❧✉çã♦✳ ( f ◦g x + 12, 5 − x, s❡✱ s❡✱ x<1 1≤x ❡ g(x) = ( x2 , 4x + 12, ❙❡ s❡✱ 4 ≤ x ≤ 16 ; −1≤x≤3 ❡ ✐♥❞✐q✉❡ s❡✉ ❞♦♠í♥✐♦✳ ❉❛ ❞❡✜♥✐çã♦ ❞❡ ❢✉♥çã♦ ( ❝♦♠♣♦st❛ t❡♠♦s✿ g(x) + 12, s❡✱ g(x) < 1 (f ◦ g)(x) = f (g(x)) = 5 − g(x), s❡✱ 1 ≤ g(x) ✐✮ s❡✱   x2 + 12,     (4x + 12) + 12, f (g(x)) =  5 − x2 ,     5 − (4x + 12), x2 < 1 ❡ 4 ≤ x ≤ 16 ⇒ ✐st♦ é s❡✱ x2 < 1 ❡ 4 ≤ x ≤ 16 s❡✱ 4x + 12 < 1 ❡ − 1 ≤ x ≤ 3 s❡✱ 1 ≤ x2 ❡ 4 ≤ x ≤ 16 s❡✱ 1 ≤ 4x + 12 ❡ − 1 ≤ x ≤ 3 (−1 < x < 1 ❡ 4 ≤ x ≤ 16)✱ ❧♦❣♦ x ∈ / R✳ ✶✶✵ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✐✐✮ ◗✉❛♥❞♦ 4x + 12 < 1 ❡ −1 ≤ x ≤ 3 x∈ / R✳ ✐✐✐✮ P❛r❛ 1 ≤ x2 ⇒ ❡ 4 ≤ x ≤ 16 ⇒ (x < − 11 4 ❡ − 1 ≤ x ≤ 3)✱ ❧♦❣♦ ⇒ [(x ≤ −1 ♦✉ 1 ≤ x) ❡ 4 ≤ x ≤ 16] ⇒ 4 ≤ x ≤ 16 ❧♦❣♦ f (g(x)) = 5 − x2 s❡ 4 ≤ x ≤ 16✳ ✐✈✮ ◗✉❛♥❞♦ ✭ 1 ≤ 4x + 12 ❡ − 1 ≤ x ≤ 3) 11 ≤x ❡ 4 (− ⇒ − 1 ≤ x ≤ 3) ⇒ −1 ≤ x ≤ 3 ❧♦❣♦ f (g(x)) = 5 − (4x + 12) = −4x − 7 s❡ −1 ≤ x ≤ 3✳ P♦rt❛♥t♦✱ (f ◦ g)(x) = ( 5 − x2 , s❡✱ 4 ≤ x ≤ 16 −4x − 7, s❡✱ − 1 ≤ x ≤ 3 ❊①❡♠♣❧♦ ✷✳✺✾✳ ❙❡❥❛ ❙♦❧✉çã♦✳ f (x) = 1 ✱ 1−x ❞❡t❡r♠✐♥❡ ❛ ❢✉♥çã♦ (f ◦ f )(x) = f (f (x)) = P♦r ♦✉tr♦ ❧❛❞♦✱ (f ◦ f ◦ f )(x)✳ x−1 1 1 1 = =1− 1 = 1 − f (x) x x 1 − 1−x (f ◦ f ◦ f )(x) = (f (f ◦ f ))(x) = f (f (f (x))) = 1 − ✐st♦ é (f ◦ f ◦ f )(x) = 1 − 1 1 1−x 1 ✱ f (x) = 1 − (1 − x) = x✳ P♦rt❛♥t♦ (f ◦ f ◦ f )(x) = x✳ ✷✳✺✳✷ ❋✉♥çã♦ ✐♥✈❡rs❛ ❙❡❥❛ f : A −→ B ✉♠❛ ❢✉♥çã♦ ❜✐❥❡t✐✈❛✱ ❞♦ ❢❛t♦ Im(f ) = B ✐st♦ s✐❣♥✐✜❝❛ q✉❡ ♣❛r❛ t♦❞♦ y ∈ B ❡①✐st❡ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ x ∈ A✱ t❛❧ q✉❡ f (x) = y ✳ ❊♥tã♦ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❢✉♥çã♦ g : B −→ A t❛❧ q✉❡ ❛ ❝❛❞❛ y ∈ B ❝♦rr❡s♣♦♥❞❛ ✉♠ ú♥✐❝♦ x ∈ A t❛❧ q✉❡ g(y) = x✱ ✐st♦ é✿ g(y) = x s❡✱ ❡ s♦♠❡♥t❡ s❡ f (x) = y ✶✶✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❉❡✜♥✐çã♦ ✷✳✶✽✳ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❋✉♥çã♦ ✐♥✈❡rs❛✳ f : A −→ B é ✉♠❛ ❢✉♥çã♦ ❜✐❥❡t✐✈❛✱ q✉❛♥❞♦ ❡①✐st❡✱ ❛ ❢✉♥çã♦ g : B −→ A t❛❧ q✉❡ f ◦ g = idB ❡ g ◦ f = idA ✱ ❞❡♥♦♠✐♥❛✲s❡ ❢✉♥çã♦ ✐♥✈❡rs❛ ❞❛ ❢✉♥çã♦ f ❡✱ é ❞❡♥♦t❛❞❛ −1 −1 ♣♦r f ✳ ■st♦ é f ◦ f = idB ❡ f −1 ◦ f = idA ♦♥❞❡ (x, y) ∈ f ❡ (y, x) ∈ f −1 . ❙❡ ❆ ❋✐❣✉r❛ ✭✷✳✷✽✮ ✐❧✉str❛ ❛ r❡❧❛çã♦ q✉❡ ❡①✐st❡ ❡♥tr❡ ❛ ❢✉♥çã♦ A ★✥ ❡ ❛ ❢✉♥çã♦ ✐♥✈❡rs❛ f −1 ✳ B f ★✥ ✲ y· x· ✛ ✧ ✦g f ✧✦ = f −1 ❋✐❣✉r❛ ✷✳✷✽✿ ❋✉♥çã♦ ✐♥✈❡rs❛ ❉♦ ❞✐❛❣r❛♠❛ ❞❛ ✐✮ ✭✷✳✷✽✮ t❡♠♦s✿ f −1 ◦ f = idA ∀ x ∈ A✳ ♦♥❞❡ ✭idA é ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ ❡♠ A✮ ✐st♦ é f −1 (f (x)) = f ◦ f −1 = idB ∀ x ∈ B✳ ♦♥❞❡ ✭idB é ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ ❡♠ B✮ ✐st♦ é f −1 (f (x)) = ❆ ❢✉♥çã♦ x, ✐✐✮ ❋✐❣✉r❛ ❆ ❢✉♥çã♦ x, ❊①❡♠♣❧♦ ✷✳✻✵✳ ❉❛❞❛ ❛ ❢✉♥çã♦ f (x) = ❙♦❧✉çã♦✳ ❙❡❥❛ y = f (x)✱ ❡♥tã♦ x−1 x+2 y= (x 6= 2) x−1 ✱ x+2 ❝❛❧❝✉❧❡ f −1 (x)✳ ❞❡✈❡♠♦s ✐s♦❧❛r x x−1 ⇒ y(x + 2) = x − 1 ⇒ x+2 1 + 2y 1 + 2y −(1 + 2y) ⇒ x = − ⇒ x= ✳ y−1 1−y 1 + 2y ▲♦❣♦✱ f −1 (y) = ✱ ❡♠ ❣❡r❛❧ ❛ ❢✉♥çã♦ ♥ã♦ 1−y ❊♥tã♦ ❡s❝r❡✈❡r y= y, t, z, ♥❡ss❛ ✐❣✉❛❧❞❛❞❡✳ yx + 2y = x − 1 f (x) = y = f (x)✳ ▼♦str❛r q✉❡✱ s❡ ❢✉♥çã♦ ✐♥✈❡rs❛ ❞❡ √ n a − xn , x > 0❀ y.x − x = ❞❡♣❡♥❞❡ ❞♦ ♣❛râ♠❡tr♦ é ✐♥❞✐❢❡r❡♥t❡ ❡t❝✱ ❝♦♠♦ ✈❛r✐á✈❡❧❀ ❛ss✐♠ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❊①❡♠♣❧♦ ✷✳✻✶✳ ⇒ t❡♠♦s q✉❡ f −1 (x) = 1 + 2x ✳ 1−x f (f (x)) = x✳ ❉❡t❡r♠✐♥❡ ❛ ❙♦❧✉çã♦✳ ✶✶✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❚❡♠♦s ❞❛ ❤✐♣ót❡s❡ x > 0✱ f (f (x)) = p n a− [f (x)]n = q n p √ a − [ n a − xn ]n = n a − [a − xn ] = x √ √ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡❥❛ y = f (x)✱ ❡♥tã♦ y = n a − xn ❛ss✐♠ x = n a − y n ✐st♦ é f −1 (y) = √ √ n a − y n ✱ s❡♥❞♦ ❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ✈❛r✐á✈❡❧ r❡s✉❧t❛ f −1 (x) = n a − xn ✷✳✺✳✸ ❘❡❧❛çã♦ ❡♥tr❡ ♦ ❣rá✜❝♦ ❞❡ f ❡ ❞❡ f −1 ❉❛ ❞❡✜♥✐çã♦ ❞❡ ❢✉♥çã♦ ✐♥✈❡rs❛ t❡♠♦s q✉❡✱ s❡ ♦ ♣♦♥t♦ P (a, b) ♣❡rt❡♥❝❡ ❛♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f ✱ ❡♥tã♦ Q(b, a) ♣❡rt❡♥❝❡ ❛♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f −1 ❡ ✈✐❝❡✲✈❡rs❛✳ ❖❜s❡r✈❡ ♥❛ ❋✐❣✉r❛ ✭✷✳✷✾✮ ❛ ✐❞❡♥t✐✜❝❛çã♦ ♥♦ ♣❧❛♥♦ ❞♦s ♣♦♥t♦s P (a, b) ❡ Q(b, a) ♥♦t❡✲s❡ q✉❡ sã♦ s✐♠étr✐❝♦s r❡s♣❡✐t♦ ❞❛ r❡t❛ ❜✐ss❡tr✐③ y = x✳ ■st♦ r❡s✉❧t❛ ❞♦ ❢❛t♦ s❡r ♦ q✉❛❞r✐❧át❡r♦ P AQB ✉♠ q✉❛❞r❛❞♦✱ ❞❡ ❧❛❞♦s AP = QB = b − a = AQ = P B ✳ ✻ y b a · · · · · ·P✳✳·(a, · · ·b)· · · · · · B(b, b) ✳✳ ✳✳❅ d′ ✳✳ ✳✳ ❅ ✳✳ ✳✳ ❅ ✳✳ d ✳✳ ❅ ✳✳ ✳✳ ✳ A(a, a) ✳· · · · · · · · · · · ❅ · · ·✳✳· Q(b, a) ✳✳ ✳✳ ✳✳ ✳✳ ✳✳ ✳ ✳ ✳✳ y=x✳ ✳ ✲ 0 a b x ❋✐❣✉r❛ ✷✳✷✾✿ ❋✐❣✉r❛ ✷✳✸✵✿ ▲♦❣♦ P ❡ Q sã♦ ♦s ✈ért✐❝❡s ♦♣♦st♦s ❞♦ q✉❛❞r❛❞♦✱ ❡ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ♥♦ q✉❛❞r❛❞♦ ❛s ❞✐❛❣♦♥❛✐s sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❡ ❝♦rt❛♠✲s❡ ♥♦ ♣♦♥t♦ ♠é❞✐♦✱ r❡s✉❧t❛ d = d′ ✱ ♦♥❞❡✿ d ❂ ❞✐stâ♥❝✐❛ ❞❡ P à ❜✐ss❡tr✐③ y = x✳ d′ ❂ ❞✐stâ♥❝✐❛ ❞❡ Q à ❜✐ss❡tr✐③ y = x ❙❡ ❝♦♥s✐❞❡r❛♠♦s ✉♠❛ ❢✉♥çã♦ f : A −→ B ❡ s✉❛ ❢✉♥çã♦ ✐♥✈❡rs❛ f −1 : B −→ A ❡♥tã♦ s❡✉s ❣rá✜❝♦s sã♦ s✐♠étr✐❝♦s r❡s♣❡✐t♦ ❞❛ ❜✐ss❡tr✐③ y = x✱ ♣♦✐s (x, y) ∈ Gf s❡ ❡ s♦♠❡♥t❡ s❡ (b, a) ∈ Gf −1 ✳ ❆ ❋✐❣✉r❛ ✭✷✳✸✵✮ r❡♣r❡s❡♥t❛ ♦s ❣rá✜❝♦s ❞❛ ❢✉♥çã♦ f ❡ s✉❛ ✐♥✈❡rs❛ f −1 ✳ ❊①❡♠♣❧♦ ✷✳✻✷✳ ❆ ❢✉♥çã♦ f : R −→ R ❞❡✜♥✐❞❛ ♣♦r f (x) = 3x+5 é ✐♥❥❡t✐✈❛✱ ❧♦❣♦ ❛❞♠✐t❡ ❢✉♥çã♦ ✐♥✈❡rs❛ f −1 : R −→ R✳ ❉❡t❡r♠✐♥❡♠♦s ❡st❛ ❢✉♥çã♦ ✐♥✈❡rs❛ f −1 ✳ ❙♦❧✉çã♦✳ ✶✶✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ Pr✐♠❡✐r♦ ♠ét♦❞♦✿ ❙❛❜❡♠♦s q✉❡ f (f −1 (y)) = y ✱ ❧♦❣♦ f (f −1 (y)) = 3f −1 (y) + 5 = y y−5 ∀ y ∈ R✱ s❡♥❞♦ ❛ ✈❛r✐á✈❡❧ y ♥❛ ❢✉♥çã♦ f −1 ✐♥❞❡♣❡♥❞❡♥t❡✱ 3 x−5 ∀ x ∈ R✳ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ❛ ❧❡tr❛ x ❡ ♦❜t❡r f −1 (x) = 3 ❞❡ ♦♥❞❡ f −1 (y) = ❙❡❣✉♥❞♦ ♠ét♦❞♦✿ ❙✉♣♦♥❤❛ y = f (x)✱ ❡♥tã♦ y = 3x + 5 ♦♥❞❡✱ ✐s♦❧❛♥❞♦ ❛ ✈❛r✐á✈❡❧ x r❡s✉❧t❛✿ x = ❧♦❣♦ f −1 (y) = y−5 3 x−5 3 ∀ y ∈ R ♦✉ f −1 (x) = y−5 ✱ 3 ∀ x ∈ R✳ ❊①❡♠♣❧♦ ✷✳✻✸✳ ❉❡t❡r♠✐♥❡ ❛ ❢✉♥çã♦ ✐♥✈❡rs❛ f −1 (x)✱ s❡ ❙♦❧✉çã♦✳ f (x + 1) = x2 − 3x + 2 ∀ x ∈ R+ ✳ ❙❡❥❛ t = x + 1✱ ❡♥tã♦ x = t − 1✱ ❧♦❣♦ f (t) = f (x + 1) = x2 − 3x + 2 = (t − 1)2 − 3(t − 1) + 2 = t2 − 5t + 6 ♦❜s❡r✈❡✱ ❛ ❢✉♥çã♦ f (t) ❡①✐st❡ ♣❛r❛ t ≥ 1✳ 2 2 ❈♦♥s✐❞❡r❡♠♦s p y = f (t) = t −5t+6 ❡♥tã♦ t −5t+6−y = 0✱ ♣❡❧❛ ❢ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛ t❡♠♦s t = 5± 25 − 4(6 − y) ✱ ❛ss✐♠ 2 25 − 4(6 − y) ≥ 0 ⇒ 1 + 4y ≥ 0 ⇒ y≥− 1 4 p 25 − 4(6 − y) 1 ♣❡❧❛ ❝♦♥❞✐çã♦ ❞❡ t✱ t❡♠♦s q✉❡ f −1 (y) = s❡♠♣r❡ q✉❡ y ≥ − ✳ 2 4 √ 1 5 1 + 4x 5 + s❡♠♣r❡ q✉❡ x ≥ − ; Im(f −1 ) = [ , +∞)✳ P♦rt❛♥t♦✱ f −1 (x) = 2 4 2 5+ ❊①❡♠♣❧♦ ✷✳✻✹✳ ❛✮ ❙✉♣♦♥❤❛ f (x) = x + 1✳ ❊①✐st❡♠ ❢✉♥çõ❡s g t❛✐s q✉❡ f ◦ g = g ◦ f ❄ ❜✮ ❙✉♣♦♥❤❛ f s❡❥❛ ✉♠❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡✳ P❛r❛ q✉❛✐s ❢✉♥çõ❡s g ❝✉♠♣r❡ q✉❡ f ◦ g = g ◦ f ❄ ❝✮ ❙✉♣♦♥❤❛ q✉❡ f ◦g = g ◦f ♣❛r❛ t♦❞❛s ❛s ❢✉♥çõ❡s g ✳ ▼♦str❡ q✉❡ f é ❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡✳ ❙♦❧✉çã♦✳ ❛✮ ❆ ❝♦♥❞✐çã♦ f ◦ g = g ◦ f s✐❣♥✐✜❝❛ q✉❡ g(x) + 1 = g(x + 1) ♣❛r❛ t♦❞♦ x ∈ R✳ ❊①✐st❡♠ ♠✉✐t❛s ❢✉♥çõ❡s g q✉❡ ❝✉♠♣r❡♠ ❡st❛ ❝♦♥❞✐çã♦✳ ✶✶✹ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❜✮ ❙✉♣♦♥❤❛ f (x) = c, ∀ x ∈ R✱ ❡♥tã♦ f ◦ g = g ◦ f s❡ ❡ s♦♠❡♥t❡ s❡ c = f (g(x)) = g(f (x)) = g(c) ✐st♦ é g(c) = c✳ ❝✮ ❙❡ f ◦ g = g ◦ f ♣❛r❛ t♦❞♦ g ✱ ❡♥tã♦ ❝✉♠♣r❡ ✐st♦ ♣❛r❛ t♦❞❛s ❛s ❢✉♥çõ❡s✱ ❡♠ ♣❛rt✐❝✉❧❛r ♣❛r❛ ❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ g(x) = c❀ ❧♦❣♦ ❞❛ ♣❛rt❡ ❜✮ s❡❣✉❡ q✉❡ f (c) = c ♣❛r❛ t♦❞♦ c✳ ❊①❡♠♣❧♦ ✷✳✻✺✳ ax + b ✭❝♦♥s✐❞❡r❛♥❞♦ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ ✐♥✈❡rs❛ ❞❛ ❢✉♥çã♦ ❤♦♠♦❣rá✜❝❛ f (x) = cx + d ad − bc 6= 0✮ t❛♠❜é♠ é ❤♦♠♦❣rá✜❝❛✳ ❙♦❧✉çã♦✳ ❙❡❥❛ y = f (x)✱ ❡♥tã♦ y = ax + b cx + d a ∀y = 6 ✳ c ❆ ✐❣✉❛❧❞❛❞❡ y = x= dy − b , a − cy ax + b −d ❡①✐st❡ s❡♠♣r❡ q✉❡ x 6= ✳ cx + d c ⇒ y(cx + d) = ax + b ⇒ x(yc − a) = b − dy ⇒ dx − b t❡♠♦s ❛ ❢✉♥çã♦ ✐♥✈❡rs❛ ❞❡ f (x)✳ a − cx x(ad − bc) = x ❞❛ ❤✐♣ót❡s❡ ad 6= bc✳ ❉❡ ♠♦❞♦ ❖❜s❡r✈❡✱ f ◦ f −1 (x) = f (f −1 (x)) = ad − bc ❛♥á❧♦❣♦ ♠♦str❛✲s❡ q✉❡ f −1 ◦ f (x) = x✳ dx − b P♦rt❛♥t♦ f −1 (x) = é ❤♦♠♦❣rá✜❝❛✳ a − cx ❉❡♥♦t❛♥❞♦ ❝♦♠ f −1 (x) = ❊①❡♠♣❧♦ ✷✳✻✻✳ ❊st✐♠❛✲s❡ q✉❡ ✉♠ ♦♣❡rár✐♦ ❞❡ ✉♠ ❡st❛❜❡❧❡❝✐♠❡♥t♦ q✉❡ ❢❛③ ♠♦❧❞✉r❛s ♣❛r❛ q✉❛❞r♦s ♣♦ss❛ ♣✐♥t❛r y ♠♦❧❞✉r❛s ❞❡♣♦✐s x ❤♦r❛s ❞♦ ✐♥í❝✐♦ ❞♦ s❡✉ tr❛❜❛❧❤♦ q✉❡ ❝♦♠❡ç❛ às 08 : 00 ❤♦r❛s ❞❛ ♠❛♥❤ã✱ ♦♥❞❡ y = 3x + 8x2 − x3 s❡ 0 ≤ x ≤ 4 ✳ ✭❛✮ ❆❝❤❡ ❛ t❛①❛ s❡❣✉♥❞♦ ❛ q✉❛❧ ♦ ♦♣❡rár✐♦ ❡st❛ ♣✐♥t❛♥❞♦ às 10 : 00 ❤♦r❛s ❞❛ ♠❛♥❤ã✳✭❜✮ ❆❝❤❡ ♦ ♥ú♠❡r♦ ❞❡ ♠♦❧❞✉r❛s ♣r♦♥t❛s ❡♥tr❡ 10 ❡ 11 : 00 ❤♦r❛s ❞❛ ♠❛♥❤ã✳ ❙♦❧✉çã♦✳ ❛✮ ❚❡♠♦s y = f (x) é ✉♠❛ ❢✉♥çã♦ q✉❡ ❞❡♣❡♥❞❡ ❞♦ t❡♠♣♦ x✳ ◆♦ ✐♥st❛♥t❡ x1 t❡♠♦s q✉❡ y = f (x1 ) = 3x1 + 8x21 − x31 ✳ ❙✉♣♦♥❤❛ ✉♠ ❧❛♣s♦ ❞❡ t❡♠♣♦ tr❛♥s❝♦rr✐❞♦ h ❞❡♣♦✐s ❞❡ x1 ✱ ❡♥tã♦ y = f (x1 + h) = 3(x1 + h) + 8(x1 + h)2 + (x1 + h)3 ✳ ❆ ❞✐❢❡r❡♥ç❛ f (x1 + h) − f (x1 ) △f = h h q✉❛♥❞♦ h ❢♦r tã♦ ♣❡q✉❡♥♦ ♣♦ssí✈❡❧✱ ❞❡t❡r♠✐♥❛ ❛ t❛①❛ s❡❣✉♥❞♦ ♦ q✉❛❧ ♦ ♦♣❡rár✐♦ ❡stá ♣✐♥t❛♥❞♦ x1 ❞❡♣♦✐s ❞❛s 08 : 00 ❞❛ ♠❛♥❤ã✳ ■st♦ é✱ △f (x1 ) = 3[(x1 + h) − x1 ] + 8[(x1 + h)2 − x21 ] − [(x1 + h)3 − x31 ] = = 3h + 8(2hx1 + h2 ) − (3hx21 + 3h2 x1 + h3 ) = h[3 + 8(2x1 + h) − (3x21 + 3hx1 + h2 )] ✶✶✺ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❡♥tã♦ △f (x1 ) h[3 + 8(2x1 + h) − (3x21 + 3hx1 + h2 )] = = h h 3 + 8(2x1 + h) − (3x21 + 3hx1 + h2 ) ◗✉❛♥❞♦ h △f (x1 ) = 3 + 8x1 − 3x21 ✳ ❆ t❛①❛ h x1 = 2 ❝♦rr❡s♣♦♥❞❡ ❛s 10 : 00 ❤♦r❛s✳ ❢♦r tã♦ ♣❡q✉❡♥♦ q✉❛♥t♦ ♦ ③❡r♦✱ t❡♠♦s s❡❣✉♥❞♦ ♦ q✉❛❧ ♦ ♦♣❡rár✐♦ ❡stá ♣✐♥t❛♥❞♦ q✉❛♥❞♦ ♣✐♥t❛♥❞♦ △f (2) = 3 + 8(2) − 3(22 ) = 7✳ P♦rt❛♥t♦✱ ❛ h às 10 : 00 ❤♦r❛s ❞❛ ♠❛♥❤ã é ❞❡ 7 q✉❛❞r♦s✳ ❙♦❧✉çã♦✳ ❜✮ ▲♦❣♦✱ t❛①❛ s❡❣✉♥❞♦ ❛ q✉❛❧ ♦ ♦♣❡rár✐♦ ❡st❛  11 : 00 ❤♦r❛s ❡❧❡ ♣✐♥t♦✉ y = 3(3) + 8(32 ) − 33 = 54 q✉❛❞r♦s✳ 2 3 ❤♦r❛s ❡❧❡ ♣✐♥t♦✉ y = 3(2) + 8(2 ) − 2 = 30 q✉❛❞r♦s✳ ▲♦❣♦ ❡♥tr❡ ❛s 10 : 00 ❞❛ ♠❛♥❤ã✱ ❡❧❡ ♣✐♥t♦✉ 54 − 30 = 24 q✉❛❞r♦s✳ ❆té ❛s 10 : 00 11 : 00 ❤♦r❛s  ❆té ❛s ❡ ❊①❡♠♣❧♦ ✷✳✻✼✳ ❙❡❥❛♠ ❛s ❢✉♥çõ❡s D(f ) ✶✳ ❡ f (x) = D(g) x2 36 − x2 ❡ ✷✳ g(x) = √ 8 − 3t. (f ◦g)(x) ❡ ❆❝❤❛r❀ f ( )(x) ❡ s❡✉s r❡s♣❡❝t✐✈♦s ❞♦♠í♥✐♦s✳ g ❙♦❧✉çã♦✳ ✶✳ D(f ) = R − {−6, 6} ✷✳ D(f ◦ g) = (−∞, 8/3] − {− ❡ D(g) = (−∞, 8/3] 28 } 3 ❡ f D( ) = (−∞, 8/3] − {−6} g 8 − 3x 28 + 3x f x2 √ ✳ ( )(x) = − g (36 − x2 ) 8 − 3x (f ◦ g)(x) = − ✶✶✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✷✲✹ ax + d ❝✉♠♣r❡ f (f (x)) = x ✶✳ P❛r❛ q✉❛✐s ♥ú♠❡r♦s r❡❛✐s a, b, c, d ❛ ❢✉♥çã♦ f (x) = cx + b ♣❛r❛ t♦❞♦ x❄ ✷✳ ❙❡ f é ✉♠❛ ❢✉♥çã♦ ❞❡ ✈❛r✐á✈❡❧ r❡❛❧ t❛❧ q✉❡ f (x − 2) = 2x2 + 1✱ ❞❡t❡r♠✐♥❛r✿ ✶✳ f (a + 2) − f (1) a−3 a 6= 3 ✷✳ f (a + 2) − f (2) a−2 a 6= 2 ✸✳ ❙❡ f (4x + 1) = x2 + 4x − 5 é ❢✉♥çã♦ r❡❛❧✱ ❛❝❤❛r f (5x)✳ ✹✳ ❙❡❥❛ f ❢✉♥çã♦ r❡❛❧ ❞❡✜♥✐❞❛ ♣♦r✿ f (x) = ( 2, s❡✱ 0 ≤ x ≤ 2 3, s❡✱ 2 < x < 3 ❡ g(x) = f (x + 2) + f (2x) ❆❝❤❛r D(g)✳ ✺✳ ❙❡❥❛ f : A −→ [0, 1]✳ ❉❡t❡r♠✐♥❡ ♦ ❞♦♠í♥✐♦ ❞❡ f s❡✿ 1. f (x) = |x+2| x+2 2. f (x) = −x2 + 4x + 12 3. f (x) = 1 + 2x 3 − 5x ✻✳ ❉❡t❡r♠✐♥❛r ♦ ❞♦♠í♥✐♦ ❞❡ ❞❡✜♥✐çã♦ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ r r √ 12 + x 3. f (x) = 9 − 6x + x2 x−5 q 2 √ √ 3x 4 4 2 4. f (x) = x − 4x + 12 + √ 5. f (x) = 1 − 4 + x2 4 −x − 20 + x2 ( | x + [|x|] | s❡✱ [|x|] é ♣❛r p 6. g(x) = x + [|x|], s❡✱ [|x|] é í♠♣❛r 1. f (x) = x−2 x−1 2.f (x) = 4 ✼✳ ❆ ❢✉♥çã♦ f (x) ❡st❛ ❞❡✜♥✐❞❛ ❝♦♠♦ s❡❣✉❡✿ ❡♠ ❝❛❞❛ ✉♠ ❞♦s ✐♥t❡r✈❛❧♦s n ≤ x < n + 1 1 ♦♥❞❡ n é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ f (x) ✈❛r✐❛ ❧✐♥❡❛r♠❡♥t❡✱ s❡♥❞♦ f (n) = −1, f (n+ ) = 0✳ 2 ❈♦♥str✉✐r ♦ ❣rá✜❝♦ ❞❡st❛ ❢✉♥çã♦✳ ✽✳ ❆ ❢✉♥çã♦ f ❡♠ R é t❛❧ q✉❡ f (2x) = 3x + 1✳ ❉❡t❡r♠✐♥❡ 2.f (3x + 1)✳ ✾✳ ❙❡♥❞♦ f ❡ g ❞✉❛s ❢✉♥çõ❡s t❛✐s q✉❡ f ◦ g(x) = 2x + 1 ❡ g(x) = 2 − x✳ ❉❡t❡r♠✐♥❡ f (x)✳ ✶✵✳ ❙❡ f (g(x)) = 5x − 2 ❡ f (x) = 5x + 4✱ ❡♥tã♦ g(x) é ✐❣✉❛❧ ❛✿ ✶✶✼ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✶✶✳ ❉❛❞❛s ❛s ❢✉♥çõ❡s f (x) = 4x + 5 ❡ g(x) = 2x − 5k ✱ ♦❝♦rr❡rá g ◦ f (x) = f ◦ g(x) s❡ ❡ s♦♠❡♥t❡ s❡ k ❢♦r ✐❣✉❛❧ ❛✿ ✶✷✳ ❙❡❥❛ f ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ❡♠ R t❛❧ q✉❡ f (x − 5) = 4x✳ ◆❡st❛s ❝♦♥❞✐çõ❡s✱ ♣❡❞❡✲s❡ ❞❡t❡r♠✐♥❛r f (x + 5)✳ ✶✸✳ ❙❡♥❞♦ f ❡ g ❞✉❛s ❢✉♥çõ❡s t❛✐s q✉❡✿ f (x) = ax + b ❡ g(x) = cx + d✳ ❙♦❜ q✉❡ ❝♦♥❞✐çõ❡s ♦❝♦rr❡rá ❛ ✐❣✉❛❧❞❛❞❡ g ◦ f (x) = f ◦ g(x)❄ ✶✹✳ ❙❡❥❛♠ f (x) = x+2 ❡ g(x) = x2 +a✱ ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ❞❡ a ❞❡ ♠♦❞♦ q✉❡ (f ◦g)(3) = (g ◦ f )(a − 1)✳ ✶✺✳ ❉❡t❡r♠✐♥❡ ❞✉❛s ❢✉♥çõ❡s f ❡ g t❛✐s q✉❡ h = gof ♥♦s s❡❣✉✐♥t❡s ❝❛s♦s✿ 1. h(x) = (x2 + 3)6 2. h(x) = 2sen2x √ 4. h(x) = x + 12 2  2x + 5 6. h(x) = x−4 3. h(x) = 3(x+ | x |) 5. h(x) = x2 + 16x + 64 7. h(x) = sen2 4x + 5sen4x + 2 ✶✻✳ ❉❛❞❛s ❛s ❢✉♥çõ❡s f (x) =| x + 1 | ❡ g(x) =| 2 − x |✳ ❉❡t❡r♠✐♥❡ f ◦ g ❡ g ◦ f ✳ ✶✼✳ ❙❡❥❛♠ f ❡ g ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ♣♦r✿ f (x) = ( 2x2 + 5x, s❡✱ x < 2 | x + 2 | −2x, s❡✱ x ≥ 2 ❆❝❤❛r : 1. f (1) + g(1) 4. f (4) g(1) g(x) = 2. f (0).g(0) 5. (f ◦ g)(−3) ( x + 4, s❡✱ x > 2 x2 − 3x, s❡✱ x ≤ 2 3. (f ◦ g)(2) 3 6. (g ◦ g)( ) 2 ✶✽✳ ❉❛❞❛ ❛ ❢✉♥çã♦ ❞❡ ♣r♦❞✉çã♦ 9p = 2q 2 ✱ ♦♥❞❡ q é ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✉♠ ✐♥s✉♠♦✱ ♦ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ ❛ ♣r♦❞✉çã♦ s❡ ❛ q✉❛♥t✐❞❛❞❡ ❞♦ ✐♥s✉♠♦ ❢♦r ❞✉♣❧✐❝❛❞❛❄ ❈♦♠♦ sã♦ ❡♥tã♦ ♦s r❡t♦r♥♦s ❞❛ ♣r♦❞✉çã♦❄ ✶✾✳ ❙❡❥❛♠ R = −2q 2 + 30q ❡ C = 3q + 72 ❛s ❢✉♥çõ❡s ❞❡ Receita ❡ Custo ♣❛r❛ ❝❡rt♦ ♣r♦❞✉t♦✳ ✭❛✮ ❉❡t❡r♠✐♥❡ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ✭❜r❡❛❦✲❡✈❡♥✮✳ ✭❜✮ ❋❛ç❛ ♦s ❣rá✜❝♦s ❞❡ C ❡ R ♥✉♠ ♠❡s♠♦ ❡✐①♦✳ ✭❝✮ ❉❡t❡r♠✐♥❡ ❛ ❢✉♥çã♦ ❧✉❝r♦ ❡ ❢❛ç❛ s❡✉ ❣rá✜❝♦✳ ✭❞✮ ❉❡t❡r♠✐♥❡ ❛ ❢✉♥çã♦ ❧✉❝r♦ ♠é❞✐♦ ❡ ❢❛ç❛ s❡✉ ❣rá✜❝♦ ♣♦r ♣♦♥t♦s t♦♠❛❞♦s ♥♦ ✐♥t❡r✈❛❧♦ ❞❡ ✈❛r✐❛çã♦ ❞❡ q ✳ √ ✷✵✳ ❙❡❥❛ P = 20 x5 ✉♠❛ ❢✉♥çã♦ q✉❡ ❞á ❛ q✉❛♥t✐❞❛❞❡ P ❞❡ ❝❡rt♦ ♣r♦❞✉t♦ q✉❡ é ♣r♦❞✉✲ ③✐❞❛ ❡♠ ❢✉♥çã♦ ❞❛ q✉❛♥t✐❞❛❞❡ x ❞❡ ❝❡rt♦ ✐♥s✉♠♦✳ ✭❛✮ ❊s❜♦ç❛r ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦✳ ✶✶✽ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❖ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ ❛ ♣r♦❞✉çã♦ P s❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✐♥s✉♠♦ ♣♦r ♠✉❧t✐♣❧✐❝❛❞❛ ♣♦r 6✳ ✭❜✮ ✷✶✳ ❯♠ ❧❛❜♦r❛tór✐♦✱ ❛♦ ❧❛♥ç❛r ✉♠ ♥♦✈♦ ♣r♦❞✉t♦ ❞❡ ❜❡❧❡③❛✱ ❡st❛❜❡❧❡❝❡ ✉♠❛ ❢✉♥çã♦ q✉❡ ❞á ❛ q✉❛♥t✐❞❛❞❡ ♣r♦❝✉r❛❞❛ y ♥♦ ♠❡r❝❛❞♦ ❡♠ ❢✉♥çã♦ ❞❛ q✉❛♥t✐❞❛❞❡ x ❞❡ ❝❛✐①❛s ❝♦♠ ❝❡rt❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❛♠♦str❛s✱ q✉❡ ❢♦r❛♠ ❞✐str✐❜✉í❞❛s ❡♥tr❡ ❞♦♥❛s✲❞❡✲❝❛s❛✳ ❆ ❢✉♥çã♦ ❡st❛❜❡❧❡❝✐❞❛ é ❞❛❞❛ ❝♦♠♦ y = 300 × (1, 3)x ✳ ✭❛✮ ◗✉❛❧ ❢♦✐ ❛ ♣r♦❝✉r❛ ❞♦ ♣r♦❞✉t♦ ❛♥t❡s ❞❛ ❞✐str✐❜✉✐çã♦ ❞❛ ❛♠♦str❛❄✳ ❊ ❛♣ós ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❞✉❛s ❝❛✐①❛s❄ ❊ ❛♣ós ❛ ❞✐str✐❜✉✐çã♦ ❞❛ q✉❛tr♦ ❝❛✐①❛s❄ ✭❜✮ ◗✉❛♥t❛s ❝❛✐①❛s ❞❛ ❛♠♦str❛ t❡♠ q✉❡ s❡r ❞✐str✐❜✉í❞❛s ♣❛r❛ q✉❡ ❛ q✉❛♥t✐❞❛❞❡ ♣r♦❝✉r❛❞❛ s❡❥❛ 3.000❄ ✭❝✮ ❊s❜♦❝❡ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦✳ ✷✷✳ ❆ ❞❡♠❛♥❞❛ ♠❡♥s❛❧ ❞❡ ✉♠ ❝❡rt♦ ♣r♦❞✉t♦ ♣♦r ❝♦♥s✉♠✐❞♦r é ❢✉♥çã♦ ❞❡ s✉❛ r❡♥❞❛✱ ❞❡ 30.000 ✱ ♦♥❞❡ x é ❛ r❡♥❞❛ ❡♠ ♠✐❧❤❛r❡s ❛❝♦r❞♦ ❝♦♠ ❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✿ q = 400 − x + 30 ❞❡ r❡❛✐s ❡ q é ❛ q✉❛♥t✐❞❛❞❡ ❞♦ ♣r♦❞✉t♦ ❡♠ ❣r❛♠❛s✳ ✭❛✮ ❋❛ç❛ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦✳ ✭❜✮ ❊ss❛ ❢✉♥çã♦ é ❝r❡s❝❡♥t❡ ♦✉ ❞❡❝r❡s❝❡♥t❡❄ ❆s t❛①❛s ❝r❡s❝❡♥t❡s ♦✉ ❞❡❝r❡s❝❡♥t❡s❄ P♦r q✉ê❄ ✭❝✮ ❊♠ q✉❡ ♣♦♥t♦ ❝♦rt❛ ♦ ❡✐①♦ ❤♦r✐③♦♥t❛❧ ❞♦s x✳ ◗✉❛❧ é ♦ s✐❣♥✐✜❝❛❞♦ ❞♦ ❢❛t♦❄ ✷✸✳ ❯♠ ❝♦♠❡r❝✐❛♥t❡ é ♦ r❡♣r❡s❡♥t❛♥t❡ ❞❡ ✈❡♥❞❛s ❞❡ ✉♠❛ ❝❡rt❛ ♠❡r❝❛❞♦r✐❛ ❡♠ ✉♠❛ ❝✐❞❛❞❡✳ ❱❡♥❞❡ ❛t✉❛❧♠❡♥t❡ 200 ✉♥✐❞❛❞❡s ❡ ♦❜s❡r✈❛ q✉❡ ❛ ♣♦r❝❡♥t❛❣❡♠ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❡ ✈❡♥❞❛s é ❞❡ 25% ❛♦ ❛♥♦✳ ✭❛✮ ❉❡t❡r♠✐♥❡ ❢✉♥çã♦ y = f (x) q✉❡ ❞á ❛ q✉❛♥t✐❞❛❞❡ q✉❡ s❡rá ✈❡♥❞✐❞❛ ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦ ❡♠ ❛♥♦s✱ ❛ ♣❛rt✐r ❞❡ ❤♦❥❡✳ ✭❜✮ ◗✉❛♥t♦ ❡st❛rá ✈❡♥❞❡♥❞♦ ❞❛q✉✐ ❛ ❞♦✐s ❛♥♦s❄ ❊ ❞❛q✉✐ ❛ q✉❛tr♦ ❛♥♦s❄✳ ❊s❜♦❝❡ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦✳ ✷✹✳ ❯♠❛ ✜r♠❛ ❞❡ s❡r✈✐ç♦s ❞❡ ❢♦t♦❝ó♣✐❛s t❡♠ ✉♠ ❝✉st♦ ✜①♦ ❞❡ ❘$800, 00 ♣♦r ♠ês ❡ ❝✉st♦s ✈❛r✐á✈❡✐s ❞❡ 0, 06 ♣♦r ❢♦❧❤❛ q✉❡ r❡♣r♦❞✉③✳ ❊①♣r❡ss❡ ❛ ❢✉♥çã♦ ❝✉st♦ t♦t❛❧ ❡♠ ❢✉♥çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ♣á❣✐♥❛s x ❝♦♣✐❛❞❛s ♣♦r ♠ês✳ ❙❡ ♦s ❝♦♥s✉♠✐❞♦r❡s ♣❛❣❛♠ 0, 1 ♣♦r ❢♦❧❤❛✳ ◗✉❛♥t❛s ❢♦❧❤❛s ❛ ✜r♠❛ t❡♠ q✉❡ ♣r♦❞✉③✐r ♣❛r❛ ♥ã♦ t❡r ♣r❡❥✉í③♦❄ ✷✺✳ ❆ ❡q✉❛çã♦ ❞❡ ❞❡♠❛♥❞❛ ❞❡ ✉♠ ❝❡rt♦ ♣r♦❞✉t♦ é q = 14 − 2p ❡ ❛ ❡q✉❛çã♦ ❞❡ ♦❢❡rt❛ q = 6p − 10✳ ❉❡t❡r♠✐♥❡ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✳ ✷✻✳ ❙❡❥❛ ❛ ❢✉♥çã♦ y = xn , x > 0✳ P❛r❛ q✉❡ ✈❛❧♦r❡s ❞❡ x ❡st❛ ❢✉♥çã♦ t❡♠ ✈❛❧♦r❡s ♠❛✐♦r❡s q✉❡ ♦s ❞❡ s✉❛ ❢✉♥çã♦ ✐♥✈❡rs❛✳ ax + b ❝♦✐♥❝✐❞❛ ❝♦♠ ✷✼✳ ◗✉❛❧ ❞❡✈❡ s❡r ❛ ❝♦♥❞✐çã♦ ♣❛r❛ q✉❡ ❛ ❢✉♥çã♦ ❤♦♠♦❣rá✜❝❛ y = cx + d s✉❛ ✐♥✈❡rs❛✳ ❙❛❜❡✲s❡ q✉❡ ad 6= bc✳ ✷✽✳ ◗✉❛❧ é ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❤♦♠♦❣rá✜❝❛ ✐❞❡♥t✐❝❛♠❡♥t❡ ❛ s✉❛ ✐♥✈❡rs❛ ❄ ✶✶✾ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✾✳ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ f (x) = x2 + 2x + c ❛ss✉♠❡ q✉❛❧q✉❡r ✈❛❧♦r r❡❛❧ s✐ 0 < c ≤ 1✳ x2 + 4x + 3c ✸✵✳ ❖ ♣❡s♦ ❛♣r♦①✐♠❛❞♦ ❞♦s ♠ús❝✉❧♦s ❞❡ ✉♠❛ ♣❡ss♦❛ é ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛ s❡✉ ♣❡s♦ ❝♦r♣♦r❛❧✳ ✭✶✳✮ ❊①♣r❡ss❡ ♦ ♥ú♠❡r♦ ❞❡ q✉✐❧♦s ❞♦ ♣❡s♦ ❛♣r♦①✐♠❛❞♦ ❞♦s ♠ús❝✉❧♦s ❞❡ ✉♠❛ ♣❡ss♦❛ ❝♦♠♦ ❢✉♥çã♦ ❞❡ s❡✉ ♣❡s♦ ❝♦r♣♦r❛❧✱ s❛❜❡♥❞♦ q✉❡ ✉♠❛ ♣❡ss♦❛ ❝♦♠ 68 kg t❡♠ ♣❡s♦ ❛♣r♦①✐♠❛❞♦ ❞❡ s❡✉s ♠ús❝✉❧♦s 27 kg ✳ ✭✷✳✮ ❆❝❤❡ ♦ ♣❡s♦ ♠✉s❝✉❧❛r ❛♣r♦①✐♠❛❞♦ ❞❡ ✉♠❛ ♣❡ss♦❛ ❝✉❥♦ ♣❡s♦ ❝♦r♣♦r❛❧ é ❞❡ 60 kg ✳ ✸✶✳ ❯♠ ❢❛❜r✐❝❛♥t❡ ✈❡♥❞❡ ❝❡rt♦ ❛rt✐❣♦ ❛♦s ❞✐str✐❜✉✐❞♦r❡s ❛ ❘$20 ♣♦r ✉♥✐❞❛❞❡ ♣❛r❛ ♣❡❞✐❞♦s ♠❡♥♦r❡s ❞❡ 50 ✉♥✐❞❛❞❡s✳ ◆♦ ❝❛s♦ ❞❡ ♣❡❞✐❞♦s ❞❡ 50 ✉♥✐❞❛❞❡s ♦✉ ♠❛✐s ✭❛té 600✮✱ ♦ ♣r❡ç♦ t❡♠ ✉♠ ❞❡s❝♦♥t♦ ❞❡ 2 ❝❡♥t❛✈♦s ✈❡③❡s ♦ ♥ú♠❡r♦ ❡♥❝♦♠❡♥❞❛❞♦✳ ◗✉❛❧ é ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡♥❝♦♠❡♥❞❛ q✉❡ ♣r♦♣♦r❝✐♦♥❛ ♠❛✐♦r ✐♥❣r❡ss♦ ♣❛r❛ ♦ ❢❛❜r✐❝❛♥t❡❄ ✸✷✳ ❉❡s❡♥❤❛r  ♦ ❣rá✜❝♦ ❡ ❞❡t❡r♠✐♥❡ ♦ ❝✉st♦ ♠é❞✐♦ ❞❛ ❢✉♥çã♦ ❞❡ ❝✉st♦ t♦t❛❧ C(q) = q+b aq ♦♥❞❡ a, b ❡ c sã♦ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s b < c✳ q+c ✸✸✳ ❯♠❛ ♠❡r❝❡❛r✐❛ ❛♥✉♥❝✐❛ ❛ s❡❣✉✐♥t❡ ♣r♦♠♦çã♦✿ ✏P❛r❛ ❝♦♠♣r❛s ❡♥tr❡ ❣❛♥❤❡ (x/10)% 100, 00 ❡ 600, 00 r❡❛✐s ❝♦♠♣r❡ (x + 100) r❡❛✐s ❡ ❞❡ ❞❡s❝♦♥t♦ ♥❛ s✉❛ ❝♦♠♣r❛✳✑ ◗✉❛❧ ❛ ♠❛✐♦r q✉❛♥t✐❛ q✉❡ s❡ ♣❛❣❛r✐❛ à ♠❡r❝❡❛r✐❛ ♥❡st❛ ♣r♦♠♦çã♦ ❄ ✸✹✳ ❈♦♥s✐❞❡r❡♠♦s ❞✉❛s ❢✉♥çõ❡s f ❡ g ❞❡✜♥✐❞❛s ♣♦r✿ f (x) =| x − 2 | + | x − 1 |    2x − 1, s❡✱ x ≤ −1 ❡ g(x) = 2, s❡✱ − 1 < x < 1   2 x, s❡✱ 1 ≤ x ❉❡t❡r♠✐♥❡ ❛s ❢✉♥çõ❡s f ◦ g ❡ g ◦ f ✳ ✶✷✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✳✻ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❖✉tr♦s t✐♣♦s ❞❡ ❢✉♥çõ❡s r❡❛✐s ✷✳✻✳✶ ❋✉♥çõ❡s ✐♠♣❧í❝✐t❛s ❙✉♣♦♥❤❛♠♦s t❡♠♦s ✉♠❛ ❡q✉❛çã♦ ❡♥✈♦❧✈❡♥❞♦ ❞✉❛s ✈❛r✐á✈❡✐s ❞✐❣❛♠♦s f (x, y) = C ♦♥❞❡ C ❡ y✱ ❞♦ t✐♣♦ é ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧✳ ●❡r❛❧♠❡♥t❡ ❡st❛ ❡q✉❛çã♦ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r x0y ✳ ❣r❛✜❝❛♠❡♥t❡ ♠❡❞✐❛♥t❡ ❛❧❣✉♠❛ ❝✉r✈❛ ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ P❡r❣✉♥t❛✿ x ❊st❛ ❝✉r✈❛ ♣♦❞❡ s❡r ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❄ y✻ ●❡r❛❧♠❡♥t❡ ✐st♦ ♥ã♦ ❛❝♦♥t❡❝❡✳ P❡r❣✉♥t❛✿ ❊①✐st❡ ✉♠ ✏ tr❡❝❤♦ ✑ ❞❛ ❝✉r✈❛ q✉❡ s❡❥❛ ♣♦ssí✈❡❧ ❡①♣r✐♠✐r x ✛ y ❝♦♠♦ ❢✉♥çã♦ ❞❡ x ✭♦✉ ❡♥tã♦ x✮❄❀ ② ❝♦♠♦ ❢✉♥çã♦ ❞❡ f : A −→ B 3 ✲ −3 ✐st♦ é ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r 3 ♣❛r❛ ❞❡t❡r♠✐♥❛❞♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ −3 ❄ ♥ú♠❡r♦s r❡❛✐s❄✳ ◗✉❛♥❞♦ ❛ r❡s♣♦st❛ é ❛✜r♠❛t✐✈❛✱ ❞✐③✲s❡ q✉❡ ❛ f : A −→ B é ❡q✉❛çã♦ f (x, y) = C ✳ ❢✉♥çã♦ ❋✐❣✉r❛ ✷✳✸✶✿ ❞❡✜♥✐❞❛ ✐♠♣❧í❝✐t❛♠❡♥t❡ ♣❡❧❛ ❊①❡♠♣❧♦ ✷✳✻✽✳ x2 + y 2 = 9✱ r❡♣r❡s❡♥t❛❞❛ ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ é ❝❡♥tr♦ (0, 0) ❡ r❛✐♦ 3 ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✭✷✳✸✶✮✳ ❙❡❥❛ ❛ ❡q✉❛çã♦ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❖❜s❡r✈❡ q✉❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♥ã♦ é ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦❀ ♠❛s ♣♦❞❡♠♦s s❡♣❛r❛r ❡♠ ✏ tr❡❝❤♦s✑ ♦ ❞♦♠í♥✐♦ ❞❡ss❛ r❡❧❛çã♦ ♣❛r❛ ♦❜t❡r ✐✮ ❆ ❢✉♥çã♦ f : [−3, 3] −→ R ❞❡✜♥✐❞❛ ♣♦r y ❝♦♠♦ ❢✉♥çã♦ ❞❡ f (x) = ❢❡rê♥❝✐❛ s✉♣❡r✐♦r ❛♦ ❡✐①♦✲x✳ ✐✐✮ ❆ ❢✉♥çã♦ f : [−3, 3] −→ R ❞❡✜♥✐❞❛ ♣♦r ❝✉♥❢❡rê♥❝✐❛ ✐♥❢❡r✐♦r ❛♦ ❡✐①♦✲x✳ ✷✳✻✳✷ √ 9 − x2 f (x) = − √ x✳ ❝✉❥♦ ❣rá✜❝♦ é ❛ s❡♠✐❝✐r❝✉♥✲ 9 − x2 ❝✉❥♦ ❣rá✜❝♦ é ❛ s❡♠✐❝✐r✲ ❋✉♥çã♦ ♣❡r✐ó❞✐❝❛ ❉❡✜♥✐çã♦ ✷✳✶✾✳ f : A −→ R é ♣❡r✐ó❞✐❝❛ x ∈ D(f )✱ t❡♠♦s✿ ✐✐✮ f (x + t) = f (x) ❉✐③❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ t 6= 0✱ t❛❧ q✉❡ ♣❛r❛ ✐✮ x + t ∈ D(f ) ❖ ♥ú♠❡r♦ t t♦❞♦ ❞❡♥♦♠✐♥❛✲s❡ ✏ ✉♠ ♣❡rí♦❞♦ ❞❡ ❖ ♠❡♥♦r ♣❡rí♦❞♦ ♣♦s✐t✐✈♦ ❝❛s♦ ❞✐③❡♠♦s q✉❡ f t ❞❡ f q✉❛♥❞♦ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ f ✑✳ q✉❛♥❞♦ ❡①✐st❛✱ ❞❡♥♦♠✐♥❛✲s❡ ✏ ♦ ♣❡rí♦❞♦ ❞❡ é ♣❡r✐ó❞✐❝❛ ❞❡ ♣❡rí♦❞♦ f ✑✱ ❡ ♥❡st❡ t✳ ✶✷✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✷✳✻✾✳ ❆ ❢✉♥çã♦ ♠❛♥t✐ss❛ f : R −→ R ❞❡✜♥✐❞❛ ♣♦r f (x) = x − [|x|] é ♣❡r✐ó❞✐❝❛ ❞❡ ♣❡rí♦❞♦ t = 1✳ ❖❜s❡r✈❡ q✉❡ f (x + 1) = (x + 1) − [|x + 1|] = x + 1 − [|x|] − 1 = x − [|x|] = f (x) ❡ ♥ã♦ ❡①✐st❡ ♦✉tr♦ ♥ú♠❡r♦ t t❛❧ q✉❡ 0 < t < 1 q✉❡ s❡❥❛ ♦ ♣❡rí♦❞♦ ❞❡ f ✱ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ♠❛♥t✐ss❛ ✐❧✉str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✷✳✸✷✮✳ r ✻y ✛ ✛ ✲ −2 −1 0 1 2 3 −2 x ❄ r ✻y r −x 1 −x 1 r −1 −1 0 ❄ ✲ r 1 2 r 3 x ❋✐❣✉r❛ ✷✳✸✸✿ ❋✐❣✉r❛ ✷✳✸✷✿ ❊①❡♠♣❧♦ ✷✳✼✵✳ ❆ ❢✉♥çã♦ ♠❛♥t✐ss❛ f : Z −→ {−1, 1} ❞❡✜♥✐❞❛ ♣♦r f (x) = (−1)x é ♣❡r✐ó❞✐❝❛ ❞❡ ♣❡rí♦❞♦ ❞♦✐s✱ s❡✉ ❣rá✜❝♦ ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✷✳✸✸✮ ✷✳✻✳✸ ❋✉♥çã♦ ❛❧❣é❜r✐❝❛ ❉❡✜♥✐çã♦ ✷✳✷✵✳ ❉✐③✲s❡ q✉❡ ✉♠❛ ❢✉♥çã♦ y = f (x) ❞❡✜♥✐❞❛ ♥✉♠ ❝♦♥❥✉♥t♦ A✱ é ❛❧❣é❜r✐❝❛ ❞❡ ❣r❛✉ n✱ q✉❛♥❞♦ ❡❧❛ é s♦❧✉çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❛❧❣é❜r✐❝❛ ❞❛ ❢♦r♠❛✿ P (x, y) = P0 (x)yn + P1 (x)yn−1 + · · · + Pn−1 (x)y + Pn (x) = 0 P❛r❛ n ∈ N, n ≥ 1 ❡ P0 (x), P1 (x), · · · , Pn−1 (x), Pn (x) ♣♦❧✐♥ô♠✐♦s ❞❡ ✈❛r✐á✈❡❧ x✳ ❊①❡♠♣❧♦ ✷✳✼✶✳ ❆ ❢✉♥çã♦ y = 2 x + x − 1 = 0✳ √ 3 x2 + 1 − x é ❛❧❣é❜r✐❝❛✱ ♣♦✐s ❡st❛ ❢✉♥çã♦ é s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ y 3 − ❊①❡♠♣❧♦ ✷✳✼✷✳ ❚♦❞♦ ♣♦❧✐♥ô♠✐♦ y = P (x) é ✉♠❛ ❢✉♥çã♦ ❛❧❣é❜r✐❝❛✱ ♦❜s❡r✈❡ q✉❡ é s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ y − P (x) = 0 ♣❛r❛ t♦❞♦ x ∈ R✳ ✶✷✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✳✻✳✹ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❋✉♥çã♦ ♣❛r✳ ❋✉♥çã♦ í♠♣❛r ❉❡✜♥✐çã♦ ✷✳✷✶✳ ❛✮ ❉✐③❡♠♦s q✉❡ ❜✮ ❉✐③❡♠♦s q✉❡ f : A −→ R é ✏ ❢✉♥çã♦ ♣❛r ✑✱ s❡ ♣❛r❛ t♦❞♦ x ∈ D(f )✱ t❡♠♦s✿ −x ∈ D(f ) ❡ f (−x) = f (x) ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✭✷✳✸✹✮ ♥❛ ❡sq✉❡r❞❛ f : A −→ R é ✏ ❢✉♥çã♦ í♠♣❛r ✑✱ s❡ ♣❛r❛ t♦❞♦ x ∈ D(f )✱ t❡♠♦s✿ −x ∈ D(f ) ❡ f (−x) = −f (x) ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✭✷✳✸✹✮ ♥❛ ❞✐r❡✐t❛✳ q✉❡ q✉❡ ❋✐❣✉r❛ ✷✳✸✹✿ ❊①❡♠♣❧♦ ✷✳✼✸✳ ❆ ❢✉♥çã♦ f (x) = x4 ✱ ♣❛r❛ x ∈ R é ❢✉♥çã♦ ♣❛r✱ ♣♦✐s ♣❛r❛ t♦❞♦ x ∈ R ❡ −x ∈ R t❡♠♦s f (−x) = (−x)4 = x4 = f (x)✳ ❊①❡♠♣❧♦ ✷✳✼✹✳ ❆ ❢✉♥çã♦ f (x) = x5 ✱ ♣❛r❛ x ∈ R é ❢✉♥çã♦ í♠♣❛r✱ ♣♦✐s ♣❛r❛ t♦❞♦ x ∈ R ❡ −x ∈ R t❡♠♦s f (−x) = (−x)5 = −x5 = −f (x)✳ ❖❜s❡r✈❛çã♦ ✷✳✽✳ ❛✮ ❖ ❣rá✜❝♦ ❞❡ t♦❞❛ ❢✉♥çã♦ í♠♣❛r é s✐♠étr✐❝❛ r❡s♣❡✐t♦ ❞♦ ♦r✐❣❡♠ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ❜✮ ❖ ❣rá✜❝♦ ❞❡ t♦❞❛ ❢✉♥çã♦ ♣❛r é s✐♠étr✐❝❛ r❡s♣❡✐t♦ ❞♦ ❡✐①♦✲y ✳ ❊①❡♠♣❧♦ ✷✳✼✺✳ ❈❧❛ss✐✜q✉❡ ❛s ❢✉♥çõ❡s ❛❜❛✐①♦ ❡♠ ♣❛r❡s✱ í♠♣❛r❡s ♦✉ s❡♠ ♣❛r✐❞❛❞❡✿ ❛✮ f (x) = 2x ❙♦❧✉çã♦✳ ❛✮ f (−x) = 2(−x) = −2x ❜✮g(x) = x2 − 1 ⇒ ❝✮ h(x) = x2 − 5x + 6 f (−x) = −f (x)✱ ✶✷✸ ♣♦rt❛♥t♦ f é í♠♣❛r✳ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❜✮ g(x) = x2 − 1 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R g(−x) = (−x)2 − 1 = x2 − 1 ⇒ ⇒ g(x) = g(−x)✱ ♣♦rt❛♥t♦ g é ♣❛r✳ ❝✮ h(x) = x2 − 5x + 6 ❈♦♠♦ ❡ h(−x) = (−x)2 − 5(−x) + 6 = x2 + 5x + 6 h(x) 6= h(−x)✱ ❡♥tã♦ h ♥ã♦ é ♣❛r❀ t❡♠♦s t❛♠❜é♠ −h(x) 6= h(−x)✱ ❧♦❣♦ h ♥ã♦ é í♠♣❛r✳ P♦r ♥ã♦ s❡r ♣❛r ♥❡♠ í♠♣❛r✱ ❝♦♥❝❧✉í♠♦s q✉❡ ✷✳✻✳✺ h é ❢✉♥çã♦ s❡♠ ♣❛r✐❞❛❞❡✳ ❋✉♥çã♦ ♠♦♥♦tô♥✐❝❛ ❉❡✜♥✐çã♦ ✷✳✷✷✳ ❙❡❥❛♠ I ✉♠ ✐♥t❡r✈❛❧♦ ❞❛ r❡t❛ R ❡ f : A −→ R ❢✉♥çã♦✱ s❡♥❞♦ I ⊆ A ❛✮ ❆ ❢✉♥çã♦ f é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ♥♦ ✐♥t❡r✈❛❧♦ I ✱ s❡ ♣❛r❛ t♦❞♦ x1 , x2 ∈ I ❝♦♠ x1 < x2 ❡♥tã♦ f (x1 ) < f (x2 )✳ ❜✮ ❯♠❛ ❢✉♥çã♦ f é ❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡ ♥♦ ✐♥t❡r✈❛❧♦ I ✱ s❡ ♣❛r❛ t♦❞♦ x1 , x2 ∈ I ❝♦♠ x1 < x2 ❡♥tã♦ f (x1 ) > f (x2 )✳ ❝✮ ❯♠❛ ❢✉♥çã♦ f é ❝r❡s❝❡♥t❡ ♥♦ ✐♥t❡r✈❛❧♦ I ✱ s❡ ♣❛r❛ t♦❞♦ x1 , x2 ∈ I ❝♦♠ x1 < x2 ❡♥tã♦ f (x1 ) ≤ f (x2 )✳ ❞✮ ❯♠❛ ❢✉♥çã♦ f é ❞❡❝r❡s❝❡♥t❡ ♥♦ ✐♥t❡r✈❛❧♦ I ✱ s❡ ♣❛r❛ t♦❞♦ x1 , x2 ∈ I ❝♦♠ x1 < x2 ❡♥tã♦ f (x1 ) ≥ f (x2 )✳ ❊①❡♠♣❧♦ ✷✳✼✻✳ ❆ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r f (x) = 5 é ❝r❡s❝❡♥t❡ ❡ ♥ã♦ ❝r❡s❝❡♥t❡ ❡♠ t♦❞♦ s❡✉ ❞♦♠í♥✐♦✱ ❡st❛ ❢✉♥çã♦ ♥ã♦ é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ♥❡♠ ❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡✳ ❊①❡♠♣❧♦ ✷✳✼✼✳ ❆ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r f (x) = 5x + 2✱ é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ❡♠ t♦❞♦ s❡✉ ❞♦♠í♥✐♦✳ ❆ ❢✉♥çã♦ g(x) = −x3 é ❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡ ❡♠ t♦❞♦ s❡✉ ❞♦♠í♥✐♦✳ y✻ ❊♠ q✉❛❧q✉❡r ✉♠ ❞♦s ❝❛s♦s✱ s❡ ❞✐③ q✉❡ ❛ ❢✉♥çã♦ f é ♠♦♥♦tô♥✐❝❛ ♥♦ ✐♥t❡r✈❛❧♦ I❀ ♥♦s ❝❛s♦s ❛✮ ❡ ❜✮ ❡❧❛ t❛♠❜é♠ s❡ ❞✐③ ♠♦♥♦tô♥✐❝❛ ❡str✐t❛ ♥♦ ✐♥t❡r✈❛❧♦ I✳ 2 ✛ ❊①❡♠♣❧♦ ✷✳✼✽✳ ❆ ❢✉♥çã♦✿ f (x) =| x2 − 9 | é ❡str✐t❛♠❡♥t❡ ❝r❡s✲ ❝❡♥t❡ ♥♦ ✐♥t❡r✈❛❧♦ [−3, 0] ∪ [3, +∞) ❡ ❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡ ♥♦ ✐♥t❡r✈❛❧♦ (−∞, −3] ∪ [0, 3]✳ ✶✷✹ −x ✲ −3 3 0 x ❄ ❋✐❣✉r❛ ✷✳✸✺✿ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❖ ❣rá✜❝♦ ❞❡st❛ ❢✉♥çã♦ f (x) ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✷✳✸✺✮✳ ❖❜s❡r✈❛çã♦ ✷✳✾✳ ❆ ❢✉♥çã♦ f : I −→ R é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡✭❞❡❝r❡s❝❡♥t❡✮✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ ❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡ ✭❝r❡s❝❡♥t❡✮✳ −f é Pr♦♣r✐❡❞❛❞❡ ✷✳✶✳ ❙❡ ❛ ❢✉♥çã♦ f : I −→ R é ❡str✐t❛♠❡♥t❡ ♠♦♥♦tô♥✐❝❛✱ ❡♥tã♦ f é ✐♥❥❡t✐✈❛✳ ❉❡♠♦♥str❛çã♦✳ f : I −→ R s❡❥❛ ❡str✐t❛♠❡♥t❡ ♠♦♥♦tô♥✐❝❛ ❡ s❡❥❛♠ a, b ∈ I ❞❡ ♠♦❞♦ q✉❡ a 6= b✳ ▲♦❣♦ a < b ♦✉ b < a✳ ❙✉♣♦♥❤❛♠♦s q✉❡ a < b ❡ f s❡❥❛ ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡✱ ❡♥tã♦ f (a) < f (b)✱ ❞❡ ♦♥❞❡ f (a) 6= f (b)✳ ❊♠ q✉❛❧q✉❡r ❞♦s ❞♦✐s ❝❛s♦s s❡❣✉❡ q✉❡ f (a) 6= f (b)✳ P♦rt❛♥t♦✱ f é ✐♥❥❡t✐✈❛✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ ❢✉♥çã♦ ✷✳✻✳✻ ❋✉♥çã♦ ❧✐♠✐t❛❞❛ ❉❡✜♥✐çã♦ ✷✳✷✸✳ ❙❡❥❛ ❛✮ f : R −→ R ❝✮ s✉♣❡r✐♦r♠❡♥t❡✏ ✱ q✉❛♥❞♦ ❡①✐st❡ M1 ∈ R t❛❧ f é ✏ ❧✐♠✐t❛❞❛ M2 ≤ f (x) ∀ x ∈ D(f )✳ ✐♥❢❡r✐♦r♠❡♥t❡ ✑✱ q✉❛♥❞♦ ❡①✐st❡ M2 ∈ R t❛❧ f (x) ≤ M1 ❉✐③❡♠♦s q✉❡ ❛ ❢✉♥çã♦ q✉❡ D(f )✳ f é ✏❧✐♠✐t❛❞❛ ∀ x ∈ D(f )✳ ❉✐③❡♠♦s q✉❡ ❛ ❢✉♥çã♦ q✉❡ ❜✮ ✉♠❛ ❢✉♥çã♦ r❡❛❧ ❝♦♠ ❞♦♠í♥✐♦ ❙❡ ✉♠❛ ❢✉♥çã♦ ❢♦r ❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡ ❡ ✐♥❢❡r✐♦r♠❡♥t❡✱ ❞✐③✲s❡ q✉❡ ❡❧❛ é M ∈ R M = max .{ | M1 |, | M2 | }✳ ✏ ❧✐♠✐t❛❞❛ ✑✱ ❡♠ ❝♦♥s❡q✉ê♥❝✐❛ t❡♠♦s q✉❡ ❡①✐st❡ M, ❞✮ ∀ x ∈ D(f )✱ s❡♥❞♦ x ∈ D(f ) t❛❧ q✉❡ | f (x) |≥ M ♣❛r❛ ❛❧❣✉♠ M ❞✐③❡♠♦s q✉❡ f (x) é ✏❢✉♥çã♦ ♥ã♦ ❧✐♠✐t❛❞❛ ✑✳ ❙❡ ❡①✐st❡ t❛❧ q✉❡ | f (x) |≤ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ ❊①❡♠♣❧♦ ✷✳✼✾✳ ✐✮ f (x) = xx ♥ã♦ é ❧✐♠✐t❛❞❛ ❡♠ R✱ ♣♦✐s ✐♠❛❣✐♥❡✱ ♣❛r❛ ✉♠ ❡❧❡♠❡♥t♦ ❜❛st❛♥t❡ ✏❣r❛♥❞❡✑ x ∈ R ❞♦ ❞♦♠í♥✐♦✱ ♥ã♦ ❝♦♥s❡❣✉✐r✐❛♠♦s ♦❜t❡r ♦✉tr♦ M ∈ R t❛❧ q✉❡ f s❡❥❛ ❆ ❢✉♥çã♦ ❧✐♠✐t❛❞❛✳ ✶✷✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ P♦rt❛♥t♦✱ ✐✐✮ ❆ ❢✉♥çã♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R f (x) = xx g(x) = senx A ⊆ D(R)✱ ❧✐♠✐t❛❞♦ ❡♠ A✳ ❙❡❥❛ é ♥ã♦ ❧✐♠✐t❛❞❛ ✭é ✐❧✐♠✐t❛❞❛✮✳ é ❧✐♠✐t❛❞❛✱ s❛❜❡✲s❡ q✉❡ q✉❡ f s❡ é ❧✐♠✐t❛❞♦ ♣❛r❛ t♦❞♦ |g(x)| ≤ 1✳ x ∈ A✱ f ❞✐③❡♠♦s q✉❡ é ✉♠ ❝♦♥❥✉♥t♦ ❉❡✜♥✐çã♦ ✷✳✷✹✳ ❙❡❥❛ ❛✮ f : R −→ R ❙❡ ✉♠❛ ❢✉♥çã♦ Im(f ) ✉♠❛ ❢✉♥çã♦ r❡❛❧ ❝♦♠ ❞♦♠í♥✐♦ f D(f )✳ ❢♦r ❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡✱ ♦ ♠❡♥♦r ❞♦s ❧✐♠✐t❡s s✉♣❡r✐♦r❡s ❞❛ sup .f (x) ❞❡♥♦♠✐♥❛✲s❡ ✏s✉♣r❡♠♦ ❞❛ ❢✉♥çã♦✏ ✱ ❡ ✐♥❞✐❝❛✲s❡ ❝♦♠✿ x∈D(f ) ❜✮ ❙❡ ❛ ❢✉♥çã♦ f é ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡✱ ♦ ♠❛✐♦r ❞♦s ❧✐♠✐t❡s ✐♥❢❡r✐♦r❡s ❞❛ ❞❡♥♦♠✐♥❛✲s❡ í♥✜♠♦ ❞❛ ❢✉♥çã♦✱ ❡ ✐♥❞✐❝❛✲s❡ ❝♦♠✿ Im(f ) inf .f (x)✳ x∈D(f ) ❊①❡♠♣❧♦ ✷✳✽✵✳ ✐✮ 1 ✱ ♦ x sup .f (x) = ∞ ❙❡❥❛ ❛ ❢✉♥çã♦ ♣♦✐s ♦ f (x) = inf .f (x) = 0✳ í♥✜♠♦ ❊st❛ ❢✉♥çã♦ ♥ã♦ t❡♠ s✉♣r❡♠♦✱ x∈D(f ) x∈D(f ) ✐✐✮ ✐✐✐✮ ❙❡❥❛ ❙❡❥❛ g : (0, 1] −→ R✱ ❡ h : (2, 6) −→ R✱ g(x) = 1 − 1 ✳ x ❞❡✜♥✐❞❛ ♣♦r ❆q✉✐✱ sup .f (x) = 0 ❡ ♦ ✐♥✜♠♦ ♥ã♦ ❡①✐st❡✳ x∈D(f ) h(x) = (x − 3)2 + 1✱ t❡♠♦s inf .f (x) = 1 ❡ x∈D(f ) sup .f (x) = 10 x∈D(f ) ❉❡✜♥✐çã♦ ✷✳✷✺✳ ❙❡❥❛ ❛✮ f : R −→ R ✉♠❛ ❢✉♥çã♦ r❡❛❧ ❝♦♠ ❞♦♠í♥✐♦ ❙❡ ♦ s✉♣r❡♠♦ ❞♦ ❝♦♥❥✉♥t♦ Im(f ) é t❛❧ q✉❡ sup .f (x) ≤ f (α), ∀ α ∈ D(f )✱ x∈D(f ) s✉♣r❡♠♦ é ❝❤❛♠❛❞♦ ❞❡ ♠á①✐♠♦ ❞❛ ❢✉♥çã♦ ❜✮ D(f )✳ ❙❡ ♦ í♥✜♠♦ ❞♦ ❝♦♥❥✉♥t♦ Im(f ) é t❛❧ q✉❡ f✱ ❡ ✐♥❞✐❝❛✲s❡ ❝♦♠✿ f (β) ≤ inf .f (x), í♥✜♠♦ é ❝❤❛♠❛❞♦ ❞❡ ♠í♥✐♠♦ ❞❛ ❢✉♥çã♦ x∈D(f ) f✱ ❡ ✐♥❞✐❝❛✲s❡ ❝♦♠✿ max .f (x) x∈D(f ) ∀ α ∈ D(f )✱ min .f (x) x∈A ♦ ✳ ♦ ✳ ❊①❡♠♣❧♦ ✷✳✽✶✳ ✶✷✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✶✳ ❆ ❢✉♥çã♦ ❝♦♥st❛♥t❡ f (x) = k, ∀ x ∈ R✭k ❝♦♥st❛♥t❡✮ é ❧✐♠✐t❛❞❛ ♦❜s❡r✈❡✱ sup .f (x) = x∈R max .f (x) = inf .f (x) = min .f (x) = k ✳ x∈R x∈R x∈R ✷✳ ❆ ❢✉♥çã♦ h(x) = x2 ❞❡✜♥✐❞❛ ♥♦ ✐♥t❡r✈❛❧♦ A = (−2, 3) é ❧✐♠✐t❛❞❛ ♦❜s❡r✈❡✱ sup .h(x) = 9 ❡ inf .h(x) = 0 = min .h(x) ♣♦ré♠♥ã♦ ❡①✐st❡ max .h(x)✳ x∈A x∈A x∈A x∈A ✸✳ ❆ ❢✉♥çã♦ g(x) = x2 ❞❡✜♥✐❞❛ ♥♦ ✐♥t❡r✈❛❧♦ A = [−2, 3] é ❧✐♠✐t❛❞❛ ♦❜s❡r✈❡✱ sup .h(x) = x∈A 9 = max .h(x) ❡ inf .h(x) = 0 = min .h(x)✳ x∈A ✷✳✻✳✼ x∈A x∈A ❋✉♥çã♦ ❡❧❡♠❡♥t❛r ❉❡✜♥✐çã♦ ✷✳✷✻✳ ❋✉♥çã♦ ❡❧❡♠❡♥t❛r✳ ❯♠❛ ❢✉♥çã♦ ❡❧❡♠❡♥t❛r é ❛q✉❡❧❛ q✉❡ ♦❜té♠✲s❡ ♠❡❞✐❛♥t❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♦♣❡r❛✲ çõ❡s ❞❡ ❛❞✐çã♦✱ s✉❜tr❛çã♦✱ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❞✐✈✐sã♦✱ ❡ ❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿ ❛s ❢✉♥çõ❡s ❝♦♥st❛♥t❡s❀ ❛ ❢✉♥çã♦ ♣♦tê♥❝✐❛ y = xn ❀ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ y = ax ❀ ❛s ❢✉♥çõ❡s ❧♦❣❛rít♠✐❝❛s❀ tr✐❣♦♥♦♠étr✐❝❛s ❡ tr✐❣♦♥♦♠étr✐❝❛s ✐♥✈❡rs❛s✳ ❙❡❥❛♠ f1 , f2 , f3 , · · · , fn ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ♥✉♠ ♠❡s♠♦ ❝♦♥❥✉♥t♦ A✱ ❡ a1 , a2 , a3 , · · · , an ♥ú♠❡r♦s r❡❛✐s s❡♥❞♦ n ∈ N✳ ❉❡✜♥✐çã♦ ✷✳✷✼✳ ❈♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ✜♥✐t❛✳ ❆ ❢✉♥çã♦ f : A −→ R ❞❡✜♥✐❞❛ ♣♦r✿ f = a1 f1 + a2 f1 + a3 f3 + · · · + an fn é ❞❡♥♦♠✐♥❛❞❛ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ✜♥✐t❛ ❞❡ f1 , f2 , f3 , · · · , fn ✳ ▲♦❣♦✱ f é ✉♠❛ ❢✉♥çã♦ ❡❧❡♠❡♥t❛r✳ ❊①❡♠♣❧♦ ✷✳✽✷✳ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ f (x) = (x + 3)n ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❝♦♠♦ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ✜♥✐t❛✳ ❉❡♠♦♥str❛çã♦✳ ❙❛❜❡✲s❡ ♣❡❧♦ ❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥ q✉❡ f (x) = (x + 3)n = n X k=0 = xn + 3nxn−1 + n k ! × xn−k · 3k = 3n−2 n(m − 1) 3n−1 n 32 n(n − 1) n−2 x + ··· + + + 3n 2! 2! 1 ✶✷✼ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❊s❝r❡✈❡♥❞♦ fk (x) = xn−k ✱ ❡ ak = 3k n k ! ✱ ❡♥tã♦ s❡❣✉❡ q✉❡ f (x) = (x + 3)n = a1 f1 (x) + a2 f2 (x) + a3 f3 (x) + · · · + an fn (x) ▲♦❣♦✱ f é ❞❡♥♦♠✐♥❛❞❛ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ✜♥✐t❛✳ ❊①❡♠♣❧♦ ✷✳✽✸✳ ❖ ♣r❡ç♦ ❛ ♣❛❣❛r ♣❡❧❛ ❧♦❝❛çã♦ ❞❡ ✉♠ ❛✉t♦♠ó✈❡❧ é ❝♦♠♣♦st♦ ❞❡ ❞✉❛s ♣❛rt❡s✿ ✉♠❛ t❛r✐❢❛ ✜①❛ ❞✐ár✐❛ ❞❡ R$40, 00 ❡ ✉♠❛ q✉❛♥t✐❛ ❞❡ R$0, 15 ♣♦r q✉✐❧ô♠❡tr♦ r♦❞❛❞♦✳ ▼♦str❡ q✉❡ ♦ ♣r❡ç♦ ❛ s❡r ♣❛❣♦ ♣❡❧❛ ❧♦❝❛çã♦ ❞❡ ✉♠ ❞❡st❡s ❛✉t♦♠ó✈❡✐s ♣♦r 5 ❞✐❛s ❡ r♦❞❛♥❞♦ 1200 km s❡rá✱ ❡♠ r❡❛✐s✱ ✐❣✉❛❧ ❛ R$380, 00✳ ❙♦❧✉çã♦✳ ❙❡❥❛ x ♦ ♥ú♠❡r♦ ❞❡ ❞✐❛s✱ ❡ y ♦s q✉✐❧ô♠❡tr♦s r♦❞❛❞♦s✱ ❡♥tã♦ ❛ ❢✉♥çã♦ q✉❡ ❞❡s❝r❡✈❡ ♦ ❢❡♥ô♠❡♥♦ é f (x, y) = 40x + 0, 15y ✱ ❧♦❣♦ q✉❛♥❞♦ x = 5 ❡ y = 1200 f (5, 1200) = 40(5) + (0, 15)(1200) = 200 + 180 = 380 ❊①❡♠♣❧♦ ✷✳✽✹✳ ❯♠ ❣r✉♣♦ ❞❡ ❡st✉❞❛♥t❡s ❞❡❞✐❝❛❞♦s á ❝♦♥❢❡✐çã♦ ❞❡ ❛rt❡s✐❛♥❛ t❡♠ ✉♠ ❣❛st♦ ✜①♦ ❞❡ ❘$600.00✱ ❡ ❡♠ ♠❛t❡r✐❛❧ ❣❛st❛ ❘$25.00 ♣♦r ✉♥✐❞❛❞❡ ♣r♦❞✉③✐❞❛✳ ❈❛❞❛ ✉♥✐❞❛❞❡ s❡rá ✈❡♥❞✐❞❛ ♣♦r ❘$175.00✳ ✶✳ ◗✉❛♥t❛s ✉♥✐❞❛❞❡s ♦s ❡st✉❞❛♥t❡s t❡rã♦ q✉❡ ✈❡♥❞❡r ♣❛r❛ ❡①✐st✐r ❡q✉✐❧í❜r✐♦❄ ✷✳ ◗✉❛♥t❛s ✉♥✐❞❛❞❡s ♦s ❡st✉❞❛♥t❡s t❡rã♦ ✈❡♥❞❡r ♣❛r❛ ♦❜t❡r ❧✉❝r♦ ❞❡ ❘$450.00❄ ❙♦❧✉çã♦✳ ❙❡❥❛♠ x ✉♥✐❞❛❞❡s ♣r♦❞✉③✐❞❛s✱ ♦ ❣❛st♦ t♦t❛❧ ♣❛r❛ ❛ ♣r♦❞✉çã♦ ❞❡st❛s ✉♥✐❞❛❞❡s é ❞❛❞❛ ♣❡❧❛ ❢✉♥çã♦ g(x) = 600 + 25x✱ s❡♥❞♦ q✉❡ ♦ ✐♥❣r❡ss♦ ♣❡❧❛ ✈❡♥❞❛ ❞❡st❛s x ✉♥✐❞❛❞❡s é ❞❛❞❛ ♣❡❧❛ ❢✉♥çã♦ f (x) = 175x ✭❛✮ P❛r❛ ❛❝♦♥t❡❝❡r ❡q✉✐❧í❜r✐♦ ❞❡✈❡♠♦s t❡r q✉❡✿ g(x) = f (x)✱ ❡♥tã♦ 600 + 25x = 175x ❞❡ ♦♥❞❡ 600 = 150x ♦ q✉❡ r❡s✉❧t❛ x = 4✳ P♦rt❛♥t♦✱ t❡♠ q✉❡ s❡r ✈❡♥❞✐❞❛s q✉❛tr♦ ✉♥✐❞❛❞❡s ♣❛r❛ ❡①✐st✐r ❡q✉✐❧í❜r✐♦✳ ✭❜✮ ❖ ❧✉❝r♦ é ❞❛❞❛ ♣❡❧❛ ❡①♣r❡ssã♦ f (x) = 450 + g(x)✱ ✐st♦ é 175x = 450 + (600 + 25x) ❞❡ ♦♥❞❡ r❡s✉❧t❛ 150x = 1050✱ ❧♦❣♦ x = 7✳ ❱❡♥❞❡♥❞♦ s❡t❡ ✉♥✐❞❛❞❡s ♦❜té♠✲s❡ ❧✉❝r♦ ❞❡ 450 r❡❛✐s✳ ✶✷✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✷✲✺ ✶✳ ❉❛❞❛ ❛ ❢✉♥çã♦ f (x) = √ ❞❡ f (x)✳ 1 x3 −1 ❞❡t❡r♠✐♥❛r s✉❛ ❢✉♥çã♦ ✐♥✈❡rs❛ f −1 (x) ❡ ❛ ✐♠❛❣❡♠ ✷✳ ▼♦str❡ q✉❡✱ ♣❛r❛ x > 0 ❛ ❡q✉❛çã♦ y+ | y | −x− | x |= 0 ❞❡t❡r♠✐♥❛ ❛ ❢✉♥çã♦ ❝✉❥♦ ❣rá✜❝♦ s❡rá ❛ ❜✐ss❡tr✐③ ❞♦ ♣r✐♠❡✐r♦ â♥❣✉❧♦ ❝♦♦r❞❡♥❛❞♦✱ ❡♥t❛♥t♦ ♣❛r❛ x ≤ 0 sã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ t♦❞♦s ♦s ♣♦♥t♦s ❞♦ t❡r❝❡✐r♦ q✉❛❞r❛♥t❡ ✭✐♥❝❧✉í❞♦s s❡✉s ♣♦♥t♦s ❞❡ ❢r♦♥t❡✐r❛✮ ❛s q✉❡ ❝✉♠♣r❡♠ ❛ ❡q✉❛çã♦ ❞❛❞❛✳ ✸✳ ❉❛❞❛s ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s r❡❛✐s✱ ❞❡t❡r♠✐♥❡ ❝❛s♦ ❡①✐st❛✱ s✉❛ ❢✉♥çã♦ ✐♥✈❡rs❛✳ 1. f (x) = x2 − 5x + 6 √ 4. h(x) = x2 − 4x + 4 x2 − 4 x+2 5. s(x) = x+ | x + 1 | 2. g(x) = 5 7 − 2x √ 6. t(x) = x + 2 − 5 3. f (x) = ✹✳ ❙❡ f (x) = x − 2a✱ ❞❡t❡r♠✐♥❛r ♦s ✈❛❧♦r❡s ❞❛ ❝♦♥st❛♥t❡ ❛ ❞❡ ♠♦❞♦ q✉❡ f (a2 ) = f −1 (a − 2)✳ 4 + 3x ✿ 1 − 3x 2. ▼♦str❡ q✉❡ f é 1 − 1✳ ✺✳ ❙❡❥❛ f : A −→ [−9, −1) ❞❡✜♥✐❞❛ ♣♦r f (x) = 1. ❉❡t❡r♠✐♥❛r A✳ 3. f é s♦❜r❡❄✳ ✻✳ ❙❡ f (x) = x + 2c ❡ f (c2 ) = f −1 (c)✱ ❛❝❤❛r ♦ ✈❛❧♦r ❞❡✿ 1. f (0) · f −1 (0) 2. f (1) ✳ f −1 (1) ✼✳ ❈♦♥str✉✐r ♦ ❣rá✜❝♦ ❡ ❞❡t❡r♠✐♥❛r ❛ ✐♠❛❣❡♠ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ ✶✳ ✷✳ ✸✳ ✹✳  2  x −4 , s❡✱ x 6= −2 f (x) = x+2  3, s❡✱ x = −2 ( | 4 − x2 |, s❡✱ | x |< 3 f (x) = 5, s❡✱ | x |≥ 3    | x + 3 |, s❡✱ − 4 ≤ x ≤ 0 f (x) = 3 − x2 , s❡✱ 0 < x ≤ 4   −2, s❡✱ | x |> 4  2   (x − 1) , s❡✱ 0 ≤ x < 2 f (x) = 10 − x2 , s❡✱ 2 ≤ x ≤ 3   −2, s❡✱ x < 0 ♦✉ x > 3 ✶✷✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✺✳ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠   − | x + 4 |,     x2 − 4x − 2, f (x) =  −x2 + 10x − 22,     −3, R s❡✱ − 8 ≤ x ≤ 2 s❡✱ 2 < x ≤ 5 s❡✱ 5 < x ≤ 8 s❡✱ | x |> 8 ✽✳ ❈♦♥str✉✐r ♦ ❣rá✜❝♦✱ ❞❡t❡r♠✐♥❛r ❛ ✐♠❛❣❡♠ ❡ ✈❡r✐✜q✉❡ s❡ ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s sã♦ ✐♥✈❡rsí✈❡✐s ✿  3   (x − 1) , s❡✱ 0 ≤ x < 2 1. f (x) = 10 − x2 , s❡✱ 2 ≤ x ≤ 3   −2, s❡✱ x < 0 ♦✉ x > 3    | x + 3 |, s❡✱ − 4 ≤ x ≤ 0 3. f (x) = 3 − x2 , s❡✱ 0 < x ≤ 4   −2, s❡✱ | x |> 4   | x + 4 |, s❡✱ − 8 ≤ x ≤ 2     x + 2, s❡✱ 2 < x ≤ 5 5. f (x) =  x3 , s❡✱ 5 < x ≤ 8     −3, s❡✱ | x |> 8   − | x + 4 |, s❡✱ − 8 ≤ x ≤ 2     x2 − 4x − 2, s❡✱ 2 < x ≤ 5 7. f (x) =  10x − x2 − 22, s❡✱ 5 < x ≤ 8     −3, s❡✱ | x |> 8 2. f (x) = 5(x+ | x + 1 |) 4. f (x) = x2 − 5x + 6 6. f (x) = ( | 4 − x2 |, s❡✱ | x |< 3 5, s❡✱ | x |≥ 3  2  x −4 , s❡✱ x 6= −2 8. f (x) = x+2  3, s❡✱ x = 2 ✾✳ ❉❡t❡r♠✐♥❡ ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B ♣❛r❛ q✉❡ ❛ ❡q✉❛çã♦ ❛ s❡❣✉✐r ❞❡t❡r♠✐♥❡ ✉♠❛ ❢✉♥çã♦ ✐♠♣❧í❝✐t❛ f : A −→ B ✳ 1. 4. x2 y 2 + =1 9 4 x+1 =y x 2. x2 − y 2 = 1 5. 3. x2 − 3y + y 2 − 9y = −8 | x | + | y |= 2 6. yx2 − x − 9y = 0 ✶✵✳ ❉❡t❡r♠✐♥❡ ✈❛❧♦r❡s ❞❡ a ❡ b ♥❛ ❡①♣r❡ssã♦ ❞❛ ❢✉♥çã♦ f (x) = ax2 + bx + 5 ♣❛r❛ ♦s q✉❛✐s s❡❥❛ ✈á❧✐❞❛ ❛ ✐❞❡♥t✐❞❛❞❡ f (x + 1) − f (x) = 8x + 3✳ ✶✶✳ ❱❡r✐✜q✉❡ s❡ ❛ ❢✉♥çã♦ ❛ s❡❣✉✐r é ♣❛r ♦ í♠♣❛r ❥✉st✐✜❝❛♥❞♦ s✉❛ r❡s♣♦st❛✳ 1. f (x) = −x3 + x 2. f (x) = x · ex + x2 3. f (x) = −x + x3 4. f (x) = 5. h(x) = 6. w(t) = x · et ex + e−x 2 x |x| 2 ✶✷✳ ❙❡ ♦ ❝♦♥❥✉♥t♦ A é s✐♠étr✐❝♦ ❡♠ r❡❧❛çã♦ à ♦r✐❣❡♠ ✭s❡ x ∈ A✱ ❡♥tã♦ −x ∈ A✮ ♣❛r❛ ✶✸✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ f : A −→ R ♣r♦✈❡ q✉❡ f (x) + f (−x) é ♣❛r✳ 2 t♦❞❛ 1. R ❛ ❢✉♥çã♦✿ 2. f (x) − f (−x) 2 é í♠♣❛r✳ ✶✸✳ ❆♣r❡s❡♥t❡ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s ❝♦♠♦ s♦♠❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ♣❛r ❡ ♦✉tr❛ í♠♣❛r✿ 1. y = x3 + 3x + 2 2. y = 1 − x3 − x4 − 2x5 ✶✹✳ ▼♦str❡ q✉❡ ♦ ♣r♦❞✉t♦ ❞❡ ❞✉❛s ❢✉♥çõ❡s ♣❛r❡s ♦✉ í♠♣❛r❡s é ✉♠❛ ❢✉♥çã♦ ♣❛r ❡✱ ♦ ♣r♦❞✉t♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ♣❛r ♣♦r ✉♠❛ í♠♣❛r é ❢✉♥çã♦ í♠♣❛r✳ n ♥❛t✉r❛❧ [0, +∞)✳ ✶✺✳ ❙❡❥❛ ✶✻✳ ❙❡❥❛ 1 x f (x) = í♠♣❛r✳ ▼♦str❡ q✉❡ f (x) = x ∈ I = (0, 1]✳ ♣❛r❛ √ n x é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ♥♦ ✐♥t❡r✈❛❧♦ P❡r❣✉♥t❛✲s❡✿ ✶✳ ❊st❛ ❢✉♥çã♦ é ❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡❄ ✷✳ ❊st❛ ❢✉♥çã♦ é ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡❄ ✸✳ ❊①✐st❡ max .f (x) x∈I ✹✳ ❊①✐st❡ ❄ min .f (x) x∈I ❄ ✶✼✳ ❆♥á❧♦❣♦ ❛♦ ❡①❡r❝í❝✐♦ ❛♥t❡r✐♦r ♣❛r❛ ❛ ❢✉♥çã♦✿ ✶✳ f (x) = x3 − x ✷✳ f (x) = x2 − 2x + 1 ✶✽✳ ▼♦str❡ q✉❡ 2x x+2 q✉❛♥❞♦ x ∈ I = [−4, 4]✳ q✉❛♥❞♦ x ∈ I = [−4, 4]✳ é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ♥♦s ✐♥t❡r✈❛❧♦s (−∞, −2) ❡ (−2, +∞)✳ ✶✾✳ ▼♦str❡ q✉❡ t♦❞❛ ❢✉♥çã♦ ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ♦✉ ❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡ é ✐♥❥❡✲ t✐✈❛✳ n R✳ ✷✵✳ ❙❡❥❛ ❡♠ ✷✶✳ ❙❡♥❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ í♠♣❛r✱ ♠♦str❡ q✉❡ f (x) = senx ❡ g(x) = log x✱ ✷✸✳ ❙❡❥❛ ✷✹✳ ❙❡ f f (x) = Ln(x)✳ ▼♦str❡ q✉❡ é ✉♠❛ ❢✉♥çã♦ t❛❧ q✉❡ ❡♥tã♦ ✷✺✳ ❙❡❥❛♠ ✶✳ ❙❡ f (2 + p) ❡ g x+1 é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ♣❛r❛ ♦ q✉❛❧ 2 x > xn π g[f ( )] 2 ♣❛r❛ t♦❞❛s ❛s x ≥ 100✳ f (x) + f (x + 1) = f (x(x + 1))✳ f (1) = a, f (p) = b ❡ f (x + y) = f (x) · f (y), ∀ x, y ∈ R✱ é ✐❣✉❛❧ ❛✿ f : A −→ B f √ n ♣❡❞❡✲s❡ ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ❞❡ n∈Z ✷✷✳ ❉❡t❡r♠✐♥❡ ♦ ♣♦ssí✈❡❧ ✈❛❧♦r ♣❛r❛ f (x) = ❡ g : B −→ R sã♦ ✐♥❥❡t✐✈❛s✱ ❡♥tã♦ ❞✉❛s ❢✉♥çõ❡s✳ ❉❡♠♦♥str❡ q✉❡✿ g◦f é ✐♥❥❡t✐✈❛❄ ✶✸✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✳ ✸✳ ✹✳ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❙❡ f ❡ g sã♦ s♦❜r❡❥❡t✐✈❛s✱ ❡♥tã♦ g ◦ f é s♦❜r❡❥❡t✐✈❛❄ ❙❡ g ◦ f é ✐♥❥❡t✐✈❛✱ ❡♥tã♦ f é ✐♥❥❡t✐✈❛✳ ❙❡ g ◦ f é s♦❜r❡❥❡t✐✈❛✱ ❡♥tã♦ g é s♦❜r❡❥❡t✐✈❛✳ ✷✻✳ ❊♠ ✉♠ ❝❡rt♦ ❝❧✉❜❡ ❞❡ ❢✉t❡❜♦❧✱ ❛ t❛①❛ ❛♥✉❛❧ ❝♦❜r❛❞❛ ❛♦s só❝✐♦s é ❞❡ ❘$300, 00 ❡ ♦ só❝✐♦ ♣♦❞❡ ✉t✐❧✐③❛r ❝❛♠♣♦ ❞❡ ❢✉t❡❜♦❧ ♣❛❣❛♥❞♦ ❘$2, 00 ♣♦r ❤♦r❛✳ ❊♠ ♦✉tr♦ ❝❧✉❜❡✱ ❛ t❛①❛ é ❘$200, 00 ❡ ❝♦❜r❛♠ ❘$3, 00 ♣♦r ❤♦r❛ ❞❡ ✉s♦ ❞♦ ❝❛♠♣♦ ❞❡ ❢✉t❡❜♦❧✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛s q✉❡stõ❡s ✜♥❛♥❝❡✐r❛s❀ q✉❡ ❝❧✉❜❡ ✈♦❝ê ❡s❝♦❧❤❡r✐❛ ❄ ✷✼✳ ❆s ❢✉♥çõ❡s ❞❡ ♦❢❡rt❛ ❡ ❞❡♠❛♥❞❛ ❞❡ ✉♠ ❝❡rt♦ ♣r♦❞✉t♦ sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ S(p) = p − 10 ❡ D(p) = 5.600p−1 ✳ ✶✳ ✷✳ ❈❛❧❝✉❧❛r ♦ ♣r❡ç♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡ ♦ ♥ú♠❡r♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❞❡ ✉♥✐❞❛❞❡s ❡♠ ♦❢❡rt❛ ❡ ❞❡♠❛♥❞❛✳ ❈♦♥str✉í❛ ❛s ❣rá✜❝♦s ❞❛s ❢✉♥çõ❡s ♥✉♠ ♠❡s♠♦ ♣❛r ❞❡ ❡✐①♦s✳ ✷✽✳ ❯♠ ♥ú♠❡r♦ ❞❡ ❞♦✐s ❛❧❣❛r✐s♠♦s ❡①❝❡❞❡ ❡♠ ✉♠❛ ✉♥✐❞❛❞❡ ♦ sê①t✉♣❧♦ ❞❛ s♦♠❛ ❞❡ s❡✉s ❛❧❣❛r✐s♠♦s ❞❡ss❡ ♥ú♠❡r♦✳ ❙❡ ❛ ♦r❞❡♠ ❞♦s ❛❧❣❛r✐s♠♦s ❞❡ss❡ ♥ú♠❡r♦ ❢♦r ✐♥✈❡rt✐❞❛✱ ♦ ♥♦✈♦ ♥ú♠❡r♦ t❡rá ♥♦✈❡ ✉♥✐❞❛❞❡s ❛ ♠❡♥♦s ❞♦ q✉❡ ♦ ♥ú♠❡r♦ ♦r✐❣✐♥❛❧✳ ❊♥❝♦♥tr❛r ♦ ♥ú♠❡r♦ ♦r✐❣✐♥❛❧✳ ✷✾✳ ❆s ❡q✉❛çõ❡s ❞❡ ♦❢❡rt❛ ❡ ❞❡♠❛♥❞❛ ♥✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❢á❜r✐❝❛ ❡stã♦ ❞❛❞❛s ♣♦r q = 24 − p ❡ q = 10 p − 20✱ ❢✉♥çõ❡s ❧✐♥❡❛r❡s ❞♦ ♣r❡ç♦✳ ❉❡t❡r♠✐♥❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡q✉✐❧í❜r✐♦✳ ✸✵✳ ❆ ❢♦❧❤❛ ❞❡ ♣❛❣❛♠❡♥t♦ ♠❡♥s❛❧ ❞❡ ✉♠❛ ❡♠♣r❡s❛ é ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ tr❛❜❛❧❤❛❞♦r❡s✱ s❛❜❡♥❞♦ q✉❡ 20 ❞♦s tr❛❜❛❧❤❛❞♦r❡s t❡♠ ✉♠❛ ❢♦❧❤❛ ❞❡ ♣❛❣❛♠❡♥t♦ ❞❡ ❘$3000, 00✳ ✶✳ ✷✳ ❊①♣r❡ss❡ ♦ ✈❛❧♦r ❞❛ ❢♦❧❤❛ ❞❡ ♣❛❣❛♠❡♥t♦ ♠❡♥s❛❧ ❝♦♠♦ ❢✉♥çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ tr❛❜❛❧❤❛❞♦r❡s❀ ◗✉❛❧ ❛ ❢♦❧❤❛ ❞❡ ♣❛❣❛♠❡♥t♦ ♣❛r❛ 18 tr❛❜❛❧❤❛❞♦r❡s❄ ✶✸✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✳✼ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❋✉♥çõ❡s tr❛♥s❝❡♥❞❡♥t❡s ❈❤❛♠❛✲s❡ ❢✉♥çã♦ tr❛♥s❝❡♥❞❡♥t❡ ❛ ❛q✉❡❧❛ ❢✉♥çã♦ q✉❡ ♥ã♦ é ❛❧❣é❜r✐❝❛✳ ❙ã♦ ❢✉♥çõ❡s tr❛♥s❝❡♥❞❡♥t❡s✿ ❛✮ ❆ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❡ s✉❛ ✐♥✈❡rs❛✱ ❛ ❢✉♥çã♦ ❧♦❣❛r✐t♠♦✳ ❜✮ ❆s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❡ s✉❛s ✐♥✈❡rs❛s✳ ✷✳✼✳✶ ❆ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ❜❛s❡ a ❉❡✜♥✐çã♦ ✷✳✷✽✳ ❙❡ a > 0 ❡ r = √ p é ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ❞❡✜♥❡✲s❡ ar = ap/q = q ap ✳ q Pr♦♣r✐❡❞❛❞❡ ✷✳✷✳ P❛r❛ q✉❛❧q✉❡r ♣❛r ❞❡ ♥ú♠❡r♦s r, s ∈ Q t❡♠♦s ✿ a) ar .as = ar+s  a r ar d) = r b 6= 0 b b b) (ar )s = ars ar e) = ar−s as c) (ab)r = ar .br ❉❡✜♥✐çã♦ ✷✳✷✾✳ ❋✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳ ❙❡❥❛ a 6= 1 ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦✳ ❆ ❢✉♥çã♦ f : R −→ R ❞❡✜♥✐❞❛ ♣♦r f (x) = ax é ❝❤❛♠❛❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ❜❛s❡ a✳ ❖ ❞♦♠í♥✐♦ ❞❡ ❡st❛ ❢✉♥çã♦ é D(f ) = R ❡ s✉❛ ✐♠❛❣❡♠ Im(f ) = R+ = (0, +∞)✳ P❛r❛ s❡✉ ❣rá✜❝♦ ❝♦♥s✐❞❡r❡♠♦s ❞♦✐s ❝❛s♦s ❝♦♠♦ s❡ ♦❜s❡r✈❛ ♥❛ ❋✐❣✉r❛ ✭✷✳✸✻✮✳ ◗✉❛♥❞♦ 0 < a < 1 ◗✉❛♥❞♦ a > 1 ❋✐❣✉r❛ ✷✳✸✻✿ ❋✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ Pr♦♣r✐❡❞❛❞❡ ✷✳✸✳ ✶✸✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❊✶✮ ❙❡ 0 < a < 1✱ ❛ ❢✉♥çã♦ f (x) = ax é ❞❡❝r❡s❝❡♥t❡ ❡♠ t♦❞♦ s❡✉ ❞♦♠í♥✐♦✳ ❊✷✮ ❙❡ a > 1✱ ❛ ❢✉♥çã♦ f (x) = ax é ❝r❡s❝❡♥t❡ ❡♠ t♦❞♦ ❡♠ s❡✉ ❞♦♠í♥✐♦✳ ❊✸✮ ❖ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ❜❛s❡ ❛ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ P (0, 1)✳ ❊✹✮ ❙❡ 0 < a < 1✱ ❡♥tã♦ ✿ ax t❡♥❞❡ ♣❛r❛ +∞ q✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ −∞✱ ❡ ax t❡♥❞❡ ♣❛r❛ −∞ q✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ +∞✳ ❊✺✮ ❙❡ a > 1 ❡♥tã♦ ✿ ax t❡♥❞❡ ♣❛r❛ +∞ q✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ +∞✱ ❡ ax t❡♥❞❡ ♣❛r❛ −∞ q✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ −∞✳ ❊✻✮ ax+z = ax .az ❡ ax−z = ax az ❊①❡♠♣❧♦ ✷✳✽✺✳ 1 1 f (x) = (ax + a−x ) ❡ g(x) = (ax − a−x ) ♠♦str❡ q✉❡✿ 2 2 ✐✮ f (x + y) = f (x)f (y) + g(x)g(y) ✐✐✮ g(x + y) = f (x)g(y) + f (y)g(x) ❙❡❥❛♠ ❙♦❧✉çã♦✳ ✐✮ ❚❡♠♦s✿ 1 f (x + y) = (ax+y + a−x−y ) 2 P♦r ♦✉tr♦ ❧❛❞♦✿ 1 f (x) · f (y) = (ax + a−x ) · 2 1 x g(x) · g(y) = (a − a− x) · 2 ▲♦❣♦ 1 y (a + a−y ) = 2 1 y (a − a−y ) = 2 ✭✷✳✶✮ 1 x+y (a + ax−y + a−x+y + a−x−y) ✳ 4 1 x+y (a − ax−y − a−x+y + a−x−y )✳ 4 1 1 f (x) · f (y) + g(x) · g(y) = (2ax+y + 2a−x−y ) = (ax+y + a−x−y ) 4 2 ✭✷✳✷✮ ❉❡ ✭✷✳✶✮ ❡ ✭✷✳✷✮ t❡♠♦s f (x + y) = f (x)f (y) + g(x)g(y) ❙♦❧✉çã♦✳  ✐✐✮ ❚❡♠♦s 1 g(x + y) = (ax+y − a−x−y ) 2 ✭✷✳✸✮ P♦r ♦✉tr♦ ❧❛❞♦❀ 1 f (x)g(y) + f (y)g(x) = (ax + a−x )(ay − a−y ) + (ay + a−y )(ax − a−x ) = 4 1 x+y x−y −x+y −a +a − a−x−y ) + (ay+x − ay−x + a−y+x − a−y−x )] = = [(a 4 1 x+y 1 (2a − 2a−x−y ) = (ax+y − a−x−y ) 4 2 ✶✸✹ ✭✷✳✹✮ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ g(x + y) = f (x)g(y) + f (y)g(x)✳ ❉❡ ✭✷✳✸✮ ❡ ✭✷✳✹✮ t❡♠♦s ✷✳✼✳✷ ❋✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ❆ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ é ❛ ❢✉♥çã♦ ✐♥✈❡rs❛ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳ E1 ❡ E2 ❝♦♥❝❧✉✐✲s❡ q✉❡ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ a > 0 ❡ a 6= 1 é ✐♥❥❡t✐✈❛ ❡♠ s❡✉ ❞♦♠í♥✐♦ R ❡ ❉❛s ♣r♦♣r✐❡❞❛❞❡s x f (x) = a q✉❛♥❞♦ ❜❛s❡ a ❞❛❞❛ ♣♦r ♣♦rt❛♥t♦ ❛❞♠✐t❡ ❢✉♥çã♦ ✐♥✈❡rs❛✱ ❝❤❛♠❛❞❛ ✏ ❋✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ❞❡ ❜❛s❡ ❛ ✑ ❡ ❡stá ❞❡✜♥✐❞❛ ♣❡❧❛ ❢✉♥çã♦ (0, +∞) −→ R t❛❧ q✉❡ ❙❡✉ ❞♦♠í♥✐♦ é g(x) = loga x✳ D(g) = (0, +∞) ❡ s✉❛ ✐♠❛❣❡♠ Im(g) = R✳ g(x) = loga x g(x) = loga x q✉❛♥❞♦ a > 1✳ ◆❛ ❋✐❣✉r❛ ✭✷✳✸✼✮ ♠♦str❛✲s❡ ♦ ❣rá✜❝♦ ❞❡ s❡ ♠♦str❛ ♦ ❣rá✜❝♦ ❞❡ g : s❡ 0 < a < 1✳ y y ✻ ✻ x ✲ ✛ −1 ✛ 1 ◗✉❛♥❞♦ ◆❛ ❋✐❣✉r❛ ✭✷✳✸✽✮ ✲ −1 0<a<1 ◗✉❛♥❞♦ 1 x a>1 ❋✐❣✉r❛ ✷✳✸✽✿ ❋✐❣✉r❛ ✷✳✸✼✿ P♦r ❞❡✜♥✐çã♦ ❞❡ ❢✉♥çã♦ ✐♥✈❡rs❛✱ t❡♠♦s✿ ✶✮ f (g(x)) = x, ∀ x ∈ (0, +∞) ✷✮ g(f (x)) = x, ∀x∈R ♦✉ ♦✉ aloga x = x ∀ x ∈ (0, +∞)✳ loga (ax ) = x ∀ x ∈ R ❊♠ r❡s✉♠♦✿ ay = x s❡✱ ❡ s♦♠❡♥t❡ s❡ y = loga x✳ P♦r ❡①❡♠♣❧♦✱ 34 = 81 s❡✱ ❡ s♦♠❡♥t❡ s❡ 4 = log3 (81) Pr♦♣r✐❡❞❛❞❡ ✷✳✹✳ ▲✶✮ ❙❡ 0 < a < 1✱ ▲✷✮ ❙❡ a > 1✱ ❋✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ❞❡ ❜❛s❡ ❛ ❢✉♥çã♦ ❛ ❢✉♥çã♦ g(x) = loga x g(x) = loga x a✳ é ❞❡❝r❡s❝❡♥t❡ ❡♠ é ❝r❡s❝❡♥t❡ ❡♠ ✶✸✺ R+ ✳ R+ ✳ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ▲✸✮ ❙❡ A, B ❡ N sã♦ ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s✱ ❡♥tã♦✿ a) c) e) loga (A × B) = loga A + loga B   B B logak (A ) = loga A k logc A logB A = logc B b)   A = loga A − loga B loga B d) loga (Ar ) = r · loga A r∈R (❋ór♠✉❧❛ ❞❡ ♠✉❞❛♥ç❛ ❞❡ ❜❛s❡) ▲✹✮ ❖ ❣rá✜❝♦ ❞❡ t♦❞❛ ❛ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ♣❛ss❛ ♣♦r P (1, 0)✳ ▲✺✮ ❙❡ 0 < a < 1✱ ❡♥tã♦✿ t❡♥❞❡ ♣❛r❛ +∞ q✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ ③❡r♦ ✭♣❡❧❛ ❞✐r❡✐t❛✮✱ ❡ t❡♥❞❡ ♣❛r❛ −∞ q✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ +∞✳ ▲✻✮ ❙❡ a > 1✱ ❡♥tã♦ ✿ loga x t❡♥❞❡ ♣❛r❛ −∞ q✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ ③❡r♦ ✭♣❡❧❛ ❞✐r❡✐t❛✮✱ ❡ loga x t❡♥❞❡ ♣❛r❛ +∞ q✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ +∞✳ ❉❡♠♦♥str❛çã♦✳ ▲✸✲✭❡✮ ❙✉♣♦♥❤❛ z = logB A✱ ❡♥tã♦ B z = A✳ ❈♦♥s✐❞❡r❛♥❞♦ ❧♦❣❛r✐t♠♦ ♥❛ ❜❛s❡ c t❡♠♦s✿ logc B z = logc A ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭▲✸✲❞✮ t❡♠♦s z · logc B = logc A✳ ▲♦❣♦ z = logc A logc A ✐st♦ é logB A = ✳ logc B logc B ❆♣❧✐❝❛çõ❡s ❊①❡♠♣❧♦ ✷✳✽✻✳ ❯♠❛ r❛♠♣❛ ♣❛r❛ ♠❛♥♦❜r❛s ❞❡ ✏s❦❛t❡✑ ❞❡ ❛❧t✉r❛ ❋✐❣✉r❛ ✭✷✳✸✾✮✳ 4m é r❡♣r❡s❡♥t❛❞❛ ♣❡❧♦ ❡sq✉❡♠❛ ❞❛ ❙❡ ❛ ♣❛rt❡ ❝✉r✈❛ ♣✉❞❡ss❡ s❡r ❛ss♦❝✐❛❞❛ ❛ ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✱ ❝♦♠♦ s❡r✐❛ ❡st❛ ❢✉♥çã♦❄ ❙♦❧✉çã♦✳ ❋✐❣✉r❛ ✷✳✸✾✿ ✶✸✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❖❜s❡r✈❡✱ ♣♦❞❡♠♦s ♦❜t❡r ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛ ❞❡ ✈❛❧♦r❡s✿ x 0m 1m 2m 3m 4m f (x) 4m 2m 1m 0, 5m 0, 25m  x−2 1 ✱ é ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳ P♦rt❛♥t♦ f (x) = 2 ❊①❡♠♣❧♦ ✷✳✽✼✳ ❉❡t❡r♠✐♥❡ ♦ ❞♦♠í♥✐♦ ❞❡ ❞❡✜♥✐çã♦ ❞❛ ❢✉♥çã♦ f (x) = log 21 ❙♦❧✉çã♦✳ ❉❛ ❞❡✜♥✐çã♦ ❞❛ ❢✉♥çã♦ ❧♦❣❛r✐t♠♦ t❡♠♦s 3 5   3 − 5x ✳ x+7 −5(x − 35 ) 3 − 5x > 0 ✐st♦ é > 0✳ ❖♥❞❡ ♦ x+7 (x + 7) ❞♦♠í♥✐♦ D(f ) = (−7, )✳  ❊①❡♠♣❧♦ ✷✳✽✽✳ ❙❡ a ❡ b sã♦ s♦❧✉çõ❡s ✐♥t❡✐r❛s ❞♦ s✐st❡♠❛✿ 2x = 210−y ❡ log2 x + log2 y = 4✱ ❡♥tã♦ 2a + 2b é ✐❣✉❛❧ ❛✿ ❙♦❧✉çã♦✳ ❈♦♠♦ a ❡ b sã♦ s♦❧✉çõ❡s ❞♦ s✐st❡♠❛ ❡♥tã♦ 2a · 2b−10 = 1 ❡ log2 a + log2 b = 4 ❞❡ ♦♥❞❡ 2a+b = 210 ❡ log2 (ab) = 4 ⇒ a + b = 10 ❡ ab = 24 = 16❀ ✐st♦ ❝✉♠♣r❡ s❡ a = 8 ❡ b = 2✳ P♦rt❛♥t♦ 2a + 2b = 28 + 22 = 260✳ ❊①❡♠♣❧♦ ✷✳✽✾✳ ❯♠ ❥✉✐③ ❞❡t❡r♠✐♥♦✉ ♦ ♣❛❣❛♠❡♥t♦ ❞❡ ✉♠❛ ✐♥❞❡♥✐③❛çã♦ ❛té ❝❡rt❛ ❞❛t❛✳ ❉❡t❡r♠✐♥♦✉ t❛♠❜é♠ q✉❡✱ ❝❛s♦ ♦ ♣❛❣❛♠❡♥t♦ ♥ã♦ ❢♦ss❡ ❢❡✐t♦✱ s❡r✐❛ ❝♦❜r❛❞❛ ✉♠❛ ♠✉❧t❛ ❞❡ R$3, 00 q✉❡ ❞♦❜r❛r✐❛ ❛ ❝❛❞❛ ♠ês ❞❡ ❛tr❛s♦✳ ❊♠ q✉❛♥t♦s ♠ês❡s ❞❡ ❛tr❛s♦ ❡ss❛ ♠✉❧t❛ s❡r✐❛ s✉♣❡r✐♦r ❛ 600.000, 00 r❡❛✐s❄ ❙♦❧✉çã♦✳ ❆ ♠✉❧t❛ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦ ❥✉✐③ ♣♦❞❡ ♣❛r❡❝❡r ♣❡q✉❡♥❛✱ s❡ ♦ ❛tr❛s♦ ♥♦ ♣❛❣❛♠❡♥t♦ ❢♦r ❞❡ ♣♦✉❝♦s ❞✐❛s✳ ▼❛s ❡❧❛ ❝r❡s❝❡ ❝♦♠ ✉♠❛ r❛♣✐❞❡③ ♠✉✐t♦ ❣r❛♥❞❡✳ ❈❤❛♠❛♥❞♦ ❞❡ x ♦ ♥ú♠❡r♦ ❞❡ ❞✐❛s ❞❡ ❛tr❛s♦ ♥♦ ♣❛❣❛♠❡♥t♦✱ ♦ ✈❛❧♦r ❞❛ ❞í✈✐❞❛ s❡rá 3x ✳ ❱❡❥❛✿ 1 ♠ês ❞❡ ❛tr❛s♦ ⇒ x = 1 ⇒ ♠✉❧t❛ = 31 = 3 2 ♠ês ❞❡ ❛tr❛s♦ ⇒ x = 2 ⇒ ♠✉❧t❛ = 32 = 9 3 ♠ês ❞❡ ❛tr❛s♦ ⇒ x = 3 ⇒ ♠✉❧t❛ = 33 = 27✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ❈♦♠♦ ✈❡♠♦s✱ ❛s ♠✉❧t❛s ❝r❡s❝❡♠ ❡♠ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛✳ ✶✸✼ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ❉❡✈❡♠♦s ❝❛❧❝✉❧❛r ❡♠ q✉❡ ♠ês ❡ss❛ ♠✉❧t❛ ❛t✐♥❣❡ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦✿ 600.000, 00 R r❡❛✐s✱ ♦✉ s❡❥❛✱ ❞❡✈❡♠♦s x 3 = 600.000, 00✳ ❆♣❧✐❝❛♥❞♦ ❧♦❣❛r✐t♠♦s✱ log 3x = log 600.000 = log(6 × 105 ) = log 6 + 5 · log 10 ❝♦♠♦ log 10 = 1, log 6 = log(2 · 3) = log 2 + log 3 ⇒ x log 3 = log 2 + log 3 + 5 P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s✱ ♥♦ 11o x−1= log 2 + 5 log 3 ⇒ x= 5, 301 = 11, 11 0, 478 ♠❡s ❞❡ ❛tr❛s♦ ❛ ♠✉❧t❛ t❡rá ♣❛ss❛❞♦ ❞❡ 600.000, 00 r❡❛✐s r❡❛✐s✳ ❊①❡♠♣❧♦ ✷✳✾✵✳ ❊♠ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❝✐❞❛❞❡ ❛ t❛①❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❡ ❞❡ 4% ❛♦ ❛♥♦✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✳ ❊♠ q✉❛♥t♦s ❛♥♦s ❛ ♣♦♣✉❧❛çã♦ ❞❡st❛ ❝✐❞❛❞❡ ✐rá ❞✉♣❧✐❝❛r✱ s❡ ❛ t❛①❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❝♦♥t✐♥✉❛r ❛ ♠❡s♠❛❄ ❙♦❧✉çã♦✳ ❙❡❥❛ P0 ❛ ♣♦♣✉❧❛çã♦ ❞♦ ❛♥♦✲❜❛s❡✳ P0 (1, 04) = P1 ✳ ❆ ♣♦♣✉❧❛çã♦ P0 (1, 04) = P2 ✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ n ❛♣ós n ❛♥♦s s❡rá P0 (1, 04) = Pn ❛♣ós ❞♦✐s ❛♥♦s s❡rá ❆ ♣♦♣✉❧❛çã♦ ❆♣ós ✉♠ ❛♥♦ s❡rá 2 ❱❛♠♦s s✉♣♦r q✉❡ ❛ ♣♦♣✉❧❛çã♦ ❞✉♣❧✐❝❛ ❡♠ r❡❧❛çã♦ ❛♦ ❛♥♦✲❜❛s❡ ❛♣ós Pn = 2P0 P0 (1, 04)n = 2P0 ⇒ ⇒ n= ⇒ n log 1, 04 = log 2 n ❛♥♦s✱ t❡♠♦s✿ ⇒ 0, 01703 log 1, 04 = ≈ 0, 0566 log 2 0, 30103 ❆ss✐♠✱ t❡♠♦s q✉❡ ❛ ♣♦♣✉❧❛çã♦ ❞✉♣❧✐❝❛ ❡♠ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✶✸✽ P0 (1, 04)0,0566 ❛♥♦s✳ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✷✲✻ ✶✳ ◆♦s s❡❣✉✐♥t❡s ❡①❡r❝í❝✐♦s r❡s♦❧✈❛ ♣❛r❛ x✳ 1. log10 10000 = x 3. logx 81 = 3 log10 0, 01 = x √ 5. eLnx = 3 4. 7. log2 x = −5 8. Lnx = −2 9. 2. log4   1 =x 256 6. x2 − 8x = log4 (256)−1 log35 x + log35 (x + 2) = 1 ✷✳ ❚r❛ç❛r ♦ ❣rá✜❝♦ ♣❛r❛ ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ x 1. y = −(6) √ 4. y = ( 3)x 7. 2. y = 4 x 5. y = π −2x log4 x2 8. log3 (x − 1)  x 5 3. y = 4 6. y = −(2−x 9. loge ex ✸✳ ❉❡t❡r♠✐♥❡ s❡ ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s ❞❛❞❛s sã♦ ✐♥✈❡rs❛s ✉♠❛ ❞❛ ♦✉tr❛ ❡s❜♦ç❛♥❞♦ s❡✉s ❣rá✜❝♦s ♥♦ ♠❡s♠♦ s✐st❡♠❛ ❞❡ ❡✐①♦s✳ ❈❛❧❝✉❧❛r s❡✉ ❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ❢✉♥çõ❡s✿ 1. f (x) = 2ex 3. f (x) = e2x+1 √ g(x) = Ln x 2. f (x) = ex + 1 g(x) = Ln(x − 1) g(x) = 1 − Ln2x 4. f (x) = e3x g(x) = Lnx−3 ✹✳ ❘❡s♦❧✈❡r ❛s s❡❣✉✐♥t❡s ❡q✉❛çõ❡s✿ 1. x = log 1 36 6 4. log25 x = 3 √ 3 7. x(x−2) = log 10 10 2. x = log3√2 cos 30o √ 3 5. x = log2x ( 25)4 = 6 √ 4 3 8. logx 10 10 = 3 √ 3. x = log23 5 2 1 6. xx−1 = 27 1 1 9. log 1 x = 4 3 2 ✺✳ ▼♦str❡ q✉❡ ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s ❞❛❞❛s sã♦ ✐♥✈❡rs❛s ✉♠❛ ❞❛ ♦✉tr❛ ❡s❜♦ç❛♥❞♦ s❡✉s ❣rá✜❝♦s ♥♦ ♠❡s♠♦ s✐st❡♠❛ ❞❡ ❡✐①♦s✳ ❈❛❧❝✉❧❛r s❡✉ ❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ❢✉♥çõ❡s✳ 1. f (x) = e2x √ g(x) = Ln x 2. f (x) = ex − 1 g(x) = Ln(x + 1) ✶✸✾ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 3. f (x) = ex−1 g(x) = 1 + Lnx  1−x ♠♦str❡ q✉❡ ✻✳ ❙❡ f (x) = Ln 1+x  f −1 x 4. f (x) = e 3 g(x) = Lnx3  ex/2 − e−x/2 ✳ (x) = − x/2 e + e−x/2  ✼✳ ❯♠❛ ❢✉♥çã♦ y = f (x) ❡st❛ ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ y 2 − 1 + log2 (x − 1) = 0✳ ❉❡t❡r♠✐♥❡ ♦ ❞♦♠í♥✐♦ ❞❡ ❞❡✜♥✐çã♦ ❞❛ ❢✉♥çã♦✱ ❡ ❞❡✜♥❛ ❛ ❢✉♥çã♦ ✐♥✈❡rs❛ f −1 (x)✳ ✽✳ ❙❡ f (x) = 4x ❡ x1 , x2 ❡ x3 sã♦ três ♥ú♠❡r♦s ❡♠ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛ ❡♥tã♦ ❞❡♠♦♥str❛r q✉❡ f (x1 ), f (x2 ) ❡ f (x3 ) ❡stã♦ ❡♠ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛✳ ◗✉❛❧ é ❛ r❛③ã♦ ❄ ✾✳ ❙✉♣♦♥❤❛ q✉❡ ❛ t ❤♦r❛s ❞❛ ♠❛❞r✉❣❛❞❛ ❛ t❡♠♣❡r❛t✉r❛ ❞❡ ✉♠❛ ❝✐❞❛❞❡ s❡❥❛✱ C(t) = t2 − + 4t + 8 ❣r❛✉s ❝❡♥tí❣r❛❞♦s✳ ❛✮ ◗✉❡ t❡♠♣❡r❛t✉r❛ t✐♥❤❛ ❛s 14 ❤♦r❛s ❄ ❜✮ ❊♠ q✉❡ 7 t❛♥t♦ ❛✉♠❡♥t♦✉ ♦✉ ❞✐♠✐♥✉✐✉ ❛ t❡♠♣❡r❛t✉r❛✱ ❡♥tr❡ 6 ❡ 7 ❤♦r❛s❄ ✶✵✳ ❙✉♣♦♥❤❛ q✉❡ ♦ ❝✉st♦ t♦t❛❧ ♣❛r❛ ❢❛❜r✐❝❛r q ✉♥✐❞❛❞❡s ❞❡ ✉♠ ❝❡rt♦ ♣r♦❞✉t♦ s❡❥❛ ❞❛❞❛ ♣❡❧❛ ❢✉♥çã♦ C(q) = q 3 − 30q 2 + 400q + 500✳ ✶✳ ❈❛❧❝✉❧❛r ♦ ❝✉st♦ ❞❡ ❢❛❜r✐❝❛çã♦ ❞❡ 20 ✉♥✐❞❛❞❡s✳ ✷✳ ❈❛❧❝✉❧❛r ♦ ❝✉st♦ ❞❡ ❢❛❜r✐❝❛çã♦ ❞❛ 20a ✉♥✐❞❛❞❡✳ ✶✶✳ ❆ ❢♦❧❤❛ ❞❡ ♣❛❣❛♠❡♥t♦ ✭F.P.✮ ❞✐ár✐❛ ❞❡ ✉♠❛ ❡q✉✐♣❡ ❞❡ tr❛❜❛❧❤♦ é ❞✐r❡t❛♠❡♥t❡ ♣r♦✲ ♣♦r❝✐♦♥❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ tr❛❜❛❧❤❛❞♦r❡s ✭T ✮✱ ❡ ✉♠❛ ❡q✉✐♣❡ ❞❡ 12 tr❛❜❛❧❤❛❞♦r❡s t❡♠ ✉♠❛ ❢♦❧❤❛ ❞❡ ♣❛❣❛♠❡♥t♦ ❞❡ ❘$540✳ ✶✳ ❊①♣r❡ss❡ ♦ ✈❛❧♦r t♦t❛❧ ❞❛ ❢♦❧❤❛ ❞❡ ♣❛❣❛♠❡♥t♦ ❞✐ár✐❛ ❝♦♠♦ ❢✉♥çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ tr❛❜❛❧❤❛❞♦r❡s✳ ✷✳ ◗✉❛❧ ❛ ❢♦❧❤❛ ❞❡ ♣❛❣❛♠❡♥t♦ ❞❡ ✉♠❛ ❡q✉✐♣❡ ❞❡ 15 tr❛❜❛❧❤❛❞♦r❡s✳ ✶✷✳ ◆✉♠❛ ❝✐❞❛❞❡ ❞❡ 70.000 ❤❛❜✐t❛♥t❡s ❛ t❛①❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❡ ✉♠❛ ❡♣✐❞❡♠✐❛ é ❝♦♥✲ ❥✉♥t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s ✐♥❢❡❝t❛❞❛s ❡ ❛♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s ♥ã♦ ✐♥❢❡❝t❛❞❛s✳✱ ✶✳ ❙❡ ❛ ❡♣✐❞❡♠✐❛ ❡st❛ ❝r❡s❝❡♥❞♦ ❛ r❛③ã♦ ❞❡ 20 ♣❡ss♦❛s ♣♦r ❞✐❛ q✉❛♥❞♦ 100 ♣❡ss♦❛s ❡stã♦ ✐♥❢❡❝t❛❞❛s✱ ❡①♣r❡ss❡ ❛ t❛①❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❛ ❡♣✐❞❡♠✐❛ ❡♠ ❢✉♥çã♦ ❞❡ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s ✐♥❢❡❝t❛❞❛s✳ ✷✳ ◗✉ã♦ rá♣✐❞♦ ❡stá s❡ ❡s♣❛❧❤❛♥❞♦ ❛ ❡♣✐❞❡♠✐❛ q✉❛♥❞♦ 400 ♣❡ss♦❛s ❡stã♦ ✐♥❢❡❝t❛❞❛s❄ ✶✹✵ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✳✼✳✸ ❋✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❊♠ ♠❛t❡♠át✐❝❛✱ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s sã♦ ❢✉♥çõ❡s ❛♥❣✉❧❛r❡s✱ ✐♠♣♦rt❛♥t❡s ♥♦ ❡st✉❞♦ ❞♦s tr✐â♥❣✉❧♦s ❡ ♥❛ ♠♦❞❡❧❛çã♦ ❞❡ ❢❡♥ô♠❡♥♦s ♣❡r✐ó❞✐❝♦s✳ P♦❞❡♠ s❡r ❞❡✜♥✐❞❛s ❝♦♠♦ r❛③õ❡s ❡♥tr❡ ❞♦✐s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❡♠ ❢✉♥çã♦ ❞❡ ✉♠ â♥❣✉❧♦✱ ♦✉✱ ❞❡ ❢♦r♠❛ ♠❛✐s ❣❡r❛❧✱ ❝♦♠♦ r❛③õ❡s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❞❡ ♣♦♥t♦s ♥♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦✳ ◆❛ ❛♥á❧✐s❡ ♠❛t❡♠át✐❝❛✱ ❡st❛s ❢✉♥çõ❡s r❡❝❡❜❡♠ ❞❡✜♥✐çõ❡s ❛✐♥❞❛ ♠❛✐s ❣❡r❛✐s✱ ♥❛ ❢♦r♠❛ ❞❡ sér✐❡s ✐♥✜♥✐t❛s ♦✉ ❝♦♠♦ s♦❧✉çõ❡s ♣❛r❛ ❝❡rt❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✳ ◆❡st❡ ú❧t✐♠♦ ❝❛s♦✱ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❡stã♦ ❞❡✜♥✐❞❛s ♥ã♦ só ♣❛r❛ â♥❣✉❧♦s r❡❛✐s✱ ❝♦♠♦ t❛♠❜é♠ ♣❛r❛ â♥❣✉❧♦s ❝♦♠♣❧❡①♦s✳ ◆♦ ♣❧❛♥♦✲xy ✭❋✐❣✉r❛ ✭✷✳✹✵✮✮ ❝♦♥s✐❞❡r❡♠♦s ❛ ❝✐r✲ ❝✉♥❢❡rê♥❝✐❛ ✉♥✐tár✐❛ ❞❡ ❝❡♥tr♦ ❛ ♦r✐❣❡♠ ❞❡ ❝♦♦r❞❡✲ ♥❛❞❛s✱ ❡❧❛ t❡♠ ♣♦r ❡q✉❛çã♦ x2 + y 2 = 1✳ ❙❡❥❛ A(1, 0) ♦ ♣♦♥t♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✜①❛❞♦ ♥❛ ♦r✐❣❡♠ ❞♦s ❛r❝♦s ♦r✐❡♥t❛❞♦s AT s♦❜r❡ ❛ ❝✐r❝✉♥❢❡✲ rê♥❝✐❛✳ ❊st❛ ♦r✐❡♥t❛çã♦ é ❛ ✉s✉❛❧✿ ♥♦ s❡♥t✐❞♦ ❛♥t✐✲ ❤♦rár✐♦✱ é ♣♦s✐t✐✈❛ ❡ ♥♦ s❡♥t✐❞♦ ❤♦rár✐♦✱ é ♥❡❣❛t✐✈❛✳ ❊st❛❜❡❧❡❝❡♠♦s ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ♦s ♥ú♠❡r♦s r❡❛✐s ❡ ♦s ♣♦♥t♦s ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦ ♠♦❞♦ s❡❣✉✐♥t❡✿ ❋✐❣✉r❛ ✷✳✹✵✿ ❆♦ ♥ú♠❡r♦ r❡❛❧ t ❝♦rr❡s♣♦♥❞❡ ♦ ♣♦♥t♦ T ❞❛ ❝✐r✲ c ♠❡❞❡ ❝✉♥❢❡rê♥❝✐❛✱ ❞❡ ♠♦❞♦ q✉❡ ♦ ❛r❝♦ ♦r✐❡♥t❛❞♦ AT | t | r❛❞✐❛♥♦s✳ ❖ ❛r❝♦ t❡♠ ♦r✐❡♥t❛çã♦ ♣♦s✐t✐✈❛ s❡ t é ♣♦s✐t✐✈♦❀ ❡ ♦r✐❡♥t❛çã♦ ♥❡❣❛t✐✈❛ s❡ t é ♥❡❣❛t✐✈♦✳ ❙❡ T (x, y) é ♦ ♣♦♥t♦ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ s❡✉ ♥ú♠❡r♦ r❡❛❧ t✱ ❛ ❛❜s❝✐ss❛ x ❝❤❛♠❛✲s❡ ❞❡✿ ❝♦ss❡♥♦ ❞❡ t ❡ s❡ ❞❡♥♦t❛ ✭cos t✮ ❡ ❛ ♦r❞❡♥❛❞❛ y ❞❡♥♦♠✐♥❛✲s❡ s❡♥♦ ❞❡ t ❡ ❞❡♥♦t❛✲s❡ sent ❡ ♦ ♣♦♥t♦ T ❡s❝r❡✈❡✲s❡ x = cos t, y = sent ♦✉ T (cos x, sent)✳ P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ 2π, ✭❧♦❣♦✱ s❡✉ r❛✐♦ é 1✮✱ π π π ❛♦ ♥ú♠❡r♦ ❝♦rr❡s♣♦♥❞❡ ♦ ♣♦♥t♦ B(0, 1)❀ ❧♦❣♦ cos = 0 ❡ sen = 1✳ 2 2 2 ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ❛♦s ♥ú♠❡r♦s π ❡ 3π ❝♦rr❡s♣♦♥❞❡ ♦ ♣♦♥t♦ A′ (−1, 0)✱ ❡♥tã♦ cos π = cos 3π = −1 ❡ senπ = sen3π = 0✳ ❉❡st❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ♣♦❞❡♠♦s ❞❡❞✉③✐r ❛s ♣r♦♣r✐❡❞❛❞❡s✱ t❛✐s ❝♦♠♦✿ Pr♦♣r✐❡❞❛❞❡ ✷✳✺✳ ✶✮ ❈♦♠♦ T (cos t, sent) é ✉♠ ♣♦♥t♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❡ t❡♠♦s ❛ r❡❧❛çã♦ ❢✉♥❞❛♠❡♥t❛❧✿ cos2 t + sen2 t = 1✳ ✷✮ ❈♦♥s✐❞❡r❛♥❞♦ q✉❡ T ✈❛r✐❛ s♦❜r❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ s✉❛ ❛❜s❝✐ss❛ ❡ s✉❛ ♦r❞❡♥❛❞❛ ✈❛r✐❛ ❡♥tr❡ −1 ❡ 1 ✐st♦ é −1 ≤ cos t ≤ 1 ❡ −1 ≤ sent ≤ 1✳ ✶✹✶ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✸✮ P❡r✐♦❞✐❝✐❞❛❞❡ ❞♦ s❡♥♦ ❡ ❝♦ss❡♥♦ ✿ ❙❡ ❛♦ ♥ú♠❡r♦ r❡❛❧ t ❝♦rr❡s♣♦♥❞❡ ♦ ♣♦♥t♦ T ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ❝♦♥s✐❞❡r❛♥❞♦ 2kπ ♣❛r❛ k ∈ Z✱ r❡♣r❡s❡♥t❛ ♦ ♥ú♠❡r♦ ❞❡ k ✈♦❧t❛s ❛♦ r❡❞♦r ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❛♦ ♥ú♠❡r♦ r❡❛❧ t + 2kπ ❝♦rr❡s♣♦♥❞❡ ♦ ♠❡s♠♦ ♣♦♥t♦ T ✱ ❧♦❣♦ sent = sen(t + 2kπ) ❡ cos t = cos(t + 2kπ)✳ ❖ ♠❡♥♦r ♥ú♠❡r♦ r❡❛❧ p > 0 ♣❛r❛ ♦ q✉❛❧ sent = sen(t + p) ❡ cos t = cos(t + p)✱ ❞❡♥♦♠✐♥❛♠♦s ♣❡rí♦❞♦ ❞♦ s❡♥♦ ❡ ❝♦ss❡♥♦✳ ✹✮ ❆♦s ♥ú♠❡r♦s r❡❛✐s t ❡ −t ❝♦rr❡s♣♦♥❞❡ ♦s ♣♦♥t♦s T ❡ T ′ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ sã♦ s✐♠étr✐❝♦s r❡s♣❡✐t♦ ❞♦ ❡✐①♦✲x ❡ ❡st❡s ♣♦♥t♦s t❡♠ ❛ ♠❡s♠❛ ❛❜s❝✐ss❛ ♣♦ré♠ s✉❛s ♦r❞❡♥❛❞❛s só ❞✐❢❡r❡♠ ♥♦ s✐♥❛❧❀ ✐st♦ é cos(−t) = cos t ❡ sen(−t) = −sent✳ ✺✮ ❆s ♣r♦♣r✐❡❞❛❞❡s ✭✐❞❡♥t✐❞❛❞❡s ✮ q✉❡ ❡st❛♠♦s ❞❡❞✉③✐♥❞♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛♦ ❧❡✐t♦r ♣♦r s✉❛ ✉t✐❧✐❞❛❞❡✳     a−b a+b sen 2 2 ✶✳ sen.a − sen.b = 2 cos ✷✳ sen(a ± b) = sena cos b ± senb cos a     a−b a+b cos cos a + cos b = 2 cos 2 2 ✸✳ ✹✳ ✺✳ ✻✳ cos(a ± b) = cos a cos b ∓ senb cos a     a+b a−b cos a − cos b = −2sen sen 2 2     a+b a−b sena + senb = 2sen cos 2 2 ✼✳ 1 sena · cos b = [sen(a + b) + sen(a − b)] 2 ✽✳ sen2 a = ✾✳ 1 sena · senb = [cos(a − b) − cos(a + b)] 2 ✶✵✳ 1 − cos 2a 2 cos2 a = 1 + cos 2a 2 1 cos a · cos b = [cos(a + b) + cos(a − b)] 2 ❉♦ ❢❛t♦ q✉❡✱ ❛ ❝❛❞❛ ♥ú♠❡r♦ r❡❛❧ x✱ ♣♦❞❡♠♦s r❡❧❛❝✐♦♥❛r ❝♦♠ ♦ s❡♥♦ ❡ ❝♦ss❡♥♦✱ ✐st♦ é ❡①✐st❡♠ senx ❡ cos x ♣❛r❛ x ∈ R ❞❡✜♥❡✲s❡✿ ✶✶✳ • • senx cos x cos x cot x = senx tan x = s❡✱ cos x 6= 0 ✐st♦ é x 6= (2k + 1) s❡✱ senx 6= 0 ✐st♦ é x 6= kπ ✶✹✷ π 2 k∈Z k∈Z 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ • sec x = 1 s❡✱ cos x 6= 0 ✐st♦ é cos x • csc x = 1 senx s❡✱ x 6= (2k + 1) senx 6= 0 ✐st♦ é x 6= kπ π 2 k∈Z k∈Z P❛r❛ ♦s ✈❛❧♦r❡s ❞❡ x✱ ♣❛r❛ ♦s q✉❛✐s ❡①✐st❛♠ tan x, cot x, sec x ❡ csc x ✈❡r✐✜❝❛♠✲s❡ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ sec2 x − tan2 x = 1 | sec x |≥ 1 ✶✳ ✸✳ ✷✳ ✹✳ csc2 x − cot2 x = 1 | csc x |≥ 1 ✷✳✼✳✸✳✶ ❋✉♥çã♦ s❡♥♦ ❆ ❢✉♥çã♦ s❡♥♦ f : R −→ R é ❞❡✜♥✐❞❛ ♣♦r✿ f (x) = senx✱ ✐st♦ é ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♥ú♠❡r♦ r❡❛❧ x ♦ ♥ú♠❡r♦ y = senx✳ ❆❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛ ❢✉♥çã♦ s❡♥♦✿ ❛✮ D(f ) = R ❜✮ Im(f ) = [−1, 1] ❆ ❢✉♥çã♦ s❡♥♦ é ♣❡r✐ó❞✐❝❛✱ s❡✉ ♣❡rí♦❞♦ é 2π ✳ ✳ ✐st♦ é✱ ❛ ❢✉♥çã♦ s❡♥♦ é í♠♣❛r ❡ s❡✉ ❣rá✜❝♦ é s✐♠étr✐❝♦ r❡s♣❡✐t♦ ❞❛ ♦r✐❣❡♠ ❡ ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✷✳✹✶✮✳ ❝✮ sen(−x) = −senx ❞✮ f (x) = senx é ♣♦s✐t✐✈❛ ♥♦ 1o ❡ 2o q✉❛❞r❛♥t❡s ✭♦r❞❡♥❛❞❛ ♣♦s✐t✐✈❛✮✳ f (x) = senx é ♥❡❣❛t✐✈❛ ♥♦ 3o ❡ 4o q✉❛❞r❛♥t❡s ✭♦r❞❡♥❛❞❛ ♥❡❣❛t✐✈❛✮✳ ✻y ✻y 1 − π2 −π 0 1 π 2 x ✲ x π − π2 −π −1 0 π 2 ✲ π −1 ❋✐❣✉r❛ ✷✳✹✶✿ ❙❡♥♦ ❋✐❣✉r❛ ✷✳✹✷✿ ❈♦ss❡♥♦✳ ✷✳✼✳✸✳✷ ❋✉♥çã♦ ❝♦ss❡♥♦ ❆ ❢✉♥çã♦ ❝♦ss❡♥♦ f : R −→ R é ❞❡✜♥✐❞❛ ♣♦r✿ f (x) = cos x ❆❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛ ❢✉♥çã♦ ❝♦ss❡♥♦✿ ❛✮ D(f ) = R Im(f ) = [−1, 1] ❜✮ cos(−x) = cos x✱ ✐st♦ é✱ ❛ ❢✉♥çã♦ ❝♦ss❡♥♦ é ♣❛r ❡ s❡✉ ❣rá✜❝♦ é s✐♠étr✐❝♦ r❡s♣❡✐t♦ ❛♦ ❡✐①♦✲y ❡ ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✷✳✹✷✮✳ ✶✹✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❝✮ ❆ ❢✉♥çã♦ ❝♦ss❡♥♦ é ♣❡r✐ó❞✐❝❛✱ s❡✉ ♣❡rí♦❞♦ é 2π ✳ é ♣♦s✐t✐✈❛ ♥♦ 1o ❡ 4o q✉❛❞r❛♥t❡ ✭❛❜s❝✐ss❛ ♣♦s✐t✐✈❛✮✳ f (x) = cos x é ♥❡❣❛t✐✈❛ ♥♦ 2o ❡ 3o q✉❛❞r❛♥t❡ ✭❛❜s❝✐ss❛ ♥❡❣❛t✐✈❛✮✳ ❞✮ f (x) = cos x ❆❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛ ❢✉♥çã♦ s❡♥♦ ❡ ❝♦ss❡♥♦✿ π ❉❡s❞❡ q✉❡ sen( + x) = cos x✱ ♦ ❣rá✜❝♦ ❞❡ y = senx tr❛♥s❢♦r♠❛✲s❡ ♥♦ ❣rá✜❝♦ ❞❡ 2 π y = cos x s❡ ❛ ♦r✐❣❡♠ s❡ ❞❡s❧♦❝❛ ❛♦ ♣♦♥t♦ ( , 0)✳ 2 ❋✉♥çã♦ ❱❛❧♦r 0 ✭③❡r♦✮ ❡♠✿ ❱❛❧♦r 1 ✭✉♠✮ ❡♠✿ senx π π + 2kπ 2 cos x π + 2kπ 2 2π ❱❛❧♦r −1 ❡♠✿ 3π + 2kπ 2 (2k + 1)π ✷✳✼✳✸✳✸ ❋✉♥çã♦ t❛♥❣❡♥t❡ ❆ ❢✉♥çã♦ r❡❛❧ f : R −→ R ❝❤❛♠❛❞❛ ✧❢✉♥çã♦ t❛♥❣❡♥t❡✧é ❞❡✜♥✐❞❛ ♣♦r✿ f (x) = tan x = senx cos x ❆s ❝❛r❛❝t❡ríst✐❝❛s ✐♠♣♦rt❛♥t❡s ❞❛ ❢✉♥çã♦ t❛♥✲ ❣❡♥t❡ sã♦ ❛s s❡❣✉✐♥t❡s✿ ❛✮ D(f ) = R − { ❜✮ π + kπ, 2 k ∈ Z }, Im(f ) = R ❆ ❢✉♥çã♦ t❛♥❣❡♥t❡ é ♣❡r✐ó❞✐❝❛✱ s❡✉ ♣❡rí♦❞♦ é π✳ ❝✮ tan(−x) = − tan x ✐st♦ é✱ ❛ ❢✉♥çã♦ t❛♥❣❡♥t❡ é í♠♣❛r ❡ s❡✉ ❣rá✜❝♦ é s✐♠étr✐❝♦ r❡s♣❡✐t♦ ❞❛ ♦r✐❣❡♠ ❝♦♠♦ s❡ ♠♦str❛ ♥❛ ❋✐❣✉r❛ ✭✷✳✹✸✮✳ ❞✮ ❋✐❣✉r❛ ✷✳✹✸✿ ❚❛♥❣❡♥t❡✳ f (x) = tan x é ♣♦s✐t✐✈❛ ♥♦ 1o ❡ 3o q✉❛❞r❛♥✲ t❡s ✭♣r♦❞✉t♦ ❞❛ ♦r❞❡♥❛❞❛ ♣❡❧❛ ❛❜s❝✐ss❛ ♣♦s✐t✐✈❛✮✳ f (x) = tan x é ♥❡❣❛t✐✈❛ ♥♦ 2o ❡ 4o q✉❛❞r❛♥t❡s ✭♣r♦❞✉t♦ ❞❛ ♦r❞❡♥❛❞❛ ♣❡❧❛ ❛❜s❝✐ss❛ ♥❡❣❛t✐✈❛✮✳ ❊①❡♠♣❧♦ ✷✳✾✶✳ ❉❛❞❛s ❛s ❢✉♥çõ❡s f (x) = senx ❡ g(x) = r❡s♣❡❝t✐✈♦s ❞♦♠í♥✐♦s ❞❡ ❞❡✜♥✐çã♦✳ ❙♦❧✉çã♦✳ √ ✶✹✹ 1 − 9x2 ✱ ❞❡t❡r♠✐♥❡ f ◦ g ❡ g ◦ f ❡ s❡✉s 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 1o √ (f ◦ g)(x) = f (g(x)) = seng(x) = sen 1 − 9x2 ✳ ❉♦ ❢❛t♦ s❡r t♦❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ 2 ♥ú♠❡r♦s r❡❛✐s ♦ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ s❡♥♦✱ t❡♠♦s D(f ◦ g) = { x ∈ R /. 1 − 9x ≥ 0 }❀ ❚❡♠♦s ✐st♦ é D(f ◦ g) = { x ∈ R /. − 2o ❚❡♠♦s 0 (g ◦ f )(x) = g(f (x)) = 1 1 ⇒ − ≤ senx ≤ ❛ss✐♠ 3 3 p 1 1 ≤x≤ } 3 3 1 − 9[f (x)]2 = 1 1 ≤ senx ≤ } 3 3 ❊①❡♠♣❧♦ ✷✳✾✷✳ ❉❛❞❛s ❛s ❢✉♥çõ❡s f (x) = tan x ❡ g(x) = r❡s♣❡❝t✐✈♦s ❞♦♠í♥✐♦s ❞❡ ❞❡✜♥✐çã♦✳ ❙♦❧✉çã♦✳ 1 − 9sen2 x✱ ❧♦❣♦ t❡♠♦s 1 − 9x2 ≥ √ ❡ (g ◦ f )(x) = √ 1 − 9sen2 x 1 − x2 ❞❡t❡r♠✐♥❡ f ◦ g ❡ g ◦ f ❡ s❡✉s π + kπ, k ∈ Z } ❡ D(g) = [−1, 1] 2 √ √ (f ◦ g)(x) = f (g(x)) = tan g(x) = tan 1 − x2 ✳ ▲♦❣♦ (f ◦ g)(x) = tan 1 − x2 ❀ ❙❛❜❡♠♦s q✉❡ ♦ ❞♦♠í♥✐♦ ❚❡♠♦s √ t❡♠♦s q✉❡ D(g ◦ f ) = { x ∈ R /. − 1o √ (f og)(x) = sen 1 − 9x2 ❡ D(f ) = R − { ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ ❞♦♠í♥✐♦✿ D(f ◦ g) = { x ∈ D(g) /. 2o √ 1 − x2 6= (g ◦ f )(x) = g(f (x)) = √ 2 1 − tan x❀ ❧♦❣♦ 1 − x2 ≥ 0 ⇒ ❚❡♠♦s q✉❡ ❆ss✐♠ t❡♠♦s q✉❡✿ π + kπ, 2 k ∈ Z }❀ ✐st♦ é D(f ◦ g) = [−1, 1]✳ p √ 1 − [f (x)]2 = 1 − tan2 x✱ −1 ≤ tan x ≤ 1✳ D(g ◦ f ) = { x ∈ D(g) /. − 1 ≤ tan x ≤ 1 } = ❡♥tã♦ (g ◦ f )(x) = S π π [kπ − , kπ + ] 4 4 k∈Z ✷✳✼✳✸✳✹ ❋✉♥çã♦ ❝♦t❛♥❣❡♥t❡ f : ❆ ✏❢✉♥çã♦ ❝♦t❛♥❣❡♥t❡✑ é ❞❡✜♥✐❞❛ ♣♦r✿ R −→ R t❛❧ q✉❡✿ ❆❧❣✉♠❛s cos x f (x) = cot x = senx ❝❛r❛❝t❡ríst✐❝❛s ❞❛ ❢✉♥çã♦ ❝♦t❛♥✲ ❣❡♥t❡✿ ❛✮ D(f ) = R − { kπ, ❜✮ cot(−x) = − cot x✱ k ∈ Z }; Im(f ) = R ✐st♦ é✱ ❛ ❢✉♥çã♦ ❝♦t❛♥✲ ❣❡♥t❡ é í♠♣❛r ❡ s❡✉ ❣rá✜❝♦ é s✐♠étr✐❝♦ r❡s✲ ♣❡✐t♦ ❞❛ ♦r✐❣❡♠ ❝♦♠♦ s❡ ♠♦str❛ ♥❛ ❋✐❣✉r❛ ✭✷✳✹✹✮✳ ❋✐❣✉r❛ ✷✳✹✹✿ ❈♦t❛♥❣❡♥t❡ ❝✮ ❆ ❝♦t❛♥❣❡♥t❡ é ❢✉♥çã♦ ♣❡r✐ó❞✐❝❛✱ s❡✉ ♣❡rí♦❞♦ é π✳ ✶✹✺ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✳✼✳✸✳✺ ❋✉♥çã♦ s❡❝❛♥t❡ 1 ➱ ❛ ❢✉♥çã♦ f : R −→ R ❞❡✜♥✐❞❛ ♣♦r✿ f (x) = sec x = csc x ❆❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛ ❢✉♥çã♦ s❡❝❛♥t❡✿ ❛✮ D(f ) = R − { ❜✮ π + kπ, 2 k ∈ Z }; Im(f ) = (−∞, −1] ∪ [1, +∞)✳ ❆ ❢✉♥çã♦ s❡❝❛♥t❡ é ♣❡r✐ó❞✐❝❛✱ s❡✉ ♣❡rí♦❞♦ é 2π ✳ ✐st♦ é✱ ❛ ❢✉♥çã♦ s❡❝❛♥t❡ é ♣❛r ❡ s❡✉ ❣rá✜❝♦ é s✐♠étr✐❝♦ r❡s♣❡✐t♦ ❛♦ ❡✐①♦✲y ❝♦♠♦ s❡ ♠♦str❛ ♥❛ ❋✐❣✉r❛ ✭✷✳✹✺✮✳ ❝✮ sec(−x) = sec x ❋✐❣✉r❛ ✷✳✹✺✿ ❙❡❝❛♥t❡ ❋✐❣✉r❛ ✷✳✹✻✿ ❈♦ss❡❝❛♥t❡ ✷✳✼✳✸✳✻ ❋✉♥çã♦ ❝♦ss❡❝❛♥t❡ 1 ➱ ❛ ❢✉♥çã♦ f : R −→ R ❞❡✜♥✐❞❛ ♣♦r✿ f (x) = csc x = ✳ sec x ❆❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛ ❢✉♥çã♦ ❝♦ss❡❝❛♥t❡✿ ❛✮ D(f ) = R − { π + kπ, ❜✮ k ∈ Z }; Im(f ) = (−∞, −1] ∪ [1, +∞)✳ ❆ ❢✉♥çã♦ ❝♦ss❡❝❛♥t❡ é ♣❡r✐ó❞✐❝❛✱ s❡✉ ♣❡rí♦❞♦ é 2π ✳ ❝✮ csc(−x) = − csc x✳ ✐st♦ é✱ ❛ ❢✉♥çã♦ ❝♦ss❡❝❛♥t❡ é í♠♣❛r ❡ s❡✉ ❣rá✜❝♦ é s✐♠étr✐❝♦ r❡s♣❡✐t♦ ❛♦ ❡✐①♦✲y ❝♦♠♦ s❡ ♠♦str❛ ♥❛ ❋✐❣✉r❛ ✭✷✳✹✻✮✳ ❊①❡♠♣❧♦ ✷✳✾✸✳ ❉❡t❡r♠✐♥❡ ❛ ár❡❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❞❛ ❜❛s❡ a✱ ❧❛❞♦ b✱ ❛❧t✉r❛ h ❡ â♥❣✉❧♦ ❞❛ ❜❛s❡ α✳ ❙♦❧✉çã♦✳ ❈♦♥s✐❞❡r❡ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❞❛ ❋✐❣✉r❛ ✭✷✳✹✼✮✳ h ❉❛ ❞❡✜♥✐çã♦ ❞❛ ❢✉♥çã♦ s❡♥♦ t❡♠♦s sena = ✱ ♦♥❞❡ h é ❛ ❛❧t✉r❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦❀ b ❧♦❣♦✱ ❝♦♠♦ ❛ ár❡❛ é✿ A = (base)(altura)✳ ▲♦❣♦✱ A = (a)(h) ⇒ A = (a)(b · senα)✳ P♦rt❛♥t♦ ❛ ár❡❛ ❞♦ tr❛♣é③✐♦ é A = ab · senα✳  ✶✹✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ b ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ❊s❝❛❞❛ 20m ✁ h α ✁✁ ✁ ✁ ✁ o a P a r e d e x ✁ 60 ❋✐❣✉r❛ ✷✳✹✼✿ R ❋✐❣✉r❛ ✷✳✹✽✿ ❊①❡♠♣❧♦ ✷✳✾✹✳ ❯♠❛ ❡s❝❛❞❛ ❡stá ❡♥❝♦st❛❞❛ ❡♠ ✉♠❛ ♣❛r❡❞❡ ❢♦r♠❛♥❞♦ ✉♠ â♥❣✉❧♦ ❞❡ 20 ❙❡ ❛ ❡s❝❛❞❛ t❡♠ 60o ❝♦♠ ♦ ❝❤ã♦✳ ♠❡tr♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦✱ q✉❡ ❛❧t✉r❛ ❞❛ ♣❛r❡❞❡ ❡❧❛ ❛t✐♥❣❡❄ ❙♦❧✉çã♦✳ ❆ ♣❛rt✐r ❞♦ ❞❡s❡♥❤♦ ❞❛ é ❝♦♥❤❡❝✐❞♦ t❡♠♦s✿ √ ❋✐❣✉r❛ 3 x = 2 20 P♦rt❛♥t♦✱ ❛ ❡s❝❛❞❛ ❛t✐♥❣❡ x ❀ ❛ss✐♠✱ ❝♦♠♦ ♦ sen60o 20 √ ⇒ x = 10 3 ⇒ x = 17, 32m✳ ♣❛r❡❞❡✳  ✭✷✳✹✽✮✱ t❡♠♦s q✉❡ √ ⇒ 2x = 20 3 17, 32 m ❞❡ ❛❧t✉r❛ ❞❛ sen60o = ❊①❡♠♣❧♦ ✷✳✾✺✳ ❉❡t❡r♠✐♥❡ ❞✉❛s ❢✉♥çõ❡s f ❡ g t❛✐s q✉❡ h(x) = sen4 4x + 5sen2 4x + 2 ♦♥❞❡ h = gof ✳ ❙♦❧✉çã♦✳ h(x) = sen4 4x + 5sen2 4x + 2 = [sen4x]4 + 5[sen4x]2 + 2✳ 4 2 ❈♦♥s✐❞❡r❡ f (x) = sen4x ❡ g(x) = x + 5x + 2✳ ❖✉tr❛s r❡❧❛çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❊♠ tr✐❣♦♥♦♠❡tr✐❛✱ ❛ ❧❡✐ ❞♦s s❡♥♦s é ✉♠❛ r❡❧❛çã♦ ♠❛t❡♠át✐❝❛ ❞❡ ♣r♦♣♦rçã♦ s♦❜r❡ ❛ ♠❡❞✐❞❛ ❞❡ tr✐â♥❣✉❧♦s ❛r❜✐trár✐♦s ❡♠ ✉♠ ♣❧❛♥♦✳ ▲❡✐ ❞♦s s❡♥♦s✿ r❛✐♦ r✱ ❞❡ ❧❛❞♦s ❊♠ ✉♠ tr✐â♥❣✉❧♦ BC ✱ AC ❡ AB ABC q✉❛❧q✉❡r✱ ✐♥s❝r✐t♦ ❡♠ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ✭❋✐❣✉r❛ ✭✷✳✹✾✮✮ q✉❡ ♠❡❞❡♠ r❡s♣❡❝t✐✈❛♠❡♥t❡ a, b ❡ c ❡ ❝♦♠ â♥❣✉❧♦s ✐♥t❡r♥♦s✱ ❡ ✈❛❧❡ ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦✿ a ▲❡✐ ❞♦s ❝♦ss❡♥♦s✿ b senA = b b senB = b senC = 2r ABC q✉❛❧q✉❡r✱ ❞❡ ❧❛❞♦s ♦♣♦st♦s ❛♦s â♥❣✉❧♦s a, b ❡ c✱ ❝♦♠♦ ✐♥❞✐❝❛ ❛ ❋✐❣✉r❛ ✭✷✳✹✾✮✱ ✈❛❧❡♠ ❛s ❊♠ ✉♠ tr✐â♥❣✉❧♦ ✐♥t❡r♥♦s ❡ ❝♦♠ ♠❡❞✐❞❛s r❡s♣❡❝t✐✈❛♠❡♥t❡ c r❡❧❛çõ❡s✿ b a2 = b2 + c2 − 2bc cos A, b b2 = a2 + c2 − 2ac cos B, ✶✹✼ b c2 = a2 + b2 − 2ab cos C 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ▲❡✐ ❞❛s t❛♥❣❡♥t❡s✿ ❊♠ tr✐❣♦♥♦♠❡tr✐❛✱ ❛ ❧❡✐ ❞❛s t❛♥❣❡♥t❡s ❡st❛❜❡❧❡❝❡ ❛ r❡❧❛çã♦ ❡♥tr❡ ❛s t❛♥✲ ❣❡♥t❡s ❞❡ ❞♦✐s â♥❣✉❧♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡ ♦s ❝♦♠✲ ♣r✐♠❡♥t♦s ❞❡ s❡✉s ❧❛❞♦s ♦♣♦st♦s✳❚❛❧ ♣r♦♣♦s✐çã♦ ❢♦✐ ❞❡s❝♦❜❡rt❛ ♣♦r ✈♦❧t❛ ❞❡ 1580✱ ♣❡❧♦ ♠❛t❡♠á✲ t✐❝♦ ❋r❛♥ç♦✐s ❱✐èt❡✳ ❋✐❣✉r❛ ✷✳✹✾✿ a, b ❡ c ♦s ❝♦♠♣r✐♠❡♥t♦s ❞♦s três ❧❛❞♦s tr✐â♥❣✉❧♦ ❡ α, β ❡ θ ✱ ♦s r❡s♣❡❝t✐✈♦s â♥❣✉❧♦s ❙❡❥❛♠ ❞♦ ♦♣♦st♦s ❛ ❡st❡s três ❧❛❞♦s✳ ❆ ❧❡✐ ❞❛s t❛♥❣❡♥t❡s ❡st❛❜❡❧❡❝❡ q✉❡ b − B) b tan 21 (α − β) tan 12 (A a−b = = b + B) b a+b tan 21 (α + β) tan 21 (A ✷✳✼✳✹ ❋✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ✐♥✈❡rs❛s ❉❡st❛❝❛♠♦s q✉❡ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s sã♦ ♣❡r✐ó❞✐❝❛s✱ ♣♦rt❛♥t♦ ♥ã♦ sã♦ ✐♥❥❡t✐✈❛s❀ ♥ã♦ ♦❜st❛♥t❡✱ r❡str✐♥❣✐♥❞♦ ❝♦♥✈❡♥✐❡♥t❡♠❡♥t❡ ♦ ❞♦♠í♥✐♦ ❞❡ ❝❛❞❛ ✉♠❛ ❞❡❧❛s✱ ♣♦❞❡♠♦s ♦❜t❡r q✉❡ s❡❥❛♠ ✐♥❥❡t✐✈❛s✳ ◆❡ss❛ r❡str✐çã♦✱ ❛ ❢✉♥çã♦ tr✐❣♦♥♦♠étr✐❝❛ ❛❞♠✐t❡ ❢✉♥çã♦ ✐♥✈❡rs❛✳ ❊st❛s r❡str✐çõ❡s sã♦ ❝❤❛♠❛❞❛s ❞❡ ✏ r❡str✐çã♦ ♣r✐♥❝✐♣❛❧ ✑✳ ✷✳✼✳✹✳✶ ❋✉♥çã♦ ❛r❝s❡♥ ❈♦♥s✐❞❡r❛♥❞♦ ❛ r❡str✐çã♦ ❞❛ ❢✉♥çã♦ s❡♥♦ ❛♦ ✐♥t❡r✈❛❧♦ π π [− , ] 2 2 t❡rí❛♠♦s q✉❡ ❡❧❛ é ❜✐❥❡t✐✈❛✱ ❡♥tr❡t❛♥t♦✱ ❡♠ ❣❡r❛❧ ❡❧❛ ♥ã♦ ♦ é ❡♠ t♦❞♦ s❡✉ ❞♦♠í♥✐♦✳ ❆ss✐♠✱ π π sen : [− , ] −→ [−1, 1] 2 2 ❋✐❣✉r❛ é ❜✐❥❡t✐✈❛✳ P♦rt❛♥t♦✱ ❛❞♠✐t❡ ❢✉♥çã♦ ✐♥✈❡rs❛ ✭ ✭✷✳✺✵✮✮ q✉❡ é ❛ ❢✉♥çã♦ ✿ π π arcsen : [−1, 1] −→ [− , ] 2 2 ❞❡ ♠♦❞♦ q✉❡✿ x = arcseny ⇔ y = senx ✷✳✼✳✹✳✷ ❋✉♥çã♦ ❛r❝❝♦s ❊♠ ❣❡r❛❧ ❛ ❢✉♥çã♦ ❝♦ss❡♥♦ ♥ã♦ é ❜✐❥❡t✐✈❛ ❡♠ t♦❞♦ s❡✉ ❞♦♠í♥✐♦✳ ❙❡ ❝♦♥s✐❞❡r❛♠♦s ❛ r❡str✐çã♦ ❞❛ ❢✉♥çã♦ ❝♦ss❡♥♦ ❛♦ ✐♥t❡r✈❛❧♦ [0, π]❀ ❡♥tã♦ t❡rí❛♠♦s q✉❡ ❡❧❛ é ❜✐❥❡t✐✈❛✳ ❆ss✐♠✱ cos : [0, π] −→ [−1, 1] é ❢✉♥çã♦ ❜✐❥❡t✐✈❛✳ ✶✹✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ π 2 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✻ y y✻ π y = arccos x y = arcsenx ✲ −1 R 0 π 2 x 1 ✲ − π2 −1 0 1 x ❋✐❣✉r❛ ✷✳✺✶✿ ❆r❝♦ ❝♦ss❡♥♦✳ ❋✐❣✉r❛ ✷✳✺✵✿ ❆r❝♦ s❡♥♦✳ ❋✐❣✉r❛ P♦rt❛♥t♦✱ ❛❞♠✐t❡ ❢✉♥çã♦ ✐♥✈❡rs❛ ✭ ✭✷✳✺✶✮✮ q✉❡ é ❛ ❢✉♥çã♦ ✿ arccos : [−1, 1] −→ [0, π] ❞❡ ♠♦❞♦ q✉❡✿ ⇒ x = arccos y y = cos x ✷✳✼✳✹✳✸ ❋✉♥çã♦ ❛r❝t❛♥ ❈❤❛♠❛✲s❡ r❡str✐çã♦ ♣r✐♥❝✐♣❛❧ ❞❛ t❛♥❣❡♥t❡ à ❢✉♥çã♦❀ ❋✐❣✉r❛ ❧♦❣♦ ❡❧❛ ❛❞♠✐t❡ ❢✉♥çã♦ ✐♥✈❡rs❛ ✭ π π tan : [− , ] −→ R 2 2 é ❜✐❥❡t✐✈❛❀ ✭✷✳✺✷✮✮ é ❛ ❢✉♥çã♦✿ π π arctan : R −→ [− , ] 2 2 ❞❡ ♠♦❞♦ q✉❡ ⇔ x = arctan y y = tan x✳ ✻ y ✻ y π 2 π y = arctan x −1 ✲ 0 1 x π 2 y = arccotx ✲ − π2 0 ❋✐❣✉r❛ ✷✳✺✷✿ ❆r❝♦ t❛♥❣❡♥t❡ x ❋✐❣✉r❛ ✷✳✺✸✿ ❆r❝♦ ❝♦t❛♥❣❡♥t❡ ✷✳✼✳✹✳✹ ❋✉♥çã♦ ❛r❝❝t❣ ❈❤❛♠❛✲s❡ r❡str✐çã♦ ♣r✐♥❝✐♣❛❧ ❞❛ ❝♦t❛♥❣❡♥t❡ à ❢✉♥çã♦❀ cot : [0, π] −→ R✳ ✶✹✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❡❧❛ é ❜✐❥❡t✐✈❛❀ ❧♦❣♦ ❡❧❛ ❛❞♠✐t❡ ❝♦♠♦ ❢✉♥çã♦ ✐♥✈❡rs❛ ✭❋✐❣✉r❛ ✭✷✳✺✸✮✮ ❛ ❢✉♥çã♦✿ arccot : R −→ [0, π] x = arccoty ⇔ y = cot x✳ ❞❡ ♠♦❞♦ q✉❡ ✷✳✼✳✹✳✺ ❋✉♥çã♦ ❛r❝s❡❝ ❈❤❛♠❛✲s❡ r❡str✐çã♦ ♣r✐♥❝✐♣❛❧ ❞❛ s❡❝❛♥t❡ à ❢✉♥çã♦✿ sec : [0, π π ) ∪ ( , π] −→ (−∞, −1] ∪ [1, +∞) 2 2 ❡st❛ ❢✉♥çã♦ é ❜✐❥❡t✐✈❛❀ ❧♦❣♦ ❡❧❛ ❛❞♠✐t❡ ❢✉♥çã♦ ✐♥✈❡rs❛✳ ❙✉❛ ❢✉♥çã♦ ✐♥✈❡rs❛ é✿ arcsec : (−∞, −1] ∪ [1, +∞) −→ [0, ❞❡ ♠♦❞♦ q✉❡ x = arcsecy ⇔ y = sec x π π ) ∪ ( , π] 2 2 ✭❋✐❣✉r❛ ✭✷✳✺✹✮✮ ❋✐❣✉r❛ ✷✳✺✹✿ ❆r❝♦ s❡❝❛♥t❡ ❋✐❣✉r❛ ✷✳✺✺✿ ❆r❝♦ ❝♦ss❡❝❛♥t❡✳ ✷✳✼✳✹✳✻ ❋✉♥çã♦ ❛r❝❝s❝ ❈❤❛♠❛✲s❡ r❡str✐çã♦ ♣r✐♥❝✐♣❛❧ ❞❛ ❝♦ss❡❝❛♥t❡ à ❢✉♥çã♦❀ π π csc : [− , 0) ∪ (0, ] −→ (−∞, −1] ∪ [1, +∞) 2 2 ❡❧❛ é ❜✐❥❡t✐✈❛❀ ❡ ❛❞♠✐t❡ ❢✉♥çã♦ ✐♥✈❡rs❛ ✭❋✐❣✉r❛ ✭✷✳✺✺✮✮ é ❛ ❢✉♥çã♦✿ π π arccsc : (−∞, −1] ∪ [1, +∞) −→ [− , 0) ∪ (0, ] 2 2 ❞❡ ♠♦❞♦ q✉❡✿ ❊①❡♠♣❧♦ ✷✳✾✻✳ ▼♦str❡ q✉❡ ❙♦❧✉çã♦✳ x = arccscy ⇒ y = csc x✳ √ cos(arcsenx) = ± 1 − x2 . ✶✺✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❙❛❜❡♠♦s q✉❡ ❛ ❢✉♥çã♦ senx ❡ arcsenx ✉♠❛ é ❢✉♥çã♦ ✐♥✈❡rs❛ ❞❛ ♦✉tr❛❀ ❧♦❣♦ sen(arcsenx) = x✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛ ✐❞❡♥t✐❞❛❞❡ tr✐❣♦♥♦♠étr✐❝❛ sen2 x + cos2 x = 1 s❡❣✉❡ ♣♦r q✉❡stã♦ ❞❡ ♥♦t❛çã♦ q✉❡ [senx]2 +[cos x]2 = 1✱ ❧♦❣♦ s❡♥❞♦ ♦ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ s❡♥♦ ❡ ❝♦ss❡♥♦ q✉❛✐sq✉❡r ♥ú♠❡r♦ r❡❛❧ ✈❡♠✱ q✉❡ [sen(arcsenx)]2 + [cos .(arcsenx)]2 = 1 ✐st♦ é x2 + [cos(arcsenx)]2 = 1 √ ❡♥tã♦ cos(arcsenx) = ± 1 − x2 ✳ ✷✳✼✳✺ ❋✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s ❈♦♥s✐❞❡r❛♥❞♦ ❞✐❢❡r❡♥t❡s tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s ❝♦♠♦ ♥❛ ❋✐❣✉r❛ ✭✷✳✺✻✮ ❡ ❝❛❧❝✉❧❛♥❞♦ ❛ r❡❧❛çã♦ ❡♥tr❡ s❡✉s ❧❛❞♦s✱ ♦❜t❡r❡♠♦s q✉❡ ❡st❛s r❡❧❛çõ❡s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ s❡✉s ❧❛❞♦s✳ ❆ss✐♠✱ s❛❜❡♠♦s q✉❡✿ senα = BC , OC cos α = OB , OC tan α = BC OB cot α = OB BC ❊✱ s✉❛s ❝♦rr❡s♣♦♥❞❡♥t❡s r❡❧❛çõ❡s ✐♥✈❡rs❛s sã♦✿ csc α = OC , BC sec α = OC , OB r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❋✐❣✉r❛ ✷✳✺✻✿ ❋✐❣✉r❛ ✷✳✺✼✿ ❆ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ 0 ❡ r❛✐♦ OA = r é ✐❣✉❛❧ ❛ πr2 ✱ ❧♦❣♦ ❛ ár❡❛ ❞❡ ✉♠ s❡t♦r ❝✐r❝✉❧❛r ❞❡ â♥❣✉❧♦ 2α é αr2 ✳ ❈♦♥s✐❞❡r❛♥❞♦ r = 1✱ ❛ ár❡❛ ❞♦ s❡t♦r ❝✐r❝✉❧❛r ❞❡ â♥❣✉❧♦ 2α é α✳ ❈❤❛♠❛♠♦s x ❛ ár❡❛ ❞♦ s❡t♦r ❝✐r❝✉❧❛r ❞❡ â♥❣✉❧♦ 2α✱ ❡♥tã♦ senx = BC, cos x = OB ❡ tan x = BC/OB ❀ r❡s✉❧t❛ q✉❡ ❛ ❡q✉❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ ✉♠ ❡ ❝❡♥tr♦ ❛ ♦r✐❣❡♠ ❞❡ ❝♦♦r❞❡♥❛❞❛s é x2 + y 2 = 1✱ ❡ ❛ ❡q✉❛çã♦ ❞❡ ✉♠❛ ❤✐♣ér❜♦❧❡ ❡q✉✐❧át❡r❛ ❞❡ r❛✐♦ ✉♠ ❡ ❝❡♥tr♦ ❛ ♦r✐❣❡♠ ❞❡ ❝♦♦r❞❡♥❛❞❛s é x2 − y 2 = 1✳ P♦❞❡♠♦s ❞❡✜♥✐r ♥❛ ❋✐❣✉r❛ ✭✷✳✺✼✮✱ ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s✿ ✶✺✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R BC OA OB • ❈♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦✿ cosh α = OA BC • ❚❛♥❣❡♥t❡ ❤✐♣❡r❜ó❧✐❝♦✿ tanh α = OB OB • ❈♦t❛♥❣❡♥t❡ ❤✐♣❡r❜ó❧✐❝♦✿ coth α = BC OA • ❙❡❝❛♥t❡ ❤✐♣❡r❜ó❧✐❝♦✿ sechα = OB OA • ❈♦ss❡❝❛♥t❡ ❤✐♣❡r❜ó❧✐❝♦✿ cschα = BC ❖❜s❡r✈❡ q✉❡ ❛s r❡❧❛çõ❡s coth α, sechα ❡ cschα sã♦ ✐♥✈❡rs❛s ❞❛s r❡❧❛çõ❡s tanh α, cosh α ❡ senhα r❡s♣❡❝t✐✈❛♠❡♥t❡✳ • ❙❡♥♦ ❤✐♣❡r❜ó❧✐❝♦✿ senhα = ❉♦ ♠❡s♠♦ ♠♦❞♦✱ ♣❛r❛ ♦ ❝❛s♦ ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❤❛❜✐t✉❛✐s✱ ❛ ár❡❛ s♦♠❜r❡❛❞❛ ❞❛ ❤✐♣ér❜♦❧❡ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠ â♥❣✉❧♦ 2α✱ ❝♦♥s✐❞❡r❛♥❞♦ OA = 1, é α✳ ❙❡❥❛ x ❛ ár❡❛ ❞♦ s❡t♦r ❝✐r❝✉❧❛r ❞❡ â♥❣✉❧♦ 2α✱ ❡♥tã♦✿ senhx = BC, cosh x = OB ❡ tanh x = AD✳ ❊♠ ❛❧❣✉♠❛s ♦❝❛s✐õ❡s ❛s ❝♦♠❜✐♥❛çõ❡s ❞❡ ex ❡ e−x ❛♣❛r❡❝❡♠ ❝♦♠ ❢r❡q✉ê♥❝✐❛❀ ❡♠ t❛✐s ♦❝❛s✐õ❡s ❛❝♦st✉♠❛✲s❡ ❛ ❡s❝r❡✈❡r ♦ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ✉t✐❧✐③❛♥❞♦ ❛s ❢✉♥çõ❡s f : R −→ R ❝❤❛♠❛❞❛s ❤✐♣❡r❜ó❧✐❝❛s✱ ❡ ❞❡✜♥✐❞❛s ❛ s❡❣✉✐r✿ • ❙❡♥♦ ❤✐♣❡r❜ó❧✐❝♦✿ f (x) = senhx = ex − e−x 2 • ❈♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦✿ f (x) = cosh x = ex + e−x 2 • ❚❛♥❣❡♥t❡ ❤✐♣❡r❜ó❧✐❝♦✿ f (x) = tanh x = cosh x ex + e−x = ex − e−x senhx 2 ex + e−x • ❈♦ss❡❝❛♥t❡ ❤✐♣❡r❜ó❧✐❝♦✿ f (x) = cschx = ✶✺✷ ∀x∈R ex − e−x senhx = x −x e +e cosh x • ❈♦t❛♥❣❡♥t❡ ❤✐♣❡r❜ó❧✐❝♦✿ f (x) = coth x = • ❙❡❝❛♥t❡ ❤✐♣❡r❜ó❧✐❝♦✿ f (x) = sechx = ∀x∈R 2 ex − e−x ∀x∈R ∀ x ∈ R − {0}✳ ∀x∈R ∀ x ∈ R − {0} 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✷✲✼ ✶✳ ❱❡r✐✜q✉❡ s❡ ❛ ❢✉♥çã♦ ❛ s❡❣✉✐r é ♣❛r ♦ í♠♣❛r ❥✉st✐✜❝❛♥❞♦ s✉❛ r❡s♣♦st❛✳ 1. f (x) = −x3 + x 4. f (x) = 5x − senx2 7. f (x) = ex + e−x 2 2. f (x) = x · senx x 5. h(x) = |x| 3. f (x) = sen3x · cos x 6. f (x) = x · et 2 9. f (x) = x4 + cos3 x 8. g(x) = 5 ✷✳ ❉❡t❡r♠✐♥❡ ❞✉❛s ❢✉♥çõ❡s f ❡ g t❛✐s q✉❡ h = gof ♥♦s s❡❣✉✐♥t❡s ❝❛s♦s✿ 1. h(x) = (x2 + 3)6 2. h(x) = 3(x+ | x |) 3. h(x) = 2sen2x 2  x−4 5. h(x) = cos2 5x + 7 cos6 5x 6. h(x) = (x2 − 8)4 4. h(x) = √ x−2  3 2x + 5 7. h(x) = 8. h(x) = (cos 4x)2 − 4(cos 4x) 9. h(x) = 2tan 2x x−4 ✸✳ ❙❡ f : A −→ Im(f ) é ♠♦♥♦tô♥✐❝❛ ❡str✐t❛✱ ❡♥tã♦ f −1 : Im(f ) −→ A é ♠♦♥♦tô♥✐❝❛ ❡str✐t❛ ❞♦ ♠❡s♠♦ t✐♣♦❄ h π πi ✳ 2 2 ✹✳ Pr♦✈❡ q✉❡ tan x é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ❡♠ − , ✺✳ ❉❛❞♦ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f (x) ✭❋✐❣✉r❛ ✷✳✺✽✮ ❡ ♦s ✈❛❧♦r❡s a ❡ b ❞❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ x✳ ❉❡t❡r✲ ♠✐♥❡ f (a) ❡ f (b) ♥♦ ❞❡s❡♥❤♦✳ ◗✉❛❧ é ❛ ✐♥t❡r♣r❡✲ t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ r❡❧❛çã♦✿ ✻y x ✲ ✛ a b F igura 2.58 f (b) − f (a) b−a h π πi ✻✳ Pr♦✈❡ q✉❡ ❛ ❢✉♥çã♦ sen : − , −→ R é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡✳ 2 2 ✼✳ ❙❡❥❛ f (x) = 2x2 + 5 1 2 + + 5x✳ ▼♦str❡ q✉❡ f (x) = f ( )✳ 2 x x x ✶✺✸ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✽✳ ❉❡t❡r♠✐♥❡ ♦ ❞♦♠í♥✐♦ ❞❡ ❞❡✜♥✐çã♦ ❞❛s ❢✉♥çõ❡s q✉❡ s❡ ✐♥❞✐❝❛♠✿ 1. y = 1 − Lnx √ 4. y = Ln x − 4  1 − 2x 4 2. y = Ln(sen(2x − 1)) 3. y = arccos 5. y = arcsen(x − 2) 6. y = Ln(Ln(x − 1))  x 1 11 − q✉❛♥❞♦ x ≤ ❡ g(x) = 1 + x q✉❛♥❞♦ 2 2 3 11 x > ✳ ❆❝❤❛r t♦❞❛s ❛s r❛í③❡s r❡❛✐s ❞❛ ❡q✉❛çã♦ [g(x)]2 = 7x + 25✳ 3 ✾✳ ❆ ❢✉♥çã♦ g(x) é ❞❡✜♥✐❞❛ ♣♦r✿ g(x) = ✶✵✳ ❆❝❤❛r ♦ ♠❛✐♦r ✈❛❧♦r ♣♦ssí✈❡❧ ♣❛r❛ n ♣❛r❛ ♦ q✉❛❧ 2x > xn ♣❛r❛ t♦❞❛s ❛s x ≥ 100, ∀ n ∈ Z✳ ✶✶✳ ❉❡t❡r♠✐♥❡ s❡ ❛s s❡❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s sã♦ ✈❡r❞❛❞❡✐r❛s✿ 1. cosh2 x + senh2 x = cosh 2x 2. 3. cosh(x + y) = cosh x. cosh y + senhy.senhx 4. 1 − coth2 x = csch2 x 5. senh(x + y) = senhx. cosh y + senhy. cosh x 7.. 2senhx. cosh x = senh2 x cosh2 x − senh2 x = 1 6. 1 − tan2 x = sech2 x ✶✷✳ ❙❡❥❛ f (x) = senx − cos x✳ ▼♦str❡ q✉❡ f (1) > 0✳ ✶✸✳ ❉❡t❡r♠✐♥❡ ♦ ♣❡rí♦❞♦ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ 1. y = 2sen(3x + 5)   x−1 3. y = − cos 2 2. y = 5 cos 2x   2t + 3 4. y = sen 6π ✶✹✳ ▼♦str❡ q✉❡ y = senhx ❡ y = tanhx sã♦ ❢✉♥çõ❡s í♠♣❛r❡s✱ ❡ y = cosh x é ❢✉♥çã♦ ♣❛r✳ ✶✺✳ ❘❡s♦❧✈❡r ❣r❛✜❝❛♠❡♥t❡ ❛ ❡q✉❛çã♦✿ 1. x = 2senx 2. x = tan x 3. 4senx = 4 − x ✶✻✳ ❯♠ ♥❛✈✐♦✱ ♥❛✈❡❣❛♥❞♦ ❡♠ ❧✐♥❤❛ r❡t❛✱ ♣❛ss❛ s✉❝❡ss✐✈❛♠❡♥t❡ ♣❡❧♦s ♣♦♥t♦s A, B ❡ C ✳ ◗✉❛♥❞♦ ♦ ♥❛✈✐♦ ❡stá ❡♠ A✱ ♦ ❝♦♠❛♥❞❛♥t❡ ♦❜s❡r✈❛ ♦ ❢❛r♦❧ ❡♠ F ✱ ❡ ❝❛❧❝✉❧❛ ♦ \ â♥❣✉❧♦ F[ AC = 30o ✳ ❆♣ós ♥❛✈❡❣❛r 4 ♠✐❧❤❛s ❛té B ✱ ✈❡r✐✜❝❛✲s❡ ♦ â♥❣✉❧♦ F BC = 75o ✳ ◗✉❛♥t❛s ♠✐❧❤❛s s❡♣❛r❛ ♦ ❢❛r♦❧ ❞♦ ♣♦♥t♦ B ❄ ✶✺✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R C ✶✼✳ ❯♠❛ t♦rr❡ t❡♠ 20 ♠❡tr♦s ❞❡ ❛❧t✉r❛✳ ❙❡ ♣✉①❛r♠♦s ✉♠ ❝❛❜♦ ❞♦ t♦♣♦ ❛♦ ❝❤ã♦ ✭❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✷✳✺✾✮✱ q✉❛❧ s❡rá ♦ ❝♦♠♣r✐♠❡♥t♦ ❛♣r♦①✐♠❛❞♦ ✭x✮ ❞♦ ❝❛❜♦❄ x 20 45o A B F igura 2.59 ✶✽✳ P❡❞r♦ ❡ ▼❛r❝♦s q✉❡ ❡stã♦ ❞✐st❛♥t❡s 2, 7 km ✉♠ ❞♦ ♦✉tr♦✱ ♦❜s❡r✈❛♠ ✉♠ ❤❡❧✐❝ó♣t❡r♦ q✉✐❡t♦ ♥♦ ❛r✱ P❡❞r♦ ✈ê ♦ ❤❡❧✐❝ó♣t❡r♦ s❡❣✉♥❞♦ ✉♠ â♥❣✉❧♦ ❞❡ 45o ❡ ▼❛r❝♦s✱ ❛♦ ♠❡s♠♦ t❡♠♣♦✱ ✈ê ♦ ❤❡❧✐❝ó♣t❡r♦ s❡❣✉♥❞♦ ✉♠ â♥❣✉❧♦ ❞❡ 60o ✳ ❆♣r♦①✐♠❛❞❛♠❡♥t❡ ❛ q✉❡ ❛❧t✉r❛ ❡st❛✈❛ ♦ ❤❡❧✐❝ó♣t❡r♦❄ ✶✾✳ ❯♠ ❛✈✐ã♦ ❧❡✈❛♥t❛ ✈ô♦ ❡ s♦❜❡ ❢❛③❡♥❞♦ ✉♠ â♥❣✉❧♦ ❞❡ 15o ❝♦♠ ❛ ❤♦r✐③♦♥t❛❧✳ ❆ q✉❡ ❛❧t✉r❛ ❡st❛✈❛ ❡ q✉❛❧ é ❛ ❞✐stâ♥❝✐❛ ♣❡r❝♦rr✐❞❛ q✉❛♥❞♦ ♣❛ss❛ ♣❡❧❛ ✈❡rt✐❝❛❧ ♣♦r ✉♠❛ ✐❣r❡❥❛ s✐t✉❛❞❛ ❛ 2 km ❞♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛❄ ✷✵✳ ❱❡r✐✜❝❛r q✉❡✱ s❡ 0 < α < π ✱ ❡♥tã♦ cot α ≥ 1 + cot α 2 π π )✳ ❉❡t❡r♠✐♥❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ✈❛❧♦r❡s ❞❡ α t❛✐s q✉❡ t♦❞❛s ❛s 2 2 √ s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ x4 − 4 48x2 + tan α = 0 ❡♠ x s❡ ❡♥❝♦♥tr❡♠ ❡♠ R✳ ✷✶✳ ❙❡❥❛ α ∈ (− , ✷✷✳ ❙❡ ♦s ❛r❝♦s ♣♦s✐t✐✈♦s α1 , α2 , α3 ❡ α4 s❡ ❡♥❝♦♥tr❛♠ ❡♥tr❡ 0 ❡ π ✱ ♠♦str❡ q✉❡ α1 + α2 2 ✶✳ senα1 + senα2 ≤ 2 · sen ✷✳ senα1 + senα2 + senα3 + senα4 ≤ 4 · sen α1 + α2 + α3 + α4 4 ✷✸✳ ❱❡r✐✜❝❛r q✉❡ ✶✳ tan 20o · tan 40o · tan 50o · tan 80o = 3. ✷✳ cos 20o − cos 80o = − cos 140o ✷✹✳ ❉❡t❡r♠✐♥❡ ♦ ♠á①✐♠♦ ♥ú♠❡r♦ ❞❡ r❛í③❡s ❞❛ ❡q✉❛çã♦ E(x) = log x − senx = 0✳ ✷✺✳ ❯♠❛ ár✈♦r❡✱ ♣❛rt✐❞❛ ♣❡❧♦ ✈❡♥t♦✱ ♠❛♥té♠ s❡✉ tr♦♥❝♦ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ s♦❧♦✱ ❢♦r♠❛♥❞♦ ❝♦♠ ❡❧❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✳ ❙❡ ❛ ♣❛rt❡ q✉❡❜r❛❞❛ ❢❛③ ✉♠ â♥❣✉❧♦ ❞❡ 60o ❝♦♠ ♦ s♦❧♦ ❡ s❡ ♦ t♦♣♦ ❞❛ ár✈♦r❡ ❡stá ❛❣♦r❛ ❞✐st❛♥❝✐❛❞♦ 10 m ❞❡ s✉❛ ❜❛s❡✱ q✉❛❧ ❡r❛ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❛ ❛❧t✉r❛ ♦r✐❣✐♥❛❧ ❞❛ ár✈♦r❡❄ ✷✻✳ ◆✉♠ tr✐â♥❣✉❧♦ △ABC ♦♥❞❡ AB = 10cm, AC = 12cm ❡ ♦ â♥❣✉❧♦ Ab é 30o ✱ ❞❡t❡r♠✐♥❡ ❛ ár❡❛ ❞❡ss❡ tr✐â♥❣✉❧♦✳ ✷✼✳ ❆ss♦❝✐❛♥❞♦ V ♣❛r❛ ❛s s❡♥t❡♥ç❛s ✈❡r❞❛❞❡✐r❛s ❡ F ♣❛r❛ ❛s ❢❛❧s❛s✱ ❛ss✐♥❛❧❡ ❛ ❛❧t❡r♥❛t✐✈❛ q✉❡ ❝♦♥tê♠ ❛ s❡q✉ê♥❝✐❛ ❝♦rr❡t❛✳ ✶✺✺ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✐✮ ❆ ❢✉♥çã♦ y = csc x · sec x é ♥❡❣❛t✐✈❛ ♥♦ 2o ❡ ♥♦ 4o q✉❛❞r❛♥t❡✳ 3π 10 5 ✱ q✉❛♥❞♦ < x < 2π ✱ ❡♥tã♦ cos x = ✳ 13 2 13 ✐✐✐✮ ❖ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ y = cot x é { x ∈ R / x ≤ kπ }✳ ✐✐✮ ✐✈✮ ❙❡ senx = − ❆ ❢✉♥çã♦ y = tan x é ♣❡r✐ó❞✐❝❛✱ ❝♦♠ ♣❡rí♦❞♦ P = π rad✳ ✷✽✳ ❆❝❤❛r ♦ ✐♥t❡r✈❛❧♦ ❞❡ ✈❛r✐❛çã♦ ❞❡ x ♣❛r❛ q✉❡ s❡❥❛ ✈á❧✐❞❛ ❛ ✐❞❡♥t✐❞❛❞❡✿ π 1. arcsenx + arccos x = 2 √ √ π 3. arcsen x + arccos x = 2   1 − x2 5. arccos = 2arccotx 1 + x2 7. 9. √ 1 − x2 = arcsenx 2. arccos 4. 1 − x2 = −arcsenx   1 − x2 arccos = −2 arctan x 1 + x2 1 arctan x = arccot − π x 1 arctan x = arccot x 6.   1+x arctan x + arctan 1 = arctan 8. 1−x   1+x + π 10. arctan x + arctan 1 = arctan 1−x arccos √ ✷✾✳ ▼♦str❡ q✉❡ ❛s s❡❣✉✐♥t❡s ❢ór♠✉❧❛s sã♦ ✈❡r❞❛❞❡✐r❛s✿ ✶✳ ✷✳ ✸✳ ✹✳ ✺✳ cos 2x · sen( 3x ) 2 x sen( 2 ) 2π 2π 2π 3 2π + x) + cos( + x) · cos( − x) + cos( − x) cos x = − cos x · cos( 3 3 3 3 4 9x x senx + sen2x + sen7x + sen8x = 4sen( ) · cos 3x · cos( ) 2 2 cos 3x · cos 5x tan x + tan 7x = tan 3x + tan 5x cos x · cos 7x π 1 1 = arctan + arctan 4 2 3 cos x + cos 2x + cos 3x = ✸✵✳ ❊①♣r❡ss❛r ❛ ár❡❛ ❞❡ ✉♠ tr❛♣é③✐♦ ✐sós❝❡❧❡s ❞❡ ❜❛s❡s a ❡ b ❝♦♠♦ ❢✉♥çã♦ ❞♦ â♥❣✉❧♦ β ❞❛ ❜❛s❡ a✳ ❈♦♥str✉✐r ♦ ❣rá✜❝♦ ♣❛r❛ a = 1 ❡ b = 3✳ ✸✶✳ ❙❡❥❛ a ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧ ♣♦s✐t✐✈❛✳ ❘❡s♦❧✈❡r ❛ ❡q✉❛çã♦ ❡♠ R✱ s❡♥❞♦ 0 < x < a✳ √ ❙✉❣❡stã♦✿ q q √ √ √ √ 2 2 a a + a − x + 3a a − a2 − x2 = 2 2x ❈♦♥s✐❞❡r❛r a · senβ = x ✶✺✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ▼✐s❝❡❧â♥❡❛ ✷✲✶ 9 − x2 ♣❛r❛ x ≥ 0, ✶✳ ❉❛❞❛ ❛ ❢✉♥çã♦ f (x) = 4 − x2 x 6= 2✿ ✶✳ ▼♦str❡ q✉❡ f é ✐♥❥❡t✐✈❛ ✷✳ ❉❡t❡r♠✐♥❡ f −1 ✸✳ ❉❡t❡r♠✐♥❡ D(f −1 ) ✹✳ ❉❡t❡r♠✐♥❡ Im(f −1 ) ✷✳ ❘❡s♦❧✈❡r ❣r❛✜❝❛♠❡♥t❡ ❛ ❡q✉❛çã♦✿ 2x − 2x = 0✳ ✸✳ ❉❡t❡r♠✐♥❡ ❢✉♥çõ❡s f t❛✐s q✉❡ f (x2 ) − f (y 2 ) + 2x + 1 = f (x + y) · f (x − y) q✉❛✐sq❡r q✉❡ s❡❥❛♠ ♦s ♥ú♠❡r♦s r❡❛✐s x, y ✳ ✹✳ ❉❛❞❛ ❛ r❡❧❛çã♦✿ R(x) = 2x3 − 5x2 − 23x✱ ❞❡t❡r♠✐♥❡ t♦❞❛s ❛s r❛í③❡s ❞❛ ✐❣✉❛❧❞❛❞❡ R(x) = R(−2)✳ ✺✳ ❉❡t❡r♠✐♥❡ t♦❞❛s ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ f (x) = f (5) s❛❜❡♥❞♦ q✉❡ ❛ r❡❧❛çã♦ f (x) = x2 − 12x + 3 é ❞❡✜♥✐❞❛ ♥♦ ✐♥t❡r✈❛❧♦ [−5, 5]✳ ✻✳ ❙❡❥❛ f (n) ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛✳ ▼♦str❡ q✉❡❀ f (n + 3) − 3f (n + 2) + 3f (n + 1) − f (n) = 0 ✼✳ ❊s❜♦ç❛r ♦ ❣rá✜❝♦ ❞♦s ♣♦♥t♦s q✉❡ ❝✉♠♣r❡♠ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s✿ ✶✳ S = { (x, y) ∈ R2 /. y ≤ 2x, y ≥ 2−x } ✷✳ S = { (x, y) ∈ R2 /. y ≤ 2−x , ✸✳ S = { (x, y) ∈ R2 /. y ≤ 3x, ✹✳ S = { (x, y) ∈ R2 /. y ≤ log4 x, ✺✳ S = { (x, y) ∈ R2 /. y ≤ log0.6 x, ✻✳ S = { (x, y) ∈ R2 /. x ≤ log3 y, ✼✳ S = { (x, y) ∈ R2 /. x ≤ 2y, y + x ≥ 0, y + x < 0, x2 + y 2 < 4 } y ≤ 2−x } x2 + y 2 ≤ 9, x2 + y 2 < 16, x2 + y 2 < 9, y − x ≥ 0, x>0} x>0} y >0} x2 + y 2 < 16 } ✽✳ ❉✐❣❛ q✉❛✐s ❞❛s ❢✉♥çõ❡s sã♦ ♣❡r✐ó❞✐❝❛s✳ ◆♦s ❝❛s♦s ❛✜r♠❛t✐✈♦s✱ ❞❡t❡r♠✐♥❡ q✉❛♥❞♦ ✶✺✼ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❡①✐st❡♠ ♦s ♣❡rí♦❞♦s✳ 2. f (x) = 1 s❡ x ∈ Z   senx, s❡✱ x ≥ π 2 4. f (x) =  cos x, s❡✱ x < π 2 6. f (t) = cos(10t) + cos[(10 + π)t] 1. f (x) = x + [|x|] ( 1, s❡✱ x ∈ Q 3. f (x) = 0, s❡✱ x ∈ R − Q 5. f (x) = cos x x + cos 3 4 ✾✳ ❖s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ♠❡❞❡♠ 1 cm ❡ 2 cm r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❈♦♥str✉✐r ♦ ❣rá✜❝♦ ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ❝♦♠♦ ❢✉♥çã♦ ❞♦ â♥❣✉❧♦ x ❝♦♠♣r❡❡♥❞✐❞♦ ❡♥tr❡ t❛✐s ❧❛❞♦s✳ ✶✵✳ ❉❡♠♦♥str❛r ❛s s❡❣✉✐♥t❡s ✐❞❡♥t✐❞❛❞❡s✿ ✶✳ ✷✳ Ln | csc x − cot x |= −Ln | csc x + cot x | √ 3Ln 3 f (x) = Ln ❙❡ f (x) = −Ln | csc x + cot x |✱ ❡♥tã♦ e senx 1 + cos x ex − e−x ex + e−x ❡ g(x) = ✳ ❉❡♠♦♥str❛r✿ 2 2       g(2x) + g(2y) x−y g x+y ·f 2. (x + y) = 1. f (x) + f (y) = 2f 2 2 f (2x) + f (2y) f     x−y x+y ·f 4. [f (x)]2 − [g(x)]2 = 1 3. g(x) + g(y) = 2g 2 2 ✶✶✳ ❙❡❥❛♠ ❛s ❢✉♥çõ❡s✱ f (x) = 5. [f (x) + g(x)]n = f (nx) + g(nx) n ∈ N 6. f (x) é ❢✉♥çã♦ ♣❛r ❡✱ g(x) é ❢✉♥çã♦ í♠♣❛r✳ h x i2 h x i2 7. f ( ) = [g(x)]2 + 1 ❡ g( ) = [f (x)]2 − 1 2 2 ✶✷✳ ▼♦str❡ q✉❡✿ ✶✳ ✸✳ √ 1 − x2 √ sec(arctan x) = 1 + x2 cos(arcsenx) = ✶✸✳ ❉❛ r❡❧❛çã♦ tan ✷✳ sen(arccos x) = √ ✹✳ csc(arccotx) = √ 1 − x2 1 + x2 α tan α + m − 1 α = ✱ ❞❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❡ tan ✳ 2 tan α + m + 1 2 ✶✹✳ ❙❡ A ❡ C r❡♣r❡s❡♥t❛♠ r❡s♣❡❝t✐✈❛♠❡♥t❡ ♦ ♠❛✐♦r ❡ ♠❡♥♦r ❞♦s â♥❣✉❧♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ t❛✐s q✉❡ s❡✉s ❧❛❞♦s ❢♦r♠❛♠ ✉♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛✳ ▼♦str❡ q✉❡✿ 4(1−cos A)(1− cos C) = cos A + cos C ✳ ✶✺✳ ❉❡♠♦♥str❡ q✉❡ ✉♠ tr✐â♥❣✉❧♦ ❝✉❥♦s â♥❣✉❧♦s ✈❡r✐✜❝❛ ❛ r❡❧❛çã♦✿ 2 cos BsenC = senA é ✐sós❝❡❧❡s✳ ✶✺✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✶✻✳ ❆❝❤❛r ♦ ❞♦♠í♥✐♦ ❞❡ ❞❡✜♥✐çã♦ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ ✶✳ y= p Ln(senx) ✷✳ y = Ln(senx) ✶✼✳ ❱❡r✐✜❝❛r ❛s s❡❣✉✐♥t❡s ❢ór♠✉❧❛s✿ ✶✳ ✸✳ π 1 1 1 = arctan + arctan + arctan 4 2 5 8 1 1 1 π = 2 arctan + arctan + 2 arctan 4 7 5 8 3 2 arctan 79 1 4 = arctan 2 3 π 1 = 5 arctan + 4 7 2. 2 arctan 4. √ ✶✽✳ ▼♦str❡ q✉❡ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f (x) = loga (x + x2 + 1) é s✐♠étr✐❝♦ r❡s♣❡✐t♦ à ♦r✐❣❡♠ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ❉❡t❡r♠✐♥❡ s✉❛ ❢✉♥çã♦ ✐♥✈❡rs❛✳ ✶✾✳ ❊s❝r❡✈❡r ❡♠ ❢♦r♠❛ ❡①♣❧í❝✐t❛ ✉♠❛ ❢✉♥çã♦ y = f (x) ❞❛❞❛ ❡♠ ❢♦r♠❛ ✐♠♣❧í❝✐t❛ ♠❡❞✐❛♥t❡ ❝❛❞❛ ✉♠❛ ❞❛s ❡q✉❛çõ❡s✿ 1. x2 + y 2 = 1 4. x3 + y 3 = a3 x2 y 2 + 2 =1 a2 b x+y 5. 2 (x2 − 2) = x3 + 7 2. 3. Ln(x) + Ln(y + 1) = 4 6. (1 + x) cos y − x2 = 0 ✷✵✳ ❙❡❥❛ f (x) = a · cos(bx + c)✳ ◗✉❛✐s ❞❡✈❡♠ s❡r ♦s ✈❛❧♦r❡s ❞❛s ❝♦♥st❛♥t❡s a, b ❡ c ♣❛r❛ ♦❜t❡r ❛ ✐❞❡♥t✐❞❛❞❡ f (x + 1) − f (x) = senx ❄ ✷✶✳ ❘❡s♦❧✈❡r ❛ ❡q✉❛çã♦✿ ✶✳ ✸✳ √ √ 2senx = 1 − sen2x − 1 + sen2x √ √ ✷senx = 1 + sen2x + 1 − sen2x ✷✳ ✷cos x = √ 1 − sen2x − √ 1 + sen2x ✷✷✳ ◆✉♠ ❝♦♥❡ ❝✐r❝✉❧❛r r❡t♦ ❝♦♠ r❛✐♦ ♥❛ ❜❛s❡ R ❡ ❛❧t✉r❛ H ❡♥❝♦♥tr❛✲s❡ ✐♥s❝r✐t♦ ✉♠ ❝✐❧✐♥❞r♦ ♠♦❞♦ q✉❡ ♦s ♣❧❛♥♦s ❡ ♦s ❝❡♥tr♦s ❞❛s ❜❛s❡s ❝✐r❝✉❧❛r❡s ❞♦ ❝♦♥❡ ❡ ❝✐❧✐♥❞r♦ ❝♦✐♥❝✐❞❡♠✳ ❉❡t❡r♠✐♥❡ ♦ r❛✐♦ ❞♦ ❝✐❧✐♥❞r♦ ♣❛r❛ q✉❡ s✉❛ s✉♣❡r❢í❝✐❡ t♦t❛❧ s❡❥❛ ♠á①✐♠❛✱ s❛❜❡✲s❡ q✉❡ H > 2R✳ ✷✸✳ ❆♣r❡s❡♥t❛r ♦ ♥ú♠❡r♦ x ❝♦♠♦ s♦♠❛ ❞❡ ❞♦✐s ♥ú♠❡r♦s t❛✐s q✉❡ ❛ s♦♠❛ ❞❡ s❡✉s q✉❛✲ ❞r❛❞♦s s❡❥❛ ❛ ♠❡♥♦r ♣♦ssí✈❡❧✳ ✷✹✳ ❈♦♠ ✉♠ ❧á♣✐s ❝✉❥❛ ♣♦♥t❛ t❡♠ 0, 02mm ❞❡ ❡s♣❡ss✉r❛✱ ❞❡s❡❥❛✲s❡ tr❛ç❛r ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f (x) = 2x ✳ ❆té q✉❡ ❞✐stâ♥❝✐❛ à ❡sq✉❡r❞❛ ❞♦ ❡✐①♦ ✈❡rt✐❝❛❧ ♣♦❞❡✲s❡ ✐r s❡♠ q✉❡ ♦ ❣rá✜❝♦ ❛t✐♥❥❛ ♦ ❡✐①♦ ❤♦r✐③♦♥t❛❧❄ ✷✺✳ ❙❡❥❛♠ a, b ∈ R t❛✐s q✉❡ a2 + b2 = 1 ❡ a 6= 1✱ ❞❡✜♥✐♠♦s tan q✉❡ E = cos α + senα > 0✱ ♠♦str❡ q✉❡ E = ✶✺✾ √ 1 + 2ab✳ b α = ✳ ❙❛❜❡♥❞♦ 2 a+1 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✷✻✳ ❯♠ ❛r❛♠❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ x ❞❡✈❡✲s❡ ❞✐✈✐❞✐r ❡♠ ❞✉❛s ♣❛rt❡s✳ ❯♠❛ ❞❡❧❛s ❡st❛rá ❞❡st✐♥❛❞❛ ♣❛r❛ ❝♦♥str✉✐r ✉♠ q✉❛❞r❛❞♦✱ ❡ ❛ ♦✉tr❛ ♣❛r❛ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦✳ ◗✉❛❧ é ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝❛❞❛ ♣❛rt❡ ♣❛r❛ q✉❡ ❛ s♦♠❛ ❞❛s ár❡❛s ❞❛s ✜❣✉r❛s ♦❜t✐❞❛s s❡❥❛ ❛ ♠❡♥♦r ♣♦ssí✈❡❧✳ ✷✼✳ ❯♠ ♣r♦❥❡t♦ ❞❡ ▲❡✐ ♣❛r❛ ❝♦❜r❛♥ç❛ ❞❡ ✐♠♣♦st♦s✱ s♦❜r❡ ❝❛rr♦s ♣r❡✈ê q✉❡ ♦ ♣r♦♣r✐❡tár✐♦ ❞❡ ✉♠ ❝❛rr♦ ♣❛❣❛rá ❘$100, 00 ♠❛✐s 7% ❞♦ ✈❛❧♦r ❡st✐♠❛❞♦ ❞♦ ❝❛rr♦✳ ❖✉tr♦ ♣r♦❥❡t♦ ♣r♦♣õ❡ q✉❡ ♦ ♣r♦♣r✐❡tár✐♦ ♣❛❣✉❡ ❘$400, 00 ♠❛✐s 2% ❞♦ ✈❛❧♦r ❡st✐♠❛❞♦ ❞♦ ❝❛rr♦✳ ❈♦♥s✐❞❡r❡ ❛♣❡♥❛s ♦s ❛s♣❡❝t♦s ✜♥❛♥❝❡✐r♦s❀ q✉❡ t✐♣♦ ❞♦ ❝♦❜r❛♥ç❛ s❡rá ♠❛✐s ❢❛✈♦rá✈❡❧ ❛♦ ♣r♦♣r✐❡tár✐♦❄ ✷✽✳ ❯♠ ✐♥✈❡st✐❞♦r ❛♣❧✐❝♦✉ ✉♠❛ q✉❛♥t✐❛ ❞❡ ❞✐♥❤❡✐r♦ ❡♠ ❛çõ❡s ✜♥❛♥❝❡✐r❛s ❝♦♠ r❡s❣❛t❡ ❛♦ tér♠✐♥♦ ❞❡ 60 ❞✐❛s ❞❛ ❛♣❧✐❝❛çã♦✳ ◆♦s ♣r✐♠❡✐r♦s 30 ❞✐❛s ❞❛ ❛♣❧✐❝❛çã♦ s♦✉❜❡ q✉❡ ♣❡r❞❡✉ 6% ❞♦ t♦t❛❧ ✐♥✈❡st✐❞♦✱ ❡ ❛♦ tér♠✐♥♦ ❞♦s 60 ❞✐❛s r❡❝✉♣❡r♦✉ 6% ❞❡ ❛q✉✐❧♦ q✉❡ r❡st♦✉ ❞♦s ♣r✐♠❡✐r♦s 30 ❞✐❛s ❞❛ ❛♣❧✐❝❛çã♦✳ ❆♦ tér♠✐♥♦ ❞♦s 60 ❞✐❛s r❡t✐r♦✉ t♦❞♦ s❡✉ ❞✐♥❤❡✐r♦ r❡❝❡❜❡♥❞♦ R$40.000, 00✳ ◗✉❛❧ ❢♦✐ ❛ q✉❛♥t✐❛ ✐♥✐❝✐❛❧ ❛♣❧✐❝❛❞❛ ♣❡❧♦ ✐♥✈❡st✐❞♦r❄ ✷✾✳ ❖❜s❡r✈❛çõ❡s ❢❡✐t❛s ❞✉r❛♥t❡ ❧♦♥❣♦ t❡♠♣♦ ♠♦str❛♠ q✉❡✱ ❛♣ó♦s ♣❡rí♦❞♦ ❞❡ ♠❡s♠❛ ❞✉r❛çã♦✱ ❛ ♣♦♣✉❧❛çã♦ ❞❛ t❡rr❛ ✜❝❛ ♠✉❧t✐♣❧✐❝❛❞❛ ♣❡❧♦ ♠❡s♠♦ ❢❛t♦r✳ ❙❛❜❡♥❞♦ q✉❡ ❡ss❛ ♣♦♣✉❧❛çã♦ ❡r❛ ❞❡ 2, 68 ❜✐❧❤õ❡s ❡♠ 1956 ❡ 3, 78 ❜✐❧❤õ❡s ❡♠ 1972✱ ♣❡❞❡✲s❡✿ ✶✳ ❖ t❡♠♣♦ ♥❡❝❡ssár✐♦ ♣❛r❛ q✉❡ ❛ ♣♦♣✉❧❛çã♦ ❞❛ t❡rr❛ ❞♦❜r❡ ❞❡ ✈❛❧♦r✳ ✷✳ ❆ ♣♦♣✉❧❛çã♦ ❡st✐♠❛❞❛ ♣❛r❛ ♦ ❛♥♦ 2012✳ ✸✳ ❊♠ q✉❡ ❛♥♦ ❛ ♣♦♣✉❧❛çã♦ ❞❛ t❡rr❛ ❡r❛ ❞❡ 1 ❜✐❧❤ã♦✳ ✸✵✳ P❛r❛ ❞❡t❡r♠✐♥❛r ❛ ✐❞❛❞❡ ❞❡ ✉♠❛ r♦❝❤❛ ❤♦❥❡ ❛ ❝✐ê♥❝✐❛ ❢♦✐ ❝❛♣❛③ ❞❡ ❞❡s❡♥✈♦❧✈❡r ✉♠❛ té❝♥✐❝❛ ❜❛s❡❛❞❛ ♥❛ ❝♦♥❝❡♥tr❛çã♦ ❞❡ ♠❛t❡r✐❛❧ r❛❞✐♦❛t✐✈♦ ❞❡♥tr♦ ❞❡❧❛✳ P❛r❛ ❛❝❤❛r ❡st❛ ❝♦♥❝❡♥tr❛çã♦ r❛❞✐♦❛t✐✈❛ ♠❛✐s ♥♦✈❛ ♥❛ r♦❝❤❛ ✉s❛♠♦s C(t) = k.3−t ❝♦♠♦ ❛ ❢ór♠✉❧❛✱ ♦♥❞❡ C(t) r❡♣r❡s❡♥t❛ ❛ ❝♦♥❝❡♥tr❛çã♦ ❞❡ ♠❛t❡r✐❛❧ r❛❞✐♦❛t✐✈♦✱ t ♦ t❡♠♣♦ ♠❡❞✐❞♦ ❡♠ ❝❡♥t❡♥❛s ❞❡ ❛♥♦s ❡ k ❛ ❝♦♥❝❡♥tr❛çã♦ ❞♦ ❡❧❡♠❡♥t♦ ♥♦ ♠♦♠❡♥t♦ ❡♠ q✉❡ ❛ r♦❝❤❛ ❢♦✐ ❢♦r♠❛❞❛✳ ❙❡ k = 4, 500 ✶✳ ◗✉❛♥t♦ t❡♠♣♦ ❞❡✈❡ t❡r ♣❛ss❛❞♦ ♣❛r❛ ♥ós ❡♥❝♦♥tr❛r ✉♠❛ ❝♦♥❝❡♥tr❛çã♦ ❞❡ 1500❄ ✷✳ ◗✉❛❧ s❡r✐❛ ♦ ❢♦❝♦ r❛❞✐♦❛t✐✈♦ ❞❡♣♦✐s ❞❡ ❞♦✐s sé❝✉❧♦s❄ ✶✻✵ 09/02/2021 ❈❛♣ít✉❧♦ ✸ ▲■▼■❚❊❙ ❆✉❣✉stí♥ ▲♦✉✐s ❈❛✉❝❤② ♥❛s❝❡✉ ♥♦ 21 ❞❡ ❛❣♦st♦ ❞❡ 1789✱ ❡♠ P❛r✐s✱ ❋r❛♥ç❛✳ ❋❛❧❡❝❡✉ ❡♠ 23 ❞❡ ♠❛✐♦ ❞❡ 1857✱ ❡♠ ❙❝❡❛✉① ✭♣ró①✐♠♦ ❞❡ P❛r✐s✮✳ ❊♠ 1802✱ ❡♥tr♦✉ ♥❛ ❊s❝♦❧❛ ❈❡♥tr❛❧ ❞♦ P❛♥t❡ã♦✱ ♦♥❞❡ ♣❛ss♦✉ ❞♦✐s ❛♥♦s ❡st✉❞❛♥❞♦ ✐❞✐♦♠❛s✳ ❊♠ 1804✱ ✐♥❣r❡ss♦✉ ♥❛ ❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❡ ❣r❛❞✉♦✉✲s❡ ❡♠ 1807✱ ♣❛r❛ ❧♦❣♦ ✐♥❣r❡ss❛r ♥❛ ❊s❝♦❧❛ ❞❡ ❊♥❣❡♥❤❛r✐❛ ❈✐✈✐❧✳ ❆✉❣✉stí♥ ❢♦✐ ❜❛st❛♥t❡ r❡❧✐❣✐♦s♦ ✭❝❛tó❧✐❝♦✮ ❡ ✐ss♦ ♦❝❛s✐♦♥♦✉✲❧❤❡ ♠✉✐t♦s ♣r♦❜❧❡♠❛s ❞❡ r❡❧❛❝✐♦♥❛♠❡♥t♦✳ ❖ ♣r✐♠❡✐r♦ ❛✈❛♥ç♦ ♥❛ ♠❛t❡♠át✐❝❛ ♠♦❞❡r♥❛ ♣♦r ❡❧❡ ♣r♦❞✉③✐❞♦ ❢♦✐ ❛ ✐♥tr♦❞✉çã♦ ❞♦ r✐❣♦r ♥❛ ❛♥á❧✐s❡ ♠❛t❡♠át✐❝❛✳ ❖ s❡❣✉♥❞♦ ❢♦✐ ♥♦ ❧❛❞♦ ♦♣♦st♦ ✲ ❝♦♠❜✐♥❛t♦r✐❛❧✳ P❛rt✐♥❞♦ ❞♦ ♣♦♥t♦ ❝❡♥tr❛❧ ❞♦ ❆✳ ▲✳ ❈❛✉❝❤② ♠ét♦❞♦ ❞❡ ▲❛❣r❛♥❣❡✱ ♥❛ t❡♦r✐❛ ❞❛s ❡q✉❛çõ❡s✱ ❈❛✉❝❤② t♦r♥♦✉✲❛ ❛❜str❛t❛ ❡ ❝♦♠❡ç♦✉ ❛ s✐st❡♠át✐❝❛ ❝r✐❛çã♦ ❞❛ t❡♦r✐❛ ❞♦s ❣r✉♣♦s✳ ❈❛✉❝❤② ❢❡③ ✐♠♣♦rt❛♥t❡s ❝♦♥tr✐❜✉✐çõ❡s à ❆♥á❧✐s❡✱ ❚❡♦r✐❛ ❞❡ ❣r✉♣♦s✱ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❞✐✈❡r❣ê♥❝✐❛ ❞❡ ❙ér✐❡s ✐♥✜♥✐t❛s✱ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✱ ❉❡t❡r♠✐♥❛♥t❡s✱ ❚❡♦r✐❛ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❡ ❛ ❋ís✐❝❛ ▼❛t❡♠át✐❝❛✱ ❡♠ 1811✱ ♠♦str♦✉ q✉❡ ♦s â♥❣✉❧♦s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ sã♦ ❞❡t❡r♠✐♥❛❞♦s ♣♦r s✉❛s ❢❛❝❡s✳ ❙✉❛ ❛❜♦r❞❛❣❡♠ ❞❛ t❡♦r✐❛ ❞❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❢♦✐ ✐♥♦✈❛❞♦r❛✱ ❞❡♠♦♥str❛♥❞♦ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♥✐❝✐❞❛❞❡ ❞❛s s♦❧✉çõ❡s✱ q✉❛♥❞♦ ❞❡✜♥✐❞❛s ❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦✳ ❊①❡r❝❡✉ ❣r❛♥❞❡ ✐♥✢✉ê♥❝✐❛ s♦❜r❡ ❛ ❢ís✐❝❛ ❞❡ ❡♥tã♦✱ ❛♦ s❡r ♦ ♣r✐♠❡✐r♦ ❛ ❢♦r♠✉❧❛r ❛s ❜❛s❡s ♠❛t❡♠át✐❝❛s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ét❡r✱ ♦ ✢✉✐❞♦ ❤✐♣♦tét✐❝♦ q✉❡ s❡r✈✐r✐❛ ❝♦♠♦ ♠❡✐♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞❛ ❧✉③✳ ●r❛ç❛s ❛ s❡✉ ❢♦r♠❛❧✐s♠♦ ♠❛t❡♠át✐❝♦✱ ❛ ❛♥á❧✐s❡ ✐♥✜♥✐t❡s✐♠❛❧ ❛❞q✉✐r❡ só❧✐❞❛s ❜❛s❡s✳ ❚❡✈❡ s❡r✐❛s ❞✐❢❡r❡♥ç❛s ♣❡ss♦❛✐s✱ ❝♦♠ ▲✐♦✉✈✐❧❧❡✱ ♣♦r ❝❛✉s❛ ❞❡ ✉♠❛ ♣♦s✐çã♦ ♥❛ ❊s❝♦❧❛ ❞❛ ❋r❛♥ç❛✳ ❈❛✉❝❤② ♣r♦❞✉③✐✉ 789 ❛rt✐❣♦s ❝✐❡♥tí✜❝♦s✳ ✸✳✶ ❱✐③✐♥❤❛♥ç❛ ❞❡ ✉♠ ♣♦♥t♦ x ∈ R✱ ✉♠ ❝♦♥❥✉♥t♦ A ⊂ R é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x s❡ ❡①✐st❡ ✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ (a, b) t❛❧ q✉❡ x ∈ (a, b) ⊆ A✳ P♦r ❡①❡♠♣❧♦✱ ♦s ❝♦♥❥✉♥t♦s A = (−1, 2] ❡ B = (−1, 1) sã♦ ✈✐③✐♥❤❛♥ç❛s ❞♦ ♣♦♥t♦ x = 0✱ ♣♦✐s A ❡ B sã♦ ❝♦♥❥✉♥t♦s q✉❡ ❝♦♥tê♠ ✐♥t❡r✈❛❧♦s ❛❜❡rt♦s ❙❡❥❛ ✶✻✶ ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❝♦♥t❡♥❞♦ x = 0✳ P❛r❛ ❡❢❡✐t♦ ❞❡ ♥♦ss♦ ❡st✉❞♦ ❞♦s ❧✐♠✐t❡s✱ q✉❛❧q✉❡r ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ ❝♦♥t❡♥❞♦ ✉♠ ♣♦♥t♦ a ❝♦♠♦ s❡✉ ♣♦♥t♦ ♠é❞✐♦ s❡rá ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ a✱ ✐st♦ ❥✉st✐✜❝❛ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿ ❉❡✜♥✐çã♦ ✸✳✶✳ ❱✐③✐♥❤❛♥ç❛✳ ❙❡❥❛ a ∈ R✱ ❝❤❛♠❛♠♦s ❞❡ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ ♦✉ ❜♦❧❛ ❛❜❡rt❛ ❞❡ ❝❡♥tr♦ a ❡ r❛✐♦ δ > 0✱ ❡ ❞❡♥♦t❛♠♦s B(a, δ)✱ ❛♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ (a−δ, a+δ)❀ ✐st♦ é✿ B(a, δ) = (a−δ, a+δ)✳ ◆❛ ❋✐❣✉r❛ ✭✸✳✶✮ ♦❜s❡r✈❛♠♦s q✉❡ ♦ ♣♦♥t♦ a é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ ✐♥t❡r✈❛❧♦ (a − δ, a + δ)✳ ✛ ✲✛ δ ✛ δ ✲ | a a−δ ✲ a+δ ❋✐❣✉r❛ ✸✳✶✿ ❊①❡♠♣❧♦ ✸✳✶✳ P❛r❛ ♦ ♥ú♠❡r♦ a = 4✱ s✉❛s ✈✐③✐♥❤❛♥ç❛ sã♦✿ (4 − δ, 4 + δ), 1 1 (4 − , 4 + ), 3 3 2 2 (4 − , 4 + ), 5 5 ··· ❡t❝ Pr♦♣r✐❡❞❛❞❡ ✸✳✶✳ ✐✮ B(a, d) = { x ∈ R /. | x − a |< δ } ✐✐✮ ❆ ✐♥t❡rs❡çã♦ ❞❡ ❞✉❛s ✈✐③✐♥❤❛♥ç❛s ❞❡ a✱ é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ a✳ ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ✸✳✷ ▲✐♠✐t❡ ❞❡ ✉♠❛ ❢✉♥çã♦ ❯♠ ❞♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❡ ❢✉♥❞❛♠❡♥t❛✐s ❞♦ ❝á❧❝✉❧♦ é ♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡✳ ❊st❡ ❝♦♥✲ ❝❡✐t♦ é tã♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♣r❡❝✐s❛r ♦✉tr♦s✱ t❛✐s ❝♦♠♦ ❝♦♥t✐♥✉✐❞❛❞❡✱ ❞❡r✐✈❛çã♦✱ ✐♥t❡❣r❛çã♦✱ ❡t❝✳ ◆♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦ t❡r❡♠♦s ✉♠❛ ✐❞❡✐❛ ❞❡ ❧✐♠✐t❡ ❞❡ ✉♠❛ ❢✉♥çã♦✳ ❊①❡♠♣❧♦ ✸✳✷✳ ❈♦♥s✐❞❡r❡ ❞✉❛s ❢✉♥çõ❡s r❡❛✐s f ❡ g ❞❡ ❣rá✜❝♦ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✭✸✳✷✮✱ ❛ss✐♠ ( x2 + 1 s❡ x 6= 2 ❡ g(x) = 3 + x ♣❛r❛ x 6= 2✳ ❞❡✜♥✐❞❛s✿ f (x) = 0 s❡ x = 2 ✶✻✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R y✻ −x✛ 5 · · · · · · ❜✳✳ ✳✳ ✳ ✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳✳ ✳ ✳ 2 0 −3 g(x) x ✲ −y ❄ ❋✐❣✉r❛ ✸✳✷✿ ❖❜s❡r✈❡ q✉❡ f (2) = 5 ❡♥t❛♥t♦ g(2) ♥ã♦ ❡①✐st❡ ✭♥ã♦ ❡st❛ ❞❡✜♥✐❞♦✮✳ ❖ ❝♦♠♣♦rt❛♠❡♥t♦ 2✱ ❞❡st❛s ❞✉❛s ❢✉♥çõ❡s ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ❡①❝❧✉✐♥❞♦ ♦ ♣♦♥t♦ x=2 é ❡①❛t❛♠❡♥t❡ ♦ ♠❡s♠♦ ❡ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ❛ss✐♠✿ ✏ P❛r❛ ✈❛❧♦r❡s ❞❡ ❡ g(x) x ♣ró①✐♠♦s ❛♦ ♣♦♥t♦ a = 2✱ x 6= 2 ❝♦♠ L = 5✑ ❛♣r♦①✐♠❛♠✲s❡ ❛♦ ♥ú♠❡r♦ ♦s ✈❛❧♦r❡s ❞❡ f (x) y✻ · · · · · · · · ·✳✳✳ · · · · · · ✳✳✳❜ ✳✳✳ ✳ ✳ ✳ · · · ✳✳✳ ✳✳✳ ✳✳✳   5+ε B(5, ε) 5  5−ε −x✛ g(x) 0 −3 ✳ ✳ ✳ ✳✳ ✳✳ ✳✳ ✳ ✳ ✳ ✳✳ ✳✳ ✳ ✳ ✳ ✳ ✳ 2 − δ 22 + δ | {z }✳ ✳ B(2, δ)✳ x ✲ −y ❄ ❋✐❣✉r❛ ✸✳✸✿ ❋✐❣✉r❛ ✸✳✹✿ ❯s❛♥❞♦ ✈✐③✐♥❤❛♥ç❛s✱ ❡st❛ ❞❡s❝r✐çã♦ ♣♦❞❡♠♦s ❡①♣r❡ss❛r ❛ss✐♠✿ ✏ P❛r❛ t♦❞❛ ✈✐③✐♥❤❛♥ç❛ t♦❞♦ x 6= 2 ❡ B(5, ε) x ∈ B(2, δ)✱ ❡♥tã♦ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ✉♠ f (x) ∈ B(5, ε)✑ 5 é ♦ ❧✐♠✐t❡ lim .f (x) = 5✳ ◗✉❛♥❞♦ ✐st♦ ♦❝♦rr❡ ❞✐③❡♠♦s q✉❡ ♣❛r❛ 2❀ ❛ ❡s❝r✐t❛ ❡♠ sí♠❜♦❧♦s é✿ ❆♥❛❧♦❣❛♠❡♥t❡ ♣❛r❛ ❛ ❢✉♥çã♦ ❖❜s❡r✈❡ q✉❡ ♦ ❧✐♠✐t❡ ❞❡ g(x) ❞❡ δ > 0✱ t❛❧ q✉❡ ♣❛r❛ ✭❋✐❣✉r❛ ✭✸✳✸✮✮✳ f (x) q✉❛♥❞♦ x t❡♥❞❡ ✭❛♣r♦①✐♠❛✲s❡✮ x→2 g(x)✱ lim .g(x) = 5 ✭❋✐❣✉r❛ ✭✸✳✹✮✮✳ x→2 ♥ã♦ ❞❡♣❡♥❞❡ ❞♦ ✈❛❧♦r ❞❡ g(2)✱ q✉❡ ♥❡st❡ ❝❛s♦ t❡♠♦s ♥♦ ♣♦♥t♦ 2 ♥ã♦ ❡①✐st❡✱ s♦♠❡♥t❡ ❞❡♣❡♥❞❡ ❞♦s ✈❛❧♦r❡s ❞❡ g q✉❛♥❞♦ ✶✻✸ x ❡st❛ ♣ró①✐♠♦ ❞♦ ♣♦♥t♦ 2✳ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❡✜♥✐çã♦ ✸✳✷✳ ❙❡❥❛ f : R −→ R ✉♠❛ ❢✉♥çã♦ ❡ x = a ✉♠ ♣♦♥t♦ q✉❡ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♣❡rt❡♥ç❛ ❛♦ ❞♦♠í♥✐♦ D(f )✱ ♣♦ré♠ t♦❞❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ a ❝♦♥tê♠ ♣♦♥t♦s ❞♦ ❞♦♠í♥✐♦ D(f )❀ ❞✐③✲s❡ q✉❡ ♦ ❧✐♠✐t❡ ❞❡ f (x) é L✱ q✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ a ❡ ❡s❝r❡✈❡✲s❡ lim .f (x) = L x→a q✉❛♥❞♦✿ • ∀ ε > 0, ∃ δ > 0 /. ∀ x ∈ D(f ), x 6= a ❡ a − δ < x < a + δ ❡♥tã♦ L − ε < f (x) < L + ε✳ ❊♠ t❡r♠♦s ❞❡ ✈❛❧♦r ❛❜s♦❧✉t♦✱ ❡st❛ ❞❡✜♥✐çã♦ é ❡q✉✐✈❛❧❡♥t❡ ❛✿ • ∀ ε > 0, ∃ δ > 0 /. ∀x ∈ D(f ), 0 <| x − a |< δ ✐♠♣❧✐❝❛ | f (x) − L |< ε ✳ ◆♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡✱ ❛♣❛r❡❝❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿ ◗✉❡ tã♦ ♣❡rt♦ ❞♦ ♣♦♥t♦ x = a ❞❡✈❡ s❡r ♦ ✈❛❧♦r ❞❡ x ♣❛r❛ q✉❡ f (x) ❞✐st❡ ❞♦ ✈❛❧♦r ❞❡ L✱ ✉♠ ♥ú♠❡r♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ ❡ ✜①❛❞♦❄ ❊①❡♠♣❧♦ ✸✳✸✳   x5 − 1 ✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✱ ❡st✐♠❡ ♦ ✈❛❧♦r ❞♦ ❧✐♠✐t❡ lim f (x) ❙❡❥❛ f (x) = 6 x→1 x −1 x 0, 901 0, 9001 0, 90001 0, 900001 1, 01 1, 001 1, 0001 1, 00001 f (x) ❙♦❧✉çã♦✳ x 0, 901 0, 9001 0, 90001 0, 900001 f (x) 0, 8735844779 0, 87393816822 0, 8739735220 0, 8739770573 x 1, 01 1, 001 1, 0001 1, 00001 f (x) 0, 8291600330 0, 8329165975 0, 8332916660 0, 8333291667 ❊①❡♠♣❧♦ ✸✳✹✳ ❙❡ lim (4x + 3) = 11✳ ◗✉❡ tã♦ ♣❡rt♦ ❞❡ 2 ❞❡✈❡ ❡st❛r x ♣❛r❛ q✉❡ | f (x) − 11 |< 0.01❄ x→2 ❙♦❧✉çã♦✳ ❉❡s❡❥❛♠♦s q✉❡✿ | f (x) − 11 |< 0.01 ✭♥♦t❡ q✉❡ ε = 0.01✮✱ ♣♦ré♠ | f (x) − 11 |=| (4x + 3) − 11 | = 4 | x − 2 |< 0.01 ⇒ | x − 2 |< 0, 01 4 ❉❡ ❡st❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ t❡♠♦s q✉❡ | x − 2 |< 0, 0025 ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ x ❡st❛ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ❞❡ 2 ❡♠ ♠❡♥♦s ❞❡ 0, 0025 ✉♥✐❞❛❞❡s✳ ✶✻✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✸✳✺✳ 2x2 − 4x ✳ x→2 x2 − 5x + 6 ❈❛❧❝✉❧❛r ♦ s❡❣✉✐♥t❡ ❧✐♠✐t❡✿ ❙♦❧✉çã♦✳ lim 2x(x − 2) 2x 2x2 − 4x = lim = lim = −4 2 x→2 (x − 2)(x − 3) x→2 x − 3 x→2 x − 5x + 6 lim x − 2 6= 0✳ 2x − 4x lim = −4✳ x→2 x2 − 5x + 6 ✐st♦ é ♣♦ssí✈❡❧ ♣❡❧♦ ❢❛t♦ 2 P♦rt❛♥t♦✱ ❖❜s❡r✈❛çã♦ ✸✳✶✳ P❛r❛ ✈❡r✐✜❝❛r ♦ ❧✐♠✐t❡ lim f (x) = L ♠❡❞✐❛♥t❡ ❛ ❉❡✜♥✐çã♦ ✭✸✳✷✮✱ ✐♥✐❝✐❛❧♠❡♥t❡ t❡♠♦s x→a q✉❡ ❡s❝r❡✈❡r |f (x) − L| = |x − a| · |g(x)|✱ ❧♦❣♦ ❞❡✈❡♠♦s ❡s❝♦❧❤❡r ✉♠ ✈❛❧♦r ✐♥✐❝✐❛❧ δ = δ1 ♣❛r❛ ❧✐♠✐t❛r |g(x)| ❞❡ t❛❧ ♠♦❞♦ q✉❡ 0 < |x − a| < δ ✐♠♣❧✐q✉❡ |g(x)| < M ❝♦♠ M ∈ R✳ ❆ss✐♠✱ 0 < |x − a| < δ ⇒ |f (x) − L| = |x − a| · |g(x)| < |x − a| · M < δM = ε✳ ε ❖ ✈❛❧♦r ❛❞❡q✉❛❞♦ ♣❛r❛ δ = min .{1, }✳ M ❊①❡♠♣❧♦ ✸✳✻✳ ❱❡r✐✜❝❛r ♠❡❞✐❛♥t❡ ❛ ❞❡✜♥✐çã♦ q✉❡ lim (3x2 + 2x + 4) = 9✳ x→1 ❙♦❧✉çã♦✳ ❆ ♠♦str❛r q✉❡ é ♣♦ssí✈❡❧ ❛❝❤❛r ✉♠ 2 | (3x + 2x + 4) − 9 |< ε δ > 0 ♣❛r❛ q✉❛❧q✉❡r ♥ú♠❡r♦ ❞❡ ♠♦❞♦ q✉❡ ε > 0✳ ❙❡❣✉❡✿ 0 <| x − 1 |< δ ✐♠♣❧✐q✉❡ | (3x2 + 2x + 4) − 9 |=| 3x2 + 2x − 5 |=| 3x + 5 | · | x − 1 |< δ | 3x + 5 | > 0 ❞❡ ♠♦❞♦ q✉❡ | x − 1 |< δ1 ♥ú♠❡r♦ M > 0 t❛❧ q✉❡ | 3x + 5 |< M ❙✉♣♦♥❤❛ ❡①✐st❛ ✉♠ δ1 é ❜✉s❝❛r❡♠♦s ✉♠ s❡♠♣r❡ q✉❡ | x − 1 |< δ1 ✱ ❡♥tã♦ −δ1 < x − 1 < δ1 ❡♥tã♦ 3(1 − δ1 ) + 5 < 3x + 5 < 3(1 + δ1 ) + 5❀ ♣♦r ❡①❡♠♣❧♦ 5 < 3x + 5 < 11 ❛ss✐♠ | 3x + 5 |< 11 δ = min .{1, ε }✳ 11 1 − δ1 < x < 1 + δ1 ❝♦♥s✐❞❡r❡ δ1 = 1 ❡ t❡r❡♠♦s ❧♦❣♦ ✭✸✳✷✮ | 3x + 5 | · | x − 1 |< δ | 3x + 5 |< 11δ = ε P♦r t❛♥t♦✱ ♣❛r❛ q✉❛❧q✉❡r ♥ú♠❡r♦ ε > 0✱ |(3x2 + 2x + 4) − 9| < ε ■st♦ ♠♦str❛ q✉❡✱ ♦ ❧✐♠✐t❡ | 3x + 5 |❀ ✐st♦ 0 <| x − 1 |< δ1 ✳ t❡♥t❛r❡♠♦s ❧✐♠✐t❛r ❈♦♠ ❡❢❡✐t♦✱ s❡ ❉❡ ✭✸✳✶✮ ❡ ✭✸✳✷✮ t❡♠♦s q✉❡ ❡ ❝♦♥s✐❞❡r❛♥❞♦ s❡♠♣r❡ q✉❡ lim (3x2 + 2x + 4) = 9 x→1 ✶✻✺ ✭✸✳✶✮ δ = min .{1, ε } 11 s❡♠♣r❡ q✉❡ t❡♠♦s✿ 0 < |x − 1| < δ é ✈❡r❞❛❞❡✐r♦✳ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❖❜s❡r✈❛çã♦ ✸✳✷✳ ❛✮ ❆♦ ❝♦♥s✐❞❡r❛r ✉♠ δ1 ♣❛rt✐❝✉❧❛r✱ ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ❛ ✈✐③✐♥❤❛♥ç❛ B(a, δ1 ) = (a − δ1 , a + δ1 ) 0 < |x − a| < δ1 ❣❡r❛❧♠❡♥t❡ ♦ δ1 é ✉♠ ✈❛❧♦r ♣❡q✉❡♥♦✱ ♣♦❞❡✲s❡ ❝♦♥s✐❞❡r❛r |x − 1| < δ1 = 1 ♣♦ré♠ ❡st❡ ou ✈❛❧♦r ♣♦❞❡ r❡s✉❧t❛r ✐♥❛❞❡q✉❛❞♦ ❡♠ ❛❧❣✉♥s ❝❛s♦s ♣❡❧♦ q✉❡ ❞❡✈❡♠♦s ❝♦♥s✐❞❡r❛r ♦✉tr♦ ♥ú♠❡r♦ ❛✐♥❞❛ ♠❡♥♦r✳ ❜✮ ❈♦♥s✐❞❡r❛r ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✈❛❧♦r ❛❜s♦❧✉t♦✿ ❙❡ | x − a |< δ ❡♥tã♦ a − δ < x < a + δ ✳ ✐✮ ✐✐✮ ❙❡ a < u < b ❡♥tã♦ | u |< max .{| a |, | b |}✳ P♦r ❡①❡♠♣❧♦✱ s❡ −4 < 3x − 9 < 2 ❡♥tã♦ |3x − 9| < 4 ♣♦✐s | − 4| = 4 = max .{ | − 4|, | 2 | }✳ ✐✐✐✮ ❝✮ ❙❡ a < u < b ❡♥tã♦✱ u2 < k 2 ♦♥❞❡ k = max .{| a |, | b |} ❙❡ δ > 0 ❝✉♠♣r❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ❧✐♠✐t❡✱ q✉❛❧q✉❡r ♦✉tr♦ δ1 q✉❡ ❝✉♠♣r❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ 0 < δ1 < δ ✱ t❛♠❜é♠ ❝✉♠♣r❡ ❛ ❞❡✜♥✐çã♦✳ ❊①❡♠♣❧♦ ✸✳✼✳ x2 − 16 ✱ ✈❡r✐✜❝❛r ♠❡❞✐❛♥t❡ ❛ ❞❡✜♥✐çã♦ q✉❡ lim .f (x) = 8✳ ❙❡❥❛ f (x) = x→4 x−4 ❙♦❧✉çã♦✳ P❛r❛ t♦❞♦ ε > 0✱ ❞❡✈❡✲s❡ ♠♦str❛r q✉❡ é ♣♦ssí✈❡❧ ❛❝❤❛r ✉♠ δ > 0 t❛❧ q✉❡ |f (x) − 8| < ε s❡♠♣r❡ q✉❡ 0 < |x − 4| < δ ✳ ❖ ❢❛t♦ 0 < |x − 4|✱ ❡q✉✐✈❛❧❡ ❛ q✉❡ x 6= 4✳ |f (x) − 8| = x2 − 8x + 16 x2 − 16 −8 = = |x − 4| < δ = ε x−4 x−4 ▲♦❣♦✱ ∀ ε > 0✱ ❡①✐st❡ δ = ε t❛❧ q✉❡ |f (x) − 8| < ε s❡♠♣r❡ q✉❡ 0 < |x − 4| < δ ✳ ❊①❡♠♣❧♦ ✸✳✽✳ ❈❛❧❝✉❧❛r ♦ ❧✐♠✐t❡ lim x→4 √ x−3−1 ✳ x−4 ❙♦❧✉çã♦✳ lim x→4 √ √ √ x−3−1 x−4 ( x − 3 − 1)( x − 3 + 1) √ √ = lim = = lim x→4 (x − 4)( x − 3 + 1) x→4 x−4 (x − 4)( x − 3 + 1) ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ x ❡st❛ s❡ ❛♣r♦①✐♠❛♥❞♦ ❛ 4✱ ♣♦❞❡♠♦s s✐♠♣❧✐✜❝❛r ♣❛r❛ ♦❜t❡r = lim √ x→4 1 = 0, 5 x−3+1 ✶✻✻ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ P♦rt❛♥t♦✱ lim √ x→4 x−3−1 = 0, 5✳ x−4 ❊①❡♠♣❧♦ ✸✳✾✳ ❉❛❞❛ ❛ ❢✉♥çã♦ f (x) = 2 3( x + 1) √ ❞❡♠♦♥str❡ q✉❡ lim .f (x) = x→4 2 ✳ 9 ❙♦❧✉çã♦✳ √ 2 2 2 2(2 − x) √ √ − = = = f (x) − 9 3( x + 1) 9 9( x + 1) P❛r❛ t♦❞♦ ε > 0✱ t❡♠✲s❡ √ √ 2(2 − x)(2 + x) 1 2 √ √ √ = < |4−x|· √ 9 9( x + 1)(2 + x) ( x + 1)(2 + x) ✭✸✳✸✮ ❙❡ | x − 4 |< δ1 ✱ ❡♥tã♦ −δ1 < x − 4 < δ1 ❧♦❣♦ 4 − δ1 < x < 4 + δ1 ✳ ❈♦♥s✐❞❡r❛♥❞♦ δ1 = 1 t❡♠♦s 3 < x < 5 ❡♥tã♦✱ √ 3+1< √ x+1< √ 5+1 √ √ ( 3 + 1) < ( x + 1) ⇒ √ s❛❜❡✲s❡ q✉❡✱ ♣❛r❛ ♥ú♠❡r♦s x ♣ró①✐♠♦s ❞❡ 4 ✈❛❧❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ 2 ≤ (2 + x)✱ ❡♥tã♦ √ √ √ 2( 3 + 1) ≤ ( x + 1)(2 + x) ⇒ 1 1 √ ≤ √ ( x + 1)(2 + x) 2( 3 + 1) √ ❖❜s❡r✈❡✱ ❡♠ ✭✸✳✸✮ s❡❣✉❡ q✉❡ | f (x) − ≤ 2 2 1 √ ≤ |≤ | 4 − x | √ 9 9 ( x + 1)(2 + x) 1 2 |x−4| δ = √ <√ =ε |4−x| √ 9 2( 3 + 1) 9( 3 + 1) 3+1 s❡♠♣r❡ q✉❡ | x − 4 |< δ ∀ ε > 0✳ √ ❆ss✐♠✱ ❝♦♥s✐❞❡r❛♥❞♦ δ = min .{1, ε( 3 + 1)} ❡ | x − 4 |< δ ✱ ❞❡♠♦♥str❛♠♦s q✉❡ lim f (x) = x→4 2 9 ❖❜s❡r✈❛çã♦ ✸✳✸✳ ✐✮ ❈❛❧❝✉❧❛r ✉♠ ❧✐♠✐t❡ é ❞✐❢❡r❡♥t❡ ❞❡ ❞❡♠♦♥str❛r ♦ ♠❡s♠♦❀ ♣❛r❛ ♦ ❝á❧❝✉❧♦✱ ✉t✐❧✐③❛♠♦s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ ❡ ❞❡ ♠♦❞♦ ❞✐r❡t♦✱ t❡♥t❛♠♦s ❝❤❡❣❛r ❛ ✉♠ r❡s✉❧t❛❞♦❀ ♥❛ ❞❡♠♦♥str❛çã♦✱ ✉t✐❧✐③❛♠♦s ❛ ❞❡✜♥✐çã♦✱ ❧♦❣♦ ❞❡✈❡♠♦s tr❛❜❛❧❤❛r ❝♦♠ ε ❡ δ ✳ ✐✐✮ ❙✉♣♦♥❤❛ q✉❡ ❡st❛♠♦s ❡st✉❞❛♥❞♦ ♦ ❧✐♠✐t❡ ❞❡ ✉♠❛ ❢✉♥çã♦ f (x) ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x = a ❡✱ x = b s❡❥❛ ♦ ♣♦♥t♦ ♠❛✐s ♣ró①✐♠♦ ❞❡ x = a✱ ♦♥❞❡ ❛ ❢✉♥çã♦ f (x) ♥ã♦ ❡stá 1 2 ❞❡✜♥✐❞❛✱ ❡♥tã♦ t❡♠♦s q✉❡ ❝♦♥s✐❞❡r❛r δ1 ≤ |a − b|✳ ✶✻✼ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❊①❡♠♣❧♦ ✸✳✶✵✳ ❈❛❧❝✉❧❛r ♦s ❧✐♠✐t❡s✿ ❙♦❧✉çã♦✳ ✐✮ ✐✮  √ 3 √   x2 + 3 x  lim  12  x→4 8− x  x−8 lim √ 3 x→8 x−2 ✐✐✮  ❖❜s❡r✈❡✱ ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❧✐♠✐t❡✿  P♦rt❛♥t♦✱ ❙♦❧✉çã♦✳ ✐✐✮ ❚❡♠♦s √ 3  √  √ 3 √   x2 + 3 x   4 2 + 3 4  lim  = = 12   12  x→4 8− 8− x 4   "√ # √ √ 3 3 16 + 6  x2 + 3 x  lim  = ✳ 12  x→4 5 8− x √ √ √ (x − 8) = [ 3 x − 2][( 3 x)2 + 2( 3 x) + 22 ]❀ "√ 3 16 + 6 5 # ❧♦❣♦✱ ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❧✐♠✐t❡✿  √   √ √ √ √ [ 3 x − 2][( 3 x)2 + 2( 3 x) + 22 ] x−8 2 3 3 √ x) + 2( x) + 22 ] = 12 = lim = lim [( lim √ 3 3 x→8 x→8 x→8 x−2 x−2  P♦rt❛♥t♦✱  x−8 = 12✳ lim √ 3 x→8 x−2 ❊①❡♠♣❧♦ ✸✳✶✶✳  ❙❡❥❛ f : N −→ N ❞❡✜♥✐❞❛ ♣♦r ❙♦❧✉çã♦✳ f (n + 1) = f (n) + 3 ❡ f (1) = 2✱ ❞❡t❡r♠✐♥❡ lim f (n)✳ n→20 f (2) = f (1) + 3✱ f (3) = f (2) + 3 = f (1) + 2(3)✱ f (4) = f (3) + 3 = f (1) + 3(3)✱ ❡♠ ❣❡r❛❧ f (n) = f (1) + 3(n − 1) = 3n − 1✱ é ✉♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛✳ ▲♦❣♦ ❚❡♠♦s lim f (n) = lim 3n − 1 = 59 n→20 P♦rt❛♥t♦✱ n→20 lim f (n) = 59✳ n→20 ❊①❡♠♣❧♦ ✸✳✶✷✳ ❙❡❥❛ f : N −→ N ❞❡✜♥✐❞❛ ♣♦r ❙♦❧✉çã♦✳ f (n + 1) = 2f (n) ❡ f (1) = 3✱ ❞❡t❡r♠✐♥❡ lim f (n)✳ n→20 f (2) = 2f (1)✱ f (3) = 2f (2) = 22 f (1)✱ f (4) = 2f (3) = 23 f (1)✱ f (n) = 2n−1 f (1)✱ é ✉♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛✳ ▲♦❣♦ ❚❡♠♦s ❡♠ ❣❡r❛❧ lim f (n) = lim 2n−1 f (1) = lim 3 × 2n−1 = 3 × 219 n→20 P♦rt❛♥t♦✱ n→20 n→20 lim f (n) = 3 × 219 ✳ n→20 ✶✻✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✸✲✶ x5 − 32 ❝♦♠♣❧❡t❛♥❞♦ ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✿ x→2 x6 − 64 ✶✳ ❊st✐♠❡ ♦ ✈❛❧♦r ❞♦ ❧✐♠✐t❡ lim x 1, 999 1, 9999 1, 99999 1, 999999 2, 01 2, 001 2, 0001 2, 00001 f (x) ✷✳ ❈❛❧❝✉❧❛r ♦ lim .g(x) ♣❛r❛ g(x) = √ x→1 x 0, 999 0, 9999 0, 99999 x+3 ❝♦♠♣❧❡t❛♥❞♦ ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✿ x2 + 15 − 4 0, 999991 1, 01 1, 001 1, 0001 1, 00001 f (x) ✸✳ ❈❛❧❝✉❧❛r ♦ lim g(x) ♣❛r❛ g(x) = x→3 x 2, 999 2, 9999 2, 99999 x−3 ❝♦♠♣❧❡t❛♥❞♦ ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✿ x2 − 9 2, 999999 3, 01 3, 001 3, 0001 3, 00001 f (x) ✹✳ ❈❛❧❝✉❧❛r lim f (x) ♣❛r❛ ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ x→a  2 x +5   , s❡✱ x > 1 x 2. f (x) = 2   x − 1 , s❡✱ x < 1 x−1 q✉❛♥❞♦ a = 1  2  x −4 , s❡✱ x 6= 2 1. f (x) = x−2  5, s❡✱ x = 2 q✉❛♥❞♦ a = 2 ✺✳ ❉❡♠♦♥str❛r q✉❡✿ ✶✳ lim x→4 2(x − 5) =2 2x − 7 ✷✳ x2 = −1 x→1 3x − 4 lim ✻✳ ❙❡❥❛ y = x2 ✳ ◗✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ 2❀ y t❡♥❞❡ ♣❛r❛ 4✳ ◗✉❛❧ é ♦ ✈❛❧♦r ♣❛r❛ δ ❡♠ 0 <| x − 2 |< δ ❀ q✉❡✱ ❞ê ♣♦r r❡s✉❧t❛❞♦ | y − 4 |< ε = 0, 001❄ 3 x2 − 1 ✳ P❛r❛ x → 2 t❡♠♦s y → ✳ ◗✉❛❧ é ♦ ✈❛❧♦r ❞❡ δ ♣❛r❛ q✉❡ | x − 2 |< δ 2 x +1 5 3 ❞ê ♣♦r r❡s✉❧t❛❞♦ | y − |< ε = 0, 1❄ 5 ✼✳ ❙❡❥❛ y = ✽✳ ❆♣❧✐❝❛♥❞♦ ❛ ❞❡✜♥✐çã♦✱ ❞❡♠♦♥str❛r ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✱ ❛❝❤❛♥❞♦ ✉♠ ✈❛❧♦r ♣❛r❛ ✉♠ δ > 0✱ ♣❛r❛ ♦ ✈❛❧♦r ❞❡ ε ❞❛❞♦✳ 1. lim (5x − 3) = 12 x→3 ε = 0, 03 ✶✻✾ 2. lim (3x + 5) = 11 ε = 0, 0012 x→2 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 3. 5. 7. 9. R ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠   x2 − 4 =4 ε = 0, 004 lim x→2 x − 2   2 3x − 2x − 1 = 4 ε = 0, 015 lim x→1 x−1 lim (7x2 − 20x + 2) = 5 ε = 0, 001   3x − 1 = −5 ε = 0, 001 lim x→−3 3x2 − 25 x→3 4. 6. 8. 10. √  1 x−1 lim = ε = 0, 015 x→1 x−1 2   2 4x − 1 lim =2 ε = 0, 07 2x − 1 x→ 21   x2 lim = 4 ε = 0, 001 x→2 7x − 13  2  x − 14 lim = 5 ε = 0, 1 x→−3 10x + 29 ✾✳ ❆♣❧✐❝❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❧✐♠✐t❡✱ ♠♦str❛r ❛s s❡❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s✿ r   2 + x + x2 1. lim (3x − x) = 10 =4 3. lim x→2 x→−2 2x + 5 x2 + 2x + 2 4 =4 6. lim 2 =2 4. lim x→0 x − 2x + 1 x→3 x − 2 √ 3 √ 3x2 − 11 2 3 7. lim 6 − x = 1 8. lim (x + 2) = −6 9. lim =− x→−2 x→1 x→5 3 3 √ x−1 x−1 x+1 =2 11. lim √ =1 12. lim √ 10. lim √ = 2 3 x→64 x→1 x→1 x x+3 x2 + 3 − 2  2  x − 3x − 4 1 3x |x| =0 13. lim = −5 14. lim 2 = 15. lim x→−5 x + 8 x→−1 x + 1 x→4 2 x−3 r √ 2 1 8 |2−x| x+1 x− 2 √ =− 16. lim = 17. lim = 18. lim x→1 3x − 1 x→7 9x − 60 x→0 2x + 2 3 3 3 2 3x + 1 k x k +x 7 3x 19. lim =1 =3 20. lim = 21. lim √ √ 4 2 x→π 6x − 5π 5 x→ 2 x + 1 x→ 2 3 + x − x  2    1 4x + 1 8−x kxk = =0 23. lim 24. lim = −5 22. lim1 x→8 64 − x2 x→−1 3x + 2 16 x→ 2 x + 1     √ 2x sgn(x2 − 1) 1 2 25. lim 4x + 1 = 1 26. lim =0 27. lim = x→0 x→0 63x − 1 x→3 x+4 7 √    16 2x − 4 k x k +2 −4x − 3 = 29. lim = −4 30. lim = −3 28. lim5 2 x→−4 x→−3 x 25 5x + 23 x+2 x→ 2 2 4+x 3 2. lim = 2 x→5 x −9 4 8 3 + 2x 5. lim1 = 9 x→ 2 5 − x 2 − u2 n s❡♥❞♦ u1 = 1✳ ❉❡t❡r♠✐♥❡ ♦s ♣r✐♠❡✐r♦s ✶✵✳ ❈♦♥s✐❞❡r❡ ❛ s✉❝❡ssã♦ un+1 = un + 2un ❡❧❡♠❡♥t♦s u2 , u3 , u4 ❡ ❝❛❧❝✉❧❡ ♦ ❧✐♠✐t❡ ❞❡ un q✉❛♥❞♦ n ❝r❡s❝❡ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✳ √ ✶✶✳ ❙❡❥❛ ❛ s✉❝❡ssã♦ ❞❡✜♥✐❞❛ ♣❡❧❛ r❡❧❛çã♦ ❞❡ r❡❝♦rrê♥❝✐❛ un = 2 + un−1 s❡♥❞♦ u1 = ❈❛❧❝✉❧❛r ♦ ❧✐♠✐t❡ ❞❛ s✉❝❡ssã♦ un q✉❛♥❞♦ n ❝r❡s❝❡ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✳ √ 2✳ ✶✷✳ ▼♦str❡ q✉❡ ❛ s❡q✉ê♥❝✐❛ un = 1 + (−1)n ♥ã♦ t❡♠ ❧✐♠✐t❡ q✉❛♥❞♦ n ❝r❡s❝❡ ✐♥❞❡✜♥✐❞❛✲ ♠❡♥t❡✳ ✶✼✵ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✸✳✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❞♦s ❧✐♠✐t❡s ❙❛❜❡♠♦s ❞❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ♥ú♠❡r♦s r❡❛✐s✿ Pr♦♣r✐❡❞❛❞❡ ✸✳✷✳ ✐✮ ✐✐✮ ❙❡❥❛ x ∈ R ❡ x ≥ 0✱ s❡ x < ε ♣❛r❛ t♦❞♦ ε > 0✱ ❡♥tã♦ x = 0✳ ◗✉❛♥❞♦ | x |< ε, ∀ε>0 ⇒ x = 0✳ ❉❡♠♦♥str❛çã♦✳ ✐✮ ✐✐✮ ❈♦♠♦ x ≥ 0✱ ❡♥tã♦ x = 0 ♦✉ x > 0✳ ❆ ♣♦ss✐❜✐❧✐❞❛❞❡ x > 0 ♥ã♦ ♣♦❞❡ ❛❝♦♥t❡❝❡r✱ ♣♦✐s s❡ x > 0 ❡♥tã♦ ❞♦ ❢❛t♦ x < ε ❡ ❝♦♠♦ ε > 0 ❡♠ ♣❛rt✐❝✉❧❛r ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ε = x ❞❡ ♦♥❞❡ ε = x < x ♦ q✉❡ é ❝♦♥tr❛❞✐tór✐♦✳ P♦r t❛♥t♦ x = 0✳ ❊①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ Pr♦♣r✐❡❞❛❞❡ ✸✳✸✳ ❯♥✐❝✐❞❛❞❡ ❞♦ ❧✐♠✐t❡✳ ◗✉❛♥❞♦ ❡①✐st❛ ♦ ❧✐♠✐t❡ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ❡st❡ ❧✐♠✐t❡ é ú♥✐❝♦✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ ε > 0 q✉❛❧q✉❡r ♥ú♠❡r♦ r❡❛❧❀ ❡ s✉♣♦♥❤❛ q✉❡ lim .f (x) = L1 ❡ lim .f (x) = L2 x→a x→a s❡♥❞♦ L1 6= L2 ✳ ❙❡rá s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡ | L1 − L2 |< ε ♣❛r❛ t♦❞♦ ε > 0✳ ❉♦ ❢❛t♦ lim .f (x) = L1 ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❧✐♠✐t❡ t❡♠♦s q✉❡✱ ❞❛❞♦ q✉❛❧q✉❡r ε > 0✱ ❡①✐st❡ x→a ✉♠ δ1 > 0 t❛❧ q✉❡ | f (x) − L1 |< ε 2 s❡♠♣r❡ q✉❡ 0 <| x − a |< δ1 ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦ ❞❛❞♦ lim .f (x) = L2 ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❧✐♠✐t❡ t❡♠♦s q✉❡✱ ❞❛❞♦ q✉❛❧q✉❡r x→a ε > 0✱ ❡①✐st❡ ✉♠ δ2 > 0 t❛❧ q✉❡ | f (x) − L2 |< ε 2 s❡♠♣r❡ q✉❡ 0 <| x − a |< δ2 ✳ ❈♦♥s✐❞❡r❡ δ = min .{ δ1 , δ2 } ❡ 0 <| x − a |< δ ❡♥tã♦ ❝✉♠♣r❡♠✲s❡ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ε ε ❡ | f (x) − L2 |< ✳ | f (x) − L1 |< 2 2 ❉❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ t❡♠♦s q✉❡✿ | L1 − L2 |=| L1 − f (x) + f (x) − L2 |≤ ≤| f (x) − L1 | + | f (x) − L2 |< ε ε + =ε 2 2 ♣❛r❛ 0 <| x − a |< δ ❆ss✐♠ ♠♦str❛♠♦s q✉❡ ♣❛r❛ t♦❞♦ ε > 0✱ s❡♥❞♦ 0 <| x − a |< δ ✈❡r✐✜❝❛✲s❡ | L1 − L2 |< ε ♦ q✉❡ ✐♠♣❧✐❝❛ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✸✳✷✮ q✉❡ L1 = L2 ✳ ✶✼✶ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ Pr♦♣r✐❡❞❛❞❡ ✸✳✹✳ ❈♦♥s❡r✈❛çã♦ ❞♦ s✐♥❛❧✳ lim .f (x) = L 6= 0✱ ❡①✐st❡ ✉♠❛ ∀ x ∈ B(a, δ) ❝♦♠ x 6= a✳ ❙❡ x→a s✐♥❛❧ ✈✐③✐♥❤❛♥ç❛ B(a, δ) t❛❧ q✉❡ f (x) ❡ L t❡♠ ♦ ♠❡s♠♦ ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ Pr♦♣r✐❡❞❛❞❡ ✸✳✺✳ lim .f (x) = L ✱ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ B(a, δ) ❡ ✉♠ ♥ú♠❡r♦ M > 0 t❛❧ q✉❡ | f (x) |< ❙❡ M, x→a ∀ x ∈ B(a, δ) s❡♥❞♦ ❉❡♠♦♥str❛çã♦✳ ❉❛ ❤✐♣ót❡s❡ lim .f (x) = L x→a ε > 0, ❉❛❞♦ x 6= a✳ t❡♠♦s q✉❡✿ ∃ δ > 0 /. ∀ x ∈ B(a, δ), ❉❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♥ú♠❡r♦s r❡❛✐s M = ε+ | L | s❛t✐s❢❛③ | f (x) − L |< ε✳ | f (x) | − | L |<| f (x) − L |< ε✱ | f (x) | − | L |< ε ❈♦♥s✐❞❡r❛♥❞♦ x 6= a | f (x) |< ε+ | L | ❧♦❣♦ t❡♠♦s q✉❡ ❡♥tã♦ | f (x) |< M ∀ x ∈ B(a, δ) ♣❛r❛ x 6= a✳ Pr♦♣r✐❡❞❛❞❡ ✸✳✻✳ f ❙❡ ❛✮ ❜✮ ❡ g sã♦ ❢✉♥çõ❡s t❛✐s q✉❡ ❝✉♠♣r❛♠ ❛s ❤✐♣ót❡s❡s✿ f (x) ≤ g(x) ∀ x ∈ B(a, δ) lim .f (x) = L ❡ x→a ❊♥tã♦ L ≤ M✱ ❝♦♠ x 6= a✳ lim .g(x) = M ✳ x→a ✐st♦ é lim .f (x) ≤ lim .g(x)✳ x→a x→a ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ Pr♦♣r✐❡❞❛❞❡ ✸✳✼✳ ❛✮ ❙✉♣♦♥❤❛♠♦s ❜✮ ❙❡ ❉♦ ❝♦♥❢r♦♥t♦✳ f (x) ≤ g(x) ≤ h(x) ♣♦ssí✈❡❧♠❡♥t❡ ♦ ♣ró♣r✐♦ a lim .f (x) = L = lim .h(x)✱ x→a x→a ♣❛r❛ t♦❞♦ ❡♥tã♦ x ♥✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ ❝♦♥t❡♥❞♦ a✱ ❡①❝❡t♦ lim .g(x) = L✳ x→a ❉❡♠♦♥str❛çã♦✳ P❡❧❛ ❤✐♣ót❡s❡ ❜✮ ♣❛r❛ ❝❛❞❛ ε>0 ❡①✐st❡♠ ♣♦s✐t✐✈♦s δ1 ❡ δ2 t❛✐s q✉❡✿ 0 <| x − a |< δ1 ⇒ L − ε < f (x) < L + ε ✭✸✳✹✮ 0 <| x − a |< δ2 ⇒ L − ε < h(x) < L + ε ✭✸✳✺✮ ✶✼✷ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ δ = min .{ δ1 , δ2 } ✱ ♣❛r❛ 0 <| x − a |< δ ❝♦♠♦ f (x) ≤ g(x) ≤ h(x)✳ ❊♥tã♦ ❈♦♥s✐❞❡r❛♥❞♦ ✭✸✳✹✮ ❡ ✭✸✳✺✮ ❡ 0 <| x − a |< δ ✐st♦ é 0 <| x − a |< δ ✐♠♣❧✐❝❛ L − ε < f (x) ≤ g(x) ≤ h(x) < L + ε ✐♠♣❧✐❝❛ L − ε < g(x) < L + ε P♦rt❛♥t♦✱ ❝✉♠♣r❡✲s❡ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ⇒ | g(x) − L |< ε lim .g(x) = L✳  x→a ❊st❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ❝♦♥❢r♦♥t♦✱ t❛♠❜é♠ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ♦ ✏Pr✐♥❝í♣✐♦ ❞♦ ❙❛♥❞✉í❝❤❡✑✳ Pr♦♣r✐❡❞❛❞❡ ✸✳✽✳ ❙❡❥❛♠ ❛✮ f ❡ g ❞✉❛s ❢✉♥çõ❡s t❛✐s q✉❡ ❝✉♠♣r❛♠ ❛s ❤✐♣ót❡s❡s✿ lim .f (x) = 0✳ x→a ❜✮ ❊①✐st❡ ❊♥tã♦ M >0 t❛❧ q✉❡ | g(x) |< M ∀ x ∈ B(a, δ) ❝♦♠ x 6= a✳ lim .f (x).g(x) = 0✳ x→a  ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❊①❡♠♣❧♦ ✸✳✶✸✳ f (x) = ax2 + bx + c ♦♥❞❡ a, b ∀ x ∈ R✳ ▼♦str❡ q✉❡ a = b = c = 0✳ ❙✉♣♦♥❤❛♠♦s q✉❡ x3 | ❡ c sã♦ ❝♦♥st❛♥t❡s t❛✐s q✉❡ | f (x) |≤| ❉❡♠♦♥str❛çã♦✳ 0 ≤| ax2 + bx + c |≤| x3 | lim | ax2 + bx + c |= c = 0✳ ❈♦♠♦ q✉❡ ❡ lim .0 = lim | x3 |= 0✱ x→0 x→0 ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✸✳✼✮ s❡❣✉❡ x→0 ❊♥tã♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r 0 ≤| ax + b |≤| x2 | lim | ax + b |= b = 0✳ ❧♦❣♦ f (x) = ax2 + bx❀ ❛ss✐♠ 0 ≤| ax2 + bx |≤| x3 | ♣❛r❛ x 6= 0✱ ∀ x ∈ R✱ ❛♣❧✐❝❛♥❞♦ ♥♦✈❛♠❡♥t❡ ❛ Pr♦♣r✐❡❞❛❞❡ ✭✸✳✼✮ r❡s✉❧t❛ x→0 0 ≤| ax |≤| x3 | ♣❛r❛ x 6= 0✱ ❧♦❣♦ 0 ≤| a |≤| x | R✱ ❛♣❧✐❝❛♥❞♦ ♥♦✈❛♠❡♥t❡ ❛ Pr♦♣r✐❡❞❛❞❡ ✭✸✳✼✮ r❡s✉❧t❛ lim | a |= a = 0✳ x→0 P♦rt❛♥t♦✱ a = b = c = 0✳ ❊♥tã♦ t❡♠♦s f (x) = ax❀ Pr♦♣r✐❡❞❛❞❡ ✸✳✾✳ ∀x ∈ Pr♦♣r✐❡❞❛❞❡s ❛❞✐❝✐♦♥❛✐s ❞♦s ❧✐♠✐t❡s✳ f ❡ g ❞✉❛s lim .g(x) = M ❡♥tã♦✿ ❙❡❥❛♠ ❛ss✐♠ ❢✉♥çõ❡s ❡ C ♥ú♠❡r♦ r❡❛❧ ❝♦♥st❛♥t❡✱ t❛✐s q✉❡ lim .f (x) = L x→a ❡ x→a ❛✮ lim .C = C x→a ✶✼✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❜✮ lim .C · f (x) = C · lim .f (x) = C · L x→a ❝✮ x→a lim [f (x) + g(x)] = lim ·f (x) + lim ·g(x) = L + M x→a ❞✮ x→a x→a lim [f (x) · g(x)] = lim ·f (x) · lim ·g(x) = L · M x→a ❡✮ x→a  lim x→a  x→a 1 1 1 = = g(x) lim ·g(x) M M 6= 0✳ s❡♠♣r❡ q✉❡ M 6= 0✳ x→a   lim ·f (x) f (x) L lim = = x→a x→a g(x) lim ·g(x) M ❢✮ s❡♠♣r❡ q✉❡ x→a ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ Pr♦♣r✐❡❞❛❞❡ ✸✳✶✵✳ ❙❡ ❛✮ lim fi (x) = Li x→a i = 1, 2, 3, . . . , n ♣❛r❛ t♦❞♦ ❡♥tã♦✿ lim [f1 (x) + f2 (x) + f3 (x) + · · · + fn (x)] = L1 + L2 + L3 + · · · + Ln x→a ❜✮ lim [f1 (x) × f2 (x) × f3 (x) × · · · × fn (x)] = L1 × L2 × L3 × · · · × Ln x→a ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ Pr♦♣r✐❡❞❛❞❡ ✸✳✶✶✳ ❙✉♣♦♥❤❛ ◗✉❛♥❞♦ lim .f (x) = L x→a n ≤ 0✱ ❡♥tã♦ L n ∈ Z✱ ❡ ❡♥tã♦✱ lim .[f (x)]n = [lim .f (x)]n = Ln ✳ x→a x→a t❡♠ q✉❡ s❡r ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳ ❉❡♠♦♥str❛çã♦✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ Pr♦♣r✐❡❞❛❞❡ ✸✳✶✷✳ ❙❡ lim f (x) = L x→a ❡ n ∈ Z✱ ❡♥tã♦✱ lim x→a ♦♥❞❡ L é ♥ú♠❡r♦ ♣♦s✐t✐✈♦ ❡ n q p √ n n f (x) = n lim f (x) = L x→a q✉❛❧q✉❡r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ♦✉ L < 0 ❡ n q✉❛❧q✉❡r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ í♠♣❛r✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❊①❡♠♣❧♦ ✸✳✶✹✳ ❈❛❧❝✉❧❛r ♦ ❧✐♠✐t❡✿ ❙♦❧✉çã♦✳  5x2 − 10x − 6 lim x→2 x3 − 10  ✳ ✶✼✹ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦  −6 5x2 − 10x − 6 = = 3✳ ❆♣❧✐❝❛♥❞♦ ❛ Pr♦♣r✐❡❞❛❞❡ ✭✸✳✾✮ ❢ ✮ ♦❜t❡♠♦s q✉❡ lim 3 x→2 x − 10 −2  ❊①❡♠♣❧♦ ✸✳✶✺✳ ❈❛❧❝✉❧❛r ♦ s❡❣✉✐♥t❡ ❧✐♠✐t❡ lim x→−1 ❙♦❧✉çã♦✳ r 3x2 − 2x + 3 ✳ x5 + 2 P❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✸✳✶✷✮ t❡♠♦s q✉❡✿ lim x→−1 ❊①❡♠♣❧♦ ✸✳✶✻✳ ❈❛❧❝✉❧❛r lim x→0 ❙♦❧✉çã♦✳ r  3x2 − 2x + 3 = x5 + 2 x3 − 3x + 1 +1 x−4 s  lim x→−1   r √ 3x2 − 2x + 3 8 2 = = 2 x5 + 2 1 ✳ P❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✸✳✾✮ ❢ ✮ r❡s✉❧t❛ q✉❡✱ lim x→0 ❊①❡♠♣❧♦ ✸✳✶✼✳ ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳  x3 − 3x + 1 +1 x−4  6x − 6 lim x→1 x2 − 3x + 2   = lim (x3 − 2x − 3) x→0 lim (x − 4) = x→0 −3 3 = −4 4 ✳ ❖❜s❡r✈❡✱ ❛♦ ❛♣❧✐❝❛r ❛ Pr♦♣r✐❡❞❛❞❡ ✭✸✳✾✮ ❢ ✮ ❞❡ q✉♦❝✐❡♥t❡ ❞❡ ❧✐♠✐t❡s t❡rí❛♠♦s ✉♠ q✉♦❝✐✲ 0 ❡♥t❡ ❞❛ ❢♦r♠❛ ♥♦ ❧✐♠✐t❡✱ s❡♥❞♦ ❡st❛ ✉♠❛ ❢♦r♠❛ ✐♥❞❡t❡r♠✐♥❛❞❛✳ ◆♦ ♣♦ssí✈❡❧✱ ♣❛r❛ ❡✈✐t❛r 0 ✐st♦ t❡♠♦s q✉❡ ❡s❝r❡✈❡r ♥✉♠❡r❛❞♦r ❡ ❞❡♥♦♠✐♥❛❞♦r ♥❛ ❢♦r♠❛ ❞❡ ❢❛t♦r❡s (x − 1) ❞♦ ♠♦❞♦ s❡❣✉✐♥t❡✿     lim x→1 x2 6(x − 1) 6x − 6 = lim x→1 (x − 1)(x − 2) − 3x + 2 ❉❡s❞❡ q✉❡ x → 1✱ ❡♥tã♦ (x − 1) → 0 ❛✐♥❞❛ (x − 1) ♥ã♦ é ③❡r♦❀ ❧♦❣♦ ♣♦❞❡♠♦s s✐♠♣❧✐✜❝❛r ♥♦ ❧✐♠✐t❡ ❛❝✐♠❛ ♣❛r❛ ♦❜t❡r✿      6(x − 1) 6 6 6x − 6 = lim = lim = = −6 lim 2 x→1 (x − 1)(x − 2) x→1 x − 2 x→1 x − 3x + 2 −1  ❖❜s❡r✈❛çã♦ ✸✳✹✳ ✐✮ ❙ã♦ ❢♦r♠❛s ✐♥❞❡t❡r♠✐♥❛❞❛s✿ 0 , 0 ∞ , ∞ ∞ − ∞, ∞0 , ✶✼✺ 00 , 0∞ , ∞∞ , 1∞ , 0·∞ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❙❡✱ ♥♦ ❝á❧❝✉❧♦ ❞❡ ❧✐♠✐t❡s ❛♣❛r❡❝❡♠ ❛❧❣✉♠❛ ❞❡st❛s ❢♦r♠❛s✱ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ❧✐♠✐t❡s ❞❡✈❡♠♦s ✉t✐❧✐③❛r ♣r♦❝❡ss♦s ♦✉ ❛rt✐❢í❝✐♦s ❝♦♠ ♦ ♣r♦♣ós✐t♦ ❞❡ ❡✈✐t❛r ❛ ❢♦r♠❛ ✐♥❞❡t❡r✲ ♠✐♥❛❞❛✳ ✐✐✮ ❙❡❥❛ n ∈ N✱ ♣❛r❛ ❛ r❛❝✐♦♥❛❧✐③❛çã♦✱ ❧❡♠❜r❡✿ a2 − b2 = (a + b) · (a − b) an − bn = (a − b) · (an−1 + an−2 b + an−3 b2 + an−4 b3 + · · · + a2 bn−3 + abn−2 + bn−1 ) ◗✉❛♥❞♦ n é í♠♣❛r✿ an + bn = (a + b) · (an−1 − an−2 b + an−3 b2 − an−4 b3 + · · · + a2 bn−3 − abn−2 + bn−1 ) ❊①❡♠♣❧♦ ✸✳✶✽✳  12 1 − lim x→2 2 − x 8 − x3 ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳ ❖❜s❡r✈❡ q✉❡✿  ✳    (4 + 2x + x2 ) 12 12 1 = lim = − − lim x→2 (2 − x)(4 + 2x + x2 ) x→2 2 − x 8 − x3 8 − x3       2x + x2 − 8 −(2 − x)(x + 4) 4 + 2x + x2 − 12 = lim = lim = = lim x→2 x→2 (2 − x)(4 + 2x + x2 ) x→2 8 − x3 8 − x3  P♦rt❛♥t♦✱  −6 1 −(x + 4) = =− = lim 2 x→2 4 + 2x + x 12 2    1 12 1 =− lim − 2 x→2 2 − x 8−x 2 ❊①❡♠♣❧♦ ✸✳✶✾✳ ❈❛❧❝✉❧❛r ♦ ❧✐♠✐t❡✿ lim x→1 "√ ❙♦❧✉çã♦✳ 2x + 1 − x−1 √ # 3 ❊st❡ ❧✐♠✐t❡ é ❞❛ ❢♦r♠❛ ✐♥❞❡t❡r♠✐♥❛❞❛ lim ♥✉♠❡r❛❞♦r t❡♠♦s✿ x→1 "√ 2x + 1 − x−1 √ # 3 0 ❀ 0 ❛ss✐♠✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣❡❧❛ ❝♦♥❥✉❣❛❞❛ ❞♦ " √ √ √ √ # ( 2x + 1 − 3)( 2x + 1 + 3) √ = lim = √ x→1 (x − 1)( 2x + 1 + 3)    √ 2 3 2(x − 1) √ √ = lim √ = = lim √ x→1 ( 2x + 1 + x→1 (x − 1)( 2x + 1 + 3 3) 3)  P♦rt❛♥t♦✱ lim x→1 "√ 2x + 1 − x−1 √ # 3 = √ 3 ✳ 3 ✶✼✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ❊①❡♠♣❧♦ ✸✳✷✵✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞♦ s❡❣✉✐♥t❡ ❧✐♠✐t❡✿ ❙♦❧✉çã♦✳ R √  1 − x2 − 3x + 3 √ lim x→1 4x2 − 3 − 1  ◆♦ ❧✐♠✐t❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♦ ♥✉♠❡r❛❞♦r ❡ ❞❡♥♦♠✐♥❛❞♦r ♣❡❧♦ ❢❛t♦r F (x) = (1 + √ √ x2 − 3x + 3)( 4x2 − 3 + 1) √ √    (1 − x2 + 3x − 3)( 4x2 − 3 + 1) (1 − x2 − 3x + 3)F (x) √ √ = lim = lim x→1 x→1 ( 4x2 − 3 − 1)F (x) (4x2 − 4)(1 + x2 − 3x + 3)  t❡♠♦s✿ " # √   −(x − 2) −(x − 1)(x − 2)( 1 + 1) 1 √ = lim = lim = x→1 x→1 4(x + 1) 8 4(x − 1)(x + 1)(1 + 1) P♦rt❛♥t♦✱ √  1 1 − x2 − 3x + 3 √ = lim 2 x→1 8 4x − 3 − 1  ❊①❡♠♣❧♦ ✸✳✷✶✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞♦ s❡❣✉✐♥t❡ ❧✐♠✐t❡✿ lim x→2 ❙♦❧✉çã♦✳ √ 3 √  x3 − 2x − 3 − 3 2x2 − 7 ✳ 2x3 + x − 18 P❛r❛ ♦ ❧✐♠✐t❡✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♦ ♥✉♠❡r❛❞♦r ❡ ❞❡♥♦♠✐♥❛❞♦r ♣❡❧♦ ❢❛t♦r √ √ √ √ 3 3 3 3 F (x) = ( x3 − 2x − 3)2 + ( x3 − 2x − 3) · ( 2x2 − 7) + ( 2x2 − 7)2 t❡♠♦s √  √  ( 3 x3 − 2x − 3 − 3 2x2 − 7) · F (x) lim = x→2 (2x3 + x − 18) · F (x)     (x3 − 2x − 3) − (2x2 − 7) (x2 − 2)(x − 2) lim = lim = x→2 x→2 (2x2 + 4x + 9)(x − 2) · F (x) (2x3 + x − 18) · F (x)  2 2 2 (x2 − 2) = = = lim 2 x→2 (2x + 4x + 9) · F (x) 25 · F (2) (25)(3) 75 √ √ 3 3  x − 2x − 3 − 3 2x2 − 7 2 lim = 3 x→2 2x + x − 18 75  P♦rt❛♥t♦✱ ❊①❡♠♣❧♦ ✸✳✷✷✳ ❈♦♠♦ ✈❛r✐❛♠ ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ q✉❛❞r❛❞❛ s❡✉s ✈❛❧♦r❡s ❝♦♥st❛♥t❡s ✭b > 0✮ ❡ ♦ ♣❛râ♠❡tr♦ a ax2 + bx + c = 0 q✉❛♥❞♦✱ b ❡ c ❝♦♥s❡r✈❛♠ t❡♥❞❡ ♣❛r❛ ③❡r♦ ❄ ❙♦❧✉çã♦✳ ❆♣❧✐❝❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛ ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ sã♦✿ x1 = −b + √ b2 − 4ac 2a ❡ ✶✼✼ x2 = −b − √ b2 − 4ac 2a 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ◗✉❛♥t♦ a → 0 ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♥❛ ❢♦r♠❛✿ P❛r❛ ❛ r❛✐③ x1 lim x1 = lim a→0 a→0 −b + √ √ (−b)2 − ( b2 − 4ac)2 b2 − 4ac √ = lim a→0 2a(−b − 2a b2 − 4ac) ⇒ c 2c √ =− a→0 −b − b b2 − 4ac lim x1 = lim a→0 P❛r❛ ❛ r❛✐③ x2 lim x2 = lim a→0 a→0 −b − √ √ b2 − 4ac (−b)2 − ( b2 − 4ac)2 √ = lim a→0 2a(−b + 2a b2 − 4ac) ⇒ 2c √ =∞ a→0 (−b + b2 − 4ac) lim x2 = lim a→0 c P♦rt❛♥t♦✱ ✉♠❛ ❞❛s r❛í③❡s ❝♦♥✈❡r❣❡ ♣❛r❛ − ❡ ❛ ♦✉tr❛ ❞✐✈❡r❣❡ ✭❛♣r♦①✐♠❛✲s❡ r❛♣✐❞❛♠❡♥t❡ b ❛♦ ✐♥✜♥✐t♦✮✳ ✶✼✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✸✲✷ ✶✳ ▼♦str❡ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✳ ✶✳ ✷✳ ⇒ x→a lim .f (x) = L ⇒ x→a lim .f (x) = L ⇒ x→a x→a ✸✳ lim | f (x) |= 0✳ lim .f (x) = 0 x→a x→a lim [f (x) − L] = 0 lim | f (x) |=| L |✳ lim .f (x) = lim .f (a + h) ✹✳ x→a h→0 ✷✳ ❆♣r❡s❡♥t❛r ✉♠ ❡①❡♠♣❧♦ ❞❡ ♠♦❞♦ q✉❡✿ ✶✳ ❊①✐st❛ lim | f (x) | ❡ ♥ã♦ ❡①✐st❛ lim ·f (x)✳ ✷✳ ❊①✐st❛ lim [f (x) + g(x)] ❡ ♥ã♦ ❡①✐st❛♠ lim ·f (x) ❡ lim ·g(x) ✳ x→a x→a x→a x→a x→a ✸✳ ❈❛s♦ ❡①✐st❛♠ ♦s ❧✐♠✐t❡s lim ·f (x) ❡ lim [f (x) + g(x)]✳ ❊①✐st❡ lim ·g(x) ❄ x→a x→a x→a ✹✳ ❈❛s♦ ❡①✐st❛♠ ♦s ❧✐♠✐t❡s lim ·f (x) ❡ lim [f (x) · g(x)]✳ ❊①✐st❡ lim ·g(x) ❄ x→a x→a x→a ✺✳ ❈❛s♦ ❡①✐st❛ lim ·f (x) ❡ lim ·g(x) ♥ã♦ ❡①✐st❡✱ ❡♥tã♦ ❡①✐st❡ lim [f (x) + g(x)] ❄ x→a x→a x→a ✻✳ ▼♦str❡ q✉❡ lim f (x) ❡①✐st❡✱ s❡ ❡ s♦♠❡♥t❡ s❡ lim f (a + h) ❡①✐st❡✳ x→a h→0 ✼✳ ▼♦str❡ q✉❡ lim f (x) ❡①✐st❡✱ s❡ ❡ s♦♠❡♥t❡ s❡ lim f (x + a) ❡①✐st❡✳ x→a x→0 ✽✳ ▼♦str❡ q✉❡ lim f (x) ❡①✐st❡✱ s❡ ❡ s♦♠❡♥t❡ s❡ lim f (x3 ) ❡①✐st❡✳ x→0 x→0 ✾✳ ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ f (x) = lim .f (x) = m2 − 17✳ x2 − mx + 3x − 3m ✱ ❞❡t❡r♠✐♥❡ ♦s ✈❛❧♦r❡s ❞❡ m t❛❧ q✉❡ x−m x→m ✶✵✳ ❙❡❥❛ ❛ ❢✉♥çã♦ f (x) = a > 0✳ x3 − 2a2 x + ax2 ✱ ❡ lim .f (x) = 2a − 5✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❡ x→1 2ax + x2 ✶✶✳ ▼♦str❡ q✉❡✱ ❛♦ ❝r❡s❝❡r n ✐♥❞❡✜♥✐❞❛♠❡♥t❡✱ ❛ s❡q✉ê♥❝✐❛ un = ❧✐♠✐t❡✳ ❆ s❡q✉ê♥❝✐❛ vn = 2n + (−2)n ♥ã♦ t❡♠ 2n 2n + (−2)n t❡♠ ❧✐♠✐t❡❄ ❏✉st✐✜❝❛r s✉❛ r❡s♣♦st❛✳ 3n ✶✼✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✶✷✳ ❈❛❧❝✉❧❛r ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ 1. 4. 7. 10.  3x2 − 17x + 20 2. lim x→4 4x2 − 25x + 36   3x − 6 √ lim 5. x→2 1 − 4x − 7   x+3 8. lim √ x→−3 x2 + 7 − 4 "p # √ 2+ 3x−2 lim 11. x→8 x−8  # √ x2 + 3 x − 2 − 4 lim p √ 3 x→2 4 − x 3x − 2 √ √ 2  b − x − b2 − a lim x→a x−a √  √ n x− na lim a>0 x→a x−a " 13. 15. 17.   x5 − 1 lim 3. x→1 x6 − 1  2  x − a2 6. lim x→a x3 − a3 √  3 x−1 lim √ 9. 4 x→1 x−1 √  x−8 lim √ 12. 3 x→64 x−4 14. 16. 18.  5x − 10 √ lim √ 5 x→2 x− 52  √  24x−4−4 lim √ 5 x→20 x + 12 − 2 # "√ 3 9x − 3 lim √ x→3 3x − 3 "√ # √ 3 x2 − 4 3 x + 4 lim x→8 (x − 8)2    2x2n + 1 − 3x−2n lim x→1 3x2n − 5 + 2x−2n # " √ √ 3x − 8 − x √ lim x→2+ 3x − 2 15 − 3x  2  x − (a − 1)x − a lim x→a x2 − (a − 2)x − 2a ✶✸✳ ❱❡r✐✜q✉❡ ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✱ ♣❛r❛ ❛s ❢✉♥çõ❡s ✐♥❞✐❝❛❞❛s✿   f (4 + h) − f (4) 1 1 lim =− s❡ f (x) = 2 h→0 h 50 x +4   f (−1 + h) − f (−1) = −16 s❡ f (x) = 8x2 lim h→0 h ✶✳ ✷✳ ✶✹✳ ❱❡r✐✜q✉❡ ♦ ❝á❧❝✉❧♦ ❞♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ 1. 3. 5. 7.   2 4 2 lim = − 2 x→2 3x − 6 2x − 5x + 2 9  4  4x + 9x3 + 3x2 + x + 3 lim =1 x→−1 3x4 + 9x3 + 9x2 + 3x   3 11 2x − 5x2 − 2x − 3 = lim 3 2 x→3 4x − 13x + 4x − 3 17 # "√ √ x3 + 3 x − 3x − 1 27 √ = lim √ 3 x→1 x + 3 3 x − 3 x2 − 1 8 2. 4. 6. 8.   7a4 x 7 − a7 = lim 3 x→a x − a3 3  √ 3 x + 27 − 3 32 = lim √ 4 x→0 27 x + 16 − 2 √   1 3− 5+x √ =− lim x→4 1 − 3 5−x   3 x3 + 6x2 + 9x = lim 3 2 x→−3 x + 5x + 3x − 9 4 ✶✺✳ ❙❡ f (2) = 6✱ ♦ q✉é ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r ❞♦ lim .f (x)❄ ❏✉st✐✜❝❛r s✉❛ r❡s♣♦st❛✳ x→2 ✶✻✳ ❙❡ lim .f (x) = 6✱ ♣♦❞❡♠♦s ♦❜t❡r ❝♦♥❝❧✉sã♦ ❛ r❡s♣❡✐t♦ ❞❡ f (2) ❏✉st✐✜❝❛r s✉❛ r❡s♣♦st❛✳ x→2       f (x) g(x) f (x) ✶✼✳ ❙❛❜❡✲s❡ q✉❡ lim = 4 ❡ lim = −6✳ Pr♦✈❡ q✉❡ lim = −1 x→1 1 − x3 x→1 1 − x2 x→1 g(x)       f (x + 2) f (x) g(x + 2) ✶✽✳ ❙❡ lim √ = 3✳ ❈❛❧❝✉❧❛r✿ lim ✳ = 8 ❡ lim x→−2 x→0 g(x) x→−2 x2 − 4 −2x − 2 ✶✽✵ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ R ♦ r❡tâ♥❣✉❧♦ q✉❡ s❡ ♦❜té♠ ❛♦ ✉♥✐r ♦s ♣♦♥t♦s ♠é❞✐♦s ❞♦s ❧❛❞♦s ❞♦ q✉❛❞r✐❧át❡r♦ ♣❡r✐♠❡tr♦ ❞❡ R ✳ Q✱ ♦ q✉❛❧ t❡♠ s❡✉s ✈ért✐❝❡s (±x, 0) ❡ (0, ±y)✳ ❈❛❧❝✉❧❡ lim+ x→0 ♣❡r✐♠❡tr♦ ❞❡ Q ✶✾✳ ❙❡❥❛ R$1.500.000, 00✱ ❡ ❡st❛ q✉❛♥t✐❛ ❢♦✐ ❞❡♣r❡❝✐❛❞❛ ♣❡❧♦ ♠ét♦❞♦ ❞❛ ❧✐♥❤❛ r❡t❛ ♣♦r 15 ❛♥♦s✱ ❛ ♣❛rt✐r ❞❡ 1985✳ ◗✉❛❧ ❢♦✐ ♦ ✈❛❧♦r ❧íq✉✐❞♦ ❞♦ ♣ré❞✐♦ ❡♠ 1993✳ ✷✵✳ ❖s ❝✉st♦s ❞❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ♣ré❞✐♦ ❞❡ ❛♣❛rt❛♠❡♥t♦s ❢♦r❛♠ ❞❡ ✷✶✳ ❙❡❥❛♠ f : [a, b] −→ R ▼♦str❡ q✉❡ ❡①✐st❡ g : [a, b] −→ R ❡ ❢✉♥çõ❡s t❛✐s q✉❡✿ δ > 0 t❛❧ q✉❡✿ ∀ 0 < |x−c| < δ f (x) > g(x), | x |< ε, ✷✸✳ ❉❡♠♦♥str❡ q✉❡✱ s❡ lim .f (x) = L 6= 0✱ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ B(a, δ) s✐♥❛❧ ∀ x ∈ B(a, δ) ❝♦♠ x 6= a✳ L t❡♠ ♦ ♠❡s♠♦ ✷✹✳ ❉❡♠♦♥str❡ q✉❡✱ s❡ ❛✮ lim .f (x) = L ❡ L ≤ M✱ ✷✺✳ ❉❡♠♦♥str❡ q✉❡✱ s❡ ❛✮ g ❡ t❛❧ q✉❡ ❝♦♠ x 6= a✳ lim .g(x) = M ✳ x→a ✐st♦ é f f (x) sã♦ ❢✉♥çõ❡s t❛✐s q✉❡ ❝✉♠♣r❛♠ ❛s ❤✐♣ót❡s❡s✿ ∀ x ∈ B(a, δ) x→a ❊♥tã♦ ❜✮ f c ∈ (a, b)✳ x = 0✳ x→a f (x) ≤ g(x) ❜✮ ⇒ x→c ✷✷✳ ❉❡♠♦♥str❡ q✉❡✱ s❡ ❡ ∀ε>0 ⇒ lim f (x) > lim g(x)✳ x→c ❡ g lim .f (x) ≤ lim .g(x)✳ x→a x→a ❞✉❛s ❢✉♥çõ❡s t❛✐s q✉❡ ❝✉♠♣r❛♠ ❛s ❤✐♣ót❡s❡s✿ lim .f (x) = 0✳ x→a ❊①✐st❡ M >0 t❛❧ q✉❡ | g(x) |< M ∀ x ∈ B(a, δ) ❝♦♠ x 6= a✳ lim .f (x).g(x) = 0✳ ❊♥tã♦ x→a f ❡ g ❞✉❛s ❢✉♥çõ❡s ❡ C ♥ú♠❡r♦ r❡❛❧ ❝♦♥st❛♥t❡✱ t❛✐s q✉❡ lim .f (x) = x→a lim .g(x) = M ❡♥tã♦✿ ✷✻✳ ❉❡♠♦♥str❡ q✉❡✱ s❡ L ❛✮ ❜✮ ❝✮ ❞✮ ❡✮ ❡ x→a lim .C = C x→a lim .C · f (x) = C · lim .f (x) = C · L x→a x→a lim [f (x) + g(x)] = lim ·f (x) + lim ·g(x) = L + M x→a x→a x→a lim [f (x) · g(x)] = lim ·f (x) · lim ·g(x) = L · M x→a x→a   1 1 1 lim = = s❡♠♣r❡ q✉❡ M 6= 0✳ x→a g(x) lim ·g(x) M x→a x→a ❢✮   lim ·f (x) f (x) L x→a lim = = x→a g(x) lim ·g(x) M ✷✼✳ ❉❡♠♦♥str❡ q✉❡✱ s❡ s❡♠♣r❡ q✉❡ x→a lim fi (x) = Li x→a ♣❛r❛ t♦❞♦ ✶✽✶ M 6= 0✳ i = 1, 2, 3, . . . , n ❡♥tã♦✿ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❛✮ ❜✮ lim [f1 (x) + f2 (x) + f3 (x) + · · · + fn (x)] = L1 + L2 + L3 + · · · + Ln x→a lim [f1 (x) × f2 (x) × f3 (x) × · · · × fn (x)] = L1 × L2 × L3 × · · · × Ln x→a lim .f (x) = L ❡ n ∈ Z✱ ❡♥tã♦✱ lim .[f (x)]n = [lim .f (x)]n = Ln ✳ x→a x→a n ≤ 0✱ ❡♥tã♦ L t❡♠ q✉❡ s❡r ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦❄ ✷✽✳ ❉❡♠♦♥str❡ q✉❡✱ s❡ ◗✉❛♥❞♦ ✷✾✳ ❉❡♠♦♥str❡ q✉❡✱ s❡ x→a lim f (x) = L x→a lim x→a ♦♥❞❡ p n ❡ n ∈ Z✱ f (x) = ❡♥tã♦ q n lim f (x) = x→a √ n L L é ♥ú♠❡r♦ ♣♦s✐t✐✈♦ ❡ n q✉❛❧q✉❡r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ♦✉ L < 0 ❡ n q✉❛❧q✉❡r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ í♠♣❛r✳ ✶✽✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✸✳✸ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ▲✐♠✐t❡s ❧❛t❡r❛✐s ❆♦ ❝❛❧❝✉❧❛r♠♦s lim .f (x)✱ ♦ ♣r♦❜❧❡♠❛ r❡❞✉③✲s❡ ❛ ❝❛❧❝✉❧❛r ♦ ♥ú♠❡r♦ L ♣❛r❛ ♦ q✉❛❧ x→a ❛♣r♦①✐♠❛♠✲s❡ ♦s ✈❛❧♦r❡s ❞❡ f (x) q✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ a✱ t❛♥t♦ ♣❛r❛ ✈❛❧♦r❡s ♠❛✐♦r❡s q✉❡ a ✭♣❡❧❛ ❞✐r❡✐t❛✮ q✉❛♥t♦ ♣❛r❛ ✈❛❧♦r❡s ( ❞❡ ♠❡♥♦r❡s q✉❡ a ✭♣❡❧❛ ❡sq✉❡r❞❛✮✳ x − 1, s❡✱ x < 2 ❈♦♥s✐❞❡r❛♥❞♦ ❛ ❢✉♥çã♦ f (x) = ✱ 5 − x, s❡✱ x ≥ 2 y ✻ ♦❜s❡r✈❛✲s❡ ♦ s❡❣✉✐♥t❡✿ 3 ❛✮ ◗✉❛♥❞♦ x ❛♣r♦①✐♠❛✲s❡ ❛ 2 ♣❡❧❛ ❞✐r❡✐t❛✱ f (x) ❛♣r♦①✐♠❛✲s❡ ❛ 3 ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✭✸✳✺✮❀ ✐st♦ é ❝❤❛♠❛❞♦ ❞❡ ❧✐♠✐t❡ ❧❛t❡r❛❧ ❞❡ f (x) q✉❛♥❞♦ x t❡♥❞❡ ❛ 2 ♣❡❧❛ ❞✐r❡✐t❛✱ ❡ ❡s❝r❡✈❡✲s❡ ❆ f (x) ❆ ❆ ❆ x ❆ ✲ 2 3❆ ❆ ❆ ❆ 1 −x✛ −2 lim .f (x) = 3 0 −y x→2+ ❜✮ ◗✉❛♥❞♦ x ❛♣r♦①✐♠❛✲s❡ ❛ 2 ♣❡❧❛ ❡sq✉❡r❞❛✱ f (x) ❄ ❛♣r♦①✐♠❛✲s❡ ❛ 1 ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✭✸✳✺✮❀ ❋✐❣✉r❛ ✸✳✺✿ ✐st♦ é ❝❤❛♠❛❞♦ ❞❡ ❧✐♠✐t❡ ❧❛t❡r❛❧ ❞❡ f (x) q✉❛♥❞♦ x t❡♥❞❡ ❛ 2 ♣❡❧❛ ❡sq✉❡r❞❛✱ ❡ ❞❡♥♦t❛❞♦ lim− .f (x) = 1✳ x→2 ❊♠ ❣❡r❛❧ t❡♠♦s ❛s s❡❣✉✐♥t❡s ❞❡✜♥✐çõ❡s✿ ❉❡✜♥✐çã♦ ✸✳✸✳ ❙❡❥❛♠ a < c ❡ f (x) ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♥♦ ✐♥t❡r✈❛❧♦ (a, c)❀ ❞✐③❡♠♦s q✉❡ L é ♦ ❧✐♠✐t❡ ❧❛t❡r❛❧ ❞❡ f (x) q✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ ❛ ♣❡❧❛ ❞✐r❡✐t❛ ❡ ❞❡♥♦t❛♠♦s lim+ .f (x) x→a ♦✉ f (a+ )❀ s❡✱ ❞❛❞♦ ε > 0, ∃δ > 0 /. ∀ x ∈ D(f ), | f (x) − L |< ε s❡♠♣r❡ q✉❡ 0 < x − a < δ✳ ❉❡✜♥✐çã♦ ✸✳✹✳ ❙❡❥❛♠ b < a ❡ f (x) ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♥♦ ✐♥t❡r✈❛❧♦ (b, a)❀ ❞✐③❡♠♦s q✉❡ L é ♦ ❧✐♠✐t❡ ❧❛t❡r❛❧ ❞❡ f (x) q✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ a ♣❡❧❛ ❡sq✉❡r❞❛ ❡ ❞❡♥♦t❛♠♦s lim− .f (x) x→a ♦✉ f (a− ) s❡✱ ❞❛❞♦ ε > 0, ∃δ > 0 /. ∀ x ∈ D(f ), | f (x) − L |< ε s❡♠♣r❡ q✉❡ 0 < a − x < δ✳ Pr♦♣r✐❡❞❛❞❡ ✸✳✶✸✳ lim .f (x) = L s❡✱ ❡ s♦♠❡♥t❡ s❡ lim+ .f (x) = lim− .f (x) = L✳ x→a ❉❡♠♦♥str❛çã♦✳ x→a x→a ❊①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ✶✽✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❖❜s❡r✈❛çã♦ ✸✳✺✳ ◆♦s s❡❣✉✐♥t❡s ❝❛s♦s ♦ ✐✮ lim .f (x) x→a ♥ã♦ ❡①✐st❡✿ ◗✉❛♥❞♦ ♥ã♦ ❡①✐st❡ ✉♠ ❞♦s ❧✐♠✐t❡s ❧❛t❡r❛✐s✳ ✐✐✮ ◗✉❛♥❞♦ ♦s ❧✐♠✐t❡s ❧❛t❡r❛✐s ❡①✐st❡♠ ❡ sã♦ ❞✐❢❡r❡♥t❡s✳ ✸✳ ◗✉❛♥❞♦ ♦ ❧✐♠✐t❡ ♥ã♦ ❢♦r ✉♠ ♥ú♠❡r♦ r❡❛❧ L✱ ✐st♦ é q✉❛♥❞♦ ♦ ❧✐♠✐t❡ ❢♦r ±∞✳ ◗✉❛♥❞♦ ❛ ❢✉♥çã♦ ❡st✐✈❡r ❞❡✜♥✐❞❛ ♣❛r❛ x < a ❡ x > a✱ ❣❡r❛❧♠❡♥t❡ ❛♦ ❝❛❧❝✉❧❛r lim .f (x) x→a é ♥❡❝❡ssár✐♦ ❝❛❧❝✉❧❛r ♦s ❧✐♠✐t❡s ❧❛t❡r❛✐s ❞❡ f (x) ❊①❡♠♣❧♦ ✸✳✷✸✳  2   2x − 1, s❡✱ x > 1 ❉❡t❡r♠✐♥❡ ♦ lim .g(x) ✱ s❡ g(x) = 1, s❡✱ x = 1 x→1   2 − x, s❡✱ x < 1 ❙♦❧✉çã♦✳ ❖❜s❡r✈❡ q✉❡✱ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x = 1 ❛ ❢✉♥çã♦ ❡st❛ ❞❡✜♥✐❞❛ ❞❡ ❞✐❢❡r❡♥t❡s ♠♦❞♦s ✭❋✐❣✉r❛ ✭✸✳✻✮✮✱ é ♣♦r ✐ss♦ q✉❡ é ♥❡❝❡ssár✐♦ ❝❛❧❝✉❧❛r ♦s ❧✐♠✐✲ t❡s ❧❛t❡r❛✐s✳ lim+ .g(x) = lim+ (2x2 − 1) = 1 ✱ ♣♦r ♦✉tr♦ ❧❛❞♦✿ x→1 x→1 lim .g(x) = lim− (2 − x) = 1✳ x→1− x→1 P♦rt❛♥t♦✱ lim .g(x) = 1 x→1 ❊①❡♠♣❧♦ ✸✳✷✹✳ ❙❡❥❛ ♠✐t❡ |x+2| ✱ 4 + 2x lim .f (x)✳ f (x) = ❞❡t❡r♠✐♥❡ s❡ ❡①✐st❡ ♦ ❧✐✲ ❋✐❣✉r❛ ✸✳✻✿ x→−2 ❙♦❧✉çã♦✳ ❈♦♠♦ | x + 2 |= ( x + 2, s❡✱ x ≥ −2 ❡♥tã♦✿ −x − 2, s❡✱ x < −2 lim + .f (x) = lim + 1 1 x+2 = lim + = , 4 + 2x x→−2 2 2 lim − .f (x) = lim − −1 −1 −x − 2 = lim − = x→−2 4 + 2x 2 2 x→−2 x→−2 x→−2 x→−2 ❖❜s❡r✈❡ q✉❡ ♦s ❧✐♠✐t❡s ❧❛t❡r❛✐s sã♦ ❞✐❢❡r❡♥t❡s✱ ❧♦❣♦ ♥ã♦ ❡①✐st❡ lim .f (x) x→−2 ❊①❡♠♣❧♦ ✸✳✷✺✳ ❖s ❝✉st♦s ❞❡ tr❛♥s♣♦rt❡ ❞❡ ♠❡r❝❛❞♦r✐❛s sã♦ ✉s✉❛❧♠❡♥t❡ ❝❛❧❝✉❧❛❞♦s ♣♦r ✉♠❛ ❢ór♠✉❧❛ q✉❡ r❡s✉❧t❛ ❡♠ ❝✉st♦s ♠❛✐s ❜❛✐①♦s ♣♦r q✉✐❧♦ à ♠❡❞✐❞❛ q✉❡ ❛ ❝❛r❣❛ ❛✉♠❡♥t❛✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ✶✽✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ x ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R s❡❥❛ ♦ ♣❡s♦ ❞❡ ✉♠❛ ❝❛r❣❛ ❛ s❡r tr❛♥s♣♦rt❛❞❛✱ ❡ ♦ ❝✉st♦ t♦t❛❧ ❡♠ r❡❛✐s✳ ❆❝❤❡ ❝❛❞❛ ✉♠ ❞♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ ❛✮    0, 85x, C(x) = 0, 75x,   0, 60x, lim .C(x) x→50 ❜✮ ❡ s❡✱ s❡✱ s❡✱ 0 < x ≤ 50 50 < x ≤ 200 200 < x lim .C(x) x→200 ❙♦❧✉çã♦✳ ❛✮ P❛r❛ ❝❛❧❝✉❧❛r ♦ ❧✐♠✐t❡✱ x→50 lim .C(x) ✱ t❡♠♦s q✉❡ ❛❝❤❛r ♦s ❧✐♠✐t❡s ❧❛t❡r❛✐s✿ lim (0, 85x) = 42, 5 ❡ x→50− ❡①✐st❡ lim .C(x)✳ lim .C(x) = x→50− lim .C(x) = lim+ (0, 75x) = 37, 5❀ s❡♥❞♦ ❞✐❢❡r❡♥t❡s ♥ã♦ x→50+ x→50 x→50 ❜✮ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✿ lim .C(x) = lim − (0, 75x) = 150 x→200− x→200 x→200+ x→200 ❡ lim .C(x) = lim + (0, 60x) = 120 ❀ ❙❡♥❞♦ ❞✐❢❡r❡♥t❡s ♦s ❧✐♠✐t❡s ❧❛t❡r❛✐s✱ ❧♦❣♦ ♥ã♦ ❡①✐st❡ lim .C(x) x→200 ❙❡ ❞❡s❡❥❛♠♦s s❛❜❡r ♦ ❝✉st♦ ❞❡ tr❛♥s♣♦rt❡ ❞❡ x = 50 q✉✐❧♦s✱ t❡rí❛♠♦s ❛ ♣❛❣❛r C(50) = (0, 80)(50) = 42, 5 r❡❛✐s✱ ❡ ❞❡ x = 200 é C(200) = (0, 75)(200) = 150 r❡❛✐s✳ ✸✳✹ ▲✐♠✐t❡s ❛♦ ✐♥✜♥✐t♦ 1 ❆ ❢✉♥çã♦ f (x) = 2 ✱ ❡stá ❞❡✜♥✐❞❛ ❞❡ t❛❧ ♠♦❞♦ q✉❡ ♦s ✈❛❧♦r❡s f (x) ✜❝❛♠ ❛r❜✐tr❛✲ x r✐❛♠❡♥t❡ ♣❡q✉❡♥♦s q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ♦s ✈❛❧♦r❡s ❞❡ x ♦s ♠❛✐s ❣r❛♥❞❡s ♣♦ssí✈❡✐s ✭❡♠ ✈❛❧♦r ❛❜s♦❧✉t♦✮✳ ❆ss✐♠✱ f é ❧♦❝❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛ ♣❛r❛ ✈❛❧♦r❡s ❡①tr❡♠❛♠❡♥t❡ ❣r❛♥❞❡s ❞❡ x✱ ♣ró①✐♠♦s ❞♦ ✐♥✜♥✐t♦✳ ❊♠❜♦r❛ ❡①✐st❛ ♦ ❧✐♠✐t❡ ❞❡ f q✉❛♥❞♦ x → ∞✱ ❡ ✐st♦ ❞❡✈❡ ✜❝❛r ❝❧❛r♦✱ ♣♦✐s ♥ã♦ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ x = a ♥❛s ❝♦♥❞✐çõ❡s ❞❛ ❉❡✜♥✐çã♦ ✭✸✳✷✮ ❞❡ ❧✐♠✐t❡s✳ ❙❡ ❡s❝r❡✈❡ 1 = 0✳ x→∞ x2 lim ❉❡✜♥✐çã♦ ✸✳✺✳ f : (a, +∞) −→ R ✱ ✉♠❛ ❢✉♥çã♦ ❡ L ∈ R✱ ❞✐③❡♠♦s q✉❡ L é ♦ ❧✐♠✐t❡ q✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ +∞ ❡ ❡s❝r❡✈❡♠♦s lim .f (x) = L s❡✱ ❡ s♦♠❡♥t❡ ❙❡❥❛ x→+∞ ε > 0✱ ❡①✐st❡ N >0 t❛❧ q✉❡ | f (x) − L |< ε s❡♠♣r❡ q✉❡ ❞❡ f (x) s❡ ❞❛❞♦ x > N✳ ❉❡✜♥✐çã♦ ✸✳✻✳ g : (−∞, b) −→ R✱ ✉♠❛ ❢✉♥çã♦ ❡ L ∈ R✱ ❞✐③❡♠♦s q✉❡ L é ♦ ❧✐♠✐t❡ ❞❡ g(x) q✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ −∞ ❡ ❡s❝r❡✈❡♠♦s lim .g(x) = L s❡✱ ❡ s♦♠❡♥t❡ s❡ ❞❛❞♦ ❙❡❥❛ x→−∞ ε>0 ✱ ❡①✐st❡ N >0 t❛❧ q✉❡ | g(x) − L |< ε ✶✽✺ s❡♠♣r❡ q✉❡ x < −N = M ✳ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❉❡st❛s ❞❡✜♥✐çõ❡s✱ ♣♦❞❡♠♦s ✐♥t❡r♣r❡t❛r q✉❡✱ ❡♠ t❛♥t♦ s❡❥❛ ♠❛✐♦r ✭♦✉ ♠❡♥♦r✮ ♦ ✈❛❧♦r ❞❡ x✱ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ f (x) ❡ L é ❝❛❞❛ ✈❡③ ♠❡♥♦r✱ ♦ q✉❛❧ s✐❣♥✐✜❝❛ q✉❡ f (x) ❛♣r♦①✐♠❛✲s❡ ❝❛❞❛ ✈❡③ ♠❛s ♣❛r❛ L ❝♦♠♦ ♦❜s❡r✈❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✸✳✼✮✳ ❋✐❣✉r❛ ✸✳✼✿ Pr♦♣r✐❡❞❛❞❡ ✸✳✶✹✳ ❙❡❥❛ n ∈ N ❡♥tã♦✿ ✐✮ lim x→+∞ ❉❡♠♦♥str❛çã♦✳ ✐✮ ✐✐✮ 1 =0 xn ✐✐✮ lim x→−∞ 1 ε 1 =0 xn 1 ε 1 < ε ❀ ❛ss✐♠✱ ❞❛❞♦ xn 1 1 ε > 0✱ ❡①✐st❡ N > 0 t❛❧ q✉❡ | n |< ε s❡♠♣r❡ q✉❡ x > N ✳ P♦rt❛♥t♦ lim n = 0✳ x→+∞ x x ❉❛❞♦ ε > 0✱ ❡①✐st❡ N = √ > 0 t❛❧ q✉❡ ♣❛r❛ x > N = √ t❡♠♦s n n 1 ε 1 ε > 0 t❛❧ q✉❡ ♣❛r❛ x < −N = − √ ❆♥❛❧♦❣❛♠❡♥t❡✳ ❉❛❞♦ ε > 0✱ ❡①✐st❡ N = √ n n 1 ε 1 x t❡♠♦s −x > √ ❡♥tã♦✱ 0 < − < n √ n ε ✐st♦ é | ▲♦❣♦✱ ❞❛❞♦ ε > 0✱ ❡①✐st❡ N > 0✱ t❡♠♦s | 1 |< ε✳ xn 1 |< ε s❡♠♣r❡ q✉❡ x < −N ✳ xn 1 = 0✳ x→−∞ xn P♦rt❛♥t♦ lim Pr♦♣r✐❡❞❛❞❡ ✸✳✶✺✳ ❙❡❥❛♠ f ❡ g ❞✉❛s ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ❡♠ (a, +∞) ❡ (b, +∞) r❡s♣❡❝t✐✈❛♠❡♥t❡❀ s❡ lim .f (x) = L ❡ lim .g(x) = M ❡♥tã♦✿ xto+∞ ❛✮ ❜✮ ❝✮ xto+∞ lim [C · f (x)] = C · L ♣❛r❛ C ❝♦♥st❛♥t❡✳ x→+∞ lim [f (x) + g(x)] = lim .f (x) + lim .g(x) = L + M xto+∞ xto+∞ xto+∞ lim [f (x) × g(x)] = lim .f (x) × lim .g(x) = L × M xto+∞ xto+∞ xto+∞ ✶✽✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❞✮ lim xto+∞  ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R  lim .f (x) f (x) L xto+∞ = = ❞❡s❞❡ q✉❡ M 6= 0✳ g(x) lim .g(x) M xto+∞ ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ◗✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ −∞ ♦❜té♠✲s❡ ♣r♦♣r✐❡❞❛❞❡s s✐♠✐❧❛r❡s ❛s ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭✸✳✶✺✮✳ ❆♣❧✐❝❛çã♦ ❞♦s ❧✐♠✐t❡s ❛♦ ✐♥✜♥✐t♦ ❊①❡♠♣❧♦ ✸✳✷✻✳ lim ❈❛❧❝✉❧❛r✿ x→+∞ ❙♦❧✉çã♦✳ ❚❡♠♦s ❞❛❞❡ lim x→+∞   3x2 − 6x + 2 x2 + 2x − 3 lim x→+∞ ❊①❡♠♣❧♦ ✸✳✷✼✳ a ✳ "  x2 3 − 3x2 − 6x + 2 = lim x→+∞ x2 1 + x2 + 2x − 3 ✭✸✳✶✹✮ ♦❜t❡♠♦s ❙✉♣♦♥❤❛✱   6 x 2 x + −
# 2 x2
3 x2 ✱ ❧♦❣♦ ❛♣❧✐❝❛♥❞♦ ❛ Pr♦♣r✐❡✲  3−0+0 3x2 − 6x + 2 = = 3✳ x2 + 2x − 3 1+0−0 √ √ lim [ ax2 + bx + c − ax2 ]✳ ♥ú♠❡r♦ ♣♦s✐t✐✈♦✱ ❝❛❧❝✉❧❛r x→+∞ ❙♦❧✉çã♦✳ √ √ lim [ ax2 + bx + c − ax2 ] = ❚❡♠♦s x→+∞ " √ # √ √ √ ( ax2 + bx + c − ax2 )( ax2 + bx + c + ax2 ) √ lim = √ x→+∞ ( ax2 + bx + c + ax2 ) = lim x→+∞  √ bx + c ax2 + bx + c + √ ax2  ❊①❡♠♣❧♦ ✸✳✷✽✳ ❙✉♣♦♥❤❛ a ♥ú♠❡r♦ ♣♦s✐t✐✈♦✱ ❝❛❧❝✉❧❛r  x b+  = lim  q x→+∞ x a + xb +
c x c x2  b  √   = 2 √a + a √ √ lim [ ax2 + bx + c + ax2 ]✳ x→+∞ ❙♦❧✉çã♦✳ √ √ lim [ ax2 + bx + c + ax2 ] = (+∞) + (+∞) = +∞✳ x→+∞ ❊①❡♠♣❧♦ ✸✳✷✾✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❡ ❙♦❧✉çã♦✳ lim x→+∞ √ 4 − x2 2x − 4  √ 4 − x2 ❡ s❡✉ ❞♦♠í♥✐♦ é ♦ ❝♦♥❥✉♥t♦ [−2, 2)✱ ✐st♦ s✐❣♥✐✜❝❛ q✉❡ 2x  √− 4 2  4−x ♥ã♦ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r lim ✳ x→+∞ 2x − 4 ❚❡♠♦s ❛ ❢✉♥çã♦ f (x) = P♦rt❛♥t♦ ♥ã♦ ❡①✐st❡ ♦ ❧✐♠✐t❡✳ ✶✽✼ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❊①❡♠♣❧♦ ✸✳✸✵✳ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ "√ 3 ❈❛❧❝✉❧❛r lim n→+∞ n2 + n n+1 R # ❙♦❧✉çã♦✳ 3 ❖❜s❡r✈❡✱ q✉❡ lim n→+∞ lim n2 + n  = lim  n→+∞ n+1 q 3 1+ √ 3 n→+∞ √ # "√ n2 · lim n n→+∞ 1 + 1 n 1 n 3 √  3 1 1+0 · = lim √ =0 n→+∞ 3 n (1 + 0) ❊①❡♠♣❧♦ ✸✳✸✶✳ lim ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞♦ s❡❣✉✐♥t❡ ❧✐♠✐t❡✿ x→∞ ❙♦❧✉çã♦✳ ❉♦ ❢❛t♦ x → ∞✱  q 1 3 1+ n 
= ⇒ n 1 + n1 n2 "√ x2 + 1 + 3x 2x − 5 ❡♥tã♦ t❡♠♦s q✉❡ ❝❛❧❝✉❧❛r ♦ ❧✐♠✐t❡ q✉❛♥❞♦ lim x→+∞ "√ # x → +∞ # # " √ x2 + 1 + 3x x( 1 + x−2 + 3) = lim = x→+∞ 2x − 5 x(2 − 5x−1 ) # √ 1 + x−2 + 3 1+0+3 =2 = lim −1 x→+∞ 2 − 5x 2−0 √ q✉❛♥❞♦ x → −∞✱ ❝♦♠♦ x2 =| x |= −x✱ ♣❛r❛ ❡ x → −∞✿ "√ P❛r❛ ♦ ❝á❧❝✉❧♦ ✈❛❧♦r❡s ♥❡❣❛t✐✈♦s ❞❡ x " # # √ | x | 1 + x−2 + 3x x2 + 1 + 3x = lim = lim x→−∞ x→−∞ 2x − 5 x(2 − 5x−1 ) " # " √ # √ √ x(− 1 + x−2 + 3) − 1 + x−2 + 3 − 1+0+3 lim = lim =1 = x→−∞ x→−∞ x(2 − 5x−1 ) 2 − 5x−1 2−0 "√ # x2 + 1 + 3x ❖s ❧✐♠✐t❡s sã♦ ❞✐❢❡r❡♥t❡s❀ ♣♦rt❛♥t♦✱ ♥ã♦ ❡①✐st❡ lim ✳ x→∞ 2x − 5 "√ ❡♥tã♦✿ ❊①❡♠♣❧♦ ✸✳✸✷✳ ❈❛❧❝✉❧❛r✿ √ lim [ 4x2 + 3x − 1 + 2x] x→−∞ ✳ ❙♦❧✉çã♦✳ √  √ 2  ( 4x + 3x − 1 + 2x)( 4x2 + 3x − 1 − 2x) 2 √ lim [ 4x + 3x − 1 + 2x] = lim = x→−∞ x→−∞ 4x2 + 3x − 1 − 2x √ ✶✽✽ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦    x(3 − x−1 ) 4x2 + 3x − 1 − 4x2 √ = lim = lim √ x→−∞ | x | x→−∞ 4x2 + 3x − 1 − 2x 4 + 3x−1 − x−2 − 2x   3 x(3 − x−1 ) √ =− lim x→−∞ x(− 4 + 3x−1 − x−2 − 2) 4 √ 3 P♦rt❛♥t♦✱ lim [ 4x2 + 3x − 1 + 2x] = − ✳ x→−∞ 4  ❊①❡♠♣❧♦ ✸✳✸✸✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞♦ s❡❣✉✐♥t❡ ❧✐♠✐t❡✿ lim [5 + x→−∞ √ 4x2 − x + 3 + 2x]✳ ❙♦❧✉çã♦✳ lim [5 + √ x→−∞ √ 4x2 − x + 3 + 2x] = lim [ 4x2 − x + 3 + 2x + 5] = x→−∞ √ = lim [ 4x2 − x + 3 + 2x] + lim 5 x→−∞ x→−∞ √   √ 2 ( 4x − x + 3 + 2x)( 4x2 − x + 3 − 2x) √ + lim 5 = = lim x→−∞ x→−∞ 4x2 − x + 3 − 2x  2    4x − x + 3 − 4x2 x(−1 + 3x−1 ) √ lim √ + 5 = lim +5= x→−∞ x→−∞ |x| 4 − x−1 + 3x−2 − 2x 4x2 − x + 3 − 2x   −1 21 x(−1 + 3x−1 ) √ +5= +5= lim −1 −2 x→−∞ −x( 4 − x −4 4 + 3x − 2) P♦rt❛♥t♦✱ lim [5 + x→−∞ √ 4x2 − x + 3 + 2x] = 21 ✳ 4 ❊①❡♠♣❧♦ ✸✳✸✹✳ ❉❡t❡r♠✐♥❡ ♦ ❧✐♠✐t❡ ❞❛s s❡❣✉✐♥t❡s s❡q✉ê♥❝✐❛s✿ ❛✮ ❜✮ ❝✮ 1 1 1 (−1)n−1 1, − , , − , · · · , ··· 2 3 4 n 2n 4 6 8 , , , ··· , ··· 3 5 7 2n − 1 q p √ p √ √ 2, 2 2, 2 2 2 · · · 2, ❙♦❧✉çã♦✳ ❛✮ ❖ t❡r♠♦ ❣❡r❛❧ ❞❛ s❡q✉ê♥❝✐❛ ❡st❛ ❞❛❞♦ ♣♦r n ♣❛r r❡s✉❧t❛ sn = 1 (−1)n−1 = lim = 0❀ n→+∞ n n→+∞ n lim −1 = 0✳ n→+∞ n (−1)n−1 , n ♣❛r❛ ♦ ❝❛s♦ ∀ n ∈ N, n í♠♣❛r n > 1✱ ❧♦❣♦ s❡ (−1)n−1 = n→+∞ n lim lim P♦rt❛♥t♦✱ (−1)n−1 =0 lim n→+∞ n ✶✽✾ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 2n ✱ ❝❛❧❝✉❧❛♥❞♦ ♦ ❧✐♠✐t❡ t❡♠♦s✿ 2n − 1 2n 2 2n lim = lim =1 = 1✳ P♦rt❛♥t♦ lim 1 n→+∞ 2n − 1 n→+∞ n→+∞ 2n − 1 2− n ❜✮ ❖❜s❡r✈❡ q✉❡ ♦ t❡r♠♦ ❣❡r❛❧ ❞❛ s❡q✉ê♥❝✐❛ é✿ an = ❝✮ ❱❡r✐✜❝❛r q✉❡ ♦ ❧✐♠✐t❡ é 2✱ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❊①❡♠♣❧♦ ✸✳✸✺✳ ▼♦str❡ q✉❡ ❉❡♠♦♥str❛çã♦✳ 1 lim+ .f ( ) = lim .f (x)✳ x→+∞ x→0 x ❙❡❥❛ L = lim .f (x)✱ ❡♥tã♦ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡st❡ ❧✐♠✐t❡ t❡♠♦s x→+∞ ∀ ǫ > 0, ∃N > 0 /. |f (x) − L| < ǫ s❡♠♣r❡ q✉❡ x > N 1 1 1 1 = M > 0✱ s❡ 0 < < x✱ ❡♥tã♦ 0 < < M ✱ ❝♦♠ ✐st♦ |f ( ) − L| < N M x x 1 ǫ✱ s❡♠♣r❡ q✉❡ 0 < x✱ ❛ss✐♠ lim+ .f ( ) = L✳ x→0 x 1 P♦rt❛♥t♦✱ lim+ .f ( ) = lim .f (x)✳ x→+∞ x→0 x P♦❞❡♠♦s s✉♣♦r ✶✾✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✸✲✸ ✶✳ ❈❛❧❝✉❧❛r ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ √ √  3   5x + 3 − 3x + 1 x 2 − a2 √ 1. lim a>0 2. lim x→1 x→a 2x2 − ax − a2 √ √  3 x −23x + 2   5 x+1−36x+1+2 x − 2x − 4x + 8 √ √ 4. lim 18 3. lim 25 x→0 x→−2 3x2 + 3x − 6 x + 1 + x+1−2 # " √ √   5 3 3 3 3 5 |x −1| h +1+ h +1+h −2 √ 5. lim 6. lim x→1 | x − 1 | + | x − 1 |2 h→0 h − h h2 + 1 " # √   2− x−1 1 − x2 p 0 < a 6= 1 7. lim 8. lim √ x→5 1 − 3 3 − x→1 (1 − ax)2 − (a − x)2 x − 1# " √ √  √ 2 x2 + 3 x − 3 − 9 x − 2x + 6 − x2 + 2x − 6 10. lim 9. lim p √ 3 x→3 x→3 x2 − 4x + 3 9 − x 4x − 3 q   √ √ # " p√ a+b 3 a + x + b + x − 2 x + −9x + 1 − 2 2  √ √ 12. lim  11. lim √ 3 x→−3 a→b 2 − x + 11 a+x− b+x q  √ 2 2 − 2x − b x + 2ax + a + 3 x3 + a−b 3  b > 0, a > 0✳ √ ✶✸✳ lim  √ a→b a+x− x+b √  √ x+a+b− a+b a > 0, b > 0 ✶✹✳ lim x→0 x ✷✳ ❙✉♣♦♥❤❛ lim− f (x) < lim+ f (x) ✭❝♦♥str✉✐r ♦ ❣rá✜❝♦✮ ▼♦str❡ q✉❡ ❡①✐st❡ ❛❧❣✉♠ δ > 0 x→a x→a t❛❧ q✉❡ f (x) < f (y) s❡♠♣r❡ q✉❡ x < a < y, | x − a |< δ ❡ | y − a |< δ ✳ ❈✉♠♣r❡✲s❡ ❛ r❡❝í♣r♦❝❛❄ ✸✳ ❱❡r✐✜q✉❡ ♦ ❝á❧❝✉❧♦ ❞♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ ✶✳ ✷✳ ✸✳ ✹✳  √  5x − 10  lim  √ √ x→20 2 5− x ! r q 8x 5    2− 1  √  400 − x2  = − 20 5   √ n x−1−1 6m 4 − x2 √ √ = ; m, n ∈ N, n 6= 0 lim m 2 x→2 n x−1−1 3− x +5 q  p q a−b b−a 3 2 2 3 3 x + 2 − x + 3 − 2x − (b − a) (b + x)2 (9x + 4)   √ = lim √ 3 a→b 12x2 a+x− 3x+b   p√ 2 | x − 3 | +26 | x + 3 | −26 3x + 33  q lim  = −69 2 x→3 4 − 2 3 x +15x−6 x+3  ✶✾✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✺✳ lim x→−5 " k 1 5 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R # p √ 3 100x + 2sgn(16 − x4 ) k + 3 x2 + 2 + x + 4 272 p = √ 189 x2 + −5x + 6 − 6 ✹✳ ❉❛r ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ♠♦♥ót♦♥❛ t❛❧ q✉❡ lim f (x) = 1✳ x→∞ ✺✳ P❛r❛ ❝❛❞❛ ✉♠ ❞♦s s❡❣✉✐♥t❡s ❡①❡r❝í❝✐♦s✱ tr❛ç❛r ♦ ❣rá✜❝♦ ❡ ❝❛❧❝✉❧❛r ♦ ❧✐♠✐t❡ ✐♥❞✐❝❛❞♦ ❝❛s♦ ❡①✐st❛❀ ❥✉st✐✜❝❛r s✉❛ r❡s♣♦st❛✳ x+ | 2 − x | 2 ( x −4 x2 , s❡✱ x ≤ 2 f (x) = 8 − 2x, s❡✱ x > 2  3 x − 2x2 − 5x + 6   , s❡✱ x < 3 x−3 √ f (x) =   x + 1 − 1, s❡✱ x ≥ 3 x+2  2   3 + x , s❡✱ x < 0 f (x) = 0, s❡✱ x = 0   2 11 − x , s❡✱ x > 0  x−5  √ , s❡✱ x ≥ 5  1 − x−4 f (x) = 2   x − 12x + 35 , s❡✱ <5 x−5  2  s❡✱ x < 2  6x − x , f (x) = 6, s❡✱ x = 2   2 2x − x − 3, s❡✱ x > 2  2  s❡✱ x < 1  1−x , f (x) = 1, s❡✱ 1 < x ≤ 2   | x − 3 |, s❡✱ x > 2 1. f (x) = 2. 3. 4. 5. 6. 7. lim .f (x) x→2+ lim .f (x) x→2− lim .f (x) x→2 lim .f (x) x→3 lim .f (x) x→0 lim .f (x) x→5 lim .f (x) x→2 lim .f (x) x→1 lim .f (x) x→2 ✻✳ ◆♦s s❡❣✉✐♥t❡s ❡①❡r❝í❝✐♦s ❞❡t❡r♠✐♥❡ s❡ ❡①✐st❡ ♦ ❧✐♠✐t❡❀ ❝❛s♦ ❝♦♥trár✐♦ ❥✉st✐✜❝❛r s✉❛ r❡s♣♦st❛✳ ✶✳ lim5 x→ 3 ✸✳ ✺✳ ✼✳ lim x→0 p | x | +[|3x|] + 4 p [|9 + x2 |] x3 − x2 + 3x − 3 x→1 x−1 x3 − 2x2 − 4x + 8 lim x→2 |x−2| lim ✷✳ ✹✳ ✻✳ ✽✳ p | x | +[|3x|] + 4 x→ 2 √ √ x+ x−1−1 √ lim x→1+ x2 − 1 3 2 x − x + 3x − 3 lim x→1 |x−1| x2 + [| x3 |] lim x→ 61 [|3x|] − 10 lim5 ✶✾✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✾✳ ✶✶✳ ✶✸✳ ✶✺✳ ✶✻✳ ✶✼✳ ✶✽✳ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ x2 + [| x3 |] lim x→6 [|2x|] + 10 ✶✵✳ lim [3x + sgn(| x2 − 1 | −1)] ✶✷✳ x→0 lim √ x→−1+ lim √ [x x→ 2 2 R √ −9x + 3 x − 2 x+1 − sgn(| x2 − 1 | −1)] 2 2 4 2 lim ✶✹✳ lim √ [x − sgn(| x − 1 | −1)] √ [x + 5 + sgn(| x − 1 | −1)] x→ 2 x→ 2 " √ # √ √ √ 9 36 3 x−1− x−1 x −1+ x−1 √ √ − lim 36 x→1+ 3x2 − 3 + x−1 x2 − 1 √ √ √ 5 5 x − 2 + 3 3 2 − x + 2 2x − 1 + 6x2 − 6 lim− 2 x→1 x √ √ x − √ √ 3 3 2 2 x − x + x + 3 x − 3x lim+ x→1 (x − 1)2 √ √ √ √ 5 5 x + 2 + 4 4 −1 − 2x + 3 3 2 + x − 2 −1 − 2x + 5x + 3 lim x→1− x2 − x ✼✳ ❈❛❧❝✉❧❛r s❡ ❡①✐st❡♠ ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ 1. 4. 7. 10. n3 − 100n2 + 1 lim n→∞ 100n2 + 15n n lim n→∞ n − 1 (n + 1)4 − (n − 1)4 lim n→∞ (n + 1)4 + (n − 1)4 √ 5 n3 + 2n − 1 lim n→∞ n+2 2. 5. 8. 11. √ n 2−1 lim √ n n→∞ 2+1 (2n + 1)2 lim n→∞ 2n2 (n + 1)2 lim n→∞ 2n2 n3 lim +n n→∞ n2 + 1 3. 6. 9. 12. 2n − 1 n→∞ 2n + 1 n+1 lim n→∞ n 2 n −1 lim n→∞ 2n2 + 1 n2 + 5 lim n→∞ n2 − 3 lim ✽✳ ❉❡♠♦♥str❛r q✉❡✿ ✶✳ ✸✳ lim .f (x) = lim− .f (−x) x→0+ ✷✳ x→0 lim f (| x |) = lim+ .f (x) x→0 x→0 lim .f (x2 ) = lim+ .f (x) x→0 x→0 ✾✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞♦s ❧✐♠✐t❡s✱ ❝❛s♦ ❡①✐st❛✿ 1. 3. 5. 7. 9. (n + 2)✦ + (n + 1)✦ lim n→∞ (n + 3)✦ (2n + 1)4 − (n − 1)4 lim n→∞ (2n + 1)4 + (n − 1)4 (n + 1)3 − (n − 1)3 n→∞ (n + 1)2 + (n − 1)2 n3 + n lim 4 n→∞ n − 3n2 + 1 n2 − 2n + 1 lim n→1 n3 − n lim 2. 4. 6. 8. 10.   1 + (1 + 2 + 3 + · · · + n) lim n→∞ n2   1 + 2 + 3 + ··· + n n lim − n→∞ n+2 2 √ √ 3 3 4 n − 2n + 1 + n + 1 √ lim √ 4 n→∞ n6 + 6n5 + 2 − 5 n7 + 3n3 + 1 1 1 1 lim + + ··· n→∞ 1 × 3 3×5 (2n − 1)(2n + 1) n−4 n+2 lim 2 + n→1 n − 5n + 4 3(n2 − 3n + 2) ✶✾✸ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 11. 13. ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ xm − 1 m, n ∈ Z x→1 xn − 1 100n3 + 3n2 lim n→∞ 0, 001n4 − 100n3 + 1 lim R (2n + 1)(3n2 + n + 2) 3n2 − n→∞ 2n + 1 4n2 1 1 − lim n→2 n(n − 2)2 n2 − 3n + 2 lim 12. 14. ✶✵✳ ❙❡ f é ✉♠❛ ❢✉♥çã♦ ❧✐♠✐t❛❞❛ ❡♠ ✐♥t❡r✈❛❧♦s ❧✐♠✐t❛❞♦s✳ ▼♦str❡ q✉❡✿ lim [f (x + 1) − f (x)] x→∞ f (x) x→∞ x ⇒ lim ✶✶✳ ❱❡r✐✜❝❛r ♦ ✈❛❧♦r ❞♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ 1. 3. 5. .7 9. ✶✶✳ ✶✷✳ 4n3 + 2n2 − 5 1 =− 2. 3 n→+∞ n + 2 − 8n 2 3n2 − 2 n2 − 4n + =∞ 4. lim n→+∞ 2n + 1 n−3 s 8n − 4 √ √ = −2 6. lim 3 n→+∞ (3 − n)( n + 2) √ 5 8. lim [ n2 − 5n + 6 − n] = − n→+∞ 2 q √ lim [ n 2n − 5n + 6 − n] = −∞ 10. lim n→+∞ r √ √ 5n3 − n2 + n − 1 =0 n→−∞ n4 − n3 − 2n + 1 2n + 3 √ =2 lim n→+∞ n + 3 n q p √ n+ n+ n+3 √ lim =1 n→+∞ n+3 √ lim [ n2 − 2n + 4 + n] = 1 n→−∞ √ ( n2 + 1 + n)2 √ lim =4 3 n→∞ n6 + 1 lim a+b =0 a + a2 n2 + b + a2 n2 − 2 a2 n2 − n→+∞ 2 √ √ 7 a7 n 7 + a + n 2 − 4 1+a √ lim √ = 5 4 n→+∞ 1−a a − 1 − a5 n5 + n4 − 25a2 + 144 lim ✶✷✳ ▼♦str❡ q✉❡ ✶✸✳ ▼♦str❡ q✉❡ lim .f (x) = lim .f (−x)✳ x→+∞ ✶✳ x→−∞ 1 lim− .f ( ) = lim .f (x) x→−∞ x→0 x ✷✳ 1 lim+ .f ( ) = lim .f (x)✳ x→+∞ x→0 x an xn + an−1 xn−1 + · · · a1 x + a0 ❡①✐st❡ s❡✱ ❡ s♦♠❡♥t❡ s❡ m ≥ n✳ x→+∞ bm xm + bm−1 xm−1 + · · · b1 x + b0 ◗✉❛❧ é ♦ ✈❛❧♦r ❞♦ ❧✐♠✐t❡ s❡ m = n❄✳ ❊ q✉❛♥❞♦ m < n ❄ ✶✹✳ ▼♦str❡ q✉❡ lim ✶✺✳ ❈❛❧❝✉❧❛r ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ ✶✳ ✷✳ ✸✳ x3 x2 − x→+∞ 2x2 − 1 2x + 1 n an x + an−1 xn−1 + · · · a1 x + a0 lim x→+∞ bm xm + bm−1 xm−1 + · · · b1 x + b0 (x + 1) + (x + 2)2 + (x + 3)3 + · · · + (x + n)n lim x→+∞ xn − nn lim ✶✾✹ n ∈ N✳ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✸✳✺ ▲✐♠✐t❡s ✐♥✜♥✐t♦s 1 ❖❜s❡r✈❡ q✉❡ ❛ ♠❡s♠❛ ❢✉♥çã♦ ❞❛ s❡çã♦ ❛♥t❡r✐♦r f (x) = 2 ✱ ❡stá ❞❡✜♥✐❞❛ ❞❡ t❛❧ ♠♦❞♦ x q✉❡ ♦s ✈❛❧♦r❡s f (x) ✜❝❛♠ ❛r❜✐tr❛r✐❛♠❡♥t❡ ❣r❛♥❞❡s✱ ❝♦♥s✐❞❡r❛♥❞♦ x ♠❛✐s ❡ ♠❛✐s ♣ró①✐♠♦ ❞❡ 0✳ ❆ss✐♠✱ f ♥ã♦ é ❧♦❝❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡♠ x = 0✱ ❡♠❜♦r❛ ♥ã♦ ❡①✐st❛ ♦ ❧✐♠✐t❡ ❞❡ f ❡♠ x = 0✱ ❡ ✐st♦ ❞❡✈❡ ✜❝❛r ❝❧❛r♦✱ ♣♦✐s ♥ã♦ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ L ∈ R ♥❛s ❝♦♥❞✐çõ❡s ❞❛ 1 ❉❡✜♥✐çã♦ ✭✸✳✷✮ ❞❡ ❧✐♠✐t❡s✳ ◆❡st❛ s✐t✉❛çã♦ s❡ ❡s❝r❡✈❡ lim 2 = ∞✳ x→0 x ❙❡❥❛ f ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♥✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I q✉❡ ❝♦♥t❡♥❤❛ ❛♦ ♥ú♠❡r♦ a✱ ♣♦❞❡♥❞♦ ♦ ♥ú♠❡r♦ a ♥ã♦ ❡st❛r ♥♦ ❞♦♠í♥✐♦ ❞❡ f ✳ ❉❡✜♥✐çã♦ ✸✳✼✳ ❉✐③❡♠♦s q✉❡ ♦ ❧✐♠✐t❡ ❞❡ f (x) é +∞ q✉❛♥❞♦ x t❡♥❞❡ ❛♦ ♣♦♥t♦ a ❡ ❡s❝r❡✈❡♠♦s lim .f (x) = +∞❀ s❡✱ ❞❛❞♦ K > 0 ✭tã♦ ❣r❛♥❞❡ ❝♦♠♦ q✉✐s❡r✮✱ ❡①✐st❡ δ > 0 t❛❧ q✉❡ x→a 0 <| x − a |< δ ✐♠♣❧✐❝❛ f (x) > K ✳ ❉❡✜♥✐çã♦ ✸✳✽✳ f (x) é −∞ q✉❛♥❞♦ x t❡♥❞❡ ❛♦ ♣♦♥t♦ a ❡ ❡s❝r❡✈❡♠♦s lim .f (x) = −∞❀ s❡✱ ❞❛❞♦ K > 0 ✭tã♦ ❣r❛♥❞❡ ❝♦♠♦ q✉✐s❡r✮✱ ❡①✐st❡ δ > 0 t❛❧ q✉❡ x→a 0 <| x − a |< δ ✐♠♣❧✐❝❛ f (x) < −K ✳ ❉✐③❡♠♦s q✉❡ ♦ ❧✐♠✐t❡ ❞❡ Pr♦♣r✐❡❞❛❞❡ ✸✳✶✻✳ 1 ✐✮ lim+ = +∞ x→0 x ✐✐✮ lim− x→0 1 = −∞ x ❉❡♠♦♥str❛çã♦✳ ✐✮ P❛r❛ q✉❛❧q✉❡r K > 0✱ ❡①✐st❡ δ = P♦rt❛♥t♦ lim+ x→0 1 = +∞✳ x ✐✐✮ P❛r❛ q✉❛❧q✉❡r K > 0✱ ❡①✐st❡ δ = P♦rt❛♥t♦ lim− x→0 1 = −∞✳ x 1 1 1 > 0 t❛❧ q✉❡ 0 < x < δ = ❀ ❡♥tã♦ > K ✳ K K x 1 1 1 > 0 t❛❧ q✉❡ −δ = − < x < 0❀ ❡♥tã♦ < −K ✳ K K x ❖s ❞♦✐s ❧✐♠✐t❡s sã♦ ❞❡♥♦t❛❞♦s ♣♦r 1 1 = +∞ ❡ − = −∞ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ + 0 0 Pr♦♣r✐❡❞❛❞❡ ✸✳✶✼✳ ❙❡ n ✐✮ é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ ❡♥tã♦✿ 1 lim+ n = +∞ x→0 x ✐✐✮ ✶✾✺ 1 lim− n = x→0 x ( +∞, −∞, s❡✱ s❡✱ n n é ♣❛r é í♠♣❛r 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❉❡✜♥✐çã♦ ✸✳✾✳ ❙❡❥❛ f ✉♠❛ ❢✉♥çã♦ ❞❡ ❞♦♠í♥✐♦ D(f )✳ ❊♥tã♦✿ ✐✮ ❙❡ D(f ) = (a, +∞) ❞❡✜♥❡✲s❡✿ ❛✮ x→+∞ lim .f (x) = +∞ ⇔ ∀ K > 0, ∃M > 0 t❛❧ q✉❡ x > M ❜✮ x→+∞ lim .f (x) = −∞ ⇔ ∀ K > 0, ∃M > 0 t❛❧ q✉❡ x > M ✐✐✮ ❙❡ D(f ) = (−∞, b) ❞❡✜♥❡✲s❡✿ ❛✮ x→−∞ lim f (x) = +∞ ⇔ ∀ K > 0, ∃M > 0 t❛❧ q✉❡ x < −M ❜✮ x→−∞ lim f (x) = −∞ ⇔ ∀ K > 0, ∃M > 0 t❛❧ q✉❡ x < −M ⇒ f (x) > K ✳ ⇒ f (x) < −K ✳ f (x) > K ✳ ⇒ f (x) < −K ✳ ⇒ ❆ ❉❡✜♥✐çã♦ ✭✸✳✾✮ ✐✮✲❛✮ s✐❣♥✐✜❝❛ q✉❡ ♣❛r❛ ✈❛❧♦r❡s ❞❡ x ♣♦s✐t✐✈♦s ♠✉✐t♦ ❣r❛♥❞❡s✱ ♦s ✈❛❧♦r❡s ❞❡ f (x) t❛♠❜é♠ sã♦ ♣♦s✐t✐✈♦s ❡ ♠✉✐t♦ ❣r❛♥❞❡s✳ ❙✐♠✐❧❛r ✐♥t❡r♣r❡t❛çã♦ ♣❛r❛ ❛s ♦✉tr❛s ❞❡✜♥✐çõ❡s✳ ❊①❡♠♣❧♦ ✸✳✸✻✳ ▼♦str❡ q✉❡ ❙♦❧✉çã♦✳ lim x2 = +∞✳ x→+∞ ❙❡❥❛ K > 0✱ ❝♦♥s✐❞❡r❛♥❞♦ M = ❊①❡♠♣❧♦ ✸✳✸✼✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞♦ ❧✐♠✐t❡✿ ❙♦❧✉çã♦✳ √ K t❡♠♦s✱ s❡ x > √ K ⇒ x2 > K ✳ √ 1+ x lim x→2+ x − 2 √ √ √ 1+ x 1 = lim+ (1 + x) · = (1 + 2)(+∞) = +∞✳ lim+ x→2 √ x→2 x−2 x−2 1+ x = +∞✳ P♦rt❛♥t♦✱ lim+ x→2 x−2 ❖❜s❡r✈❛çã♦ ✸✳✻✳ P♦r ❝♦♠♦❞✐❞❛❞❡ ❡s❝r❡✈❡♠♦s ♦ sí♠❜♦❧♦ ∞ ✭✐♥✜♥✐t♦✮ ❝♦♠ ♦ s✐❣♥✐✜❝❛❞♦ s❡❣✉✐♥t❡✿ lim .f (x) = x→a ∞ s❡✱ ❡ s♦♠❡♥t❡ s❡ lim .|f (x)| = +∞✳ x→a Pr♦♣r✐❡❞❛❞❡ ✸✳✶✽✳ ❙❡❥❛♠ a ∈ R ❛s ❢✉♥çõ❡s f (x), g(x) ❡ C 6= 0 ♥ú♠❡r♦ r❡❛❧ ✜①♦✱ t❛✐s q✉❡ lim .f (x) = 0 x→a ❡ lim .g(x) = C ❡♥tã♦✿ x→a ✐✮ ❙❡ C > 0 ❡ f (x) → 0 ❛tr❛✈és ❞♦s ✈❛❧♦r❡s ♣♦s✐t✐✈♦s ❞❡ f (x)✱ ❡♥tã♦ x→a lim ✶✾✻ g(x) = +∞✳ f (x) 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ g(x) = −∞✳ x→a f (x) ✐✐✮ ❙❡ C > 0 ❡ f (x) → 0 ❛tr❛✈és ❞♦s ✈❛❧♦r❡s ♥❡❣❛t✐✈♦s ❞❡ f (x)✱ ❡♥tã♦ lim g(x) = −∞✳ x→a f (x) ✐✐✐✮ ❙❡ C < 0 ❡ f (x) → 0 ❛tr❛✈és ❞♦s ✈❛❧♦r❡s ♣♦s✐t✐✈♦s ❞❡ f (x)✱ ❡♥tã♦ lim g(x) = +∞✳ x→a f (x) ✐✈✮ ❙❡ C < 0 ❡ f (x) → 0 ❛tr❛✈és ❞♦s ✈❛❧♦r❡s ♥❡❣❛t✐✈♦s ❞❡ f (x)✱ ❡♥tã♦ lim ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❆ Pr♦♣r✐❡❞❛❞❡ C = 0+ ✐✮ (  ✭✸✳✶✽✮ ♣♦❞❡♠♦s r❡s✉♠✐r ❞♦ ♠♦❞♦ s❡❣✉✐♥t❡✿ +∞, −∞, C>0 C<0 s❡✱ s❡✱ C = 0− ✐✐✮ ( +∞, −∞, s❡✱ s❡✱ C<0 C>0 Pr♦♣r✐❡❞❛❞❡ ✸✳✶✾✳ ❙❡❥❛♠ ❛✮ f ❡ g ❞✉❛s ❢✉♥çõ❡s r❡❛✐s t❛✐s q✉❡✿ lim .f (x) = ±∞ x→±∞ ❡ lim .g(x) = ±∞ x→±∞ lim [f (x) · g(x)] = +∞ lim [f (x) + g(x)] = ±∞ ❡♥tã♦✿ x→±∞ ❡ x→±∞ ❜✮ lim .f (x) = ±∞, L>0 x→±∞ ❡ ❝✮ lim [f (x) · g(x)] = +∞ ❞✮ lim .f (x) = ±∞, ❡♥tã♦✿ lim .g(x) = ±∞ ❡♥tã♦✿ lim [f (x)+g(x)] = ±∞ x→±∞ L<0 ❡ lim [f (x) · g(x)] = ±∞ x→±∞ lim [f (x)+g(x)] = ±∞ x→±∞ x→±∞ lim .f (x) = −∞, x→±∞ ❡✮ lim .g(x) = ±∞ x→±∞ x→±∞ x→±∞ ❡ ❡ lim .f (x) = L, x→±∞ ❡ lim .g(x) = +∞ x→±∞ L 6= 0 ❡ ❡♥tã♦✿ lim .g(x) = ±∞ x→±∞ lim [f (x) · g(x)] = −∞ x→±∞ ❡♥tã♦✿ f (x) = 0✳ x→±∞ g(x) lim ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❆♦ s✉❜st✐t✉✐r ❛ ❡①♣r❡ssã♦ x→a ❆ ❡st❛s ♣r♦♣r✐❡❞❛❞❡s ♣❡r♠❛♥❡❝❡♠ ✈á❧✐❞❛s✳ Pr♦♣r✐❡❞❛❞❡ x → ±∞ ♣♦r  ✭✸✳✶✾✮ ♣♦❞❡♠♦s r❡s✉♠✐r✱ ✉s❛♥❞♦ ♦s s❡❣✉✐♥t❡s sí♠❜♦❧♦s ♣❛r❛ K ❝♦♥st❛♥t❡ ✶✾✼ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳ i) K + (+∞) = +∞ ii) K + (−∞) = −∞ iii (+∞) + (+∞) = +∞ iv) (−∞) + (−∞) = −∞ v) (+∞) · (+∞) = +∞ vii) (+∞) · (−∞) = −∞ ( +∞, ix) K · (+∞) = −∞, ( +∞, xi) K · (−∞) = −∞, vi) (−∞) · (−∞) = +∞ K =0 viii) ±∞ ( +∞, s❡✱ n ∈ N é ♣❛r x) (−∞)n = −∞, s❡✱ n ∈ N é í♠♣❛r s❡✱ K > 0 s❡✱ K < 0 s❡✱ K < 0 s❡✱ K > 0 ❊①❡♠♣❧♦ ✸✳✸✽✳ ❙❡❥❛ f (x) = ❙♦❧✉çã♦✳ 5x4 + 1 ✱ x2 + x − 2 ❝❛❧❝✉❧❛r lim .f (x), lim+ .f (x) x→1− x→1 ❡ lim .f (x)✳ x→1 6 ❆♦ s✉❜st✐t✉✐r♠♦s x = 1 ❡♠ f (x)✱ ♦❜s❡r✈❛♠♦s q✉❡ t❡♠♦s ❛ ❢♦r♠❛ ♦ q✉❛❧ ✐♥❞✐❝❛ q✉❡ 0 ♦ ❝á❧❝✉❧♦ ❞♦s três ❧✐♠✐t❡s é ✐♥✜♥✐t♦✳ P❛r❛ ❞❡t❡r♠✐♥❛r ♦ s✐♥❛❧ ❞❡ ∞(+∞ ♦✉ −∞) ❞❡✈❡♠♦s ❝❛❧❝✉❧❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❢✉♥çã♦ ♣❛r❛ ✈❛❧♦r❡s ♣ró①✐♠♦s ❛ x = 1✳ ✐✮ ✐✐✮ lim [5x4 + 1] = 6 x→1 lim [x2 + x − 2] = 0 x→1 P❛r❛ x < 1 ✭♣ró①✐♠♦ ❛ 1✮ t❡♠♦s (x − 1) < 0 ❡ (x + 2) > 0❀ ❧♦❣♦ ♦ ♣r♦❞✉t♦ (x − 1).(x + 2) < 0✱ ❛ss✐♠ lim− (x + 2)(x − 1) = 0− ✳ x→1 ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣❛r❛ x > 1 ✭♣ró①✐♠♦ ❛ 1✮ t❡♠♦s (x − 1) > 0 ❡ (x + 2) > 0❀ ❧♦❣♦ ♦ ♣r♦❞✉t♦ (x − 1).(x + 2) > 0✱ ❛ss✐♠ lim+ (x + 2)(x − 1) = 0+ ✳ x→1 ❊♥tã♦✿ ❛✮ lim .f (x) = lim x→1− x→1− ❜✮ lim .f (x) = lim x→1+ x→1+ ❝✮ lim .f (x) = lim x→1 x→1      6 5x4 + 1 = lim− − = −∞✳ 2 x→1 x +x−2 0    6 5x4 + 1 = lim+ + = +∞✳ 2 x→1 x +x−2 0 5x4 + 1 = +∞✳ ❡♥tã♦ lim .f (x) = +∞✳ x→1 x2 + x − 2 ❊①❡♠♣❧♦ ✸✳✸✾✳ ❈❛❧❝✉❧❛r ♦ ❧✐♠✐t❡ lim x→3−  3x + 1 2 x −x−6  ✳ ✶✾✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❙♦❧✉çã♦✳ lim x→3−       3x + 1 10 3x + 1 = lim− = lim− = −∞ x→3 x→3 x2 − x − 6 (x − 3)(x + 2) (x − 3) · 5 ❊①❡♠♣❧♦ ✸✳✹✵✳ lim ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞♦ s❡❣✉✐♥t❡ ❧✐♠✐t❡✿ x→−3 ❙♦❧✉çã♦✳  −5x − 81 (x + 3)(x − 1)  ✳ ❈❛❧❝✉❧❡♠♦s ♦s ❧✐♠✐t❡s ❧❛t❡r❛✐s✿ lim x→−3+        −5x − 81 1 −96 −5x − 81 = lim + = .(+∞) = (+∞) x→−3 (x + 3)(x − 1) x−1 x+3 −4       1 −96 −5x − 81 −5x − 81 lim = lim − = .(−∞) = (−∞) x→−3− (x + 3)(x − 1) x→−3 x−1 x+3 −4   −5x − 81 = ∞✳ P♦rt❛♥t♦✱ lim x→−3 (x + 3)(x − 1)  ❊①❡♠♣❧♦ ✸✳✹✶✳  x2 − 5x + 4 lim √ x→3 x2 − 5x + 6 ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞♦ s❡❣✉✐♥t❡ ❧✐♠✐t❡✿ ❙♦❧✉çã♦✳ ◆♦ ❝á❧❝✉❧♦ ❞❡ ❧✐♠✐t❡s ❧❛t❡r❛✐s q✉❛♥❞♦ lim x→3+  lim ✳ t❡♠♦s✿    1 x2 − 5x + 4 2 √ = lim+ [x − 5x + 4] √ = (−2).(+∞) = −∞ x→3 x2 − 5x + 6 x2 − 5x + 6 ◗✉❛♥❞♦ x→3− x → 3+   x → 3− t❡♠♦s✿    x2 − 5x + 4 1 1 2 √ = lim+ [x − 5x + 4] √ = (−2).( √ ) = ∄ 2 2 x→3 x − 5x + 6 x − 5x + 6 0−   x2 − 5x + 4 lim √ = ∄✱ x→3 x2 − 5x + 6 P♦rt❛♥t♦✱ ✭♥ã♦ ❡①✐st❡✮✳ ❊①❡♠♣❧♦ ✸✳✹✷✳ ❈❛❧❝✉❧❛r✱ P (x) x→±∞ Q(x) lim ♦♥❞❡ P (x) ❡ Q(x) sã♦ ♣♦❧✐♥ô♠✐♦s ❞❡ ❣r❛✉ n ❡ m r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙♦❧✉çã♦✳   P (x) a0 xn + a1 xn−1 + a2 xn−2 + · · · + an−1 x + an = lim = lim x→±∞ Q(x) x→±∞ b0 xm + b1 xm−1 + b2 xm−2 + · · · + bm−1 x + bm = lim x→±∞ " xn (a0 + xm (b0 + a1 x b1 x + + a2 x2 an−1 + xann ) xn−1 bm + xbm−1 m−1 + xm + ··· + bm−2 x2 + ··· ✶✾✾ # = lim x→±∞  a0 xn b0 x m  ⇒ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠    ∞, P (x)  a0 , = lim b0 x→±∞ Q(x)    0, ❊①❡♠♣❧♦ ✸✳✹✸✳ ❈❛❧❝✉❧❛r ♦ ❧✐♠✐t❡ ❙♦❧✉çã♦✳ P♦rt❛♥t♦✱ lim x→+∞  6x3 − 2x + 1 5x2 − 3  s❡✱ n>m s❡✱ n=m s❡✱ n<m R ✳   3  2 + x13 ) x (6 − x2 6x3 − 2x + 1 = = lim lim x→+∞ x→+∞ 5x2 − 3 x2 (5 − x32 )   x(6 − 0 + 0) +∞ lim = = +∞ x→+∞ (5 − 0) 5   3 6x − 2x + 1 = +∞✳ lim x→+∞ 5x2 − 3  ❊①❡♠♣❧♦ ✸✳✹✹✳ ❈❛❧❝✉❧❛r ♦ ❧✐♠✐t❡ lim x→2+ √ 3 8 − x3 x2 − 4  ✳ ❙♦❧✉çã♦✳ lim x→2+ s s  3 2 8 − x3 8 − x 3 (2 − x)(4 + 2x + x ) 3 = lim = lim = x→2+ x2 − 4 (x2 − 4)3 x→2+ (x + 2)3 (x − 2)3 √ 3 s P♦rt❛♥t♦✱ ✸✳✻ ✸✳✻✳✶ (4 + 2x + x2 ) = lim+ 3 = lim x→2 −(x + 2)3 (2 − x)2 x→2+  √ 3 8 − x3 = −∞✳ lim x→2+ x2 − 4 s 3 12 = −∞ −64(x − 2)2 ▲✐♠✐t❡ ❞❡ ❢✉♥çõ❡s tr❛♥s❝❡♥❞❡♥t❡s ▲✐♠✐t❡s tr✐❣♦♥♦♠étr✐❝♦s P❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ❧✐♠✐t❡s tr✐❣♦♥♦♠étr✐❝♦s ❝♦♥s✐❞❡r❡♠♦s ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✳ Pr♦♣r✐❡❞❛❞❡ ✸✳✷✵✳ 1. 4. lim .senx = 0   tan x =1 lim x→0 x x→0 ❉❡♠♦♥str❛çã♦✳ 2. 5. lim . cos x = 1   1 − cos x lim =0 x→0 x x→0 3. 6. h senx i =1  x  1 − cos x 1 lim = 2 x→0 x 2 lim x→0 ✶✳ ✷✵✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R π ❆ ❢✉♥çã♦ s❡♥♦ ✈❡r✐✜❝❛ | senx |≤| x | ♣❛r❛ t♦❞♦ x ∈ (0, )✳ 2 ▼♦str❛r❡✐ q✉❡✱ ♣❛r❛ t♦❞♦ ε > 0✱ ❡①✐st❡ δ > 0 t❛❧ q✉❡ | senx |< ε s❡♠♣r❡ q✉❡ 0 <| x |< δ ✳ π ❙❡❥❛ ε > 0 q✉❛❧q✉❡r ❡ ❝♦♥s✐❞❡r❡ δ1 = ε ❡ δ = min .{ δ1 , }❀ 2 ❧♦❣♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ 0 <| x |< δ ✈❡r✐✜❝❛✲s❡ q✉❡ | senx |<| x |< δ ≤ ε ❋✐❣✉r❛ ✸✳✽✿ ■st♦ é✱ | senx |< ε✳ P♦rt❛♥t♦ lim .senx = 0✳ x→0 ❉❡♠♦♥str❛çã♦✳ ✷✳ p ❖❜s❡r✈❡ q✉❡✱ lim . cos x = lim . 1 − (senx)2 = x→0 ❉❡♠♦♥str❛çã♦✳ ✸✳ x→0 q 1 − [lim .senx]2 = 1✳ x→0 d ≤ AT ✳ ❉❛ ❋✐❣✉r❛ ✭✸✳✽✮ t❡♠♦s ❛s ❞❡s✐❣✉❛❧❞❛❞❡s✿ BB ′ ≤ ❆r❝♦AC π ❊♥tã♦ senx < x < tan x✱ s❡♥❞♦ ❛ ❢✉♥çã♦ senx ♣♦s✐t✐✈❛ ♥♦ ✐♥t❡r✈❛❧♦ (0, ) t❡♠♦s 2 x 1 senx 1 < < ❧♦❣♦✱ cos x < < 1 ❛♣❧✐❝❛♥❞♦ ♦ ❧✐♠✐t❡✱ ♣❡❧❛ ♣❛rt❡ senx cos x x ✷✳ ❞❡ ❡st❛ ♣r♦♣r✐❡❞❛❞❡ ❡ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ s❛♥❞✉í❝❤❡ s❡❣✉❡✲s❡ q✉❡✿ lim+ x→0 senx =1 x ✭✸✳✻✮ ❙❡❥❛ x = −t✱ ❡♥tã♦ q✉❛♥❞♦ x → 0− t❡♠♦s t → 0+ ✱ ❛ss✐♠✿ lim− x→0 −sent lim+ ✱ ❡♥tã♦✿ t→0 −t senx sen(−t) = lim+ = t→0 x (−t) senx sent = lim+ =1 t→0 x→0 x t senx ❉❡ ✭✸✳✻✮ ❡ ✭✸✳✼✮ s❡❣✉❡✲s❡ q✉❡ lim =1 x→0 x ❉❡♠♦♥str❛çã♦✳ ✹✳ ❉❡♠♦♥str❛çã♦✳ ✺✳ lim− ✭✸✳✼✮ tan x 1 senx senx 1 ❚❡♠♦s lim = lim · = lim · lim =1 x→0 x→0 x→0 x→0 x x cos x x cos x ❉❡ ✐❞❡♥t✐❞❛❞❡s tr✐❣♦♥♦♠étr✐❝❛s t❡♠♦s✿ 1 − cos x 1 + cos x 1 − cos x = lim · = x→0 x→0 x x 1 + cos x lim senx 0 senx sen2 x = lim · = 1. = 0 x→0 x x→0 x(1 + cos x) 1 + cos x 2 lim ❉❡♠♦♥str❛çã♦✳ ✻✳ ✷✵✶ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 1 − cos x 1 + cos x 1 − cos x = lim · = x→0 x→0 x2 x2 1 + cos x lim  h senx i2  1 1 1 sen2 x lim 2 = lim = 1. = x→0 x (1 + cos x) x→0 x 1 + cos x 2 2 ✸✳✻✳✷ ▲✐♠✐t❡s ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ✐♥✈❡rs❛s P❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦s ❧✐♠✐t❡s ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ✐♥✈❡rs❛s✱ é ♥❡❝❡ssár✐♦ ❝♦♥✲ s✐❞❡r❛r ♦s ❧✐♠✐t❡s q✉❡ s❡ ♠❡♥❝✐♦♥❛♠ ♥❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ Pr♦♣r✐❡❞❛❞❡ ✸✳✷✶✳ ❛✮ π lim . arccos x = + x→0 2 arctan x lim =1 x→0 x π lim . arctan x = + x→+∞ 2 ❜✮ lim .arcsenx = 0 x→0 arcsenx ❝✮ lim =1 x→0 x π ❡✮ lim . arctan x = − x→−∞ 2 ❞✮ ❢✮ ❉❡♠♦♥str❛çã♦✳ ❛✮ ❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s✿ π 2 t≤ ▲♦❣♦✱ ❝✮ ✱ ❡♥tã♦ x = sent ✱ s❡ x→0 t❡♠♦s t = arcsenx ♦♥❞❡ −1 ≤ x ≤ 1 ❡ − π ≤ 2 t → 0✳ lim .arcsenx = lim .t = 0. x→0 t→0 ❋❛③❡♥❞♦ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ❝♦♠♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞❛ ♣❛rt❡ ❛✮ t❡♠♦s lim t = 1. t→0 sent x→0 arcsenx = x lim ❊①❡♠♣❧♦ ✸✳✹✺✳ ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳ sen6x ✳ x→0 x lim 6x = t❀ ❡♥tã♦ q✉❛♥❞♦ x → 0✱ t❡r❡♠♦s q✉❡ t = 6x → 6 · sen6x sen6x sent sen6x = lim = 6. lim = 6. lim = 6(1) = 6✳ lim x→0 x→0 6x t→0 t x→0 x 6x ❈♦♥s✐❞❡r❡ ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s 0 ❛ss✐♠✱ ❊①❡♠♣❧♦ ✸✳✹✻✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞♦ ❧✐♠✐t❡ ❙♦❧✉çã♦✳ ❖❜s❡r✈❡ q✉❡ ax → 0 ❡ bx → 0✱ senax x→0 senbx lim ✳ senax a.bx a senax = lim · = lim · lim x→0 senbx x→0 x→0 senbx a.bx b senax ax ✱ q✉❛♥❞♦ senbx bx x → 0 t❡♠♦s ❛ss✐♠ r❡s✉❧t❛ q✉❡✿ a · x→0 b lim senax ax senbx bx h senax ax→0 ax lim a = .h b lim bx→0 ✷✵✷ i a a i = .1 = senbx b b bx 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ P♦rt❛♥t♦✱ a senax = ✳ x→0 senbx b  sen2 (sen3x) lim x→0 1 − cos(sen4x) ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳ R lim ❊①❡♠♣❧♦ ✸✳✹✼✳ ◗✉❛♥❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ x→0 ✈❛r✐á✈❡❧✱ s❡❣✉❡ q✉❡  ✳ t = sen3x → 0 t → 0 ❡ r → 0 ❡♥tã♦✿ t❡♠♦s ❡ r = sen4x → 0❀ ❧♦❣♦ ❢❛③❡♥❞♦ ♠✉❞❛♥ç❛ ❞❡ h i2 2 sen(sen3x) (sen3x) sen3x sen (sen3x) [sen(sen3x)] i= h lim = lim = lim 1−cos(sen4x) x→0 1 − cos(sen4x) x→0 1 − cos(sen4x) x→0 2 (sen 4x) sen2 4x   2 2 h h i2 i2 sent sen3x lim 9 lim 9 3x sen3x 9 9[1]2 [1]2 t→0 t 3x→0 3x h h i= i · = lim  = · 1 = i h  2 2 2 1−cos(sen4x) x→0 r 16[1] 8 lim 1−cos 16 sen4x 2 16 lim sen4x 4x sen2 4x r2 4x r→0  sen3x 2 h sen(sen3x) i2 4x→0 P♦rt❛♥t♦✱ lim x→0   9 sen2 (sen3x) = ✳ 1 − cos(sen4x) 8 ❊①❡♠♣❧♦ ✸✳✹✽✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞♦ ❧✐♠✐t❡ ❙♦❧✉çã♦✳  cos x − cos(sen4x) lim x→0 x2  ✳     cos x − cos(sen4x) 1 − cos(sen4x) 1 − cos x lim = lim = − x→0 x→0 x2 x2 x2  1 − cos(sen4x) = lim x→0 x2  sen4x sen4x 2 − lim x→0 1 − cos x = x2  2 1 1 − cos(sen4x) sen4x − = = lim 2 x→0 sen 4x x 2   2  sen4x 1 1 15 1 1 − cos(sen4x) · lim − = · 16(1)2 − = = lim 2 4x→0 sen4x→0 sen 4x x 2 2 2 2   cos x − cos(sen4x) 15 P♦rt❛♥t♦✱ lim = . 2 x→0 x 2  ❊①❡♠♣❧♦ ✸✳✹✾✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞♦ ❧✐♠✐t❡ ❙♦❧✉çã♦✳  arcsen(x − 2) lim x→2 x2 − 2x ✷✵✸   ✳ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❚❡♠♦s ❛♣❧✐❝❛♥❞♦ ❛ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ Pr♦♣r✐❡❞❛❞❡ R ✭✸✳✷✵✮ ❝✮ q✉❡✿    arcsen(x − 2) 1 1 1 arcsen(x − 2) = lim = (1) · = . · lim 2 x→2 x→2 x − 2x x−2 x 2 2    arcsen(x − 2) 1 = . lim 2 x→2 x − 2x 2 P♦rt❛♥t♦ ❊①❡♠♣❧♦ ✸✳✺✵✳ lim ❈❛❧❝✉❧❛r x→0+ ❙♦❧✉çã♦✳  √  arcsen(3x) tan x √ ✳ x csc x − cot x ❉♦ ❢❛t♦ s❡r ❛ t❛♥❣❡♥t❡ ♣♦s✐t✐✈❛ q✉❛♥❞♦ < 0✱ ❧♦❣♦ ♥ã♦ t❡♠ s❡♥t✐❞♦ 3x → 0+ ❡ ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭✸✳✷✵✮ √ x → 0+ ❡♥tã♦ ❡①✐st❡ tan x❀ ♣❛r❛ ♦ ❝❛s♦ x < 0 ❧✐♠✐t❡ x → 0− ✳ ❖❜s❡r✈❡ q✉❡ q✉❛♥❞♦ x → 0+ t❡♠♦s tan x ♦ ❡♥tã♦ ❝✮ s❡❣✉❡✿ lim x→0+  √   r 3 · arcsen(3x) tan x arcsen(3x) tan x √ = lim+ = x→0 (3x) csc x − cot x x csc x − cot x = 3(1) lim+ x→0 P♦rt❛♥t♦✱ ✸✳✻✳✸ r lim x→0+ tan x = 3. lim+ x→0 csc x − cot x  s x· √  √ arcsen(3x) tan x √ = 3 2. x csc x − cot x tan x x  x csc x 1−cos 2 x = 3. s √ 1 1 = 3 2 1( 2 ) ▲✐♠✐t❡ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❡ ❧♦❣❛rít♠✐❝❛ ❈♦♥s✐❞❡r❡ ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s s❡♠ ❞❡♠♦♥str❛çã♦✿ ✶✳ ✸✳ nn =0 n→+∞ n! lim ✷✳ h i lim Ln[f (x)] = Ln lim f (x) n→a n→a ❊①❡♠♣❧♦ ✸✳✺✶✳ ❈❛❧❝✉❧❛r lim x→0 √ x ✹✳ √ n lim n→+∞  lim n→+∞ n=1 n 1 =e 1+ n ♦♥❞❡ e ≈ 2, 71828182 · · · 1 + x✳ ❙♦❧✉çã♦✳ 1 → +∞✱ ❢❛③❡♥❞♦ x  n √ 1 x r❡s✉❧t❛✿ lim 1 + x = lim 1 + = e✳ x→0 n→+∞ n √ P♦rt❛♥t♦✱ lim x 1 + x = e✳ ◗✉❛♥❞♦ x → 0✱ ❡♥tã♦ n= ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ ♥♦ ❧✐♠✐t❡ ♦r✐❣✐♥❛❧ x→0 ✷✵✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✸✳✺✷✳ ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳ Ln(1 + x) ✳ x→0 x lim   1 1 Ln(1 + x) x x = lim Ln(1 + x) = Ln lim (1 + x) = Lne = 1. ❚❡♠♦s✿ lim x→0 x→0 x→0 x Ln(1 + x) P♦rt❛♥t♦✱ lim = 1✳ x→0 x ❊①❡♠♣❧♦ ✸✳✺✸✳ ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳ lim n→+∞ h 1+ a in ✱ s❡♥❞♦ a > 0 ♥ú♠❡r♦ r❡❛❧ q✉❛❧q✉❡r✿ n n → +∞✱ ❧♦❣♦ a " m  a  n #a  h 1 a in 1 a 1+ = lim = ea lim 1 + 1+ n = lim m→+∞ n→+∞ n→+∞ n m a ❙❡ n → +∞✱ ❡♥tã♦ m = P♦rt❛♥t♦✱ lim n→+∞ h 1+ a in = ea n  ❊①❡♠♣❧♦ ✸✳✺✹✳ ah − 1 = Ln(a)✳ h→0 h ❱❡r✐✜❝❛r ❛ s❡❣✉✐♥t❡ ✐❣✉❛❧❞❛❞❡✿ ❙♦❧✉çã♦✳ lim ❙❡❥❛ s = ah − 1✱ ❡♥tã♦ h · Ln(a) = Ln(s + 1)✱ q✉❛♥❞♦ h → 0 t❡♠♦s s → 0✱ ♥♦ ❧✐♠✐t❡✿ s s · Ln(a) s · Ln(a) s · Ln(a) ah − 1 = lim = lim = lim = lim h s→0 Ln(s + 1) h→0 h h→0 h · Ln(a) h→0 Ln(a ) h→0 h ⇒ lim ah − 1 1 = Ln(a) = Ln(a) · h→0 h lim Ln(s+1) s lim s→0 ✐st♦ ♣❡❧♦ ❊①❡♠♣❧♦ ✭✸✳✺✷✮✳ ah − 1 = Ln(a) h→0 h P♦rt❛♥t♦✱ lim ❊①❡♠♣❧♦ ✸✳✺✺✳ ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳ lim x→∞  f (x) 1+ x x s❡♥❞♦ lim .f (x) ✉♠ ♥ú♠❡r♦ ✜♥✐t♦✳ x→∞ f (x) ✱ ♣❡❧♦ ❢❛t♦ s❡r lim .f (x) ✉♠ ♥ú♠❡r♦ r❡❛❧ ✜♥✐t♦✱ q✉❛♥❞♦ x → ∞, m → 0✳ x→∞ x  x f (x) ❈♦♥s✐❞❡r❡ y = 1 + ❡♥tã♦ ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❧♦❣❛r✐t♠♦✿ x ❙❡❥❛ m = Ln(y) = x · Ln(1 + m) = Ln(1 + m) f (x) · Ln(1 + m) = f (x) · m m ✷✵✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠  f (x) lim Ln(y) = lim Ln 1 + x→∞ x→∞ x ♣❡❧♦ ❊①❡♠♣❧♦ ✭✸✳✺✷✮ s❡❣✉❡✿ x = lim f (x) · x→∞ Ln(1 + m) m R ⇒   Ln(1 + m) Ln lim y = lim f (x) · lim = lim f (x).(1) = lim f (x) x→∞ x→∞ m→0 x→∞ x→∞ m   lim .f (x) ▲♦❣♦ Ln lim y = lim .f (x) ⇒ lim y = ex→∞ ✳ x→∞ x→∞  x→∞ x f (x) lim .f (x) = ex→∞ ✳ P♦rt❛♥t♦✱ lim 1 + x→∞ x ❊①❡♠♣❧♦ ✸✳✺✻✳ (1 + α)n − 1 ✳ α→0 α lim ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳ ❙❡❥❛ m = (1 + α)n − 1✱ ❡♥tã♦ Ln(m + 1) = n · Ln(1 + α)❀ q✉❛♥❞♦✱ α → 0, m → 0✱ ❧♦❣♦ ♥♦ ❧✐♠✐t❡ t❡♠♦s✿ (1 + α)n − 1 m m · n · Ln(1 + α) = lim . = lim . = α→0 α→0 α α→0 α · n · Ln(1 + α) α lim    Ln(1 + α) m m · n · Ln(1 + α) ·n= = lim = lim . α→0 α→0 α · Ln(1 + m) α Ln(1 + m)  Ln(1 + α)  · α→0 α ❆♣❧✐❝❛♥❞♦ r❡s✉❧t❛❞♦ ❞♦ ❊①❡♠♣❧♦ ✭✸✳✺✷✮✱ = lim (1 + α)n − 1 = (1)(1)n = n✳ α→0 α P♦rt❛♥t♦✱ lim  1   ·n = n✳ Ln(1 + m) lim m→0 m ❖❜s❡r✈❛çã♦ ✸✳✼✳ P❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦s ❧✐♠✐t❡s ❞❛ ❢♦r♠❛ 1o ❈❛s♦ ✿ 2o ❈❛s♦ ✿ g(x) x→a B =A ✳ x→a ❙❡ ❡①✐st❡♠ lim .f (x) = A 6= 1 ❡ lim .g(x) = B s❡♥❞♦ B = ±∞✱ ❡♥tã♦ ♦ ❧✐♠✐t❡✿ x→a lim [f (x)]g(x) x→a ✳ ❝♦♥s✐❞❡r❡ ♦s s❡❣✉✐♥t❡s ❝❛s♦s✿ ❙❡ ❡①✐st❡♠ lim .f (x) = A ❡ lim .g(x) = B ❡ sã♦ ✜♥✐t♦s✱ ❡♥tã♦ ♦ ❧✐♠✐t❡ lim [f (x)] x→a lim [f (x)]g(x) x→a x→a   +∞, s❡✱ A > 1 ❡ B = +∞     −∞, s❡✱ A > 1 ❡ B = −∞ =  0, s❡✱ 0 < A < 1 ❡ B = +∞     +∞, s❡✱ 0 < A < 1 ❡ B = −∞ ✷✵✻ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 3o ❈❛s♦ ✿ ❙❡ lim .f (x) = 1 ❡ lim .g(x) = ±∞❀ ♥❡st❛ ❝❛s♦ 1±∞ é ✉♠❛ ❢♦r♠❛ ✐♥❞❡t❡r✲ x→a x→a ♠✐♥❛❞❛❀ ❧♦❣♦ t❡♠♦s q✉❡ ❞❡✜♥✐r h(x) = f (x) − 1 ❞❡ ♠♦❞♦ q✉❡ lim .h(x) = 0❀ ❧♦❣♦ ih(x).g(x) x→a h 1 lim .h(x).g(x) g(x) g(x) = ex→a ✳ lim [f (x)] = lim [1 + h(x)] = lim [1 + h(x)] h(x) x→a x→a ❊①❡♠♣❧♦ ✸✳✺✼✳ ❈❛❧❝✉❧❛r✿ x→a  x2 − 25 lim x→5 x−5 ❛✮ (x−3) ❜✮ lim x→+∞  3x + 2 x−4 (x+5) ✳ ❙♦❧✉çã♦✳ ❛✮ ❆♣❧✐❝❛♥❞♦ 1 ❝❛s♦ ❞❛ ❖❜s❡r✈❛çã♦ ✭✸✳✼✮✱ t❡♠♦s✿ o  x2 − 25 P♦rt❛♥t♦ lim x→5 x−5 ❜✮ (x−3)  x2 − 25 lim x→5 x−5 (x−3) = 102 = 100✳ = 100✳ ❆♣❧✐❝❛♥❞♦ ♦ s❡❣✉♥❞♦ ❝❛s♦ ❞❛ ❖❜s❡r✈❛çã♦ ✭✸✳✼✮✱ t❡♠♦s✿ lim x→+∞ P♦rt❛♥t♦ lim x→+∞ ❊①❡♠♣❧♦ ✸✳✺✽✳ ❈❛❧❝✉❧❛r✿ ❛✮  3x + 2 x−4    3x + 2 =3 ❡ x−4 (x+5) mx − nx lim x→0 x lim (x + 5) = +∞ x→+∞ = +∞✳    amx − 1 lim nx x→0 a −1 ❜✮  ❝✮  ex−1 − ax−1 lim x→1 x2 − 1  ❙♦❧✉çã♦✳ ❛✮ P❡❧♦ ❊①❡♠♣❧♦ ✭✸✳✺✹✮✱ ♦❜s❡r✈❡ q✉❡✱   x   (m − 1) − (nx − 1) mx − nx = lim = lim x→0 x→0 x x   mx − 1 nx − 1 m
lim − = Ln(m) − Ln(n) = Ln x→0 x x n P♦rt❛♥t♦✱ ❜✮   mx − nx m
✳ lim = Ln x→0 x n ◗✉❛♥❞♦ x → 0 ❡♥tã♦ mx → 0 ❡ nx → 0✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿  amx − 1  amx −1  h mx i amx − 1 m Ln(a) m  mx  mx lim nx = lim ·lim anx = · =  anx − 1  = lim −1 x→0 a x→0 nx x→0 nx x→0 −1 n Ln(a) n nx nx   h mx i    amx − 1 m P♦rt❛♥t♦✱ lim nx = ✳ x→0 a −1 n ✷✵✼ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦  x−1   (e − 1) − (ax−1 − 1) ex−1 − ax−1 = lim = lim x→1 x→1 x2 − 1 x2 − 1  ❝✮ ❚❡♠✲s❡✿    x−1   (ex−1 − 1) − (ax−1 − 1) e − 1 ax−1 − 1 = lim = lim − 2 = x→1 x→1 x2 − 1 x2 − 1 x −1  x−1  x−1   x−1    1 e −1 a −1 1 1 e − 1 ax−1 − 1 = lim · = lim − lim = − x→1 x + 1 x−1 x−1 2 x→1 x − 1 2 x→1 x − 1 ❋❛③❡♥❞♦ y =x−1 ❡♥tã♦ q✉❛♥❞♦ x → 1, y → 0❀ ❧♦❣♦  x−1  x−1  y  y     e −1 a −1 e −1 a −1 1 1 1 1 lim − lim = lim − lim = 2 x→1 x − 1 2 x→1 x − 1 2 y→0 y 2 y→0 y P♦rt❛♥t♦✱ 1 1 = − [Ln(e) − Ln(a)] = [1 − Ln(a)] 2 2  x−1  1 e − ax−1 = [1 − Ln(a)]✳ lim 2 x→1 x −1 2  ❆♣❧✐❝❛çõ❡s ❞✐✈❡rs❛s ❞❡ ❧✐♠✐t❡s ❊①❡♠♣❧♦ ✸✳✺✾✳ ❈❛❧❝✉❧❛r✿  sena + sen3x lim x→0 sena − sen3x ❛✮ 1  sen3x ❜✮ lim x→+∞  x3 + 3x2 + 2x − 1 x3 + 2x − 5 x+1 ❙♦❧✉çã♦✳ ❛✮ ❊st❡ ❧✐♠✐t❡ é ❞♦ 3o ❈❛s♦ s❛ ❖❜s❡r✈❛çã♦ ✭✸✳✼✮ ✱ ❝♦♥s✐❞❡r❡ y = 1 sen3x ✱ ♦❜s❡r✈❡ q✉❡✿ x → 0, (sen3x) → 0 ❡ y → ∞✳ ▲♦❣♦ ✿  1  1    1+ sena + sen3x sen3x sena + sen3x sen3x lim = lim = lim x→0 sena − sen3x sen3x→0 sena − sen3x sen3x→0 1 − = lim y→∞ " 1+ 1− 1 y·sena 1 y·sena #y = lim y→∞ "" 1+ 1− ❉♦ ❢❛t♦ 2sen3x sena − sen3x ❡ 1 #y·sena # sena  e1 = −1 e sena + sen3x sena − sen3x lim g(x) = 0✳ ❖✉tr♦ ♠♦❞♦ ❞❡ r❡s♦❧✈❡r é ❝♦♥s✐❞❡r❛♥❞♦ h(x) = f (x) − 1 = 1 y·sena 1 y·sena f (x) = ❡ 1  sena sen3x sena sen3x sena 1  sen3x 2 = e sena . g(x) = 1 ✱ ❡♥tã♦ sen3x y→0 2 2 = ✳ sena − sen3x sena  1  2 sena + sen3x sen3x = e sena . lim x→0 sena − sen3x lim .h(x)g(x) = lim y→0 P♦rt❛♥t♦✱ y→0 ✷✵✽ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❜✮ ❖❜s❡r✈❡✱ x3 + 2x − 5 3x2 + 4 3x2 + 4 x3 + 3x2 + 2x − 1 = + = 1 + ✱ x3 + 2x − 5 x3 + 2x − 5 x3 + 2x − 5 x3 + 2x − 5 lim x→+∞ ❙❡❥❛♠ h(x) = ▲♦❣♦ ♣❡❧♦ 3o  x3 + 3x2 + 2x − 1 x3 + 2x − 5 3x2 + 4 x3 + 2x − 5 ❈❛s♦ ❞❛ lim .h(x)(x+1) ex→+∞ P♦rt❛♥t♦✱ = lim x→+∞  # √ Ln 5 cos 8x lim x→0 5x2 ❜✮  q  12 x √ lim 4 − 3 cos x x→0 # √ √ 1 Ln 5 cos 8x = lim · 2 · Ln 5 cos 8x lim 2 x→0 5x x→0 5x x→0 . g(x) = x + 1 ❝♦♠♦ lim .h(x) = 0 ❡ lim .g(x) = ∞✳ x→+∞ x→+∞  3 x+1 x + 3x2 + 2x − 1 ❖❜s❡r✈❛çã♦ ✭✸✳✼✮ s❡❣✉❡ q✉❡✱ lim = x→+∞ x3 + 2x − 5 " lim x+1 ❡ " ❈❛❧❝✉❧❛r✿ ❛✮ ❛✮ ❚❡♠♦s✿ 3x2 + 4 1+ 3 x + 2x − 5 = e3 ✳ x+1  3 x + 3x2 + 2x − 1 = e3 ✳ lim x→+∞ x3 + 2x − 5 ❊①❡♠♣❧♦ ✸✳✻✵✳ ❙♦❧✉çã♦✳ x+1 ❡♥tã♦ " # √ 5 h √ 5 1 Ln cos 8x == Ln lim ( cos 8x) 5x2 2 x→0 5x ❆♣❧✐❝❛♥❞♦ ❛ ♣❛rt❡ 3a ❞❛ ❖❜s❡r✈❛çã♦ ✭✸✳✼✮ q✉❛♥❞♦ ⇒ i   1 = Ln  lim (cos 8x) 25x2  x→0 f (x) = cos 8x✱ ♦❜s❡r✈❡ f (x) → 1✳ 1 ✱ ♣♦✐s , g(x) → +∞ q✉❛♥❞♦ x → 0✳ g(x) = 25x2  i cos 8x2  h i h 1 64 25x g(x) = Ln lim [1 + cos 8x] cos 8x ▲♦❣♦ Ln lim .f (x) = Ln[e− 50 ]✳ x→0 x→0 " √ # Ln 5 cos 8x 64 P♦rt❛♥t♦✱ lim ✳ = − x→0 5x2 50 ❙❡❥❛ h(x) = cos 8x − 1 ❜✮ ❙❡❥❛♠ h(x) = 3(1 − √ (4 − 3 cos x) − 1 √ x→0 P♦rt❛♥t♦✱ lim x→0 1 3 ✱ ❡♥tã♦ lim h(x).g(x) = 2 x→0 2x 8 lim .h(x) = 0 ❡ lim .g(x) = +∞✱ t❡♠♦s✿ cos x) s❡♥❞♦ lim q ❡ q ❡ g(x) = x→0 4 − 3 cos x 4 − 3 cos x h(x) = x→0 √ √ ❡ ❝♦♠♦  12 x  12 x  1  √ 3 = lim 4 − 3 cos x 2x2 = e 8 x→0 3 = e8 ✳ ✷✵✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✸✳✻✶✳ n · sen(n!) ✳ n→∞ n2 + 2 lim ❉❡t❡r♠✐♥❡ ♦ ❝á❧❝✉❧♦ ❞♦ ❧✐♠✐t❡✿ ❙♦❧✉çã♦✳ P❛r❛ t♦❞♦ n∈N s❛❜❡✲s❡ q✉❡ n > 0, ∀ n ∈ N✱ n2 + 2 n n · sen(n!) n − 2 ≤ ≤ 2 ✳ 2 n +2 n +2 n +2 −1 ≤ senn! ≤ 1✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞♦ s❡♥♦ t❡♠♦s q✉❡ ❝♦♠♦ ❡♥tã♦ ❈❛❧❝✉❧❛♥❞♦ ♦ ❧✐♠✐t❡✿ − lim n→∞ n2 P♦rt❛♥t♦✱ n n · sen(n!) n ≤ lim ≤ lim 2 2 n→∞ n + 2 + 2 n→∞ n + 2 ⇒ n · senn! ≤0 n→∞ n2 + 2 0 ≤ lim n · sen(n!) = 0✳ n→∞ n2 + 2 lim ❊①❡♠♣❧♦ ✸✳✻✷✳ ❈❛❧❝✉❧❛r✿ ❙♦❧✉çã♦✳ ❙❡❥❛ senπx ✳ x→1 sen3πx lim y = x−1✱ ❧♦❣♦ y → 0 q✉❛♥❞♦ x → 1✳ ❙❡❣✉❡ senπx senπ(y + 1) = lim x→1 sen3πx y→0 sen3π(y + 1) L = lim πsen(πy) 1 sen(πy) sen(πy) cos π + senπ cos(πy) πy = = lim = lim L = lim y→0 sen(3πy) πy→0 3π · sen(3πy) y→0 sen(3πy) cos(3π) + sen(3π) cos(3πy) 3 3πy P♦rt❛♥t♦✱ 1 senπx = x→1 sen3πx 3 lim ❊①❡♠♣❧♦ ✸✳✻✸✳ ❈❛❧❝✉❧❛r ♦ ❧✐♠✐t❡ ✿ ❙♦❧✉çã♦✳ tan πx . x→−2 x + 2 lim ❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧✿ = tan πy − tan 2π tan πy − 0 senπy = = tan πy = 1 + tan πy · tan 2π 1 + tan πy · 0 cos πy 1 senπy senπy tan πx = lim = lim · = x→−2 y · cos πy y→0 x→−2 x + 2 y cos πy y → 0✱ t❡♠♦s πy → 0✱ ❧♦❣♦✿ ◆♦ ❧✐♠✐t❡✿ ◗✉❛♥❞♦ y = x + 2✱ ❡♥tã♦✿ tan πx = tan π(y − 2) = lim π senπy π tan πx = lim · lim =1· =π πy→0 πy πy→0 cos πy x→−2 x + 2 1 lim P♦rt❛♥t♦✱ lim x→−2 tan πx = π✳ x+2 ✷✶✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✸✲✹ ✶✳ ❱❡r✐✜❝❛r ♦ ❝á❧❝✉❧♦ ❞♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ 1. 3. 5. 7. 9. x+2 lim+ 2 = +∞ x→2 x − 4 √ 16 − x2 = −∞ lim− x→4 x−4 2x2 − 5x − 3 = −∞ lim+ x→1 x−1 3x2 − 9x − 6 lim− 2 = +∞ x→2 x +x−6 √ 5 x6 √ = −∞ lim √ x→+∞ 7 x − 7 x4 2. 4. 6. 8. 10. √ x2 − 9 lim+ = +∞ x→3 x−3   1 3 =∞ lim − 2 x→2 x − 2 x −4 3x3 + 2x2 − 1 lim = −∞ x→−∞ 2x2 − 3x + 5 5x3 + 1 lim+ = +∞ x→20 20x3 − 800x   1 1 lim = +∞ − 2 x→1 1 − x x − 2x − 1 ✷✳ ❈❛❧❝✉❧❛r ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ 1. 3. 5. 7.  2x x+1 x lim − − 2 x→+∞ 3x − 2 4x 6x − 1 √ √ 6 6 x7 + 3 x lim √ √ x→+∞ 5 5 x4 + 4 x √ lim 4(x x2 + 1 − x2 )  lim− x→2 x→−∞ 4. 6. 3x2 − 9x − 6 x2 + x − 6 x5 √ 5 7 x + 3 x8 √ lim (x x2 + 1 − x2 ) x→−∞ √ 4 − x2 lim 2 x→2 x + 1 √ 3x + 4x2 − x3 + x lim x→−∞ x2 + 5x + 1 lim 2. x→+∞ √ 3 8. √ 5 1 x ✸✳ ▼♦str❡ q✉❡✱ lim+ .f (x) = ∞ s❡✱ ❡ s♦♠❡♥t❡ s❡ lim .f ( ) = ∞✳ x→+∞ x→0 ✹✳ ❉❡t❡r♠✐♥❡ ❝♦♥st❛♥t❡s a ❡ b t❛✐s q✉❡✿ ✶✳ ✸✳  x3 + 1 √ 2 + x + 2 − ax = 0 lim x→+∞ x2 + 1 i h√ x2 − x + 1 − ax − b = 0 lim  ✷✳ lim x→+∞   x2 + 1 − ax − b = 0 x+1 x→+∞ 1 + 2x ✺✳ ◗✉❛♥❞♦ x → 0 t❡♠♦s y = → ∞✳ ◗✉❡ ❝♦♥❞✐çõ❡s ❞❡✈❡ ❝✉♠♣r✐r x ♣❛r❛ q✉❡ x 4 t❡♥❤❛♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ | y |> 10 ❄ x é ✐♥✜♥✐t❛♠❡♥t❡ ❣r❛♥❞❡ q✉❛♥❞♦ x → 3✳ ◗✉❛❧ ❞❡✈❡ ✻✳ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ y = x−3 s❡r ♦ ✈❛❧♦r ❞❡ x ♣❛r❛ q✉❡ ❛ ♠❛❣♥✐t✉❞❡ | y | s❡❥❛ ♠❛✐♦r q✉❡ 1000❄ ✷✶✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✼✳ ❱❡r✐✜❝❛r q✉❡✿ ✶✳ arcsenx = arctan √ x 1 − x2 ✷✳ arctan x − arctan y = arctan x−y ✳ 1 + xy πx ✽✳ ❙❡❥❛♠ f (x) = sen +cos(arctan x) ❡ g(x) = sec(2−x)−tan(arcsec(−x))✳ ❈❛❧❝✉❧❛r 4 f (1) − g(2)✳ ✾✳ ◆♦ s❡♥t✐❞♦ ❞❛ ❉❡✜♥✐çã♦ ✭✸✳✼✮✿ ✶✳ ✷✳ 1 = +∞✳ x→3 (x − 3)2 ❉❡♠♦♥str❛r q✉❡✿ lim ❉❡♠♦♥str❡ q✉❡✿ s❡ g(x) > β > 0 ♣❛r❛ t♦❞♦ x✱ ❡ s❡ lim g(x) = 0✱ ❡♥tã♦ x→a 1 = +∞✳ lim x→a g(x) ✶✵✳ ❯♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ✐sós❝❡❧❡s ❝✉❥❛ ❜❛s❡ ❡st❛ ❞✐✈✐❞✐❞❛ ❡♠ 2n ♣❛rt❡s ✭q✉❛❞r❛❞♦s✮ t❡♠ ✐♥s❝r✐t♦ ✉♠❛ ✜❣✉r❛ ❡s❝❛❧♦♥❛❞❛ s❡❣✉♥❞♦ ❛ ❋✐✲ ❣✉r❛ (3.9)✳ ❉❡♠♦♥str❡ q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ❡ ❛ ✜❣✉r❛ ❡s❝❛❧♦♥❛❞❛ é ✐♥✜♥✐t❡s✐♠❛❧ q✉❛♥❞♦ n ❝r❡s❝❡ ✐♥✜♥✐t❛♠❡♥t❡✳ ❅ ❅ ❅ ❅ ✳✳ · · · ✳✳ ✳ ✳ ❅ ❅ ❅ ❅ F igura 3.9 ✶✶✳ P❛r❛ ♦s s❡❣✉✐♥t❡s ❡①❡r❝í❝✐♦s ❡s❜♦ç❛r ♦ ❣rá✜❝♦ ♥♦ ✐♥t❡r✈❛❧♦ [−2π, 2π] 1. f (x) = cos  πkxk 2   πx  2. f (x) = 2 cos 2  πx  4. f (x) = sen 2 6. f (x) = sen2 | x | π  8. f (x) = 2 tan + senx 2 3. f (x) = sen(π k x k) 5. f (x) =| sen | x ||  π 7. f (x) = sen x − 4 ✶✷✳ ❱❡r✐✜❝❛r ♦ ❝á❧❝✉❧♦ ❞♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ 1. 3. 5. 7. tan x − senx = 0.5 x→0 x3 9 1 − cos 3x = lim x→0 1 − cos 4x 16 2 1−x 2 lim = x→1 senπx π a2 1 − cos ax = lim x→0 x2 2 lim 2. 4. 6. 8. tan ax − tan3 x =a a 6= 1 x→0 tan x 1−a x − senax lim = b 6= −1, a 6= 0 x→0 x + senbx 1+b 1 sen(π − x) lim = x→π x(π − x) π cos x =1 limπ π x→ 2 ( − x) 2 lim ✷✶✷ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 9. 11. 13. 15. 17. 19. √ 3 1 − 2 cos x =− limπ x→ 3 π − 3x 3 3 1 − x3 lim = x→1 sen(1 − x2 ) 2 √ 2 2x √ =2 lim x→0 tan x sec x − 1 x6 lim =4 x→0 (tan x − senx)2 tan(1 + cos x) = −1 lim x→π cos(tan x) − 1 π − 2 arccos x lim =2 x→0 x π2 1 + cos πx = x→1 x2 − 2x + 1 2 sen(1 − x) lim √ = −2 x→1 x−1 lim 10. 12. lim .4x · cot 4x = 1 14. x→0  πx  π 2 π = . tan x→0 x 2 2 √ 1 sen( x2 + 4 − 2) lim = 2 x→0 x 4 arcsen5x lim =5 x→0 arctan x lim 16. 18. 20. ✶✸✳ ❈♦♥s✐❞❡r❡ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ❞❡ ❧❛❞♦ a✳ ❙✉❛s três ❛❧t✉r❛s s❡r✈❡♠ ♣❛r❛ ❣❡r❛r ✉♠ ♥♦✈♦ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡ n ✈❡③❡s✳ ❉❡t❡r♠✐♥❡ ♦ ❧✐♠✐t❡ ❞❛ s♦♠❛ ❞❛s ár❡❛s ❞❡ t♦❞♦s ♦s tr✐â♥❣✉❧♦s q✉❛♥❞♦ n → +∞✳ ✶✹✳ ❯♠ ❝ír❝✉❧♦ ❞❡ r❛✐♦ r t❡♠ ✐♥s❝r✐t♦ ✉♠ q✉❛❞r❛❞♦❀ ❡st❡ t❡♠ ✐♥s❝r✐t♦ ✉♠ ❝ír❝✉❧♦ ♦ q✉❛❧ t❡♠ ✐♥s❝r✐t♦ ✉♠ q✉❛❞r❛❞♦✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡ n ✈❡③❡s✳ ❉❡t❡r♠✐♥❡ ♦ ❧✐♠✐t❡ ❞❛s s♦♠❛ ❞❛s ár❡❛s ❞❡ t♦❞♦s ♦s q✉❛❞r❛❞♦s q✉❛♥❞♦ n → +∞✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ ♣❛r❛ ❛ s♦♠❛ ❞❛s ár❡❛s ❞❡ t♦❞♦s ♦s ❝ír❝✉❧♦s✳ ✶✺✳ ▼♦str❡ q✉❡✱ s❡ f ❡ g sã♦ ❞✉❛s ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ❡♠ (a, +∞) ❡ (b, +∞) r❡s♣❡❝t✐✲ ✈❛♠❡♥t❡❀ s❡ lim .f (x) = L ❡ lim .g(x) = M ❡♥tã♦✿ xto+∞ xto+∞ lim [C · f (x)] = C · L ♣❛r❛ C ❝♦♥st❛♥t❡✳ ❛✮ x→+∞ ❜✮ lim [f (x) + g(x)] = lim .f (x) + lim .g(x) = L + M x→+∞ ❝✮ x→+∞ x→+∞ lim [f (x) × g(x)] = lim .f (x) × lim .g(x) = L × M x→+∞ ❞✮ lim x→+∞ x→+∞   x→+∞ lim .f (x) L f (x) x→+∞ = = ❞❡s❞❡ q✉❡ M 6= 0✳ g(x) lim .g(x) M x→+∞ ✶✻✳ ❈❛❧❝✉❧❛r ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ 1. 4. 7. cos x limπ x→ 2 (π − 2x) x lim √ x→0 1 − cos x lim (sec x − tan x) x→ π2 2. 5. 8. lim x. tan x→+∞ a x senx + x x→ 4 x (1 − senx)3 limπ x→ 2 (1 + cos 2x)3 limπ ✷✶✸ 3. 6. 9. lim x→0 1− √ cos x x2 arctan 3x x→0 arcsen4x 1 − cos 2x limπ π x→ 3 sen(x − ) 3 lim 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 10. 12. 14. 16. 18. 20. ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ √ x4 − x4 sen2 x lim x→0 1 − cos x x.sen(sen2x) lim x→0 1 − cos(sen4x) √ 2 − cos x − cos x lim x→0 x2   1 2 − lim x→0 sen2 x 1 − cos x tan(h + x) − tan h lim x→0 x sec(h + x) − sec h lim x→0 x R √ √ x( 1 + cos x − 2) √ lim x→0 1 − cos x sen(h + x) − senh lim x→0 x cot(h + x) − cot h lim x→0 x tan ax lim x→0 (1 + cos ax)(sec ax) cos(h + x) − cos h lim x→0 x 100sen3x + 200 cos x lim x→+∞ x 11. 13. 15. 17. 19. 21. ✶✼✳ ❈❛❧❝✉❧❛r ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ 1. 4. 7. senx x→∞ x Ln cos x lim x→0 x2 lim   1 ] lim x [1 − cos x→∞ x 2 ax − a−x a>0 ✶✵✳ lim x x→∞ a + a−x 1 − cos(1 − cos x) ✶✷✳ lim x→0 x ax a>0 ✶✹✳ lim x→∞ ax + 1 √ ✶✽✳ ❱❡r✐✜❝❛r q✉❡ lim x x = 1✳ 2. 5. 8. arctan x x→∞ x arcsenx
lim x→1 tan πx 2 √ lim senx cos x lim x→0 ✶✶✳ ✶✸✳ ✶✺✳ 3. 6. 9. x + senx x→∞ x + cos x  x 1 lim 1 + n x→+∞ x senx  senx  x−senx lim x→0 x lim q √ √ lim x( x2 + x4 + 1 − x 2) x→∞ lim √ x lim √ x x→0 x→0 cos x + senx cos x + asenbx x→+∞ ✶✾✳ ▼♦str❡ q✉❡ s❡ lim f (x) = ∞✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ {xn }n∈N+ ❞❡ ♥ú♠❡r♦s x→∞ r❡❛✐s t❛✐s q✉❡ f (xn ) > n✳ ✷✶✹ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ▼✐s❝❡❧â♥❡❛ ✸✲✶ ✶✳ ❙✉♣♦♥❤❛✲s❡ q✉❡ ❛s ❢✉♥çõ❡s f (x) ❡ g(x) tê♠ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ ❡ s❡ ✧P❛r❛ ❝❛❞❛ ε > 0 ❡ t♦❞♦ x ∈ R❀ s❡ 0 <| x−2 |< sen2 ( 0 <| x − 2 |< ε2 ✱ ❡♥tã♦ | g(x) − 4 |< ε✳✧ ε2 )+ε✱ ❡♥tã♦ | f (x)−2 |< ε 9 P❛r❛ ❝❛❞❛ ε > 0 ❛❝❤❛r ✉♠ δ > 0 ❞❡ ♠♦❞♦ q✉❡✱ ♣❛r❛ t♦❞♦ x ∈ R ✿ ✶✳ ❙❡ 0 <| x − 2 |< δ ✱ ❡♥tã♦ | f (x) + g(x) − 6 |< ε✳ ✷✳ ❙❡ 0 <| x − 2 |< δ ✱ ❡♥tã♦ | f (x) · g(x) − 8 |< ε✳ ✸✳ ❙❡ 0 <| x − 2 |< δ ✱ ❡♥tã♦ 1 1 < ε✳ − g(x) 4 ✹✳ ❙❡ 0 <| x − 2 |< δ ✱ ❡♥tã♦ f (x) 1 − < ε✳ g(x) 2 f (x) = 0✳ x→0 x ✷✳ ▼♦str❡ q✉❡✱ s❡ lim f (x) = 0 ✱ ❡♥tã♦ lim x→0 ✸✳ ▼♦str❡ q✉❡✿ ✶✳ lim+ f (x) = lim− f (−x) x→0 ✸✳ ✷✳ x→0 2 ✹✳ lim f (x ) = lim+ f (x) x→0 ✹✳ ❙❡❥❛ f (x) = x→0 ( lim f (| x |) = lim+ f (x) x→0 lim x→0− x→0 f ( x1 ) = lim f (x) x→−∞ 0, s❡✱ x ∈ Q 1, s❡✱ x ∈ I = R − Q ▼♦str❡ q✉❡ ♥ã♦ ❡①✐st❡ ♦ ❧✐♠✐t❡ lim .f (x)✱ q✉❛❧q✉❡r q✉❡ s❡❥❛ a ∈ R✳ x→a ✺✳ ❙❡❥❛ f (x) = ( x, s❡✱ x ∈ Q −x, s❡✱ x ∈ I = R − Q ▼♦str❡ q✉❡ ♥ã♦ ❡①✐st❡ ♦ ❧✐♠✐t❡ ✱ ♣❛r❛ q✉❛❧q✉❡r a 6= 0✳ f (x) f (ax) = L ❡ a 6= 0✱ ❡♥tã♦ lim = aL x→0 x x→0 x ✻✳ ▼♦str❡ q✉❡ s❡ lim (ax + 1)n ✼✳ ❈❛❧❝✉❧❛r lim ✳ ❈♦♥s✐❞❡r❡ s❡♣❛r❛❞❛♠❡♥t❡ ♦s ❝❛s♦s ❡♠ q✉❡ n s❡❥❛✿ ❛✮ ✉♠ x→∞ xn + A ✐♥t❡✐r♦ ♣♦s✐t✐✈♦❀ ❜✮ ✉♠ ✐♥t❡✐r♦ ♥❡❣❛t✐✈♦❀ ❝✮ ③❡r♦✳ ✷✶✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✽✳ ❈❛❧❝✉❧❛r ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ 1. ex−2 − e2−x x→2 sen(x − 2) lim 4. 1 − cos x − x→0 x4 7. xn lim x→+∞ ex lim 2. x2 2 n∈N 5. 8. limπ x→ 2 tan x tan 3x 3. tan x − x 6. x→0 x − senx " r #4x 15 lim cos 9. x→0 x lim ex−3 + e3−x − 2 x→3 1 − cos(x − 3) lim Lnx x→+∞ xα lim xα lim x→+∞ ex α > 0, α ∈ /N α > 0, α ∈ /N ✾✳ ▼♦str❡ ❛tr❛✈és ❞❡ ✉♠ ❡①❡♠♣❧♦ q✉❡ s❡ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ {xn }n∈N+ ❞❡ ♥ú♠❡r♦s r❡❛✐s t❛✐s q✉❡ f (xn ) > n✱ ❡♥tã♦ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❡①✐st❡ ♦ ❧✐♠✐t❡ ❞❡ f (xn ) q✉❛♥❞♦ n → +∞✳ ✶✵✳ ❙❡❥❛♠ f : [a, b] −→ R ❡ g : [a, b] −→ R ❢✉♥çõ❡s t❛✐s q✉❡✿ lim x→c f (x) =1 g(x) ⇒ lim f (x) = lim g(x) x→c x→c ♣❛r❛ c ∈ (a, b)✳ ✶✶✳ ❉❡♠♦♥str❡ q✉❡✱ lim .f (x) = L s❡✱ ❡ s♦♠❡♥t❡ s❡ lim+ .f (x) = lim− .f (x) = L✳ x→a x→a x→a ✶✷✳ ❯♠ ❡q✉✐♣❛♠❡♥t♦ ❢♦✐ ❝♦♠♣r❛❞♦ ♣♦r ❘$20.000 ❡ ❡s♣❡r❛✲s❡ q✉❡ s❡✉ ✈❛❧♦r ✜♥❛❧ ❞❡♣♦✐s ❞❡ 10 ❛♥♦s ❞❡ ✉s♦ s❡❥❛ R$1.500✳ ❙❡ ♦ ♠ét♦❞♦ ❞❛ ❧✐♥❤❛ r❡t❛ ❢♦r ✉s❛❞♦ ♣❛r❛ ❞❡♣r❡✲ ❝✐❛r ♦ ❡q✉✐♣❛♠❡♥t♦ ❞❡ ❘$20.000 ❛ ❘$1.500 ❡♠ 10 ❛♥♦s✱ q✉❛❧ ♦ ✈❛❧♦r ❧íq✉✐❞♦ ❞♦ ❡q✉✐♣❛♠❡♥t♦ ❞❡♣♦✐s ❞❡ 6 ❛♥♦s❄✳ ◗✉❛♥❞♦ ♦ ✈❛❧♦r ❞♦ ❡q✉✐♣❛♠❡♥t♦ é 0 ✭③❡r♦✮ r❡❛✐s❄ ✷✶✻ 09/02/2021 ❈❛♣ít✉❧♦ ✹ ❈❖◆❚■◆❯■❉❆❉❊ ❑❛r❧ ❚❤❡♦❞♦r ❲✐❧❤❡❧♠ ❲❡✐❡rstr❛ss ♥❛s❝❡✉ ❡♠ ❖st❡♥✲ ❢❡❧❞✱ ♥♦ ❞✐str✐t♦ ❞❡ ▼ü♥st❡r✱ ❆❧❡♠❛♥❤❛✱ ♥♦ 1815✱ ❡ ❢❛❧❡❝❡✉ ❡♠ ❇❡r❧✐♥✱ ❡♠ 14 ❈♦♠ 19 31 ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ ❞❡ ♦✉t✉❜r♦ ❞❡ 1897✳ ❛♥♦s✱ ✐♥❣r❡ss♦✉ ❛♦ ■♥st✐t✉t♦ ❈❛tó❧✐❝♦ ❞❡ P❛❞❡r❜♦r♥✳ ❙✉❛ ❛t✉❛çã♦ ♥❛ ❊s❝♦❧❛ ❢♦✐ ❜r✐❧❤❛♥t❡✱ ❝♦♥q✉✐st❛♥❞♦✱ ❝♦♠ r❡❣✉❧❛✲ r✐❞❛❞❡ ❡s♣❛♥t♦s❛✱ t♦❞♦s ♦s ♣rê♠✐♦s q✉❡ ❛❧♠❡❥❛✈❛✳ ▼❛tr✐❝✉❧♦✉✲s❡ ♥❛ ❊s❝♦❧❛ ❞❡ ▼ü♥st❡r✱ ❡♠ ❞❡r♠❛♥♥ s❡ q✉❡ 13 1839✱ ❝♦♥❤❡❝❡♥❞♦ ❛❧✐ ❈❤r✐st♦♣❤ ●✉✲ (1798−1851)✱ ❡s♣❡❝✐❛❧✐st❛ ❡♠ ❢✉♥çõ❡s ❡❧í♣t✐❝❛s✳ ❈♦♥t❛✲ ❛❧✉♥♦s ❝♦♠♣❛r❡❝❡r❛♠ à ❛✉❧❛ ✐♥❛✉❣✉r❛❧ ❞❡ ●✉❞❡r♠❛♥♥ ❡ q✉❡ à s❡❣✉♥❞❛ ❛✉❧❛ só ❝♦♠♣❛r❡❝❡✉ ❲❡✐❡rstr❛ss✳ ❊♠ ❚❤❡♦❞♦r ❲❡✐❡rstr❛ss 1841✱ ❲❡✐❡rstr❛ss ❛♣r❡s❡♥t♦✉✲s❡ ♣❛r❛ ♦s ❡①❛♠❡s ✜♥❛✐s✱ ❝♦♠♣♦st♦s ❞❡ ✉♠❛ ♣❛rt❡ ❡s❝r✐t❛ ❡ ✉♠❛ ♣❛rt❡ ♦r❛❧✳ P❛r❛ ♦ ❡①❛♠❡ ❡s❝r✐t♦✱ três t❡♠❛s ❢♦r❛♠ s✉❣❡r✐❞♦s✳ ❯♠ ❞♦s ♣r♦❜❧❡♠❛s ❡r❛ ❡①✲ tr❡♠❛♠❡♥t❡ ❝♦♠♣❧✐❝❛❞♦✿ ✏ ❉❡t❡r♠✐♥❛r ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s ❡♠ sér✐❡ ❞❡ ♣♦tê♥❝✐❛s ❞❛s ❢✉♥çõ❡s ❡❧í♣✲ t✐❝❛s✑✳ ❑❛r❧✱ ❞❡♣♦✐s ❞❡ ✉♠ ❛♥♦ ❞❡ tr❛❜❛❧❤♦s✱ ❝♦♥s❡❣✉✐✉ r❡s♦❧✈ê✲❧♦✱ r❡❝❡❜❡♥❞♦ ❡❧♦❣✐♦s❛s r❡❢❡rê♥❝✐❛s ❞❡ ●✉❞❡r♠❛♥♥✳ P❛ss❛♥❞♦ ❡♠ s❡❣✉✐❞❛✱ ♣❡❧♦ ❡①❛♠❡ ♦r❛❧✱ ❲❡✐❡rstr❛ss ♦❜t❡✈❡ ❛✜♥❛❧✱ s❡✉ tít✉❧♦ ❞❡ ♣r♦❢❡ss♦r✱ ❛❝♦♠♣❛♥❤❛❞♦ ❞❡ ✉♠ ❝❡rt✐✜❝❛❞♦ ❡s♣❡❝✐❛❧✱ ♣♦r ✏ s✉❛s ❝♦♥tr✐❜✉✐çõ❡s à ♠❛t❡♠át✐❝❛✳✑ ❊♠ 1842✱ ❲❡✐❡rstr❛ss ❢♦✐ ♣r♦❢❡ss♦r ❛✉①✐❧✐❛r ❞❡ ♠❛t❡♠át✐❝❛ ❡ ❢ís✐❝❛ ♥♦ Pr♦✲●②♠♥❛s✐✉♠ ❞❡ ❉❡✉ts❝❤✲❑rö♥❡✱ ♥❛ Prúss✐❛ ❖r✐❡♥t❛❧✳ ❙❡✐s ❛♥♦s ♠❛✐s t❛r❞❡✱ ❢♦✐ tr❛♥s❢❡r✐❞♦ ♣❛r❛ ♦ ✐♥st✐t✉t♦ ❞❡ ❇r❛✉♥s❜❡r❣✱ ♦♥❞❡ ♣❡r♠❛♥❡❝❡✉ ❞❡ 1848 ❛ 1854✳ ❖ ❝❛tá❧♦❣♦ ❞❛ ❡s❝♦❧❛✱ ❞♦ ❛♥♦ ❞❡ 1848✱ ❝♦♥té♠ ✉♠ tr❛❜❛❧❤♦ ❞❡ ❲❡✐❡rstr❛ss ✏ ❈♦♥tr✐❜✉✐çõ❡s ♣❛r❛ ❛ t❡♦r✐❛ ❞❛s ✐♥t❡❣r❛✐s ❆❜❡❧✐❛♥❛s✑✱ q✉❡ ❝❡rt❛♠❡♥t❡ ❤á ❞❡ t❡r ♣r♦✈♦❝❛❞♦ ♦ ❡s♣❛♥t♦ ❞❡ s❡✉s ❝♦❧❡❣❛s✳ ❋♦✐ ♥♦♠❡❛❞♦ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠át✐❝❛ ❞❛ ❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❡ ❇❡r❧✐♠ ❡♠ ❥✉❧❤♦ ❞❡ 1856✳ ❖ ❡st✉❞♦ ❞❛ ♠❛t❡♠át✐❝❛✱ ❡♠ ♠♦❧❞❡s ♠❛✐s ♦✉ ♠❡♥♦s ✐♥t✉✐t✐✈♦s✱ s♦❢r❡✉ ✉♠ sér✐♦ ❝❤♦q✉❡ ♥♦ ♠♦♠❡♥t♦ ❡♠ q✉❡ ❲❡✐❡rstr❛ss ✐♥✈❡♥t♦✉✿ ✏❯♠❛ ❝✉r✈❛ ❝♦♥tí♥✉❛ q✉❡ ♥ã♦ ❛❞♠✐t✐❛ t❛♥❣❡♥t❡ ❡♠ q✉❛❧q✉❡r ❞❡ s❡✉s ♣♦♥t♦s✑✳ ❲❡✐❡rstr❛ss ❞❡❞✉③ ♦ s✐st❡♠❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s R ❛ ♣❛rt✐r ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❉❡❞❡❦✐♥❞ ✉t✐❧✐③❛ ♦s ✏ ❝♦rt❡s✑✱ ❡♥t❛♥t♦ q✉❡ ❲❡✐❡rstr❛ss ❡♠♣r❡❣❛ ❛s ❝❧❛ss❡s ❞❡ r❛❝✐♦♥❛✐s✳ ❆s ❞✉❛s t❡♦r✐❛s ❡stã♦ s✉❥❡✐t❛s à ♠❡s♠❛ ❝rít✐❝❛ q✉❡ ♦s ❧ó❣✐❝♦s ❛♣❧✐❝❛♠ às ✐❞❡✐❛s ❞❡ ❈❛♥t♦r✳ ❲❡✐❡rstr❛ss r❡♣r❡s❡♥t❛ ✉♠❛ ❡s♣é❝✐❡ ❞❡ sí♥t❡s❡ ❞♦ ♠♦✈✐♠❡♥t♦ ❡♠ ❢❛✈♦r ❞❡ ♠❛✐♦r r✐❣♦r ♥❛ ♠❛t❡♠át✐❝❛✳ ✷✶✼ ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✹✳✶ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈♦♥❝❡✐t♦s ❜ás✐❝♦s ■♥t✉✐t✐✈❛♠❡♥t❡✱ ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ♥✉♠ ✐♥t❡r✈❛❧♦ (a, b) ⊆ R ♣♦❞❡ s❡r ❞❡s❡♥❤❛❞♦ s❡♠ ❧❡✈❛♥t❛r ♦ ❧á♣✐s ❞♦ ♣❛♣❡❧ ♣❛r❛ ❡ss❡ ✐♥t❡r✈❛❧♦ (a, b)✳ ◆❛s ❢✉♥çõ❡s ❞❡s❝♦♥tí✲ ♥✉❛s✱ ❡st❡ ❣rá✜❝♦ é ✐♥t❡rr♦♠♣✐❞♦ ♥♦s ♣♦♥t♦s ❞❡ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡✳ ❉❡❝♦rr❡ ❞✐st♦ q✉❡ ✉♠❛ ❢✉♥çã♦ é ❝♦♥tí♥✉❛ s❡ ❛ ♣❡q✉❡♥❛s ✈❛r✐❛çõ❡s ❞❡ ❡❧❡♠❡♥t♦s ❞♦ s❡✉ ❞♦♠í♥✐♦ ❝♦rr❡s♣♦♥❞❡♠ ♣❡✲ q✉❡♥❛s ✈❛r✐❛çõ❡s ♥❛s ✐♠❛❣❡♥s ❞❡st❡s ❡❧❡♠❡♥t♦s✳ ◆♦s ♣♦♥t♦s ♦♥❞❡ ❛ ❢✉♥çã♦ ♥ã♦ é ❝♦♥tí♥✉❛✱ ❞✐③❡♠♦s q✉❡ ❛ ❢✉♥çã♦ é ❞❡s❝♦♥tí♥✉❛✱ ♦✉ q✉❡ s❡ tr❛t❛ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡✳ ❆ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ ❢✉♥çõ❡s é ✉♠ ❞♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ❞❛ t♦♣♦❧♦❣✐❛✶ ✳ ❙❡❥❛♠ f ❡ g ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ♥✉♠ ♠❡s♠♦ ✐♥t❡r✈❛❧♦✱ s❡❣✉♥❞♦ ♦s ❣rá✜❝♦s ♠♦str❛❞♦s ♥❛ ❋✐❣✉r❛ ✭✹✳✶✮✳ ❋✐❣✉r❛ ✹✳✶✿ ❖❜s❡r✈❡✲s❡ q✉❡ ❡st❛s ❢✉♥çõ❡s tê♠ ❝♦♠♣♦rt❛♠❡♥t♦s ❞✐st✐♥t♦s ♥♦ ♣♦♥t♦ x = a✳ ❊♥t❛♥t♦ ♦ ❣rá✜❝♦ ❞❡ f ✈❛r✐❛ ❝♦♥t✐♥✉❛♠❡♥t❡ ♥❛s ♣r♦①✐♠✐❞❛❞❡s ❞❡ x = a✱ ✭♥ã♦ t❡♠ ❢✉r♦s✮❀ ♦ ❣rá✜❝♦ ❞❡ g ❛♣r❡s❡♥t❛ ✉♠ s❛❧t♦ ♥♦ ♣♦♥t♦ ❞❡ ❛❜s❝✐ss❛ x = a✳ ❆ ♣r♦♣r✐❡❞❛❞❡ q✉❡ t❡♠ ❛ ❢✉♥çã♦ f ✱ ❞❡ t❡r ♦ ❣rá✲ ✜❝♦ ✈❛r✐❛♥❞♦ ❝♦♥t✐♥✉❛♠❡♥t❡ ♥❛s ♣r♦①✐♠✐❞❛❞❡s ❞♦ ♣♦♥t♦ x = a✱ ♣♦❞❡ s❡r ❞❡s❝r✐t❛ ❞♦ ♠♦❞♦ s❡❣✉✐♥t❡✿ ∀ ε > 0, s❡ ❛❝♦♥t❡❝❡ ∃ δ > 0 /. ∀ x ∈ D(f ) ❋✐❣✉r❛ ✹✳✷✿ a − δ < x < a + δ ✱ ❡♥tã♦✿ f (a) − ε < f (x) < f (a) + ε ✭✹✳✶✮ ●❡♦♠❡tr✐❝❛♠❡♥t❡✱ s✐❣♥✐✜❝❛ q✉❡✱ s❡ x ❡st❛ ♣ró①✐♠♦ ❞❡ a ❡♥tã♦ f (x) ❡st❛ ♣ró①✐♠♦ ❞❡ f (a)✱ ✐st♦ é lim .f (x) = f (a) ✭❋✐❣✉r❛ ✭✹✳✷✮✮✳ x→a ❆ ❡①♣r❡ssã♦ ✭✹✳✶✮ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❞♦ ♠♦❞♦ s❡❣✉✐♥t❡✿ ✶ ❆ t♦♣♦❧♦❣✐❛ é ♦ r❛♠♦ ❞❛ ♠❛t❡♠át✐❝❛ ❞❡❞✐❝❛❞❛ ❛♦ ❡st✉❞♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s só❧✐❞♦s q✉❡ ♣❡r♠❛♥❡❝❡♠ ✐♥❛❧t❡r❛❞♦s ♣♦r tr❛♥s❢♦r♠❛çõ❡s ❝♦♥tí♥✉❛s✳ ✷✶✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ P❛r❛ t♦❞♦✱ ❙❡ ❛ ❢✉♥çã♦ ε > 0, f (x) ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ∃ δ > 0/. | x − a |< δ ✐♠♣❧✐❝❛ ❝✉♠♣r❡ ❡st❛ ❝♦♥❞✐çã♦✱ ❞✐③❡♠♦s q✉❡ | f (x) − f (a) |< ε✳ f é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ x = a✳ ❉❡✜♥✐çã♦ ✹✳✶✳ ❙❡❥❛ y = f (x) ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♥♦ ❝♦♥❥✉♥t♦ A ⊆ R✱ ❡ a ∈ A❀ ❞✐③✲s❡✱ q✉❡ f é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ x = a✱ s❡ s❛t✐s❢❛③ ❛s três ❝♦♥❞✐çõ❡s ✿ ✐✮ ❊①✐st❡ ✐✐✮ ❊①✐st❡ x→a lim .f (x)✳ f (x)✳ ✐✐✐✮ x→a lim .f (x) = f (a)✳ ❙❡ ❛❧❣✉♠❛ ❞❛s três ❝♦♥❞✐çõ❡s ♥ã♦ s❡ ❝✉♠♣r❡✱ ❞✐③❡♠♦s q✉❡ f é ❞❡s❝♦♥tí♥✉❛ ❡♠ x = a✳ ❊①❡♠♣❧♦ ✹✳✶✳ ❉❡t❡r♠✐♥❡ s❡ ❛ ❢✉♥çã♦ f (x) é ❝♦♥tí♥✉❛ ❡♠ x = 3✿     x2 − 9 x2 − 2x − 3 f (x) =  3   2 ❙♦❧✉çã♦✳ ✐✮ ✐✐✮ ✐✐✐✮ f (3) = s❡✱ 0 < x < 5, x 6= 3 x=3 s❡ 3 ✳ 2 3 (x + 3)(x − 3) x2 − 9 = lim = ✱ x→3 (x + 1)(x − 3) x→3 x2 − 2x − 3 2 lim .f (x) = lim x→3 lim .f (x) = x→3 P♦rt❛♥t♦✱ f (x) ❡①✐st❡ ♦ ❧✐♠✐t❡✳ 3 ✳ 2 é ❝♦♥tí♥✉❛ ❡♠ x = 3✳ ❊①❡♠♣❧♦ ✹✳✷✳ ❙✉♣♦♥❤❛ q✉❡ ♦ ❝✉st♦ ❞❡ tr❛♥s♣♦rt❡ ❞❡ t❛①❛ ♣♦st❛❧ s❡❥❛✿ R$0, 30 ❛té 300 ❣r❛♠❛s✱ ❡ R$1, 70 s❡ ♦ ♣❡s♦ ❢♦r ♠❛✐♦r q✉❡ 300 ❣r❛♠❛s ❡ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ 500 ❣r❛♠❛s✳ ❙❡ x ❣r❛♠❛s r❡♣r❡s❡♥t❛ ♦ ♣❡s♦ ❞❡ ✉♠❛ ❝❛rt❛ (0 < x ≤ 500)✱ ❡①♣r❡ss❡ ❛ t❛①❛ ♣♦st❛❧ ❝♦♠♦ ❢✉♥çã♦ ❞❡ x✳ ❙♦❧✉çã♦✳ ❚❡♠♦s f (x) = 0, 30x s❡ 0 < x ≤ 300❀ f (x) = 1, 70x f (x) = ( 0, 30x, 1, 70x, ♦❜s❡r✈❡ q✉❡ ❛ ❢✉♥çã♦ ♥ã♦ é ❝♦♥tí♥✉❛ ❡♠ s❡✱ s❡✱ s❡ 300 < x ≤ 500❀ ✐st♦ é✿ 0 < x ≤ 300 ; 300 < x ≤ 500 x = 300✳ ✷✶✾ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❊①❡♠♣❧♦ ✹✳✸✳ ❉❛❞❛ ❛ ❢✉♥çã♦✿  2   x − 6x + 1, f (x) = 2x + 6,   3 x − 15, ❉❡t❡r♠✐♥❡ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ f ❡♠ s❡✱ s❡✱ s❡✱ x=2 ❡ 1<x≤2 2<x≤3 3<x<5 x = 3✳ ❙♦❧✉çã♦✳ P❛r❛ ♦ ♣♦♥t♦ x = 2✳ ❡①✐st❡✳ ✐✮ f (2) = −7 P❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ lim .f (x) é ♥❡❝❡ssár✐♦ ❝❛❧❝✉❧❛r ♦s ❧✐♠✐t❡s ❧❛t❡r❛✐s✳ ✐✐✮ x→2 lim− .f (x) = lim− (x2 − 6x + 1) = −7; x→2 x→2 lim .f (x) = lim+ (2x + 6) = 10 x→2+ x→2 P♦rt❛♥t♦✱ ♥ã♦ ❡①✐st❡ lim .f (x)❀ ❛ss✐♠✱ f (x) ♥ã♦ é ❝♦♥tí♥✉❛ ❡♠ x = 2✳ x→2 P❛r❛ ♦ ♣♦♥t♦ x = 3✳ ✐✮ f (3) = 12 ❡①✐st❡✳ P❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ lim .f (x) é ♥❡❝❡ssár✐♦ ❝❛❧❝✉❧❛r ♦s ❧✐♠✐t❡s ❧❛t❡r❛✐s✳ ✐✐✮ x→3 lim .f (x) = lim− (2x + 6) = 12; x→3− x→3 lim .f (x) = lim+ (x3 − 15) = 12 x→3+ x→3 P♦rt❛♥t♦✱ lim .f (x) = 12❀ ❡①✐st❡✳ x→3 ✐✐✐✮ lim .f (x) = 12 = f (3)✳ x→3 ❖❜s❡r✈❛çã♦ ✹✳✶✳ ✐✮ ❙✉♣♦♥❤❛ f (x) ❞❡s❝♦♥tí♥✉❛ ❡♠ x = a✱ ❞❡ ♠♦❞♦ q✉❡ ❡①✐st❛♠ f (a) ∈ R ❡ lim .f (x) x→a ♣♦ré♠ lim .f (x) 6= f (a) ✱ ❡♥tã♦ ❞✐③✲s❡ q✉❡ ❛ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ é ❡✈✐tá✈❡❧ ♦✉ r❡♠♦✈í✈❡❧ ❀ x→a ♣♦✐s ♣♦❞❡♠♦s r❡❞❡✜♥✐r ❛ ❢✉♥çã♦ f (x) ❞❡ ♠♦❞♦ q✉❡ lim .f (x) = f (a) ✱ ✐st♦ é ❛ ❢✉♥çã♦ x→a f r❡❞❡✜♥✐❞❛ r❡s✉❧t❛ s❡r ❝♦♥tí♥✉❛ ❡♠ x = a✳ ✐✐✮ ❙❡ ❛ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❡♠ x = a ♥ã♦ é ❡✈✐tá✈❡❧ ♦✉ r❡♠♦✈í✈❡❧✱ ❝❤❛♠❛✲s❡ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❡ss❡♥❝✐❛❧ ❀ ❡st❡ ❝❛s♦ ♦❝♦rr❡ q✉❛♥❞♦ lim .f (x) ♥ã♦ ❡①✐st❡ ♦✉ ♥ã♦ é ✜♥✐t♦✳ x→a ❊①❡♠♣❧♦ ✹✳✹✳ ❉❡t❡r♠✐♥❡ ♦s ♣♦♥t♦s ❞❡ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❢✉♥çã♦✿ ❙♦❧✉çã♦✳ ✷✷✵ f (x) = 6x + 24 ✳ + 3x − 4 x2 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❖❜s❡r✈❡ q✉❡ x2 + 3x − 4 = (x + 4)(x − 1)✳ ❖ ❞❡♥♦♠✐♥❛❞♦r ❞❛ ❢✉♥çã♦ é ③❡r♦ q✉❛♥❞♦ x = −4 ♦✉ x = 1✱ ❡ss❡s sã♦ ♦s ♣♦ssí✈❡✐s ♣♦♥t♦s ❞❡ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡✱ ♣♦✐s f ♥ã♦ ❡st❛ 6 ❡ lim .f (x) = ∞✳ ❞❡✜♥✐❞❛ ♥❡ss❡s ♣♦♥t♦s ❡ ♦s ❧✐♠✐t❡s r❡s♣❡❝t✐✈♦s sã♦✿ lim .f (x) = − x→1 x→−4 5 ❆ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❡♠ x = 1 é ❡ss❡♥❝✐❛❧ ❡ ♥♦ ♣♦♥t♦ x = −4 é ❡✈✐tá✈❡❧❀ ♣❛r❛ ♦s ❞❡♠❛✐s ✈❛❧♦r❡s ❞❡ x ❛ ❢✉♥çã♦ é ❝♦♥tí♥✉❛✳  6x + 24 s❡✱ x 6= −4 + 3x − 4 P♦❞❡♠♦s r❡❞❡✜♥✐r ❛ ❢✉♥çã♦ f (x) ❛ss✐♠✿ g(x) =   −6 s❡✱ x = −4 5 ❖❜s❡r✈❡ q✉❡ g(x) é ❝♦♥tí♥✉❛ ❡♠ x = −4✱ ❡♥t❛♥t♦ ❛ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❡♠ x = 1 é   x2 ❡ss❡♥❝✐❛❧✳ P❛r❛ ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s✱ ❛❧❣✉♠❛s ✈❡③❡s é út✐❧ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✱ ❡q✉✐✈❛❧❡♥t❡ à ❉❡✜♥✐çã♦ ✭✹✳✶✮✳ ❉❡✜♥✐çã♦ ✹✳✷✳ ❙❡❥❛ y = f (x) ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♥♦ ❝♦♥❥✉♥t♦ A ⊆ R✱ ❡ a ∈ A❀ ❞✐③✲s❡✱ q✉❡ f é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ x = a✱ s❡✿ ❉❛❞♦ ε > 0, ∃ δ > 0/. x ∈ B(a, δ)✱ ❡♥tã♦ f (x) ∈ B(f (a), ε)❀ ♦✉ ❉❛❞♦ ε > 0, ∃ δ > 0/. |x − a| < δ ⇒ |f (x) − f (a)| < ε✳ ❉❡✜♥✐çã♦ ✹✳✸✳ ❈♦♥t✐♥✉✐❞❛❞❡ ♥✉♠ ❝♦♥❥✉♥t♦ ❯♠❛ ❢✉♥çã♦ f : A −→ R✱ ❞✐③✲s❡ q✉❡ é ❝♦♥tí♥✉❛ ♥♦ ❝♦♥❥✉♥t♦ B ⊆ A s❡✱ ❡ s♦♠❡♥t❡ s❡ é ❝♦♥tí♥✉❛ ❡♠ x = a, ∀ a ∈ B ✳ ❊①❡♠♣❧♦ ✹✳✺✳ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ é ❝♦♥tí♥✉❛ ❡♠ t♦❞♦ s❡✉ ❞♦♠í♥✐♦✳ ❙♦❧✉çã♦✳ ❙❡❥❛♠ k ∈ R ✉♠❛ ❝♦♥st❛♥t❡✱ ❡ f : A −→ R ❞❡✜♥✐❞❛ ♣♦r f (x) = k ∀ x ∈ A✱ ❡♥tã♦ f (a) = k ∀ a ∈ A✳ ▲♦❣♦✱ ❞❛❞♦ ε > 0 ❡①✐st❡ δ > 0✱ ♣♦✐s | f (x) − f (a) |=| k − k |=| (x − a) + (a − x) |≤ 2 | x − a |< ε ⇒ δ= ε 2 P♦rt❛♥t♦✱ s❡♥❞♦ x = a ✉♠ ❡❧❡♠❡♥t♦ ❛r❜✐trár✐♦✱ f (x) = k é ❝♦♥tí♥✉❛ ♥♦ ❝♦♥❥✉♥t♦ A✳ ❊①❡♠♣❧♦ ✹✳✻✳ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ f (x) = x2 é ❝♦♥tí♥✉❛ ❡♠ t♦❞♦ s❡✉ ❞♦♠í♥✐♦✳ ❙♦❧✉çã♦✳ ✷✷✶ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❙❡❥❛ f : A −→ R ❞❡✜♥✐❞❛ ♣♦r f (x) = x2 ∀ x ∈ A✱ ❡♥tã♦ f (a) = a2 ♣❛r❛ x = a✱ ♦♥❞❡ a ∈ A❀ ❛ss✐♠ ❞❛❞♦ ε > 0✱ ❡①✐st❡ δ > 0 t❛❧ q✉❡✿ | f (x) − f (a) |=| x2 − a2 |=| x − a | · | x + a |< | x − a | (| x | + | a |) ✭✹✳✷✮ ❙❡ a = 0 ❛ ❞❡s✐❣✉❛❧❞❛❞❡ é ✐♠❡❞✐❛t❛✳ P♦r ♦✉tr♦ ❧❛❞♦ s❡❥❛ a 6= 0✱ q✉❛♥❞♦ | x − a |< δ ❡ ❞❛ ♣r♦♣r✐❡❞❛❞❡ || x | − | a || < | x − a | < δ s❡❣✉❡✲s❡ q✉❡ | x | < δ+ | a |✱ ❝♦♥s✐❞❡r❡ ✉♠ δ1 = < |a| ✱ ❡♥tã♦ t❡♠♦s | x |< 2 3|a| |a| +|a|= 2 2 ✭✹✳✸✮ ❉❡ ✭✹✳✷✮ ❡ ✭✹✳✸✮ s❡❣✉❡ q✉❡✿ | f (x) − f (a) | < | x − a | (| x | + | a |) <| x − a | .( 3|a| 5|a| + | a |) < |x−a|< ε 2 2 2ε |a| , } t❡♠♦s✱ ♣❛r❛ t♦❞♦ ε > 0 ❝✉♠♣r❡✲s❡ | f (x) − 2 5|a| f (a) |< ε s❡♠♣r❡ q✉❡ | x − a |< δ ✳ ❈♦♥s✐❞❡r❛♥❞♦ δ = min .{ Pr♦♣r✐❡❞❛❞❡ ✹✳✶✳ ❙❡❥❛♠ f (x) ❡ g(x) ❞✉❛s ❢✉♥çõ❡s r❡❛✐s ❡ ❝♦♥tí♥✉❛s ❡♠ x=a ❡ k ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧✱ ❡♥tã♦✿ ✐✮ k · f (x) é ❝♦♥tí♥✉❛ ❡♠ x = a✳ ✐✐✮ (f ± g)(x) é ❝♦♥tí♥✉❛ ❡♠ x = a✳ ✐✐✐✮ (f · g)(x) é ❝♦♥tí♥✉❛ ❡♠ x = a✳ ✐✈✮ | f | (x) é ❝♦♥tí♥✉❛ ❡♠ x = a✳   f (x) é ❝♦♥tí♥✉❛ ❡♠ x = a✱ ❞❡s❞❡ q✉❡ g(a) 6= 0✳ ✈✮ g ✭✐✐✮ ❉❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ f (x) ❡ g(x) t❡♠♦s lim .f (x) = f (a) ❡ lim .g(x) = g(a) x→a x→a ❞❛ ❞❡✜♥✐çã♦ ❞❛ ❢✉♥çã♦ (f ± g)(x) = f (x) ± g(x) ❡ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❧✐♠✐t❡ ❞❛ s♦♠❛✱ s❡❣✉❡✲s❡ q✉❡✿ ❉❡♠♦♥str❛çã♦✳ lim (f ± g)(x) = lim [f (x) ± g(x)] = lim .f (x) ± lim .g(x) = f (a) ± g(a) = (f ± g)(a) x→a x→a x→a x→a P♦rt❛♥t♦ ❛ ❢✉♥çã♦ (f ± g)(x) é ❝♦♥tí♥✉❛ ❡♠ x = a✳  ❆s ♦✉tr❛s ♣r♦♣r✐❡❞❛❞❡s ♠♦str❛♠✲s❡ ❛♣❧✐❝❛♥❞♦ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❧✐♠✐t❡✱ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❖❜s❡r✈❛çã♦ ✹✳✷✳ ❆ r❡❝í♣r♦❝❛ ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭✹✳✶✮ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ é ✈❡r❞❛❞❡✐r❛ ❝♦♠♦ s❡ ♠♦str❛ ♥♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✳ ✷✷✷ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❊①❡♠♣❧♦ ✹✳✼✳ ❆s ❢✉♥çõ❡s r❡❛✐s f (x), g(x) ❡ h(x) ❞❡✜♥✐❞❛s ♣♦r✿ g(x) = ( 1, s❡✱ x ≤ 0 0, s❡✱ x > 0 h(x) = f (x) = ( ( 0, s❡✱ x ≤ 0 1, s❡✱ x > 0 −1, s❡✱ x ≤ 0 1, s❡✱ x > 0 ♥ã♦ sã♦ ❝♦♥tí♥✉❛s ❡♠ x = 0✳ P♦ré♠✱ ♣❛r❛ t♦❞♦ x ∈ R t❡♠♦s f (x) + g(x) = 1, f (x) · g(x) = 0 ❡ | h(x) |= 1✱❛ss✐♠✱ ❡st❛s três ú❧t✐♠❛s ❢✉♥çõ❡s sã♦ ❝♦♥tí♥✉❛s ❡♠ t♦❞♦ R✳ Pr♦♣r✐❡❞❛❞❡ ✹✳✷✳ ✐✮ f : R −→ R ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥ô♠✐❝❛✱ ✐st♦ é f (x) = a0 xn + a1 xn−1 + a2 xn−2 + · · · + an−1 x + an , a0 6= 0 ❡♥tã♦ f (x) é ❝♦♥tí♥✉❛ ∀ x ∈ R✳ ❙❡❥❛ ✐✐✮ f : R −→ R ❙❡❥❛ ✉♠❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧✱ ✐st♦ é✿ f (x) = ❡♥tã♦ f (x) a0 xn + a1 xn−1 + a2 xn−2 + · · · + an−1 x + an b0 xm + b1 xm−1 + b2 xm−2 + · · · + bm−1 x + bm é ❝♦♥tí♥✉❛ ♥♦ ❝♦♥❥✉♥t♦✿ { x ∈ R/. b0 xm + b1 xm−1 + b2 xm−2 + · · · + bm−1 x + bm 6= 0 } ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❊①❡♠♣❧♦ ✹✳✽✳ ❉❡t❡r♠✐♥❛r ♦s ✈❛❧♦r❡s ❞❡ x2 − 1 f (x) = 2 x −9 ❛✮ x✱ ♣❛r❛ ♦s q✉❛✐s ❛s ❢✉♥çõ❡s ❞❛❞❛s s❡❥❛♠ ❝♦♥tí♥✉❛s✿ ❜✮ g(x) =| x2 − 16 | ❝✮ h(x) = x5 (x + 3)7 ❙♦❧✉çã♦✳ ✭❛✮ ❚❡♠♦s f (x) é ❢✉♥çã♦ r❛❝✐♦♥❛❧ ❡ s❡✉ ❞♦♠í♥✐♦ é D(f ) = { x ∈ R/. x 6= ±3 }❀ ❧♦❣♦ ❡❧❛ é ❝♦♥tí♥✉❛ ❡♠ D(f )✳ ✭❜✮ ❆ Pr♦♣r✐❡❞❛❞❡ ✭✹✳✶✮✲ ✈✮✱ ❣❛r❛♥t❡ q✉❡ g(x) =| x2 − 16 | s❡❥❛ ❝♦♥tí♥✉❛ ♣❛r❛ t♦❞♦ x ∈ R✳ ✭❝✮ ❆ ❢✉♥çã♦ h(x) = x5 (x + 3)7 é ♣♦❧✐♥ô♠✐❝❛✱ ❡♥tã♦ ❡❧❛ é ❝♦♥tí♥✉❛ ♣❛r❛ t♦❞♦ x ∈ R✳ Pr♦♣r✐❡❞❛❞❡ ✹✳✸✳ f : A −→ R ❡ g : B −→ R ❢✉♥çõ❡s r❡❛✐s t❛✐s q✉❡ Im(f ) ⊆ B ✱ s❡♥❞♦ f x = a ❡ g ❝♦♥tí♥✉❛ ❡♠ y = f (a)✱ ❡♥tã♦ g ◦ f é ❝♦♥tí♥✉❛ ❡♠ x = a✳ ❈♦♥s✐❞❡r❡ ❝♦♥tí♥✉❛ ❡♠ ✷✷✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❉❡♠♦♥str❛çã♦✳ ❆ ♠♦str❛r q✉❡ ❞❛❞♦ ε > 0, ∃ δ > 0 /. | g(f (x)) − g(f (a)) |< ε s❡♠♣r❡ q✉❡ | x − a |< δ ✳ ❈♦♠ ❡❢❡✐t♦✱ ❞♦ ❢❛t♦ g ❝♦♥tí♥✉❛ ❡♠ f (a) = b t❡♠♦s✱ ❞❛❞♦ ε > 0 ∃ δ1 > 0/. s❡ y ∈ B, | g(y) − g(b) |< ε s❡♠♣r❡ q✉❡✿ ✭✹✳✹✮ | y − b |< δ1 P♦r ♦✉tr♦ ❧❛❞♦✱ f é ❝♦♥tí♥✉❛ ❡♠ x = a✱ ❡♥tã♦ ❞❛❞♦ ε1 > 0✱ ❡♠ ♣❛rt✐❝✉❧❛r ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ε1 = δ1 ✱ ❡①✐st❡ δ > 0 t❛❧ q✉❡ s❡ x ∈ A, | f (x) − f (a) |< δ1 s❡♠♣r❡ q✉❡ ✭✹✳✺✮ | x − a |< δ ❉♦ ❢❛t♦ Im(f ) ⊆ B ♣♦❞❡♠♦s ❡❢❡t✉❛r ❛ ❝♦♠♣♦s✐çã♦ ❡♥tr❡ ❛s ❢✉♥çõ❡s g ❡ f ♣❛r❛ ♦❜t❡r (g ◦ f )(x) = g(f (x)) ♣❛r❛ t♦❞♦ x ∈ A ❡ y = f (x)❀ ❡♥tã♦ ❞❡ ✭✹✳✹✮ ❡ ✭✹✳✺✮ ♦❜té♠✲s❡ q✉❡✱ ❞❛❞♦ ε > 0, ∃ δ > 0 /. s❡ x ∈ A, | g(y) − g(b) |=| g(f (x)) − g(f (a)) |< ε s❡♠♣r❡ q✉❡ | y − b |=| f (x) − f (a) |< δ1 s❡♠♣r❡ q✉❡ | x − a |< δ ✳ P♦rt❛♥t♦✱ ❞❛❞♦ ε > 0, ∃ δ > 0/. s❡ x ∈ A, | g(f (x)) − g(f (a)) |< ε s❡♠♣r❡ q✉❡ | x − a |< δ ✳ Pr♦♣r✐❡❞❛❞❡ ✹✳✹✳ ❙❡❥❛♠ ✐✮ f : A −→ R ❡ ✐✐✮ lim .f (x) = b x→a ❡♥tã♦ g : B −→ R g ❢✉♥çõ❡s r❡❛✐s t❛✐s q✉❡ ❝♦♥tí♥✉❛ ❡♠ Im(f ) ⊆ B ❡✿ y = b✳ lim g(f (x)) = g(lim .f (x)) = g(b) x→a x→a ❉❡♠♦♥str❛çã♦✳ ❉❡✜♥✐♠♦s h(x) = ( f (x), s❡✱ x 6= a 0, s❡✱ x = a ❞❛ ❤✐♣ót❡s❡ ✐✮ t❡♠♦s h é ❝♦♥tí♥✉❛ ❡♠ x = a❀ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✹✳✸✮ ❛ ❢✉♥çã♦ goh é ❝♦♥tí♥✉❛ ❡♠ x = a✱ ✐st♦ é✿ lim (g ◦ f )(x) = (g ◦ h)(a) = g(h(a)) = g(b) = g(lim ·f (x))✳ x→a x→a P♦r ♦✉tr♦ ❧❛❞♦✱ ❛s ❢✉♥çõ❡s f ❡ h sã♦ ❞✐❢❡r❡♥t❡s s♦♠❡♥t❡ ♥♦ ♣♦♥t♦ x = a✱ ❡♥tã♦ lim (g ◦ x→a h)(x) = lim (g ◦ f )(x)✳ x→a P♦rt❛♥t♦✱ lim (g ◦ f )(x) = lim ·g(f (x)) = g(lim ·f (x))✳ x→a x→a x→a ✷✷✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✹✲✶ ✶✳ ▼♦str❡ ✉t✐❧✐③❛♥❞♦ ε ❡ δ q✉❡ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ ✐♥❞✐❝❛❞♦✳ 1. f (x) = −5x + 6, 3. f (x) = x4 , 2. f (x) = 3x2 + 5, a = −2 a=3 4. f (x) = x2 + 5x + 6, a=1 a = −1 B(a, r) ❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ M > 0 | f (x) − f (a) |≤ M | x − a |, ∀ x ∈ B(a, r)✳ ▼♦str❡ ✷✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❝✉♠♣r❡ ❛ ❝♦♥❞✐çã♦✿ ❝♦♥tí♥✉❛ ❡♠ x = a✳ ✸✳ ▼♦str❡ q✉❡ s❡ ✹✳ ▼♦str❡ q✉❡ lim .f (x) = L > 0✱ x→a f (x) = [|x|] ❡♥tã♦ x = a✱ é ❝♦♥tí♥✉❛ ❡♠ t♦❞♦ x=a fi é ❝♦♥tí♥✉❛ ❡♠ x = a✳ ✷✳ f1 × f2 × f3 × · · · × fn é ❝♦♥tí♥✉❛ ❡♠ x = a✳ ✻✳ P❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✳ ✶✳ ✷✳ ✸✳ ✹✳ ✺✳ f (x) = f (x) = ( é ♦♥❞❡ a ∈ R − Z✳ i = 1, 2, 3, · · · n sã♦ ❢✉♥çõ❡s ❡♥tã♦✿ f1 + f2 + f3 + · · · + fn ( f x→a ✶✳ ✐♥❞✐❝❛❞♦s✳ q✉❡ p √ lim . n f (x) = n L✳ ✺✳ ❯s❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦✱ ♠♦str❡ q✉❡ s❡✿ ❝♦♥tí♥✉❛s ❡♠ t❛❧ q✉❡ 3x − 3 2 x2 1− | x | s❡✱ s❡✱ s❡✱ s❡✱ x 6= 1 x=1 ❉❡t❡r♠✐♥❡ s❡ ❡❧❛ é ❝♦♥tí♥✉❛ ♥♦s ♣♦♥t♦s a=1 x ≥ −1 x < −1 a = −1  2  s❡✱ x<1  1−x f (x) = a = 1, a = −1 1− | x | s❡✱ x > 1   1 s❡✱ x=1    x + 2 s❡✱ − 2 ≤ x ≤ −1 f (x) = a = 1, a = −1 1 s❡✱ −1<x<1   2 − x s❡✱ 1 ≤ x  2 x −x−2   s❡✱ x 6= ±2 | x2 − 4 | a = 2, a = −2 f (x) =   4 s❡✱ x = ±2 3 ✷✷✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✻✳ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠   s❡✱ − 3 ≤ x ≤ 0  −1 f (x) = x − 1 s❡✱ 0 < x < 2   5 − x2 s❡✱ 2 ≤ x R a=0 a=2 ✼✳ ❉❛r ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ❢✉♥çã♦ f ❞❡✜♥✐❞❛ ❡♠ R q✉❡ ♥ã♦ s❡❥❛ ❝♦♥tí♥✉❛ ❡♠ ♥❡♥❤✉♠ ♣♦♥t♦ x ∈ R✱ ♣♦ré♠ q✉❡✱ | f (x) | s❡❥❛ ❝♦♥tí♥✉❛ ❡♠ t♦❞♦ R✳ ✽✳ P❛r❛ ♦s s❡❣✉✐♥t❡s ❡①❡r❝í❝✐♦s✱ ❞❡t❡r♠✐♥❡ s❡ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r ✉♠ ♥ú♠❡r♦ L ♣❛r❛ q✉❡ ❛ ❢✉♥çã♦ f s❡❥❛ ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ x = a✳ ◆♦ ❝❛s♦ ❛✜r♠❛t✐✈♦ ❞❡t❡r♠✐♥❡ L✱ ❝❛s♦ ❝♦♥trár✐♦ ❥✉st✐✜❝❛r s✉❛ r❡s♣♦st❛✳ ✶✳ ✷✳ ✸✳ ✹✳ ✺✳ ✻✳ ✼✳ ✽✳  2  x − 3x − 4 s❡✱ x 6= 4 f (x) = a = 4✳ x−4  L s❡✱ x = 4   s❡✱ x > 0  |x| 2 f (x) = a = 0✳ 1 − x s❡✱ x < 0   L s❡✱ x = 0  2  s❡✱ | x |< 1  1−x f (x) = a = ±1✳ | x | −1 s❡✱ | x |> 1   L s❡✱ | x |= 1  √  x−2 , s❡✱ x 6= 4 a = 4✳ f (x) = x−4  L s❡✱ x = 4    | x | −2 s❡✱ | x |< 2 f (x) = a = 2, a = −2✳ 4 − x2 s❡✱ | x |> 2   L s❡✱ | x |= 2  2   Sgn(9 − x ) s❡✱ | x |> 4 f (x) = a = 4✳ | x2 − 16 | −1 s❡✱ | x |< 4   L s❡✱ | x |= 4  2  | x − 2x − 3 | , s❡✱ x 6= 3 a = 3✳ f (x) = x−3  L, s❡✱ x = 3 ( 4 − x2 s❡✱ | x |< 2 f (x) = a = 2, a = −2✳ L s❡✱ | x |> 2 ✾✳ ❉❡t❡r♠✐♥❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❞❡ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❢✉♥çã♦ y = f (x)✿   0     x f (x) =  −x2 + 4x − 2     4−x ✷✷✻ s❡✱ s❡✱ s❡✱ s❡✱ x<0 0≤x<1 1≤x<3 x≥3 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✶✵✳ ❊st✉❞❡ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❢✉♥çã♦ f (x) = π 1 ♥♦ ♣♦♥t♦ x = ✳ 2 + 2tan x 2 ✶✶✳ ❊st✉❞❡ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❢✉♥çã♦ g(x) = sen( x1 ) √ ♥♦ ♣♦♥t♦ x = 0✳ 1+ xe R ✶✷✳ P❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x = a✱ ❛❝❤❛r ✉♠❛ ❢✉♥çã♦ q✉❡ s❡❥❛ ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ x = a✱ ♣♦ré♠ q✉❡ ♥ã♦ s❡❥❛ ❝♦♥tí♥✉❛ ❡♠ ♥❡♥❤✉♠ ♦✉tr♦ ♣♦♥t♦✳ ✶✸✳ ❙✉♣♦♥❤❛ f (x) ❝✉♠♣r❡ f (x + y) = f (x) + f (y)✱ ❡ q✉❡ f s❡❥❛ ❝♦♥tí♥✉❛ ❡♠ x = 0✳ ▼♦str❡ q✉❡ f é ❝♦♥tí♥✉❛ ❡♠ x = a ∀ a ∈ R✳ 1 1 1 ✶✹✳ ❉❡t❡r♠✐♥❡ ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ❡♠ t♦❞♦ R q✉❡ s❡❥❛ ❞❡s❝♦♥tí♥✉❛ ❡♠ 1, , , , · · · 2 3 4 ❡ q✉❡ s❡❥❛ ❝♦♥tí♥✉❛ ♥♦s ❞❡♠❛✐s ♣♦♥t♦s✳ ✶✺✳ ❙✉♣♦♥❤❛ f é ✉♠❛ ❢✉♥çã♦ q✉❡ ❝✉♠♣r❡ | f (x) |≥| x | ∀ x ∈ R✳ ❉❡♠♦♥str❛r q✉❡ f é ❝♦♥tí♥✉❛ ❡♠ x = 0 ✭❧❡♠❜r❡ q✉❡ f (0) t❡♠ q✉❡ s❡r 0✮✳ ✶✳ ✷✳ ❉❛r ✉♠ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ❢✉♥çã♦ f q✉❡ ♥ã♦ s❡❥❛ ❝♦♥tí♥✉❛ ❡♠ ♥❡♥❤✉♠ x = a✳ ❙✉♣♦♥❤❛✲s❡ q✉❡ g s❡❥❛ ❝♦♥tí♥✉❛ ❡♠ x = 0, ▼♦str❡ q✉❡ f é ❝♦♥tí♥✉❛ ❡♠ x = 0✳ ✸✳ g(0) = 0 ❡ | f (x) |≤| g(x) | ∀x ∈ R✳ ✶✻✳ ❖s r❛✐♦s ❞❡ três ❝✐❧✐♥❞r♦s s✉♣❡r♣♦st♦s ♠❡❞❡♠ 3, 2 ❡ 1 ♠❡tr♦s r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆s ❛❧t✉r❛s ❞❡ ❝❛❞❛ ✉♠ ❞♦s ❝✐❧✐♥❞r♦s é 5m✳ ❊①♣r❡ss❛r ❛ ár❡❛ ❞❛ s❡çã♦ tr❛♥s✈❡rs❛❧ ❞♦ ❝♦r♣♦ ❣❡r❛❞♦ ❝♦♠♦ ❢✉♥çã♦ ❞❛ ❞✐stâ♥❝✐❛ q✉❡ r❡❧❛❝✐♦♥❛ ❛ s❡çã♦ ❡ ❛ ❜❛s❡ ✐♥❢❡r✐♦r ❞♦ ❝✐❧✐♥❞r♦ q✉❡ ♦❝✉♣❛ ❛ ♣❛rt❡ ❜❛✐①❛ ❞♦ ❝♦r♣♦✳ ❙❡rá ❡st❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛❄ ❈♦♥str✉✐r ♦ ❣rá✜❝♦✳ ✶✼✳ ❈♦♠♦ ❞❡✈❡♠♦s ❡❧❡❣❡r ♦ ♥ú♠❡r♦ ( α ♣❛r❛ q✉❡ ❛ ❢✉♥çã♦ f (x) s❡❥❛ ❝♦♥tí♥✉❛ ❡♠ R❄ x + 1, s❡✱ x ≤ 1 ❈♦♥str✉✐r s❡✉ ❣rá✜❝♦✳ f (x) = ✳ α, s❡✱ x > 1 ✶✽✳ ❉❡t❡r♠✐♥❡ ♦s ♥ú♠❡r♦s A ❡ B ❞❡ ♠♦❞♦ q✉❡ ❛ ❢✉♥çã♦ g(x) s❡❥❛ ❝♦♥tí♥✉❛ ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s R✳  π  −2senx s❡✱ x ≤ −    π π 2 Asenx + B s❡✱ − < x < g(x) = 2π 2     cos x s❡✱ x ≥ 2 ✶✾✳ ❉❡t❡r♠✐♥❡ ♦ (✈❛❧♦r ❞❡ a ❞❡ ♠♦❞♦ q✉❡ ❛ ❢✉♥çã♦ g(x) s❡❥❛ ❝♦♥tí♥✉❛ ❡♠ t♦❞❛ ❛ r❡t❛ x + 2 s❡✱ x ≤ 3 r❡❛❧✳ g(x) = ax + 7 s❡✱ x > 3 ✷✷✼ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✵✳ ❉❡t❡r♠✐♥❡ ♦s ✈❛❧♦r❡s ( ❞❡ b ❡ c ❞❡ ♠♦❞♦ q✉❡ ❛ ❢✉♥çã♦ f (x) s❡❥❛ ❝♦♥tí♥✉❛ ❡♠ t♦❞❛ ❛ x+1 s❡✱ 1 < x < 3 r❡t❛ r❡❛❧✳ f (x) = 2 x + bx + c s❡✱ | x − 2 |≥ 1 ✷✶✳ ❙❡ lim .f (x) ❡①✐st❡✱ ♣♦ré♠ é ❞✐❢❡r❡♥t❡ ❞❡ f (a)✱ ❞✐③❡♠♦s q✉❡ f t❡♠ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ x→a ❡✈✐tá✈❡❧ ❡♠ x = a ✳ 1 ♣❛r❛ x 6= 0✳ ❆ ❢✉♥çã♦ f t❡♠ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❡✈✐tá✈❡❧ ❡♠ x = 0 x 1 ❄ ◗✉❡ ❛❝♦♥t❡❝❡ s❡ f (x) = x · sen ♣❛r❛ x 6= 0 ❡ f (0) = 1 ❄ x ✷✳ ❙✉♣♦♥❤❛ q✉❡ g t❡♥❤❛ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❡✈✐tá✈❡❧ ❡♠ x = a✳ ❙❡❥❛ h(x) = g(x) ♣❛r❛ x 6= a ❡ s❡❥❛ h(a) = lim .g(x)✳ ▼♦str❡ q✉❡ h é ❝♦♥tí♥✉❛ ❡♠ x = a✳ ✶✳ ❙❡ f (x) = sen x→a ✸✳ p 1 p s❡ é ✉♠❛ ❢r❛çã♦ ✐rr❡❞✉tí✈❡❧✳ ◗✉❛❧ é ❛ q q q ❢✉♥çã♦ g ❞❡✜♥✐❞❛ ♣♦r g(x) = lim .f (y) ❙❡❥❛ f (x) = 0 s❡ x ∈ Q✱ ❡ f ( ) = y→x ✷✷✳ ◆✉♠❛ ❝♦♠✉♥✐❞❛❞❡ ❞❡ 8.000 ♣❡ss♦❛s✱ ❛ r❛③ã♦ s❡❣✉♥❞♦ ❛ q✉❛❧ ✉♠ ❜♦❛t♦ s❡ ❡s♣❛❧❤❛ é ❝♦♥❥✉♥t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s q✉❡ ♦✉✈✐r❛♠ ♦ ❜♦❛t♦ ❡ ❛♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s q✉❡ ♥ã♦ ♦ ♦✉✈✐r❛♠✳ ✶✳ ❙❡ ♦ ❜♦❛t♦ ❡stá s❡ ❡s♣❛❧❤❛♥❞♦ ❛ ✉♠❛ r❛③ã♦ ❞❡ 20 ♣❡ss♦❛s ♣♦r ❤♦r❛ q✉❛♥❞♦ 200 ♣❡ss♦❛s ♦ ♦✉✈✐r❛♠✱ ❡①♣r❡ss❡ ❛ t❛①❛ s❡❣✉♥❞♦ ♦ q✉❛❧ ♦ ❜♦❛t♦ ❡st❛ s❡ ❡s♣❛❧❤❛♥❞♦ ❝♦♠♦ ❢✉♥çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s q✉❡ ♦ ♦✉✈✐r❛♠✳ ✷✳ ◗✉ã♦ rá♣✐❞♦ ♦ ❜♦❛t♦ ❡stá s❡ ❡s♣❛❧❤❛♥❞♦ q✉❛♥❞♦ 500 ♣❡ss♦❛s ♦ ♦✉✈✐r❛♠❄ ✷✸✳ ❯♠❛ ❞❡t❡r♠✐♥❛❞❛ ❧❛❣♦❛ ♣♦❞❡ s✉♣♦rt❛r ✉♠ ♠á①✐♠♦ ❞❡ 14.000 ♣❡✐①❡s✱ ❡ ❛ t❛①❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❡❧❡s é ❝♦♥❥✉♥t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ♥ú♠❡r♦ ♣r❡s❡♥t❡ ❡ à ❞✐❢❡r❡♥ç❛ ❡♥tr❡ 14.000 ❡ ❛ q✉❛♥t✐❞❛❞❡ ❡①✐st❡♥t❡✳ ❛✮ ❙❡ f (x) ♣❡✐①❡s ♣♦r ❞✐❛ ❢♦r ❛ t❛①❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ q✉❛♥❞♦ ❤♦✉✈❡r x ♣❡✐①❡s✱ ❡s❝r❡✈❛ ✉♠❛ ❢✉♥çã♦ q✉❡ ❞❡✜♥❛ f (x)✳ ❜✮ ▼♦str❡ q✉❡ f (x) é ❝♦♥tí♥✉❛ ❡♠ t♦❞♦ s❡✉ ❞♦♠í♥✐♦✳ ✷✷✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✹✳✷ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈♦♥t✐♥✉✐❞❛❞❡ ❡♠ ✐♥t❡r✈❛❧♦s ❉❡✜♥✐çã♦ ✹✳✹✳ ❯♠❛ ❢✉♥çã♦ f : (a, b) −→ R é ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ (a, b) ✱ s❡ é ❝♦♥tí♥✉❛ ❡♠ t♦❞♦ x ∈ (a, b)✳ ❊①❡♠♣❧♦ ✹✳✾✳ ❆s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s✱ tr✐❣♦♥♦♠étr✐❝❛s✿ s❡♥♦ ❡ ❝♦s❡♥♦✱ ❛s ❡①♣♦♥❡♥❝✐❛✐s ❡ ♦s ❧♦❣❛r✐t✲ ♠♦s sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❡♠ s❡✉s r❡s♣❡❝t✐✈♦s ❞♦♠í♥✐♦s ❞❡ ❞❡✜♥✐çã♦✳ ❆ ♣❛rá❜♦❧❛✱ ❝♦♠♦ ❢✉♥çã♦ ♣♦❧✐♥ô♠✐❝❛✱ é ✉♠ ❡①❡♠♣❧♦ ❞❡ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♠ t♦❞♦ s❡✉ ❞♦♠í♥✐♦ R✳ ❉❡✜♥✐çã♦ ✹✳✺✳ ❛✮ ❯♠❛ ❢✉♥çã♦ f : (a, b) −→ R é ❝♦♥tí♥✉❛ ♣❡❧❛ ❞✐r❡✐t❛ ❞❡ x = a✱ s❡ lim+ .f (x) = x→a f (a)✳ ❜✮ ❯♠❛ ❢✉♥çã♦ f : (a, b) −→ R é ❝♦♥tí♥✉❛ ♣❡❧❛ ❡sq✉❡r❞❛ x = b✱ s❡ lim− .f (x) = x→b f (b)✳ ❉❡✜♥✐çã♦ ✹✳✻✳ ❯♠❛ ❢✉♥çã♦ f : (a, b] −→ R é ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ (a, b]✱ s❡ ❝✉♠♣r❡ ❛s ❞✉❛s ❝♦♥❞✐çõ❡s✿ 1a f é ❝♦♥tí♥✉❛ ❡♠ (a, b)✳ 2a f é ❝♦♥tí♥✉❛ ♣❡❧❛ ❡sq✉❡r❞❛ ❡♠ x = b✳ ❉❡✜♥✐çã♦ ✹✳✼✳ ❯♠❛ ❢✉♥çã♦ f : [a, b) −→ R é ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ [a, b)✱ s❡ ❝✉♠♣r❡ ❛s ❞✉❛s ❝♦♥❞✐çõ❡s✿ 1a f é ❝♦♥tí♥✉❛ ❡♠ (a, b)✳ 2a f é ❝♦♥tí♥✉❛ ♣❡❧❛ ❞✐r❡✐t❛ ❡♠ x = a✳ ❉❡✜♥✐çõ❡s ❛♥á❧♦❣❛s ♣♦❞❡♠♦s ♦❜t❡r ♣❛r❛ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ ❢✉♥çõ❡s ❡♠ ✐♥t❡r✈❛❧♦s ❞❛ ❢♦r♠❛ (−∞, b] ❡ [a, +∞) ✷✷✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❡✜♥✐çã♦ ✹✳✽✳ ❯♠❛ ❢✉♥çã♦ ❝♦♥❞✐çõ❡s✿ f : [a, b] −→ R é ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ 1a f é ❝♦♥tí♥✉❛ ❡♠ (a, b)✳ 2a f é ❝♦♥tí♥✉❛ ♣❡❧❛ ❞✐r❡✐t❛ ❡♠ x = a✳ 3a f é ❝♦♥tí♥✉❛ ♣❡❧❛ ❡sq✉❡r❞❛ ❡♠ x = b✳ [a, b] ✱ s❡ ❝✉♠♣r❡ ❛s três ❊①❡♠♣❧♦ ✹✳✶✵✳ f (x) = [|x|], x ∈ R✱ ❡①✐st❡ lim .f (x)✳ ❙❡❥❛ ♥ã♦ ♠♦str❡ q✉❡ f é ❝♦♥tí♥✉❛ ♣❡❧❛ ❞✐r❡✐t❛ ❡♠ t♦❞♦ n∈Z ❡ q✉❡ x→n ❙♦❧✉çã♦✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ f (x) = [|x|]✱ t❡♠♦s x ∈ [n, n + 1)✱ ❡♥tã♦ [|x|] = n ❧♦❣♦ lim+ .f (x) = x→n lim+ [|x|] = lim+ n = n = f (n) ❛ss✐♠✱ f é ❝♦♥tí♥✉❛ ♣❡❧❛ ❞✐r❡✐t❛ ❞❡ x = n✳ x→n x→n P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ t♦❞♦ x ∈ [n − 1, n) t❡♠♦s f (x) = [|x|] = n − 1✱ ❧♦❣♦ lim− .f (x) = x→n lim− [|x|] = lim− (n − 1) = n − 1✳ x→n x→n ❈♦♠♦ ♦s ❧✐♠✐t❡s ❧❛t❡r❛✐s sã♦ ❞✐st✐♥t♦s ❡♥tã♦ ♥ã♦ ❡①✐st❡ lim .f (x)✳ x→n ❊①❡♠♣❧♦ ✹✳✶✶✳ ❯♠ ❢❛❜r✐❝❛♥t❡ ♣♦❞❡ ♦❜t❡r ✉♠ ❧✉❝r♦ ❞❡ ❘$30, 00 ❡♠ ❝❛❞❛ ✐t❡♠ s❡ ♥ã♦ ♠❛✐s ❞❡ ✐t❡♥s ❢♦r❡♠ ♣r♦❞✉③✐❞♦s ♣♦r s❡♠❛♥❛✳ ❖ ❧✉❝r♦ ❡♠ ❝❛❞❛ ✐t❡♠ ❜❛✐①❛ ❛❝✐♠❛ ❞❡ ❘$0, 30 1.000 ♣❛r❛ t♦❞♦ ✐t❡♠ ❛✮ ❙❡ x ✐t❡♥s ❢♦r❡♠ ♣r♦❞✉③✐❞♦s ♣♦r s❡♠❛♥❛✱ ❡①♣r❡ss❡ ♦ ❧✉❝r♦ s❡♠❛♥❛❧ ❞♦ ❝♦♠♦ ❢✉♥çã♦ ❞❡ x✳ ❙✉♣♦♥❤❛ ❧✉❝r♦ ♥ã♦ ♥❡❣❛t✐✈♦✳ ❜✮ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ ❞❛ 1.000✳ ❢❛❜r✐❝❛♥t❡ ♣❛rt❡ ❛✮ é ❝♦♥tí♥✉❛ ❡♠ x = 1.000❀ ♣♦rt❛♥t♦ ❝♦♥tí♥✉❛ ❡♠ t♦❞♦ s❡✉ ❞♦♠í♥✐♦✳ ❙♦❧✉çã♦✳ ❙❡❥❛ L(x) ♦ ❧✉❝r♦ s❡♠❛♥❛❧ ❛ ❝❛❞❛ x ✐t❡♥s ♣r♦❞✉③✐❞♦s✱ ❡♥tã♦ t❡♠♦s L(x) = 30x s❡ 0 ≤ x < 1000 ❡ L(x) = (30 − 0, 30)x s❡ 0 ≤ x < 1.000✳ ▲♦❣♦ L(x) = ( 30x, s❡✱ 0 ≤ x < 1.000 ✳ 29, 7x, s❡✱ x ≥ 1.000 P♦rt❛♥t♦ ❛ ❢✉♥çã♦ ♥ã♦ é ❝♦♥tí♥✉❛ ❡♠ x = 1.000✳ ❊①❡♠♣❧♦ ✹✳✶✷✳ ❉❡t❡r♠✐♥❡ ♦s ✐♥t❡r✈❛❧♦s ❞❡ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❢✉♥çã♦✿ ❙♦❧✉çã♦✳ ✷✸✵ f (x) = r x2 − 9 ✳ 25 − x2 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❖ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ sã♦ t♦❞♦s ♦s ♥ú♠❡r♦s r❡❛✐s ♣❛r❛ ♦s q✉❛✐s ❛ r❛✐③ q✉❛❞r❛❞❛ ❞❡ 2 x −9 x2 − 9 s❡❥❛ ✉♠ ♥ú♠❡r♦ r❡❛❧✱ r❡s♦❧✈❡♥❞♦ ≥ 0 s❡❣✉❡ q✉❡ ♦ ❞♦♠í♥✐♦ 25 − x2 25 − x2 D(f ) = { x ∈ R /. x ∈ (−5, −3] ∪ [3, 5) } ❊st✉❞♦ ❞❛ ❝♦♥t✐♥✉✐❞❛❞❡ ♥♦ ✐♥t❡r✈❛❧♦ (−5, −3]✳ ✐✮ f é ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ (−5, −3)✳ ✐✐✮ lim .f (x) = 0 = f (−3)✳ x→−3− P♦rt❛♥t♦ ✱ f é ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ (−5, −3]✳ ❊st✉❞♦ ❞❛ ❝♦♥t✐♥✉✐❞❛❞❡ ♥♦ ✐♥t❡r✈❛❧♦ [3, 5)✳ ✐✮ f é ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ (3, 5)✳ ✐✐✮ lim .f (x) = 0 = f (3)✳ x→3+ P♦rt❛♥t♦✱ f é ❝♦♥tí♥✉❛ ❡♠ [3, 5) ✹✳✷✳✶ ❋✉♥çõ❡s ❝♦♥tí♥✉❛s ❡♠ ✐♥t❡r✈❛❧♦s ❢❡❝❤❛❞♦s Pr♦♣r✐❡❞❛❞❡ ✹✳✺✳ f : R −→ R ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡ lim xn = x✱ ❡♥tã♦ lim f (xn ) = f (x)✳ ❈♦♥s✐❞❡r❡ q✉❡ n→+∞ s❡❥❛ xn ✉♠❛ s✉❝❡ssã♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s t❛✐s n→+∞ ❉❡♠♦♥str❛çã♦✳ ❉❛ ❞❡✜♥✐çã♦ ❞❡ ❧✐♠✐t❡s ❛♦ ✐♥✜♥✐t♦s t❡♠♦s✱ s❡ lim xn = x❀ ❡♥tã♦ ❞❛❞♦ ε1 > 0✱ ❡①✐st❡ n→+∞ N > 0 t❛❧ q✉❡ | xn − x |< ε1 s❡♠♣r❡ q✉❡ N > n✳ ❙❡♥❞♦ f ❝♦♥tí♥✉❛ ❡♠ R ❡♠ ♣❛rt✐❝✉❧❛r ♥♦ ♥ú♠❡r♦ xn ∈ R✱ ❡♥tã♦ lim f (xn ) = f (x) xn →x ❞❡st❛ ❞❡✜♥✐çã♦ t❡♠♦s✱ ∀ε > 0, ∃δ > 0 t❛❧ q✉❡ | f (xn )−f (x) |< ε s❡♠♣r❡ q✉❡ | xn −x |< δ ✳ ❋❛③❡♥❞♦ δ = ε1 , ∀ ε > 0, N > 0 ❡ ∃δ > 0 t❛❧ q✉❡ | f (xn ) − f (x) |< ε s❡♠♣r❡ q✉❡ | xn − x |< δ q✉❛♥❞♦ N > n✳ ■st♦ é ∀ ε > 0, ∃ N > 0 t❛❧ q✉❡ | f (xn ) − f (x) |< ε1 s❡♠♣r❡ q✉❡ N > n✳ P♦rt❛♥t♦✱ lim f (xn ) = f (x)✳ n→+∞ ❚❡♦r❡♠❛ ✹✳✶✳ ❚❡♦r❡♠❛ ❞❡ ❇♦❧③❛♥♦✳ f : R −→ R✱ é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♠ [a, b] ♠❡♥♦s ✉♠ ♣♦♥t♦ c ∈ (a, b) t❛❧ q✉❡ f (c) = 0✳ ❙❡ ❡ f (a) · f (b) < 0✱ ❡♥tã♦ ❡①✐st❡ ♣❡❧♦ ❉❡♠♦♥str❛çã♦✳ ✷✸✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ f (a) · f (b) < 0 f (b) > 0✳ ❉❛ ❤✐♣ót❡s❡ f (a) < 0 ❡ ❡♥tã♦ f (a) ❡ f (b) tê♠ s✐♥❛✐s ❝♦♥trár✐♦s✳ ❙✉♣♦♥❤❛♠♦s q✉❡ a+b ✱ s❡ f (m) = 0✱ ❡st❛ ♣r♦♣r✐❡❞❛❞❡ ❡stá ♠♦str❛❞❛✳ 2 ❙✉♣♦♥❤❛♠♦s f (m) 6= 0✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ✐♥t❡r✈❛❧♦ [a1 , b1 ] ⊂ [a, b] b1 = m✱ t❛❧ q✉❡ f (a1 ) < 0 ❡ f (b1 ) > 0✳ ❙❡❥❛ R m= a1 + b1 ✱ 2 ❝♦♠ a1 = m ♦✉ f (m1 ) = 0✱ ❡st❛ ♣r♦♣r✐❡❞❛❞❡ ❡st❛ ♠♦str❛❞❛✳ ❆♣ós ❞❡ r❡♣❡t✐r ❡st❡ ♣r♦❝❡ss♦ ✉♠ ♥ú♠❡r♦ n ❞❡ ✈❡③❡s❀ t❡♠♦s q✉❡ ❡①✐st❡ ✉♠ ✐♥t❡r✈❛❧♦ [an , bn ] ⊂ [a, b] t❛❧ b−a ✳ q✉❡ f (an ) < 0 ❡ f (bn ) > 0❀ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ♣♦♥t♦s an ❡ bn é 2n ❆♣ós r❡✐t❡r❛❞❛s ✈❡③❡s ❡st❡ ♣r♦❝❡ss♦✱ ❝♦♥str✉í♠♦s ✉♠❛ s✉❝❡ssã♦ ♥ã♦ ❞❡❝r❡s❝❡♥t❡ a ≤ a1 ≤ a2 ≤ a3 ≤ a4 ≤ · · · ✱ ❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡❀ s❡❥❛ lim .an = c1 ✳ ❙❡❥❛ m1 = s❡ n→+∞ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ ❝♦♥str✉í♠♦s ✉♠❛ s✉❝❡ssã♦ ♥ã♦ ❝r❡s❝❡♥t❡ ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡ b1 ≥ b2 ≥ b3 ≥ b 4 ≥ · · · ❀ b≥ lim .bn = c2 ✳   b−a P♦r ♦✉tr♦ ❧❛❞♦✱ lim [bn − an ] = lim = 0 ❡♥tã♦ lim .bn = lim .an = c2 = n→+∞ n→+∞ n→+∞ n→+∞ 2n c1 = c✳ ❈♦♠♦ P❡❧❛ f (xn ) f (x) é ❝♦♥tí♥✉❛ ❡♠ Pr♦♣r✐❡❞❛❞❡ t❡♠ ❧✐♠✐t❡ n→+∞ x = c✱ t❡♠♦s lim .f (x) = f (c)✳ x→c ✭✹✳✺✮ s❛❜❡✲s❡ q✉❡ s❡ ✉♠❛ s✉❝❡ssã♦ f (c)❀ ▲♦❣♦✱ ♣❛r❛ t♦❞♦ f (c) ≥ 0✳ s❡❥❛ ❡♥tã♦ {xn } t❡♠ ❧✐♠✐t❡ c✱ ❡♥tã♦ ❛ s✉❝❡ssã♦ lim f (an ) = f (c) = lim f (bn )✳ n→+∞ n→+∞ n ∈ N ❛ ❞❡s✐❣✉❛❧❞❛❞❡ f (an ) < 0 ✐♠♣❧✐❝❛ f (c) ≤ 0 P♦rt❛♥t♦✱ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ c ∈ (a, b) t❛❧ q✉❡ ❡ f (bn ) > 0✱ ✐♠♣❧✐❝❛ f (c) = 0✳  ❖❜s❡r✈❡ ❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡st❡ t❡♦r❡♠❛✿ ❖ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ q✉❡ ✉♥❡ ♦s ♣♦♥t♦s P (a, f (a)) ❡ Q(b, f (b)) ♦♥❞❡ f (a) ❡ f (b) sã♦ ❞❡ s✐♥❛✐s ❝♦♥trár✐♦s✱ ❝♦rt❛ ♦ ❡✐①♦✲x ❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ ✑✳ ✭❋✐❣✉r❛ ✭✹✳✸✮✮✳ ✏ ❋✐❣✉r❛ ✹✳✸✿ ❋✐❣✉r❛ ✹✳✹✿ ✷✸✷ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❆ ❝♦♥❞✐çã♦ ❞❡ s❡r f ❝♦♥tí♥✉❛ ❡♠ [a, b] é ♥❡❝❡ssár✐❛❀ ❛ ❋✐❣✉r❛ ✭✹✳✹✮ ♠♦str❛ q✉❡ s❡ f é ❞❡s❝♦♥tí♥✉❛ ❡♠ [a, b] ❛ ♣r♦♣r✐❡❞❛❞❡ ♥❡♠ s❡♠♣r❡ ✈❡r✐✜❝❛✲s❡✳ ❊①❡♠♣❧♦ ✹✳✶✸✳ ▼♦str❡ q✉❡✱ s❡ ✐✮ f : R −→ R é ❝♦♥tí♥✉❛ ❡ ❝✉♠♣r❡✿ lim f (x) = K > 0 x→+∞ ✐✐✮ lim f (x) = N < 0 x→−∞ ❊♥tã♦ ❡①✐st❡ x0 ∈ R t❛❧ q✉❡ f (x0 ) = x0 ✳ ❉❡♠♦♥str❛çã♦✳ ❉❛ ❤✐♣ót❡s❡ ✐✮ t❡♠♦s q✉❡ ∀ ε1 > 0, ∃ M1 > 0 ✭s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✮ t❛❧ q✉❡✱ s❡ x > M1 ⇒ |f (x) − K| < ε1 ❀ ❧♦❣♦ x > M1 ⇒ K − ε1 < f (x) < K + ε1 ✳ ❈♦♠♦ tr❛t❛✲s❡ ❞❡ q✉❛❧q✉❡r ε1 > 0✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♣♦r ❡①❡♠♣❧♦ ε1 = 10−100 ✱ ❛ss✐♠✱ s❡ x > M1 ⇒ K − 10−100 < f (x) < K + 10−100 ✳ ❆ ❞❡✜♥✐çã♦ ❞❡ ❧✐♠✐t❡ ❛♦ ✐♥✜♥✐t♦ ❣❛r❛♥t❡ ❛✐♥❞❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ M1′ > K + 10−100 t❛❧ q✉❡ x2 > M1′ ♣❛r❛ ❛❧❣✉♠ x2 ∈ R✳ ▲♦❣♦✱ s❡ x2 > M1′ ⇒ f (x2 ) < K + 10−100 < M1′ < x2 ⇒ f (x2 ) < x2 ✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✳ ❉❛ ❤✐♣ót❡s❡ ✐✐✮ t❡♠♦s q✉❡ ∀ ε2 > 0, ∃ M2 < 0 ✭s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✮ t❛❧ q✉❡✱ s❡ x < M2 ⇒ |f (x) − N | < ε2 ❀ ❧♦❣♦ x < M2 ⇒ N − ε2 < f (x) < N + ε2 ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ε2 = 10−100 ✱ ❛ss✐♠✱ s❡ x < M2 ⇒ N −10−100 < f (x) < N + 10−100 ✳ ❆ ❞❡✜♥✐çã♦ ❞❡ ❧✐♠✐t❡ ❛ ♠❡♥♦s ✐♥✜♥✐t♦ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ M2′ < N − 10−100 t❛❧ q✉❡ x1 < M2′ ♣❛r❛ ❛❧❣✉♠ x1 ∈ R✳ ▲♦❣♦✱ s❡ x1 < M2′ ⇒ x1 < M2′ < N − 10−100 < f (x1 ) ⇒ x1 < f (x1 )✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ❢✉♥çã♦ g : [x1 , x2 ] −→ R ❞❡✜♥✐❞❛ ♣♦r g(x) = f (x) − x✱ ❧♦❣♦ ❝♦♠♦ f é ❝♦♥tí♥✉❛✱ t❡♠♦s q✉❡ g é ❝♦♥tí♥✉❛ ❡♠ [x1 , x2 ]✱ ❛✐♥❞❛ ♠❛✐s✱ t❡♠♦s q✉❡ g(x1 ) = f (x1 )−x1 > 0 ❡ g(x2 ) = f (x2 ) − x2 < 0✳ ❖ ❚❡♦r❡♠❛ ❞❡ ❇♦❧③❛♥♦ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ x0 ∈ [x1 , x2 ] t❛❧ q✉❡ g(x0 ) = 0 ⇒ f (x0 ) = x0 ✳ P♦rt❛♥t♦✱ ❡①✐st❡ x0 ∈ R t❛❧ q✉❡ f (x0 ) = x0 ✳ Pr♦♣r✐❡❞❛❞❡ ✹✳✻✳ ❙❡ f ❉❛ ❧✐♠✐t❛çã♦ ❣❧♦❜❛❧✳ é ❝♦♥tí♥✉❛ ❡♠ [a, b]✱ ❡♥tã♦ f é ❧✐♠✐t❛❞❛ ❡♠ [a, b]✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❝♦♥❥✉♥t♦ A = { x ∈ [a, b] /. f é ❧✐♠✐t❛❞❛ }✱ ♦❜s❡r✈❡ q✉❡ A 6= ∅ ♣♦✐s s❡♥❞♦ f ❝♦♥tí♥✉❛ ❡♠ a ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❧✐♠✐t❛çã♦ ❧♦❝❛❧✱ ❡①✐st❡ M > 0 ❡ δ > 0 t❛❧ q✉❡ | f (x) |< M, x ∈ [a, a + δ] ✐st♦ é a + δ ∈ A✳ ✷✸✸ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ■♥✈❡rs❛♠❡♥t❡ ❈♦♠♦ A é ❧✐♠✐t❛❞♦ ❛❞♠✐t❡ s✉♣r❡♠♦✳ ❙❡❥❛ c = sup .A✱ ❡✈✐❞❡♥t❡♠❡♥t❡ c ≤ b✳ ❙✉♣♦✲ ♥❤❛♠♦s q✉❡ c < b✱ ❡♥tã♦ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❧✐♠✐t❛çã♦ ❧♦❝❛❧✱ ∃ M1 > 0 ❡ δ1 > 0 t❛❧ q✉❡ | f (x) |< M1 , ∀ x ∈ [c − δ1 , c + δ1 ]✳ ❈♦♠♦ f é ❧✐♠✐t❛❞❛ ❡♠ [a, c − δ1 ] ♣❛r❛ ❛❧❣✉♠ M2 > 0✱ ❝♦♥s✐❞❡r❛♥❞♦ M3 = max .{M1 , M2 } t❡♠♦s | f (x) |< M3 , ∀ x ∈ [a, c + δ1 ] ♦♥❞❡ c + δ1 ∈ A ♦ q✉❛❧ ❝♦♥tr❛❞✐③ ♦ ❢❛t♦ ❞❡ q✉❡ c = sup .A❀ ♣♦rt❛♥t♦ c ♥ã♦ é ❡str✐t❛♠❡♥t❡ ♠❡♥♦r q✉❡ b✳ ❈♦♠♦ c ≤ b s❡❣✉❡✲s❡ q✉❡ c = b✳ P❡❧♦ ✉♠ r❛❝✐♦❝í♥✐♦ ❛♥á❧♦❣♦✱ ❝♦♠♦ f é ❝♦♥tí♥✉❛ ❡♠ b✱ ❡❧❛ é ❧✐♠✐t❛❞❛ ❡♠ [b − δ2 , b] ♣❛r❛ ❛❧❣✉♠ δ2 > 0 ❡ s❡♥❞♦ ❧✐♠✐t❛❞♦ ❡♠ [a, b − δ2 ] ✭✐st♦ ♣❡❧♦ ❛♥t❡r✐♦r✮ s❡❣✉❡✲s❡ q✉❡ f é ❧✐♠✐t❛❞❛ ❡♠ [a, b]✳ ◆♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦ ♠♦str❛✲s❡ q✉❡ s❡ f ♥ã♦ é ❝♦♥tí♥✉❛ ❡♠ [a, b] ❛ ❢✉♥çã♦ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ é ❧✐♠✐t❛❞❛✳  ❊①❡♠♣❧♦ ✹✳✶✹✳ ❙❡❥❛ ❋✐❣✉r❛ ✹✳✺✿   1 , s❡✱ 0 ≤ x < 3 3−x f (x) =  1, s❡✱ x = 3 ❆ ❋✐❣✉r❛ ✭✹✳✺✮ ♠♦str❛ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f ✱ ♦❜s❡r✈❡ q✉❡ f ♥ã♦ é ❧✐♠✐t❛❞❛✳ ❚❡♦r❡♠❛ ✹✳✷✳ ❚❡♦r❡♠❛ ❞❡ ❲❡✐❡rstr❛ss✳ f é ❝♦♥tí♥✉❛ ❡♠ [a, b]✱ ❡♥tã♦ ❡❧❛ ♣♦ss✉✐ ✉♠ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❡ ✉♠ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ [a, b]❀ ✐st♦ é✱ ❡①✐st❡♠ x1 , x2 ∈ [a, b] t❛✐s q✉❡✿ ❙❡ ❡♠ m = f (x1 ) = min .{ f (x) /. x ∈ [a, b] }✳ M = f (x2 ) = max .{ f (x) /. x ∈ [a, b] }❀ ♦✉✱ f (x1 ) ≤ f (x) ≤ f (x2 ) ∀ x ∈ [a, b]✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ f é ❝♦♥tí♥✉❛ ❡♠ [a, b]✱ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✹✳✻✮✱ ♦ ❝♦♥❥✉♥t♦ A = { f (x) /. x ∈ [a, b] } é ❧✐♠✐t❛❞♦ ♥ã♦ ✈❛③✐♦❀ ❡♥tã♦ A ❛❞♠✐t❡ ✉♠ s✉♣r❡♠♦ M ❡ ✉♠ í♥✜♠♦ m ❀ ✐st♦ é m ≤ f (x) ≤ M ∀ x ∈ [a, b]✳ ❆ ♠♦str❛r q✉❡ ❡①✐st❡ x1 ∈ [a, b] t❛❧ q✉❡ f (x1 ) = m ✐st♦ é m = min .A✳ ❙✉♣♦♥❤❛♠♦s ✭♣❡❧♦ ❛❜s✉r❞♦✮ q✉❡ ∀ x ∈ [a, b] t❡♠♦s f (x) > m ♦✉ f (x) − m > 0✳ ❆ ❢✉♥çã♦ g : [a, b] −→ R ❞❡✜♥✐❞❛ ♣♦r g(x) = 1 é ❝♦♥tí♥✉❛ ❞♦ ❢❛t♦ s❡r ♦ f (x) − m q✉♦❝✐❡♥t❡ ❞❡ ❞✉❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❝♦♠ ❞❡♥♦♠✐♥❛❞♦r ❞✐st✐♥t♦ ❞❡ ③❡r♦✳ P❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✹✳✻✮ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ L > 0 t❛❧ q✉❡ 1 <L f (x) − m ✷✸✹ x ∈ [a, b]✱ ❧♦❣♦ f (x) − m > 1 ✐st♦ L 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 1 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ 1 R 1 é f (x) > m + x ∈ [a, b]❀ ♣♦ré♠ m + > m✱ ♦ q✉❛❧ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ♣♦✐s m + L L L é ✉♠ ❧✐♠✐t❡ ✐♥❢❡r✐♦r ♠❛✐♦r q✉❡ ♦ í♥✜♠♦ m✳ P♦rt❛♥t♦ ❝♦♥❝❧✉í♠♦s q✉❡ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ x1 ∈ [a, b] t❛❧ q✉❡ f (x1 ) = m = min A✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ ♠♦str❛✲s❡ q✉❡ ❡①✐st❡ x2 ∈ [a, b] t❛❧ q✉❡ f (x2 ) = M ✳ ❊st❡ ú❧t✐♠♦ t❡♦r❡♠❛ ♥♦s ♠♦str❛ q✉❡ t♦❞❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ f ✱ ❞❡✜♥✐❞❛ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ ❡ ❧✐♠✐t❛❞♦ [a, b]✱ ❛ss✉♠❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ✈❛❧♦r ♠í♥✐♠♦ m = f (x1 ) ❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ✈❛❧♦r ♠á①✐♠♦ M = f (x2 ✮✳ ❊①❡♠♣❧♦ ✹✳✶✺✳ 1 ❈♦♥s✐❞❡r❡♠♦s ❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ f : (0, 1] −→ R✱ ❞❡✜♥✐❞❛ ♣♦r f (x) = ♣❛r❛ t♦❞♦ x x ∈ (0, 1]✳ ❈♦♠♦ f ((0, 1]) = [1, +∞)✱ ♥ã♦ ❡①✐st❡ x2 ∈ (0, 1] t❛❧ q✉❡ f (x) ≤ f (x2 ) ♣❛r❛ t♦❞♦ x ∈ (0, 1]✳ ◆♦t❡♠♦s q✉❡✱ ❛♣❡s❛r ❞❡ (0, 1] s❡r ❧✐♠✐t❛❞♦✱ ❡❧❡ ♥ã♦ é ❢❡❝❤❛❞♦✳ ❊①❡♠♣❧♦ ✹✳✶✻✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ f : (0, 1) −→ R✱ ❞❡✜♥✐❞❛ ♣♦r f (x) = x ♣❛r❛ t♦❞♦ x ∈ (0, 1)✳ ❈♦♠♦ f ((0, 1)) = (0, 1)✱ ♥ã♦ ❡①✐st❡♠ x1 , x2 ∈ (0, 1) t❛✐s q✉❡ f (x1 ) ≤ f (x) ≤ f (x2 ) ♣❛r❛ t♦❞♦ x ∈ (0, 1)✳ ◆♦t❡♠♦s q✉❡✱ ❛♣❡s❛r ❞❡ (0, 1) s❡r ❧✐♠✐t❛❞♦✱ ❡❧❡ ♥ã♦ é ❢❡❝❤❛❞♦ ❊①❡♠♣❧♦ ✹✳✶✼✳ ❙❡❥❛ f : [a, b] −→ R ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♠ [a, b] t❛❧ q✉❡ f (x) > 0 ♣❛r❛ t♦❞♦ x ∈ [a, b]✳ ❊♥tã♦ ❡①✐st❡ δ > 0 t❛❧ q✉❡ f (x) ≥ δ ♣❛r❛ t♦❞♦ x ∈ [a, b]✳ ❉❡ ❢❛t♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✭✹✳✷✮ ❡①✐st❡ x1 ∈ [a, b] t❛❧ q✉❡ f (x1 ) ≤ f (x) ♣❛r❛ t♦❞♦ x ∈ [a, b]✳ ❈♦♠♦ f (x1 ) > 0✱ ❜❛st❛ t♦♠❛r δ = f (x1 ) ♣❛r❛ ❝♦♥❝❧✉✐r ❛ ✈❛❧✐❞❛❞❡ ❞❛ ♥♦ss❛ ❛✜r♠❛çã♦✳ ❉♦ ✈❛❧♦r ✐♥t❡r♠❡❞✐ár✐♦✳ ❙❡ f é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♠ [a, b], m ❡ M sã♦ ♦ ♠í♥✐♠♦ ❡ ♠á①✐♠♦ ❞❡ f ❡♠ [a, b] r❡s♣❡❝t✐✈❛♠❡♥t❡ ❡ d é t❛❧ q✉❡ m < d < M ✱ ❡♥tã♦ ❡①✐st❡ c ∈ (a, b) t❛❧ q✉❡ f (c) = d✳ ❚❡♦r❡♠❛ ✹✳✸✳ ❉❡♠♦♥str❛çã♦✳ P❡❧♦ ❚❡♦r❡♠❛ ✭✹✳✷✮✱ ❡①✐st❡♠ x1 , x2 ∈ [a, b] t❛✐s q✉❡ f (x1 ) = m ❡ f (x2 ) = M ✳ ❆ ❢✉♥çã♦ g(x) = f (x) − d é ❝♦♥tí♥✉❛ ❡♠ [a, b]❀ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ♥♦ ✐♥t❡r✈❛❧♦ ❞❡ ❡①tr❡♠♦s x1 ❡ x2 ✳ ❖❜s❡r✈❡ q✉❡ g(x1 ) = f (x1 ) − d = m − d < 0 ❡ g(x2 ) = f (x2 ) − d = M − d > 0✱ ❧♦❣♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✭✹✳✶✮ ❡①✐st❡ c ♥♦ ✐♥t❡r✈❛❧♦ ❞❡ ❡①tr❡♠♦s x1 ❡ x2 t❛❧ q✉❡ g(c) = 0✱ ✐st♦ é f (c) = d✳ ✷✸✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✹✳✶✽✳ ❖ ♣♦❧✐♥ô♠✐♦ p(x) = x3 + x − 1 ♣♦ss✉✐ ✉♠❛ r❛✐③ ♥♦ ✐♥t❡r✈❛❧♦ (0, 1)✳ ❉❡ ❢❛t♦✱ t❡♠♦s p(0) = −1 < 0 ❡ p(1) = 1 > 0✳ ❈♦♠♦ p(x) é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ (0, 1)✱ s❡❣✉❡ ❞♦ t❡♦r❡♠❛ ❞♦ ✈❛❧♦r ✐♥t❡r♠é❞✐♦ q✉❡ ❡①✐st❡ x ∈ (0, 1) t❛❧ q✉❡ p(x) = 0✳ ❊①❡♠♣❧♦ ✹✳✶✾✳ ❉❛❞❛ ❛ ❢✉♥çã♦ f (x) = x−1 ✱ ❞❡t❡r♠✐♥❡ ✉♠ ✈❛❧♦r c ∈ [0, 2] ❞♦ ❚❡♦r❡♠❛ x2 + 1 ✭✹✳✸✮ ❞♦ ✈❛❧♦r ✐♥t❡r♠❡❞✐ár✐♦✱ ❡ ✈❡r✐✜q✉❡ ❛ ✈❛❧✐❞❛❞❡ ❞♦ r❡s✉❧t❛❞♦✳ ❙♦❧✉çã♦✳ ❚❡♠♦s q✉❡ − 1 ✱ 2 f (0) = −1 ❡ f (2) = ❧♦❣♦ ❞❡✈❡♠♦s ❞❡t❡r♠✐♥❛r x0 1 ✳ 5 ❈♦♥s✐❞❡r❡♠♦s ✉♠ ✈❛❧♦r ❡♥tr❡ −1 ❡ 1 ✱ ♣♦r ❡①❡♠♣❧♦ 5 ♥❛ ✐❣✉❛❧❞❛❞❡✿ x−1 1 =− 2 x +1 2 ❉❡ ♦♥❞❡ ♦❜t❡♠♦s x 1 − ∈ [−1, 51 ]✳ 2 √ √ √ = −1± 2✱ ♦ ✈❛❧♦r x0 = −1+ 2 ∈ [0, 2] ❞❡ ♠♦❞♦ q✉❡ f (−1+ 2) = ◆ã♦ ❝♦♥s✐❞❡r❡♠♦s ♦ ✈❛❧♦r x = −1 − √ 2 ♣❡❧♦ ❢❛t♦ ♥ã♦ ♣❡rt❡♥❝❡r ❛♦ ✐♥t❡r✈❛❧♦ [0, 2]✳ Pr♦♣r✐❡❞❛❞❡ ✹✳✼✳ ❙❡ n é í♠♣❛r✱ ❡♥tã♦ q✉❛❧q✉❡r ❡q✉❛çã♦ xn + an−1 xn−1 + an−2 xn−2 + · · · + a1 x + a0 = 0 ♣♦ss✉✐ ✉♠❛ r❛✐③ r❡❛❧✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ h an−1 an−2 an−3 a2 a1 a0 i f (x) = xn 1 + + 2 + 3 + · · · + n−2 + n−1 + n x x x x x x ♦❜s❡r✈❡ q✉❡✿ a2 a1 a0 an−1 an−2 an−3 + 2 + 3 + · · · + n−2 + n−1 + n ≤ x x x x x x an−1 an−3 a2 a1 a0 an−2 + + · · · + n−2 + n−1 + n ✭✹✳✻✮ + 2 3 x x x x x x ❊s❝♦❧❤❡♠♦s ✉♠ x q✉❡ ❝✉♠♣r❡ ♦ s❡❣✉✐♥t❡✿ | x |> 1, | x |> 2n | an−1 |, | x |> 2n | an−2 |, · · · , | x |> 2n | a1 |, | x |> 2n | a0 |✱ ≤ ❡♥tã♦ | xk |>| x | ❉❡ ✭✹✳✻✮ ❡ an−1 an−2 an−3 + 2 + 3 x x x ≤ 1 | an−k | an−k an−k = < < k x x 2n | an−k | 2n a2 a1 a0 + · · · + n−2 + n−1 + n ≤ x x x 1 1 1 1 1 1 + + ··· + + + = 2n 2n 2n 2n 2n 2 ✷✸✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ▲♦❣♦✱ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R a2 a1 a0 1 1 an−1 an−2 an−3 − < + 2 + 3 + · · · + n−2 + n−1 + n ≤ ❡ 2 x x x x x x 2 1 a2 a1 a0 an−1 an−2 an−3 <1+ + 2 + 3 + · · · + n−2 + n−1 + n 2 x x x x x x ❙✉♣♦♥❤❛ ✉♠ x1 > 0✱ ❡♥tã♦✿   a2 an−1 an−2 an−3 a1 a0 x21 2 ≤ x1 1 + + 2 + 3 + · · · + n−2 + n−1 + n = f (x1 ) 0≤ 2 x1 x1 x1 x1 x1 x1 ❉❡ ♠♦❞♦ q✉❡ f (x1 ) > 0✳ P♦r ♦✉tr♦ ❧❛❞♦✱ q✉❛♥❞♦ x2 < 0✱ ❡♥tã♦ xn2 < 0 ✭n é í♠♣❛r✮ ❡ ✿   x22 a1 a0 an−1 an−2 an−3 a2 2 0≥ ≥ x2 1 + + 2 + 3 + · · · + n−2 + n−1 + n = f (x2 ) 2 x2 x2 x2 x2 x2 x2 ❞❡ ♠♦❞♦ q✉❡ f (x2 ) < 0✳ ❖❜s❡r✈❡ q✉❡ f (x1 )·f (x2 ) < 0✱ ❛♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✭✹✳✶✮ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ x ∈ [x1 , x2 ] ❞❡ ♠♦❞♦ q✉❡ f (x) = 0✳ ❊①❡♠♣❧♦ ✹✳✷✵✳ ❖✉tr❛ ❞❡♠♦♥str❛çã♦ ❞❛ Pr♦♣r✐❡❞❛❞❡ ▼♦str❡ q✉❡ s❡✱ a1 x + a0 = 0 n é í♠♣❛r✱ ❡♥tã♦ q✉❛❧q✉❡r ❡q✉❛çã♦ ♣♦ss✉✐ ✉♠❛ r❛✐③ r❡❛❧✳ ❉❡♠♦♥str❛çã♦✳ h a a a ✭✹✳✼✮ xn + an−1 xn−1 + an−2 xn−2 + · · · + a a a i n−3 2 1 0 ❙❡❥❛ f (x) = xn 1 + n−1 + n−2 + 3 + · · · + n−2 + n−1 + n 2 x x x x x x f (x) = xn · g(x)✱ ♦♥❞❡ g(x) = 1 + ♦❜s❡r✈❡ q✉❡✿ a2 a1 a0 an−1 an−2 an−3 + 2 + 3 + · · · + n−2 + n−1 + n x x x x x x P♦r ♦✉tr♦ ❧❛❞♦✱ lim g(x) = 1 ❧♦❣♦ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❧✐♠✐t❡ ❛♦ ✐♥✜♥✐t♦✱ t❡♠♦s ∀ ε > x→∞ ∃ n > 0 t❛❧ q✉❡ | g(x) − 1 |< ε, s❡♠♣r❡ q✉❡ | x |> n✳ 1 1 s❡♠♣r❡ q✉❡ | x |> n✳ ▲♦❣♦ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❝♦♥s✐❞❡r❡ ε = ✱ ❛ss✐♠ | g(x) − 1 |< 2 2 1 3 < g(x) < ✳ 2 2 ❉♦ ❢❛t♦ | x |> n ✱ ❡①✐st❡ x1 > 0 ❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❡st❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣♦r xn1 > 0 3xn xn t❡♠♦s q✉❡ 0 < 1 < xn1 · g(x1 ) = f (x1 ) < 1 ⇒ 0 < f (x1 )✳ 2 2 ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ❞♦ ❢❛t♦ | x |> n✱ ❡①✐st❡ x2 < 0 t❛❧ q✉❡ ❞♦ ❢❛t♦ n í♠♣❛r xn2 < 0 ⇒ xn 3xn 0 > 2 > xn2 · g(x2 ) = f (x2 ) > 2 ⇒ f (x2 ) < 0✳ 2 2 P♦rt❛♥t♦✱ ♣❛r❛ x1 , x2 ∈ R t❛❧ q✉❡✱ f (x1 ) · f (x2 ) < 0 ♣❡❧❛ ❚❡♦r❡♠❛ ✭✹✳✶✮ ❡①✐st❡ c ∈ R t❛❧ q✉❡ f (c) = 0✳ 0, ✷✸✼ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R P❛r❛ ❡st❡ ❡①❡♠♣❧♦✱ ❧❡♠❜r❡ q✉❡ D(f ) = R✳ Pr♦♣r✐❡❞❛❞❡ ✹✳✽✳ ❙❡ n é ♣❛r ❡ f (x) = xn + an−1 xn−1 + an−2 xn−2 + · · · + a1 x + a0 = 0✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ y t❛❧ q✉❡ f (y) ≤ f (x) ♣❛r❛ t♦❞♦ x ∈ R✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ M = max .{ 1, 2n | an−1 |, 2n | an−2 |, · · · , 2n | a1 |, 2n | a0 | }✱ ❡♥tã♦ ♣❛r❛ t♦❞♦ | x |≥ M t❡♠♦s 1 a2 a1 a0 an−1 an−2 an−3 <1+ + 2 + 3 + · · · + n−2 + n−1 + n 2 x x x x x x ❉♦ ❢❛t♦ n ♣❛r✱ xn ≥ 0 ♣❛r❛ t♦❞♦ x✱ ❞❡ ♠♦❞♦ q✉❡✿ h x2 a2 a1 a0 i an−1 an−2 an−3 2 0≤ ≤x 1+ + 2 + 3 + · · · + n−2 + n−1 + n = f (x) 2 x x x x x x s❡♠♣r❡ q✉❡ | x |≥ M ✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ♥ú♠❡r♦ f (0)✱ ❡ s❡❥❛ b > 0 ✉♠ ♥ú♠❡r♦ t❛❧ q✉❡ bn ≥ 2f (0) ❡ t❛♠❜é♠ b > M ✳ xn bn ≥ ≥ f (0)✳ 2 2 (−b)n xn ≥ ≥ f (0)❀ ❧♦❣♦ ✱ s❡ x ≥ b ♦✉ x ≤ −b ❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ x ≤ −b ❡♥tã♦ f (x) ≥ 2 2 ❡♥tã♦ f (x) ≥ f (0)✳ ❆♣❧✐❝❛♥❞♦ ❛ Pr♦♣r✐❡❞❛❞❡ ✭✹✳✼✮ ♣❛r❛ ❛ ❢✉♥çã♦ f (x) ♥♦ ✐♥t❡r✈❛❧♦ [−b, b]✱ ❡①✐st❡ y t❛❧ q✉❡✿ ❙❡ − b ≤ x ≤ b, ❡♥tã♦ f (y) ≤ f (x) ✭✹✳✼✮ ❊♥tã♦ s❡ x ≥ b t❡♠♦s f (x) ≥ ❊♠ ♣❛rt✐❝✉❧❛r f (x) ≤ f (0)✳ ❉❡st❡ ♠♦❞♦ ❙❡ x ≥ b ♦✉ x ≤ −b ❡♥tã♦ f (x) ≥ f (0) ≥ f (y) ✭✹✳✽✮ ❈♦♠❜✐♥❛♥❞♦ ✭✹✳✼✮ ❡ ✭✹✳✽✮ t❡♠♦s f (y) ≤ f (x) ♣❛r❛ t♦❞♦ x ∈ R✳ ❊①❡♠♣❧♦ ✹✳✷✶✳ ▼♦str❡ q✉❡✱ s❡ f : [0; 1] −→ R ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♠ [0, 1] t❛❧ q✉❡ f (x) ∈ [0, 1] ♣❛r❛ t♦❞♦ x ∈ [0, 1]✳ ❊♥tã♦ ❡①✐st❡ x ∈ [0, 1] t❛❧ q✉❡ f (x) = x✱ ♦✉ s❡❥❛✱ f ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ ✜①♦✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ f (0) = 0 ♦✉ f (1) = 1✱ ♥❛❞❛ ❛ ♠♦str❛r✳ ❙✉♣♦♥❤❛♠♦s q✉❡ f (0) 6= 0 ❡ f (1) 6= 1 ❡♥tã♦ ❝♦♠♦ f (0) ≥ 0 ❡ f (1) ≤ 1✱ ♥❡❝❡ss❛r✐❛✲ ♠❡♥t❡ f (0) > 0 ❡ f (1) < 1✳ ❉❡✜♥❛♠♦s g(x) = f (x) − x ♣❛r❛ t♦❞♦ x ∈ [0, 1] ❡♥tã♦ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✹✳✶✮ s❡❣✉❡✲s❡ g é ❝♦♥tí♥✉❛✳ ❈♦♠♦ g(1) = f (1) − 1 < 0 < f (0) − 0 = g(0)✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ✈❛❧♦r ✐♥t❡r♠❡❞✐ár✐♦✱ ❡①✐st❡ x ∈ (0, 1) t❛❧ q✉❡ g(x) = 0✱ ✐st♦ é f (x) = x✳ ✷✸✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✹✲✷ ✶✳ ❉❛❞❛ ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✱ ❞❡t❡r♠✐♥❡ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ♥♦s ✐♥t❡r✈❛❧♦s ✐♥❞✐❝❛❞♦s✿ ✶✳  | 16 − x4 |     4 − x2 f (x) =  −8    8 s❡ x 6= ±2 s❡ x = −2 ♥♦s ✐♥t❡r✈❛❧♦s✿ s❡ x = 2 (−∞, −2); (−∞, −2]; (−2, 2); [−2,  3 2   |x +x −x−1|   x2 − 3x + 2 ✷✳ f (x) = −4     4 2); [−2, 2]; (−2, 2]; [2, +∞); (2, +∞)✳ s❡ x 6= 1 ❡ x 6= 2 s❡ x = 1 s❡ x = 2 ♥♦s ✐♥t❡r✈❛❧♦s✿ (−∞, 1); (−∞, 1]; (1, 2); [1, 2]; [2, +∞); (2, +∞)✳ p ✸✳ f (x) = | x | −[|x|] ❡♠ (0, 1], [0, 1], [1, 3]✳ ✹✳ f (x) = (x − 1)[|x|] ❡♠ [0, 2]✳ ✷✳ P❛r❛ ♦s s❡❣✉✐♥t❡s ❡①❡r❝í❝✐♦s✱ ❡st❛❜❡❧❡❝❡r s❡ ❛ ❢✉♥çã♦ é ❝♦♥tí♥✉❛ ♥♦s ✐♥t❡r✈❛❧♦s ✐♥❞✐✲ ❝❛❞♦s✳ ❈♦♥str✉✐r ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦✳ ✶✳ ✷✳ ✸✳ ✹✳ ✺✳ ✻✳ x+2 ❡♠ (2, 4)✳ − 3x − 10  x−6  s❡ x 6= 4 2 x − 2x − 8 f (x) = ❡♠ (−1, 6)  −2 s❡ x = 4  x+4   s❡ x 6= ±4   x2 − 16 1 ❡♠ (−5, 5)✳ f (x) = s❡ x = −4 −   8   2 s❡ x = 4  2  s❡ − 1 < x ≤ 2  x − 6x + 1 f (x) = ❡♠ (−1, 5). 2x − 6 s❡ 2 < x ≤ 3   2 4x − 3 − x s❡ 3 < x < 5 ( x−4 s❡ − 1 < x ≤ 2 f (x) = ❡♠ (−1, 5)✳ x2 − 6 s❡ 2 < x < 5  (x − 1) | x − 2 |   s❡ 0 < x < 4, x 6= 1 | x2 − 1 | f (x) = ❡♠ (0, 4)   1 s❡ x = 1 2 f (x) = x2 ✷✸✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✸✳ ❉❡t❡r♠✐♥❡ ♦s ✈❛❧♦r❡s ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ a ❡ b R ❞❡ t❛❧ ♠♦❞♦ q✉❡ ❝❛❞❛ ✉♠❛ ❞❛s ❢✉♥çõ❡s s❡❥❛ ❝♦♥tí♥✉❛ ❡♠ s❡✉ ❞♦♠í♥✐♦✳ ✶✳ ✷✳ ✸✳   s❡ x < −2  x + 2a f (x) = 3ax + b s❡ −2≤x≤1   6x − 2b s❡ x>1  3  s❡✱ − ≤x≤0  b   2  | 2x2 − 3x − 9 | 3 f (x) = , s❡✱ x < − ♦✉ x>3 2  2x − 3x − 9 2   3   a s❡✱ 0≤x≤ 2 √  3 3 − 3x + 3   s❡ x<8  3  a(√  x − 2) f (x) = ab, s❡✱ x=8    2   s❡ x>8 b· | 2x − 7 | ✹✳ ❉❡t❡r♠✐♥❡ ♦ ✐♥t❡r✈❛❧♦ ❞❡ ❝♦♥t✐♥✉✐❞❛❞❡ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿   1 − x + [|x|] h 1 i 1. f (x) =  x s❡ s❡ x≥0 2. f (x) = x<0 5. f (x) =| x − [|x|]+ | [|1 − x|] |  1   s❡ x < 0    x x2 s❡ 0 ≤ x < 5 7. f (x) =  2  x − 4x − 5   s❡ x > 5  |x−5|  3  s❡ x ≤ −1  x + 3x + 3 9. f (x) = |x−2| s❡ − 1 < x ≤ 4   8x − x2 − 15 s❡ x > 4 ✶✳ f (x) = sgn(x) ❡ ✷✳ f (x) = sgn(x) ❡ ✸✳ ✹✳ x+ | x | 2 ( 1 s❡ f (x) = 0 s❡ f (x) = R ♣❛r❛ ❛s ❢✉♥çõ❡s f og 4. f (x) = r 8. f (x) = p 16 − x2 x−6 x2 − 16 x−6 q √ 3 6. f (x) = 4 − x − 2 3. f (x) =| 1 − x + [|x|] − [|1 − x|] | ✺✳ ❆♥❛❧✐s❡ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❡♠ r 10. f (x) = ❡ gof | x | −[|x|] | 4x − 3 | −1 [|3 − 4x|] ✱ s❡✿ g(x) = x − x3 ✳ ❡ g(x) = 1 + x − [|x|]✳ ( x s❡ x<0 g(x) = ✳ x2 s❡ x≥0 ( 2 | x |≤ 1 ❡ g(x) = 2 − x2 | x |> 1 ✷✹✵ s❡ s❡ | x |> 2 | x |≤ 2 ✳ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✻✳ ❉❛r ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ❡♠ [0, 1] q✉❡ ♥ã♦ t❡♥❤❛ ♠á①✐♠♦ ♥❡♠ ♠í♥✐♠♦ ❡♠ t❛❧ ✐♥t❡r✈❛❧♦ ✳ ✼✳ ❙❡ f (x) = x4 −5x+3✱ ❧♦❝❛❧✐③❛r ✉♠ ✐♥t❡r✈❛❧♦ [a, b] ♦♥❞❡ t❡♥❤❛ ✉♠❛ r❛✐③ r❡❛❧✱ ❥✉st✐✜q✉❡ s✉❛ r❡s♣♦st❛✳ x2 + 1 ✱ ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r q✉❡ ❝✉♠♣r❡ ♦ ❚❡♦r❡♠❛ ✭✹✳✸✮ ✭❞♦ ✈❛❧♦r ✐♥t❡r♠❡✲ ✽✳ ❙❡❥❛ f (x) = x ❞✐ár✐♦✮ ♣❛r❛ d = 3✱ ❡♠ [1, 6]✳ ✾✳ ❙❡❥❛ f : [a; b] −→ R ❝♦♥tí♥✉❛ ❡♠ [a; b]✳ ❯s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ❲❡✐❡rstr❛ss ♠♦str❡ q✉❡ ❡①✐st❡ C > 0 t❛❧ q✉❡ |f (x)| ≤ C ♣❛r❛ t♦❞♦ x ∈ [a; b]✳ ✶✵✳ ❈♦♥s✐❞❡r❡ ✉♠ ✐♥t❡r✈❛❧♦ ♥ã♦ tr✐✈✐❛❧ I ⊂ R ❡ ✉♠❛ ❢✉♥çã♦ f : I −→ R ❝♦♥tí♥✉❛ ❡♠ I ✳ ▼♦str❡ q✉❡ f (I) = { f (x) /. x ∈ I } é ✉♠ ✐♥t❡r✈❛❧♦✳ sen(x2 ) /. x ∈ [−1, 2] }✳ ▼♦str❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ T é ✉♠ ✐♥t❡r✈❛❧♦ ✶✶✳ ❙❡❥❛ T = { 4 x +1 ❢❡❝❤❛❞♦ ❡ ❧✐♠✐t❛❞♦✳ ❙✉❣❡stã♦✿ ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ f : [−1, 2] −→ R✱ ❞❡✜♥✐❞❛ ♣♦r f (x) = sen(x2 ) ✳ x4 + 1 ✶✷✳ ▼♦str❡ q✉❡ ❛ ❡q✉❛çã♦ x5 + 3x − 2 = 0 t❡♠ ✉♠❛ r❛✐③ ♥♦ ✐♥t❡r✈❛❧♦ (0, 1)✳ 1 ✳ +2 1 ❞❡✜♥✐❞❛ ♥♦ ✐♥t❡r✈❛❧♦ [0, 1]✳ ❙✉❣❡stã♦✿ ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ f (x) = x5 − 4 x +2 ✶✸✳ ▼♦str❡ q✉❡ ❡①✐st❡ x ∈ (0, 1) t❛❧ q✉❡ x5 = x4 π 2 ✶✹✳ ▼♦str❡ q✉❡ ❡①✐st❡ x ∈ ( , π) t❛❧ q✉❡ senx = x − 1✳ π 2 ❙✉❣❡stã♦✿ ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ f (x) = senx − x + 1 ❞❡✜♥✐❞❛ ♥♦ ✐♥t❡r✈❛❧♦ ( , π)✳ ✶✺✳ ❙❡❥❛ f : [0, 1] −→ R ❝♦♥tí♥✉❛ ❡♠ [0, 1] t❛❧ q✉❡ f (0) > 0 ❡ f (1) < 1✳ ▼♦str❡ q✉❡ √ ❡①✐st❡ x ∈ (0, 1) t❛❧ q✉❡ f (x) = x✳ senπx s❡❥❛ ❞❡✜♥✐❞❛ ♥♦ ✐♥t❡r✈❛❧♦ (0, 1)✳ ❉❡✜♥✐r x(x − 1) f ❡♠ x = 0 ❡ x = 1 ❞❡ ♠♦❞♦ q✉❡ f s❡❥❛ ❝♦♥tí♥✉❛ ❡♠ [0, 1]✳ ✶✻✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ ❢✉♥çã♦ f (x) = ✶✼✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ ❢✉♥çã♦ f ❡stá ❞❡✜♥✐❞❛ ♥♦ ✐♥t❡r✈❛❧♦ (0, 1) ♣♦r f (x) = ❘❡❞❡✜♥✐r f ♣❛r❛ q✉❡ s❡❥❛ ❝♦♥tí♥✉❛ ❡♠ [0, 1]✳ 1 − cos(2πx) ✳ x2 (1 − x)2 ✶✽✳ ❯♠❛ ❡❞✐t♦r❛ ✈❡♥❞❡ 10.000 ❧✐✈r♦s ❞❡ ♠❛t❡♠át✐❝❛ ❛♣❧✐❝❛❞❛ q✉❛♥❞♦ ♦ ♣r❡ç♦ ✉♥✐tár✐♦ é ❞❡ ❘$15, 00✱ ❛ ❡❞✐t♦r❛ ❞❡t❡r♠✐♥♦✉ q✉❡ ♣♦❞❡ ✈❡♥❞❡r 2.000 ✉♥✐❞❛❞❡s ❛ ♠❛✐s ❝♦♠ ✉♠❛ r❡❞✉çã♦ ❞❡ ❘$3, 00 ♥♦ ♣r❡ç♦ ✉♥✐tár✐♦✳ ❆❝❤❡ ❛ ❡q✉❛çã♦ ❞❡ ❞❡♠❛♥❞❛✱ s✉♣♦♥❞♦✲❛ ❧✐♥❡❛r✱ ❡ tr❛❝❡ ♦ ❣rá✜❝♦ r❡s♣❡❝t✐✈♦✳ ✷✹✶ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✶✾✳ ◆✉♠❛ ♣❡q✉❡♥❛ ❝✐❞❛❞❡✱ ❝♦♠ ♣♦♣✉❧❛çã♦ ❞❡ 5.000 ❤❛❜✐t❛♥t❡s✱ ❛ t❛①❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❡ ✉♠❛ ❡♣✐❞❡♠✐❛ ✭❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s ✐♥❢❡❝t❛❞❛s✮ é ❝♦♥❥✉♥t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s ✐♥❢❡❝t❛❞❛s ❡ ❛♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s ♥ã♦ ✐♥❢❡❝t❛❞❛s✳ ✭❛✮ ❙❡ ❛ ❡♣✐❞❡♠✐❛ ❡stá ❝r❡s❝❡♥❞♦ à r❛③ã♦ ❞❡ 9 ♣❡ss♦❛s ♣♦r ❞✐❛ q✉❛♥❞♦ 100 ♣❡ss♦❛s ❡stã♦ ✐♥❢❡❝t❛❞❛s✱ ❡①♣r❡ss❡ ❛ t❛①❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❛ ❡♣✐❞❡♠✐❛ ❝♦♠♦ ❢✉♥çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s ✐♥❢❡❝t❛❞❛s✳ ✭❜✮ ◗✉ã♦ rá♣✐❞♦ ❡stá s❡ ❛❢❛st❛♥❞♦ ❛ ❡♣✐❞❡♠✐❛✱ q✉❛♥❞♦ 200 ♣❡ss♦❛s ❡stã♦ ✐♥❢❡❝t❛❞❛s ❄ ✷✹✷ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ▼✐s❝❡❧â♥❡❛ ✹✲✶ ✶✳ ❉❡t❡r♠✐♥❡ q✉❛✐s ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s ❡stã♦ ❧✐♠✐t❛❞❛s s✉♣❡r✐♦r ❡ ✐♥❢❡r✐♦r♠❡♥t❡ ♥♦ ✐♥t❡r✈❛❧♦ ✐♥❞✐❝❛❞♦❀ ❡ q✉❛✐s ❞❡❧❛s ❛❧❝❛♥ç❛♠ s❡✉s ✈❛❧♦r❡s ❞❡ ♠á①✐♠♦ ♦✉ ♠í♥✐♠♦✳ ✶✳ f (x) = x2 ❡♠ (−1, 1) ✷✳ g(x) = x3 ❡♠ (−1, 1) ✸✳ h(x) = x2 ❡♠ R ✹✳ f (x) = x2 ❡♠ [0, +∞) ✷✳ P❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✱ ❞❡t❡r♠✐♥❡ ✉♠ ✐♥t❡✐r♦ n t❛❧ q✉❡ f (x) = 0 ♣❛r❛ ❛❧❣✉♠ x ∈ [n, n + 1]✳ ✶✳ f (x) = x3 − x + 3 ✷✳ g(x) = x5 + 5x4 + 2x + 1 ✸✳ f (x) = x5 + x + 1 ✹✳ f (x) = 4x2 − 4x + 1 ✸✳ ✶✳ ▼♦str❡ q✉❡ s❡ f é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♠ [a, b]✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ g q✉❡ é ❝♦♥tí♥✉❛ ❡♠ R✱ ❡ q✉❡ ❝✉♠♣r❡ g(x) = f (x) ∀ x ∈ [a, b]✳ ✭❙✉❣❡stã♦✿ ❝♦♥s✐❞❡r❡ ✉♠❛ ❢✉♥çã♦ g(x) ❝♦♥st❛♥t❡ ❡♠ (−∞, a] ∪ [b, +∞)✳ ❖❜s❡r✈❡ q✉❡ ❛ ❛✜r♠❛çã♦ ❡♠ ✭✶✳✮ ❞❡st❡ ✐t❡♠ é ❢❛❧s❛ s❡ s✉❜st✐t✉✐r♠♦s ♦ ✐♥t❡r✈❛❧♦ [a, b] ♣♦r (a, b)✳ ❏✉st✐✜❝❛r✳ ✷✳ ✹✳ ❙❡❥❛ f : [0, 4] → R ❞❛❞❛ ♣♦r f (x) = ✶✳ ✷✳ 2x − x2 ✱ ♣❡❞❡✲s❡✿ x + x2 Pr♦✈❛r q✉❡ x = 4 é ♦ ♣♦♥t♦ ♠í♥✐♠♦ ❞❡ f ✐st♦ é f (4) ≤ f (x)✱ ♣❛r❛ t♦❞♦ x ∈ [0, 4]✳ Pr♦✈❛r q✉❡ ∃ x2 ∈ [0, 2] t❛❧ q✉❡ f (x2 ) é ♦ ✈❛❧♦r ♠á①✐♠♦ ❞❡ f ✱ ✐st♦ é f (x2 ) ≥ ∀ x ∈ [0, 4]   sen 1 , s❡✱ x 6= 0 ❀ ✺✳ ❙❡❥❛ f (x) = x  1, s❡✱ x = 0 f (x), 1 f t❡♠ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❡✈✐tá✈❡❧ ❡♠ x = 0❄✳ ❊ q✉❛♥❞♦ s❡ s✉❜st✐t✉✐ f (x) = x.sen( ) x ♣❛r❛ x 6= 0 ❄ ✻✳ ❙❡❥❛ f : [a, b] −→ R ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ♥ã♦ ❝♦♥st❛♥t❡ ❡♠ [a, b]✳ Pr♦✈❛r q✉❡ Im(f ) = [m, M ] ♦♥❞❡ m = min .f (x) ❡ M = max .f (x)✳ x∈[a, b] x∈[a, b] ✼✳ Pr♦✈❛r q✉❡ ♦ ♣♦❧✐♥ô♠✐♦ P (x) = 4x3 − 14x2 + 14x − 3 t❡♠ três r❛í③❡s r❡❛✐s ❞✐❢❡r❡♥t❡s✳ ✽✳ ❙✉♣♦♥❤❛♠♦s q✉❡ f : [0, 1] −→ [0, 1] s❡❥❛ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ Pr♦✈❛r q✉❡ ❡①✐st❡ c ∈ [0, 1] t❛❧ q✉❡ f (c) = c ✷✹✸ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✾✳ ▼♦str❡ q✉❡ ❡①✐st❡ ❛❧❣✉♠ ♥ú♠❡r♦ ✶✳ x179 + 163 = 119 1 + + sen2 x x∈R R t❛❧ q✉❡✿ ✷✳ x2 senx = x − 1✳ ✶✵✳ ❉❡t❡r♠✐♥❡ q✉❛✐s ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s ❡stã♦ ❧✐♠✐t❛❞❛s s✉♣❡r✐♦r ❡ ✐♥❢❡r✐♦r♠❡♥t❡ ♥♦ ✐♥t❡r✈❛❧♦ ✐♥❞✐❝❛❞♦❀ ❡ q✉❛✐s ❞❡❧❛s ❛❧❝❛♥ç❛♠ s❡✉s ✈❛❧♦r❡s ❞❡ ♠á①✐♠♦ ♦✉ ♠í♥✐♠♦✳ ✶✳ ✷✳ ✸✳ ✹✳ ✺✳ f (x) = ( x2 , a + 2,   0, f (x) = 1  , q   1, f (x) = 1  , q s❡✱ s❡✱ s❡✱ s❡✱ x<a x≥a ❡♠ x∈I=R−Q p x= é ❢r❛çã♦ q [−a − 1, a + 1] ❡♠ ✐rr❡❞✉tí✈❡❧ x∈I=R−Q p s❡✱ x= é ❢r❛çã♦ ✐rr❡❞✉tí✈❡❧ q √ g(x) = sen2 (cos x + 1 + a2 ) ❡♠ [0, a3 ]✳ h(x) = |[x|] [0, 1] s❡✱ ❡♠ ❡♠ [0, 1] [0, a]✳ ✷✹✹ 09/02/2021 ❈❛♣ít✉❧♦ ✺ ❉❊❘■❱❆❉❆❙ ✑❋❡r♠❛t ♦ ✈❡r❞❛❞❡✐r♦ ✐♥✈❡♥t♦r ❞♦ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ···✑ ▲❆P▲❆❈❊ P✐❡rr❡ ❉❡ ❋❡r♠❛t ♥❛s❝❡✉ ❡♠ ❇❡❛✉♠♦♥t ♥❛ ❋r❛♥ç❛✱ ♥♦ ❛♥♦ ❞❡ 1601✱ ❡ ❢❛❧❡❝❡✉ ❡♠ ❈❛str❡s✱ ❡♠ 12 ❞❡ ❥❛♥❡✐r♦ ❞❡ 1665✳ ❈❡❞♦✱ ♠❛✲ ♥✐❢❡st♦✉ ✐♥t❡r❡ss❡ ♣❡❧♦ ❡st✉❞♦ ❞❡ ❧í♥❣✉❛s ❡str❛♥❣❡✐r❛s✱ ❧✐t❡r❛t✉r❛ ❝❧áss✐❝❛✱ ❝✐ê♥❝✐❛ ❡ ♠❛t❡♠át✐❝❛❀ ❢♦✐ ❡❞✉❝❛❞♦ ❡♠ ❝❛s❛✳ ❚rês ❛♥♦s ❞❡♣♦✐s ❞❡ s❡ ❢♦r♠❛r ❡♠ ❞✐r❡✐t♦ ♣❡❧❛✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❖r❧é❛♥s✱ t♦r♥♦✉✲s❡ ❝♦♥s❡❧❤❡✐r♦ ❞♦ P❛r❧❛♠❡♥t♦ ❞❡ ❚♦❧♦✉s❡✱ ❡♠ 1634❀ ❡r❛ ♠✉✐t♦ ♦❝✉♣❛❞♦❀ ❡♠ s✉❛s ❤♦r❛s ❧✐✈r❡s t❡✈❡ t❡♠♣♦ ♣❛r❛ s❡ ❞❡❞✐❝❛r à ❧✐t❡r❛t✉r❛ ❝❧áss✐❝❛✱ ✐♥❝❧✉s✐✈❡ ❝✐ê♥❝✐❛ ❡ ♠❛t❡♠át✐❝❛✳ ❊♠ 1629✱ ❡❧❡ ❝♦♠❡ç♦✉ ❛ ❢❛③❡r ❞❡s❝♦❜❡rt❛s ❞❡ ✐♠♣♦rtâ♥❝✐❛ ❝❛✲ ♣✐t❛❧ ❡♠ ♠❛t❡♠át✐❝❛✳ ◆❡ss❡ ❛♥♦✱ ❡❧❡ ❝♦♠❡ç♦✉ ❛ ♣r❛t✐❝❛r ✉♠ ❞♦s P✳ ❋❡r♠❛t ❡s♣♦rt❡s ❢❛✈♦r✐t♦s ❞♦ t❡♠♣♦✿ ❛ ✏r❡st❛✉r❛çã♦✑ ❞❡ ♦❜r❛s ♣❡r❞✐❞❛s ❞❛ ❛♥t✐❣✉✐❞❛❞❡✱ ❝♦♠ ❜❛s❡ ❡♠ ✐♥❢♦r♠❛çõ❡s ❡♥❝♦♥tr❛❞❛s ♥♦s tr❛✲ t❛❞♦s ❝❧áss✐❝♦s ♣r❡s❡r✈❛❞♦s✳ ❋❡r♠❛t s❡ ♣r♦♣ôs ❛ r❡❝♦♥str✉✐r ♦s ❧✉❣❛r❡s ♣❧❛♥♦s ❞❡ ❆♣♦❧ô♥✐♦✱ ❜❛s❡❛❞♦ ❡♠ ❛❧✉sõ❡s ❝♦♥t✐❞❛s ♥❛ ❈♦✲ ❧❡çã♦ ▼❛t❡♠át✐❝❛ ❞❡ P❛♣✉s✳ ❙✉❛s ♦❜r❛s ❝♦♥s✐st❡♠ ❡♠ ❛rt✐❣♦s ✐s♦❧❛❞♦s✳ ❙❡✉s r❡s✉❧t❛❞♦s ♠❛✐s ✐♠♣r❡ss✐♦♥❛♥t❡s ❢♦r❛♠ ❡♥❝♦♥tr❛❞♦s ❞❡♣♦✐s ❞❡ s✉❛ ♠♦rt❡✳ ❋✉♥❞❛❞♦r ❞❛ ♠♦❞❡r♥❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ❋❡r♠❛t ❛♥t❡❝✐♣♦✉✲s❡ ❛ ❉❡s❝❛rt❡s✱ ❞❡s❝♦❜r✐✉ ❡♠ 1636 ♦ ♣r✐♥❝í♣✐♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛✳ ❋❡r♠❛t ♥ã♦ ❝♦♥❝♦r❞♦✉ ❝♦♠ ❉❡s❝❛rt❡s ❡ ❞❡✉ ê♥❢❛s❡ ❛♦ ❡s❜♦ç♦ ❞❡ s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s ✐♥❞❡t❡r✲ ♠✐♥❛❞❛s ❛♦ ✐♥✈❡③ ❞❡ à ❝♦♥str✉çã♦ ❣❡♦♠étr✐❝❛ ❞❛s s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s ❞❡t❡r♠✐♥❛❞❛s✳ ❋❡r♠❛t ❧✐♠✐t♦✉ s✉❛ ❡①♣♦s✐çã♦ ♥♦ ❝✉rt♦ tr❛t❛❞♦ ✐♥t✐t✉❧❛❞♦ ✏■♥tr♦❞✉çã♦ ❛♦s ❧✉❣❛r❡s ♣❧❛♥♦s ❡ só❧✐✲ ❞♦s✑✳ P❡rt❡♥❝❡ ❛ ❋❡r♠❛t ❛ ❢❛♠♦s❛ ❝♦♥❥❡t✉r❛ s♦❜r❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❡♠ ♥ú♠❡r♦s ✐♥t❡✐r♦s xn + y n = z n ✱ ♣❛r❛ n ∈ N✱ ❞❡♠♦♥str❛❞❛ ❡♠ 1993✳ ♣❛r❛ ❛ ❡q✉❛çã♦ ✷✹✺ ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✺✳✶ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈♦♥❝❡✐t♦s ❜ás✐❝♦s ❯♠ ❞♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ é ♦ ❞❛ ❞❡r✐✈❛❞❛✳ ❆s ❝✐ê♥❝✐❛s ❡♠ ❣❡r❛❧ t✐✈❡r❛♠ ❣r❛♥❞❡ ✐♠♣✉❧s♦ ❡♠ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ♣❡❧❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❝♦♥❝r❡t♦s✳ ❖s ❞♦✐s ♣r♦❜❧❡♠❛s ♣rát✐❝♦s s❡❣✉✐♥t❡s sã♦ ♦s q✉❡ ♣r♦♣✐❝✐❛r❛♠ ❛ ❝r✐❛çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡r✐✈❛❞❛✿ ✶✳ ❉❡t❡r♠✐♥❛r ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛ ✉♠❛ ❝✉r✈❛ ♥✉♠ ♣♦♥t♦ ❞❛❞♦✳ ✷✳ ❉❛❞❛ ❛ ❧❡✐ ❤♦rár✐❛ ❞♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✈✐♥❝✉❧❛❞❛ ❛ ✉♠❛ r❡t❛✳ ■st♦ é✱ ✉♠❛ ❡q✉❛çã♦ s = f (t) q✉❡ ❞á ❛ ♣♦s✐çã♦ ❞❛ ♣❛rtí❝✉❧❛ s♦❜r❡ ❛ r❡t❛ ❡♠ ❝❛❞❛ ✐♥st❛♥t❡ t✱ ❞❡t❡r♠✐♥❛r ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ♣❛rtí❝✉❧❛ ❡♠ ❝❛❞❛ ✐♥st❛♥t❡✳ ❉❡ ✐♥í❝✐♦✱ ❛s ❞❡✜♥✐çõ❡s ♥ã♦ t✐♥❤❛♠ ♣r❡❝✐sã♦✳ ❏á ❡♠ 1.629 P✐❡rr❡ ❋❡r♠❛t ❢❛③✐❛ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❞♦ ♣r✐♠❡✐r♦ ♣r♦❜❧❡♠❛✱ t❡♥❞♦ ❡♥❝♦♥tr❛❞♦ ✉♠❛ ♠❛♥❡✐r❛ ❞❡ ❝♦♥str✉✐r t❛♥❣❡♥t❡s ❛ ✉♠❛ ♣❛rá❜♦❧❛✱ ❡ q✉❡ ❝♦♥t✐♥❤❛ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ❛ ✐❞❡✐❛ ❞❡ ❞❡r✐✈❛❞❛✳ ❇❡♠ ♠❛✐s t❛r❞❡✱ s❡ ♣❡r❝❡❜❡✉ q✉❡ ♦s ❞♦✐s ♣r♦❜❧❡♠❛s t✐♥❤❛♠ ❛❧❣♦ ❡♠ ❝♦♠✉♠ ❡ q✉❡ ❛ ✐❞❡✐❛ ❣❡r❛❧ q✉❡ ♣❡r♠✐t✐r✐❛ r❡s♦❧✈ê✲❧♦s ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❧❡✈❛r✐❛ ❛ ♥♦çã♦ ❞❡ ❞❡r✐✈❛❞❛ ♥✉♠ ♣♦♥t♦✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s✱ ❛❧é♠ ❞❡ ❢❛❝✐❧✐t❛r ♦ ❡st✉❞♦ ❞❡ ❝✉r✈❛s ❥á ❝♦♥❤❡❝✐❞❛s✱ ♣❡r♠✐t✐✉ ❛ ✏ ❝r✐❛çã♦ ✑ ❞❡ ♥♦✈❛s ❝✉r✈❛s✱ ✐♠❛❣❡♥s ❣❡♦♠étr✐❝❛s ❞❡ ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ♣♦r r❡❧❛çõ❡s ❡♥tr❡ ✈❛r✐á✈❡✐s✳ ❊♥q✉❛♥t♦ s❡ ❞❡❞✐❝❛✈❛ ❛♦ ❡st✉❞♦ ❞❡ ❛❧❣✉♠❛s ❞❡st❛s ❢✉♥çõ❡s✱ ❋❡r♠❛t ❞❡✉ ❝♦♥t❛ ❞❛s ❧✐♠✐t❛çõ❡s ❞♦ ❝♦♥❝❡✐t♦ ❝❧áss✐❝♦ ❞❡ r❡t❛ t❛♥❣❡♥t❡ ❛ ✉♠❛ ❝✉r✈❛ ❝♦♠♦ s❡♥❞♦ ❛q✉❡❧❛ q✉❡ ❡♥❝♦♥tr❛✈❛ ❛ ❝✉r✈❛ ♥✉♠ ú♥✐❝♦ ♣♦♥t♦✳ ❚♦r♥♦✉✲s❡ ❛ss✐♠ ✐♠♣♦rt❛♥t❡ r❡❢♦r♠✉❧❛r t❛❧ ❝♦♥✲ ❝❡✐t♦ ❡ ❡♥❝♦♥tr❛r ✉♠ ♣r♦❝❡ss♦ ❞❡ tr❛ç❛r ✉♠❛ t❛♥❣❡♥t❡ ❛ ✉♠ ❣rá✜❝♦ ♥✉♠ ❞❛❞♦ ♣♦♥t♦ ✲ ❡st❛ ❞✐✜❝✉❧❞❛❞❡ ✜❝♦✉ ✻y ❝♦♥❤❡❝✐❞❛ ♥❛ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛ ❝♦♠♦ ♦ ✏ Pr♦✲ ❜❧❡♠❛ ❞❛ ❚❛♥❣❡♥t❡ ✑✳ ❋❡r♠❛t r❡s♦❧✈❡✉ ❡st❛ ❞✐✜❝✉❧❞❛❞❡ ✲ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ♠✉✐t♦ s✐♠♣❧❡s✿ ♣❛r❛ ❞❡t❡r♠✐♥❛r ✉♠❛ t❛♥❣❡♥t❡ ❛ ✉♠❛ ❝✉r✈❛ ♥✉♠ ♣♦♥t♦ P x ❝♦♥s✐❞❡r♦✉ ♦✉tr♦ Q s♦❜r❡ ❛ ❝✉r✈❛❀ ❝♦♥s✐❞❡r♦✉ ❛ r❡t❛ P Q s❡❝❛♥t❡ à ❝✉r✈❛✳ ❙❡❣✉✐❞❛♠❡♥t❡ ❢❡③ ❞❡s❧✐③❛r Q ❛♦ ❧♦♥❣♦ ❞❛ ❝✉r✈❛ ❡♠ ❞✐r❡çã♦ ❛ P ✱ ♦❜t❡♥❞♦ ❞❡st❡ ♠♦❞♦ r❡t❛s P Q q✉❡ s❡ ❛♣r♦①✐♠❛✈❛♠ ❞❡ ✉♠❛ r❡t❛ t ❛ q✉❡ P✳ ❋❡r♠❛t ❝❤❛♠♦✉ P ✑ ✭❋✐❣✉r❛ ✭✺✳✶✮✮✳ ♣♦♥t♦ ❋✐❣✉r❛ ✺✳✶✿ ✏ ❛ r❡t❛ t❛♥❣❡♥t❡ à ❝✉r✈❛ ♥♦ ♣♦♥t♦ ▼❛✐s✱ ❋❡r♠❛t ♥♦t♦✉ q✉❡ ♣❛r❛ ❝❡rt❛s ❢✉♥çõ❡s✱ ♥♦s ♣♦♥t♦s ♦♥❞❡ ❛ ❝✉r✈❛ ❛ss✉♠✐❛ ✈❛❧♦r❡s ❡①tr❡♠♦s✱ ❛ t❛♥❣❡♥t❡ ❛♦ ❣rá✜❝♦ ❞❡✈✐❛ s❡r ✉♠❛ r❡t❛ ❤♦r✐③♦♥t❛❧✱ ❥á q✉❡ ❛♦ ❝♦♠♣❛r❛r ♦ ✈❛❧♦r P (x, f (x)) ❝♦♠ ♦ ✈❛❧♦r ❛ss✉♠✐❞♦ ♥♦ ♦✉tr♦ ♣♦♥t♦ Q(x + E, f (x + E)) ♣ró①✐♠♦ ❞❡ P ✱ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ f (x + E) ❡ f (x) ❡r❛ ♠✉✐t♦ ♣❡q✉❡♥❛✱ q✉❛s❡ ♥✉❧❛✱ q✉❛♥❞♦ ❝♦♠♣❛r❛❞❛ ❝♦♠ ♦ ✈❛❧♦r ❞❡ E ✱ ❞✐❢❡r❡♥ç❛ ❞❛s ❛❜s❝✐ss❛s ❞❡ Q ❡ P ✳ ❛ss✉♠✐❞♦ ♣❡❧❛ ❢✉♥çã♦ ♥✉♠ ❞❡ss❡s ♣♦♥t♦s ✷✹✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❆ss✐♠✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡t❡r♠✐♥❛r ❡①tr❡♠♦s ❡ ❞❡ ❞❡t❡r♠✐♥❛r t❛♥❣❡♥t❡s ❛ ❝✉r✈❛s ♣❛ss❛♠ ❛ ❡st❛r ✐♥t✐♠❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦s✳ ❊st❛s ✐❞❡✐❛s ❝♦♥st✐t✉ír❛♠ ♦ ❡♠❜r✐ã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ✏ ❉❡r✐✲ ✈❛❞❛ ✑ ❡ ❧❡✈♦✉ ▲❛♣❧❛❝❡ ❛ ❝♦♥s✐❞❡r❛r ✏❋❡r♠❛t ♦ ✈❡r❞❛❞❡✐r♦ ✐♥✈❡♥t♦r ❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧✑✳ ❈♦♥t✉❞♦✱ ❋❡r♠❛t ♥ã♦ ❞✐s♣✉♥❤❛ ❞❡ ♥♦t❛çã♦ ❛♣r♦♣r✐❛❞❛ ❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡ ♥ã♦ ❡st❛✈❛ ❛✐♥❞❛ ❝❧❛r❛♠❡♥t❡ ❞❡✜♥✐❞♦✳ ◆♦ sé❝✉❧♦ XVII✱ ▲❡✐❜♥✐t③ ❛❧❣❡❜r✐③❛ ♦ ❈á❧❝✉❧♦ ■♥✜♥✐t❡s✐♠❛❧✱ ✐♥tr♦❞✉③✐♥❞♦ ♦s ❝♦♥❝❡✐t♦s ❞❡ ✈❛r✐á✈❡❧✱ ❝♦♥st❛♥t❡ ❡ ♣❛râ♠❡tr♦✱ ❜❡♠ ❝♦♠♦ ❛ ♥♦t❛çã♦ dy ♣❛r❛ ❞❡s✐❣♥❛r ❛ ♠❡♥♦r ♣♦ssí✈❡❧ ❞❛s ❞✐❢❡r❡♥ç❛s ❡♠ ❆ss✐♠✱ ❡♠❜♦r❛ só ♥♦ sé❝✉❧♦ XIX x ❡ ❡♠ dx ❡ y✳ ❈❛✉❝❤② ❤❛❥❛ ✐♥tr♦❞✉③✐❞♦ ❢♦r♠❛❧♠❡♥t❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡ ❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡r✐✈❛❞❛✱ ♥♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ XVII✱ ❝♦♠ ▲❡✐❜♥✐t③ ❡ ◆❡✇t♦♥✱ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ t♦r♥❛✲s❡ ✉♠ ✐♥str✉♠❡♥t♦ ❝❛❞❛ ✈❡③ ♠❛✐s ✐♥❞✐s♣❡♥sá✈❡❧✱ ♣❡❧❛ s✉❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❛♦s ♠❛✐s ❞✐✈❡rs♦s ❝❛♠♣♦s ❞❛ ❝✐ê♥❝✐❛✳ ❉❡✜♥✐çã♦ ✺✳✶✳ P♦♥t♦ ❞❡ ❛❝✉♠✉❧❛çã♦✳ ❖ ♣♦♥t♦ ❧✐♠✐t❡ ♦✉ ♣♦♥t♦ ❞❡ ❛❝✉♠✉❧❛çã♦✱ é ✉♠ ♣♦♥t♦ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ q✉❡ ♣♦❞❡ s❡r ❛♣r♦①✐♠❛❞♦ tã♦ ❜❡♠ q✉❛♥t♦ s❡ q✉❡✐r❛ ♣♦r ✐♥✜♥✐t♦s ♦✉tr♦s ♣♦♥t♦s ❞♦ ❝♦♥❥✉♥t♦ ✳ ✺✳✷ ❉❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ❙❡❥❛ ❛ ❢✉♥çã♦ f : A −→ R✱ ❡ ✉♠ ♣♦♥t♦ ❞❡ ❛❝✉♠✉❧❛çã♦ a ∈ A✳ f ♣❛ss❛ ❞♦ ♣♦♥t♦ a ∈ A ♣❛r❛ ❛♦ ♣♦♥t♦ x ∈ A✱ s♦❢r❡♥❞♦ ✉♠ ❛❝rés❝✐♠♦ ♦✉ ✐♥❝r❡♠❡♥t♦ ∆x = x − a✱ ♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ✈❛❧♦r❡s ❞❛❞♦s ♣❡❧❛ ❢✉♥çã♦ ♣❛ss❛♠ ❞❡ f (a) ♣❛r❛ f (a + ∆x)✱ s♦❢r❡♥❞♦ t❛♠❜é♠ ✉♠ ✐♥❝r❡♠❡♥t♦ ◗✉❛♥❞♦ ❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❢✉♥çã♦ ∆y = f (x) − f (a) = f (a + ∆x) − f (a) ❉❡✜♥✐çã♦ ✺✳✷✳ ❚❛①❛ ♠é❞✐❛ ❞❡ ✈❛r✐❛çã♦✳ ❈❤❛♠❛✲s❡ ✏t❛①❛ ♠é❞✐❛ ❞❡ ✈❛r✐❛çã♦✑ ❞❛ ❢✉♥çã♦ ❡♥t❡✿ s❡♥❞♦ ❡st❛ ❢✉♥çã♦ f r❡❧❛t✐✈❛ ❛♦ ♣♦♥t♦ f (x) − f (a) f (a + ∆x) − f (a) ∆y = = ∆x x−a ∆x ❞❡✜♥✐❞❛ ❡♠ t♦❞♦ x ∈ A✱ ❡①❝❡t♦ ♣♦ss✐✈❡❧♠❡♥t❡ a∈A ❡♠ ❛♦ q✉♦❝✐✲ x = a✳ ❊①❡♠♣❧♦ ✺✳✶✳ ❙❡❥❛ ❛ ❢✉♥çã♦ a = 3✳ f (x) = x2 ❚❡♠♦s✿ ❛ q✉❛❧✱ ♣❛r❛ x 6= 3✱ ❝♦♥str✉❛♠♦s ❛ t❛①❛ ♠é❞✐❛ ❞❡ ✈❛r✐❛çã♦ r❡❧❛t✐✈❛ ❛♦ ♣♦♥t♦ x2 − 3 2 (x − 3)(x + 3) ∆y = = ∆x x−3 (x − 3) ♣♦❞❡ s❡r ❡s❝r✐t❛ ✭✺✳✶✮ ∆y = x + 3✳ ∆x ✷✹✼ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R 0 ◆♦t❡✲s❡ q✉❡✱ s❡ ✜③❡r♠♦s x = 3 ❡♠ ✭✺✳✶✮✱ ♦❜t❡♠♦s ❛ ❢♦r♠❛ ✐♥❞❡t❡r♠✐♥❛❞❛ ✳ ❊♥tr❡✲ 0 t❛♥t♦✱ ♣♦❞❡ s❡r q✉❡ ❡①✐st❛ ♦ ❧✐♠✐t❡ ❞❛ r❛③ã♦ ✭✺✳✶✮ q✉❛♥❞♦ x → 3 ♦✉ q✉❛♥❞♦ ∆x → 0 ❡ ❡ss❡ ❧✐♠✐t❡ s❡❥❛ ✜♥✐t♦✳ ❉❡✜♥✐çã♦ ✺✳✸✳ ❉❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ❡♠ ✉♠ ♣♦♥t♦✳ ❙❡❥❛ f : A −→ R ✉♠❛ ❢✉♥çã♦✱ ❞✐③❡♠♦s q✉❡ f é ❞❡r✐✈á✈❡❧ ♥♦ ♣♦♥t♦ ❞❡ ❛❝✉♠✉❧❛çã♦ a ∈ A✱ q✉❛♥❞♦ ♦ s❡❣✉✐♥t❡ ❧✐♠✐t❡ ❡①✐st❡ ❡✱ é ✜♥✐t♦✿ ∆y f (a + ∆x) − f (a) = lim ∆x→0 ∆x ∆x→0 ∆x ✭✺✳✷✮ lim ◗✉❛♥❞♦ f s❡❥❛ ❞❡r✐✈á✈❡❧ ❡♠ x = a✱ ♦ ❧✐♠✐t❡ ✭✺✳✷✮ é ❝❤❛♠❛❞♦ ❞❡r✐✈❛❞❛ ❞❡ f ♥♦ ♣♦♥t♦ a✱ ❡ é ✐♥❞✐❝❛❞♦ ❝♦♠ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s✿ f ′ (a); Df (a) ♦✉ ♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ❏✳ ▲✳ ▲❛❣r❛♥❣❡✱ ❆✳ ▲✳ ❈❛✉❝❤②✱ ❡ ●✳ ❲✳ ▲❡✐❜♥✐t③✳ df (a) ❞❡✈✐❞❛s✱ r❡s✲ dx ❖❜s❡r✈❛çã♦ ✺✳✶✳ ❆ ❉❡✜♥✐çã♦ ✭✺✳✷✮ é ❡q✉✐✈❛❧❡♥t❡ ❛✿ f ′ (a) = lim x→a f (a + ∆x) − f (a) f (x) − f (a) = lim ∆x→0 x−a ∆x ❉❡✜♥✐çã♦ ✺✳✹✳ ❋✉♥çã♦ ❞❡r✐✈❛❞❛✳ ❙❡❥❛ f : R −→ R ✉♠❛ ❢✉♥çã♦✱ ❞❡s✐❣♥❡♠♦s ♣♦r B = { x ∈ R /. f ′ (x) ❡①✐st❛ }✱ s❡ B 6= ∅ ❛ ❢✉♥çã♦✿ f ′ : B ⊆ R −→ R ′ x 7→ f (x) ❞❡✜♥✐❞❛ ❡♠ B é ❞❡♥♦♠✐♥❛❞❛ ❢✉♥çã♦ ❞❡r✐✈❛❞❛ ❞❡ f ✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ♣r✐♠❡✐r❛ df ✳ ❞❡r✐✈❛❞❛ ❞❡ f ✱ ❡ é ✐♥❞✐❝❛❞❛ ❝♦♠ ✉♠❛ ❞❛s ♥♦t❛çõ❡s ✿ f ′ ; Df ; dx ❊①❡♠♣❧♦ ✺✳✷✳ ❉❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡✳ Pr♦✈❡ q✉❡ ❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ f (x) = k ♦♥❞❡ k ∈ R✱ é ❞❡r✐✈á✈❡❧ ❡♠ t♦❞♦ ♣♦♥t♦ a ∈ R ❡ f ′ (a) = 0✳ ❙♦❧✉çã♦✳ P❛r❛ t♦❞♦ a ∈ R t❡♠♦s✿ lim x→a k−k f (x) − f (a) = lim = 0✱ ✐st♦ é✱ f ′ (a) = 0 ∀ x ∈ R✳ x→a x−a x−a P♦rt❛♥t♦✱ s✉❛ ❢✉♥çã♦ ❞❡r✐✈❛❞❛ é f ′ (x) = 0, ∀ x ∈ R✳ ❊①❡♠♣❧♦ ✺✳✸✳ ❉❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ ❛✜♠✳ ✷✹✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ Pr♦✈❛r q✉❡ ❛ ❢✉♥çã♦ f (x) = cx + d ′ f (a) = c✳ (c, d ∈ R, c 6= 0✮ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R a∈R ❡✱ é ❞❡r✐✈á✈❡❧ ❡♠ t♦❞♦ ❙♦❧✉çã♦✳ ❈♦♠ ❡❢❡✐t♦✱ ♣❛r❛ t♦❞♦ a ∈ R t❡♠♦s✿ (cx + d) − (ca + d) c(x − a) f (x) − f (a) = lim = lim =c x→a x→a x − a x→a x−a x−a f ′ (a) = lim ❆ss✐♠✱ ♦❜t❡♠♦s ❞❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ f (x) = cx + d✱ ❡ ❛ ❢✉♥çã♦ f ′ (x) = c, P♦rt❛♥t♦✱ f ′ (a) = c✳ ❊①❡♠♣❧♦ ✺✳✹✳ ❉❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ ▼♦str❡ q✉❡✱ s❡ f (x) = x2 ✱ ❡♥tã♦ ∀ x ∈ R✳ f (x) = x2 ✳ f é ❞❡r✐✈á✈❡❧ ❡♠ t♦❞♦ ❙♦❧✉çã♦✳ x∈R ❡ t❡♠♦s f ′ (x) = 2x ❚❡♠♦s✱ ♣❛r❛ t♦❞♦ x ∈ R ❡ h = ∆x✿ f (x + h) − f (x) (x + h)2 − x2 = lim = lim (2x + h) = 2x h→0 h→0 h→0 (x + h) − x h f ′ (x) = lim P♦rt❛♥t♦✱ f ′ (x) = 2x, ❊①❡♠♣❧♦ ✺✳✺✳ ❉❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ ′ f (x) = nx ∀ x ∈ R✳ f (x) = xn n−1 f (x) = xn ✳ (n ∈ N, n 6= 2) é ❞❡r✐✈á✈❡❧ ❡♠ t♦❞♦ x∈R ❡ t❡♠♦s ❙♦❧✉çã♦✳ f (x + ∆x) − f (x) (x + ∆x)n − xn = lim = ∆x→0 ∆x→0 ∆x ∆x P❛r❛ t♦❞♦ x ∈ R t❡♠♦s✿ f ′ (x) = lim [(x + ∆x) − x][(x + ∆x)n−1 + x(x + ∆x)n−2 + · · · + xn−2 (x + ∆x) + xn−1 ] = ∆x→0 ∆x = lim lim [(x + ∆x)n−1 + x(x + ∆x)n−2 + · · · + xn−2 (x + ∆x) + xn−1 ] = nxn−1 ∆x→0 ✐st♦ é✱ f ′ (x) = nxn−1 ✳ ❊①❡♠♣❧♦ ✺✳✻✳ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ ❙♦❧✉çã♦✳ f (x) =| x | ♥ã♦ é ❞❡r✐✈á✈❡❧ ❡♠ x=0 |x| f (x) − f (0) = lim ✳ x→0 x−0 x |x| |x| =1 ❡ lim− = −1✳ ❉❛ ❞❡✜♥✐çã♦ ❞♦ ✈❛❧♦r ❛❜s♦❧✉t♦✱ s❡❣✉❡✿ lim+ x→0 x→0 x x P♦rt❛♥t♦✱ ♥ã♦ ❡①✐st❡ f ′ (0)✱ ♣♦ré♠ ✈❡r✐✜❝❛✲s❡ q✉❡ f é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♠ x = 0✳ ❉❛ ❞❡✜♥✐çã♦ ❞❛ ❞❡r✐✈❛❞❛ f ′ (0) = lim x→0 ✷✹✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✺✳✼✳ ❉❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳ Pr♦✈❡ q✉❡ ❛ ❢✉♥çã♦ f (x) = ax ♣❛r❛ a>0 ❡ f ′ (x) = ax Lna✳ a 6= 1 é ❞❡r✐✈á✈❡❧ ❡♠ t♦❞♦ x ∈ R✱ ❡ t❡♠♦s ❙♦❧✉çã♦✳ ah − 1 ah − 1 ax+h − ax = lim ax · = ax · lim ✱ ❡ s❡♥❞♦✱ h→0 h→0 h→0 h h h ah − 1 ax+h − ax ♣❡❧♦ ❧✐♠✐t❡ ♥♦tá✈❡❧ ❞♦ ❊①❡♠♣❧♦ ✭✸✳✺✹✮✱ lim = Lna✱ s❡❣✉❡✲s❡ q✉❡ lim = h→0 h→0 h h x a · Lna✳ P❛r❛ t♦❞♦ x ∈ R t❡♠♦s✿ lim P♦rt❛♥t♦ ✱ f (x) = ax é ❞❡r✐✈á✈❡❧ ❡ t❡♠♦s f ′ (x) = ax · Lna✳ ◆♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❡♠ q✉❡ a = e t❡rí❛♠♦s✱ f ′ (x) = ex ✱ ♣♦✐s Lne = 1✳ ❊①❡♠♣❧♦ ✺✳✽✳ ❉❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ Pr♦✈❡ q✉❡ s❡ f (x) = senx✱ ❡♥tã♦ sen x✳ f ′ (x) = cos x✳ ❙♦❧✉çã♦✳ sen(x + h) − senx) f (x + h) − f (x) = lim = h→0 h→0 h h ❚❡♠♦s✿ lim   senx(cos h − 1) senh · cos x senx · cos h + senh · cos x − senx = lim + = lim h→0 h→0 h h h cos h − 1 senh + cos x · lim = (senx)(0) + (cos x)(1) = cos x h→0 h→0 h h = senx · lim ▲♦❣♦✱ ❛ ❢✉♥çã♦ ❞❡r✐✈❛❞❛ ♣❛r❛ f (x) = senx é ❛ ❢✉♥çã♦ f ′ (x) = cos x✳ ✺✳✷✳✶ ❘❡t❛ t❛♥❣❡♥t❡✳ ❘❡t❛ ♥♦r♠❛❧ ❈♦♥s✐❞❡r❡ ✉♠❛ ❝✉r✈❛ C ✱ ❡ ✉♠ ♣♦♥t♦ ✜①♦ P ❡♠ t❛❧ ❝✉r✈❛✱ ❡ s❡❥❛ ✉♠❛ r❡t❛ s❡❝❛♥t❡ q✉❡ ❝♦rt❛ à ❝✉r✈❛ C ♥♦s ♣♦♥t♦s P ❡ Q✱ ♦♥❞❡ ❡ P 6= Q ❡ ♦ ♣♦♥t♦ Q ∈ C ✳ ◗✉❛♥❞♦ Q ❛♣r♦①✐♠❛✲s❡ ✐♥❞❡✜♥✐❞❛♠❡♥t❡ ❛♦ ♣♦♥t♦ P ✱ ❛tr❛✈és ❞❛ ❝✉r✈❛ C ✱ ❛ s❡❝❛♥t❡ ♦❝✉♣❛rá ❞✐✈❡rs❛s ♣♦✲ s✐çõ❡s✳ ❙❡✱ ❝♦♠ ❛ ❛♣r♦①✐♠❛çã♦ ✐❧✐♠✐t❛❞❛ ❞♦ ♣♦♥t♦ Q ❛tr❛✈és ❞❛ ❝✉r✈❛ C ♣❛r❛ ♦ ♣♦♥t♦ P ✱ ❛ s❡❝❛♥t❡ t❡♥❞❡ ❛ ♦❝✉♣❛r ❛ ♣♦s✐çã♦ ❞❡ ✉♠❛ r❡t❛ ❞❡♥♦♠✐♥❛❞❛ LT ✱ ❝❤❛♠❛✲ s❡ ❛ ❡st❛ ú❧t✐♠❛ ❞❡ r❡t❛ t❛♥❣❡♥t❡ à ❝✉r✈❛ C ♥♦ ♣♦♥t♦ P ✱ ❝♦♠♦ ✐♥❞✐❝❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✺✳✷✮✳ ❋✐❣✉r❛ ✺✳✷✿ ❙❡❥❛ f : R −→ R✱ ❢✉♥çã♦ ❞❡r✐✈á✈❡❧ ❡♠ x = a❀ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ ❞❡r✐✈❛❞❛ f ′ (a) t❡♠♦s ❛s s❡❣✉✐♥t❡s ❞❡✜♥✐çõ❡s✿ ✷✺✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❉❡✜♥✐çã♦ ✺✳✺✳ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❘❡t❛ t❛♥❣❡♥t❡✳ ❆ r❡t❛ t❛♥❣❡♥t❡ ❛♦ ❣rá✜❝♦ ❞❡ y = f (x) ♥♦ ♣♦♥t♦ P (a, f (a)) t❡♠ ♣♦r ❡q✉❛çã♦✿ LT : y − f (a) = f ′ (a)(x − a) ❉❡✜♥✐çã♦ ✺✳✻✳ ❘❡t❛ ♥♦r♠❛❧✳ ❆ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ ❞❡ f ❡♠ P✱ P (a, f (a)) ❡ é ♣❡r♣❡♥❞✐❝✉❧❛r à r❡t❛ t❛♥❣❡♥t❡ ♥♦ ❣rá✜❝♦ é ❝❤❛♠❛❞❛ ✏❘❡t❛ ♥♦r♠❛❧ ❛♦ ❣rá✜❝♦ ❞❡ f ♥♦ ♣♦♥t♦ P ✑✳ ✭❋✐❣✉r❛ ✭✺✳✸✮✮✳ ❙❡ f ′ (a) 6= 0 ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ ♥♦r♠❛❧ é ❞❛❞❛ ♣♦r✿ LN : y − f (a) = − 1 f ′ (a) (x − a)✳ ❙❡ f ′ (a) = 0✱ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ ♥♦r♠❛❧ é✿ LN : x = a✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ ❞❛ t❛♥❣❡♥t❡ AP ✱ ❝♦♠✲ ♣r❡❡♥❞✐❞♦ ❡♥tr❡ ♦ ♣♦♥t♦ ❞❡ t❛♥❣ê♥❝✐❛ ❡ ♦ ❡✐①♦ x✱ é ❝❤❛✲ ♠❛❞♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❛ t❛♥❣❡♥t❡✱ ❡ é ❞❡♥♦t❛❞♦ ♣♦r T✳ ❆ ♣r♦❥❡çã♦ ❞❡ AP s♦❜r❡ ♦ ❡✐①♦ x✱ ✐st♦ é AB é ❋✐❣✉r❛ ✺✳✸✿ ❝❤❛♠❛❞♦ s✉❜t❛♥❣❡♥t❡✱ ❡ s❡✉ ❝♦♠♣r✐♠❡♥t♦ ❞❡♥♦t❛✲s❡ ❝♦♠ ST ✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ ❞❛ ♥♦r♠❛❧ P C ✱ ❝♦♠♣r❡❡♥❞✐❞♦ ❡♥tr❡ ♦ ♣♦♥t♦ ❞❡ t❛♥❣ê♥❝✐❛ ❡ ♦ ❡✐①♦ x✱ é ❝❤❛♠❛❞♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♥♦r♠❛❧✱ ❡ é ❞❡♥♦t❛❞♦ ❝♦♠ N ✳ ❆ ♣r♦❥❡çã♦ ❞❡ P C s♦❜r❡ ♦ ❡✐①♦ x✱ é ❝❤❛♠❛❞♦ s✉❜♥♦r♠❛❧ ❡ s❡✉ ❝♦♠♣r✐♠❡♥t♦ ❞❡♥♦t❛✲s❡ ❝♦♠ SN ✳ ❉❛ ❋✐❣✉r❛ ✭✺✳✸✮ t❡♠♦s✿ f (a) f (a) = ′ tan α f (a) • ST =| AB |= • q f (a) p ′ (f (a))2 + 1 T =| AP |= (f (a))2 + ST2 = ′ f (a) • SN =| BC |=| f (a) · tan α |=| f (a) · f ′ (a) | • N =| P C |= p p 2 = f (a) · (f ′ (a))2 + 1 (f (a))2 + SN ➚ ❧✉③ ❞❡st❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛✱ ♣♦❞❡♠♦s ❞❡✜♥✐r✿ r(∆x) := f (a + ∆x) − f (a) − f ′ (a)∆x ✷✺✶ ✭✺✳✸✮ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❞❡ ♦♥❞❡✱ ❡♠ ✈✐rt✉❞❡ ❞❛ ❞❡✜♥✐çã♦ ❞❛ ❞❡r✐✈❛❞❛ ❡ ❞❛ ✐❣✉❛❧❞❛❞❡ ✭✺✳✸✮✱ s❡❣✉❡ q✉❡ r(∆x) =0 ∆x→0 ∆x lim ✭✺✳✹✮ ❱❡❥❛♠♦s t❛❧ ❢❛t♦ ❣❡♦♠❡tr✐❝❛♠❡♥t❡✳ ❖❜s❡r✈❡✱ ❛ ♠❡❞✐❞❛ q✉❡ ∆x → 0✱ ♦ ♣♦♥t♦ a + ∆x t❡♥❞❡ ♣❛r❛ ♦ ♣♦♥t♦ a✱ ❛s r❡t❛s s❡❝❛♥t❡s ❛♦s ♣♦♥t♦s (a, f (a)) ❡ (a + ∆x, f (a + ∆x)) t❡♥❞❡♠ à r❡t❛ t❛♥❣❡♥t❡ ♥♦ ♣♦♥t♦ (a, f (a)) ❡ ♦ r❡st♦ r(∆x) t❡♥❞❡ ♠♦❞✉❧❛r♠❡♥t❡ ♣❛r❛ ③❡r♦✳ ◆♦t❡♠♦s q✉❡ ♦ ♣r♦❞✉t♦ f ′ (a) · ∆x ♣♦❞❡ s❡r ❡♥❝❛r❛❞♦✱ ❛ ♠❡❞✐❞❛ q✉❡ ∆x ✈❛r✐❛ ❡♠ R ✱ ❝♦♠♦ ✉♠❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r T : R −→ R✱ ❞❡✜♥✐❞❛ ♣♦r T (∆x) = f ′ (a) · ∆x ✭q✉❡ ❞❡♣❡♥❞❡ ❞♦ ♣♦♥t♦ a✮ ❋✐❣✉r❛ ✺✳✹✿ ❖ ❝♦♥❝❡✐t♦ ❞❡ ❞❡r✐✈❛❞❛ ❛ss♦✲ ❞❡ ♠♦❞♦ q✉❡ ❞❡✜♥✐♥❞♦✲s❡✱ ❝♦♠♦ ❡♠ ✭✺✳✸✮✱ ❝✐❛❞♦ à ❡①✐stê♥❝✐❛ ❞❡ ❛♣❧✐❝❛çõ❡s ❧✐♥❡❛r❡s✳ r(∆x) := f (a + ∆x) − f (a) − T (∆x) t❡♠♦s r(∆x) = 0✳ ∆x→0 ∆x lim ◆♦ ❝❛s♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡r✐✈❛❞❛ à ❧✉③ ❞♦ ❡①♣♦st♦ ❛❝✐♠❛ ♣♦❞❡ ♥ã♦ ❛❥✉❞❛r ♠✉✐t♦✳ ❈♦♥t✉❞♦✱ q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ❢✉♥çõ❡s r❡❛✐s ❞❡ ♠❛✐s ❞❡ ✉♠❛ ✈❛r✐á✈❡❧✱ ❡st❛ ♥♦✈❛ ♠❛♥❡✐r❛ ❞❡ ❝♦♥❝❡❜❡r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡r✐✈❛❞❛ é ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛✱ ❝♦♥❢♦r♠❡ s❡rá ❛❜♦r❞❛❞♦ ♣♦st❡r✐♦r♠❡♥t❡ ❡♠ ❞✐s❝✐♣❧✐♥❛ ❈á❧❝✉❧♦ ❞❡ ❱ár✐❛s ❱❛r✐á✈❡✐s✳ ❊①❡♠♣❧♦ ✺✳✾✳ ❉❛❞❛ ❛ ❢✉♥çã♦ g(x) = x2 + 3x − 2✱ ♦❜t❡r ❛s ❡q✉❛çõ❡s ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❡ r❡t❛ ♥♦r♠❛❧ ❛♦ ❣rá✜❝♦ ❞❡ f ♥♦ ♣♦♥t♦ P (2, 8) ❡ ❞❡t❡r♠✐♥❡ ♦s ❝♦♠♣r✐♠❡♥t♦s ❞❛ r❡t❛ t❛♥❣❡♥t❡✱ ♥♦r♠❛❧✱ s✉❜t❛♥❣❡♥t❡ ❡ s✉❜♥♦r♠❛❧✳ ❙♦❧✉çã♦✳ ❈♦♠♦ g(2) = 8✱ ❡♥tã♦ P (2, 8) ♣❡rt❡♥❝❡ ❛♦ ❣rá✜❝♦ ❞❡ g(x)✳ P♦r ♦✉tr♦ ❧❛❞♦✱ g ′ (x) = 2x+3✱ ❧♦❣♦ g ′ (2) = 7✱ ❛ss✐♠ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ♣❡❞✐❞❛ é✿ LT : y − 8 = 7(x − 2) ✐st♦ é 7x − y = 6✳ ❖ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ ♥♦r♠❛❧ é m = − 1 1 = − ❡ s✉❛ ❡q✉❛çã♦✱ LN : y − 8 = g ′ (2) 7 1 − (x − 2) ✐st♦ é✿ LN : x + 7y = 58✳ 7 √ 40 √ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞❛ t❛♥❣❡♥t❡ é✿ T = 2❀ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♥♦r♠❛❧ é ✿ N = 40 2❀ ♦ 7 √ 8 ❝♦♠♣r✐♠❡♥t♦ ❞❛ s✉❜t❛♥❣❡♥t❡ é ✿ ST = ❡✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ s✉❜♥♦r♠❛❧ é ✿ SN = 40 2✳ 7 ✷✺✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✺✳✶✵✳ ❙❡❥❛ f (x) = x2 − x − 2✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛♦ ❣rá✜❝♦ ❞❡ f q✉❡ s❡❥❛ ♣❛r❛❧❡❧❛ à r❡t❛ L : x + y = 8✳ ❙♦❧✉çã♦✳ ❖ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ L é m = −1✳ ❖ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ ❛ ❞❡t❡r♠✐♥❛r ❡ f ′ (x) = 2x − 1✳ ❉❡s❞❡ q✉❡ ❛s r❡t❛s t❡♠ q✉❡ s❡r ♣❛r❛❧❡❧❛s✱ f ′ (x) = −1 ♦ q✉❡ ✐♠♣❧✐❝❛ 2x − 1 = −1 ❧♦❣♦ x = 0 ❡ ♦ ♣♦♥t♦ ❞❡ t❛♥❣ê♥❝✐❛ ❛❝♦♥t❡❝❡ ❡♠ P (0, f (0)) ✐st♦ é ❡♠ P (0, −2)✳ P♦rt❛♥t♦✱ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ ♣❡❞✐❞❛ é✿ y − (−2) = −1(x − 0) ✐st♦ é x + y = −2✳ ❊①❡♠♣❧♦ ✺✳✶✶✳ ❆ r❡t❛ L ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s P (4, 5) ❡ Q(9, 11) ❡✱ é ♥♦r♠❛❧ ❛♦ ❣rá✜❝♦ ❞❡ h(x) = x2 − 4 ❡♠ R(a, h(a))✳ ❉❡t❡r♠✐♥❡ R ❡ ❛ ❡q✉❛çã♦ ❞❡ L✳ ❙♦❧✉çã♦✳ ❆♣❧✐❝❛♥❞♦ ❞❡r✐✈❛❞❛s✱ ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ ♥♦r♠❛❧ L é ✿m = − 1 h′ (a) =− 1 ✳ 2a P♦r ♦✉tr♦ ❧❛❞♦✱ ❛♣❧✐❝❛♥❞♦ ❛ ❞❡✜♥✐çã♦✱ ❞❛❞♦s ♦s ♣♦♥t♦s P ❡ Q ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ 6 11 − 5 = ✳ 9−4 5  2 1 6 5 5 551 551 −5 ▲♦❣♦ − = ⇒ a=− ❡ h(a) = ❀ ❡♥tã♦ R(− , − ) −4 = − 2a 5 12 12 144 12 144 6 ❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ L é ✿ y − 5 = (x − 4)✳ 5 P♦rt❛♥t♦✱ L : 6x − 5y = −1✳ r❡t❛ ♥♦r♠❛❧ L é ❞❛❞♦ ♣♦r m = ✺✳✸ ❉❡r✐✈❛❞❛s ❧❛t❡r❛✐s ❙❡❥❛ f : A −→ R ✉♠❛ ❢✉♥çã♦✱ ❡ ♦ ♣♦♥t♦ ❞❡ ❛❝✉♠✉❧❛çã♦ a ∈ A✳ ❉❡✜♥✐çã♦ ✺✳✼✳ ❉❡r✐✈❛❞❛ á ❡sq✉❡r❞❛✳ ❉✐③✲s❡ q✉❡ f é ❞❡r✐✈á✈❡❧ à ❡sq✉❡r❞❛ ♥♦ ♣♦♥t♦ x = a✱ q✉❛♥❞♦ ❡①✐st❡ ❡ é ✜♥✐t♦ ♦ ❧✐♠✐t❡✿ lim− x→a f (x) − f (a) x−a ❊st❡ ❧✐♠✐t❡ é ❝❤❛♠❛❞♦ ❞❡r✐✈❛❞❛ ❞❡ f à ❡sq✉❡r❞❛ ❞♦ ♣♦♥t♦ x = a✱ ❡ ✐♥❞✐❝❛❞♦ ❝♦♠ ✉♠❛ ❞❛s ♥♦t❛çõ❡s✿ f ′ (a− ); Df (a− ); df − (a )✳ dx ✷✺✸ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❡✜♥✐çã♦ ✺✳✽✳ ❉❡r✐✈❛❞❛ à ❞✐r❡✐t❛✳ ❉✐③✲s❡ q✉❡ f é ❞❡r✐✈á✈❡❧ à ❞✐r❡✐t❛ ♥♦ ♣♦♥t♦ x = a q✉❛♥❞♦ ❡①✐st❡ ❡ é ✜♥✐t♦ ♦ ❧✐♠✐t❡✿ f (x) − f (a) x−a lim+ x→a ❊st❡ ❧✐♠✐t❡ é ❝❤❛♠❛❞♦ ❞❡r✐✈❛❞❛ ❞❡ f à ❞✐r❡✐t❛ ❞♦ ♣♦♥t♦ x = a ❡ ✐♥❞✐❝❛❞♦ ❝♦♠ ✉♠❛ ❞❛s ♥♦t❛çõ❡s✿ f ′ (a+ ); df + (a )✳ dx Df (a+ ); ❊①❡♠♣❧♦ ✺✳✶✷✳ ❈❛❧❝✉❧❡ ❛s ❞❡r✐✈❛❞❛s ❧❛t❡r❛✐s ♥♦ ♣♦♥t♦ a = 0 ❞❛ ❢✉♥çã♦✿ f (x) = ( s❡ x ≤ 0 s❡ x > 0 x x2 ❙♦❧✉çã♦✳ ❉❛ ❉❡✜♥✐çã♦ ✭✺✳✻✮✱ t❡♠♦s q✉❡ lim− x→0 t❛♥t♦✱ f ′ (0− ) = 1✳ x−0 f (x) − f (0) = lim− = lim− ·1 = 1✳ P♦r x→0 x→0 x−0 x x2 − 0 f (x) − f (0) = lim+ = lim+ ·x = 0✳ x→0 x→0 x→0 x−0 x ′ + ▲♦❣♦✱ f (0 ) = 0✳ ◆ã♦ ❡①✐st❡ ❞❡r✐✈❛❞❛ ❞❡ f (x) ♥♦ ♣♦♥t♦ x = 0✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❡❧❛ ❉❡✜♥✐çã♦ ✭✺✳✼✮ lim+ ❊①❡♠♣❧♦ ✺✳✶✸✳ ❈❛❧❝✉❧❡ ❛s ❞❡r✐✈❛❞❛s ❧❛t❡r❛✐s ❞❛ ❢✉♥çã♦ f (x) =| x | ♥♦ ♣♦♥t♦ x = 0✳ ❙♦❧✉çã♦✳ P❡❧♦ ♠♦str❛❞♦ ♥♦ ❊①❡♠♣❧♦ ✭✺✳✻✮✱ r❡s✉❧t❛ q✉❡✿ f (x) − f (0) |x| = = x−0 x ( 1 −1 s❡ x > 0 s❡ x < 0 ❧♦❣♦✱ f ′ (0− ) = −1 ❡ f ′ (0+ ) = 1❀ ♣♦rt❛♥t♦ f (x) =| x | ♥ã♦ é ❞❡r✐✈á✈❡❧ ❡♠ x = 0✳ ❊①❡♠♣❧♦ ✺✳✶✹✳ Pr♦✈❡ q✉❡ ❛ ❢✉♥çã♦ f (x) = x = 0✳ ( x 1 s❡ x ≥ 0 ♥ã♦ é ❞❡r✐✈á✈❡❧ à ❡sq✉❡r❞❛ ♥♦ ♣♦♥t♦ s❡ x < 0 ❙♦❧✉çã♦✳ ❉❡✱ ❢❛t♦ t❡♠♦s lim− x→0 f (x) − f (0) 1−0 1 = lim− = lim− = −∞✱ ❡ ❛ ❢✉♥çã♦ ♥ã♦ é x→0 x − 0 x→0 x x−0 ❞❡r✐✈á✈❡❧ à ❡sq✉❡r❞❛✱ ♣♦rq✉❡ ♦ ❧✐♠✐t❡ ❧❛t❡r❛❧ à ❡sq✉❡r❞❛ ♥ã♦ é ✜♥✐t♦ ✭é ✐♥✜♥✐t♦✮✳ ✷✺✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R Pr♦♣r✐❡❞❛❞❡ ✺✳✶✳ ❙❡❥❛ f : R −→ R ✉♠❛ ❢✉♥çã♦✱ f é ❞❡r✐✈á✈❡❧ ❡♠ x = a s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡♠ ❡ sã♦ ✐❣✉❛✐s ❛s ❞❡r✐✈❛❞❛s ❧❛t❡r❛✐s f ′ (a− ) ❡ f ′ (a+ )✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ✺✳✹ ❉❡r✐✈❛❜✐❧✐❞❛❞❡ ❡ ❝♦♥t✐♥✉✐❞❛❞❡ Pr♦♣r✐❡❞❛❞❡ ✺✳✷✳ ❙❡ ✉♠❛ ❢✉♥çã♦ y = f (x) é ❞❡r✐✈á✈❡❧ ♥♦ ♣♦♥t♦ x = a✱ ❡♥tã♦ ❡❧❛ é ❝♦♥tí♥✉❛ ❡♠ x = a✳ ❉❡♠♦♥str❛çã♦✳ f (x) − f (a) ❡①✐st❡ ❡✱ é ✜♥✐t♦✳ x→a x−a P♦r ❤✐♣ót❡s❡✱ f é ❞❡r✐✈á✈❡❧ ❡♠ x = a❀ ❡♥tã♦ f ′ (a) = lim P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ t♦❞♦ x ∈ D(f ), f (x) − f (a) = x 6= a✱ ❛ s❡❣✉✐♥t❡ ✐❞❡♥t✐❞❛❞❡ é ✈á❧✐❞❛✿ f (x) − f (a) · (x − a) x−a ❊♥tã♦ ❝❛❧❝✉❧❛♥❞♦ ♦ ❧✐♠✐t❡ ❡♠ [f (x) − f (a)] q✉❛♥❞♦ x → a✱ ❡ ❛♣❧✐❝❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ♣r♦❞✉t♦ ❞❡ ❧✐♠✐t❡s ❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ f ′ (a)✱ t❡♠♦s✿ f (x) − f (a) · (x − a) = f ′ (a).0 = 0 x→a x−a lim [f (x) − f (a)] = lim x→a ✐st♦ é lim [f (x) − f (a)] = 0✳ x→a P♦rt❛♥t♦✱ lim f (x) = f (a) ❀ ✐st♦ é f é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ x = a✳ x→a ❖❜s❡r✈❛çã♦ ✺✳✷✳ ❆ r❡❝í♣r♦❝❛ ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✷✮ ♥ã♦ é ✈❡r❞❛❞❡✐r❛✱ ✐st♦ é✱ ✉♠❛ ❢✉♥çã♦ ♣♦❞❡ s❡r ❝♦♥tí✲ ♥✉❛ ♥✉♠ ♣♦♥t♦✱ s❡♠ q✉❡ s❡❥❛ ❞❡r✐✈á✈❡❧ ♥❡ss❡ ♣♦♥t♦✳ ❯♠ ❡①❡♠♣❧♦ é ❞❛❞♦ ♣❡❧❛ ❢✉♥çã♦ f (x) =| x | q✉❡ é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ x = 0✱ ♣♦ré♠ ♥ã♦ é ❞❡r✐✈á✈❡❧ ♥❡ss❡ ♣♦♥t♦ ✭✈❡❥❛ ♦ ❊①❡♠♣❧♦ ✭✺✳✻✮✮✳ ❖✉tr♦ ❡①❡♠♣❧♦ é ❞❛❞♦ ♣❡❧❛ ❢✉♥çã♦ f (x) = ( x x2 s❡ x ≤ 0 s❡ x > 0 ❡❧❛ é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ x = 0✱ ♣♦ré♠ ♥ã♦ é ❞❡r✐✈á✈❡❧ ♥❡ss❡ ♣♦♥t♦✳ ❊①❡♠♣❧♦ ✺✳✶✺✳ ❆♥❛❧✐s❛r ❛ ❞❡r✐✈❛❜✐❧✐❞❛❞❡ ❡♠ x = 2 ♣❛r❛ ❛ ❢✉♥çã♦ f (x) ❞❡✜♥✐❞❛ ♣♦r✿ ✷✺✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ f (x) = ( ❙♦❧✉çã♦✳ 2 − x2 x2 − 4x + 4 R s❡ x ≤ 2 s❡ x > 2 ❆ ❢✉♥çã♦ é ❝♦♥tí♥✉❛ ❡♠ x = 2✱ ♣♦ré♠ ♥ã♦ é ❞❡r✐✈á✈❡❧ ❡♠ x = 2❀ ♦❜s❡r✈❡ q✉❡ ❛s ❞❡r✐✈❛❞❛s ❧❛t❡r❛✐s sã♦ ❞✐❢❡r❡♥t❡s✿ f ′ (2− ) = lim− x→2 f ′ (2+ ) = lim+ x→2 ❊①❡♠♣❧♦ ✺✳✶✻✳ (2 − x2 ) − (2 − 22 ) f (x) − f (2) = lim− = −4 x→2 x−2 x−2 (x2 − 4x + 4) − (2 − 22 ) f (x) − f (2) = lim+ = +∞ x→2 x−2 x−2 ❉❡t❡r♠✐♥❡ ✈❛❧♦r❡s a ❡ b ♣❛r❛ q✉❡ ❡①✐st❛ f ′ (1) s❡✿ f (x) = ❙♦❧✉çã♦✳ ( ax + b, s❡✱ x ≥ 1 x2 , s❡✱ x < 1 ❈♦♠♦ f ′ (1) ❡①✐st❡✱ ❡♥tã♦ f é ❝♦♥tí♥✉❛ ❡♠ x = 1❀ ✐st♦ é f (1) = 1 = a + b ❡ f ′ (1− ) = f ′ (1+ )✱ ❝♦♠♦ f ′ (1− ) = 2 ❡ f (1+ ) = a ♦❜té♠✲s❡ q✉❡ a = 2✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ b = −1✳ ❊①❡♠♣❧♦ ✺✳✶✼✳ ❉❡t❡r♠✐♥❡ s❡ ❛ ❢✉♥çã♦ f (x) = ❙♦❧✉çã♦✳ ( x2 , s❡✱ x é r❛❝✐♦♥❛❧ é ❞❡r✐✈á✈❡❧ ❡♠ x = 0✳ 0, s❡✱ x é ✐rr❛❝✐♦♥❛❧ ❉❛ ❞❡✜♥✐çã♦ ❞❡ ❢✉♥çã♦ ❞❡r✐✈á✈❡❧ ♥♦ ♣♦♥t♦ x = 0 t❡♠♦s✿ f (h) − 02 f (h) f (h) − f (0) = lim = lim h→0 h→0 h h→0 h h f ′ (0) = lim ♣♦ré♠✱ f (h) = h ( h, s❡✱ h é r❛❝✐♦♥❛❧ ❧♦❣♦✱ é ❞❡r✐✈á✈❡❧ ❡♠ x = 0 ❡ ❡♠ q✉❛✐sq✉❡r ❞♦s 0, s❡✱ h é ✐rr❛❝✐♦♥❛❧ f (h) =0✳ h→0 h P♦rt❛♥t♦✱ f ′ (0) = 0✳ ❞♦✐s ❝❛s♦s lim ❊①❡♠♣❧♦ ✺✳✶✽✳ ❉❡t❡r♠✐♥❡ s❡ ❛ ❢✉♥çã♦ f (x) ❛ss✐♠ ❞❡✜♥✐❞❛ ✿ f (x) = ( x, s❡✱ x ≥ 0 1, s❡✱ x < 0 é ❞❡r✐✈á✈❡❧ ❡♠ x = 0✳ ❙♦❧✉çã♦✳ ✷✺✻ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✷✮ t❡♠♦s✿ ❡♠ x = a✱ ❡♥tã♦ ❡❧❛ ♥ã♦ é ❞❡r✐✈á✈❡❧ ❡♠ x = a✑✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛ r❡❝í♣r♦❝❛ ❞❛ ❖❜s❡r✈❡ q✉❡ ❛ ❢✉♥çã♦ ✺✳✹✳✶ ✏ ❙❡ ✉♠❛ ❢✉♥çã♦ ♥ã♦ é ❝♦♥tí♥✉❛ f (x) ♥ã♦ é ❝♦♥tí♥✉❛ ❡♠ x = 0❀ ❧♦❣♦ ❡❧❛ ♥ã♦ é ❞❡r✐✈á✈❡❧ ❡♠ x = 0✳ ❘❡❣r❛s ❞❡ ❞❡r✐✈❛çã♦ Pr♦♣r✐❡❞❛❞❡ ✺✳✸✳ ❙❡❥❛♠ f ❡ g ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ♥✉♠ ❝♦♥❥✉♥t♦ A ⊆ R ❡ ❞❡r✐✈á✈❡✐s ❡♠ a ∈ A✱ ❡ k ∈ R ✉♠❛ ❝♦♥st❛♥t❡✳ f 1 s❡ g(a) 6= 0✱ sã♦ ❞❡r✐✈á✈❡✐s ❡♠ x = a✱ ❊♥tã♦✱ ❛s ❢✉♥çõ❡s kf, f + g ✱ ❡ t❛♠❜é♠ ❡ g g ❡ t❡♠♦s✿ ✐✮ (kf )′ (a) = kf ′ (a)✳ ✐✐✮ (f + g)′ (a) = f ′ (a) + g ′ (a)✳ ✐✐✐✮ (f g)′ (a) = f ′ (a)g(a) + f (a)g ′ (a)✳  ′ g ′ (a) 1 s❡♠♣r❡ q✉❡ g(a) 6= 0✳ (a) = − g (g(a))2  ′ f f ′ (a)g(a) − f (a)g ′ (a) s❡♠♣r❡ q✉❡ g(a) 6= 0✳ (a) = g (g(a))2 ✐✈✮ ✈✮ ❉❡♠♦♥str❛çã♦✳ ✭✐✮ ❉♦ ❢❛t♦ s❡r k ✉♠❛ ❝♦♥st❛♥t❡ t❡♠♦s✿   (kf )(x) − (kf )(a) kf (x) − kf (a) f (x) − f (a) (kf ) (a) = lim = = lim = lim k · x→a x→a x→a x−a x−a x−a ′  f (x) − f (a) = k · lim = k · f ′ (a) x→a x−a  P♦rt❛♥t♦✱ kf é ❞❡r✐✈á✈❡❧ ❡♠ x = a✱ ❡ (kf )′ (a) = kf ′ (a)✳ ❉❡♠♦♥str❛çã♦✳ ✭✐✐✮   (f + g)(x) − (f + g)(a) f (x) − f (a) g(x) − g(a) (f + g) (a) = lim = = lim + x→a x→a x−a x−a x−a ′ ❡ ❝♦♠♦ f ❡ g sã♦ ❞❡r✐✈á✈❡✐s ❡♠ x = a✱ g(x) − g(a) f (x) − f (a) + lim = f ′ (a) + g ′ (a) x→a x→a x−a x−a lim P♦rt❛♥t♦✱ f +g é ❞❡r✐✈á✈❡❧ ❡♠ x=a ❡ (f + g)′ (a) = f ′ (a) + g ′ (a) ❉❡♠♦♥str❛çã♦✳ ✭✐✐✐✮ ✷✺✼ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❚❡♠♦s✱ ❛❞✐❝✐♦♥❛♥❞♦ ❡ s✉❜str❛✐♥❞♦ f (a) · g(x) (f · g)(x) − (f · g)(a) = x→a x−a (f g)′ (a) = lim   f (x) · g(x) − f (a) · g(x) + f (a) · g(x) − f (a) · g(a) = lim x→a x−a   f (x) − f (a) g(x) − g(a) = lim · g(x) + f (a) · x→a x−a x−a f ❈♦♠♦ g ❡ sã♦ ❞❡r✐✈á✈❡✐s ❡♠ x = a✱ ❡❧❛s sã♦ ❝♦♥tí♥✉❛s ❡♠ x = a❀ ✭✺✳✺✮ ❧♦❣♦✱ ❡♠ ✭✺✳✺✮ ✿  g(x) − g(a) f (x) − f (a) = · g(x) + f (a) · (f g) (a) = lim x→a x−a x−a  ′    g(x) − g(a) f (x) − f (a) · lim g(x) + f (a) · lim = f ′ (a)g(a) + f (a)g ′ (a) lim x→a x→a x→a x−a x−a  P♦rt❛♥t♦✱ (f.g)(x) g x=a ❡ (f.g)′ (a) = f ′ (a)g(a) + f (a)g ′ (a)✳ ✭✐✈✮ ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ é ❞❡r✐✈á✈❡❧ ❡♠ é ❞❡r✐✈á✈❡❧ ❡♠ x = a✱ é ❝♦♥tí♥✉❛ ❡♠ x=a ❡ s❡♥❞♦✱ ♣♦r ❤✐♣ót❡s❡ g(a) 6= 0✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❝♦♥s❡r✈❛çã♦ ❞♦ s✐♥❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛✱ ❡①✐st❡ ✉♠❛ ❜♦❧❛ B(a, r)✱ t❛❧ q✉❡ ♣❛r❛ q✉❛❧q✉❡r ✐st♦ s❡❣✉❡ q✉❡ g(x) 6= 0 ▲♦❣♦✱ ♣❛r❛ ❡♠ x ∈ B(a, r)❀  ′ 1 (a) = lim x→a g = lim − x→a P♦rt❛♥t♦✱ 1 g ❉❡♠♦♥str❛çã♦✳ g(x)−g(a) g(a)·g(x) x−a B(a, r)✳ x ∈ B(a, r)✱ t❡♠♦s g(x) t❡♠ ♦ ♠❡s♠♦ s✐♥❛❧ g(a)❀ ❞❡ t❡♠♦s✿   1 g (x) −   x−a 1 g (a) = lim x→a 1 g(x) − 1 g(a) x−a = lim x→a g(a)−g(x) g(a)·g(x) x−a = 1 g(x) − g(a) 1 · lim = −g ′ (a) · x→a x→a g(x) · g(a) x−a (g(a))2 = − lim é ❞❡r✐✈á✈❡❧ ❡♠ x = a✱ ❡ t❡♠✲s❡✿ ✭✈✮  ′ 1 1 · g ′ (a)✳ (a) = − 2 g (g(a)) f 1 = f · ❡✱ ♣♦r ❤✐♣ót❡s❡ f ❡ g ❞❡r✐✈á✈❡✐s ❡♠ x = a✱ ❧♦❣♦ ♣♦r ✭✐✈✮ ❞❡st❛ g g f 1 é ❞❡r✐✈á✈❡❧ ♣r♦♣r✐❡❞❛❞❡ s❡❣✉❡ q✉❡ ✱ ✭♣♦✐s g(a) 6= 0✮ é ❞❡r✐✈á✈❡❧❀ ❞❡ ✭✐✐✐✮ s❡❣✉❡✲s❡ q✉❡ ❡ g g ❡♠ x = a✱ ❛ss✐♠✿ ❖❜s❡r✈❡ q✉❡✱ ′  ′  −g ′ (a) 1 1 f = + f (a) · (a) = f · (a) = f ′ (a) · g g g(a) (g(a))2 ✷✺✽ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ = f ′ (a) · g(a) − f (a) · g ′ (a) (g(a))2 ❊①❡♠♣❧♦ ✺✳✶✾✳ ❉❛❞❛ ❛ ❢✉♥çã♦ f (x) = (x2 − 3x)2 ❞❡t❡r♠✐♥❡ f ′ (x)✳ ❙♦❧✉çã♦✳ f (x) = (x2 − 3x)2 = (x2 − 3x)(x2 − 3x)✱ ❡♥tã♦ ❛♣❧✐❝❛♥❞♦ ❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✸✮ ✐✐✐✮ s❡❣✉❡ f ′ (x) = (2x − 3)(x2 − 3x) + (x2 − 3x)(2x − 3) = 2(x2 − 3x)(2x − 3)✳ Pr♦♣r✐❡❞❛❞❡ ✺✳✹✳ ❙❡❥❛♠ f1 , f2 , · · · , fn ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ♥✉♠ ♠❡s♠♦ ❝♦♥❥✉♥t♦ A✱ ❡ ❞❡r✐✈á✈❡✐s ❡♠ x = a ∈ A ❡♥tã♦✿ ✐✮ f1 + f2 + · · · + fn é ❞❡r✐✈á✈❡❧ ❡♠ x = a ❡ t❡♠♦s✿ (f1 + f2 + · · · + fn )′ (a) = f1′ (a) + f2′ (a) + · · · + fn′ (a)✳ ✐✐✮ f1 × f2 × · · · × fn é ❞❡r✐✈á✈❡❧ ❡♠ x = a ❡ t❡♠♦s✿ (f1 × f2 × · · · × fn )′ (a) = = f1′ (a) × f2 (a) · · · fn (a) + f1 (a) × f2′ (a) · · · fn (a) + · · · + f1 (a) × f2 (a) · · · fn′ (a)✳ ❉❡♠♦♥str❛çã♦✳ ✭✐✮ ❆ ❞❡♠♦♥str❛çã♦ é ❢❡✐t❛ ♣♦r ✐♥❞✉çã♦ ✜♥✐t❛ ✳ ❉❡ ❢❛t♦ ✱ ♣❛r❛ n = 2 ❡❧❛ é ✈❡r❞❛❞❡✐r❛ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✸✮ ✭✐✐✮✱ ✐st♦ é ✱ s❡ f1 ❡ f2 sã♦ ❞❡r✐✈á✈❡✐s ❡♠ x = a❀ ❡♥tã♦ f1 + f2 é ❞❡r✐✈á✈❡❧ ❡♠ x = a ❡ t❡♠♦s (f1 + f2 )′ (a) = f1′ (a) + f2′ (a)✳ ❙✉♣♦♥❤❛ ♣❛r❛ n = p ✈❡r❞❛❞❡✐r❛✱ ✐st♦ é✱ (f1 +f2 +· · ·+fp )′ (a) = f1′ (a)+f2′ (a)+· · ·+fp′ (a)✱ ♠♦str❡♠♦s ♣❛r❛ n = p + 1✳ P❛r❛ n = p + 1✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r f1 + f2 + · · · + fp + fp+1 = (f1 + f2 + · · · + fp ) + fp+1 ✳ ❊✱ ❝♦♠♦ g = f1 + f2 + · · · + fp é ❞❡r✐✈á✈❡❧ ❡♠ x = a ✭ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✮ ❡ t❛♠❜é♠ fp+1 s❡❣✉❡✲s❡ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✸✮✭✐✐✮ q✉❡✿ (f1 + f2 + · · · + fp + fp+1 )′ (a) = (f1 + f2 + · · · + ′ ′ fp )′ (a) + fp+1 (a) = f1′ (a) + f2′ (a) + · · · + fp′ (a) + fp+1 (a)✳ ▲♦❣♦✱ ❡❧❛ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ n ∈ N✳ ❉❡♠♦♥str❛çã♦✳ ✭✐✐✮ ❊①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❊①❡♠♣❧♦ ✺✳✷✵✳ ❉❛❞❛ f (x) = 3x5 + x4 − x3 + 1 ❝❛❧❝✉❧❡✿ ❙♦❧✉çã♦✳ ❛✮ ❛✮ f ′ (x); b)f ′ (1)✳ P❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✸✮ ♣❛rt❡ ✭✐✮ ❡ ✭✐✐✮ t❡♠♦s✿ ✷✺✾ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ f ′ (x) = (3x5 )′ + (x4 )′ + (−x3 )′ + (1)′ = 3(x5 )′ + 4x3 − (x3 )′ + 0 = = 15x4 + 4x3 − 3x2 = 15x4 + 4x3 − 3x2 ❙♦❧✉çã♦✳ ❜✮ ➱ ✉♠❛ s✉❜st✐t✉✐çã♦ ❞✐r❡t❛✱ f ′ (1) = 15(1)4 + 4(1)3 − 3(1)2 = 16✳ ❊①❡♠♣❧♦ ✺✳✷✶✳ f (x) = (x2 + x + 1) · x3 ❉❛❞❛ ❙♦❧✉çã♦✳ ❝❛❧❝✉❧❛r f ′ (x)✳ ❆♣❧✐❝❛♥❞♦ ❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✸✮ ♣❛rt❡ ✭✐✐✐✮ ❡ ✭✐✮ t❡♠♦s✿ f ′ (x) = (x2 + x + 1)′ · x3 + (x2 + x + 1) · (x3 )′ = = (2x + 1 + 0) · x3 + (x2 + x + 1) · 3x2 = x2 (2x2 + x + 3x2 + 3x + 3) = = x2 (5x2 + 4x + 3) P♦rt❛♥t♦✱ f ′ (x) = x2 (5x2 + 4x + 3)✳ ❊①❡♠♣❧♦ ✺✳✷✷✳ ❙❡ f (x) = x ❡ ❙♦❧✉çã♦✳ ❚❡♠♦s✿ ▲♦❣♦✱ g(x) =| x |✱ f ′ (x) = 1✱ ❝❛❧❝✉❧❛r ♣❛r❛ t♦❞♦ x∈R (f + g)′ (x) = f ′ (x) + g ′ (x) = (f + g)′ (x)✳ ❡ ( g ′ (x) = ( 2, 0, x≥0 x<0 s❡✱ s❡✱ 1, −1, s❡✱ s❡✱ x≥0 x<0 ✳ ❊①❡♠♣❧♦ ✺✳✷✸✳ ❉❛❞❛ f (x) = ❙♦❧✉çã♦✳ ❙❡♥❞♦ 1 ✱ xn f (x) = 1 xn x ∈ R − {0} ♣❛r❛ ♣❛r❛ n ∈ N❀ ❡ n ∈ N✱ ❝❛❧❝✉❧❡ f ′ (x)✳ t❡♠♦s ♣♦r ❛♣❧✐❝❛çõ❡s ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✸✮ ✭✐✈✮✱ ♣❛r❛ n ′ 0 − (x ) −n · xn−1 t♦❞♦ x ∈ R − {0}✿ f (x) = = = −nx−n−1 ✳ n 2 2n (x ) x −n ′ −n−1 P♦rt❛♥t♦✱ f (x) = −n · x = n+1 ✳ x ′ ❊①❡♠♣❧♦ ✺✳✷✹✳ ❉❛❞❛ ❙♦❧✉çã♦✳ f (x) = x+2 , 1−x x 6= 1✱ ❝❛❧❝✉❧❡ f ′ (x)✳ ✷✻✵ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❚❡♠♦s✱ ♣♦r ❛♣❧✐❝❛çã♦ ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✸✮✲✭✈✮✱ ♣❛r❛ x 6= 1✿ (x + 2)′ (1 − x) − (x + 2)(1 − x)′ 1 · (1 − x) − (x + 2)(−1) 3 = = 2 2 (1 − x) (1 − x) (1 − x)2 3 ′ P♦rt❛♥t♦✱ f (x) = (1 − x)2 f ′ (x) = ❊①❡♠♣❧♦ ✺✳✷✺✳ ❉❛❞❛ ❛ ❢✉♥çã♦ f (x) = ❙♦❧✉çã♦✳ x · ex ✱ 1 + x2 ❝❛❧❝✉❧❛r f ′ (x)✳ ❆♣❧✐❝❛♥❞♦ ❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✸✮✲✭✈✮ ❡ ♦ ❊①❡♠♣❧♦ ✭✺✳✼✮✱ ✈❡♠✿ f ′ (x) = (x · ex )′ (1 + x2 ) − x · ex (1 + x2 )′ (1 · ex + x · ex )(1 + x2 ) − x · ex · 2x = = (1 + x2 )2 (1 + x2 )2 = P♦rt❛♥t♦✱ ex (1 + x − x2 + x3 ) ex + x · ex + x2 ex + x3 ex − 2x2 ex = (1 + x2 )2 (1 + x2 )2 f ′ (x) = ex (1 + x − x2 + x3 ) (1 + x2 )2 ❖❜s❡r✈❛çã♦ ✺✳✸✳ ❛✮ ◗✉❛♥❞♦ n ∈ Z ❡ f (x) = xn ✱ ❡♥tã♦ f ′ (x) = n.xn−1 ✳ ❜✮ ❊♠ ❣❡r❛❧✱ s❡ c é ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ❡ f (x) = xc ✱ ❡♥tã♦ ❛ ❞❡r✐✈❛❞❛ f ′ (x) = c · xc−1 ✳ P♦r ❡①❡♠♣❧♦✱ s❡ f (x) = √ 5 x✱ ❡♥tã♦ f ′ (x) = 1√ 5 x−4 ✳ 5 ❊①❡♠♣❧♦ ✺✳✷✻✳ ❉❛❞❛ ❛ ❢✉♥çã♦ ❙♦❧✉çã♦✳ f (x) = (x2 − 2x)3 ✱ ❞❡t❡r♠✐♥❡ f ′ (x)✳ f (x) = (x2 − 2x)3 = (x2 − 2x)(x2 − 2x)(x2 − 2x) ❧♦❣♦ f ′ (x) = (x2 − 2x)′ (x2 − 2x)(x2 − 2x) + (x2 − 2x)(x2 − 2x)′ (x2 − 2x) + (x2 − 2x)(x2 − 2x)(x2 − 2x)′ ✱ ✐st♦ é✿ f ′ (x) = (2x − 2)(x2 − 2x)(x2 − 2x) + (x2 − 2x)(2x − 2)(x2 − 2x) + (x2 − 2x)(x2 − 2x)(2x − 2) = 3(2x − 2)(x2 − 2x)2 = 6(x − 1)(x2 − 2x)2 ✳ P♦rt❛♥t♦✱ f ′ (x) = 6(x − 1)(x2 − 2x)2 ✳ ❆♣❧✐❝❛♥❞♦✲s❡ ❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✸✮✲✭✐✐✐✮ t❡♠♦s✿ ❊①❡♠♣❧♦ ✺✳✷✼✳ ❉❛❞❛ ❛ ❢✉♥çã♦ ❙♦❧✉çã♦✳ ax5 + bx4 + c ✱ f (x) = √ a2 + b 2 + c 2 ❞❡t❡r♠✐♥❡ f ′ (x)✳ ❆♣❧✐❝❛♥❞♦✲s❡ ❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✸✮✭✐✐✮ ❡ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ 1 f (x) = √ · (ax5 + bx4 + c)✱ 2 2 2 a +b +c 5ax4 + 4bx3 ✳ P♦rt❛♥t♦✱ f ′ (a) = √ a2 + b2 + c2 ❡♥tã♦ a, b ❡ c sã♦ ❝♦♥st❛♥t❡s✱ t❡♠♦s✿ 1 f ′ (x) = √ · (5ax4 + 4bx3 )✳ 2 2 2 a +b +c ✷✻✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✺✳✹✳✷ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❡r✐✈❛❞❛ ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ❙❡❥❛ f : R −→ R ✉♠❛ ❢✉♥çã♦✱ ❡ B = { x ∈ D(f ) /. f é ❞❡r✐✈á✈❡❧ ❡♠ x } = 6 ∅✳ ❆ ❢✉♥çã♦ f ❞❡✜♥✐❞❛ ❡♠ B é ❝❤❛♠❛❞❛ ❢✉♥çã♦ ❞❡r✐✈❛❞❛ ❞❡ f (x) ♦✉ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛ ❞❡ f (x) ❡ é ❞❡♥♦t❛❞❛ ♣❡❧❛ ❢✉♥çã♦ f ′ (x)✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡♠ B ♣❛r❛ ♦ q✉❛❧ f ′ (x) ❛❞♠✐t❛ ❞❡r✐✈❛❞❛❀ ✐st♦ é ♣❛r❛ ♦ q✉❛❧ (f ′ )′ (x) ❡①✐st❛✳ ❆ ❞❡r✐✈❛❞❛ ❞❛ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛ ❞❡ f ′ (x) é ❝❤❛♠❛❞❛ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛ ❞❡ f (x) ❡ ✐♥❞✐❝❛❞❛ ❝♦♠ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s✿ f ′′ (x), d2 f (x) , dx2 Dx2 f (x), d2 y dx2 s❡ y = f (x) ◗✉❛♥❞♦ f ′′ (a) ❡①✐st❡✱ ❞✐③❡♠♦s q✉❡ f (x) é ❞✉❛s ✈❡③❡s ❞❡r✐✈á✈❡❧ ❡♠ x = a ❡ ♦ ♥ú♠❡r♦ f ′′ (a) é ❝❤❛♠❛❞♦ ❞❡ ✲ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛ ❞❡ f ❡♠ x = a✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡♠ B ♣❛r❛ ♦ q✉❛❧ f ′′ (x) ❛❞♠✐t❛ ❞❡r✐✈❛❞❛❀ ✐st♦ é ♣❛r❛ ♦ q✉❛❧ (f ′′ )′ (x) ❡①✐st❛✳ ❆ ❞❡r✐✈❛❞❛ ❞❛ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛ ❞❡ f (x) é ❝❤❛♠❛❞❛ ❞❡ t❡r❝❡✐r❛ ❞❡r✐✈❛❞❛ ❞❡ f (x)✱ ❡ ✐♥❞✐❝❛❞❛ ❝♦♠ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s✿ f ′′′ (x), d3 f (x) , dx3 Dx3 f (x), d3 y dx3 s❡ y = f (x) ◗✉❛♥❞♦ f ′′′ (a) ❡①✐st❡✱ ❞✐③❡♠♦s q✉❡ f (x) é três ✈❡③❡s ❞❡r✐✈á✈❡❧ ❡♠ x = a ❡ ♦ ♥ú♠❡r♦ f ′′′ (a) é ❝❤❛♠❛❞♦ ❞❡ t❡r❝❡✐r❛ ❞❡r✐✈❛❞❛ ❞❡ f ❡♠ x = a✳ ❉❡r✐✈❛♥❞♦ s✉❝❡ss✐✈❛♠❡♥t❡ ❛ ❢✉♥çã♦ f (x) ✭s❡♠♣r❡ q✉❡ s❡❥❛ ♣♦ssí✈❡❧✮✱ ♦❜té♠✲s❡ ❛ n✲ és✐♠❛ ❞❡r✐✈❛❞❛ ♦✉ ❞❡r✐✈❛❞❛ ❞❡ ♦r❞❡♠ n ❞❛ ❢✉♥çã♦ f (x)✱ ❡ ✐♥❞✐❝❛✲s❡ ❝♦♠ ❛❧❣✉♠❛ ❞❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s✿ f (n) (x), dn f (x) , dxn Dxn f (x), dn y dxn s❡ y = f (x) Pr♦♣r✐❡❞❛❞❡ ✺✳✺✳ ❋ór♠✉❧❛ ❞❡ ▲❡✐❜♥✐t③✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛s ❢✉♥çõ❡s s✉❜❝♦♥❥✉♥t♦ A ❞❡ ♥ú♠❡r♦s r❡❛✐s✳ ❡ t❡♠♦s✿ = ··· + f (x) ! n f (n) (x) · g(x) + 0 g(x) s❡❥❛♠ ❞❡r✐✈á✈❡✐s ❛té ❛ ♦r❞❡♠ n ♥✉♠ ♠❡s♠♦ ❊♥tã♦ y = f (x) · g(x) é ❞❡r✐✈á✈❡❧ ❛té ❛ ♦r❞❡♠ n ❡♠ A ❡ dn y = [f (x) · g(x)](n) = dxn ! ! n n f (n−1) (x) · g ′ (x) + f (n−2) (x) · g”(x) + · · · 1 2 ! n f ′′ (x) · g (n−2) (x) + n−2 ! n f ′ (x) · g (n−1) (x) + n−1 ✷✻✷ ! n f (x) · g (n) (x) n 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❊①❡♠♣❧♦ ✺✳✷✽✳ ❉❛❞❛ ❛s ❢✉♥çõ❡s ❙♦❧✉çã♦✳ ✭✐✮ f (x) =| 5x2 − 3x + 9 | f (x) =| 5x2 − 3x + 9 |= ( 10x − 3, f ′ (x) = −(10x − 3), ❙♦❧✉çã♦✳ ( s❡✱ g(x) = 5x 5x2 − 3x + 9 −(5x2 − 3x + 9) ❝❛❧❝✉❧❛r✿ ✐✮ f ”(x) ✐✐✮ g”(x)✳ 5x2 − 3x ≥ −9 s❡✱ 5x2 − 3x < −9 ( 10 s❡✱ 5x2 − 3x ≥ −9 5x2 − 3x ≥ −9 ❡ f ′′ (x) = −10 s❡✱ 5x2 − 3x < −9 5x2 − 3x < −9 s❡✱ ✭✐✐✮ g(x) = 5x ♣❡❧♦ ❊①❡♠♣❧♦ 5x · Ln5 · Ln5 ❛ss✐♠ g ′′ (x) = 5x · (Ln5)2 ✳ P❛r❛ ❛ ❢✉♥çã♦ ❊①❡♠♣❧♦ ✺✳✷✾✳ h(x) = ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ ❡ ❙♦❧✉çã♦✳ x ✱ 3x − 1 s❡✱ g ′ (x) = 5x · Ln5✱ ✭✺✳✼✮ t❡♠♦s ❞❡t❡r♠✐♥❡ ❧♦❣♦ h(n) (x)✳ −1 1 ✱ ❡♥tã♦ h′ (x) = = −(3x − 1)−2 ✳ 2 3 (3x − 1) h′′ (x) = −(−2)(3)(3x − 1)−3 , h”′ (x) = −(−2)(−3)(3)2 (3x − 1)−4 , (−1)4 · 4! · 33 −(−2)(−3)(−4)(3)3 (3x − 1)−5 ✐st♦ é h(4) (x) = (3x − 1)5 ❙✉♣♦♥❤❛ x 6= ▼♦str❛✲s❡ ♣♦r ✐♥❞✉çã♦ q✉❡✱ ✺✳✹✳✸ I ❡ J f : I −→ J (−1)n · n! · 3n−1 ✳ (x) = (3x − 1)n+1 f ✉♠❛ ❢✉♥çã♦ ♠♦♥ót♦♥❛ ✭❝r❡s❝❡♥t❡ ♦✉ ❞❡❝r❡s❝❡♥t❡✮ ❡str✐t❛ ❡ s♦❜r❡❥❡t✐✈❛ g : J −→ I ✐♥t❡r✈❛❧♦s r❡❛✐s✳ ❊♥tã♦ ❡①✐st❡✱ ❛ ❢✉♥çã♦ ✐♥✈❡rs❛ Pr♦♣r✐❡❞❛❞❡ ✺✳✻✳ ❙❡ h (n) h(4) (x) = ❉❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ ✐♥✈❡rs❛ ❙❡❥❛ ❡ g”(x) = ❡ ❛♠❜❛s sã♦ ❝♦♥tí♥✉❛s✳ ❘❡❣r❛ ❞❛ ❞❡r✐✈❛❞❛ ❞❡ ❢✉♥çã♦ ✐♥✈❡rs❛✳ é ❞❡r✐✈á✈❡❧ ❡♠ x=b∈I 1 1 g ′ (a) = ′ = ′ ✳ f (b) f (g(a)) ❡ f ′ (b) 6= 0✱ ❡♥tã♦✱ g é ❞❡r✐✈á✈❡❧ ❡♠ a = f (b) ❡ t❡♠♦s✿ ❉❡♠♦♥str❛çã♦✳ y 6= a g(y) − g(a) 6= 0 ❡ ✿ ❈♦♠ ❡❢❡✐t♦✱ ❝♦♠♦ ♣❛r❛ ❛ss✐♠ t❡r❡♠♦s ❝♦rr❡s♣♦♥❞❡ g(y) − g(a) = y−a g(y) 6= g(a)✱ 1 = y−a g(y) − g(a) ✷✻✸ ♣♦✐s g é ♠♦♥ót♦♥❛ ❡str✐t❛✱ 1 f (x)−f (b) x−b 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R P❛ss❛♥❞♦ ❛♦ ❧✐♠✐t❡ q✉❛♥❞♦ y → a✱ ❝♦♠♦ x = g(y) → b = g(a)✱ ♣♦✐s g é ❞❡r✐✈á✈❡❧❀ g(y) − g(a) = y→a y−a (b) ❡✱ s❡♥❞♦ ♣♦r ❤✐♣ót❡s❡ lim f (x)−f = f ′ (b) 6= 0✱ s❡❣✉❡✲s❡✿ g ′ (a) = lim x−b x→b 1 1 1 1 = ′ = ′ ✳ lim = y−a y→a f (x) − f (b) f (b) f (g(a)) lim g(y) − g(a) x→b x−b ❊①❡♠♣❧♦ ✺✳✸✵✳ ❉❛❞❛ ❛ ❢✉♥çã♦ g(x) = √ n n ∈ Z, x, ❙♦❧✉çã♦✳ ❝❛❧❝✉❧❡ g ′ (x)✳ √ ❆ ❢✉♥çã♦ g(x) = n x✱ ❞❡✜♥✐❞❛ ♣♦r g : R −→ R s❡ n é í♠♣❛r ♦✉ ❣✿ g : R+ −→ R+ s❡ √ n é ♣❛r✳ ❊♠ q✉❛❧q✉❡r ❝❛s♦ y = g(x) = n x s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x = f (y) = y n ✳ ❈♦♠♦ ❥á ❡st✉❞❛♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ s❡ f (y) = y n ❡♥tã♦ f ′ (y) = ny n−1 ❡ f ′ (y) 6= 0✳ ▲♦❣♦✱ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✻✮✱ g ′ (x) = 1 f ′ (y) = 1 1 = √ n n−1 ny n( x)n−1 1 ♣❛r❛ x 6= 0✳ ❊st❡ r❡s✉❧t❛❞♦ ♣♦❞❡ s❡r ♣♦st♦ s♦❜ ❢♦r♠❛ ❞❡ ❡①♣♦❡♥t❡ ✐st♦ é✱ g(x) = x n ❡♥tã♦   1−n 1 1 1 = ·x n ✳ g (x) = · n−1 n x n n ′ ❊①❡♠♣❧♦ ✺✳✸✶✳ ❉❛❞❛ ❛ ❢✉♥çã♦ g(x) = loga x ✱ ♣❛r❛ ❙♦❧✉çã♦✳ x ∈ R+ ✱ ❝❛❧❝✉❧❡ g ′ (x)✳ ❚❡♠♦s ✿ y = g(x) = loga x s❡✱ ❡ s♦♠❡♥t❡ s❡ x = f (y) = ay ✳ ❉❛❞♦ f (y) = ay ✱ ♣❡❧♦ ❊①❡♠♣❧♦ ✭✺✳✼✮ s❡❣✉❡ q✉❡ f ′ (y) = ay Lna 6= 0 q✉❛♥❞♦ ay > 0 ❡ 1 1 = · Lna x · Lna 1 1 q✉❛♥❞♦ x > 0✳ ◆♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❡♠ q✉❡ g(x) = Lnx t❡♠♦s q✉❡✿ g ′ (x) = = ✱ x · Lne x ❧❡♠❜r❡ q✉❡ Lne = loge e = 1✳ a > 0✱ ❧♦❣♦ ♣❡❧❛ r❡❣r❛ ❞❡ ❞❡r✐✈❛❞❛ ❞❡ ❢✉♥çã♦ ✐♥✈❡rs❛ g ′ (x) = 1 f ′ (y) = ay Pr♦♣r✐❡❞❛❞❡ ✺✳✼✳ f : A −→ R é ❞❡r✐✈á✈❡❧ ♥♦ ♣♦♥t♦ a ∈ A✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ N (h)✱ t❛❧ f (a + h) − f (a) = f ′ (a) · h + N (h) · h ♣❛r❛ t♦❞♦ x = a + h ∈ A ❡ N (h) = 0 = N (0)✳ ❙❡ q✉❡✿ ❉❡♠♦♥str❛çã♦✳ f (a + h) − f (a) = f ′ (a) ❛ss✐♠✱ h→0  h ❉❡ ❢❛t♦✱ s❡♥❞♦ f ❞❡r✐✈á✈❡❧ ❡♠ x = a t❡♠♦s✿ lim  f (a + h) − f (a) − f ′ (a) = 0✳ h→0 h   f (a + h) − f (a) , s❡✱ h = 6 0 ❉❡✜♥✐♠♦s✿ N (h) = h  0, s❡✱ h = 0 ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♥❛ ❢♦r♠❛ lim ✷✻✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❚❡♠✱ ♣❛r❛ h 6= 0, N (h) · h = f (a + h) − f (a) − f ′ (a) · h✳ P♦rt❛♥t♦✱ f (a + h) − f (a) = f ′ (a) · h + N (h) · h✳ ✺✳✹✳✹ ❘❡❣r❛ ❞❛ ❝❛❞❡✐❛ Pr♦♣r✐❡❞❛❞❡ ✺✳✽✳ f : A −→ R ❡ g : B −→ R ❢✉♥çõ❡s t❛✐s q✉❡ Im(f ) ⊆ B ✳ ❙❡ f é ❞❡r✐✈á✈❡❧ ❡♠ x = a ∈ A ❡ g é ❞❡r✐✈á✈❡❧ ❡♠ b = f (a) ∈ B ✱ ❡♥tã♦✱ g ◦ f é ❞❡r✐✈á✈❡❧ ❡♠ x = a ❡ t❡♠♦s✿ (g ◦ f )′ (a) = g ′ (f (a)) · f ′ (a)✳ ❙❡❥❛♠ ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✱ é s✉✜❝✐❡♥t❡ ❛♣❧✐❝❛r ❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✼✮ ❊①❡♠♣❧♦ ✺✳✸✷✳ ❉❛❞❛ ❛ ❢✉♥çã♦ g(x) = √ ❙♦❧✉çã♦✳ P❛r❛ ❛ ❢✉♥çã♦ g(x) = √ x2 − 15 ❝❛❧❝✉❧❛r✿ g ′′ (x)✳ x2 − 15 t❡♠♦s g ′ (x) = √ √ ′′ g (x) = x ✱ ❧♦❣♦✿ − 15 x2 2 x2 − 15 − √xx2 −15 −15 √ = √ ( x2 − 15)2 ( x2 − 15)3 15 ✳ ( x2 − 15)3 ❛ss✐♠✱ g ′′ (x) = − √ ❊①❡♠♣❧♦ ✺✳✸✸✳ ❉❛❞❛ F (x) = (x3 + 1)2 ✱ ❝❛❧❝✉❧❡ F ′ (x) ❙♦❧✉çã♦✳ ❖❜s❡r✈❛♥❞♦ q✉❡ F (x) = (x3 +1)·(x3 +1) ♣♦❞❡♠♦s ❛♣❧✐❝❛r ❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✸✮✱ ♦❜t❡♥❞♦ ✿ F ′ (x) = (x3 + 1)′ (x3 + 1) + (x3 + 1)(x3 + 1)′ = (3x2 )(x3 + 1) + (x3 + 1)(3x2 ) = 6x2 (x3 + 1) ❊①❡♠♣❧♦ ✺✳✸✹✳ ❉❛❞❛ ❙♦❧✉çã♦✳ F (x) = (x2 + 4x − 2)100 ✱ ❝❛❧❝✉❧❡ F ′ (x)✳ ❆ ❢✉♥çã♦ F (x) é ❝♦♠♣♦st❛ g ◦ f ❞❛s ❢✉♥çõ❡s g(y) = y 100 ❡ f (x) = x2 + 4x − 2❀ ❞❡s❞❡ q✉❡ g ′ (y) = 100y 99 ❡ f ′ (x) = 2x + 4✱ s❡❣✉❡✲s❡ q✉❡ F ′ (x) = 100(f (x)) · (4x − 2) = 100(x2 + 4x − 2)99 (2x + 4) P♦rt❛♥t♦✱ F ′ (x) = 200(x2 + 4x − 2)99 (x + 2)✳ ✷✻✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✺✳✸✺✳ ❉❛❞❛ F (x) = ax 3 −x2 +1 ✱ ❝❛❧❝✉❧❡ F ′ (x)✳ ❙♦❧✉çã♦✳ ❆ ❢✉♥çã♦ F (x) é ❝♦♠♣♦st❛ g ◦ f ❞❛s ❢✉♥çõ❡s g(y) = ay ❡ f (x) = x3 − x2 + 1✱ ❧♦❣♦ g ′ (y) = ay · Lna ❡ f ′ (x) = 3x2 − 2x✳ ❆ss✐♠✱ F ′ (x) = (g ◦ f )′ (x) = g ′ (f (x)) · f ′ (x) = 3 −x2 +1 [af (x) · Lna](3x2 − 2x) = ax P♦rt❛♥t♦✱ F ′ (x) = ax ❊①❡♠♣❧♦ ✺✳✸✻✳ p ❉❛❞❛ F (x) = x q = 3 −x2 +1 √ q xp ✱ · [Lna](3x2 − 2x) · (3x2 − 2x) · [Lna]✳ ❝❛❧❝✉❧❡ F ′ (x)✳ ❙♦❧✉çã♦✳ ❚❡♠♦s q✉❡ F (x) é ❛ ❝♦♠♣♦st❛ gof ❞❛s ❢✉♥çõ❡s g(y) = √ q y ❡ f (x) = xp ✱ ❡♥tã♦ 1 1q −1 y ❡ f ′ (x) = p · xp−1 ✳ ❆ss✐♠✱ F ′ (x) = (g ◦ f )′ (x) = g ′ (f (x)) · f ′ (x) = q 1 1 1 1 p p−q [f (x)] q −1 · pxp−1 = [xp ] q −1 · pxp−1 = x p ✳ q q q p−q p ′ P♦rt❛♥t♦✱ F (x) = q x p ✳ g ′ (y) = ❊①❡♠♣❧♦ ✺✳✸✼✳ ❉❛❞❛ ❛ ❢✉♥çã♦ ❙♦❧✉çã♦✳ F (x) = loga (2x3 + 4x2 − 1) ❝❛❧❝✉❧❡ F ′ (x)✳ ❆ ❢✉♥çã♦ F (x) é ❝♦♠♣♦st❛ g ◦ f ❞❛s ❢✉♥çõ❡s g(y) = loga y ❡ f (x) = 2x3 + 4x2 − 1 ❡ t❡♠♦s g ′ (y) = 1 ❡ f ′ (x) = 6x2 + 8x✳ yLna 1 · (6x2 + 8x)✳ 2 − 1)Lna + 4x   2x(3x + 4) 1 ′ P♦rt❛♥t♦✱ F (x) = ✳ Lna 2x3 + 4x2 − 1 ▲♦❣♦✱ F ′ (x) = g ′ (f (x)) · f ′ (x) = ✺✳✹✳✺ (2x3 ❉❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ✐♠♣❧í❝✐t❛ ◆♦s ♣r♦❜❧❡♠❛s ❞❡ ❛♣❧✐❝❛çã♦✱ ♥❡♠ s❡♠♣r❡ é ♣♦ssí✈❡❧ ❛❝❤❛r ✉♠❛ s♦❧✉çã♦ q✉❡ ❞❡s❝r❡✈❛ ✉♠ ♠♦❞❡❧♦ ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❡♠ t❡r♠♦s ❞❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✳ ❆❧❣✉♠❛s ✈❡③❡s ❛ ❢✉♥çã♦ é ❞❛❞❛ ❡♠ ❢♦r♠❛ ✐♠♣❧í❝✐t❛ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿ x4 − x3 y + 3xy 2 − y 3 = 0 ❆q✉✐ y é ✉♠❛ ❢✉♥çã♦ q✉❡ ❞❡♣❡♥❞❡ ❞❡ x✱ ♠❛✐s ♥ã♦ ❡stá ❞❛❞❛ ♥❛ ❢♦r♠❛ ❡①♣❧í❝✐t❛ ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❞❡ x❀ ✐st♦ é y = f (x)✳ ✷✻✻ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❙❡❥❛ E(x, y) = 0 ✉♠❛ ❡q✉❛çã♦ ❞❡ ✈❛r✐á✈❡✐s x ❡ y ✳ ❙❡ ❛♦ s✉❜st✐t✉✐r y ♣♦r f (x) ❛ ❡q✉❛çã♦ tr❛♥s❢♦r♠❛✲s❡ ♥✉♠❛ ✐❞❡♥t✐❞❛❞❡ ❡♥tã♦ ❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r y = f (x) é ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ ✐♠♣❧í❝✐t❛ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ E(x, y) = 0✳ P♦r ❡①❡♠♣❧♦✱ s✉♣♦♥❤❛♠♦s ❛ ❡q✉❛çã♦ E(x, y) = y 2 −x−2 = 0 ❞❡t❡r♠✐♥❛ ✐♠♣❧✐❝✐t❛♠❡♥t❡ √ √ ❛s ❢✉♥çõ❡s y = x + 2 ❡ y = − x + 2✱ ♣♦❞❡♠♦s s✉♣♦r y = f (x)✱ ❡♥tã♦ ♥❛ ❡q✉❛çã♦ E(x, y) = 0 r❡s✉❧t❛✿ [f (x)]2 − x − 2 = 0 ♦♥❞❡ [f (x)]2 = x + 2✳ ❉❡r✐✈❛♥❞♦ ❡♠ r❡❧❛çã♦ à ✈❛r✐á✈❡❧ x t❡♠♦s 2f (x) · f ′ (x) = 1 ❛ss✐♠ f ′ (x) = f ′ (x) = √ 1 1 ♦✉ y ′ = √ ✳ x+2 x+2 1 ✳ ▲♦❣♦✱ 2f (x) d 2 (y ) = dx √ 1 d(x + 2) ❡♥tã♦ 2y · y ′ = 1 ❛ss✐♠ y ′ = ✳ ❙❡ ❝♦♥s✐❞❡r❛♠♦s ❛ ✐❣✉❛❧❞❛❞❡ y = − x + 2 ♦ dx 2y ❊st❡ r❡s✉❧t❛❞♦ ♣♦❞❡♠♦s ♦❜t❡r s❡♠ s✉❜st✐t✉✐r y ♣♦r f (x)✱ ♦❜s❡r✈❡ q✉❡ r❡s✉❧t❛❞♦ ♣❡r♠❛♥❡❝❡ ✈á❧✐❞♦✳ ❊♠ ❣❡r❛❧✱ s❡ ❛ ❡q✉❛çã♦ E(x, y) = 0 ❞❡✜♥❡ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ❛ ❢✉♥çã♦ y = f (x)✱ ♣❛r❛ dy é s✉✜❝✐❡♥t❡ ❞❡r✐✈❛r ❛ ❡q✉❛çã♦ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ✈❛r✐á✈❡❧ y ❝♦♠♦ ❢✉♥çã♦ ❞❡ x ❡ ❞❛ ♦❜t❡r dx ❡q✉❛çã♦ r❡s✉❧t❛♥t❡ ✐s♦❧❛r ❛ ✈❛r✐á✈❡❧ y ❀ ✐st♦ é✿ dE dy dx = − dE dx dy ❊①❡♠♣❧♦ ✺✳✸✽✳ ❆s s❡❣✉✐♥t❡s ❢✉♥çõ❡s ❞❡✜♥❡♠ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ✉♠❛ ❢✉♥çã♦ y = f (x)✱ ❞❡t❡r♠✐♥❡ ❛ ❞❡r✐✲ ✈❛❞❛ y ′ ✳ x2 + y 2 = 6 ❛✮ ❝✮ ❜✮ 4x2 − 16y 2 − 64 = 0 ❞✮ y 2 − 5x − 8 = 0 xy 2 − x2 y − y 3 = 9x ❙♦❧✉çã♦✳ ✭❛✮ √ ❖❜s❡r✈❡ q✉❡ y = ± 6 − x2 ❡✱ ♥❛ ❡q✉❛çã♦ x2 + y 2 = 6 ❛♦ ❞❡r✐✈❛r ❡♠ r❡❧❛çã♦ à ✈❛r✐á✈❡❧ x x ✳ x r❡s✉❧t❛ 2x + 2y · y ′ = 0✱ ♦♥❞❡ y ′ = − ✱ ✐st♦ é y ′ = ∓ √ 2 ❙♦❧✉çã♦✳ y ✭❜✮ 6−x P❛r❛ ❡ ❡q✉❛çã♦ y 2 − 5x − 8 = 0 s❡❣✉❡✲s❡ q✉❡ 2yy ′ − 5 = 0✱ ❧♦❣♦ y ′ = √ 5 y = ± 5x + 8 ❡♥tã♦ y ′ = ± √ 2 5x + 8 5 ✱ ❝♦♠♦ 2y ❙♦❧✉çã♦✳ ✭❝✮ x ❆♦ ❞❡r✐✈❛r ❛ ❡q✉❛çã♦ 4x2 − 16y 2 − 64 = 0✱ r❡s✉❧t❛ 8x − 32y · y ′ = 0✱ ♦♥❞❡ y ′ = ❡ s✉❜st✐t✉✐♥❞♦ 2y = ± √ x2 x − 16 s❡❣✉❡✲s❡ q✉❡ y = ± √ 2 x2 − 16 4y ′ ❙♦❧✉çã♦✳ ✭❞✮ ❉❡r✐✈❛♥❞♦ ❛ ❡q✉❛çã♦ xy 2 −x2 y −y 3 = 9x✱ t❡♠♦s (y 2 +2xyy ′ )−(2xy +x2 y ′ )−3y 2 ·y ′ = 9 ❡♥tã♦ y ′ (2xy − x2 − 3y 2 ) = 9 − y 2 + 2xy ✳ ✷✻✼ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ P♦rt❛♥t♦✱ y ′ = ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R 9 − y 2 + 2xy ✳ 2xy − x2 − 3y 2 ❊①❡♠♣❧♦ ✺✳✸✾✳ ❉❛❞❛ ❛ ❡q✉❛çã♦ ❙♦❧✉çã♦✳ x5 + y 5 − 2xy = 0 ❡ y = f (x)✱ ❞❡t❡r♠✐♥❡ f ′ (1)✳ ❉❡r✐✈❛♥❞♦ ✐♠♣❧✐❝✐t❛♠❡♥t❡✱ 5x4 + 5y 4 · y ′ − 2y − 2xy ′ = 0 ♦♥❞❡ y ′ (5y 4 − 2x) = 2y − 5x4 ✳ ◆❛ ❡q✉❛çã♦ ♦r✐❣✐♥❛❧ q✉❛♥❞♦ x = 1 t❡♠♦s y = 1 ❡ y ′ = 2(1) − 5(1)4 = −1✳ 5(1)4 − 2(1) P♦rt❛♥t♦✱ f ′ (1) = −1✳ ❊①❡♠♣❧♦ ✺✳✹✵✳ 2y − 5x4 ✱ ❡♥tã♦ f ′ (1) = 5y 4 − 2x √ √ 2 2 ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ♥♦ ♣♦♥t♦ ( ,− ) à ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ 2 2 ♥❛ ♦r✐❣❡♠ ❡ r❛✐♦ ❛ ✉♥✐❞❛❞❡✳ ❙♦❧✉çã♦✳ ❆ ❡q✉❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ é ❞❛❞❛ ♣♦r x2 + y 2 = 1✳ ❙❛❜❡♠♦s q✉❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ t❛♥❣❡♥t❡ ♥✉♠ ♣♦♥t♦ à ❝✐r❝✉♥❢❡rê♥❝✐❛✱ é ❞❛❞❛ ♣❡❧♦ ✈❛❧♦r ❞❡ s✉❛ ❞❡r✐✈❛❞❛ ♥❡ss❡ ♣♦♥t♦✳ ❉❡r✐✈❛♥❞♦ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ❡ ❡q✉❛çã♦ ❞❛ ❝✉r✈❛✱ t❡♠♦s q✉❡✿ 2x + 2y dy =0 dx ⇒ dy x =− dx y √ √ 2 dy 2 2 = − 2√ = 1 ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ ♦ ♣♦♥t♦ ( , − ) r❡s✉❧t❛ q✉❡ 2 2 dx − 22 √ √ √ 2 2 ▲♦❣♦✱ y − (− ) = 1(x − ) ⇒ y = x − 2✳ 2 2 √ P♦rt❛♥t♦✱ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ ♣❡❞✐❞❛ é✱ y = x − 2✳ √ ✷✻✽ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❊①❡r❝í❝✐♦s ✺✲✶ ✶✳ ❆♣❧✐❝❛♥❞♦ ❛ ❞❡✜♥✐çã♦✱ ❝❛❧❝✉❧❛r ❛ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s ❡ ✐♥❞✐❝❛r s❡✉ ❞♦♠í♥✐♦✳ 1. f (x) = 6x2 − 5x + 2 4. f (x) = √ ✷✳ ❉❛❞❛ f (x) = √ 1 x+3 x✱ ❝❛❧❝✉❧❡ ✿ 2x + 3 3x − 2 x 6. f (x) = 3−x 2. f (x) = x3 − 3x2 5. f (x) = ✶✳ √ 3. f (x) = 16 − x2 f (2)❀ ✷✳ f ′ (x)✳ ✸✳ ❉❛❞❛ f (x) = x2 + 4x − 5✱ ❝❛❧❝✉❧❡ f ′ (−1)✳ 1 x ✹✳ ❉❛❞❛ f (x) = , x 6= 0✱ ❝❛❧❝✉❧❡✿ f ′ (2)❀ ✶✳ f ′ (x)✳ ✷✳ ✺✳ ❉❡t❡r♠✐♥❡ q✉❛✐s ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s sã♦ ❞❡r✐✈á✈❡✐s ♥♦s ♣♦♥t♦s ✐♥❞✐❝❛❞♦s✿ ✶✳ ✷✳ f (x) = f (x) = ( ( ( x + 3, s❡✱ x ≤ 3 −x + 5, s❡ x > 3 a=3 x2 − 9, s❡✱ x < 3 √ x + 3, s❡ x ≥ 3 a=3 (1 − x)2 , s❡✱ x ≥ 1 √ 1 − x, s❡ x < 1 ✸✳ f (x) = ✹✳ f (x) =| x2 − 4 | a=2   s❡✱ x < 0  |x+2| 2 f (x) = 2−x , s❡✱ 0 ≤ x < 2   2 x − 4x + 2, s❡ 2 ≤ x ✺✳ a=1 a=0 ❡ a = 2✳ ✻✳ ▼♦str❡ q✉❡✿ ✭✶✳✮ ❙❡ f é ❢✉♥çã♦ ♣❛r✱ ❡♥tã♦ f ′ (x) = −f ′ (−x)✳ ✭✷✳✮ ❙❡ f é ❢✉♥çã♦ í♠♣❛r✱ ❡♥tã♦ f ′ (x) = f ′ (−x)✳ ✼✳ ❉❡✜♥❡✲s❡ ♦ â♥❣✉❧♦ ❡♥tr❡ ❛s ❝✉r✈❛s y = f1 (x) ❡ y = f2 (x) ♥♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ M (x0 , y0 )✱ ❛♦ ♠❡♥♦r â♥❣✉❧♦ ❝♦♠♣r❡❡♥❞✐❞♦ ❡♥tr❡ ❛s t❛♥❣❡♥t❡s r❡s♣❡❝t✐✈❛s ♥♦ ♣♦♥t♦ M ✳ ❊st❡ â♥❣✉❧♦ é ❞❡t❡r♠✐♥❛❞♦ ♣❡❧❛ ❢ór♠✉❧❛ s❡❣✉✐♥t❡✿ tan ϕ = ✶✳ f2′ (x0 ) − f1′ (x0 ) ✳ 1 + f1′ (x0 ) · f2′ (x0 ) ❉❡t❡r♠✐♥❡ ♦ â♥❣✉❧♦ q✉❡ ❢♦r♠❛ ❝♦♠ ♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ❛ t❛♥❣❡♥t❡ à ❝✉r✈❛ y= 2x5 x3 − tr❛ç❛❞❛ ♥♦ ♣♦♥t♦ x = 1✳ 3 9 ✷✻✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✷✳ ❉❡t❡r♠✐♥❡ ♦ â♥❣✉❧♦ ❝♦♠♣r❡❡♥❞✐❞♦ ❡♥tr❡ ❛s ♣❛rá❜♦❧❛s ✸✳ ❉❡t❡r♠✐♥❡ ♦ â♥❣✉❧♦ ❡♥tr❡ ❛ ♣❛rá❜♦❧❛ y = 4 − x2 y = 8 − x2 ❡ R y = x2 ✳ ❡ ♦ r❛✐♦ ✈❡t♦r ❞♦ ♣♦♥t♦ M (1, 3) ❞❡st❛ ❧✐♥❤❛✳ ✽✳ P❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s ❞❡t❡r♠✐♥❡ ❛ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛✳ 1. f (x) = 3 x4 |x| 1 + x2 1 − x2 4. f (x) = 1 + x2 2. f (x) = r x−1 3. f (x) = (x − 1) 3 x+1 s 4 − x2 5. f (x) = (1 + x2 )3 p 6. f (x) = (5 − x) 7 (x + 5)6 x3 8. f (x) = p (1 − x2 )3 7. f (x) = x2 | x |3 4x + 6 x2 + 3x + 4 r √ 2 f (x) = 2x + x p f (x) = 3 (x3 − | x |3 )2 s √ 1− x √ f (x) = 1+ x x √ f (x) = a2 · a2 + x2 10. f (x) =| x2 − 9 | √ √ 1+x+ 1−x √ 12. f (x) = √ 1+x− 1−x √ a2 x 14. f (x) = x x2 − a2 − √ x 2 − a2 1p 1p 16. f (x) = n (1 + x3 )8 − 3 (1 + x3 )5 8 5 √ √ 18. f (x) = ( x + 1 + x − 1)4 9. f (x) = √ 11. 13. 15. 17. ✾✳ ❉❛❞❛ ❛ ❢✉♥çã♦✿ f (x) = ( x, 0, s❡✱ s❡✱ x 6= 0 x=0 ✶✳ Pr♦✈❛r q✉❡ ❡❧❛ é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ ♣❡❞❡✲s❡✿ x = 0✳ ✷✳ ❈❛❧❝✉❧❛r ❛s ❞❡r✐✈❛❞❛s ❧❛t❡r❛✐s ❞❡ss❛ ❢✉♥çã♦ ♥♦ ♣♦♥t♦ ✶✵✳ ❙✉♣♦♥❤❛ ❛ ❢✉♥çã♦ ✶✳ ✷✳ ✶✶✳ ❙❡❥❛ y = f (x) s❡❥❛ ❞❡r✐✈á✈❡❧ ❡♠ x✳ x = 0✳ ▼♦str❡ ♦ s❡❣✉✐♥t❡✿ f (x + h) − f (x − h) h→0 2h f (x + h) − f (x − k) f ′ (x) = lim + k+h→0 h+k f ′ (x) = lim g(x) = xn ❡ 0 ≤ k ≤ n❀ ♠♦str❡ q✉❡✿ n! xn−k ✳ (n − k)! f é ❞❡r✐✈á✈❡❧ ❡♠ x = a✱ ❡♥tã♦ | f (x) | t❛♠❜é♠ f (a) 6= 0✳ ❉❛r ✉♠ ❡①❡♠♣❧♦ q✉❛♥❞♦ f (a) = 0✳ ✶✷✳ ▼♦str❡ q✉❡ s❡ s❡♠♣r❡ q✉❡ g (k) (x) = ✷✼✵ é ❞❡r✐✈á✈❡❧ ❡♠ x=a 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✶✸✳ P❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s 1 x f (x) = 17 f (x) = ✶✳ ✸✳ ✶✹✳ ❉❡t❡r♠✐♥❡ ❞❡t❡r♠✐♥❡ f (f ′ (xc))✳ f (x) = x2 ✷✳ f (x) = 17x ✹✳ f ′ (x) f (x)✱ R ❡♠ tér♠✐♥♦s ❞❡ g ′ (x) s❡✿ 1. f (x) = g(x + g(a)) 2. f (x) = g(x. · g(a)) 4. f (x) = g(x)(x − a) 5. f (x) = g(a)(x − a) 3. f (x) = g(x + g(x)) 6. f (x + 3) = g(x2 ) ✶✺✳ ❉❡t❡r♠✐♥❡ ❛s ❞❡r✐✈❛❞❛s ❞❛s ❢✉♥çõ❡s ✐♥✈❡rs❛s ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ 1 x+1 1. f (x) = x2 2. g(x) = 3x2 − x 3. h(x) = 4. f (x) = (x + 2)2 5. g(x) = 6. h(x) = (x2 − 1)2 ✶✻✳ ❙❡❥❛ t = 2 − 3s + 3s2 ✱ ❞❡t❡r♠✐♥❡ ds dt ✶✼✳ ❙❡❥❛ x = y 3 − 4y + 1✳ ❉❡t❡r♠✐♥❡ dx ✳ dy x x−1 ♠❡❞✐❛♥t❡ s ✶✽✳ ❉❡t❡r♠✐♥❡ ❛ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛ ✐♠♣❧í❝✐t❛ ♣❛r❛ ❛s ❢✉♥çõ❡s x2 y 2 + 2 =1 a2 b 4 4. x + y 4 = x2 y 2 1. 2. √ x+ √ y= √ a 5. xy = y x y = f (x) ✳ 3. x3 − y 3 = 3axy p √ √ 3 3 x2 + 3 y 2 = a2 6. ✶✾✳ ◗✉❡ â♥❣✉❧♦ ❢♦r♠❛ ❝♦♠ ♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ❝♦♠ ❛ r❡t❛ t❛♥❣❡♥t❡ à ❝✉r✈❛ tr❛ç❛❞❛ ♥♦ ♣♦♥t♦ ❝♦♠ ❛❜s❝✐ss❛ x = x0 ❄ ✷✵✳ ❊s❝r❡✈❡r ❛s ❡q✉❛çõ❡s ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❡ ♥♦r♠❛❧ à ❝✉r✈❛ ♣♦♥t♦ y= 2x5 x2 − ✱ 3 9 x2 + 2xy 2 + 3y 4 = 6 ♥♦ M (1, −1)✳ ✷✶✳ ▼♦str❡ q✉❡ ❛ t❛♥❣❡♥t❡ à ❡❧✐♣s❡ xx0 yy0 + 2 = 1✳ a2 b x2 y 2 + = 1 ♥♦ ♣♦♥t♦ M (x0 , y0 ) é ❞❛❞❛ ♣❡❧❛ ✐❣✉❛❧❞❛❞❡ a2 b2 ✷✷✳ ▼♦str❡ q✉❡ ❛ t❛♥❣❡♥t❡ à ❤✐♣ér❜♦❧❡ ✐❣✉❛❧❞❛❞❡ xx0 yy0 − 2 = 1✳ a2 b y2 x2 − =1 a2 b2 ✷✸✳ ❉❡t❡r♠✐♥❡ ❛s ❡q✉❛çõ❡s ❞❛s t❛♥❣❡♥t❡s á ❤✐♣ér❜♦❧❡ ❝✉❧❛r❡s á r❡t❛ 2x + 4y − 3 = 0 ✷✼✶ ♥♦ ♣♦♥t♦ M (x0 , y0 ) x2 y 2 − =1 2 7 é ❞❛❞❛ ♣❡❧❛ q✉❡ s❡❥❛♠ ♣❡r♣❡♥❞✐✲ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✷✹✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛♦ ❣rá✜❝♦ f (x) = R 1 1 ❞❡ ♥♦ ♣♦♥t♦ (6, )✳ x 6 8 ✷✺✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛♦ ❣rá✜❝♦ ❞❡ g(x) = q✉❡ ♣❛ss❛ ♣❡❧♦ 1+x ♣♦♥t♦ (−3, −4)✳ ❈♦♠♣❛r❡ ❝♦♠ ♦ ❊①❡r❝í❝✐♦ ✭✷✹✮ ❡ ❡♥❝♦♥tr❡ ✉♠❛ ❡①♣❧✐❝❛çã♦ r❛③♦á✈❡❧ ♣❛r❛ ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❡ss❛ r❡t❛✳ ✷✻✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ à ❤✐♣ér❜♦❧❡ ❞❡ ❡q✉❛çã♦ 2x2 − 3y 2 − 12 = 0✱ √ ♥♦ ♣♦♥t♦ (2 3, 2) ✷✼✳ ❈❛❧❝✉❧❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ ♥♦r♠❛❧ ❛♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ g(x) = ♥♦ ♣♦♥t♦ (3, g(3)) ✳ ✷✽✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ ♥♦r♠❛❧ à ❝✉r✈❛ y = x2 √ 3 x2 − 1✱ 8 ✱ ♥♦ ♣♦♥t♦ ❞❡ ❛❜s❝✐ss❛ 2✳ +4 ✷✾✳ ❉❡t❡r♠✐♥❡ ❛ ❞❡❝❧✐✈✐❞❛❞❡ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛♦ ❣rá✜❝♦ ❞❡ 2x3 y − x2 + 2xy − y 3 = −1✱ ♥♦ ♣♦♥t♦ (1, 2) ✳ ✸✵✳ ❉❡t❡r♠✐♥❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ à ❝✉r✈❛ y = à r❡t❛ 8x − 4y − 1 = 0✳ √ x ❞❡ ♠♦❞♦ q✉❡ ❡❧❛ s❡❥❛ ♣❛r❛❧❡❧❛ 3 3 ✸✶✳ ▼♦str❡ q✉❡ ❛ r❡t❛ ♥♦r♠❛❧ à ❝✉r✈❛ x3 + y 3 = 3xy ♥♦ ♣♦♥t♦ ( , )✱ ♣❛ss❛ ♣❡❧❛ ♦r✐❣❡♠ 2 2 ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ✸✷✳ ❉❡ 1988 ❛ 2000✱ ❛ r❡❝❡✐t❛ ✭❡♠ ♠✐❧❤õ❡s ❞❡ r❡❛✐s✮ ❞❡ ✉♠❛ ❝♦♠♣❛♥❤✐❛ t✐♥❤❛ ❝♦♠♦ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ R(t) = 0, 87t4 − 15, 82t3 + 147, 96t2 − 542, 75t + 784, 93✱ ♦♥❞❡ t = 5 ❝♦rr❡s♣♦♥❞❡ ❛ 1988✳ ◗✉❛❧ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❛ r❡❝❡✐t❛ ❞❛ ❝♦♠♣❛♥❤✐❛ ❡♠ 1993❄ ✸✸✳ ❯♠❛ ❡♠♣r❡s❛ ❞❡ ❡❧❡trô♥✐❝♦s ✉t✐❧✐③❛ 600 ❝❛✐①❛s ❞❡ tr❛♥s✐st♦r❡s ♣♦r ❛♥♦✳ ❖ ❝✉st♦ ❞❡ ❛r♠❛③❡♥❛♠❡♥t♦ ❞❡ ✉♠❛ ❝❛✐①❛ ♣♦r ✉♠ ❛♥♦ é ❞❡ 45 ❝❡♥t❛✈♦s ❞❡ r❡❛❧✱ ❡ ❣❛st♦s ❡♠ tr❛♥s♣♦rt❡ ❞❡ ❡♥✈✐♦ é ❞❡ R$30, 00 ♣♦r ♦r❞❡♠✳ ◗✉❛♥t❛s ❝❛✐①❛s ❞❡✈❡rá ♣❡❞✐r ❛ ❡♠♣r❡s❛ ❡♠ ❝❛❞❛ ❡♥✈✐♦ ♣❛r❛ ♠❛♥t❡r ♦ ❝✉st♦ t♦t❛❧ ♠í♥✐♠♦❄ ✸✹✳ ❯♠ ♣r♦❞✉t♦r ❞❡ ❧❛r❛♥❥❛s ❡♠ ●♦✐â♥✐❛ ❡st✐♠❛ q✉❡✱ s❡ ♣❧❛♥t❛ 60 ❧❛r❛♥❥❡✐r❛s ♥✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ár❡❛✱ ❛ ♣r♦❞✉çã♦ ♠é❞✐❛ ♣♦r ár✈♦r❡ s❡r✐❛ ❞❡ 400 ❧❛r❛♥❥❛s ♣♦r ár✈♦r❡✳ ❖ ♣r♦❞✉t♦r s❛❜❡ t❛♠❜é♠ q✉❡ ♣♦r ❝❛❞❛ ár✈♦r❡ ❛❞✐❝✐♦♥❛❧ ❞❡ ❧❛r❛♥❥❛ ♣❧❛♥t❛❞♦ ♥❛ ♠❡s♠❛ ár❡❛✱ ❛ ♠é❞✐❛ ❞❡ ♣r♦❞✉çã♦ ❞❡ ❝❛❞❛ ár✈♦r❡ ❞✐♠✐♥✉✐ ❡♠ 4 ❧❛r❛♥❥❛s✳ ◗✉❛♥t❛s ár✈♦r❡s ❞❡✈❡ ♣❧❛♥t❛r ♦ ♣r♦❞✉t♦r ♣❛r❛ ♠❛①✐♠✐③❛r ❛ ♣r♦❞✉çã♦ t♦t❛❧❄ ✷✼✷ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✺✳✺ ❉❡r✐✈❛❞❛ ❞❡ ❢✉♥çõ❡s tr❛♥s❝❡♥❞❡♥t❡s ✺✳✺✳✶ ❉❡r✐✈❛❞❛ ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❆s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s sã♦ ❞❡r✐✈á✈❡✐s ❡♠ s❡✉s r❡s♣❡❝t✐✈♦s ❞♦♠í♥✐♦s ❡ t❡♠♦s ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ Pr♦♣r✐❡❞❛❞❡ ✺✳✾✳ f ′ (x) = cos x✳ ❛✮ ❙❡ f (x) = senx✱ ❜✮ ❙❡ f (x) = cos x✱ ❡♥tã♦ f ′ (x) = −senx✳ ❝✮ ❙❡ f (x) = tan x✱ ❡♥tã♦ f ′ (x) = sec2 x✳ ❞✮ ❙❡ f (x) = cot x✱ ❡♥tã♦ f ′ (x) = − csc2 x✳ ❡✮ ❙❡ f (x) = sec x✱ ❡♥tã♦ f ′ (x) = tan x · sec x✳ ❢✮ ❙❡ f (x) = csc x✱ ❡♥tã♦ f ′ (x) = − cot x · csc x✳ ❉❡♠♦♥str❛çã♦✳ ❡♥tã♦ ✭❛✮ ❈♦♥s✐❞❡r❡ f (x) = senx ❡♥tã♦✱ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❞❡r✐✈❛❞❛ t❡♠♦s✿ sen(x + h) − senx senx · cos h + senh · cos x − senx = lim h→0 h→0 h h f ′ (x) = lim cos h − 1 senh senx(cos h − 1) + senh · cos x = senx · lim + cos x · lim = h→0 h→0 h h→0 h h = (senx).(0) + (cos x)(1) = lim P♦rt❛♥t♦ s❡✱ f (x) = senx✱ ❡♥tã♦ f ′ (x) = cos x ❉❡♠♦♥str❛çã♦✳ ✭❜✮ ❈♦♥s✐❞❡r❡ f (x) = cos x ❡♥tã♦✱ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❞❡r✐✈❛❞❛ t❡♠♦s✿ cos x · cos h − senh · senx − cos x cos(x + h) − cos x = lim h→0 h→0 h h f ′ (x) = lim cos h − 1 senh cos x(cos h − 1) − senh · senx = cos x · lim − senx · lim = h→0 h→0 h→0 h h h = lim (cos x).(0) − (senx)(1) P♦rt❛♥t♦ s❡✱ f (x) = cos x✱ ❡♥tã♦ f ′ (x) = −senx ❉❡♠♦♥str❛çã♦✳ ✭❝✮ ✷✼✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❚❡♠♦s senx ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❞❡r✐✈❛❞❛ cos x ′ (senx) cos x − senx(cos x)′ ✱ ✐st♦ é f ′ (x) = cos2 x f (x) = tan x✱ ❞✉❛s ❢✉♥çõ❡s✱ r❡s✉❧t❛ ❡♥tã♦ f (x) = f ′ (x) = P♦rt❛♥t♦ s❡✱ f (x) = tan x✱ ❞♦ q✉♦❝✐❡♥t❡ ❞❡ cos2 x + sen2 x 1 = = sec2 x 2 cos x cos2 x ❡♥tã♦ f ′ (x) = sec2 x✳ Pr♦♣r✐❡❞❛❞❡ ✺✳✶✵✳ ❙❡❥❛ u = u(x) ❢✉♥çã♦ ❞❡r✐✈á✈❡❧ ❡♠ x✱ ❡♥tã♦✿ ❛✮ ❙❡ f (x) = sen[u(x)]✱ ❡♥tã♦ f ′ (x) = {cos[u(x)]} · u′ (x). ❜✮ ❙❡ f (x) = cos[u(x)]✱ ❡♥tã♦ f ′ (x) = −{sen[u(x)]} · u′ (x)✳ ❝✮ ❙❡ f (x) = tan g[u(x)]✱ ❞✮ ❙❡ f (x) = cot[u(x)]✱ ❡♥tã♦ f ′ (x) = −{csc2 [u(x)]} · u′ (x)✳ ❡✮ ❙❡ f (x) = sec[u(x)]✱ ❡♥tã♦ f ′ (x) = {tan[u(x)] · sec[u(x)]} · u′ (x)✳ ❢✮ ❙❡ f (x) = csc[u(x)]✱ ❡♥tã♦ f ′ (x) = −{cot[u(x)] · csc[u(x)]} · u′ (x)✳ ❡♥tã♦ f ′ (x) = {sec2 [u(x)]} · u′ (x)✳ ❊①❡♠♣❧♦ ✺✳✹✶✳ ❉❡t❡r♠✐♥❡ ❛ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛ ♣❛r❛ ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ ❛✮ ❞✮ ❙♦❧✉çã♦✳ f (x) = sen2 (5x − 3) x · senx f (x) = 1 + x2 ❜✮ ❡✮ g(x) = cos2 (a − x) 4 ❝✮ h(x) = sen3 3 h(x) = sen x · cos x✳ 3 ❛✮ f (x) = sen2 (5x − 3)✱ 5sen(10x − 6) ❜✮ g(x) = cos2 (a − x)✱ ❡♥tã♦ g ′ (x) = −{2 cos(a − x)sen(a − x)}(−1)✱ ✐st♦ é g ′ (x) = sen(2a − 2x)✳ x 1 x   x x ′ 2 x ✱ ❡♥tã♦ h (x) = {3sen cos }· ✱ ❧♦❣♦ h′ (x) = sen2 cos h(x) = sen3 3 3 3 3 3 3 x · senx f (x) = ❡♥tã♦✿ 1 + x2 (1 + x2 )[x · senx]′ − (1 + x2 )′ x · senx f ′ (x) = = (1 + x2 )2 ❝✮ ❞✮ ♦♥❞❡ ❡♥tã♦ f ′ (x) = 2sen(5x − 3) · cos(5x − 3) · 5❀ x ✐st♦ é f ′ (x) = (1 + x2 )[senx + x · cos x] − (2x)[x · senx] = (1 + x2 )2 2 2 (1 − x )senx + (1 + x )x · cos x f ′ (x) = (1 + x2 )2 ✷✼✹ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ h(x) = sen4 x · cos3 x✱ ❡♥tã♦ h′ (x) = [4sen3 x · cos x] cos3 x + sen4 x[3 cos2 x · (−senx)]❀ ′ 3 2 2 2 ✐st♦ é h (x) = sen x cos x(4 cos x − 3sen x)✳ ❡✮ ❊①❡♠♣❧♦ ✺✳✹✷✳ p 4 ❙❡❥❛♠ ❛s ❢✉♥çõ❡s✿ sec √ x✳ ❉❡t❡r♠✐♥❡ 1 f (x) = tan3 x + sec2 x − , x f ′ (1), g ′ (0) ❡ h′ (1)✳ g(x) = sen(tan x + sec x) ❡ h(x) = ❙♦❧✉çã♦✳ 1 ′ 2 2 ✱ ❡♥tã♦ f (x) = 3 tan x · sec x + 2 sec x · x 1 1 tan x · sec x + 2 ❛ss✐♠ f ′ (x) = tan x · sec2 x(3 tan x + 2) + 2 ❡ f ′ (1) = tan 1 · sec2 1 · x x 1 2 (3 tan 1 + 2) + 2 = tan 1 · sec 1 · (3 tan 1 + 2) + 1✳ 1 ′ 2 P♦rt❛♥t♦✱ f (1) = tan 1 · sec 1 · (3 tan 1 + 2) + 1 ❛✮ ❉❛❞❛ ❛ ❢✉♥çã♦ f (x) = tan3 x + sec2 x − ❜✮ P❛r❛ ❛ ❢✉♥çã♦ g(x) = sen(tan x + sec x) t❡♠♦s g ′ (x) = [cos(tan x + sec x)] · (sec2 x + sec x · tan x) ⇒ g ′ (x) = sec x · [cos(tan x + sec x)] · (sec x + tan x) ⇒ g ′ (0) = sec 0 · [cos(tan 0 + sec 0)] · (sec 0 + tan 0) = cos(1) g ′ (0) = cos 1✳ p √ h(x) = 4 sec x✱ P♦rt❛♥t♦✱ ❝✮ ❙❡ 1 h (x) = 4 ′ P♦rt❛♥t♦✱ ✺✳✺✳✷ ❡♥tã♦ p √ √ q 4 √ −3 √ √ sec x · tan x 4 √ (sec x) [sec x · tan x] = 8 x p √ √ 4 sec x · tan x ′ √ h (1) = ✳ 8 x ❉❡r✐✈❛❞❛ ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ✐♥✈❡rs❛s Pr♦♣r✐❡❞❛❞❡ ✺✳✶✶✳ ❆s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ✐♥✈❡rs❛s sã♦ ❞❡r✐✈á✈❡✐s ❡♠ s❡✉ ❞♦♠í♥✐♦ ❡ t❡♠♦s✿ f ′ (x) = √ 1 , 1 − x2 ❛✮ ❙❡ f (x) = arcsenx✱ ❜✮ ❙❡ f (x) = arccos x✱ ❡♥tã♦ f ′ (x) = − √ ❝✮ ❙❡ f (x) = arctan x✱ ❡♥tã♦ f ′ (x) = ❞✮ ❙❡ f (x) = arccotx✱ ❡♥tã♦ f ′ (x) = − ❡♥tã♦ | x |< 1✳ 1 , 1 − x2 1 , 1 + x2 1 , 1 + x2 ✷✼✺ | x |< 1✳ x ∈ R✳ x ∈ R✳ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 1 √ , | x | x2 − 1 ❡✮ ❙❡ f (x) = arcsecx✱ ❡♥tã♦ f ′ (x) = ❢✮ ❙❡ f (x) = arccscx✱ ❡♥tã♦ f ′ (x) = − | x |> 1✳ 1 √ , | x | x2 − 1 | x |> 1✳ ✭❛✮ ❉❡♠♦♥str❛çã♦✳ π π ]✳ 2 2 p dx ❉❛ ✐❣✉❛❧❞❛❞❡ y = arcsenx s❡❣✉❡ q✉❡ x = seny ❡ = cos y = 1 − sen2 y ♦♥❞❡✱ dy p √ 2 2 dx = 1 − sen y · dy = 1 − x · dy ✳ dy 1 1 P♦rt❛♥t♦✱ ✱ ✐st♦ é f ′ (x) = √ ♣❛r❛ | x |< 1✳ =√ dx 1 − x2 1 − x2 ❙❡❥❛ f (x) = arcsenx✱ ❡ y = f (x)✱ ❡♥tã♦ x ∈ [−1, 1] ❡ y ∈ [− , ✭❡✮ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ f (x) = y = arcsecx✱ ❡♥tã♦ ❞❛ ❞❡✜♥✐çã♦ ❞❛ ❢✉♥çã♦ tr✐❣♦♥♦♠étr✐❝❛ ✐♥✈❡rs❛✱ x ∈ (−∞, −1] ∪ [1, +∞) ❡ π π y ∈ [0, ) ∪ ( , π] 2 2 P♦❞❡♠♦s ❡s❝r❡✈❡r x = sec y ✱ ❞❡r✐✈❛♥❞♦ ❡♠ r❡❧❛çã♦ à ✈❛r✐á✈❡❧ y s❡❣✉❡ p dx = sec y · tan y = sec y · sec2 y − 1 dy p p π ) ❡ tan y = sec2 y − 1 ✐st♦ é dx = x · sec2 y − 1 · dy ❀ s❡ 2 p p π x ∈ (−∞, −1] ❡♥tã♦ y ∈ ( , π] ❡ tan y = − sec2 y − 1✱ ❧♦❣♦ dx = sec y· sec2 y − 1·dy =| 2 √ x | · x2 − 1 · dy ✳ dy 1 √ P♦rt❛♥t♦✱ = , | x |> 1✳ dx | x | x2 − 1 s❡ x ∈ [1, +∞) ❡♥tã♦ y ∈ [0, Pr♦♣r✐❡❞❛❞❡ ✺✳✶✷✳ ❙❡❥❛ u = u(x) ❢✉♥çã♦ ❞❡r✐✈á✈❡❧ r❡s♣❡✐t♦ à ✈❛r✐á✈❡❧ ❛✮ ❙❡ f (x) = arcsen[u(x)]✱ ❡♥tã♦ ❜✮ ❙❡ f (x) = arccos[u(x)]✱ ❡♥tã♦ x✱ ❡♥tã♦✿ u′ (x) f ′ (x) = p ✳ 1 − [u(x)]2 u′ (x) ✳ f ′ (x) = − p 1 − [u(x)]2 ❝✮ ❙❡ f (x) = arctan[u(x)]✱ ❡♥tã♦ u′ (x) f (x) = ✳ 1 + [u(x)]2 ❞✮ ❙❡ f (x) = arccot[u(x)]✱ ❡♥tã♦ f ′ (x) = − ❡✮ ❙❡ f (x) = arcsec[u(x)]✱ ❡♥tã♦ f ′ (x) = ′ u′ (x) ✳ 1 + [u(x)]2 u′ (x) p ✳ | u(x) | [u(x)]2 − 1 ✷✼✻ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❢✮ ❙❡ f (x) = arccsc[u(x)]✱ ❊①❡♠♣❧♦ ✺✳✹✸✳ ❡♥tã♦ f ′ (x) = − √ f (x) = arcsen 1 − x2 ❉❛❞❛ ❛ ❢✉♥çã♦ ❙♦❧✉çã♦✳ ❈♦♥s✐❞❡r❡✲s❡ ❛ ❢✉♥çã♦ f (x) = arcsen[u(x)]✱ u(x) = √ u′ (x) p ✳ | u(x) | [u(x)]2 − 1 q✉❛♥❞♦ 1 − x2 ❡♥tã♦ | x |≤ 1❀ ❞❡t❡r♠✐♥❡ f ′ (x)✳ x ✱ ❧♦❣♦ ❝♦♥s✐❞❡r❛♥❞♦ 1 − x2 ✐♥❞❡♣❡♥❞❡♥t❡ x s❡❣✉❡✲s❡ u′ (x) = − √ ❡ ❞❡r✐✈❛♥❞♦ ❡♠ r❡❧❛çã♦ à ✈❛r✐á✈❡❧ 1 u′ (x) =p · u′ (x) f ′ (x) = p 2 2 1 − [u(x)] 1 − [u(x)] ⇒ 1 x x 1 √ √ √ f ′ (x) = − q · = − · √ 1 − x2 1 − x2 x2 1 − [ 1 − x2 ] 2 q✉❛♥❞♦ 0 <| x |< 1❀ f ′ (x) = − ✐st♦ é ❊①❡♠♣❧♦ ✺✳✹✹✳ ❙❡❥❛♠ y = cos(x2 + y 3 ) ❡ x √ | x | 1 − x2 y = f (x)✱ q✉❛♥❞♦ ❞❡t❡r♠✐♥❡ 0 <| x |< 1✳ y′✳ ❙♦❧✉çã♦✳ du u(x, y) = x2 + y 3 ✱ ❡♥tã♦ y = cos[u(x, y)] ❡ y ′ = −sen[u(x, y)]. ✳ dx ❡q✉❛çã♦ u(x, y) = 0 ❞❡t❡r♠✐♥❛ ❛ ❢✉♥çã♦ ✐♠♣❧í❝✐t❛ y = f (x)✱ ❧♦❣♦ ❈♦♥s✐❞❡r❡ ❆ y ′ = −sen[u(x, y)] · [2x + 3y 2 · y ′ ] = −2x · sen(x2 + y 3 ) − 3y 2 · y ′ · sen(x2 + y 3 ) ♦♥❞❡ y′ = −2x · sen(x2 + y 3 ) ✳ 1 + 3y 2 · sen(x2 + y 3 ) ❊①❡♠♣❧♦ ✺✳✹✺✳ ❉❛❞❛ ❛ ❢✉♥çã♦ ❙♦❧✉çã♦✳✳ ❖❜s❡r✈❡ q✉❡ u′ (x) =  3a2 x − x3 g(x) = arctan a(a2 − 3x2 )  3a2 x − x3 u(x) = a(a2 − 3x2 )   ✱ ❞❡t❡r♠✐♥❡ g ′ (x)✳ ❧♦❣♦✱ ❞❡r✐✈❛♥❞♦ ❡♠ r❡❧❛çã♦ ❛ x t❡♠♦s✿ 3a(x4 + 2a2 x2 + a4 ) (3a2 − 3x2 )[a(a2 − 3x2 )] − (3a2 x − x3 )(−6ax) = a2 (a2 − 3x2 )2 a2 (a2 − 3x2 )2 P♦r ♦✉tr♦ ❧❛❞♦ g ′ (x) = 1+ h g(x) = arctan[u(x)]✱ 1 3a2 x−x3 a(a2 −3x2 ) i2 · ❡♥tã♦ g ′ (x) = 1 · u′ (x) 1 + [u(x)]2 ✐st♦ é 3a(x4 + 2a2 x2 + a4 ) 3a(x2 + a2 )2 = a2 (a2 − 3x2 )2 a2 (a 2 − 3x2 )2 + (3a2 x − x3 )2 ✷✼✼ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❆ss✐♠ ✺✳✺✳✸ g ′ (x) = 3a ✳ a2 + x2 ❉❡r✐✈❛❞❛ ❞❛s ❢✉♥çõ❡s✿ ❊①♣♦♥❡♥❝✐❛❧ ❡ ❧♦❣❛rít♠✐❝❛ Pr♦♣r✐❡❞❛❞❡ ✺✳✶✸✳ ❆s ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛❧ ❡ ❧♦❣❛rít♠✐❝❛ sã♦ ❞❡r✐✈á✈❡✐s ❡♠ s❡✉s ❝♦rr❡s♣♦♥❞❡♥t❡s ❞♦♠í♥✐♦s✱ ❡ t❡♠♦s✿ ❛✮ ❙❡ f (x) = ax , x ∈ R✱ ❡♥tã♦ f ′ (x) = ax · Lna, ❜✮ ❙❡ f (x) = ex , x ∈ R✱ ❡♥tã♦ f ′ (x) = ex , ❝✮ ❙❡ f (x) = loga x, ❞✮ ❙❡ f (x) = Lnx, ❡✮ ❙❡ f (x) = Ln | x |, ❉❡♠♦♥str❛çã♦✳ x > 0✱ x > 0✱ f ′ (x) = ❡♥tã♦ ❡♥tã♦ x 6= 0✱ ❡♥tã♦ ∀ x ∈ R✳ 1 , x · Lna 1 , x f ′ (x) = ∀ x ∈ R✳ f ′ (x) = ∀x > 0✳ ∀ x > 0✳ 1 , x ∀ x 6= 0✳ ✭❛✮ f (x) = ax , x ∈ R ❡♥tã♦ a > 0 ♦✉ a 6= 1✳ a − ax ah − 1 lim = ax · lim = ax Lna✳ h→0 h→0 h h ❙❡ ❉♦ ❊①❡♠♣❧♦ 5.7 f ′ (x) = t❡♠♦s✿ x+h ❉❡♠♦♥str❛çã♦✳ ❙❡ ♣❛rt❡ ✭❜✮ f (x) = ex , x ∈ R✱ ❡♥tã♦ é ✉♠ ❛✮ t❡♠♦s f ′ (x) = ex · Lne = ex ✳ ❉❡♠♦♥str❛çã♦✳ ❙❡ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ f (x) = loga x, x > 0✱ f ′ (x) = ❡♥tã♦ ♣❡❧♦ ♠♦str❛❞♦ ♥❛ ✭❝✮ ❡♥tã♦ y = loga x ♠❡♥t❡ ❡st❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ❡♠ r❡❧❛çã♦ ❛ ✐st♦ é✿ a = e✱ x s❡ ❡ s♦♠❡♥t❡ s❡ x = ay ❞❡r✐✈❛♥❞♦ ✐♠♣❧✐❝✐t❛✲ 1 = ay · y ′ · Lna✱ t❡♠♦s✿ 1 ✳ x · Lna ❧♦❣♦ y′ = ay 1 · Lna Pr♦♣r✐❡❞❛❞❡ ✺✳✶✹✳ ❙❡ u = u(x) ❛✮ ❙❡ f (x) = au(x) ✱ ❡♥tã♦ f ′ (x) = au(x) · Lna · u′ (x)✳ ❜✮ ❙❡ f (x) = eu(x) ✱ ❡♥tã♦ f ′ (x) = eu(x) · u′ (x)✳ ❝✮ ❙❡ f (x) = loga [u(x)], ❞✮ ❙❡ f (x) = Ln[u(x)], ❡ v = v(x) sã♦ ❢✉♥çõ❡s ❞❡r✐✈á✈❡✐s r❡s♣❡✐t♦ à ✈❛r✐á✈❡❧ u(x) > 0✱ u(x) > 0✱ ❡♥tã♦ ❡♥tã♦ f ′ (x) = f ′ (x) = ✷✼✽ x✱ t❡♠♦s✿ 1 · u′ (x)✳ u(x) · Lna 1 · u′ (x)✳ u(x) 09/02/2021 ❀ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❡✮ ❙❡ f (x) = [u(x)]v(x) ✱ ❉❡♠♦♥str❛çã♦✳ ❡♥tã♦✿ f ′ (x) = [u(x)]v(x) [v ′ (x) · Ln[u(x)] + v(x) ′ · u (x)] u(x) ✭❡✮ ❆ ❞❡♠♦♥str❛çã♦ ❞❡ ✭❛✮✱ ✭❜✮✱ ✭❝✮ ❡ ✭❞✮ é ✐♠❡❞✐❛t❛✳ ❙❡❥❛ f (x) = u(x)v(x) ❡♥tã♦ f (x) = eLn[u(x)] f ′ (x) = eLn[u(x)] = eLn[u(x)] v(x) v(x) v(x) · (v(x) · Ln[u(x)])′ = v ′ (x) · Ln[u(x)] + ❆ss✐♠✱ f ′ (x) = [u(x)]v(x) [v ′ (x) · Ln[u(x)] + ❊①❡♠♣❧♦ ✺✳✹✻✳ ❉❡t❡r♠✐♥❡ ❛ ❞❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ ❙♦❧✉çã♦✳ = ev(x)·Ln[u(x)] ✱ ❧♦❣♦ y= r v(x) ′ · u (x). u(x) v(x) ′ · u (x)]✳ u(x) x(x − 1) ✳ x−2 1 2 ❉❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ t❡♠♦s q✉❡ Lny = [Lnx+Ln(x−1)−Ln(x−2)]❀ 1 1 1 1 y′ = [ + − ]✳ y 2 x x−1 x−2 1 1 x2 − 4x + 2 y 1 ✳ − ]✱ ✐st♦ é y ′ = p P♦rt❛♥t♦✱ y ′ = [ + 2 x x−1 x−2 2 x(x − 1)(x − 2)3 ❞❡r✐✈❛♥❞♦ ❊①❡♠♣❧♦ ✺✳✹✼✳ ❉❡t❡r♠✐♥❡ ❛ ❞❡r✐✈❛❞❛ ❞❛ s❡❣✉✐♥t❡ ❢✉♥çã♦✿ ❙♦❧✉çã♦✳  1 y = 1+ x x ✳   1 ❊♠ ♠♦❞♦ ❞❡ ❧♦❣❛rít♠✐❝❛ t❡♠♦s Lny = x · Ln 1 + = x[Ln(x + 1) − Lnx]✱ ❝❛❧❝✉❧❛♥❞♦ x     y′ 1 1 1 1 ❛ ❞❡r✐✈❛❞❛ ♣r✐♠❡✐r❛ = [Ln(x + 1) − Lnx] + x · − = Ln 1 + − ✳ y x+1 x x 1+x   x    1 1 1 1 1 ′ P♦rt❛♥t♦✱ y = y[Ln 1 + − − ]= 1+ ]✳ · [Ln 1 + x 1+x x x 1+x ✺✳✺✳✹ ❉❡r✐✈❛❞❛ ❞❛s ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s ❖ ❣rá✜❝♦ ❞❡ ✉♠❛ ❝✉r✈❛ y = f (x) é ❝♦♠♣♦st♦ ♣❡❧♦s ♣♦♥t♦s P (x, y) ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✳ P♦r ❡①❡♠♣❧♦✱ ❛ tr❛❥❡tór✐❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❡♠ ♠♦✈✐♠❡♥t♦ ♥♦ ♣❧❛♥♦ é ❞❡s❝r✐t❛ ♣♦r ✉♠ ♣❛r ❞❡ ❡q✉❛çõ❡s ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦ ♥❛ ❢♦r♠❛ x = x(t) ❡ y = y(t)❀ ❡st❛s ❡q✉❛çõ❡s ❞❡s❝r❡✈❡♠ ♦ ♠❡❧❤♦r ♦ ♠♦✈✐♠❡♥t♦✱ ❛ ♣♦s✐çã♦ ❞❛ ♣❛rtí❝✉❧❛ (x, y) = (x(t), y(t)) ❡♠ q✉❛❧q✉❡r ✐♥st❛♥t❡ t✳ P♦r ❡①❡♠♣❧♦✱ x = cos t, y = sent, 0 ≤ t ≤ 2π ❞❡s❝r❡✈❡ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♥♦ s❡♥t✐❞♦ ❛♥t✐✲❤♦rár✐♦✳ ✷✼✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❡✜♥✐çã♦ ✺✳✾✳ ❈✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛✳ ❙❡ x ❡ y ✱ sã♦ ❞❛❞❛s ❝♦♠♦ ❢✉♥çõ❡s x = x(t) ❡ y = y(t) ❛♦ ❧♦♥❣♦ ❞❡ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ ✈❛❧♦r❡s ❞❡ t✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s (x, y) = (x(t), y(t)) ❞❡✜♥✐❞♦ ♣♦r ❡ss❛s ❡q✉❛çõ❡s é ✉♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛✳ ❆s ❡q✉❛çõ❡s sã♦ ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s ♣❛r❛ ❛ ❝✉r✈❛✳ t ❆ ✈❛r✐á✈❡❧ é ✉♠ ♣❛râ♠❡tr♦ ♣❛r❛ ❛ ❝✉r✈❛✱ ❡ s❡✉ ❞♦♠í♥✐♦ é ❝❤❛♠❛❞♦ ✏✐♥t❡r✈❛❧♦ ❞♦ ♣❛râ♠❡tr♦✑✱ q✉❛♥❞♦ é ♦ ♣♦♥t♦ ✜♥❛❧✳ a ≤ t ≤ b✱ ♦ ♣♦♥t♦ (x(a), y(a)) é ♦ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❡ ♦ ♣♦♥t♦ (x(b), y(b)) ❯♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ❞❡r✐✈á✈❡✐s ❡♠ t✳ x = x(t) ❡ y = y(t) s❡rá ❞❡r✐✈á✈❡❧ ❡♠ t✱ s❡ x ❡ y ❢♦r❡♠ ❊st❛s ❞❡r✐✈❛❞❛s ♥✉♠ ♣♦♥t♦ ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❛ r❡❣r❛ ❞❛ ❝❛❞❡✐❛ dy dy dx = · dt dx dt ❖❜s❡r✈❛çã♦ ✺✳✹✳ ❈✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛✳ ❙❡ ❛s três ❞❡r✐✈❛❞❛s ❡①✐st❡♠ ❡ dx 6= 0✱ ❡♥tã♦ dt dy/dt dy = dx dx/dt ❊①❡♠♣❧♦ ✺✳✹✽✳ d2 y ❉❡t❡r♠✐♥❡ 2 ✱ ❡♠ ❢✉♥çã♦ ❞❡ t✱ s❡ x = 2t − t2 ❡ y = 3t − t3 dt ❙♦❧✉çã♦✳ ❚❡♠✲s❡ q✉❡ ❖❜s❡r✈❡ q✉❡ dy dy/dt = = dx dx/dt   d2 y d dy = = dt2 dt dt y′ = 3d = 4 dt P♦rt❛♥t♦✱  1+t 1−t 3(1 − t2 ) 3(1 + t) = ✳ 2(1 − t) 2   3(1 + t)   d dy/dx d   2 =  ✱ dt dx/dt dt 2(1 − t)  = ❡♥tã♦ 3 3 2 = · 2 4 (1 − t) 2(1 − t)2 3 d2 y = ✳ dt2 2(1 − t)2 ✷✽✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✺✲✷ ✶✳ P❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✱ ❞❡t❡r♠✐♥❡ s✉❛ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛ ❡♠ r❡❧❛çã♦ à ✈❛r✐á✈❡❧ x✳ y = sen2 (3 − 5x) x · senx y= 1 + x2 tan x y= sen2x sen2x y= tan x [sen(nx)]m y= [cos(mx)]n 1. 4. 7. 10. 13. y = sen(cos x) 16. y= ✶✾✳ 2. y = cos2 (x − a) 5. y = sen4 x · cos3 x 6. senx + cos x y= 9. senx − cos x y = sen(nx) · senn x 12. 8. 11. 14. y= cot x − 1 tan x + 1 15. 17. y = sen2 x + cos2 x 18. sec(1 − x) sec(1 − x) + tan(1 − x) hxi y = sen3 3 tan x − 1 y= tan x + 1 sec x − tan x y= sec x + tan x 1 + cos 2x y= 1 − cos 2x csc x + cot x y= csc √ x − cot x cos 2x + 1 y= 2 3. ✷✵✳ y= √ √ 1 − senx− 1 + senx ✷✳ ❉❡t❡r♠✐♥❡ ❝♦♥st❛♥t❡s A ❡ B ❞❡ ♠♦❞♦ q✉❡ y = A · sen3x − B · cos 3x✱ ❝✉♠♣r❛ ❛ ✐❣✉❛❧❞❛❞❡✿ y ′ + 5y = 18 cos 3x✳ ✸✳ ❉❡t❡r♠✐♥❡ ❛ ❞❡r✐✈❛❞❛ ✐♠♣❧í❝✐t❛ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ 1. y = cos(x − y) 2. 3. cot(xy) + xy = 0 4. 5. 7. 9. cos(xy) = y · tan(xy) y = sen(cos(x2 + y 2 )) cos(x + y) = y · senx 6. 8. 10. tan y = 3x2 + tan(x + y) r r x y − =2 y x y = sen2 x + cos2 y sen(x + y) + sen(x − y) = 1 y = sen(x + y) ✹✳ ❉❡s❡♥❤❛r ♦ ❣rá✜❝♦ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ 1. 4. y = x · arctan x 2. y = arcsen(x2 + 3x − 10) 5. y = x − 2 arctan x 3. √ 6. y = arccos x y = arcsec(x2 ) √ y = arccos 1 − x2 arcsenx ✱ ❝✉♠♣r❡ ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ✿ (1 − x2 ).y ′ − xy − 1 = 0❄ 1 − x2 ✺✳ ❆ r❡❧❛çã♦ y = √ ❏✉st✐✜q✉❡ s✉❛ r❡s♣♦st❛✳   x2 · sen 1 , s❡✱ x 6= 0 ′ ✻✳ ❈❛❧❝✉❧❛r f (x) ❡ s❡✉ ❞♦♠í♥✐♦ ♣❛r❛ ❛ ❢✉♥çã♦✿ f (x) = x2  0, s❡✱ x = 0 ✷✽✶ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✼✳ ❉❡r✐✈❛r y = Ln(x) ❡♠ r❡❧❛çã♦ ❛ u = esenx ✳ ✽✳ ❉❡t❡r♠✐♥❡ ❛ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿  x y = arctan √ 2  1 − x 1 y = arcsec 2 2x − 1  tan x 1 y = √ arctan √ 2   2 hxi 2 + arctan y = arctan x 2 5 tan x2 + 4 2 y = arctan 3 hyi 3 xy = arctan x x = arcsen(1 − y) r r x y + =8 y x  1. 3. 5. 7. 9. 11. 13. 15.  2x y = arctan 1 − x2  √  √ x y = (x + a) · arctan − ax a  b + a cos x y = arccos a + b cos x h x i √a2 − x2 y = arcsen + x a 3senx y = arctan 4 + 5 cos x p   2 2 x + y = b · arctan xy  2. 4. 6. 8. 10. 12. arccos(xy) = arcsen(x + y) r r y x + =2 x y 14. 16. ✾✳ ❉❡t❡r♠✐♥❡ ❡①♣r❡ssõ❡s ❝♦♠✉♥s ♣❛r❛ ❛s ❞❡r✐✈❛❞❛s ❞❡ ♦r❞❡♠ 1. y = senax + cos bx 2. y = sen2 x 4. y = sen4 x + cos4 x 5. y = x2 n ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ 1 ax + b x 6. y = 2 x −1 3. y = 1 − 3x + 2 1 f (x) = x2 · sen s❡ x 6= 0 ❡ f (0) = 0✳ ❙✉♣♦♥❤❛♠♦s q✉❡ g(x) ❡ h(x) s❡❥❛♠ x ′ 2 ′ ❢✉♥çõ❡s t❛✐s q✉❡✿ h (x) = sen (sen(x + 1)), h(0) = 3, g (x) = f (x + 1) ❡ g(0) = 0✳ ✶✵✳ ❙❡❥❛ ❆❝❤❛r✿ ❛✮ (f oh)′ (0) ✶✶✳ ❉❡t❡r♠✐♥❡ √ dy dx ❜✮ (gof )′ (0) ❝✮ k ′ (x2 )✱ ✶✷✳ ❙❡❥❛ √ √ f (x) = 0 s❡ x ∈ Q a∈R k(x) = h(x2 )✳ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ x2 xarcsen( x) + 1 − √  √ 4 tan x + 1 − 2 tan x √ 3. y = Ln √ 4 tan x + 1 + 2 tan x p 5. y = Ln[x.senx + cos x + (x · senx + cos x)2 + 1] 1. y = ♦♥❞❡ ❡ p q s❡ x∈ / Q✳ ▼♦str❡ q✉❡ ✷✽✷ f  2Ln2 senx + 3 2. y = Ln 2Ln2 senx − 3 r 1−x 4. y = arctan 1+x  ♥ã♦ é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ ♥❡♥❤✉♠ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✶✸✳ ❙❡❥❛♠ ❛s ❢✉♥çõ❡s y = x3 · Ln(x) ❡ z = Ln(x)✳ ❊st❛❜❡❧❡ç❛ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ y (n) ❡ z (n−3) ♣❛r❛ n ≥ 4✳ ✶✹✳ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ y = x−3 ❝✉♠♣r❡ ❛ r❡❧❛çã♦✿ 2(y ′ )2 = (y − 1)y”✳ x+4 ✶✺✳ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ y = (x2 − 1)n ❝✉♠♣r❡ ❛ r❡❧❛çã♦✿ (x2 − 1)y (n+2) + 2xy (n+1) − 2(n + 1)y (n) = 0, ∀n ∈ N, n≥2 ✶✻✳ ❙❡❥❛♠ f (x) ❡ g(x) ❢✉♥çõ❡s ❞❡ x✳ ❈♦♥s✐❞❡r❡ ❛s s❡❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s✿ y = f (x) − g ′ (x), z = g(x) + f ′ (x), Y = f ′ (x)senx − g ′ (x) cos x ❡ Z = f ′ (x) cos x + g ′ (x)senx✳         dy dx ▼♦str❡ q✉❡ ✈❡r✐✜❝❛✲s❡ ❛ ✐❞❡♥t✐❞❛❞❡✿ 2 + dz dx 2 =+ dY dx 2 + 2 dZ dx ✳ √ ✶✼✳ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ y = (x+ x2 + 1)k ❝✉♠♣r❡ ❛ r❡❧❛çã♦✿ (x2 +1)y ′′ +x·y ′ −k 2 ·y = 0✳ ✶✽✳ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ y = A · sen(̟t + ̟0 ) + B · cos(̟t + ̟0 ) ♦♥❞❡ A, B, ̟ ❡ ̟0 sã♦ ❝♦♥st❛♥t❡s❀ ❝✉♠♣r❡ ❛ r❡❧❛çã♦✿ d2 y + ̟ 2 y = 0✳ dt2 ✶✾✳ ▼♦str❡ q✉❡ s❡ ax2 + 2bxy + cy 2 + 2gx + 2f y + h = 0 t❡♠♦s✿ dy ax + by + g =− dx bx + cy + f ✶✳ A d2 y = 2 dx (bx + cy + f )3 ✷✳ ❖♥❞❡ A é ❝♦♥st❛♥t❡ q✉❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ x ❡ y ✳ d ✷✵✳ ❙❡❥❛♠ u, v, z três ❢✉♥çõ❡s ❞❡ ✈❛r✐á✈❡❧ x t❛✐s q✉❡✿ y = dx    dz dy d u y· = 0✳ −z· ▼♦str❡ q✉❡ dx dx dx  dy u· dx  d ,z = dx   dz u· ✳ dx ✷✶✳ ❱❡r✐✜❝❛r q✉❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ D(x) ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ x ✭é ✉♠❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡✮✿ D(x) = cos(x + a) sen(x + a) 1 cos(x + b) sen(x + b) 1 cos(x + c) sen(x + c) 1 ✷✷✳ ❉❡t❡r♠✐♥❡ ❛ ❞❡r✐✈❛❞❛ n✲és✐♠❛ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ y= ✶✳ 1−x 1+x ✷✳ y= 3x + 2 x2 − 4 ✷✸✳ ❉❡t❡r♠✐♥❡ ❛s ❞❡r✐✈❛❞❛s n✲és✐♠❛ ♣❛r❛ ❛s ❢✉♥çõ❡s✿ 1. 4. y = Ln(x + 1) 2. y = sen2 x 5. y = arctan(x) 3. 4 4 y = sen x + cos x 6. ✷✽✸ ✸✳ y= mx + p x2 − a2 y = sen3 x + cos3 x y = senx · sen2x · sen3x 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦   x2 · sen 1 , s❡✱ x 6= 0 ✷✹✳ Pr♦✈❡ q✉❡ ❛ ❢✉♥çã♦✿ f (x) = é ❞❡r✐✈á✈❡❧ ❡♠ x = 0 ❡ x  0, s❡✱ x = 0 f ′ (0) = 0✳ ✷✺✳ ❙❡ y = a · cos x + b · senx ❡ z = a · senx + b · cos x✱ ❞❡t❡r♠✐♥❡ ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ y (m) z (n) = y (n) z (m) ✱ ♦♥❞❡ m, n ∈ N✳   x · sen 1 , s❡✱ x 6= 0 ♥ã♦ é ❞❡r✐✈á✈❡❧ ♥❡♠ à ❡s✲ ✷✻✳ Pr♦✈❡ q✉❡ ❛ ❢✉♥çã♦✿ f (x) = x  0, s❡✱ x = 0 q✉❡r❞❛ ♥❡♠ à ❞✐r❡✐t❛ ♥♦ ♣♦♥t♦ x = 0✳ ✷✼✳ ▼♦str❡ ♣♦r r❡❝♦rrê♥❝✐❛ q✉❡ ❛ ❞❡r✐✈❛❞❛ ❞❡ ♦r❞❡♠ n ❞❡✿ n−1 √ x é (n) ✶✳ y=x ✷✳ y = ex·cos α · cos(x · sen(α)) ✸✳ y = eax · sen(bx + c) · e y √ x e xn+1 é y (n) y (n) = ex·cos α · cos[x · senα + n · α] p b = (a2 + b2 )n · eax [sen(bx + c) + n · arctan( )] a x 1 π − arctan( )✱ ❛ n✲és✐♠❛ ❞❡r✐✈❛❞❛ ❞❡ y = 2 2 a a + x2 sen(n + 1)θ = (−1)n · n! p ✳ a (a2 + x2 )n+1 ✷✽✳ ▼♦str❡ q✉❡✱ q✉❛♥❞♦ θ = y (n) é = (−1) n é ✷✾✳ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ y = ex +2e2x s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧✿ y ′′′ −6y ′′ +11y ′ = 6y ✳ ✸✵✳ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ y = x3 s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧✿ y (v) + y (iv) + y ′′′ + y ′′ + y ′ + y = x3 + 3x2 + 6x + 6✳ ✸✶✳ ❈❛❧❝✉❧❛r ❛ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛ ❡♠ x = 0 ♣❛r❛ ❛ ❢✉♥çã♦ p f (x) = 3 Ln(arcsenx + arccos x)2 + √ √ 5 x2 − 1 x2 − 5x + 4 ✸✷✳ ❯♠ ❝❧✉❜❡ ✉♥✐✈❡rs✐tár✐♦ ❧❡✈❛♥t❛ ❢✉♥❞♦s ✈❡♥❞❡♥❞♦ ❜❛rr❛s ❞❡ ❝❤♦❝♦❧❛t❡ ❛ ❘$1, 00 ❝❛❞❛✳ ❖ ❝❧✉❜❡ ♣❛❣❛ ❘$0, 60 ♣♦r ❝❛❞❛ ❜❛rr❛ ❡ t❡♠ ✉♠ ❝✉st♦ ❛♥✉❛❧ ✜①♦ ❞❡ ❘$250, 00✳ ❊s❝r❡✈❛ ♦ ❧✉❝r♦ L ❝♦♠♦ ❢✉♥çã♦ ❞❡ x✱ ♥ú♠❡r♦ ❞❡ ❜❛rr❛s ❞❡ ❝❤♦❝♦❧❛t❡ ✈❡♥❞✐❞❛s✳ ▼♦str❡ q✉❡ ❛ ❞❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ ❧✉❝r♦ é ❝♦♥st❛♥t❡ ❡ q✉❡ é ✐❣✉❛❧ ❛♦ ❧✉❝r♦ ♦❜t✐❞♦ ❡♠ ❝❛❞❛ ❜❛rr❛ ✈❡♥❞✐❞❛✳ ✸✸✳ ❆ r❡❝❡✐t❛ R ✭❡♠ ♠✐❧❤õ❡s ❞❡ r❡❛✐s✮ ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❡♠♣r❡s❛ ❞❡ 1.989 ❛ 1.993 ❛❞♠✐t❡ ♦ ♠♦❞❡❧♦ R(t) = −5, 1t3 + 25, 6t2 − 29, 3t + 45, 2✱ ♦♥❞❡ t = 0 r❡♣r❡s❡♥t❛ ♦ t❡♠♣♦ ❡♠ 1.989✳ ❛✮ ❆❝❤❛r ❛ ✐♥❝❧✐♥❛çã♦ ❞♦ ❣rá✜❝♦ ❡♠ 1.990 ❡ ❡♠ 1.989✳ ❜✮ ◗✉❛✐s sã♦ ❛s ✉♥✐❞❛❞❡s ❞❡ ✐♥❝❧✐♥❛çã♦ ❞♦ ❣rá✜❝♦❄ ✷✽✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✺✳✻ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❆♣r♦①✐♠❛çã♦ ❧♦❝❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ ❙❡❥❛ f ✉♠❛ ❢✉♥çã♦ ❞❡r✐✈á✈❡❧ ♥♦ ♣♦♥t♦ x = a ❡ ❝♦♥s✐❞❡r❡♠♦s ❛ ❢✉♥çã♦ ❛✜♠ ❞❡✜♥✐❞❛ ♣♦r✿ Tm (x) = f (a) + m(x − a) ♦♥❞❡ m é ♥ú♠❡r♦ r❡❛❧✳ ❚♦❞❛ ❢✉♥çã♦ ❛✜♠ Tm (x) ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x = a✱ é ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ❛ ❢✉♥çã♦ f (x)✱ ♥♦ s❡♥t✐❞♦ q✉❡ ♦ ❡rr♦ ❝♦♠❡t✐❞♦ ♥❡ss❛ ❛♣r♦①✐♠❛çã♦ t❡♥❞❡ ❛ ③❡r♦ q✉❛♥❞♦ (a + ∆x) → a ♦✉ ∆x → 0 ✭❋✐❣✉r❛ ✭✺✳✺✮✮✳ ❉❡ ❢❛t♦✱ s❡ ❡①♣r❡ss❛♠♦s ❡st❡ ❡rr♦ ❡♠ t❡r♠♦s ❞❡ ∆x ❡ ❢❛③❡♥❞♦ E(∆x) ❝♦♠♦✿ E(∆x) = f (a + ∆x) − Tm (a + ∆x) ❈♦♠♦ f é ❞❡r✐✈á✈❡❧ ❡♠ x = a✱ ❡♥tã♦ f é ❝♦♥tí♥✉❛ ❡♠ x = a ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡✿ ❋✐❣✉r❛ ✺✳✺✿ lim [f (a + ∆x) − Tm (a + ∆x)] = lim E(∆x) = 0 ∆x→0 ∆x→0 ✐st♦ s✐❣♥✐✜❝❛ q✉❡ ♣❛r❛ ✈❛❧♦r❡s ♣❡q✉❡♥♦s ❞❡ ∆x t❡♠♦s f (a + ∆x) ❜❛st❛♥t❡ ♣ró①✐♠♦ ❞❡ Tm (a + ∆x) Pr♦♣r✐❡❞❛❞❡ ✺✳✶✺✳ ❙❡ f é ❞❡r✐✈á✈❡❧ ❡♠ x=a ❡ E(∆x) = f (a + ∆x) − Tm (a + ∆x) ❡♥tã♦ E(∆x) =0 ∆x→0 ∆x lim s❡✱ ❡ s♦♠❡♥t❡ s❡ m = f ′ (x)✳ ❉❡♠♦♥str❛çã♦✳ (⇒) P♦r ❤✐♣ót❡s❡ E(∆x) =0 ❡ ∆x→0 ∆x lim Tm (a + ∆x) = f (a) + m(∆x)✱ ❡♥tã♦ f (a + ∆x) − f (a) − m(∆x) f (a + ∆x) − f (a) = lim − lim ·m = f ′ (a) − m ∆x→0 ∆x→0 ∆x→0 ∆x ∆x lim P♦r t❛♥t♦ m = f ′ (a)✳ (⇐) ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ♣♦r ❤✐♣ót❡s❡ m = f ′ (a)✱ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❞❡r✐✈❛❞❛ ♥✉♠ ♣♦♥t♦✱ t❡♠♦s q✉❡ ♦ ❧✐♠✐t❡ ✿ f (a + ∆x) − f (a) − f ′ (a)∆x f (a + ∆x) − f (a) − f ′ (a) = lim = ∆x→0 ∆x→0 ∆x ∆x 0 = lim ✷✽✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R f (a + ∆x) − f (a) − m(∆x) E(∆x) = lim =0 ∆x→0 ∆x→0 ∆x ∆x lim ❖❜s❡r✈❛çã♦ ✺✳✺✳ ❉❡st❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✶✺✮✱ ♦❜s❡r✈❛♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❢✉♥çã♦ ❛✜♠ q✉❡ ❛♣r♦①✐♠❛ E(∆x) = 0✳ ∆x→0 ∆x ❛ f (x) ❝♦♠ ❛ ❝♦♥❞✐çã♦ lim ❊st❛ ❛♣r♦①✐♠❛çã♦ é ❡①❛t❛♠❡♥t❡ ❛ r❡t❛ t❛♥❣❡♥t❡ à ❝✉r✈❛ f (x) ♥♦ ♣♦♥t♦ x = a✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ q✉❛❧q✉❡r ❢✉♥çã♦ ❞❡r✐✈á✈❡❧ ♥♦ ♣♦♥t♦ x = a✱ ♣♦❞❡ s❡r ❛♣r♦①✐♠❛❞❛ ❧♦❝❛❧♠❡♥t❡ ♣♦r ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉ ✉♠✳ ❊①❡♠♣❧♦ ✺✳✹✾✳ ◆✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞♦ ♣♦♥t♦ x = 3✱ ❞❡t❡r♠✐♥❡ ♦ ♣♦❧✐♥ô♠✐♦ q✉❡ ❛♣r♦①✐♠❛ ❧♦❝❛❧♠❡♥t❡ à √ ❢✉♥çã♦ g(x) = x2 − 1✳ ❙♦❧✉çã♦✳ √ x 3 ✱ ❡♥tã♦ g ′ (3) = √ ✱ ❛ss✐♠ ♦ x2 − 1 8 √ 3 ♣♦❧✐♥ô♠✐♦ q✉❡ ❛♣r♦①✐♠❛ é P (x) = g(3) + g ′ (3)(x − 3) ✐st♦ é P (x) = 8 + √ (x − 3)✳ 8 ❖❜s❡r✈❡ q✉❡ P (3, 01) = 2, 8391 ❡ g(3, 01) = 2, 8390 ♦ ❡rr♦ E(0, 01) = 0, 0001 é ♠í♥✐♠♦✳ P❛r❛ ❛ ❢✉♥çã♦ g(x) = ✺✳✻✳✶ x2 − 1 t❡♠♦s g ′ (x) = √ ❋✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ ❉❡✜♥✐çã♦ ✺✳✶✵✳ ❙❡❥❛ f : A −→ R ✉♠❛ ❢✉♥çã♦ ❡ a ∈ A ✉♠ ♣♦♥t♦ ❞❡ ❛❝✉♠✉❧❛çã♦ ❞❡ A✳ ❙❡ ❞✐③ q✉❡ f é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ x = a✱ s❡✿ f (a + ∆x) = f (a) + m · ∆x + ∆x · ε(∆x) ✭✺✳✻✮ ♦♥❞❡ ε(∆x) → 0 s❡ ∆x → 0 ❡ ε(0) = 0✳ ❆ ❡①♣r❡ssã♦ m · ∆x ❞❛ ✐❣✉❛❧❞❛❞❡ ✭✺✳✻✮ ❞❡♥♦♠✐♥❛✲s❡ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ f ♥♦ ♣♦♥t♦ x = a✱ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ✐♥❝r❡♠❡♥t♦ ∆x ❡ ❞❡♥♦t❛✲s❡ d(a, ∆x) ♦✉ s✐♠♣❧❡s♠❡♥t❡ df (a)✳ ❊♠ ❣❡r❛❧ ❛ df (x) ❝❤❛♠❛✲s❡ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ f (x)✳ Pr♦♣r✐❡❞❛❞❡ ✺✳✶✻✳ ❙❡ f é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ x = a✱ ❛ ❝♦♥st❛♥t❡ m q✉❡ ❛♣❛r❡❝❡ ♥❛ ❉❡✜♥✐çã♦ ✭✺✳✽✮ é ú♥✐❝❛✳ ❉❡♠♦♥str❛çã♦✳ ✷✽✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛♠ m 6= m1 ❡ ε1 (∆x) t❛✐s q✉❡✿ f (a + ∆x) = f (a) + m1 · ∆x + ∆x · ε1 (∆x) lim ε1 (∆x) =✵✳ ❈♦♠ ∆x→0 R ✭✺✳✼✮ ❙✉❜str❛✐♥❞♦ ✭✺✳✻✮ ❞❡ ✭✺✳✼✮ ♦❜té♠✲s❡ ✿ 0 = (m − m1 )∆x + [ε(∆x) − ε1 (∆x)]∆x ∆x 6= 0, m − m1 = ε(∆x) − ε1 (∆x)✱ ♥♦ ❧✐♠✐t❡ ❛ ❛♠❜♦s ∆x → 0 ♦❜t❡♠♦s m = m1 ✱ ✐st♦ s✐❣♥✐✜❝❛ q✉❡ ❛ ❝♦♥st❛♥t❡ é ú♥✐❝❛✳ P❛r❛ ♦s ♠❡♠❜r♦s q✉❛♥❞♦ Pr♦♣r✐❡❞❛❞❡ ✺✳✶✼✳ ❆ ❢✉♥çã♦ f (x) é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ x=a s❡✱ ❡ s♦♠❡♥t❡ s❡ f é ❞❡r✐✈á✈❡❧ ♥♦ ♣♦♥t♦ x = a✳ ❉❡♠♦♥str❛çã♦✳ ⇒✮ ✭ P♦r ❤✐♣ót❡s❡✱ f (x) é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ x = a✱ ❡♥tã♦ f (a + ∆x) = f (a) + m(∆x) + ∆x · ε(∆x) m é ❝♦♥st❛♥t❡ ❡ lim ε(∆x) ❂ ✵✱ ❞✐✈✐❞✐♥❞♦ ♣♦r ∆x 6= 0 ❡ ❝❛❧❝✉❧❛♥❞♦ ♦ ❧✐♠✐t❡ q✉❛♥❞♦ ∆x→0 ∆x → 0 ♦❜té♠✲s❡✿ ❝♦♠♦ f (a + ∆x) − f (a) = lim [m − ε(∆x)] = m ∆x→0 ∆x→0 ∆x lim m = f ′ (a)❀ ✐st♦ é f ✭⇐✮ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ❡♥tã♦ é ❞❡r✐✈á✈❡❧ ❡♠ x = a✳ é ❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✼✮✳ ❊①❡♠♣❧♦ ✺✳✺✵✳ ❙❡❥❛ f (x) = x ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡✱ ❝❛❧❝✉❧❛r ♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ f (x)✳ ❙♦❧✉çã♦✳ df (x) = f ′ (x) · ∆x ❝♦♠♦ f ′ (x) = 1 ❡ f (x) = x✱ ■st♦ s✐❣♥✐✜❝❛ q✉❡ ♦ ✏✐♥❝r❡♠❡♥t♦ ❞❛ ✈❛r✐á✈❡❧ ❝✐❛❧ dx = ∆x✳ ✐♥❞❡♣❡♥❞❡♥t❡ x(∆x) é ✐❣✉❛❧ ♦❜t❡♠♦s ❛ s❡✉ ❞✐❢❡r❡♥✲ dx✑✳ ❊①❡♠♣❧♦ ✺✳✺✶✳ ❙❡❥❛ f (x) = ❢✭①✮❄ 1 3 x✱ 4 ❝❛❧❝✉❧❛r ♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ f ♥♦ ♣♦♥t♦ x = 2❀ ◗✉❛❧ é ♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ❙♦❧✉çã♦✳ 3 d(2, ∆x) = f ′ (2) · ∆x✱ s❡♥❞♦ f ′ (x) = x2 ✱ 4 x = 2✱ é d(2, ∆x) = f ′ (2) · ∆x = 3∆x✳ ❚❡♠♦s ❞❡ f ❡♠ ✷✽✼ ❧♦❣♦ f ′ (2) = 3❀ ❛ss✐♠ ♦ ❞✐❢❡r❡♥❝✐❛❧ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 3 2 P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ f (x) é df (x) = f ′ (x) · ∆x = x · ∆x✳ ❖❜s❡r✈❛çã♦ ✺✳✻✳ ❈♦♥s✐❞❡r❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❛♥t❡r✐♦r❡s✱ s❡ df (a) = f ′ (a)·dx ❛✮ ❜✮ y = f (x) t❡♠♦s✿ dy = df (x) = f ′ (x)·dx ❝✮ dy = f ′ (x)✳ dx ✺✳✻✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ Pr♦♣r✐❡❞❛❞❡ ✺✳✶✽✳ ❙❡❥❛♠ u = f (x) ❡ v = g(x) ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡ c ✉♠❛ ❝♦♥st❛♥t❡✱ ❡♥tã♦✿ a) d(c) = 0. b) d(cu) = cd(u) c) d(u + v) = d(u) + d(v)  u  v · d(u) − u · d(v) d = v v2 d) d(u.v) = u.d(v) + v.d(u) e) ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❊①❡♠♣❧♦ ✺✳✺✷✳ ❙❡❥❛ f (x) = √ x2 + 5 ✱ ❞❡t❡r♠✐♥❡ df ✳ ❙♦❧✉çã♦✳ √ x · dx . x2 + 5 ❉♦ ❢❛t♦ df (x) = f ′ (x) · dx t❡♠♦s df (x) = ( x2 + 5)′ · dx = √ ❊①❡♠♣❧♦ ✺✳✺✸✳ f (x) = x2 + 3✱ ❞❡t❡r♠✐♥❡ ∆f ❡ df q✉❛♥❞♦ x = 2 E(∆x) q✉❛♥❞♦ ✉t✐❧✐③❛♠♦s df ♣❛r❛ ❛♣r♦①✐♠❛r ∆f ❄ ❉❛❞♦ ❡rr♦ ❡ ∆x = dx = 0, 5✳ ◗✉❛❧ é ♦ ❙♦❧✉çã♦✳ P❛r❛ a = 2 ❡ ∆x = 0, 5 t❡♠♦s✿ ∆f = f (a + ∆x) − f (a) = f (2, 5) − f (2) = 2, 25 df (2, 5) = f ′ (2)dx = 2(2) · (0, 5) = 2 ▲♦❣♦✱ E(∆x) = ∆f − df = 2, 25 − 2 = 0, 25✳ ✺✳✻✳✸ ❙✐❣♥✐✜❝❛❞♦ ❣❡♦♠étr✐❝♦ ❞♦ ❞✐❢❡r❡♥❝✐❛❧ ❘❡❡s❝r❡✈❡♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ♦❜t❡♠♦s✿ f (a + ∆x) = f (a) + f ′ (a) · ∆x + ∆x · ε(∆x) ✷✽✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ♦♥❞❡✿ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R   f (a + ∆x) − f (a) , s❡✱ ∆x 6= 0 ε(∆x) = ∆x  0, s❡✱ ∆x = 0 ■st♦ s✐❣♥✐✜❝❛ q✉❡ s❡ f é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ x = a✱ ❡ q✉❡ f é ❧♦❝❛❧♠❡♥t❡ ❛♣r♦①✐♠❛❞❛ ♣♦r s✉❛ r❡t❛ t❛♥❣❡♥t❡✿ Tm (x) = f (a) + f ′ (a)(x − a) ❙❡❥❛♠ P (a, f (a)) ❡ Q(a + ∆x, f (a + ∆x)) ♦s ♣♦♥t♦s s♦❜r❡ ♦ ❣rá✜❝♦ ❞❡ f ✭❋✐❣✉r❛ ✭✺✳✻✮✮✳ ❆ r❡t❛ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ y q✉❡ ♣❛ss❛ ♣♦r Q ✐♥t❡r❝❡♣t❛ à r❡t❛ t❛♥❣❡♥t❡ Tm (x) ♥♦ ♣♦♥t♦ S ❡ à r❡t❛ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ x q✉❡ ♣❛ss❛ ♣♦r P ❛ ✐♥t❡r❝❡♣t❛ ♥♦ ♣♦♥t♦ R✳ ❋✐❣✉r❛ ✺✳✻✿ RS ❚❡♠♦s tan α = ✱ ♣♦ré♠ P R = ∆x = dx ❡ tan α = f ′ (a) ♦♥❞❡✱ RS = f ′ (a)dx = PR d(f, ∆x)✳ ❆ss✐♠ ♦❜té♠✲s❡ q✉❡ ∆x → 0 ❡ ∆y ≈ dy ✳ P♦rt❛♥t♦✱ f (a + ∆x) ≈ f (a) + f ′ (a)dx✳ ❖❜s❡r✈❛çã♦ ✺✳✼✳ ❙❡ y = f (x)✱ ∆y = f (a + ∆x) ≈ f (a) ❡ ∆y ≈ dy ❞❡❞✉③✐♠♦s q✉❡ dy ✈❛r✐❛çã♦ q✉❡ s♦❢r❡ ❛ ❢✉♥çã♦ f q✉❛♥❞♦ x ✈❛r✐❛ ❞❡ a ❛té a + ∆x✳ s❡♥❞♦ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❛ é ❊①❡♠♣❧♦ ✺✳✺✹✳ ❊st✐♠❛✲s❡ ❡♠ 12cm ♦ r❛✐♦ ❞❡ ✉♠❛ ❡s❢❡r❛✱ ❝♦♠ ✉♠ ❡rr♦ ♠á①✐♠♦ ❞❡ 0, 006cm✳ ❊st✐♠❡ ♦ ❡rr♦ ♠á①✐♠♦ ♥♦ ❝á❧❝✉❧♦ ❞♦ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛✳ ❙♦❧✉çã♦✳ 4 ❙❡❥❛ r ♦ r❛✐♦ ❞❛ ❡s❢❡r❛✱ s❡✉ ✈♦❧✉♠❡ é ❞❛❞♦ ♣♦r V (r) = πr3 ❀ ❞❡♥♦t❛♥❞♦ dr ♦ ❞✐❢❡r❡♥❝✐❛❧ 3 ❞♦ r❛✐♦❀ t❡♠♦s dV = V ′ (r)dr✱ ✐st♦ é dV = 4πr2 dr✱ ❢❛③❡♥❞♦ r = 12, dr = ±0, 06✱ ❛ss✐♠✱ dV = 4π(12)2 (±0, 006) = ±10, 857cm3 ✳ ❖ ❡rr♦ ♠á①✐♠♦ ♥❛ ♠❡❞✐❞❛ ❞♦ ✈♦❧✉♠❡✱ ❞❡✈✐❞♦ ❛♦ ❡rr♦ ♥❛ ♠❡❞✐❞❛ ❞♦ r❛✐♦ é 10, 857cm3 ✳ ❊①❡♠♣❧♦ ✺✳✺✺✳ ❆♣r♦①✐♠❛r ♠❡❞✐❛♥t❡ ❞✐❢❡r❡♥❝✐❛✐s ❛ r❛✐③ q✉✐♥t❛ ❞❡ 3127✳ ❙♦❧✉çã♦✳ √ √ ❙❡❥❛ f (x) = 5 x ❡ a = 3125✱ t❡♠♦s f (a) = 5 3125 = 5✳ ❙❡ a + ∆x = 3127✱ ❡♥tã♦ ∆x = 2 = dx✳ ❈♦♠♦ f (a + ∆x) ≈ f (a) + f ′ (a)dx ❡♥tã♦✱ f (3127) ≈ f (3125) + f ′ (3125).(2)✳ ■st♦ é √ 5 3127 ≈ √ 5 1 ) = 5 + 0, 0032 = 5, 0032 3125 + 2( √ 5 5 31254 ✷✽✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ P♦rt❛♥t♦ √ 5 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R 3.127 ≈ 5, 0032✳ ❉❡✜♥✐çã♦ ✺✳✶✶✳ ❙❡ ❡①✐st❡ ❡rr♦ ♥❛ ♠❡❞✐❞❛ ❞❡ ✉♠ ❡①♣❡r✐♠❡♥t♦ q✉❡ ❞❡s❝r❡✈❡ ✉♠❛ ❢✉♥çã♦ y = f (x)✱ ❞❡✜♥❡✲s❡✿ dy ❡rr♦ ♥❛ ♠❡❞✐❞❛ = ❊rr♦ r❡❧❛t✐✈♦ = ✈❛❧♦r ♠é❞✐♦ f (a) ❖ ❡rr♦ ♣❡r❝❡♥t✉❛❧ é ♦ ❡rr♦ r❡❧❛t✐✈♦ ♠✉❧t✐♣❧✐❝❛❞♦ ♣♦r 100❀ ✐st♦ é dy · 100%✳ f (a) P♦r ❡①❡♠♣❧♦✱ s❡ ❛ ♠❡❞✐❞❛ ❞❡ ✉♠ ❝♦♠♣r✐♠❡♥t♦ ❛❝✉s❛ 25cm✳ ❈♦♠ ✉♠ ♣♦ssí✈❡❧ ❡rr♦ ❞❡ 0, 1cm✱ ❡♥tã♦ ♦ ❡rr♦ r❡❧❛t✐✈♦ é 0, 1 = 0, 004✳ 25 ❖ s✐❣♥✐✜❝❛❞♦ ❞❡st❡ ♥ú♠❡r♦ é q✉❡ ♦ ❡rr♦✱ é ❡♠ ♠é❞✐❛ ❞❡ 0, 004cm ♣♦r ❝❡♥tí♠❡tr♦✳ ❊①❡♠♣❧♦ ✺✳✺✻✳ ❆ ❛❧t✉r❛ ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❜❛s❡ q✉❛❞r❛❞❛ é 15cm✳ ❙❡ ♦ ❧❛❞♦ ❞❛ ❜❛s❡ ♠✉❞❛ ❞❡ 10 ♣❛r❛ 10.02cm✱ ✉s❛♥❞♦ ❞✐❢❡r❡♥❝✐❛✐s ❝❛❧❝✉❧❛r ❛ ♠✉❞❛♥ç❛ ❛♣r♦①✐♠❛❞❛ ❞❡ s✉❡ ✈♦❧✉♠❡✳ ❙♦❧✉çã♦✳ ❖ ✈♦❧✉♠❡ ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ é V = x2 h✱ ♦♥❞❡ ❛ ❛❧t✉r❛ h = 15 é ❝♦♥st❛♥t❡ ❡ x ❧❛❞♦ ❞❛ ❜❛s❡ q✉❛❞r❛❞❛ é ✈❛r✐á✈❡❧❀ ❡♥tã♦✱ V = 15x2 ❡ dV = 30x · dx✳ P❛r❛ ♥♦ss♦ ❝❛s♦ x = 10 ❡ dx = ±0, 02❀ ❧♦❣♦ dV = ±6cm3 ✳ ❖ ✈♦❧✉♠❡ s♦❢r❡ ❛♣r♦①✐♠❛✲ ❞❛♠❡♥t❡ ✉♠ ❛✉♠❡♥t♦ ❞❡ 6cm3 ✳ ❖ ❡rr♦ r❡❧❛t✐✈♦ é ✺✳✼ dV dV 30x · dx = 0, 004 ❡ ♦ ❡rr♦ ♣❡r❝❡♥t✉❛❧ é = · 100% = 0, 4%✳ 2 V 15x V ❚❡♦r❡♠❛ s♦❜r❡ ❢✉♥çõ❡s ❞❡r✐✈á✈❡✐s ❙❡❥❛ f : R −→ R ❢✉♥çã♦ r❡❛❧ ❝♦♠ ❞♦♠í♥✐♦ D(f )✱ ❡ a ∈ D(f )✳ ❉❡✜♥✐çã♦ ✺✳✶✷✳ ❉✐③❡♠♦s q✉❡ f ❛♣r❡s❡♥t❛ ✉♠ ♠á①✐♠♦ ❛❜s♦❧✉t♦ ❡♠ x = a✱ s❡ f (x) ≤ f (a) ∀ x ∈ D(f )✳ ❖ ✈❛❧♦r f (a) é ❝❤❛♠❛❞♦ ✲ ♠á①✐♠♦ ❛❜s♦❧✉t♦ ❞❡ f ✳ ❉❡✜♥✐çã♦ ✺✳✶✸✳ ❉✐③❡♠♦s q✉❡ f ❛♣r❡s❡♥t❛ ✉♠ ♠í♥✐♠♦ ❛❜s♦❧✉t♦ ❡♠ x = a✱ s❡ f (a) ≤ f (x) ∀ x ∈ D(f )✳ ❖ ✈❛❧♦r f (a) é ❝❤❛♠❛❞♦ ✲ ♠í♥✐♠♦ ❛❜s♦❧✉t♦ ❞❡ f ✳ ✷✾✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❡✜♥✐çã♦ ✺✳✶✹✳ ❉✐③❡♠♦s q✉❡ f ❛♣r❡s❡♥t❛ ✉♠ ✲ ♠á①✐♠♦ r❡❧❛t✐✈♦ ✲ ♦✉ ✲ ♠á①✐♠♦ ❧♦❝❛❧ ✲ ❡♠ x = a✱ δ > 0 t❛❧ q✉❡ f (x) ≤ f (a) ∀ x ∈ B(a, δ) = (a − δ, a + δ) ⊆ D(f )✳ ♥ú♠❡r♦ f (a) é ❝❤❛♠❛❞♦ ✲ ♠á①✐♠♦ r❡❧❛t✐✈♦ ✲ ♦✉ ✲ ♠á①✐♠♦ ❧♦❝❛❧ ❞❡ f s❡ ❡①✐st❡ ❖ ✭❋✐❣✉r❛ ✭✺✳✼✮✮✳ ❉❡✜♥✐çã♦ ✺✳✶✺✳ ❉✐③❡♠♦s q✉❡ f ❛♣r❡s❡♥t❛ ✉♠ ♠í♥✐♠♦ r❡❧❛t✐✈♦ ♦✉ ♠í♥✐♠♦ ❧♦❝❛❧ ❡♠ x = a✱ s❡ ❡①✐st❡ δ > 0 t❛❧ q✉❡ f (a) ≤ f (x) ∀ x ∈ B(a, δ) = (a − δ, a + δ) ⊆ D(f )✳ ❖ ♥ú♠❡r♦ f (a) é ❝❤❛♠❛❞♦ ✲ ♠í♥✐♠♦ r❡❧❛t✐✈♦ ✲ ♦✉ ✲ ♠í♥✐♠♦ ❧♦❝❛❧ ❞❡ f ✳ ✭❋✐❣✉r❛ ✭✺✳✽✮✮ ❋✐❣✉r❛ ✺✳✼✿ ❋✐❣✉r❛ ✺✳✽✿ ❊①❡♠♣❧♦ ✺✳✺✼✳ ❙❡❥❛ ❙♦❧✉çã♦✳ f (x) = √ 16 − x2 ✱ ❞❡t❡r♠✐♥❡ s❡✉s ✈❛❧♦r❡s ❞❡ ♠á①✐♠♦ ❡ ♠í♥✐♠♦ ❛❜s♦❧✉t♦s✳ ❖ ❉♦♠í♥✐♦ ❞❡ f (x) é D(f ) = [−4, 4] ❡ s❡✉ ❣rá✜❝♦ é ✉♠❛ s❡♠✐❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ 4✳ ❊①✐st❡ ♠á①✐♠♦ ❛❜s♦❧✉t♦ ❡♠ x = 0; f (0) = 4 é ♦ ♠á①✐♠♦ ❛❜s♦❧✉t♦✱ ❡ ♦ ♠í♥✐♠♦ ❛❜s♦❧✉t♦ ❡♠ x = −4 ♦✉ x = 4; f (4) = 0 é ♦ ♠í♥✐♠♦ ❛❜s♦❧✉t♦✳ ❖❜s❡r✈❛çã♦ ✺✳✽✳ • ❙❡ f (c) é ♦ ✈❛❧♦r ❞❡ ♠í♥✐♠♦ ♦✉ ♠á①✐♠♦✱ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ❡①tr❡♠♦ ❞❡ f ♦✉ ✈❛❧♦r ❡①tr❡♠♦ ❞❡ f ✱ ❛ss✐♠ ♣♦❞❡r❡♠♦s ❢❛❧❛r ❞❡ ❡①tr❡♠♦s ❛❜s♦❧✉t♦s ♦✉ ❡①tr❡♠♦s r❡❧❛t✐✈♦s✳ ❖ ♣♦♥t♦ x = c é ❝❤❛♠❛❞♦ ❞❡ ♣♦♥t♦ ❞❡ ❡①tr❡♠♦✳ • ❙❡ f (c) é ✉♠ ❡①tr❡♠♦ r❡❧❛t✐✈♦✱ ❡♥tã♦ x = c é ✉♠ ♣♦♥t♦ ❞♦ ✐♥t❡r✐♦r ❞♦ D(f ) ✐st♦ é ❡①✐st❡ δ > 0 t❛❧ q✉❡ B(c, δ) ⊆ D(f )✳ ❊st❛ ❝♦♥❞✐çã♦ ✈❡r✐✜❝❛✲s❡ ♥❡❝❡ss❛r✐❛♠❡♥t❡ s❡ ✷✾✶ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ f (c) é ✉♠ ❡①tr❡♠♦ ❛❜s♦❧✉t♦✱ ❥á q✉❡ ♦ ❡①tr❡♠♦ ❛❜s♦❧✉t♦ ♣♦❞❡ ♦❝♦rr❡r ♥✉♠ ♣♦♥t♦ q✉❡ ♥ã♦ é ♣♦♥t♦ ✐♥t❡r✐♦r ❞♦ ❞♦♠í♥✐♦✳ ❊①❡♠♣❧♦ ✺✳✺✽✳ ❙❡❥❛ ❛ ❢✉♥çã♦ f (x) = | 3x | s❡✉ ❣rá✜❝♦ ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✺✳✾✮✳ 2 + x2 ❖❜s❡r✈❡ q✉❡ f (−1) = f (1) = 1 é ♠á①✐♠♦ ❧♦❝❛❧ ❡ ❛❜s♦❧✉t♦✱ f (0) = 0 é ♦ ♠í♥✐♠♦ ❧♦❝❛❧ ❡ ❛❜s♦❧✉t♦✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❡①tr❡♠♦✱ s❡ t❡♠♦s ❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ f (x) = k ♣❛r❛ t♦❞♦ x ∈ R✱ ❡♥tã♦ x é ✉♠ ♣♦♥t♦ ❞❡ ❡①tr❡♠♦ ❛❜s♦❧✉t♦ ❡ r❡❧❛t✐✈♦✱ k é s❡✉ ♠á①✐♠♦ ❛❜s♦❧✉t♦✱ ♠á①✐♠♦ r❡❧❛t✐✈♦✱ ♠í♥✐♠♦ ❛❜s♦❧✉t♦ ❡ ♠í♥✐♠♦ r❡❧❛t✐✈♦✳ ❋✐❣✉r❛ ✺✳✾✿ ❋✐❣✉r❛ ✺✳✶✵✿ ❊①❡♠♣❧♦ ✺✳✺✾✳ ❙❡❥❛  x2   − ,    2   −x, f (x) =  1,    2    x − 3, 2 2 s❡✱ −2≤x<0 s❡✱ 0 ≤ x < 1 s❡✱ x = 1 s❡✱ 1 < x < 3 ❖❜s❡r✈❛♥❞♦ ♦ ❣rá✜❝♦ ❞❡st❛ ❢✉♥çã♦ ✭❋✐❣✉r❛ ✭✺✳✶✵✮✮✱ t❡♠♦s✿ • f (−2) = −2 é ♦ ♠í♥✐♠♦ ❛❜s♦❧✉t♦❀ ♥ã♦ t❡♠ ♠á①✐♠♦ ❛❜s♦❧✉t♦✳ • f (0) = 0 ❡ f (1) = 2 sã♦ ♠á①✐♠♦s r❡❧❛t✐✈♦s✳ Pr♦♣r✐❡❞❛❞❡ ✺✳✶✾✳ ❙❡❥❛ f : R −→ R ❢✉♥çã♦ r❡❛❧ t❛❧ q✉❡✿ ❛✮ f (c) é ✉♠ ❡①tr❡♠♦ r❡❧❛t✐✈♦ ❞❡ f ✳ ✷✾✷ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❜✮ f ❊♥tã♦ t❡♠ ❞❡r✐✈❛❞❛ ❡♠ x = c✳ ′ f (c) = 0✳ ❉❡♠♦♥str❛çã♦✳ P♦❞❡♠♦s s✉♣♦r f (c) s❡❥❛ ♠á①✐♠♦ ❧♦❝❛❧✳ ◆❡st❡ ❝❛s♦ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ B(c, δ) ⊆ D(f )✱ t❛❧ q✉❡ f (x) ≤ f (c), ∀ x ∈ B(c, δ)✳ ❊♥tã♦✿ ❙❡ x < c ❡ x ∈ B(c, δ) ⇒ f (x) ≤ f (c) ❡ ❙❡ c < x ❡ x ∈ B(c, δ) ⇒ f (x) ≤ f (c) ❡ f (x) − f (c) ≥0 x−c f (x) − f (c) ≤0 x−c ✭✺✳✽✮ ✭✺✳✾✮ f (x) − f (c) ≥0 x→c x−c f (x) − f (c) ≤0 ❉❡ ✭✺✳✾✮ t❡♠♦s f ′ (c+ ) = lim+ x→c x−c ❉♦ ❢❛t♦ f (x) t❡r ❞❡r✐✈❛❞❛ ❡♠ x = c✱ ❡st❡s ❧✐♠✐t❡s sã♦ ✐❣✉❛✐s✱ ❡♥tã♦ f ′ (c− ) = 0 = f ′ (c+ )❀ ✐st♦ é f ′ (c) = 0✳ ❉❡ ✭✺✳✽✮ t❡♠♦s f ′ (c− ) = lim− ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ ♠♦str❛✲s❡ q✉❛♥❞♦ f (c) s❡❥❛ ♠í♥✐♠♦ ❧♦❝❛❧✳ ❖❜s❡r✈❛çã♦ ✺✳✾✳ ❛✮ ❆ ✭✺✳✶✾✮ ❛✜r♠❛ q✉❡✱ s❡ f (c) é ✉♠ ❡①tr❡♠♦ r❡❧❛t✐✈♦ ❞❡ f ✱ ❡ s❡ f t❡♠ ❞❡r✐✈❛❞❛ ❡♠ x = c✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡ f ′ (c) = 0❀ ✐st♦ s✐❣♥✐✜❝❛ q✉❡ ❛ r❡t❛ t❛♥❣❡♥t❡ à ❝✉r✈❛ y = f (x) é ❤♦r✐③♦♥t❛❧ ♥♦ ♣♦♥t♦ P (c, f (c))✳ ❜✮ ❖ ❢❛t♦ f ′ (c) = 0 ♥ã♦ ✐♠♣❧✐❝❛ q✉❡ x = c s❡❥❛ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✉♠ ♣♦♥t♦ ❞❡ ❡①tr❡♠♦✳ Pr♦♣r✐❡❞❛❞❡ ❊①❡♠♣❧♦ ✺✳✻✵✳ ❙❡❥❛ f (x) = (x − 2)3 ◆ã♦ ♦❜st❛♥t❡✱ x=2 ∀ ∈ R❀ ❡♥tã♦ f ′ (x) = 3(x − 2)2 ❡ f ′ (2) = 0✳ ♥ã♦ é ♣♦♥t♦ ❞❡ ❡①tr❡♠♦ r❡❧❛t✐✈♦ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✭✺✳✶✶✮✳ ❋✐❣✉r❛ ✺✳✶✶✿ ✷✾✸ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❡✜♥✐çã♦ ✺✳✶✻✳ P♦♥t♦ ❝rít✐❝♦✳ ❙❡❥❛ f : R −→ R ❢✉♥çã♦ r❡❛❧ ❞❡ ❞♦♠í♥✐♦ D(f ) ❡ a ∈ D(f )❀ ♦ ♣♦♥t♦ x = a é ❝❤❛♠❛❞♦ ♣♦♥t♦ ❝rít✐❝♦ ♦✉ ♣♦♥t♦ s✐♥❣✉❧❛r ❞❡ f s❡❀ f ′ (a) = 0 ♦✉✱ s❡ ♥ã♦ ❡①✐st❡ f ′ (a)✳ ❖❜s❡r✈❛çã♦ ✺✳✶✵✳ ❉❛ ❖❜s❡r✈❛çã♦ ✭✺✳✾✮✱ ✉♠❛ ❢✉♥çã♦ f ♣♦❞❡ t❡r ❡①tr❡♠♦s r❡❧❛t✐✈♦s ♥♦s ♣♦♥t♦s ❝rít✐❝♦s❀ ❡✱ ♣❛r❛ ❝❛❧❝✉❧❛r ❡st❡s ♣♦♥t♦s é s✉✜❝✐❡♥t❡ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ f ′ (x) = 0✱ ♦✉ ❛ q✉❡ r❡s✉❧t❛ ❞❡ ❝♦♥s✐❞❡r❛r q✉❡ f ′ (x) ♥ã♦ ❡①✐st❛✳ ❊①❡♠♣❧♦ ✺✳✻✶✳ ❉❡t❡r♠✐♥❡ ♦s ♣♦♥t♦s ❝rít✐❝♦s ♣❛r❛ ❝❛❞❛ ✉♠❛ ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ ❝✮ x 5 + 5 x √ √ 3 h(x) = 9 x5 + 12 3 x ❡✮ g(x) = senx ❛✮ f (x) = ❜✮ g(x) = 3|x| 1 + x2 ❞✮ f (x) = 1 (2x3 + 3x2 − 36x + 6) 12 ❙♦❧✉çã♦✳ ❛✮ x2 − 25 1 5 x 5 = 0✱ + ✱ ❡♥tã♦ f ′ (x) = − 2 ✱ q✉❛♥❞♦ f ′ (x) = 0 t❡♠♦s 5 x 5 x 5x2 ❧♦❣♦ sã♦ ♣♦♥t♦s ❝rít✐❝♦s✿ x = 5 ❡ x = −5✳ ◗✉❛♥❞♦ x = 0 ♦ ♥ú♠❡r♦ f ′ (0) ♥ã♦ ❡①✐st❡✱ ♣♦ré♠ x = 0 ♥ã♦ é ♣♦♥t♦ ❝rít✐❝♦ ♣♦r ♥ã♦ ♣❡rt❡♥❝❡r ❛♦ ❞♦♠í♥✐♦ ❞❡ f ✳ ❚❡♠♦s f (x) = ❙♦❧✉çã♦✳ ❜✮ 3|x| g(x) = 1 + x2 q✉❛♥❞♦ ♥ã♦ ❡①✐st❛ ❡♥tã♦ g ′ (x)   2 3x 1 − x g ′ (x) = | x | (1 + x2 )2 P❛r❛ ❛ ❢✉♥çã♦ g ′ (x) = 0 ⇒ x = ±1 ❡✱ t❡♠♦s ❙ã♦ ♣♦♥t♦s ❝rít✐❝♦s ♣❛r❛ ❛ ❙♦❧✉çã♦✳ ❝✮ x = 0✳ ❢✉♥çã♦ g(x)✱ q✉❛♥❞♦ √ √ 3 h(x) = 9 x5 + 12 3 x ♦s ♥ú♠❡r♦s x = 1, x = −1 ❡ x = 0✳ t❡♠♦s √ √ 3 3 h′ (x) = 15 x2 + 4 x−2 ✐st♦ é √ 3 x4 + 4 15 ′ √ ✳ h (x) = 3 x2 h′ (0) ♥ã♦ x = 0✳ ❖❜s❡r✈❡ q✉❡ ♣♦♥t♦ ❝rít✐❝♦ é ❡①✐st❡✱ ❡ ♥ã♦ ❡①✐st❡ ♥ú♠❡r♦ r❡❛❧ t❛❧ q✉❡ h′ (x) = 0✱ ❧♦❣♦ ♦ ú♥✐❝♦ ❙♦❧✉çã♦✳ ❞✮ ✷✾✹ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ f (x) = 1 (2x3 + 3x2 − 36x + 6) ❡♥tã♦ 12 1 1 f ′ (x) = (x2 + x − 6) = (x − 2)(x + 3) = 0 2 2 ✐♠♣❧✐❝❛ q✉❡ ♦s ú♥✐❝♦s ♣♦♥t♦s ❝rít✐❝♦s sã♦ x = −3 ❡ x = 2✳ ❙♦❧✉çã♦✳ ❡✮ g(x) = senx t❡♠♦s g ′ (x) = cos x❀ q✉❛♥❞♦ g ′ (x) = 0 t❡♠♦s x = k ∈ Z✳ ❙ã♦ ♣♦♥t♦s ❝rít✐❝♦s ❞❡ g(x) ♦s ♥ú♠❡r♦s x = (2k + 1)π ♣❛r❛ t♦❞♦ 2 (2k + 1)π ♣❛r❛ t♦❞♦ k ∈ Z✳ 2 ❚❡♦r❡♠❛ ✺✳✶✳ ❞❡ ❘♦❧❧❡✳ ✭1652 − 1719✮✳ ❙❡❥❛ ❛✮ f f : [a, b] −→ R ❝♦♥tí♥✉❛ ❡♠ ✉♠❛ ❢✉♥çã♦ q✉❡ ❝✉♠♣r❡✿ [a, b]✳ ❜✮ f ❝✮ f (a) = f (b) = 0✳ t❡♠ ❞❡r✐✈❛❞❛ ❡♠ ❊♥tã♦ ❡①✐st❡ c ∈ (a, b) (a, b)✳ t❛❧ q✉❡ f ′ (c) = 0✳ ❉❡♠♦♥str❛çã♦✳ ❉❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❢✉♥çã♦ ❡♠ [a, b]✱ s❡❣✉❡ q✉❡ ❛ ❢✉♥çã♦ t❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠ ♠í♥✐♠♦ ❡ ✉♠ ♠á①✐♠♦ ❛❜s♦❧✉t♦ ❡♠ [a, b]❀ ✐st♦ é ❡①✐st❡♠ c1 ❡ c2 ❡♠ [a, b] t❛✐s q✉❡ f (c1 ) = m = min .f (x) x∈[a, b] ❡ f (c2 ) = M = max .f (x) x∈[a, b] ❙❡ c1 ∈ (a, b)✱ ♣❡❧❛ ❤✐♣ót❡s❡ ❜✮ ❡ ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✶✾✮ t❡♠♦s f ′ (c1 ) = 0 ❡ ❡st❛ ♣r♦♣r✐❡❞❛❞❡ ❡st❛rá ♠♦str❛❞❛ s❡♥❞♦ c = c1 ❀ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦ s❡ c2 ∈ (a, b)✳ ❘❡st❛ ♠♦str❛r ♦ ❝❛s♦ q✉❡ c1 ❡ c2 s❡❥❛♠ ♦s ❡①tr❡♠♦s ❞♦ ✐♥t❡r✈❛❧♦ [a, b]✳ ❙✉♣♦♥❤❛♠♦s q✉❡ c1 = a ❡ c2 = b ✭♦✉ c1 = b ❡ c2 = a✮✱ ❛ ❤✐♣ót❡s❡ ❝✮ ✐♥❞✐❝❛ q✉❡ f (a) = f (b) = 0✱ ✐st♦ s✐❣♥✐✜❝❛ q✉❡ m = M = 0 ❡ f (x) = 0 ∀ x ∈ [a, b]❀ ❧♦❣♦ f ′ (x) = 0 ∀ x ∈ [a, b] ❡✱ ❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ✈❡r❞❛❞❡✐r❛✳ ❖❜s❡r✈❛çã♦ ✺✳✶✶✳ ❆ ❚❡♦r❡♠❛ ✺✳✼✳✶ ✭✺✳✶✮ s❡❣✉❡ s❡♥❞♦ ✈á❧✐❞❛ s❡ ❛ ❤✐♣ót❡s❡ ❝✮ é s✉❜st✐t✉í❞❛ ♣♦r f (a) = f (b)✳ ■♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ t❡♦r❡♠❛ ❞❡ ❘♦❧❧❡ ❖ t❡♦r❡♠❛ ❞❡ ❘♦❧❧❡ t❡♠ s✐❣♥✐✜❝❛❞♦ ❣❡♦♠étr✐❝♦ ✐♠❡❞✐❛t♦✳ ❆s ❤✐♣ót❡s❡s ❞✐③❡♠ q✉❡ ♦ ❣rá✜❝♦ ❞❡ f é ❝♦♥tí♥✉♦ ♥♦ ✐♥t❡r✈❛❧♦ [a, b] ❡ t❡♠ r❡t❛s t❛♥❣❡♥t❡ ❡♠ t♦❞♦ ♦s ♣♦♥t♦s ❝♦♠ ❛❜s❝✐ss❛s ❡♠ (a, b) ❡✱ s❡ A(a, f (a)) ❡ B(b, f (b)) sã♦ ♦s ♣♦♥t♦s ✷✾✺ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❝♦♠✱ f (a) = f (b)✱ ❡♥tã♦ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ P (c, f (c)) ❝♦♠ P ❞✐❢❡r❡♥t❡ ❞❡ A ❡ B ♥♦ q✉❛❧ ❛ r❡t❛ t❛♥❣❡♥t❡ é ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦✲x ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✭✺✳✶✷✮✳ ❋✐❣✉r❛ ✺✳✶✷✿ ❊①❡♠♣❧♦ ✺✳✻✷✳ x2 − 9x ❉❛❞❛ ❛ ❢✉♥çã♦ f (x) = ✈❡r✐✜❝❛r s❡ ❝✉♠♣r❡ ♦ t❡♦r❡♠❛ ❞❡ ❘♦❧❧❡✳ x−3 ❙♦❧✉çã♦✳ ❖❜s❡r✈❡ q✉❡ f (0) = f (9) = 0✱ ♣♦ré♠ ❛ ❢✉♥çã♦ f ♥ã♦ é ❝♦♥tí♥✉❛ ❡♠ x = 3✳ ▲♦❣♦✱ ♥ã♦ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ t❡♦r❡♠❛ ❞❡ ❘♦❧❧❡✱ ✐st♦ ♥ã♦ s✐❣♥✐✜❝❛ q✉❡ ♥ã♦ ❡①✐st❛ ✉♠ ✈❛❧♦r ❞❡♥tr♦ ❞♦ ✐♥t❡r✈❛❧♦ ♣❛r❛ ♦ q✉❛❧ s✉❛ ❞❡r✐✈❛❞❛ s❡❥❛ ✐❣✉❛❧ ❛ ③❡r♦✳ ❊①❡♠♣❧♦ ✺✳✻✸✳ ❉❛❞❛ ❛ ❢✉♥çã♦ f (x) = [0, 3]✳ ❙♦❧✉çã♦✳ ✐✮ ✐✐✮ ✐✐✐✮ √ 3 √ x4 −3 3 x✱ ✈❡r✐✜❝❛r s❡ ❝✉♠♣r❡ ♦ t❡♦r❡♠❛ ❞❡ ❘♦❧❧❡ ♥♦ ✐♥t❡r✈❛❧♦ f é ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ [0, 3]✳ f ′ (x) = 4√ 1 3 x− √ ❀ ✐st♦ é✱ f t❡♠ ❞❡r✐✈❛❞❛ ♥♦ ✐♥t❡r✈❛❧♦ (0, 3)✳ 3 3 x2 f (0) = f (3) = 0✳ ❊♥tã♦✱ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ❘♦❧❧❡✱ ❡①✐st❡ c ∈ (0, 3) t❛❧ q✉❡ f ′ (c) = 0✱ ✐st♦ é f ′ (c) = 1 3 4√ 3 c− √ = 0 ♦♥❞❡ c = ✳ 3 2 3 4 c ❊①❡♠♣❧♦ ✺✳✻✹✳ ❖ ❝✉st♦ C(x) ❞❡ ♣❡❞✐❞♦ ❞❡ ✉♠❛ ♠❡r❝❛❞♦r✐❛ é ❞❛❞❛ ♣❡❧❛ ❢✉♥çã♦✿ C(x) = 10(x2 + x + 3) x(x + 3) ♦♥❞❡ C(x) é ♠❡❞✐❞♦ ❡♠ ♠✐❧❤❛r❡s ❞❡ r❡❛✐s ❡ x é ♦ t❛♠❛♥❤♦ ❞♦ ♣❡❞✐❞♦ ♠❡❞✐❞♦ ❡♠ ❝❡♥t❡♥❛s✳ ✷✾✻ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✭❛✮ ❱❡r✐✜q✉❡ q✉❡ C(3) = C(6)✳ ✭❜✮ ❙❡❣✉♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❘♦❧❧❡✱ ❛ t❛①❛ ✈❛r✐❛çã♦ ❞❡ ❝✉st♦ ❞❡✈❡ s❡r ③❡r♦ ♣❛r❛ ❛❧❣✉♠ ♣❡❞✐❞♦ ♥♦ ✐♥t❡r✈❛❧♦ [3, 6]✳ ❉❡t❡r♠✐♥❡ ♦ t❛♠❛♥❤♦ ❞❡ss❡ ♣❡❞✐❞♦✳ ❙♦❧✉çã♦✳ ❛✮ ❖❜s❡r✈❡ q✉❡ C(3) = 10(32 + 3 + 3) 150 25 10(62 + 6 + 3) 450 = = ❡ C(6) = = = 3(3 + 3) 18 3 6(6 + 3) 54 25 ✱ ❧♦❣♦ C(3) = C(6)✳ 3 ❜✮ ❆ ❢✉♥çã♦ ❝✉st♦ C(x) é ❝♦♥tí♥✉❛ ❡♠ t♦❞♦ s❡✉ ❞♦♠í♥✐♦ ✭ x > 0✮✱ ❡♠ ♣❛rt✐❝✉❧❛r ♥♦ ✐♥t❡r✈❛❧♦ [3, 6]✱ s✉❛ ❞❡r✐✈❛❞❛ é  2x2 − 6x − 9 C (x) = 10 x2 (x + 3)2 ′  ❡①✐st❡ ♥♦ ✐♥t❡r✈❛❧♦ (3, 6)❀ ❧♦❣♦ ❡①✐st❡ c ∈ (3, 6) t❛❧ q✉❡ C ′ (c) = 0✳  2c2 − 6c − 9 =0 ■st♦ é 10 c2 (c + 3)2  2c2 − 6c − 9 = 0 ⇒ √ ⇒ c= 6± √ 4 108 ✳ 6 + 10, 4 = 4, 1 ❛♣r♦①✐♠❛❞❛♠❡♥t❡✳ 4 4 ◗✉❛♥❞♦ ♦ ♣❡❞✐❞♦ ❢♦r ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ♠❛✐♦r q✉❡ 4, 1 ❝❡♥t❡♥❛s ✭410 ✉♥✐❞❛❞❡s✮✱ ❛ t❛①❛ ❈♦♠♦ c ∈ (3, 6) ⇒ c= 6+ 108 = ❞❡ ✈❛r✐❛çã♦ ❞❡ ❝✉st♦ ❞❡✈❡ s❡r ③❡r♦✳ ❉♦ ❱❛❧♦r ▼é❞✐♦ ✲ ❚✳❱✳▼✳ ❙❡❥❛ f : [a, b] −→ R ✉♠❛ ❢✉♥çã♦ q✉❡ ❝✉♠♣r❡✿ ❚❡♦r❡♠❛ ✺✳✷✳ ❛✮ f ❝♦♥tí♥✉❛ ❡♠ [a, b]✳ ❜✮ f t❡♠ ❞❡r✐✈❛❞❛ ❡♠ (a, b)✳ ❊♥tã♦ ❡①✐st❡ c ∈ (a, b) t❛❧ q✉❡ f ′ (c) = ❉❡♠♦♥str❛çã♦✳ f (b) − f (a) ✳ b−a ❙❡❥❛ m ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s A(a, f (a)) ❡ B(b, f (b)) ❡ f (b) − f (a) ✱ ❡ g(x) = f (a)+m·(x−a)✳ b−a ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ ❛✉①✐❧✐❛r F (x) = f (x) − g(x)✱ ✐st♦ é F (x) = f (x) − f (a) − m(x − a) ∀ x ∈ [a, b]✳ ❖❜s❡r✈❡ q✉❡ F (x) ❝✉♠♣r❡ ❛s ❝♦♥❞✐çõ❡s ❞♦ ❚❡♦r❡♠❛ ❞❡ ❘♦❧❧❡ ♥♦ ✐♥t❡r✈❛❧♦ [a, b]✱ ♣♦✐s F é ❝♦♥tí♥✉❛ ❡♠ [a, b]✱ é ❞❡r✐✈á✈❡❧ ❡♠ (a, b) ❡ F (a) = F (b) = 0✳ f (b) − f (a) ✳  ❊♥tã♦ ❡①✐st❡ c ∈ (a, b) t❛❧ q✉❡ F ′ (c) = 0✱ ✐st♦ é F ′ (c) = f ′ (c) = m = b−a g(x) ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s A ❡ B ✱ ❡♥tã♦ m = ❊st❡ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ t❛♠❜é♠ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚❡♦r❡♠❛ ❞❡ ♦✉ ❞❡ ▲❛❣r❛♥❣❡✳ ✷✾✼ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✺✳✼✳✷ ■♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦✳ ❖ ❣rá✜❝♦ ❞❡ f (x) ♥♦ ✐♥t❡r✈❛❧♦ [a, b] t❡♠ ❛ ♣r♦✲ ♣r✐❡❞❛❞❡ ❞❡ s❡r ❝♦♥tí♥✉♦ ❡♠ [a, b] ❡ ♣♦ss✉✐ r❡t❛s t❛♥✲ ❣❡♥t❡s ❡♠ t♦❞♦s s❡✉s ♣♦♥t♦s ❞❡ ❛❜s❝✐ss❛s ❡♠ (a, b) ❡♥tã♦ ♦ ❚✳❱✳▼✳ ❛✜r♠❛ q✉❡ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ P (c, f (c)) ❝♦♠ P ❞✐❢❡r❡♥t❡ ❞❡ A(a, f (a)) ❡ B(b, f (b)) ♥❛ q✉❛❧ ❛ r❡t❛ t❛♥❣❡♥t❡ é ♣❛r❛❧❡❧❛ à ❝♦r❞❛ ✭❋✐❣✉r❛ ✭✺✳✶✸✮✮✳ Pr♦♣r✐❡❞❛❞❡ ✺✳✷✵✳ ❙❡❥❛ f : [a, b] −→ R ❛✮ f ❝♦♥tí♥✉❛ ❡♠ ❜✮ f t❡♠ ❞❡r✐✈❛❞❛ ❡♠ ❊♥tã♦ f ❋✐❣✉r❛ ✺✳✶✸✿ ✉♠❛ ❢✉♥çã♦ q✉❡ ❝✉♠♣r❡✿ [a, b]✳ (a, b) é ❝♦♥st❛♥t❡ ❡♠ ❡ [a, b]✱ f ′ (x) = 0 ✐st♦ é ∀ x ∈ (a, b)✳ f (x) = k ❉❡♠♦♥str❛çã♦✳ ∀ x ∈ [a, b]✳ ❙❡❥❛ x ∈ (a, b) ✉♠ ❡❧❡♠❡♥t♦ ❛r❜✐trár✐♦ ❡ k ∈ R ✉♠❛ ❝♦♥st❛♥t❡✳ ❆s ❝♦♥❞✐çõ❡s ❞♦ ❚✳❱✳▼✳ sã♦ ✈❡r✐✜❝❛❞❛s ♥♦ ✐♥t❡r✈❛❧♦ [a, x] ⊆ [a, b]✱ ❧♦❣♦ ❡①✐st❡ c ∈ (a, x) t❛❧ q✉❡ f (x) − f (a) = f ′ (c)(x − a)✳ ❉❛ ❤✐♣ót❡s❡ ❜✮ s❡❣✉❡ f ′ (c) = 0✱ ❧♦❣♦ f (x) − f (a) = 0 ✐st♦ é f (x) = f (a) = k ✱ ♣♦✐s x é ❛r❜✐trár✐♦ ❡♠ (a, b) ❛ss✐♠ f (x) = k ∀ x ∈ [a, b)✳ ❉❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ f ❡♠✱ [a, b] s❡❣✉❡ q✉❡ f (x) = k ∀ x ∈ [a, b]✳ Pr♦♣r✐❡❞❛❞❡ ✺✳✷✶✳ f t❡♠ ❞❡r✐✈❛❞❛ ❡♠ (a, b) ❡ f ′ (x) = 0 t♦❞♦ x ∈ (a, b) ♦♥❞❡ k é ❝♦♥st❛♥t❡✳ ❙❡ ✉♠❛ ❢✉♥çã♦ f (x) = k ♣❛r❛ ♣❛r❛ t♦❞♦ x ∈ (a, b)✱ ❡♥tã♦ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❖❜s❡r✈❛çã♦ ✺✳✶✷✳ ❙❡ ♦ ✐♥t❡r✈❛❧♦ ♥ã♦ é ❛❜❡rt♦✱ ❛ Pr♦♣r✐❡❞❛❞❡ f (x) = [|x|] = n, ∀ x ∈ (R − Z)✳ ❡①❡♠♣❧♦✱ ♣❛r❛ ❛ ❢✉♥çã♦ ′ f (x) = 0 ✭✺✳✷✶✮ ♥❡♠ s❡♠♣r❡ é ✈❡r❞❛❞❡✐r❛✳ ∀ x ∈ [n, n + 1), ∀ n ∈ Z✱ P♦r s✉❛ ❞❡r✐✈❛❞❛ ❊st❡ ❡①❡♠♣❧♦ ♠♦str❛ q✉❡✱ s❡ ❛ ❞❡r✐✈❛❞❛ é ③❡r♦ ♥✉♠ ❞❡t❡r♠✐♥❛❞♦ ❝♦♥❥✉♥t♦✱ ❡♥tã♦ ❛ ❢✉♥çã♦ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ é ❝♦♥st❛♥t❡ ❡♠ t❛❧ ❝♦♥❥✉♥t♦✳ Pr♦♣r✐❡❞❛❞❡ ✺✳✷✷✳ ❙❡❥❛♠ ❛✮ f ❡ f g ❡ g : [a, b] −→ R ❝♦♥tí♥✉❛s ❡♠ ❢✉♥çõ❡s q✉❡ s❛t✐s❢❛③❡♠✿ [a, b]✳ ✷✾✽ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❜✮ f ❡ g ❞❡r✐✈á✈❡✐s ❡♠ (a, b) ❡ f ′ (x) = g ′ (x) ∀ x ∈ (a, b)✳ ❊♥tã♦ f (x) = g(x) + k ∀ x ∈ [a, b] ♦♥❞❡ k é ✉♠❛ ❝♦♥st❛♥t❡✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ h(x) = f (x) − g(x), ∀ x ∈ [a, b] ❡♥tã♦ h é ❝♦♥tí♥✉❛ ❡♠ [a, b] ❡ t❡♠ ❞❡r✐✈❛❞❛ ❡♠ (a, b) ❡ h′ (x) = f ′ (x) − g ′ (x) = 0 ∀ x ∈ (a, b) ❡ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✷✶✮ h(x) = k ∀ x ∈ [a, b] ♦♥❞❡ k é ❝♦♥st❛♥t❡✳ P♦rt❛♥t♦ f (x) = g(x) + k ∀ x ∈ [a, b]✳ ❖❜s❡r✈❛çã♦ ✺✳✶✸✳ ❆ Pr♦♣r✐❡❞❛❞❡ ✭✺✳✷✷✮ ✐♥❞✐❝❛ q✉❡ s❡ f ❡ g sã♦ ❢✉♥✲ çõ❡s ❞❡r✐✈á✈❡✐s ♥♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I ⊆ R ❡ f ′ (x) = g ′ (x) ❡♠ I ✱ ❡♥tã♦ s❡✉s ❣rá✜❝♦s sã♦ ❝✉r✈❛s ♣❛r❛❧❡❧❛s ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✭✺✳✶✹✮✳ ❋✐❣✉r❛ ✺✳✶✹✿ ❊①❡♠♣❧♦ ✺✳✻✺✳ ❙❡❥❛ f (x) = x3 − x2 , ❙♦❧✉çã♦✳ x ∈ [−1, 3]✱ ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r q✉❡ ❝✉♠♣r❡ ♦ ❚✳❱✳▼✳ ❆ ❢✉♥çã♦ f (x) é ✉♠ ♣♦❧✐♥ô♠✐♦✱ ❧♦❣♦ ❡❧❛ é ❝♦♥tí♥✉❛ ❡♠ [−1, 3] ❡ ❝♦♠ ❞❡r✐✈❛❞❛ ❡♠ (−1, 3) ❡ f ′ (x) = 3x2 − 2x✳ ❊♠ ✈✐rt✉❞❡ ❞♦ ❚✳❱✳▼✳ ❡①✐st❡ c ∈ (−1, 3) t❛❧ q✉❡ f ′ (c) = 3c2 − 2c ♦♥❞❡ 3c2 − 2c = 5 ⇒ 5 c = −1 ❡ c = ✳ 3 5 3 P♦rt❛♥t♦✱ ♦ ✈❛❧♦r q✉❡ ❝✉♠♣r❡ ♦ ❚✳❱✳▼✳ é c = ✳ ❊①❡♠♣❧♦ ✺✳✻✻✳ ❱❡r✐✜❝❛r s❡ ♦ ❚✳❱✳▼✳ ♣♦❞❡♠♦s ❛♣❧✐❝❛r à ❢✉♥çã♦ f (x) ♥♦ ✐♥t❡r✈❛❧♦ [0, 2] ♦♥❞❡✿ f (x) = ( 6 − 3x2 s❡✱ x ≤ 1 3x−2 s❡✱ x > 1 ❙♦❧✉çã♦✳ ◆♦ ✐♥t❡r✈❛❧♦ [0, 1] ❛ ❢✉♥çã♦ é ♣♦❧✐♥ô♠✐❝❛✱ ❡ ♥♦ ✐♥t❡r✈❛❧♦ (1, 2] ❛ ❢✉♥çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❛ss✐♠✱ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ f (x) ♥♦ ✐♥t❡r✈❛❧♦ [0, 2] é ✐♠♣♦rt❛♥t❡ ❞❡t❡r♠✐♥❛r ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❡♠ x = 1✳ ❖❜s❡r✈❡ q✉❡ f (1+ ) = f (1− ) = 3 ❡ lim f (x) = 3✱ x→1 ❧♦❣♦ f é ❝♦♥tí♥✉❛ ❡♠ x = 1✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❡♠ [0, 2]✳ ✷✾✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ P♦r ♦✉tr♦ ❧❛❞♦✱ f ′ (x) = ( R −6x s❡✱ x ≤ 1 −6x−3 s❡✱ x > 1 ❡ f ′ (1+ ) = f ′ (1− ) = −6✱ ❡♥tã♦ f é ❞❡r✐✈á✈❡❧ ❡♠ (0, 2)✳ f (2) − f (0) = 2−0 21 − ✳ ❖❜s❡r✈❡ q✉❡ f ′ (1+ ) = f ′ (1− ) = f ′ (1) = −6 ❡♥tã♦ c < 1 ♦✉ c > 1✱ ♠❛✐s❀ s❡ 8 7 21 ′ ⇒ c= ∈ (0, 2)✳ f (x) = −6x ♣❛r❛ x < 1 ❡♥tã♦ f ′ (c) = −6c = − 8 16 21 ⇒ c= P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ f ′ (x) = −6x−3 ♣❛r❛ x > 1✱ ❡♥tã♦ f ′ (c) = −6c−3 = − 8 r 3 16 ∈ (0, 2)✳ 7 r 7 16 P♦rt❛♥t♦✱ ♦s ✈❛❧♦r❡s q✉❡ ✈❡r✐✜❝❛♠ ♦ ❚✳❱✳▼✳ sã♦ ❡ 3 ✳ 16 7 ❈♦♠♦ f ❝✉♠♣r❡ ❛s ❝♦♥❞✐çõ❡s ❞♦ ❚✳❱✳▼✳ ✱ ❡①✐st❡ c ∈ [0, 2] t❛❧ q✉❡ f ′ (c) = ❊①❡♠♣❧♦ ✺✳✻✼✳ 1 ♠♦str❡ q✉❡ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ♥ú♠❡r♦ r❡❛❧ c ♥♦ ✐♥t❡r✈❛❧♦ x−4 g(6) − g(2) (2, 6) t❛❧ q✉❡ g ′ (c) = ✳ ❉❡t❡r♠✐♥❡ s❡ ✐ss♦ ❝♦♥tr❛❞✐③ ♦ ❚✳❱✳▼✳ ❥✉st✐✜q✉❡ s✉❛ 4 ❉❛❞❛ ❛ ❢✉♥çã♦ g(x) = r❡s♣♦st❛✳ ❙♦❧✉çã♦✳ ❖ ❚✳❱✳▼✳ ❞✐③ q✉❡✱ s❡ g é ❝♦♥tí♥✉❛ ❡♠ [2, 6] ❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ (2, 6) ❡♥tã♦ ❡①✐st❡ g(6) − g(2) ✳ c ∈ (2, 6) t❛❧ q✉❡ g ′ (c) = 4 ❖❜s❡r✈❡ q✉❡ g(x) ♥ã♦ é ❞❡r✐✈á✈❡❧ ❡♠ (2, 6) ❡♠ ♣❛rt✐❝✉❧❛r ❡♠ x = 4 t❡♠♦s g ′ (x) = 1 − ♥ã♦ ❡①✐st❡✳ (x − 4)2 P♦r ♦✉tr♦ ❧❛❞♦✱ g ♥ã♦ é ❝♦♥tí♥✉❛ ❡♠ [2, 6]✱ ❡♠ ♣❛rt✐❝✉❧❛r ❡♠ x = 4✳ P♦rt❛♥t♦ ♥ã♦ s❡ ❝♦♥tr❛❞✐③ ♦ ❚✳❱✳▼✳ ❙✉♣♦♥❞♦ q✉❡ ❡①✐st❛ c ∈ (2, 6) t❛❧ q✉❡ g ′ (c) = − g(6) − g(2) 1 = 2 (c − 4) 4 ⇒ − 1 1 = 2 (c − 4) 4 ■st♦ ú❧t✐♠♦ é ✉♠ ❛❜s✉r❞♦✳ ✸✵✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✺✲✸ ✶✳ P❛r❛ ♦s s❡❣✉✐♥t❡s ❡①❡r❝í❝✐♦s✱ ❞❡t❡r♠✐♥❡ s❡ ❝✉♠♣r❡ ♦ t❡♦r❡♠❛ ❞❡ ❘♦❧❧❡ ♣❛r❛ ❛s ❢✉♥çõ❡s ❞❛❞❛s ♥♦ ✐♥t❡r✈❛❧♦ ✐♥❞✐❝❛❞♦✱ s❡ ❢♦r ❛ss✐♠✱ ❞❡t❡r♠✐♥❡ ♦s ✈❛❧♦r❡s q✉❡ ♦ ❝✉♠♣r❡♠✳ 1. f (x) = x2 − 4x √ 5 3. f (x) = 1 − x4 5. f (x) = x2 + 4x ❡♠ ❡♠ ❡♠ [0, 4] [−1, 1] [−4, 0] 2. f (x) = x2 − 4x + 3 4. f (x) = x4 − 5x2 + 4 ❡♠ ❡♠ 6. f (x) = 4x3 + x2 − 4x − 1 [1, 3] [−2, 2] ❡♠ 1 [− , 1] 4 ✷✳ P♦❞❡✲s❡ ❛♣❧✐❝❛r ♦ t❡♦r❡♠❛ ❞❡ ❘♦❧❧❡ ♣❛r❛ ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s ❄ x2 − 4x x−2 x+1 3. f (x) = ❡♠ [2, 4] x−1 π 5π ] 5. f (x) = Ln(senx) ❡♠ [ , 6 6 7. g(x) = x3 + 4x2 − 7x − 11 ❡♠ [−1, 2] √ 9. g(x) = 3 x2 − 3x + 2 ❡♠ [1, 2] 1. f (x) =  1 5 2 3    x − 2 x − 6x + 2 , f (x) = | x2 − 4 |,    4 x − x3 − 3x + 6, ✶✶✳ 2. 4. f (x) = x2 − 3x 3x2 − 2x + 4 8. f (x) = x−2 senx 10. g(x) = 4 ❡♠ [0, π] 6. s❡✱ x ≤ −1 s❡✱ | x |< 1 x≥1 s❡✱ x2 − 4x x+2 2 f (x) = x + 2x − 5 f (x) = ❡♠ [−2, 2]✳ a0 xn +a1 xn−1 +a2 xn−2 +· · ·+an−1 x = 0 n−1 t❡♠ ✉♠❛ r❛✐③ ♣♦s✐t✐✈❛ x = x0 ✱ ❡♥tã♦ ❛ ❡q✉❛çã♦ na0 x + (n − 1)a1 xn−2 + (n − 2)a2 xn−3 + · · · + an−1 = 0 t❛♠❜é♠ t❡♠ ✉♠❛ r❛✐③ ♣♦s✐t✐✈❛✱ s❡♥❞♦ ❡st❛ ♠❡♥♦r q✉❡ x0 ✳ ✸✳ ▼♦str❡ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ ❙❡ ❛ ❡q✉❛çã♦ 2 − x2 t❡♠ ✈❛❧♦r❡s ✐❣✉❛✐s ♥♦s ❡①tr❡♠♦s ❞♦ ✐♥t❡r✈❛❧♦ [−1, 1]✳ ▼♦str❡ f (x) = 4 x ′ ❞❡r✐✈❛❞❛ f (x) ♥ã♦ s❡ r❡❞✉③ ❛ ③❡r♦ ❡♠ [−1, 1] ❡ ❡①♣❧✐❝❛r ♣♦r q✉❡ ♥ã♦ ❝✉♠♣r❡ ✹✳ ❆ ❢✉♥çã♦ q✉❡ ❛ ♦ t❡♦r❡♠❛ ❞❡ ❘♦❧❧❡✳ g(x) =| x − 2 | t❡♠ ✈❛❧♦r❡s ✐❣✉❛✐s ♥♦s ❡①tr❡♠♦s ❞♦ ✐♥t❡r✈❛❧♦ [2 − a, 2 + a] a > 0✳ ▼♦str❡ q✉❡ ❛ g ′ (x) ♥ã♦ s❡ r❡❞✉③ ❛ ③❡r♦ ❡♠ [2 − a, 2 + a] ❡ ❡①♣❧✐❝❛r ♣♦r ✺✳ ❆ ❢✉♥çã♦ ♣❛r❛ q✉❡ ♥ã♦ ❝✉♠♣r❡ ♦ t❡♦r❡♠❛ ❞❡ ❘♦❧❧❡✳ f (x) = 1 + xm (x − 1)n ♦♥❞❡ m ❡ n sã♦ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✳ ❙❡♠ ❝❛❧❝✉❧❛r ′ ♠♦str❡ q✉❡ ❛ ❡q✉❛çã♦ f (x) = 0 t❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ r❛✐③ ♥♦ ✐♥t❡r✈❛❧♦ ✻✳ ❙❡❥❛ ❛ ❢✉♥çã♦ ❛ ❞❡r✐✈❛❞❛✱ (0, 1)✳ ✸✵✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✼✳ ▼♦str❡ q✉❡ ❛ ❡q✉❛çã♦ x3 − 3x + c = 0 ♥ã♦ ♣♦❞❡ t❡r r❛í③❡s ❞✐❢❡r❡♥t❡s ♥♦ ✐♥t❡r✈❛❧♦ (0, 1)✳ ✽✳ P❛r❛ ♦ s❡❣✉✐♥t❡s ❡①❡r❝í❝✐♦s ❞❡t❡r♠✐♥❛r s❡ ♦ ❝❛s♦ ❛✜r♠❛t✐✈♦ ✈❡r✐✜❝❛r✳ ❚✳❱✳▼✳ 1. f (x) = x2 + 2x ❡♠ [−2, 0] 3. f (x) = 2x3 − x2 5. f (x) =| 9 − 4x2 | 2. f (x) = ❡♠ [−2, 2] ✶✷✳ ✶✸✳ ✶✹✳ ✶✺✳ ✶✻✳ f (x) = xn   f (x) =     f (x) =   ( f (x) = √ x2 + 9 ❡♠ [0, 4] 4. f (x) =| 4 − x2 | 3 3 2 2 ❡♠ [−2, 2] 6. f (x) = Lnx ❡♠ [1, e] ❡♠ [− , ] x3 ❡♠ [−9, −4] 7. f (x) = 4 x −4 x2 ❡♠ [−1, 2] 9. f (x) = 4+ | x | ✶✶✳ é ❛♣❧✐❝á✈❡❧ ♥♦ ✐♥t❡r✈❛❧♦ ❞❛❞♦❀ x+1 ❡♠ [2, 4] x−1 | x |3 ❡♠ [−2, 2] 10. f (x) = 1 + x6 8. f (x) = ❡♠ [0, a] n > 0 a > 0 4 , s❡✱ x ≤ −1 x2 8 − 4x2 , s❡✱ x > −1 ❡♠ [−2, 0] 3 − x2 , s❡✱ x < 1 2 1 , s❡✱ x ≥ 1 x ❡♠ [0, 2] 2x + 3 s❡✱ x < 3 15 − 2x s❡✱ x ≥ 3 ❡♠ [−1, 5]  2  s❡✱ x < 2  | x −√9 | ❡♠ [−4, 12] f (x) = s❡✱ 2 ≤ x < 11 5+2 x−2   2 11 + (x − 11) , s❡✱ x > 11   x2 + 4 s❡✱ − 2 ≤ x < 0   4 − x3 s❡✱ 0 ≤ x < 1 f (x) = ❡♠ [−2, 2]  6   , s❡✱ 1 ≤ x ≤ 2 x2 + 1 ✾✳ ❉❡t❡r♠✐♥❡ ♦s ♣♦♥t♦s ❝rít✐❝♦s ❞❛s ❢✉♥çõ❡s ❞♦ ❡①❡r❝í❝✐♦ ❛♥t❡r✐♦r✳ ✶✵✳ ❙❡♠ ❝❛❧❝✉❧❛r ❛ ❞❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ f (x) = (x − 1)(x − 2)(x − 3)(x − 4)✱ ❡st❛❜❡❧❡❝❡r q✉❛♥t❛s r❛í③❡s t❡♠ ❛ ❡q✉❛çã♦ f ′ (x) = 0 ❡ ✐♥❞✐❝❛r ❡♠ q✉❡ ✐♥t❡r✈❛❧♦s s❡ ❡♥❝♦♥tr❛♠✳ ✶✶✳ ▼♦str❡ q✉❡ ❛ q✉❡ ❛ ❡q✉❛çã♦ f (x) = xn + px + q ♥ã♦ ♣♦❞❡ t❡r ♠❛✐s ❞❡ ❞♦✐s r❛í③❡s r❡❛✐s q✉❛♥❞♦ n é ♣❛r❀ ❡ ♠❛✐s ❞❡ três r❛í③❡s q✉❛♥❞♦ n é í♠♣❛r✳ ✶✷✳ P❛r❛ ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✱ ❞❡t❡r♠✐♥❡ ♦ ♣♦❧✐♥ô♠✐♦ T (x) ❞❡ ❣r❛✉ um q✉❡ ❛♣r♦①✐♠❡ ❧♦❝❛❧♠❡♥t❡ ❛ f (x) ♥♦ ♣♦♥t♦ ✐♥❞✐❝❛❞♦ ❡ ♦❜t❡r ✈❛❧♦r❡s q✉❡ s❡ ✐♥❞✐❝❛♠✿ ✸✵✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R √ √ 15 + x2 + 3 x ❡♠ x = 64, f (67), T (67)✳ x f (x) = 2 ❡♠ x = 2, f (1.68), T (1.68)✳ x +1 f (x) = x2 + 4x + 5 ❡♠ x = 5, f (5.8), T (5.8)✳ f (x) = ✶✳ ✷✳ ✸✳ ✶✸✳ P❛r❛ ❛s ❢✉♥çõ❡s s❡❣✉✐♥t❡s✳ ∆f, df, ❆❝❤❛r E(x) = ∆f − df ❡ ♣❛r❛ ♦s ✈❛❧♦r❡s ✐♥❞✐❝❛❞♦s✿ ✶✳ f (x) = x2 + 5x, ✷✳ f (x) = x3 + 5x2 − 3x + 2, x = 2, ∆x = 0.01✳ x x = 0, ∆x = 0.1✳ f (x) = x+1 1 x = 5, ∆x = 0.01✳ f (x) = √ x−1 x2 x = 1, ∆x = 0.3✳ f (x) = 3 x +1 ✸✳ ✹✳ ✺✳ x = −1, ∆x = 0.02✳ ✶✹✳ P❛r❛ ♦s s❡❣✉✐♥t❡s ❡①❡r❝í❝✐♦s ❛❝❤❛r ♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❛ ❢✉♥çã♦✿ √ 2. f (x) = x2 + 2 x − 1 r t+1 4. f (t) = t−1 4 · sgn(x − 1) √ 6. f (x) = x2 − 1 1. f (x) = 3x3 + 5x2 + 2 2ax 3. f (x) = (x + 1)3 3kx 5. f (x) = √ x+1 ✶✺✳ ❯s❛♥❞♦ ❞✐❢❡r❡♥❝✐❛✐s ❞❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ✐♥❞✐❝❛❞♦✳ 4 2 f (x) = x + 2x − 3, f (−2.97)✳ r x+1 f (0.1)✳ f (x) = 3 x−1 f (x) = x3 + 5x2 − x + 1, f (0.003)✳ ✶✳ ✸✳ ✺✳ ✶✻✳ ❖ ❞✐â♠❡tr♦ ❞❡ ✉♠❛ ❡s❢❡r❛ é ✷✳ ✹✳ √ 5 + 3x f (x) = x+1 √ 4x + 1 f (x) = 2 x +1 f (2.024)✳ f (1.91)✳ 9cm ❛♦ ♠é❞✐♦ ✐♥tr♦❞✉③✲s❡ ✉♠ ♣♦ssí✈❡❧ ❡rr♦ ❞❡ ±0.05cm✳ ◗✉❛❧ é ♦ ❡rr♦ ♣❡r❝❡♥t✉❛❧ ♣♦ssí✈❡❧ ♥♦ ❝á❧❝✉❧♦ ❞♦ ✈♦❧✉♠❡❄ ✶✼✳ ❈❛❧❝✉❧❛r ♦ ✈❛❧♦r ❛♣r♦①✐♠❛❞♦ ♣❛r❛ ❛s s❡❣✉✐♥t❡s ❡①♣r❡ssõ❡s✿ 1. √ 37, 5 4. √ 82 + √ 4 82 √ 3 9, 12 √ 1 5. 3 63 + √ 3 2 63 2. p 1 3 (8, 01)4 + (8, 01)2 − √ 3 8, 01 √ 5 1020 6. 3. a > b ♠❡❞✐❛♥t❡ ♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦✱ ♠♦str❡ ✈❛❧✐❞❛❞❡ ❞❛s ❞❡s✐❣✉❛❧❞❛❞❡s✿ nbn−1 (a − b) < an − bn < nan−1 (a − b) s❡ n > 1❀ ❡ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ♦♣♦st❛s s❡ n < 1✳ ✶✽✳ P❛r❛ ✸✵✸ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✶✾✳ ❯s❛♥❞♦ ❞✐❢❡r❡♥❝✐❛✐s ❞❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❡ x ♣❛r❛ ♦s q✉❛✐s✿ ✶✳ √ x+1− √ x < 0, 01✳ ✷✳ √ 4 x+1− √ 4 x < 0, 002✳ ✷✵✳ ❯♠ ♣♦♥t♦ ♠♦✈✐♠❡♥t❛✲s❡ ♥❛ ♠❡t❛❞❡ s✉♣❡r✐♦r ❞❛ ❝✉r✈❛ y 2 = x + 1✱ ❞❡ ♠♦❞♦ q✉❡ dx √ dy = 2x + 1✳ ❉❡t❡r♠✐♥❡ q✉❛♥❞♦ x = 4✳ dt dt ✷✶✳ ▼❡❞✐❛♥t❡ ♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦✱ ♠♦str❡ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s✿ hai a − b a−b ≤ Ln ≤ a b b a−b ✷✳ ≤ tan a − tan b ≤ cos2 b  2   3+x s❡ 4 ✷✷✳ ❙❡❥❛ f (x) =   1 s❡ x ✶✳ ✶✳ ✷✳ s❡♥❞♦ 0 < b ≤ a✳ a−b cos2 a π 2 s❡♥❞♦ 0 < b ≤ a < ✳ x≤1 x≥1 ❉❡s❡♥❤❛r ♦ ❣rá✜❝♦ ❞❡ y = f (x) ♣❛r❛ x ∈ [0, 2]✳ ❱❡r✐✜❝❛r s❡ s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s ❞♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦✳ ❙❡ s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s ❞♦ ❚❱▼✱ ❞❡t❡r♠✐♥❛r ❡ss❡s ✈❛❧♦r❡s✳ ✷✸✳ ❙❡❥❛ f : R −→ R ✉♠❛ ❢✉♥çã♦✳ ❙❡ ❞✐③ q✉❡ x = c é ✉♠ ♣♦♥t♦ ✜①♦ ❞❡ f ✱ s❡ f (c) = c✳ ✶✳ ❉❡t❡r♠✐♥❡ ♦s ♣♦♥t♦s ✜①♦s ❞❡ f (x) = x3 − 8x✳ ✷✳ ❱❡r✐✜❝❛r s❡ f (x) = x2 + x + 1 t❡♠ ♣♦♥t♦s ✜①♦s✳ ✸✳ ❙✉♣♦♥❤❛ y = f (x) ∀ x ∈ R t❡♥❤❛ ❞❡r✐✈❛❞❛ f ′ (x) 6= 1 ∀ x ∈ R✳ ▼♦str❡ q✉❡ f ❛❞♠✐t❡ ♥♦ ♠á①✐♠♦ ✉♠ ♣♦♥t♦ ✜①♦✳ ✷✹✳ ▼♦str❡ q✉❡ s❡ ✉♠❛ ❢✉♥çã♦ é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ R ❡ f ′ (x) < 1 ∀ x ∈ R✱ ❡♥tã♦ f t❡♠ ♥♦ ♠á①✐♠♦ ✉♠ ♣♦♥t♦ ✜①♦✳ ✷✺✳ ❆s ✈❛r✐á✈❡✐s x, y, z sã♦ t♦❞❛s ❢✉♥çõ❡s ❞❡ t ❡ ❝✉♠♣r❡♠ ❛ r❡❧❛çã♦✿ x3 − 2xy + y 2 + 2xz + 2xz 2 + 3 = 0✳ ❆❝❤❛r dx dy dz q✉❛♥❞♦ x = 1, y = 2 s❡ =3 ❡ = 4 ♣❛r❛ t♦❞♦ t✳ dt dt dt ✷✻✳ ❊st✐♠❛✲s❡ ❡♠ ✉♠ ♠❡tr♦ ♦ ❧❛❞♦ ❞❡ ✉♠ q✉❛❞r❛❞♦✱ ❝♦♠ ✉♠ ❡rr♦ ♠á①✐♠♦ ❞❡ 0, 005cm✳ ❯s❛♥❞♦ ❞✐❢❡r❡♥❝✐❛✐s ❡st✐♠❡ ♦ ❡rr♦ ♠á①✐♠♦ ♥♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛✳ ◗✉❛✐s sã♦ ♦ ❡rr♦ r❡❧❛t✐✈♦ ❡ ♣❡r❝❡♥t✉❛❧ ❛♣r♦①✐♠❛❞♦❄ ✷✼✳ ❆ ár❡❛ ❧❛t❡r❛❧ ❞❡ ✉♠ ❝♦♥❡ r❡t♦ ❝✐r❝✉❧❛r ❡ ❛❧t✉r❛ h ❡ r❛✐♦ ❞❛ ❜❛s❡ r é ❞❛❞❛ ♣♦r √ AL = πr r2 + h2 ✳ P❛r❛ ❞❡t❡r♠✐♥❛❞♦ ❝♦♥❡✱ r = 6cm ❡ ❛ ♠❡❞✐❞❛ ❞❛ ❛❧t✉r❛ h ❛❝✉s❛ 8cm ❝♦♠ ✉♠ ❡rr♦ ♠á①✐♠♦ ❞❡ 0, 01cm❀ ❞❡t❡r♠✐♥❡ ♦ ❡rr♦ ♠á①✐♠♦ ♥❛ ♠❡❞✐❞❛ ❞❛ ár❡❛ ❧❛t❡r❛❧✳ ◗✉❛❧ ♦ ❡rr♦ ♣❡r❝❡♥t✉❛❧ ❛♣r♦①✐♠❛❞♦❄ ✸✵✹ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ▼✐s❝❡❧â♥❡❛ ✺✲✶  2  x + (a − 3)x − 3a , ✶✳ ❆ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r f (x) = x−3  1 s❡ x 6= 3 s❡ x = 3 t♦❞❛ ❛ r❡t❛ r❡❛❧✳ ✶✳ ◗✉❛❧ ♦ ✈❛❧♦r ❞❡ a❄ ✷✳ é ❞❡r✐✈á✈❡❧ ❡♠ ◗✉❛❧ ♦ ✈❛❧♦r ❞❡ f ′ (3)❄ f (x) − f (2) = 0✳ ◗✉❛✐s ❞❛s s❡❣✉✐♥t❡s x→2 x−2 ✷✳ ❙✉♣♦♥❤❛ q✉❡ f é ✉♠❛ ❢✉♥çã♦ ♣❛r❛ ♦ q✉❛❧ lim ♣r♦♣♦s✐çõ❡s sã♦ ✈❡r❞❛❞❡✐r❛s✱ q✉❛✐s ♣♦❞❡♠ s❡r ✈❡r❞❛❞❡✐r❛s ❡ q✉❛✐s ♥❡❝❡ss❛r✐❛♠❡♥t❡ sã♦ ❢❛❧s❛s❄ ✶✳ f ′ (2) = 2 ✹✳ f ❡s ❝♦♥tí♥✉❛ ❡♥ x = 0 ✷✳ ✸✳ f (2) = 0 ✺✳ lim f (x) = f (2) x→2 f ❡s ❝♦♥tí♥✉❛ ❡♠ x = 2✳ ✸✳ ❙✉♣♦♥❤❛ q✉❡ f ❡ g s❡❥❛♠ ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ♣❛r❛ ❛s q✉❛✐s ✈❡r✐✜❝❛♠✲s❡ ❛s s❡❣✉✐♥✲ t❡s ❝♦♥❞✐çõ❡s✿ ❛✮ f (0) = 0 ❡ g(0) = 1 ❜✮ f ′ (x) = g(x) ❡ g ′ (x) = −f (x)✳ ✶✳ ❙❡❥❛ h(x) = f 2 (x) + g 2 (x)✳ ❈❛❧❝✉❧❛r h′ (x) ❡ ✉t✐❧✐③❛r ❡st❡ r❡s✉❧t❛❞♦ ♣❛r❛ ♠♦str❛r q✉❡ f 2 (x) + g 2 (x) = 1 ♣❛r❛ t♦❞♦ x✳ ✷✳ ❙✉♣♦♥❤❛ q✉❡ F ❡ G sã♦ ♦✉tr♦ ♣❛r ❞❡ ❢✉♥çõ❡s q✉❡ ❝✉♠♣r❡♠ ❛s ❝♦♥❞✐çõ❡s ❛✮ ❡ ❜✮ ❡ s❡❥❛ k(x) = [F (x) − f (x)]2 + [G(x) − g(x)]2 ✳ ❈❛❧❝✉❧❛r k ′ (x) ❡ ✉t✐❧✐③❛r ❡st❡ r❡s✉❧t❛❞♦ ♣❛r❛ ❞❡❞✉③✐r q✉❛❧ é ❛ r❡❧❛çã♦ ❡♥tr❡ f (x) ❡ F (x) ❡ ❡♥tr❡ g(x) ❡ G(x)✿ ✸✳ ▼♦str❡ ✉♠ ♣❛r ❞❡ ❢✉♥çõ❡s q✉❡ ❝✉♠♣r❡♠ ❛s ❝♦♥❞✐çõ❡s ❛✮ ❡ ❜✮✳ P♦❞❡♠ ❡①✐st✐r ♦✉tr❛s✳ ❏✉st✐✜❝❛r s✉❛ r❡s♣♦st❛✳ ✹✳ ❉❡t❡r♠✐♥❡ t♦❞❛s ❛s ❢✉♥çõ❡s f ❞❛ ❢♦r♠❛ f (x) = ax3 + bx2 + cx + d ❝♦♠ a 6= 0 q✉❡ ✈❡r✐✜❝❛♠ f ′ (−1) = f ′ (1) = 0✳ ❆❧❣✉♠❛ ❞❛s ❢✉♥çõ❡s ❞❡t❡r♠✐♥❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡ ✈❡r✐✜❝❛ f (0) = f (1)❄ ❏✉st✐✜❝❛r s✉❛ r❡s♣♦st❛✳ ✺✳ ❙❡❥❛ f : R → R ❢✉♥çã♦ ❞❡r✐✈á✈❡❧❀ ❡ s❡❥❛♠ a ❡ b ❞✉❛s r❛í③❡s ❞❛ ❞❡r✐✈❛❞❛ f ′ (x) ❞❡ ♠♦❞♦ q✉❡ ❡♥tr❡ ❡❧❛s ♥ã♦ ❡①✐st❛ ♦✉tr❛ r❛✐③ ❞❡ f ′ (x)✳ ❉❡t❡r♠✐♥❡ s❡ ♣♦❞❡ ♦❝♦rr❡r ❛❧❣✉♠❛ ❞❛s s❡❣✉✐♥t❡s ♣♦ss✐❜✐❧✐❞❛❞❡s✿ ✶✳ ❊♥tr❡ a ❡ b ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠❛ r❛✐③ ❞❡ f (x)✳ ✷✳ ❊♥tr❡ a ❡ b ❡①✐st❡ só ✉♠❛ r❛✐③ ❞❡ f (x)✳ ✸✳ ❊♥tr❡ a ❡ b ❡①✐st❡♠ ❞♦✐s ♦✉ ♠❛✐s r❛í③❡s ❞❡ f (x)✳ ✸✵✺ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✻✳ ▼♦str❡ q✉❡ ❛ ❡q✉❛çã♦ x + xsenx − x2 = 0 t❡♠ ❡①❛t❛♠❡♥t❡ ❞✉❛s r❛í③❡s r❡❛✐s✳ ✼✳ ❯s❛r q✉❡ y = et cos t, E= dy = et cos t − et sent, dt dy d2 y − 2 + 2t ✳ ▲♦❣♦ s✐♠♣❧✐✜❝❛r✳ dt2 dt d2 y = −2et sent ♣❛r❛ s✉❜st✐t✉✐r ❡♠✿ dt2 ✽✳ ❉❡t❡r♠✐♥❡✿ ✶✳ ✸✳ ✺✳ d3 y dx3 d2 y dx2 s❡♥❞♦ y = sen(3x) ✷✳ f ′′′ (0) s❡♥❞♦ f (x) = senx cos x s❡♥❞♦ y = Ln(x2 − 3x) ✹✳ f ′′ (x) s❡♥❞♦ f (x) = ex+x 2 ❚♦❞❛s ❛s ❞❡r✐✈❛❞❛s ❞❛ ❢✉♥çã♦ f ❞❡✜♥✐❞❛ ♣♦r f (x) = 8x4 + 5x3 − x2 + 7 ✻✳ d3 y dx3 s❡♥❞♦ y = 2senx + 3 cos x − x3 ✾✳ ❙✉♣♦♥❞♦ q✉❡ ❛s ❢✉♥çõ❡s ❛❜❛✐①♦ ❞❡✜♥❡♠ ✐♠♣❧✐❝✐t❛♠❡♥t❡ y ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❞❡ x✱ ❞❡t❡r♠✐♥❡ ❛ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛ y ′ ✶✳ ✸✳ ✺✳ x4 + 2y 3 − 4xy = 0 x2 y 2 + 8x = y − 1 y 2 + cos(2xy) = y ✹✳ (x + y)2 − (x − y)2 = x3 − y 3 x2 y + sen2y = π ✻✳ y 2 + x2 = xy ✷✳ ✶✵✳ ❯♠❛ ❢r❡♥t❡ ❢r✐❛ ❛♣r♦①✐♠❛✲s❡ ❞❛ ❯❋❚✳ ❆ t❡♠♣❡r❛t✉r❛ é z ❣r❛✉s t ❤♦r❛s ❛ ♠❡✐❛ ♥♦✐t❡ ❡ z = 0, 1(400 − 40t + t2 ) 0 ≤ t ≤ 12✳ ✭❛✮ ❆❝❤❡ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ♠é❞✐❛ ❞❡ z ❡♠ r❡❧❛çã♦ ❛ t ❡♥tr❡ 5 ❡ 6 ❤♦r❛s ❞❛ ♠❛♥❤ã❀ ✭❜✮ ❆❝❤❡ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❡ z ❡♠ r❡❧❛çã♦ ❛ t às 5 ❤♦r❛s ❞❛ ♠❛♥❤❛✳ ✶✶✳ ❙❡ A cm2 é ❛ ár❡❛ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❡ s cm é ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ s❡✉ ❧❛❞♦✱ ❛❝❤❡ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ♠é❞✐❛ ❞❡ A ❡♠ r❡❧❛çã♦ ❛ s q✉❛♥❞♦ s ♠✉❞❛ ❞❡✿ ✭❛✮ 4, 00 ❛ 4, 60❀ ✭❜✮ 4, 00 ❛ 4, 30 ❀ ✭❝✮ 4, 00 ❛ 4, 10 ❀ ✭❞✮ ◗✉❛❧ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ✐♥st❛♥tâ♥❡❛ ❞❡ A ❡♠ r❡❧❛çã♦ ❛ s q✉❛♥❞♦ s = 4, 00❄ ✶✷✳ ❯♠ t❛♥q✉❡ ❞❡ á❣✉❛ t❡♠ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❝♦♥❡ ❝✐r❝✉❧❛r r❡t♦ ✐♥✈❡rt✐❞♦✱ ❞❡ ❛❧t✉r❛ 12 ♣és ❡ r❛✐♦ ❞❛ ❜❛s❡ 6 ♣és✳ ❇♦♠❜❡✐❛✲s❡ á❣✉❛ ❛ r❛③ã♦ ❞❡ 10gal ♣♦r ♠✐♥✉t♦✳ ❉❡t❡r♠✐♥❛r ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❛ r❛③ã♦ ❝♦♠ ❛ q✉❛❧ ♦ ♥í✈❡❧ ❞❡ á❣✉❛ s♦❜❡ ❛♦ t❛♥q✉❡ q✉❛♥❞♦ ❛ ♣r♦❢✉♥❞✐❞❛❞❡ é 3 ♣és ✭1 gal ≈ 0.1337♣és 3 ✮✳ ✶✸✳ ❯♠❛ ❡♠♣r❡s❛ ✐♥tr♦❞✉③ ✉♠ ♥♦✈♦ ♣r♦❞✉t♦ ♥♦ ♠❡r❝❛❞♦ ❝✉❥❛s ✈❡♥❞❛s sã♦ ❞❛❞❛s ♣♦r✿ 200(2t + 1) ♦♥❞❡ S(t) é ❛ q✉❛♥t✐❞❛❞❡ ✈❡♥❞✐❞❛ ❞✉r❛♥t❡ ♦s t ♣r✐♠❡✐r♦s ♠❡s❡s✳ t+2 ✭❛✮ ❊♥❝♦♥tr❡ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ♠é❞✐❛ ❞❡ S(t) ❛♦ ❧♦♥❣♦ ❞♦ ♣r✐♠❡✐r♦ ❛♥♦✳ ✭❜✮ ❊♠ q✉❡ ♠ês S ′ (t) é ✐❣✉❛❧ à t❛①❛ ❞❡ ✈❛r✐❛çã♦ ♠é❞✐❛ ❞✉r❛♥t❡ ♦ ♣r✐♠❡✐r♦ ❛♥♦❄ S(t) = ✶✹✳ ❆♦ ❡sq✉❡♥t❛r ✉♠ ❞✐s❝♦ ❞❡ ♠❡t❛❧✱ s❡✉ ❞✐â♠❡tr♦ ✈❛r✐❛ ❛ r❛③ã♦ ❞❡ 0.01cm/min✳ ◗✉❛♥❞♦ ♦ ❞✐â♠❡tr♦ ❡st❛ ❝♦♠ 5 ♠❡tr♦s✱ ❝♦♠ q✉❡ r❛③ã♦ ❡st❛ ✈❛r✐❛♥❞♦ ❛ ár❡❛ ❞❡ ✉♠❛ ❞❡ s✉❛s ❢❛❝❡s❄ ✸✵✻ 09/02/2021 ❈❛♣ít✉❧♦ ✻ ❆P▲■❈❆➬Õ❊❙ ❉❆❙ ❉❊❘■❱❆❉❆❙ ●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ▲❡✐❜♥✐t③ ♥❛s❝❡✉ ♥♦ ❡♠ ▲❡✐♣③✐❣ ✭❆❧❡♠❛♥❤❛✮✱ ❡ ❢❛❧❡❝❡✉ ❡♠ ❊♠ 1661✱ q✉❛♥❞♦ t✐♥❤❛ 15 14 1 ❞❡ ❥✉❧❤♦ ❞❡ 1646✱ 1716✳ ❞❡ ♥♦✈❡♠❜r♦ ❞❡ ❛♥♦s✱ ✐♥❣r❡ss♦✉ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ▲❡✐♣③✐❣ ❡✱ ❛♦s ❞❡③❡ss❡t❡✱ ❡♠ 1663✱ ❥á ❤❛✈✐❛ ❛❞q✉✐r✐❞♦ ♦ s❡✉ ❞✐♣❧♦♠❛ ❞❡ ❜❛❝❤❛r❡❧✳ ❊st✉❞♦✉ ❚❡♦❧♦❣✐❛✱ ❉✐r❡✐t♦✱ ❋✐❧♦s♦✜❛ ❡ ▼❛✲ t❡♠át✐❝❛✱ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡✳ P❛r❛ ♠✉✐t♦s ❤✐st♦r✐❛❞♦r❡s✱ ▲❡✐❜♥✐t③ é t✐❞♦ ❝♦♠♦ ♦ ú❧t✐♠♦ ❡r✉❞✐t♦ q✉❡ ♣♦ss✉í❛ ❝♦♥❤❡❝✐♠❡♥t♦ ✉♥✐✈❡rs❛❧✳ ❋♦✐ ✉♠ ❞♦s ♣r✐✲ ♠❡✐r♦s✱ ❞❡♣♦✐s ❞❡ P❛s❝❛❧✱ ❛ ✐♥✈❡♥t❛r ✉♠❛ ♠áq✉✐♥❛ ❞❡ ❝❛❧❝✉❧❛r✳ ●✳ ▲❡✐❜♥✐t③ ■♠❛❣✐♥♦✉ ♠áq✉✐♥❛s ❞❡ ✈❛♣♦r✱ ❡st✉❞♦✉ ✜❧♦s♦✜❛ ❝❤✐♥❡s❛ ❡ t❡♥t♦✉ ♣r♦♠♦✈❡r ❛ ✉♥✐❞❛❞❡ ❞❛ ❆❧❡♠❛♥❤❛✳ ❆♦s 20 ❛♥♦s ❞❡ ✐❞❛❞❡✱ ❥á ❡st❛✈❛ ♣r❡♣❛r❛❞♦ ♣❛r❛ r❡❝❡❜❡r ♦ tít✉❧♦ ❞❡ ❞♦✉t♦r ❡♠ ❞✐r❡✐t♦✳ ❊st❡ ❧❤❡ ❢♦✐ r❡❝✉s❛❞♦ ♣♦r s❡r ❡❧❡ ♠✉✐t♦ ❥♦✈❡♠✳ ❉❡✐①♦✉ ❡♥tã♦ ▲❡✐♣③✐❣ ❡ ❢♦✐ r❡❝❡❜❡r ♦ s❡✉ tít✉❧♦ ❞❡ ❞♦✉t♦r ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❆❧t❞♦r❢✱ ❡♠ ◆✉r❡♠❜❡r❣✳ ❆ ♣❛rt✐r ❞❛í✱ ▲❡✐❜♥✐t③ ❡♥tr♦✉ ♣❛r❛ ❛ ✈✐❞❛ ❞✐♣❧♦♠át✐❝❛ ♥❛ ❝♦rt❡ ❞❡ ❍❛♥ô✈❡r✱ ❛♦ s❡r✈✐ç♦ ❞♦s ❞✉q✉❡s✱ ✉♠ ❞♦s q✉❛✐s s❡ t♦r♥♦✉ r❡✐ ❞❡ ■♥❣❧❛t❡rr❛✱ s♦❜ ♦ ♥♦♠❡ ❞❡ ❏♦r❣❡ I✳ ❈♦♠♦ r❡♣r❡s❡♥t❛♥t❡ ❣♦✈❡r♥❛♠❡♥t❛❧ ✐♥✢✉❡♥t❡✱ ❡❧❡ t❡✈❡ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ✈✐❛❥❛r ♠✉✐t♦ ❞✉r❛♥t❡ t♦❞❛ ❛ s✉❛ ✈✐❞❛✳ ❊♠ 1672✱ ❢♦✐ ♣❛r❛ P❛r✐s ♦♥❞❡ ❝♦♥❤❡❝❡✉ ❍✉②❣❡♥s✱ q✉❡♠ ❧❤❡ s✉❣❡r✐✉ ❛ ❧❡✐t✉r❛ ❞♦s tr❛t❛❞♦s ❞❡ 1658✱ ❞❡ ❇❧❛✐s❡ P❛s❝❛❧✱ s❡ q✉✐s❡ss❡ t♦r♥❛r✲s❡ ✉♠ ♠❛t❡♠át✐❝♦✳ ❊♠ 1673✱ ✈✐s✐t♦✉ ▲♦♥❞r❡s✱ ♦♥❞❡ ❛❞q✉✐r✐✉ ✉♠❛ ❝ó♣✐❛ ❞♦ ✏▲❡❝t✐♦♥❡s ●❡♦♠❡tr✐❝❛❡✑✱ ❞❡ ■s❛❛❝ ❇❛rr♦✇✱ ❡ t♦r♥♦✉✲s❡ ♠❡♠❜r♦ ❞❛ ❘♦②❛❧ ❙♦❝✐❡t②✳ ❋♦✐ ❞❡✈✐❞♦ ❛ ❡ss❛ ✈✐s✐t❛ ❛ ▲♦♥❞r❡s✱ ♦♥❞❡ ❛♣❛✲ r❡❝❡r❛♠ r✉♠♦r❡s q✉❡ ▲❡✐❜♥✐t③ t❛❧✈❡③ t✐✈❡ss❡ ✈✐st♦ ♦ tr❛❜❛❧❤♦ ❞❡ ◆❡✇t♦♥✱ q✉❡ ♣♦r s✉❛ ✈❡③ ♦ t❡r✐❛ ✐♥✢✉❡♥❝✐❛❞♦ ♥❛ ❞❡s❝♦❜❡rt❛ ❞♦ ❈á❧❝✉❧♦✱ ❝♦❧♦❝❛♥❞♦ ❡♠ ❞ú✈✐❞❛ ❛ ❧❡❣✐t✐♠✐❞❛❞❡ ❞❡ s✉❛s ❞❡s❝♦❜❡rt❛s r❡❧❛❝✐♦♥❛❞❛s ❛♦ ❛ss✉♥t♦✳ ❆ ♣r♦❝✉r❛ ❞❡ ✉♠ ♠ét♦❞♦ ✉♥✐✈❡rs❛❧✱ ❛tr❛✈és ❞♦ q✉❛❧ ♣✉❞❡ss❡ ♦❜t❡r ❝♦♥❤❡❝✐♠❡♥t♦s✱ ❢❛③❡r ✐♥✲ ✈❡♥çõ❡s ❡ ❝♦♠♣r❡❡♥❞❡r ❛ ✉♥✐❞❛❞❡ ❡ss❡♥❝✐❛❧ ❞♦ ✉♥✐✈❡rs♦✱ ❢♦✐ ♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❛ s✉❛ ✈✐❞❛✳ ❆ ✏❙❝✐❡♥t✐❛ ●❡♥❡r❛❧✐s✑ q✉❡ q✉❡r✐❛ ❝♦♥str✉✐r t✐♥❤❛ ♠✉✐t♦s ❛s♣❡❝t♦s✱ ❡ ✈ár✐♦s ❞❡❧❡s ❧❡✈❛r❛♠ ▲❡✐❜♥✐t③ ❛ ❞❡s❝♦❜❡rt❛s ♥❛ ♠❛t❡♠át✐❝❛✳ ❆ ♣r♦❝✉r❛ ❞❡ ✉♠❛ ✏❝❤❛r❛❝t❡r✐st✐❝❛ ❣❡♥❡r❛❧✑ ❧❡✈♦✉✲♦ ❛ ♣❡r♠✉t❛çõ❡s✱ ❝♦♠❜✐♥❛çõ❡s ❡ à ❧ó❣✐❝❛ s✐♠❜ó❧✐❝❛❀ ❛ ♣r♦❝✉r❛ ❞❡ ✉♠❛ ✏❧í♥❣✉❛ ✉♥✐✈❡rs❛❧✐s✑✱ ♥❛ q✉❛❧ t♦❞♦s ♦s ❡rr♦s ❞❡ r❛❝✐♦❝í♥✐♦ ♣✉❞❡ss❡♠ ❛♣❛r❡❝❡r ❝♦♠♦ ❡rr♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s✱ ♦ ❧❡✈♦✉ ♥ã♦ só à ❧ó❣✐❝❛ s✐♠❜ó❧✐❝❛✱ ♠❛s t❛♠❜é♠ ❛ ♠✉✐t❛s ✐♥♦✈❛çõ❡s ♥❛ ♥♦t❛çã♦ ♠❛t❡♠át✐❝❛✳ ▲❡✐❜♥✐t③ ❢♦✐ ✉♠ ❞♦s ♠❛✐♦r❡s ✐♥✈❡♥t♦r❡s ❞❡ sí♠❜♦❧♦s ♠❛t❡♠át✐❝♦s✳ ✸✵✼ ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✻✳✶ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❱❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛✳ ❆❝❡❧❡r❛çã♦ ✐♥st❛♥tâ♥❡❛✳ ❯♠❛ ❞❛s ✉t✐❧✐❞❛❞❡s ❞❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ é ❛ ❞❡s❝r✐çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠ ♦❜❥❡t♦ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ r❡t❛❀ t❛❧ ♠♦✈✐♠❡♥t♦ é ❝❤❛♠❛❞♦ ♠♦✈✐♠❡♥t♦ r❡t✐❧í♥❡♦✳ ❙❡ ✉t✐❧✐③❛♠♦s ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s✱ ❝♦♥✈❡♥❝✐♦♥❛❧♠❡♥t❡ s❡ ♦ ♦❜❥❡t♦ s❡ ♠♦✈✐♠❡♥t❛ ♣❛r❛ ❛ ❞✐r❡✐t❛ ✭♦✉ ♣❛r❛ ❝✐♠❛✮✱ s✉❛ ❞✐r❡çã♦ é ♣♦s✐t✐✈❛ ❛♦ ♣❛ss♦ q✉❡✱ s❡ ♦ ♠♦✈✐♠❡♥t♦ é ♣❛r❛ ❛ ❡sq✉❡r❞❛ ✭♦✉ ♣❛r❛ ❜❛✐①♦✮✱ s✉❛ ❞✐r❡çã♦ é ♥❡❣❛t✐✈❛✳ ◗✉❛♥❞♦ ✉♠❛ ❢✉♥çã♦ S(t) ❞♦ t❡♠♣♦✱ ❡❧❛ é ❝❤❛♠❛❞❛ ❞á ❛ ♣♦s✐çã♦ ✭r❡❧❛t✐✈❛ à ♦r✐❣❡♠✮ ❞❡ ✉♠ ♦❜❥❡t♦ ❝♦♠♦ ❢✉♥çã♦ ❢✉♥çã♦ ♣♦s✐çã♦✳ ❙❡✱ ❞✉r❛♥t❡ ✉♠ ♣❡rí♦❞♦ ∆t ❞❡ t❡♠♣♦✱ ♦ ♦❜❥❡t♦ ∆S(t) = S(t + ∆t) − S(t) ✐st♦ é ❛ ✈❛r✐❛çã♦ ❞❛ ❞✐stâ♥❝✐❛✱ ❡♥tã♦ ❛ t❛①❛ ❞❡ ∆S(t) ❀ ❡st❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ♠é❞✐❛ é ❝❤❛♠❛❞❛ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛✳ ♠é❞✐❛ é✿ ∆t s❡ ❞❡s❧♦❝❛✿ ✈❛r✐❛çã♦ ❉❡✜♥✐çã♦ ✻✳✶✳ ❙❡ S(t) ❞á ❛ ♣♦s✐çã♦ ♥♦ ✐♥st❛♥t❡ t ❞❡ ✉♠ ♦❜❥❡t♦ s❡ ♠♦✈❡♥❞♦ ❡♠ ❧✐♥❤❛ r❡t❛✱ ❡♥tã♦ ❛ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛ ❞♦ ♦❜❥❡t♦ ♥♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ❱❡❧♦❝✐❞❛❞❡ ♠é❞✐❛ = [t, t + ∆t] é ❞❛❞♦ ♣♦r✿ S(t + ∆t) − S(t) ∆S(t) = ∆t ∆t ❊①❡♠♣❧♦ ✻✳✶✳ 40m✱ s✉❛ ❛❧t✉r❛ h ♥♦ ✐♥st❛♥t❡ t é ❞❛❞❛ é ❞❛❞❛ ♣❡❧❛ ❢✉♥çã♦ S(t) = −4, 9t + 40✱ ♦♥❞❡ S(t) é ♠❡❞✐❞♦ ❡♠ ♠❡tr♦s ❡ t ❡♠ s❡❣✉♥❞♦s✳ ❉❡t❡r♠✐♥❡ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ♠é❞✐❛ ♥♦s ✐♥t❡r✈❛❧♦s✿ ❛✮ [1, 1.1]; ❜✮ [1, 1.5]; ❝✮ [1, 2]✳ ❯♠ ♦❜❥❡t♦ ❝❛✐ ❞❡ ✉♠❛ ❛❧t✉r❛ ❞❡ 2 ❙♦❧✉çã♦✳ h = S(t)✳ ❯s❛♥❞♦ ❛ ❡q✉❛çã♦ S(t) t = 1, t = 1.4 ❡ t = 2 s❡❣✉♥❞♦s ♥❛ t❛❜❡❧❛✿ ❚❡♠♦s ❛ ❛❧t✉r❛ ✐♥st❛♥t❡s✿ ❛✮ P❛r❛ ♦ ✐♥t❡r✈❛❧♦ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ❛s ❛❧t✉r❛s ♥♦s t 1 1, 1 1, 5 2 S(t) 35, 1 34, 1 29 20, 4 [1, 1.1] ♦ ♦❜❥❡t♦ ❝❛✐ ❞❡ ✉♠❛ ❛❧t✉r❛ ❞❡ 35, 1m ♣❛r❛ 34, 1m ❡ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ♠é❞✐❛ é✿ S(t + ∆t) − S(t) 34, 1 − 35, 1 ∆S(t) = = = −10 m/s ∆t ∆t 1, 1 − 1 ❜✮ P❛r❛ ♦ ✐♥t❡r✈❛❧♦ [1, 1.5] ♦ ♦❜❥❡t♦ ❝❛✐ ❞❡ ✉♠❛ ❛❧t✉r❛ ❞❡ 35, 1m ♣❛r❛ 29m ❡ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ♠é❞✐❛ é✿ ∆S(t) S(t + δt) − S(t) 29 − 35, 1 = = = −12, 2 m/s ∆t ∆t 1, 5 − 1 ✸✵✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❝✮ P❛r❛ ♦ ✐♥t❡r✈❛❧♦ [1, 2] ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ♦ ♦❜❥❡t♦ ❝❛✐ ❞❡ ✉♠❛ ❛❧t✉r❛ ❞❡ 35, 1m ♣❛r❛ 20, 4m ❡ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ♠é❞✐❛ é✿ S(t + δt) − S(t) 20, 4 − 35, 1 ∆S(t) = = = −14, 7 m/s ∆t ∆t 2−1 ❖❜s❡r✈❡✱ ❛ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛ ♥❡st❡ ❡①❡♠♣❧♦ é ♥❡❣❛t✐✈❛✱ ❧♦❣♦ ♦ ♦❜❥❡t♦ s❡ ♠♦✈✐♠❡♥t❛ ♣❛r❛ ❛❜❛✐①♦✳ ✻✳✶✳✶ ❱❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛ ❉❡✜♥✐çã♦ ✻✳✷✳ ❙❡ S(t) ❱❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛✳ ❞❡t❡r♠✐♥❛ ❛ ♣♦s✐çã♦ ♥♦ ✐♥st❛♥t❡ ❡♥tã♦ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ♦❜❥❡t♦ ♥♦ t ❞❡ ✉♠ ♦❜❥❡t♦ ✐♥st❛♥t❡ t é ❞❛❞❛ ♣♦r✿ s❡ ♠♦✈❡♥❞♦ ❡♠ ❧✐♥❤❛ r❡t❛✱ S(t + ∆t) − S(t) ∆t→0 ∆t V ′ (t) = lim ✭✻✳✶✮ ❊①❡♠♣❧♦ ✻✳✷✳ ❉❡t❡r♠✐♥❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛ q✉❛♥❞♦ ❢✉♥çã♦ ❞❡ ♣♦s✐çã♦ é ❞❛❞❛ ♣♦r ♠❡tr♦s✳ t = 2✱ ❞❡ ✉♠ ♦❜❥❡t♦ ❡♠ q✉❡❞❛ ❧✐✈r❡ ❝✉❥❛ S(t) = 200 − 32t2 ♦♥❞❡ V ′ (t) = −64t❀ V ′ (2) = −(64)(2) = −128m/s✳ t é ❞❛❞♦ ❡♠ s❡❣✉♥❞♦s ❡ S(t) ❡♠ ❙♦❧✉çã♦✳ P❡❧❛ ❡①♣r❡ssã♦ ✭✻✳✶✮ t❡♠♦s ❊①❡♠♣❧♦ ✻✳✸✳ ❧♦❣♦ ❯♠❛ ♣❛rtí❝✉❧❛ s❡ ♠♦✈✐♠❡♥t❛ ❡♠ ❧✐♥❤❛ r❡t❛ ❤♦r✐③♦♥t❛❧ ✭♣♦s✐t✐✈❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✮ s❡❣✉♥❞♦ ❛ r❡❧❛çã♦ s = t3 − 3t2 − 9t + 5✳ ❊♠ q✉❡ ✐♥t❡r✈❛❧♦s ❞❡ t❡♠♣♦ ❛ ♣❛rtí❝✉❧❛ ♠♦✈✐♠❡♥t❛✲s❡ ♣❛r❛ ❛ ❞✐r❡✐t❛❀ ❡ ❡♠ q✉❛✐s ♣❛r❛ ❛ ❡sq✉❡r❞❛❄ ❙♦❧✉çã♦✳ ❆ ♣❛rtí❝✉❧❛ ♠♦✈✐♠❡♥t❛✲s❡ ♣❛r❛ ❛ ❞✐r❡✐t❛ q✉❛♥❞♦ ❛ ✈❡❧♦❝✐❞❛❞❡ é ♣♦s✐t✐✈❛❀ ❡ ♣❛r❛ ❛ ❡sq✉❡r❞❛ q✉❛♥❞♦ ❛ ✈❡❧♦❝✐❞❛❞❡ é ♥❡❣❛t✐✈❛✳ ❆ ✈❡❧♦❝✐❞❛❞❡ é ❞❛❞❛ ♣❡❧❛ ❢✉♥çã♦ t❛❜❡❧❛ ♣❛r❛ ❛ ❢✉♥çã♦ v(t)✿ t v(t) ❙❡ t < −1✱ −2 + s′ (t) = v(t) = 3t2 − 6t − 9✳ −1 0 ❈♦♥str✉í♠♦s ❛ s❡❣✉✐♥t❡ 1 3 4 − 0 + ❛ ✈❡❧♦❝✐❞❛❞❡ é ♣♦s✐t✐✈❛ ❡ ♦ ♠♦✈✐♠❡♥t♦ é ♣❛r❛ ❛ ❞✐r❡✐t❛❀ s❡ ✈❡❧♦❝✐❞❛❞❡ é ♥❡❣❛t✐✈❛ ❡ ♦ ♠♦✈✐♠❡♥t♦ é ♣❛r❛ ❛ ❡sq✉❡r❞❛❀ s❡ t > 3✱ −1 < t < 3✱ ❛ ❛ ✈❡❧♦❝✐❞❛❞❡ é ♣♦s✐t✐✈❛ ❡ ♦ ♠♦✈✐♠❡♥t♦ é ♣❛r❛ ❛ ❞✐r❡✐t❛✳ ✸✵✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❖ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ❛ ❞✐r❡✐t❛ ❡ ♦ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ❛ ❡sq✉❡r❞❛✱ ❡♥tã♦ s❡♣❛r❛❞♦s ♣♦r ✐♥st❛♥t❡s ❞❡ ✈❡❧♦❝✐❞❛❞❡ ♥✉❧❛✳ ❊①❡♠♣❧♦ ✻✳✹✳ 20m ❞❡ ❛❧t✉r❛ ❡ 3 ✉♠❛ ❜❛s❡ ❝♦♠ 10m ❞❡ r❛✐♦✳ ❆ á❣✉❛ ✏✢✉✐✑ ♥♦ t❛♥q✉❡ ❛ ✉♠❛ t❛①❛ ❞❡ 5m /s✳ ❈♦♠ q✉❡ ✈❡❧♦❝✐❞❛❞❡ ♦ ♥í✈❡❧ ❞❛ á❣✉❛ ❡st❛rá s❡ ❡❧❡✈❛♥❞♦ q✉❛♥❞♦ s✉❛ ♣r♦❢✉♥❞✐❞❛❞❡ ❢♦r ❞❡ 8m❄✳ ❯♠ t❛♥q✉❡ t❡♠ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❝♦♥❡ ✐♥✈❡rt✐❞♦ ✭❋✐❣✉r❛ ✭✻✳✶✮✮ ❝♦♠ ❙♦❧✉çã♦✳ ❙❡❥❛♠ h ❛ ♣r♦❢✉♥❞✐❞❛❞❡✱ r ♦ r❛✐♦ ❞❛ ❜❛s❡ ❞♦ ❝♦♥❡ V ♦ ✈♦❧✉♠❡ ❞❛ á❣✉❛ ♥♦ ✐♥st❛♥t❡ t❀ q✉❡r❡♠♦s ❛❝❤❛r dh dV s❛❜❡♥❞♦ q✉❡ é 5m3 /s✳ dt dt 1 2 πr h ♦♥❞❡ ❖ ✈♦❧✉♠❡ ❞❛ á❣✉❛ é ❞❛❞♦ ♣♦r V = 3 t♦❞❛s ❛s ♠❡❞✐❞❛s ❞❡♣❡♥❞❡♠ ❞♦ t❡♠♣♦ t❀ ♣♦r s❡♠❡✲ r 10 10 ❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s = ♦✉ r = h✱ ❧♦❣♦✿ h 20 20 1 1
2 1 V = π h h = πh3 ❡✱ ✉t✐❧✐③❛♥❞♦ ❞✐❢❡r❡♥❝✐❛✐s 3 2 12 1 dV = πh2 dh✳ 4 ❉✐✈✐❞✐♥❞♦ ❡st❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ♣♦r dt✱ ♦❜té♠✲s❡ dV 1 dh = πh2 ❡♥tã♦ q✉❛♥❞♦ h = 8m s❡❣✉❡ 5 m3 /s = dt 4 dt dh 20 5 1 π (8m)2 ⇒ = m/s = m/s = 4 dt 64π 16π 0, 0995 m/s✳ ❡ ❋✐❣✉r❛ ✻✳✶✿ P♦rt❛♥t♦✱ s♦❜❡ ♦ ♥í✈❡❧ ❞❛ á❣✉❛ ♥♦ ✐♥st❛♥t❡ ❡♠ q✉❡ ❛ ♣r♦❢✉♥❞✐❞❛❞❡ ❞❛ á❣✉❛ é ❞❡ ❝♦♠ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ✻✳✶✳✷ 8m 0, 0995m/s ❆❝❡❧❡r❛çã♦ ✐♥st❛♥tâ♥❡❛ ❆ ❛❝❡❧❡r❛çã♦ é ✉♠❛ ♠❡❞✐❞❛ ❞❛ ✈❛r✐❛çã♦ ❞❛ ✈❡❧♦❝✐❞❛❞❡✳ ◗✉❛♥❞♦ ✉♠❛ ♣❛rtí❝✉❧❛ t❡♠ ♠♦✈✐♠❡♥t♦ r❡t✐❧í♥❡♦ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ❝♦♥st❛♥t❡✱ ❛ ❛❝❡❧❡r❛çã♦ é ♥✉❧❛ ✭③❡r♦✮✳ P♦r ❡①❡♠♣❧♦✱ ❡♠ ✉♠❛ ❝♦♠♣❡t✐çã♦ ❞❛ ❋ór♠✉❧❛ ✶✱ ♦s ✈❡í❝✉❧♦s ♣❛ss❛♠ ♣❡❧♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ✉♥✐❢♦r♠❡✱ ❞✐❣❛♠♦s ❞❡ 300 km/h✱ 200km/h✳ ❖✐t♦ s❡❣✉♥❞♦s ❛♣ós ✉♠ ❞❡ ❡❧❡s ❡stá ❝♦rr❡♥❞♦ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ❛ ❛❝❡❧❡r❛çã♦ ♠é❞✐❛ ❞❡ss❡ ❛✉t♦ é✿ 300 − 200 = 12, 5 (km/h)/seg 8 ❆s ✉♥✐❞❛❞❡s ♣❛r❡❝❡♠ ❜❛st❛♥t❡ ❡str❛♥❤❛s ❞❡s❞❡ q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❡stá ❡①♣r❡ss❛ ❡♠ km/h ❡ ♦ t❡♠♣♦ ❡♠ segundos✱ tr❛♥s❢♦r♠❛♥❞♦ ✸✶✵ km/h ♣❛r❛ m/seg ✱ t❡♠♦s q✉❡ ❛ ❛❝❡❧❡r❛çã♦ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ♠é❞✐❛ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❞❡ss❡ ❛✉t♦ é✿ 300 − 200 = 12, 5 (km/h)/s = 12, 5 (1000m/3600seg)/seg = 3, 472 m/seg 2 8 ❉❡✜♥✐çã♦ ✻✳✸✳ ❙❡ S(t) ❆❝❡❧❡r❛çã♦ ✐♥st❛♥tâ♥❡❛✳ ❞á ❛ ♣♦s✐çã♦ ♥♦ ✐♥st❛♥t❡ t ❞❡ ✉♠ ♦❜❥❡t♦ s❡ ♠♦✈❡♥❞♦ ❡♠ ❧✐♥❤❛ r❡t❛✱ ❡♥tã♦ ❛ ❛❝❡❧❡r❛çã♦ ✐♥st❛♥tâ♥❡❛ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❛ ❛❝❡❧❡r❛çã♦ t é ❞❛❞❛ ♣♦r✿ ′ a(t) = v (t)✱ ♦♥❞❡ v(t) a(t) ❞♦ ♦❜❥❡t♦ ♥♦ ✐♥st❛♥t❡ é ❛ ✈❡❧♦❝✐❞❛❞❡ ♥♦ ✐♥st❛♥t❡ t✳ ❊①❡♠♣❧♦ ✻✳✺✳ 80km/h ❡ ♦ ❈♦♠ q✉❡ ✈❡❧♦❝✐❞❛❞❡ ❛✉♠❡♥t❛ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❛♠❜♦s 3hs ❉♦✐s ❝❛rr♦s ♣❛rt❡♠ ❛♦ ♠❡s♠♦ t❡♠♣♦ ❞❡ ✉♠ ♣♦♥t♦ ♦✉tr♦ ♣❛r❛ ♦ ♥♦rt❡ ❛ 45km/h✳ A✱ ✉♠ ♣❛r❛ ♦ ♦❡st❡ ❛ ❞❡♣♦✐s ❞❛ s❛í❞❛❄ ❙♦❧✉çã♦✳ 80t ✻ ❙✉♣♦♥❤❛ t❡♥❤❛♠ ♣❡r❝♦rr✐❞♦ t ❤♦r❛s✱ s❡❣✉♥❞♦ ❛ ❋✐✲ ❣✉r❛ ✭✻✳✷✮ ❡ ❛♣❧✐❝❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ t❡♠♦s p ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❡❧❡s✿ d(t) = (80t)2 + (45t)2 = √ 5t 337✳ ❆ ✈❡❧♦❝✐❞❛❞❡ ❝♦♠ q✉❡ ❛✉♠❡♥t❛ ❛ ❞✐stâ♥❝✐❛ √ ❡♥tr❡ ❡❧❡s é d′ (t) = 5 337km/h = 91, 78km/h✳ ✓ ✓ ✓ ✛45t ✓ ✓ ✓d(t) ✓ ✓ ❋✐❣✉r❛ ✻✳✷✿ ❊①❡♠♣❧♦ ✻✳✻✳ ❉❡t❡r♠✐♥❡ ❛ ❛❝❡❧❡r❛çã♦ ❞❡ ✉♠ ♦❜❥❡t♦ ❡♠ q✉❡❞❛ ❧✐✲ ✈r❡ ❝✉❥❛ ❢✉♥çã♦ ♣♦s✐çã♦ é✿ ❙♦❧✉çã♦✳ S(t) = −4, 9t2 + 40✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛✱ s❛❜❡♠♦s q✉❡ v(t) = −9, 8t❀ ♣♦rt❛♥t♦ ❛ ❛❝❡✲ ❧❡r❛çã♦ é a(t) = −9, 8m/s2 ✳ ❊st❛ ❛❝❡❧❡r❛çã♦ ❞❡♥♦t❛❞❛ ♣♦r g é ❞❡✈✐❞❛ à ❣r❛✈✐❞❛❞❡❀ s❡✉ ✈❛❧♦r ❡①❛t♦ ❞❡♣❡♥❞❡ ❞♦ ❧✉❣❛r ❞❛ ♣♦s✐çã♦ ❞♦ ❡①♣❡r✐♠❡♥t♦✳ ❊♠ ❣❡r❛❧ ❛ ♣♦s✐çã♦ ❞❡ ✉♠ ♦❜❥❡t♦ ❡♠ q✉❡❞❛ ❧✐✈r❡ ✭❞❡s♣r❡③❛♥❞♦ ❛ r❡s✐stê♥❝✐❛ ❞♦ ❛r✮ s♦❜ ❛ ✐♥✢✉ê♥❝✐❛ ❞❛ ❣r❛✈✐❞❛❞❡ é S(t) = gt2 +v0 t+s0 ✱ ♦♥❞❡ g é ❛ ❣r❛✈✐❞❛❞❡ ❞❛ t❡rr❛✱ v0 é ❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥✐❝✐❛❧ ❡ s0 é ❛ ❛❧t✉r❛ ✐♥✐❝✐❛❧✳ ❊①❡♠♣❧♦ ✻✳✼✳ r ❡♠ % ❞❡ ✉♠ ❛❧✉♥♦ ♥❡st❡ ❡①❛♠❡ ❞❡ ❞✉❛s ❤♦r❛s s❡❥❛ r(t) = 300t(2−t)✳ P❡❞❡✲s❡✿ ❛✮ ❊♠ q✉❡ ♠♦♠❡♥t♦ ❛✉♠❡♥t❛ ♦ ❞✐♠✐♥✉❡ ❙✉♣♦♥❤❛♠♦s q✉❡ ♦ r❡♥❞✐♠❡♥t♦ ❞❛❞❛ ♣❡❧❛ ❢✉♥çã♦ ♦ r❡♥❞✐♠❡♥t♦❄ ❜✮ ❊♠ q✉❡ ♠♦♠❡♥t♦ ♦ r❡♥❞✐♠❡♥t♦ é ♥✉❧♦❄ ❝✮ ❊♠ q✉❡ ✐♥st❛♥t❡ s❡ ♦❜tê♠ ♦ ♠❛✐♦r r❡♥❞✐♠❡♥t♦❄ ◗✉❛❧ é ❛q✉❡❧❡❄ ❙♦❧✉çã♦✳ ✸✶✶ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❛✮ ❚❡♠♦s t ∈ [0, 2]✱ ❛ ❞❡r✐✈❛❞❛ r′ (t) = 300(2 − 2t) q✉❛♥❞♦ t = 1 t❡♠♦s r(1) = 300 ❡ r′ (t) < 0 ❡♠ [1, 2]✳ ❖ r❡♥❞✐♠❡♥t♦ ❛✉♠❡♥t❛ ♥❛ ♣r✐♠❡✐r❛ ❤♦r❛ ❞❛ ♣r♦✈❛✱ ❡ ❞✐♠✐♥✉❡ ♥❛ s❡❣✉♥❞❛ ❤♦r❛ ❞❛ ♣r♦✈❛✳ ❜✮ ❖ r❡♥❞✐♠❡♥t♦ é ♥✉❧♦ ❡①❛t❛♠❡♥t❡ ♥♦ ✐♥í❝✐♦ ❡ ♥♦ ✜♥❛❧ ❞❛ ♣r♦✈❛✳ ❝✮ ❖ ♠❛✐♦r r❡♥❞✐♠❡♥t♦ s❡ ♦❜tê♠ ❡①❛t❛♠❡♥t❡ ✉♠❛ ❤♦r❛ ❛♣ós ✐♥✐❝✐❛❞♦ ❛ ♣r♦✈❛✳ ❖ ♠❛✐♦r r❡♥❞✐♠❡♥t♦ é 300✳ ❊①❡♠♣❧♦ ✻✳✽✳ ❯♠❛ ♣❛rtí❝✉❧❛ ♠♦✈✐♠❡♥t❛✲s❡ ❡♠ ❧✐♥❤❛ r❡t❛ s❡❣✉♥❞♦ ❛ r❡❧❛çã♦✿ 132✱ s é ❛ ❞✐stâ♥❝✐❛✱ ❡♠ ♠❡tr♦s ❡ t é ♦ t❡♠♣♦ t = 2 ❄ ❊ q✉❛❧ é ❛ ❛❝❡❧❡r❛çã♦ q✉❛♥❞♦ t = 3 ❄ ♦♥❞❡ q✉❛♥❞♦ S = 3t3 − 16t2 + 108t + ❡♠ s❡❣✉♥❞♦s✳ ◗✉❛❧ é ❛ ✈❡❧♦❝✐❞❛❞❡ ❙♦❧✉çã♦✳ ❙❡❥❛ v(t) ❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛✱ ❡♥tã♦ v(t) = 9t2 − 32t + 108❀ q✉❛♥❞♦ t = 2 ♦❜t❡♠♦s v(2) = 80 ✐st♦ s✐❣♥✐✜❝❛ q✉❡✱ ❛ ✈❡❧♦❝✐❞❛❞❡ q✉❛♥❞♦ t = 2 é 80 m/seg ✳ ❆ ❛❝❡❧❡r❛çã♦ é ❞❛❞❛ ♣❡❧❛ r❡❧❛çã♦ a(t) = 18t − 32✱ q✉❛♥❞♦ t = 3 t❡♠♦s q✉❡ a(3) = 22✱ s✐❣♥✐✜❝❛ q✉❡ ❛ ❛❝❡❧❡r❛çã♦ ♥♦ ✐♥st❛♥t❡ t = 3 é 22 m/seg 2 ✳ ✸✶✷ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❊①❡r❝í❝✐♦s ✻✲✶ ✶✳ ❆ ❛❧t✉r❛ ❞❡ ✉♠❛ ❜♦❧❛ t s❡❣✉♥❞♦s ❞❡♣♦✐s ❞❡ s❡✉ ❧❛♥ç❛♠❡♥t♦ ✈❡rt✐❝❛❧ é ❞❛❞❛ ♣❡❧❛ ❢✉♥çã♦✿ h(t) = −16t2 + 48t + 32✳ ✭❛✮ ❱❡r✐✜q✉❡ q✉❡ h(1) = h(2)✳ ✭❜✮ ❙❡❣✉♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❘♦❧❧❡✱ ❞❡t❡r♠✐♥❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛ ♥♦ ✐♥t❡r✈❛❧♦ [1, 2]✳ 10(x2 + x + 2) ✷✳ ❖ ❝✉st♦ C(x) ❞❡ ♣❡❞✐❞♦ ❞❡ ✉♠❛ ♠❡r❝❛❞♦r✐❛ é ❞❛❞❛ ♣♦r ✿ C(x) = x2 + 2x ♦♥❞❡ C é ♠❡❞✐❞♦ ❡♠ ♠✐❧❤❛r❡s ❞❡ r❡❛✐s ❡ x é ♦ t❛♠❛♥❤♦ ❞♦ ♣❡❞✐❞♦ ♠❡❞✐❞♦ ❡♠ ❝❡♥t❡♥❛s✳ ✭❛✮ ❱❡r✐✜q✉❡ q✉❡ C(4) = C(6)✳ ✭❜✮ ❙❡❣✉♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❘♦❧❧❡✱ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❡ ❝✉st♦ ❞❡✈❡ s❡r ③❡r♦ ♣❛r❛ ❛❧❣✉♠ ♣❡❞✐❞♦ ♥♦ ✐♥t❡r✈❛❧♦ [4, 6]✳ ❉❡t❡r♠✐♥❡ ♦ t❛♠❛♥❤♦ ❞❡ss❡ ♣❡❞✐❞♦✳  2 x +x+1   s❡✱ x < 1   x+a  3 ✸✳ ❙❡❥❛ ❛ ❢✉♥çã♦ r❡❛❧✿ f (x) = x3 + bx2 − 5x + 3 s❡✱ 1≤x≤  2   3  x+2  s❡✱ x > x2 − 9 2 3 ✶✳ ❙✉♣♦♥❤❛ f s❡❥❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ✐♥t❡r✈❛❧♦ (−∞, )❀ ❞❡t❡r♠✐♥❡ a ❡ b✳ 2 ✷✳ ❆❝❤❛r ❛ n✲és✐♠❛ ❞❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ f ❡♠ x = 2✳ ✹✳ ❯♠ ❛✈✐ã♦ ❛ ✉♠❛ ❛❧t✉r❛ ❞❡ 3000m ❡stá ✈♦❛♥❞♦ ❤♦r✐③♦♥t❛❧♠❡♥t❡ ❛ 500km/h✱ ❡ ♣❛ss❛ ❞✐r❡t❛♠❡♥t❡ s♦❜r❡ ✉♠ ♦❜s❡r✈❛❞♦r✳ ❉❡t❡r♠✐♥❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❝♦♠ q✉❡ s❡ ❛♣r♦①✐♠❛ ❞♦ ♦❜s❡r✈❛❞♦r ♥♦ ✐♥st❛♥t❡ ❡♠ q✉❡ ❡stá ❛ 5.000m ❞♦ ❞❡❧❡✳ ✺✳ ❯♠ t❛♥q✉❡ t❡♠ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❝♦♥❡ ❝♦♠ ♦ ✈ért✐❝❡ ♣❛r❛ ❛❜❛✐①♦ ❡ ♠❡❞❡ 12 m ❞❡ ❛❧t✉r❛ ❡ 12 m ❞❡ ❞✐â♠❡tr♦✳ ❇♦♠❜❡✐❛✲s❡ á❣✉❛ à r❛③ã♦ ❞❡ 4 m3 /min✳ ❈❛❧❝✉❧❛r ❛ r❛③ã♦ ❝♦♠ q✉❡ ♦ ♥í✈❡❧ ❞❡ á❣✉❛ s♦❜❡✿ ❛✮ ◗✉❛♥❞♦ ❛ á❣✉❛ t❡♠ 2 m ❞❡ ♣r♦❢✉♥❞✐❞❛❞❡✳ ❜✮ ◗✉❛♥❞♦ ❛ á❣✉❛ t❡♠ 8 m ❞❡ ♣r♦❢✉♥❞✐❞❛❞❡✳ hxi ✻✳ ❊s❝r❡✈❡r ❛s ❡q✉❛çõ❡s ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❡ ♥♦r♠❛❧ à ❝❛t❡♥ár✐❛ y = cosh ✱ ♥♦ ♣♦♥t♦ 2 x = 2Ln2✳ ✼✳ ◆✉♠ ✐♥st❛♥t❡ ❞❛❞♦✱ ♦s ❝❛t❡t♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡t♦ sã♦ 8cm ❡ 6cm✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖ ♣r✐♠❡✐r♦ ❝❛t❡t♦ ❞❡❝r❡s❝❡ à r❛③ã♦ ❞❡ 1cm ♣♦r ♠✐♥✉t♦✱ ❡ ♦ s❡❣✉♥❞♦ ❝r❡s❝❡ à r❛③ã♦ ❞❡ 2cm ♣♦r ♠✐♥✉t♦✳ ❈♦♠ q✉❡ ✈❡❧♦❝✐❞❛❞❡ ❝r❡s❝❡ ❛ ár❡❛ ❞❡♣♦✐s ❞❡ ❞♦✐s ♠✐♥✉t♦s❄ ✽✳ ❯♠❛ ❜♦❧❛ ❡♥❝❤❡✲s❡ ❞❡ ❛r ❛ r❛③ã♦ ❞❡ 15cm3 /sg ✳ ❈♦♠ q✉❡ ✈❡❧♦❝✐❞❛❞❡ ❡st❛ ❝r❡s❝❡♥❞♦ ♦ ❞✐â♠❡tr♦ ❞❡♣♦✐s ❞❡ 5 s❡❣✉♥❞♦s❄ ❙✉♣♦r q✉❡ ♦ ❞✐â♠❡tr♦ é ③❡r♦ ♥♦ ✐♥st❛♥t❡ ③❡r♦✳ ✸✶✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✾✳ ❯♠ ❝♦r♣♦ ❡♠ q✉❡❞❛ ❧✐✈r❡ ♣❡r❝♦rr❡ ✉♠❛ ❞✐stâ♥❝✐❛ D q✉❡ ✈❛r✐❛ ❝♦♠ ♦ t❡♠♣♦ s❡❣✉♥❞♦ ❛ ❡q✉❛çã♦✿ D(t) = 4, 9t2 ✭❞✐stâ♥❝✐❛ ❡♠ ♠❡tr♦s ❡ t ❡♠ s❡❣✉♥❞♦s✮✳ ❛✮ ❈❛❧❝✉❧❛r ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❡ d ✭❞✐stâ♥❝✐❛✮ ❡♠ r❡❧❛çã♦ ❛ t ❡♥tr❡ t1 ❡ t2 ♥♦s s❡❣✉✐♥t❡s ✐♥t❡r✈❛❧♦s✿ (1s, 1.5s), (1s, 1.3s)✳ ❜✮ ❈❛❧❝✉❧❛r ❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛ ♥♦ ✐♥st❛♥t❡ t = 1S ✳ ✶✵✳ ❯♠❛ ❡s❝❛❞❛ ❝♦♠ 6m ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❡stá ❛♣♦✐❛❞❛ ❡♠ ✉♠❛ ♣❛r❡❞❡ ✈❡rt✐❝❛❧✳ ❙❡ ❛ ❜❛s❡ ❞❛ ❡s❝❛❞❛ ❝♦♠❡ç❛ ❛ s❡ ❞❡s❧✐③❛r ❤♦r✐③♦♥t❛❧♠❡♥t❡✱ à r❛③ã♦ ❞❡ 0, 6m/s✱ ❝♦♠ q✉❡ ✈❡❧♦❝✐❞❛❞❡ ♦ t♦♣♦ ❞❛ ❡s❝❛❞❛ ❞❡s❝❡ ❛ ♣❛r❡❞❡✱ q✉❛♥❞♦ ❡stá ❛ 4m ❞♦ s♦❧♦❄ ✶✶✳ ❯♠ ♣♦♥t♦ s❡ ♠♦✈❡ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ ❝✉r✈❛ y = √ 1 + x2 ❞❡ ♠♦❞♦ q✉❡ dy q✉❛♥❞♦ x = 3✳ dt dx = 4✳ ❆❝❤❛r dt ✶✷✳ ❆ ❛❧t✉r❛ ❞❡ ✉♠ ♦❜❥❡t♦ t s❡❣✉♥❞♦s ❛♣ós ❞❡ s❡r ❧❛r❣❛❞♦ ❛ 150m ❞♦ s♦❧♦ é ❞❛❞❛ ♣❡❧❛ ❢✉♥çã♦✿ f (t) = 150 − 4, 9t✳ ✭❛✮ ❊♥❝♦♥tr❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛ ❞♦ ♦❜❥❡t♦ ❞✉r❛♥t❡ ♦s três ♣r✐♠❡✐r♦s s❡❣✉♥❞♦s✳ ✭❜✮ ▼❡❞✐❛♥t❡ ♦ ❚✳❱✳▼✳ ✈❡r✐✜❝❛r q✉❡ ❡♠ ❛❧❣✉♠ ✐♥st❛♥t❡ ❞✉r❛♥t❡ ♦s três ♣r✐♠❡✐r♦s s❡❣✉♥❞♦s ❞❡ q✉❡❞❛✱ ❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛ é ✐❣✉❛❧ à ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛✳ ❊♥❝♦♥tr❡ ❡ss❡ ✐♥st❛♥t❡✳ ✶✸✳ ❯♠❛ ❜♦❧❛ ❞❡ ❜✐❧❤❛r é ❛t✐♥❣✐❞❛ ❡ ♠♦✈❡✲s❡ ❡♠ ❧✐♥❤❛ r❡t❛✳ ❙❡ Scm é ❛ ❞✐stâ♥❝✐❛ ❞❛ ❜♦❧❛ ❞❡ s✉❛ ♣♦s✐çã♦ ✐♥✐❝✐❛❧ ❡♠ t s❡❣✉♥❞♦s✱ ♦♥❞❡ S = 100t2 + 100t✳ ❙❡ vcm/s é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❜♦❧❛ ✱ ❡♥tã♦ v é ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❡ S ❝♦♠ r❡❧❛çã♦ ❛ t✳ ❙❡ ❛ ❜♦❧❛ ❜❛t❡ ♥❛ t❛❜❡❧❛ ❛ 39cm ❞❛ ♣♦s✐çã♦ ✐♥✐❝✐❛❧ ✱❝♦♠ q✉❡ ✈❡❧♦❝✐❞❛❞❡ ❡❧❛ ❜❛t❡ ♥❛ t❛❜❡❧❛❄ ✶✹✳ ❯♠ ❢♦❣✉❡t❡ é ❧❛♥ç❛❞♦ ✈❡rt✐❝❛❧♠❡♥t❡ ♣❛r❛ ❝✐♠❛✱ ❡ ❡stá S ♠❡tr♦s ❛❝✐♠❛ ❞♦ s♦❧♦✱ t s❡❣✉♥❞♦s ❛♣ós ♦ ❧❛♥ç❛♠❡♥t♦✱ ♦♥❞❡ S = 560t − 16t2 é ❛ ❞✐r❡çã♦ ♣♦s✐t✐✈❛ ♣❛r❛ ❝✐♠❛✳ ❙❡ v m/s é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❢♦❣✉❡t❡✱ ❡♥tã♦ v é ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❡ S ❡♠ r❡❧❛çã♦ ❛ t ✳ ✭❛✮ ❆❝❤❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❢♦❣✉❡t❡ 2seg ✳ ❛♣ós ♦ ❧❛♥ç❛♠❡♥t♦❀ ✭❜✮ ❙❡ ❛ ❛❧t✉r❛ ♠á①✐♠❛ é ❛t✐♥❣✐❞❛ q✉❛♥❞♦ ❛ ✈❡❧♦❝✐❞❛❞❡ é ③❡r♦✱ ❛❝❤❡ q✉❛♥t♦ t❡♠♣♦ ❞❡♠♦r❛ ♣❛r❛ ♦ ❢♦❣✉❡t❡ ❛t✐♥❣✐r s✉❛ ❛❧t✉r❛ ♠á①✐♠❛✳ ✶✺✳ ❯♠ ❝❛rr♦ t❡♠ q✉❡ s❡ tr❛s❧❛❞❛r ❞♦ ♣♦♥t♦ A ❛té ♦ ♣♦♥t♦ B ✭✈❡r ❋✐❣✉r❛ ✮✳ ❖ ♣♦♥t♦ B s❡ ❡♥❝♦♥tr❛ ❛ 36❦♠ ❞❡ ✉♠❛ ❡str❛❞❛ r❡t❛✳ ❙♦❜r❡ ❛ ❡str❛❞❛ ♦ ❝❛rr♦ ♣❡r❝♦rr❡ ❛ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡ ❝♦♥st❛♥t❡ ❞❡ 100❦♠✴❤✱ ❡♥t❛♥t♦ q✉❡ ♥♦ t❡rr❡♥♦ s✉❛ ✈❡❧♦❝✐❞❛❞❡ é ❞❡ 80❦♠✴❤✳ ◗✉❛❧ é ♦ ♣❡r❝♦rr✐❞♦ q✉❡ ♦ ❝♦♥❞✉t♦r ❞❡✈❡ s❡❣✉✐r ♣❛r❛ q✉❡ ♦ t❡♠♣♦ ❡♠ ✐r ❞❡ A ❛té B s❡❥❛ ♦ ♠í♥✐♠♦❄ ◗✉❛❧ ♦ t❡♠♣♦ q✉❡ ❞❡♠♦r❛ ♣❛r❛ ♣❡r❝♦rr❡r ❞❡ A ❛té B ❄ ❇ ❆✛ 100 ❦♠ ✲ ✳✳ ✳✳ ✳✳ 36 ❦♠ ✳✳ ✳✳ ✳✳ ❊str❛❞❛ ✸✶✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✻✳✷ ❊st✉❞♦ ❞♦ ❣rá✜❝♦ ❞❡ ❢✉♥çõ❡s ❊st✉❞❛r❡♠♦s ❛♣❧✐❝❛çõ❡s s♦❜r❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❞❡r✐✈❛çã♦✱ ♦❜t❡♥❞♦ ♥♦✈❛s ♣r♦♣r✐❡❞❛✲ ❞❡s q✉❡ ♥♦s ♣❡r♠✐t✐r❛♠ ❡st✉❞❛r ❛ ✈❛r✐❛çã♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❞❡t❡r♠✐♥❛♥❞♦ ✐♥t❡r✈❛❧♦s ❞❡ ❝r❡s❝✐♠❡♥t♦ ♦✉ ❞❡❝r❡s❝✐♠❡♥t♦✱ ♣♦♥t♦s ❞❡ ❡①tr❡♠♦✱ ✐♥t❡r✈❛❧♦s ❞❡ ❝♦♥❝❛✈✐❞❛❞❡ ❡ ♣♦♥t♦s ❞❡ ✐♥✢❡①ã♦✳ ✻✳✷✳✶ ❋✉♥çã♦ ❝r❡s❝❡♥t❡✳ ❋✉♥çã♦ ❞❡❝r❡s❝❡♥t❡ ❉❡✜♥✐çã♦ ✻✳✹✳ ❙❡❥❛ f : R −→ R ✉♠❛ ❢✉♥çã♦✱ ❡ ❛✮ ❉✐③❡♠♦s q✉❡ f (x) é I ⊆ D(f )✳ I q✉❛♥❞♦✱ ❝r❡s❝❡♥t❡ ❡♠ t❡♠♦s f (x1 ) ≤ f (x2 ✮✳ ∀ x1 , x2 ∈ I ❝♦♠ x1 < x2 ❜✮ ❉✐③❡♠♦s q✉❡ f (x) é ❞❡❝r❡s❝❡♥t❡ ❡♠ I q✉❛♥❞♦✱ ∀ x1 , x2 ∈ I ❝♦♠ x1 < x2 t❡♠♦s f (x1 ) ≥ f (x2 ✮✳ ❝✮ ❉✐③❡♠♦s q✉❡ f (x) é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ❡♠ I q✉❛♥❞♦✱ x1 < x2 t❡♠♦s f (x1 ) < f (x2 ✮✳ ❞✮ ❉✐③❡♠♦s q✉❡ f (x) é ❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡ ❡♠ ∀ x1 , x2 ∈ I ❝♦♠ I q✉❛♥❞♦✱ ❝♦♠ x1 < x2 t❡♠♦s f (x1 ) > f (x2 ✮✳ ∀ x 1 , x2 ∈ I Pr♦♣r✐❡❞❛❞❡ ✻✳✶✳ ❙✉♣♦♥❤❛ t❡♠♦s✿ f : [a, b] −→ R s❡❥❛ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♠ [a, b] ❝♦♠ ❞❡r✐✈❛❞❛ ❡♠ (a, b)✱ ✐✮ ❙❡ f ′ (x) > 0, ∀ x ∈ (a, b)❀ ❡♥tã♦ f é ❝r❡s❝❡♥t❡ ❡♠ [a, b]✳ ✐✐✮ ❙❡ f ′ (x) < 0, ∀ x ∈ (a, b)❀ ❡♥tã♦ f é ❞❡❝r❡s❝❡♥t❡ ❡♠ [a, b]✳ ❉❡♠♦♥str❛çã♦✳ ✭✐✮ ❙❡❥❛♠ x1 , x2 ∈ [a, b] ❝♦♠ x1 < x2 ✳ ❆s ❝♦♥❞✐çõ❡s ✭❛✮ ❡ ✭❜✮ ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭✻✳✶✮ sã♦ ✈❡r✐✜❝❛❞❛s ♥♦ s✉❜✐♥t❡r✈❛❧♦ [x1 , x2 ] ❞❡ [a, b]❀ ❧♦❣♦✱ ♣❡❧♦ ❚✳❱✳▼✳ ❊①✐st❡ c ∈ (x1 , x2 ) t❛❧ q✉❡ f (x2 ) − f (x1 ) = (x2 − x1 ) · f ′ (c)✳ ❈♦♠♦ c ∈ (x1 , x2 )✱ ❡♥tã♦ c ∈ (a, b)❀ ❧♦❣♦✱ ♣❡❧❛ ❤✐♣ót❡s❡ f ′ (c) > 0✱ ❡ ❝♦♠♦ x2 − x1 > 0 s❡❣✉❡✱ f (x2 ) − f (x1 ) = (x2 − x1 ) · f ′ (c) > 0✳ ▲♦❣♦✱ ∀ x1 , x2 ∈ [a, b] ❝♦♠ x1 < x2 t❡♠♦s f (x1 ) < f (x2 ) ❡ f é ❝r❡s❝❡♥t❡ ❡♠ [a, b]✳ ❉❡♠♦♥str❛çã♦✳ ✭✐✐✮ ❊①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳  ✸✶✺ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ❞❡ ❡①tr❡♠♦ ❝♦♠ ❛ ❞❡r✐✈❛❞❛ 1a ✳ ❙❡❥❛ y = f (x) ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ B(c, δ) ❞♦ ♣♦♥t♦ x = c✱ ❝♦♥tí♥✉❛ ❡♠ B(c, δ) ❡ ❝♦♠ ❞❡r✐✈❛❞❛ ❡♠ B(c, δ)✱ ❡①❝❡t♦ ♣♦ss✐✈❡❧♠❡♥t❡ ❡♠ x = c ❡♥tã♦✿ ❛✮ ❙❡ f ′ (x) > 0, ∀ x ∈ (c − δ, c) ❡ f ′ (x) < 0, ∀ x ∈ (c, c + δ)✱ ❡♥tã♦ f (c) é ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❧♦❝❛❧ ❞❡ f ✳ ❜✮ ❙❡ f ′ (x) < 0, ∀ x ∈ (c − δ, c) ❡ f ′ (x) > 0, ∀ x ∈ (c, c + δ)✱ ❡♥tã♦ f (c) é ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧ ❞❡ f ✳ Pr♦♣r✐❡❞❛❞❡ ✻✳✷✳ ❉❡♠♦♥str❛çã♦✳ ✭❛✮ ❉❛s ❤✐♣ót❡s❡s ❡ ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭✻✳✶✮✱ s❡❣✉❡ q✉❡ f é ❝r❡s❝❡♥t❡ ❡♠ (c−δ, c) ❡ ❞❡❝r❡s❝❡♥t❡ ❡♠ (c, c + δ)❀ ❧♦❣♦ f (x) ≤ f (c) ∀ x ∈ B(c, δ) ❡ ❞❡❞✉③✲s❡ ❞❛ ❉❡✜♥✐çã♦ ✭✻✳✹✮ q✉❡ f (c) é ✉♠ ♠á①✐♠♦ ❧♦❝❛❧ ❞❡ f ✳  ❉❡♠♦♥str❛çã♦✳✭❜✮ ❊①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳  ❈r✐tér✐♦ ❞❛ ❞❡r✐✈❛❞❛ 1a ✳ ❆ Pr♦♣r✐❡❞❛❞❡ ✭✻✳✷✮ ♣❡r♠✐t❡ ❡st❛❜❡❧❡❝❡r ♦ s❡❣✉✐♥t❡ ❝r✐tér✐♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦s ♠á①✐✲ ♠♦s ♦✉ ♠í♥✐♠♦s r❡❧❛t✐✈♦s ❞❡ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❖❜s❡r✈❛çã♦ ✻✳✶✳ 1o ❉❡t❡r♠✐♥❛r ♦s ♣♦♥t♦s ❝rít✐❝♦s ❞❡ f ✳ 2o ❙❡ c é ✉♠ ♣♦♥t♦ ❝rít✐❝♦✱ ❞❡✈❡✲s❡ ❞❡t❡r♠✐♥❛r ♦ s✐♥❛❧ ❞❡ f ′ (x)✱ ♣r✐♠❡✐r♦ ♣❛r❛ ✈❛❧♦r❡s ♣ró①✐♠♦s à ❡sq✉❡r❞❛ ❞❡ c ❡ ❧♦❣♦ ♣❛r❛ ✈❛❧♦r❡s à ❞✐r❡✐t❛ ❞❡ c✳ 3o ❙❡ ♦ s✐♥❛❧ ♠✉❞❛ ❞❡ + ♣❛r❛ − ✱ ❡♥tã♦ f (c) é ♠á①✐♠♦ r❡❧❛t✐✈♦❀ ❡ s❡ ♦ s✐♥❛❧ ♠✉❞❛ ❞❡ − ♣❛r❛ + ❀ ❡♥tã♦ f (c) é ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ r❡❧❛t✐✈♦✳ 4o ❙❡ ♥ã♦ ❡①✐st❡ ♠✉❞❛♥ç❛ ❞❡ s✐♥❛❧✱ ❡♥tã♦ ♥ã♦ ❡①✐st❡ ♥❡♠ ♠á①✐♠♦ ♥❡♠ ♠í♥✐♠♦ r❡❧❛t✐✈♦ ❡♠ x = c✳ ❈♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ❞❡ ❡①tr❡♠♦ ❝♦♠ ❛ ❞❡r✐✈❛❞❛ 2a ❙❡❥❛ y = f (x) ✉♠❛ ❢✉♥çã♦ ❝♦♠ ❞❡r✐✈❛❞❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❝♦♥tí♥✉❛ ❡♠ ✉♠❛ ✈✐③✐✲ ♥❤❛♥ç❛ B(c, δ) ❞❡ x = c✱ ❞❡ ♠♦❞♦ q✉❡ f ′ (c) = 0 ❡ f ′ (c) 6= 0 ❡♥tã♦✿ Pr♦♣r✐❡❞❛❞❡ ✻✳✸✳ ✐✮ ✐✐✮ ❙❡ f ′′ (c) > 0✱ ❡♥tã♦ f (c) é ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧ ❞❡ f ✳ ❙❡ f ′′ (c) < 0✱ ❡♥tã♦ f (c) é ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❧♦❝❛❧ ❞❡ f ✳ ❉❡♠♦♥str❛çã♦✳ ✭❛✮ f ′ (c + h) f ′ (c + h) − f ′ (c) = lim ✱ ♣♦✐s h→0 h→0 h h ❉❛ ❞❡✜♥✐çã♦ ❞❡ ❞❡r✐✈❛❞❛ s❡❣✉❡✱ f ′′ (c) = lim f ′ (c) = 0✳ ✸✶✻ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ f ′ (c + h) P♦r ❤✐♣ót❡s❡ f ′′ (x) é ❝♦♥tí♥✉❛ ❡♠ x = c ❡ f ′′ (c) > 0✱ ❡♥tã♦ lim > 0✱ ❧♦❣♦ h→0 h ′ t❡♠♦s ♣❛r❛ h > 0✱ ✭s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✮ f (x) > 0, ∀ x ∈ (c, c + δ)✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ♣❛r❛ h < 0 ✭s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✮ t❡♠♦s f ′ (c + h) < 0 ♦ q✉❡ ✐♠♣❧✐❝❛ f ′ (x) < 0, ∀ x ∈ (c − δ, c)❀ ❛♣❧✐❝❛♥❞♦ ❛ Pr♦♣r✐❡❞❛❞❡ ✭✻✳✷✮ ♣❛r❛ ❛ ❢✉♥çã♦ f ′ (x) r❡s✉❧t❛ q✉❡ f (c) é ✉♠ ♠í♥✐♠♦ ❧♦❝❛❧ ❞❡ f ✳  ❉❡♠♦♥str❛çã♦✳ ✭❜✮ ❊①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❖❜s❡r✈❛çã♦ ✻✳✷✳ ❆ Pr♦♣r✐❡❞❛❞❡ ❈r✐tér✐♦ ❞❛ ❞❡r✐✈❛❞❛ ✭✻✳✸✮ 2a ✳ ♣❡r♠✐t❡ ❡st❛❜❡❧❡❝❡r ♦ s❡❣✉✐♥t❡ ❝r✐tér✐♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦s ♠á①✐✲ ♠♦s ♦✉ ♠í♥✐♠♦s r❡❧❛t✐✈♦s ❞❡ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❉❡t❡r♠✐♥❛r ♦s ♣♦♥t♦s ❝rít✐❝♦s ❞❡ f ✳ ❉❡t❡r♠✐♥❛r ❛ ❞❡r✐✈❛❞❛ s❡❣✉♥❞❛ ❞❡ f ✳ P❛r❛ ❝❛❞❛ ♣♦♥t♦ x = c ❝rít✐❝♦ ❞❡t❡r♠✐♥❛r f ′′ (c)✳ ❙❡ f ′′ (c) é ♣♦s✐t✐✈♦✱ ❡♥tã♦ f (c) é ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ r❡❧❛t✐✈♦✳ 5o ❙❡ f ′′ (c) é ♥❡❣❛t✐✈♦✱ ❡♥tã♦ f (c) é ♣♦♥t♦ ❞❡ ♠á①✐♠♦ r❡❧❛t✐✈♦✳ 6o ❙❡ f ′′ (c) é ③❡r♦ ♦✉ ♥ã♦ ❡①✐st❡✱ ♦ ❝r✐tér✐♦ é ✐♥❝♦♥s✐s✲ t❡♥t❡✳ 1o 2o 3o 4o ❋✐❣✉r❛ ✻✳✸✿ ❊①❡♠♣❧♦ ✻✳✾✳ ❉❡t❡r♠✐♥❡ ♦s ✐♥t❡r✈❛❧♦s ❞❡ ❝r❡s❝✐♠❡♥t♦ ❡ ♦s ❡①tr❡♠♦s r❡❧❛t✐✈♦s ❞❛ ❢✉♥çã♦ ❙♦❧✉çã♦✳ f (x) = x3 − 3x2 ✳ ❚❡♠♦s f ′ (x) = 3x(x − 2)❀ q✉❛♥❞♦ f ′ (x) = 0 r❡s✉❧t❛ x = 0 ❡ x = 2 ❛ss✐♠✱ 0 ❡ 2 sã♦ ♣♦♥t♦s ❝rít✐❝♦s✳ ❆♣❧✐❝❛♥❞♦ ❛ Pr♦♣r✐❡❞❛❞❡ ✭✻✳✶✮ ❝♦♥str✉❛♠♦s ❛ t❛❜❡❧❛✳ ■♥t❡r✈❛❧♦s ❙✐♥❛❧ ❞❡ f ′ (x) ❈♦♠♣♦rt❛♠❡♥t♦ (−∞, 0) + ❝r❡s❝❡♥t❡ (0, 2) − ❞❡❝r❡s❝❡♥t❡ (2, +∞) + ❝r❡s❝❡♥t❡ ❊①tr❡♠♦s f (0) = 0 ♠á①✳ r❡❧❛t✳ f (2) = −4 ♠í♥✳ r❡❧❛t✳ ❖ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ é ♠♦str❛❞❛ ♥❛ ❋✐❣✉r❛ ✭✻✳✸✮✳ ❊①❡♠♣❧♦ ✻✳✶✵✳ ❉❡t❡r♠✐♥❡ ♦s ✐♥t❡r✈❛❧♦s ❞❡ ❝r❡s❝✐♠❡♥t♦ ❡ ♦s ❡①tr❡♠♦s r❡❧❛t✐✈♦s ❞❛ ❢✉♥çã♦ ❙♦❧✉çã♦✳ ✸✶✼ g(x) = 6 x + ✳ x 6 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R (x − 6)(x + 6) ❚❡♠♦s ♦ D(g) = R = {0}, g ′ (x) = q✉❛♥❞♦ g ′ (x) = 0 ♦❜té♠♦s ♦s ♣♦♥t♦s 6x2 ❝rít✐❝♦s x = 6 ❡ x = −6❀ ♦ ♣♦♥t♦ x = 0 ♥ã♦ é ♣♦♥t♦ ❝rít✐❝♦ ♣♦r ♥ã♦ ♣❡rt❡♥❝❡r ❛♦ ❞♦♠í✲ ♥✐♦ D(g)❀ ♣♦ré♠ ❞❡✈❡♠♦s ❝♦♥s✐❞❡r❛r✲❧✵ ♣♦r s❡r ♣♦♥t♦ ❞❡ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡✳ ❈♦♥s✐❞❡r❡✲s❡ ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✿ ■♥t❡r✈❛❧♦s ❙✐♥❛❧ ❞❡ f ′ (x) ❈♦♠♣♦rt❛♠❡♥t♦ (−∞, −6) + ❝r❡s❝❡♥t❡ (−6, 0) − ❞❡❝r❡s❝❡♥t❡ (0, 6) − (6, +∞) + ❊①❡♠♣❧♦ ✻✳✶✶✳ ❙❡❥❛ ❛ ❢✉♥çã♦ ❙♦❧✉çã♦✳ f (x) = p 3 x2 (x + 3)✱ ❞❡❝r❡s❝❡♥t❡ ❝r❡s❝❡♥t❡ ❊①tr❡♠♦s f (−6) = −2 ♠á①✳ r❡❧❛t✳ f (6) = 2 ♠í♥✳ r❡❧❛t✳ ❞❡t❡r♠✐♥❡ ♦s ♣♦♥t♦s ❞❡ ❡①tr❡♠♦s r❡❧❛t✐✈♦s✳ ❖ ❞♦♠í♥✐♦ D(f ) = R✱ ❡ f ′ (x) = p 3 x+2 x(x + 3)2 ✱ q✉❛♥❞♦ f ′ (x) = 0 t❡♠♦s ♦s ♣♦♥t♦s ❝rít✐❝♦s sã♦✿ 0, −2 ❡ −3✳ ❖❜s❡r✈❡✱ ❡♠ x = −3 ❡ x = 0 ❛ ❞❡r✐✈❛❞❛ ♥ã♦ ❡①✐st❡ ✭é ✐♥✜♥✐t❛✮✳ ❆♣❧✐❝❛♥❞♦ ❛ Pr♦♣r✐❡❞❛❞❡ ✭✻✳✷✮✱ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦s ✐♥t❡r✈❛❧♦s ❞❡ ❝r❡s❝✐♠❡♥t♦ ♦✉ ❞❡❝r❡s❝✐♠❡♥t♦✱ s❡❣✉♥❞♦ ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✱ t❡♠♦s✿ ■♥t❡r✈❛❧♦s ❙✐♥❛❧ ❞❡ f ′ (x) ❈♦♠♣♦rt❛♠❡♥t♦ (−∞, −3) + ❝r❡s❝❡♥t❡ (−3, −2) + ❞❡❝r❡s❝❡♥t❡ (−2, 0) − ❝r❡s❝❡♥t❡ (0, +∞) + ❝r❡s❝❡♥t❡ ❊①tr❡♠♦s f (−2) = √ 3 4 ♠á①✳ r❡❧❛t✳ f (0) = 0 ♠í♥✳ r❡❧❛t✳ ❊①❡♠♣❧♦ ✻✳✶✷✳ ❯♠❛ ❡♠♣r❡s❛ ❛♣✉r♦✉ q✉❡ s✉❛ r❡❝❡✐t❛ t♦t❛❧ ✭❡♠ r❡❛✐s✮ ❝♦♠ ❛ ✈❡♥❞❛ ❞❡ ✉♠ ♣r♦❞✉t♦ ❛❞✲ ♠✐t❡ ❝♦♠♦ ♠♦❞❡❧♦ R = −x3 +450x2 +52.500x✱ ♦♥❞❡ x é ♦ ♥ú♠❡r♦ ❞❡ ✉♥✐❞❛❞❡s ♣r♦❞✉③✐❞❛s✳ ◗✉❛❧ ♦ ♥í✈❡❧ ❞❡ ♣r♦❞✉çã♦ q✉❡ ❣❡r❛ ❛ r❡❝❡✐t❛ ♠á①✐♠❛❄ ❙♦❧✉çã♦✳ ❚❡♠♦s R = −x3 + 450x2 + 52.500x✱ ❧♦❣♦ R′ = −3x2 + 900x + 52500❀ r❡s♦❧✈❡♥❞♦ R′ (x) = −3x2 + 900x + 52500 = 0 ⇒ −3(x2 − 300x − 17500) = 0 ⇒ −3(x − 350)(x + 50) = 0 ⇒ x = 350 ♦✉ x = −50✳ ❖❜s❡r✈❡✱ R′′ (x) = −6x + 900 ⇒ R′′ (350) < 0✱ ❛ss✐♠ q✉❛♥❞♦ x = 350 ♦ ♥í✈❡❧ ❞❡ ♣r♦❞✉çã♦ ❣❡r❛ ❛ r❡❝❡✐t❛ ♠á①✐♠❛✳ ✸✶✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✻✳✶✸✳ ❉❡t❡r♠✐♥❡ ♦s ✐♥t❡r✈❛❧♦s ❞❡ ❝r❡s❝✐♠❡♥t♦ ❡ ♦s ❡①tr❡♠♦s r❡❧❛t✐✈♦s ❞❛ ❢✉♥çã♦ 3 f (x) = 2 x − 3x − 9x + 2✳ ❙♦❧✉çã♦✳ f ′ (x) = 3x2 − 6x − 9 = 3(x − 3)(x + 1)✱ ❡♠ ✈✐rt✉❞❡ f ′ (x) = 0 ✐♠♣❧✐❝❛ x = 3 ❡ x = −1✱ ❡ s❡❣✉♥❞♦ ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✿ ❖❜s❡r✈❡✱ ■♥t❡r✈❛❧♦s ❙✐♥❛❧ ❞❡ f ′ (x) ❈♦♠♣♦rt❛♠❡♥t♦ (−∞, −1) + ❝r❡s❝❡♥t❡ (−1, 3) − ❞❡❝r❡s❝❡♥t❡ (3, +∞) + ❝r❡s❝❡♥t❡ ❞❛ ✭✻✳✷✮ Pr♦♣r✐❡❞❛❞❡ ❊①tr❡♠♦s f (−1) = 7 ♠á①✳ r❡❧❛t✳ f (3) = −25 ♠í♥✳ r❡❧❛t✳ ❊①❡♠♣❧♦ ✻✳✶✹✳ ❙❡❥❛ ❛ ❢✉♥çã♦ f (x) = ❙♦❧✉çã♦✳ ❚❡♠♦s x2 (x − 2)2 ❞❡t❡r♠✐♥❡ ♦s ❡①tr❡♠♦s r❡❧❛t✐✈♦s✳ D(f ) = R − {2}, f ′ (x) = ■♥t❡r✈❛❧♦s ❙✐♥❛❧ ❞❡ −4x (x − 2)2 f ′ (x) ♦ ú♥✐❝♦ ♣♦♥t♦ ❝rít✐❝♦ é ❈♦♠♣♦rt❛♠❡♥t♦ (−∞, 0) − ❞❡❝r❡s❝❡♥t❡ (0, 2) + ❞❡❝r❡s❝❡♥t❡ (2, +∞) − ❝r❡s❝❡♥t❡ ❆ r❡t❛ ♠♦str❛ ❛ x=2 x = 2✳ ▲♦❣♦✿ ❊①tr❡♠♦s f (0) = 0 ♠í♥✳ r❡❧❛t✳ é ❛ssí♥t♦t❛ ✈❡rt✐❝❛❧ ❞❛ ❝✉r✈❛ ❝♦♠♦ ✭✻✳✹✮✳ ❋✐❣✉r❛ ❊①❡♠♣❧♦ ✻✳✶✺✳ ❙❡❥❛ a > 0✱ ♠♦str❡ q✉❡ ♦ ♠á①✐♠♦ ❛❜s♦❧✉t♦ ❞❛ ❢✉♥çã♦✿ f (x) = 1 1 − 1+ | x | 1+ | x − a | é 2+a 1+a ❙♦❧✉çã♦✳ ❋✐❣✉r❛ ✻✳✹✿ ▲❡♠❜r❡✱ s❡ g ′ (x) = g(x) =| x |✱ x ❀ |x| ❡♥tã♦✿ ❧♦❣♦ t❡♠♦s f ′ (x) = (x − a) x 1 1 · · − ✱ 2 2 (1+ | x − a |) | x − a | (1+ | x |) | x | ✸✶✾ ♦ q✉❡ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✐♠♣❧✐❝❛ q✉❡ ❛ ❞❡r✐✈❛❞❛ ♥ã♦ ❡①✐st❡ ❡♠ x = 0 ❡ ❡♠ x = a✳ 1 (x − a) x 1 a · · = ♦♥❞❡ x = ❀ 2 2 (1+ | x − a |) | x − a | (1+ | x |) | x | 2 a ❡ a ❛ss✐♠ ♦s ♣♦♥t♦s ❝rít✐❝♦s sã♦ 0, 2 ◗✉❛♥❞♦ f ′ (x) = 0 ❡♥tã♦ ■♥t❡r✈❛❧♦s ❙✐♥❛❧ ❞❡ f ′ (x) ❈♦♠♣♦rt❛♠❡♥t♦ (−∞, 0) a (0, ) 2 a ( , a) 2 (a, +∞) + ❝r❡s❝❡♥t❡ − ❞❡❝r❡s❝❡♥t❡ + ❝r❡s❝❡♥t❡ − ❞❡❝r❡s❝❡♥t❡ ❊①tr❡♠♦s ♠á①✳ ❧♦❝❛❧ ❡♠ f (0) a 2 ♠á①✳ ❧♦❝❛❧ ❡♠ f (a) ♠í♥✳ ❧♦❝❛❧ ❡♠ f ( ) 2+a a 4 2+a 1 = ; f( ) = ❡ f (a) = ✱ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ 1+a 1+a 2 2+a 1+a a f é ❝♦♥tí♥✉❛ ❡ ❞♦ ❢❛t♦ f ( ) < f (0) = f (a) ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ♠á①✐♠♦ ❛❜s♦❧✉t♦ ❞❡ f (x) é 2 2+a f (a) = ✳ 1+a ❚❡♠♦s f (0) = 1 + ❊①❡♠♣❧♦ ✻✳✶✻✳ ❉❡t❡r♠✐♥❡ ♦s ✈❛❧♦r❡s ❞❡ a, b ❡ c ❞❡ ♠♦❞♦ q✉❡ ❛ ❢✉♥çã♦ f (x) = ax4 + bx2 + c t❡♥❤❛ 1 ❡①tr❡♠♦ r❡❧❛t✐✈♦ ❡♠ x = ✱ ❡ q✉❡ ❛ ❡q✉❛çã♦ ❞❛ t❛♥❣❡♥t❡ ♥♦ ♣♦♥t♦ ❞❡ ❛❜s❝✐ss❛ x = −1 2 s❡❥❛ 2x − y + 4 = 0✳ ❙♦❧✉çã♦✳ ❆ ❞❡r✐✈❛❞❛ f ′ (x) = 4ax3 + 2bx ❝♦♠♦ x = 1 1 1 é ♣♦♥t♦ ❝rít✐❝♦ t❡♠♦s f ′ ( ) = 4a( )3 + 2 2 2 a 1 2b( ) = 0 ❛ss✐♠ + b = 0✳ 2 2 P♦r ♦✉tr♦ ❧❛❞♦✱ m = 2 é ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ t❛♥❣❡♥t❡ q✉❛♥❞♦ x = −1✱ ❡♥tã♦ f ′ (−1) = −4a − 2b = 2✳ ◆❛ r❡t❛ t❛♥❣❡♥t❡✱ q✉❛♥❞♦ x = −1t❡♠♦s y = 2 ❡ ♥❛ ❢✉♥çã♦✱ f (−1) = a + b + c = 2✳ a ❘❡s♦❧✈❡♥❞♦ ❛s três ✐❣✉❛❧❞❛❞❡s✿ + b = 0, −4a − 2b = 2 ❡ a + b + c = 2 s❡❣✉❡ 2 1 7 2 ❡ c= ✳ a=− , b= 3 3 3 1 7 2 P♦rt❛♥t♦✱ f (x) = − x4 + x2 + ✳ 3 3 3 ❖❜s❡r✈❛çã♦ ✻✳✸✳ ❈r✐tér✐♦ ♣❛r❛ ♦s ❡①tr❡♠♦s ❛❜s♦❧✉t♦s ❞❡ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ♥✉♠ ✐♥✲ t❡r✈❛❧♦ ❢❡❝❤❛❞♦✳ ❙❡ f é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ [a, b]✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❲❡✐❡rstr❛ss✱ f ❛♣r❡✲ s❡♥t❛ ❡①tr❡♠♦s ❛❜s♦❧✉t♦s✳ P❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ s❡✉s ❡①tr❡♠♦s✱ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ❡st❡s ♣♦❞❡♠ ❡st❛r ♥♦s ❡①tr❡♠♦s ❞♦ ✐♥t❡r✈❛❧♦✱ é s✉✜❝✐❡♥t❡ ❛❞✐❝✐♦♥❛r ♦s ♣♦♥t♦s a ❡ b ❛♦s ♣♦♥t♦s ❝rít✐❝♦s ❞❡ f ✱ ❧♦❣♦ ❝♦♠♣❛r❛r ♦s ✈❛❧♦r❡s q✉❡ f ❛ss✉♠❡ ❡♠ ❝❛❞❛ ✉♠ ❞❡st❡s ♣♦♥t♦s ❝rít✐❝♦s✱ ♦ ♠❛✐♦r é ♦ ♠á①✐♠♦ ❛❜s♦❧✉t♦ ❡ ♦ ♠❡♥♦r ♦ ♠í♥✐♠♦ ❛❜s♦❧✉t♦✳ ✸✷✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✻✳✶✼✳ ❉❡t❡r♠✐♥❡ ♦s ✈❛❧♦r❡s ♠á①✐♠♦s ❡ ♠í♥✐♠♦s ❛❜s♦❧✉t♦s ❞❛ ❢✉♥çã♦ 10 ❡♠ [0, 4]✳ f (x) = x3 + 3x2 − 24x − ❙♦❧✉çã♦✳ ❖❜s❡r✈❡✱ f (x) é ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ [0, 4]✱ ❡ f ′ (x) = 3(x − 2)(x + 4)❀ ❧♦❣♦ s❡✉s ♣♦♥t♦s ❝rít✐❝♦s sã♦ 2 ❡ −4✳ P♦r ♦✉tr♦ ❧❛❞♦✱ f (2) = −38 ❡ −4 ∈ / [0, 4]✳ P♦rt❛♥t♦✱ ♠❡❞✐❛♥t❡ ♦ ❣rá✜❝♦ t❡♠♦s q✉❡ f (2) = −38 é ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❛❜s♦❧✉t♦ ♥♦ ✐♥t❡r✈❛❧♦ [0, 4]✳ ❊①❡♠♣❧♦ ✻✳✶✽✳ ❙❡ g(x) = − ❙♦❧✉çã♦✳ 4|x| ✱ 1 + x2 ❞❡t❡r♠✐♥❡ ♦s ✈❛❧♦r❡s ♠á①✐♠♦s ❡ ♠í♥✐♠♦s ❛❜s♦❧✉t♦s ❚❡♠♦s g(x) ❝♦♥tí♥✉❛ ❡♠ [−4, 2] ❡ g ′ (x) = −4, −1, 0, 1 ❡ 2✳ 4 | x | (x2 − 1) ♦s ♣♦♥t♦s ❝rít✐❝♦s sã♦ | x | (1 + x2 )2 8 16 P♦r ♦✉tr♦ ❧❛❞♦✱ g(−4) = − , g(−1) = −2, g(0) = 0, g(1) = −2 ❡ g(2) = − ✳ 17 5 P♦rt❛♥t♦✱ ♦ ✈❛❧♦r ♠á①✐♠♦ ❛❜s♦❧✉t♦ é 0 = g(0)✱ ❡ ♦ ✈❛❧♦r ♠í♥✐♠♦ ❛❜s♦❧✉t♦ é −2 = g(−1) = g(1)✳ ❊①❡♠♣❧♦ ✻✳✶✾✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ♠á①✐♠♦ ❞❛ ❢✉♥çã♦ y = senx sen2x✳ ❙♦❧✉çã♦✳ ❉❡s❞❡ q✉❡ sen2x = 2senx cos x✱ t❡♠♦s y = senx sen2x = 2 cos x sen2 x = 2 cos x(1 − cos2 x)✳ ❈♦♥s✐❞❡r❡ z = cos x✱ ❧♦❣♦ −1 ≤ z ≤ 1✳ ❆ ❢✉♥çã♦ g(z) = z − z 3 = z(1 − z 2 ) ❛ss✉♠❡ ✈❛❧♦r❡s ♥❡❣❛t✐✈♦s ♥♦ ✐♥t❡r✈❛❧♦ −1 ≤ z < 0✱ é ✐❣✉❛❧ ❛ ③❡r♦ s❡ z = 0✱ ❡ ❛ss✉♠❡ ✈❛❧♦r❡s ♣♦s✐t✐✈♦s ♥♦ ✐♥t❡r✈❛❧♦ 0 < z ≤ 1✳ 1 ◗✉❛♥❞♦ g(z) = z(z − z 2 ) ❡♥tã♦ g ′ (z) = 1 − 3z 2 ✱ ❢❛③❡♥❞♦ g ′ (z) = 0 s❡❣✉❡ z = ± √ sã♦ 3 1 ♣♦♥t♦s ❝rít✐❝♦s❀ g(z) t❡♠ ✈❛❧♦r ♠á①✐♠♦ r❡❧❛t✐✈♦ ❡♠ z = √ ✳ 3 ▲♦❣♦ ❛ ❢✉♥çã♦ y = senxsen2x ❛❧❝❛♥ç❛ s❡✉ ✈❛❧♦r ♠á①✐♠♦ ♥♦s ♣♦♥t♦s ♥♦s q✉❛✐s z = 4 1 cos x = √ ❡st❡ ✈❛❧♦r ❛❝♦♥t❡❝❡ q✉❛♥❞♦ x = √ ≈ 0, 777✳ 3 3 3 ❊①❡♠♣❧♦ ✻✳✷✵✳ ♥♦ s  β a β−1 β ✱ ▼♦str❡ q✉❡ ❛ ❢✉♥çã♦ f (x) = x −ax ❛❧❝❛♥ç❛ s❡✉ ✈❛❧♦r ♠í♥✐♠♦ ✐❣✉❛❧ ❛ (1−β) β s  a β−1 ♣♦♥t♦ x = ✱ s❡♠♣r❡ q✉❡ a > 0, β > 1, x > 0✳ β ❙♦❧✉çã♦✳ ✸✷✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❢✉♥çã♦ f (x) = xβ − ax s❡❣✉❡ f ′ (x) = βxβ−1 − a✱ q✉❛♥❞♦ f ′ (x) = 0✱ ❡♥tã♦ x = s❉❛   a β−1 ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ ❞❡r✐✈❛❞❛ s❡❣✉♥❞❛ ❞❡ f (x) é f ′′ (x) = β(β − 1)xβ−2 > 0 ♣❡❧❛ β ❤✐♣ót❡s❡ ❞❡ β ✳ ❖ ❝r✐tér✐♦ ❛✜r♠❛r q✉❡ f (x) ❛t✐♥❣❡ s❡✉ ✈❛❧♦r ♠í♥✐♠♦ ✐❣✉❛❧ s❞❛ ❞❡r✐✈❛❞❛ s❡❣✉♥❞❛ ♣❡r♠✐t❡ s ❛ (1 − β)  β a ✱ ♥♦ ♣♦♥t♦ x = β β−1 β−1   a ✳ β Pr♦♣r✐❡❞❛❞❡ ✻✳✹✳ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍♦❧❞❡r✳ ❙❡ 1 1 + = 1, p q p > 1, x>0 ❡ y > 0✱ t❡♠♦s xy ≤ xp y q + ✳ p q ❉❡♠♦♥str❛çã♦✳ P❡❧♦ ✭✻✳✷✵✮✱ s❡ β > 1, a > 0 ❡ x > 0✱ ♣❛r❛ ❛ ❢✉♥çã♦ f (x) = xβ − ax t❡♠♦s ❊①❡♠♣❧♦ q✉❡ f (x) ≥ f  rh i β−1 a β ✱ ✐st♦ é s  β a β−1 β x − ax > (1 − β) β ✭✻✳✷✮ ❈♦♥s✐❞❡r❛♥❞♦ ♥❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡ β = p ❡ a = py ✱ ❡♥❝♦♥tr❛♠♦s ❡♠ ✭✻✳✷✮ xp − (py)x > r (1 − p) p−1 h ip py p √ = (1 − p) p−1 y p ✳ 1 1 1 p−1 1 + = 1 r❡s✉❧t❛ = 1 − = p q q p p p q xp xq p x − pyx ≥ − y ❞❡ ♦♥❞❡ xy ≤ + ✳ q p q ❈♦♠♦ ⇒ q = p p , p − 1 = ✱ ❡♥tã♦ p−1 q ❊①❡♠♣❧♦ ✻✳✷✶✳ ❙❡❥❛ a > 0✱ ❛t✐♥❣❡ q✉❛♥❞♦ ❙♦❧✉çã♦✳ ♠♦str❡ q✉❡ ♦ ✈❛❧♦r ♠á①✐♠♦ ❞❛ ❢✉♥çã♦ x= 2+a ✳ 1+a f (x) = 1 1 + 1+ | x | 1+ | x − a |  1 1  + , s❡✱ x < 0    x 1+a−x  1− 1 1 ❚❡♠♦s f (x) = + , s❡✱ 0 < x < a  1+x 1+a−x   1   1 + , s❡✱ a < x 1+x 1−a+x ❞❡ ♦♥❞❡  1 1   + , s❡✱ x < 0   − x)2 (1 + a − x)2   (1 −1 1 + , s❡✱ 0 < x < a f ′ (x) = 2 (1 + x) (1 + a − x)2    −1 1    − , s❡✱ a < x 2 (1 + x) (1 − a + x)2 ✸✷✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❖❜s❡r✈❡✱ f (x) ❝r❡s❝❡ ♥♦ ✐♥t❡r✈❛❧♦ (−∞, 0) ❡ ❞❡❝r❡s❝❡ ♥♦ ✐♥t❡r✈❛❧♦ [a, +∞)✱ ❧♦❣♦ ♦ ♠á①✐♠♦ ❞❡ f (x) ❛❝♦♥t❡❝❡ ♥♦ ✐♥t❡r✈❛❧♦ [0, a]✳ a ◗✉❛♥❞♦ f ′ (x) = 0✱ ♣❛r❛ x ∈ (0, a) ❡♥tã♦ (1 + x)2 − (1 − x + a)2 = 0 ⇒ x= ✳ 4 2+a 2+a a < = f (0) = f (a)✱ ♦ ♠á①✐♠♦ é ✳ ❈♦♠♦ f ( ) = 2 2+a 1+a 1+a 2 ❊①❡♠♣❧♦ ✻✳✷✷✳ ❯♠ ❝♦♠❡r❝✐❛♥t❡ ✈❡♥❞❡ 2.000 ✉♥✐❞❛❞❡s ♣♦r ♠ês ❛♦ ♣r❡ç♦ ❞❡ R$10, 00 ❝❛❞❛✳ ❊❧❡ ♣♦❞❡ ✈❡♥❞❡r ♠❛✐s 250 ✉♥✐❞❛❞❡s ♣♦r ♠ês ♣❛r❛ ❝❛❞❛ R$0, 25 ❞❛ r❡❞✉çã♦ ♥♦ ♣r❡ç♦✳ ◗✉❛❧ ♦ ♣r❡ç♦ ✉♥✐tár✐♦ q✉❡ ♠❛①✐♠✐③❛rá ❛ r❡❝❡✐t❛❄ ❙♦❧✉çã♦✳ ❙❡❥❛ q ♦ ♥ú♠❡r♦ ❞❡ ✉♥✐❞❛❞❡s ✈❡♥❞✐❞❛s ❡♠ ✉♠ ♠ês✱ ❝♦♥s✐❞❡r❡♠♦s p ♦ ♣r❡ç♦ ✉♥✐tár✐♦✱ ❡ R ❛ r❡❝❡✐t❛ ♠❡♥s❛❧✱ s✉♣♦♥❞♦ ❡♠ ❝♦♥❞✐çõ❡s ❞❡ ❧✐✈r❡ ❝♦♥❝♦rrê♥❝✐❛✱ ❛ r❡❝❡✐t❛ é ❞❛❞❛ ♣♦r R = qp❀ q✉❛♥❞♦ p ♣r❡ç♦ p = 10 t❡♠♦s q = 2.000 ❡✱ q✉❛♥❞♦ p = 10, 00 − 0, 25 = 9, 75 t❡♠♦s q✉❡ q = 2.250✳ ❈♦♠ ❡st❛ ✐♥❢♦r♠❛çã♦ ♣♦❞❡♠♦s ♦❜t❡r ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s (10, 2.000) ❡ (9, 75; 2250) ♦♥❞❡ m= 10 − 9, 75 p − 10 = q − 2.000 2.000 − 2.250 ❧♦❣♦ p = −0, 001q +12❀ ❝♦♥s✐❞❡r❛♥❞♦ ❡st❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ♥❛ ❡q✉❛çã♦ ❞❛ r❡❝❡✐t❛ ♦❜t❡♠♦s R = q(−0, 001 × q + 12) ⇒ R′ = 12 − 0, 002q ⇒ q = 6.000 ♦❜s❡r✈❡✱ R′′ = −0, 002 < 0✳ ◗✉❛♥❞♦ q = 6.000 ♦ ♥í✈❡❧ ❞❡ ♣r♦❞✉çã♦ ♣r♦♣♦r❝✐♦♥❛ r❡❝❡✐t❛ ♠á①✐♠❛❀ ♥❡st❡ ❝❛s♦ p = 12 − 0, 001(6.000) = 6 r❡❛✐s✳ ✻✳✷✳✷ ❆ssí♥t♦t❛s ❊♠ ♠❛t❡♠át✐❝❛✱ ✉♠❛ ❛ssí♥t♦t❛ ❞❡ ✉♠❛ ❝✉r✈❛ C é ✉♠ ♣♦♥t♦ ♦✉ ✉♠❛ ❝✉r✈❛ ❞❡ ♦♥❞❡ ♦s ♣♦♥t♦s ❞❡ C s❡ ❛♣r♦①✐♠❛♠ à ♠❡❞✐❞❛ q✉❡ s❡ ♣❡r❝♦rr❡ C ✳ ◗✉❛♥❞♦ C é ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ❡♠ ❣❡r❛❧ ♦ t❡r♠♦ ❛ssí♥t♦t❛ r❡❢❡r❡✲s❡ ❛ ✉♠❛ r❡t❛✳ ❈♦♥s✐❞❡r❡♠♦s ✉♠❛ ❝✉r✈❛ q✉❛❧q✉❡r C ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ y = f (x)✱ ❡ ✉♠ ♣♦♥t♦ A q✉❡ s❡ ♠♦✈✐♠❡♥t❛ ❛♦ ❧♦♥❣♦ ❞❡ss❛ ❝✉r✈❛✳ ❉❡✜♥✐çã♦ ✻✳✺✳ ❉✐③❡♠♦s q✉❡ ♦ ♣♦♥t♦ A ∈ R2 t❡♥❞❡ ✭❝♦♥✈❡r❣❡✮ ❛♦ ✐♥✜♥✐t♦ s❡✱ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦ ♣♦♥t♦ A ❡ ❛ ♦r✐❣❡♠ ❞❡ ❝♦♦r❞❡♥❛❞❛s (0, 0) t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦ ✭❛ ❞✐stâ♥❝✐❛ ❝r❡s❝❡ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✮ ✸✷✸ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❡✜♥✐çã♦ ✻✳✻✳ ❙❡❥❛ A ✉♠ ♣♦♥t♦ q✉❡ s❡ ♠♦✈✐♠❡♥t❛ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ ❝✉r✈❛ y = f (x) ❡ d ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ A ❡ ✉♠❛ r❡t❛ L✳ ❙❡ ❛❝♦♥t❡❝❡ q✉❡ ♦ ♣♦♥t♦ A t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦ ❡✱ ❛ ❞✐stâ♥❝✐❛ d t❡♥❞❡ ❛ ③❡r♦✱ ❞✐③❡♠♦s q✉❡ ❛ r❡t❛ L é ✉♠❛ ❞❡ ❛ssí♥t♦t❛ ❞❛ ❝✉r✈❛ y = f (x)❀ ✐st♦ é lim d(A, L) = 0✳ ✭❋✐❣✉r❛ ✭✻✳✺✮✮ A→+∞ ❋✐❣✉r❛ ✻✳✺✿ Pr♦♣r✐❡❞❛❞❡ ✻✳✺✳ ❆ r❡t❛ x = a é ✉♠❛ ❛ssí♥t♦t❛ ✈❡rt✐❝❛❧ ❞❛ ❝✉r✈❛ y = f (x) s❡ ❝✉♠♣r❡ ✉♠ ❞♦s s❡❣✉✐♥t❡s ❡♥✉♥❝✐❛❞♦s✿ 1o 2o 3o lim .f (x) = ±∞ x→a lim .f (x) = +∞ ✭❋✐❣✉r❛ ✭✻✳✻✮✮ x→a+ lim .f (x) = −∞ ✭❋✐❣✉r❛ ✭✻✳✼✮✮ x→a− ❆ ❞❡♠♦♥str❛çã♦ ♦❜tê♠✲s❡ ❝♦♠ ❢❛❝✐❧✐❞❛❞❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❛ssí♥t♦t❛✳ lim .f (x) = −∞ lim .f (x) = +∞ x→a+ x→a− ❋✐❣✉r❛ ✻✳✻✿ ❋✐❣✉r❛ ✻✳✼✿ ✸✷✹ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ Pr♦♣r✐❡❞❛❞❡ ✻✳✻✳ ❆ r❡t❛ y=k é ✉♠❛ ❛ssí♥t♦t❛ ❤♦r✐③♦♥t❛❧ ❞❛ ❝✉r✈❛ y = f (x) s❡ ❝✉♠♣r❡ ✉♠ ❞♦s s❡❣✉✐♥t❡s ❡♥✉♥❝✐❛❞♦s✿ 1o 2o 3o lim .f (x) = k x→∞ lim .f (x) = k ✭❋✐❣✉r❛ ✭✻✳✽✮✮ x→−∞ lim .f (x) = k ✭❋✐❣✉r❛ ✭✻✳✾✮✮ x→+∞ ❆ ❞❡♠♦♥str❛çã♦ é ó❜✈✐❛✳ lim .f (x) = k lim .f (x) = k x→−∞ x→+∞ ❋✐❣✉r❛ ✻✳✽✿ ❋✐❣✉r❛ ✻✳✾✿ Pr♦♣r✐❡❞❛❞❡ ✻✳✼✳ ❆ r❡t❛ y = mx + b, m 6= 0 é ✉♠❛ ❛ssí♥t♦t❛ ♦❜❧íq✉❛ ❞❛ ❝✉r✈❛ y = f (x) s❡ ❡ s♦♠❡♥t❡ s❡ ❝✉♠♣r❡ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿ 1 o 2o   f (x) =m lim x→+∞ x   f (x) lim =m x→−∞ x ❉❡♠♦♥str❛çã♦✳ ❡ ❡ lim [f (x) − mx] = b x→+∞ lim [f (x) − mx] = b x→−∞ ✭❋✐❣✉r❛ ✭✻✳✶✵✮✮ ✭❋✐❣✉r❛ ✭✻✳✶✶✮✮ ✐✮ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ ❝✉r✈❛ y = f (x) t❡♥❤❛ ✉♠❛ ❛ssí♥t♦t❛ ♦❜❧íq✉❛ ❞❡ ❡q✉❛çã♦ y = mx+b✳ ❙❡❥❛ A(x, f (x)) ♦ ♣♦♥t♦ q✉❡ s❡ ♠♦✈✐♠❡♥t❛ ❛♦ ❧♦♥❣♦ ❞❛ ❝✉r✈❛ y = f (x) ❀ ❡ C(x, mx + b) ♦ ♣♦♥t♦ ❞❛ ❛ssí♥t♦t❛ ❞❡ ❛❜s❝✐ss❛ x✳ ❉❛ ❞❡✜♥✐çã♦ ❞❡ ❛ssí♥t♦t❛ t❡♠♦s q✉❡ lim AB = 0❀ ♣♦ré♠ x→+∞ AB = AC · cos α ❡ AC =| f (x) − (mx + b) | ♦♥❞❡ cos α é ✉♠❛ ❝♦♥st❛♥t❡ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳ ✸✷✺ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❋✐❣✉r❛ ✻✳✶✵✿ ❋✐❣✉r❛ ✻✳✶✶✿ ⇒ lim AB = 0 ▲♦❣♦✱ x→+∞ P♦rt❛♥t♦✱ t❡♠♦s lim AB = 0 x→+∞ lim [f (x) − mx] = b✳ lim [f (x) − (mx + b)] = 0 é ♦✉tr♦ ❧❛❞♦✱ ❞❡t❡r♠✐♥❡♠♦s m x→+∞ P♦r ó❜✈✐♦ q✉❡ ❛ r❡t❛ ❡ lim [f (x) − (mx + b)] = 0 ❡♥tã♦ ❞❡✈❡ ❛❝♦♥t❡❝❡r lim x→+∞  lim x→+∞ ❛ss✐♠✱ lim x→+∞ ❙❡♥❞♦ lim [f (x) − mx] = b✳ ♦✉tr♦ ❧❛❞♦✱ s❡ m ❡ b sã♦ ♦❜t❡♠♦s P♦r  f (x) = m✳ x lim ❞❡ é ✉♠❛ ❛ssí♥t♦t❛✳ lim x→+∞   f (x) (mx + b) ·x=0 − x x ♣♦✐s x → +∞ ❞❡ ♦♥❞❡✱    b f (x) − m − lim =0 x→+∞ x x m ❝♦♥❤❡❝✐❞♦ ❡ ❝♦♥s✐❞❡r❛♥❞♦ lim [f (x) − (mx + b)] = 0 x→+∞ x→+∞ x→+∞ ❡♥tã♦ ⇒  f (x) (mx + b) =0 − x x  y = mx + b b✳ x→+∞  lim [f (x) − (mx + b)] = 0✳ x→+∞ x→+∞ ❘❡❝í♣r♦❝❛♠❡♥t❡ ✭⇐✮ ❙❡ ⇒  ♥ú♠❡r♦s q✉❡ ❝✉♠♣r❡♠ ❛s ❝♦♥❞✐çõ❡s  f (x) =m x lim [f (x) − (mx + b)] = 0 x→+∞ ❡ lim [f (x) − mx] = b x→+∞ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ ❛ r❡t❛ y = mx + b é ✉♠❛ ❛ssí♥t♦t❛ y = f (x)✳ ❖❜s❡r✈❛çã♦ ✻✳✹✳ ❘❡s♣❡✐t♦ à Pr♦♣r✐❡❞❛❞❡ ✭✻✳✼✮ é ♥❡❝❡ssár✐♦ ♦ s❡❣✉✐♥t❡✿ ✐✮ ❙❡ ❛♦ ❝❛❧❝✉❧❛r ♦s ✈❛❧♦r❡s m ❡ b ✭q✉❛♥❞♦ x → +∞✮ ✉♠ ❞♦s ❧✐♠✐t❡s ♥ã♦ ❡①✐st❡✱ ❛ ❝✉r✈❛ ♥ã♦ ❛♣r❡s❡♥t❛ ❛ssí♥t♦t❛ ♦❜❧íq✉❛ à ❞✐r❡✐t❛✳ ❘❡s✉❧t❛❞♦ s✐♠✐❧❛r ♦❜tê♠✲s❡ q✉❛♥❞♦ x → −∞✳ ✸✷✻ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✐✐✮ ❙❡ m = 0 ❡ b é ✐♥✜♥✐t♦✱ ❛ ❛ssí♥t♦t❛ é ❤♦r✐③♦♥t❛❧✳ ❊①❡♠♣❧♦ ✻✳✷✸✳ ❉❡t❡r♠✐♥❡ ❛s ❛ssí♥t♦t❛s ❞❛ ❝✉r✈❛ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛s ❡q✉❛çõ❡s✿ ❛✮ f (x) = ❙♦❧✉çã♦✳❛✮ x2 + 4 √ + 3x x−2 ❜✮ g(x) = 5x2 − 8x + 3 x+5 ❖ ❞♦♠í♥✐♦ D(f ) = R − {2}✳   x2 + 4 √ 3 ❖ ❝á❧❝✉❧♦ ❞♦ ❧✐♠✐t❡ lim + x = ±∞✱ ❧♦❣♦ x = 2 é ❛ssí♥t♦t❛ ✈❡rt✐❝❛❧✳ x→2 x − 2  2  x +4 √ 3 ❖❜s❡r✈❡✱ lim + x = ±∞✱ ❧♦❣♦ ♥ã♦ t❡♠ ❛ssí♥t♦t❛ ❤♦r✐③♦♥t❛❧✳ x→+∞ x − 2 P❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ❛ssí♥t♦t❛ ♦❜❧íq✉❛ ✿  2  √ 3 x x +4 f (x) = lim + =1=m lim x→+∞ x(x − 2) x→+∞ x x  2  x +4 √ 3 lim [f (x) − mx] = lim + x − x = +∞✱ ❧♦❣♦ ♥ã♦ ❡①✐st❡ ❛ssí♥t♦t❛ ♦❜❧í✲ x→+∞ x→+∞ x − 2 q✉❛✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ♥ã♦ ❡①✐st❡ ❛ssí♥t♦t❛ ♦❜❧íq✉❛ q✉❛♥❞♦ x → −∞✳ ❙♦❧✉çã♦✳❜✮ ❖ ❞♦♠í♥✐♦ D(g) = R − {−5}✳ P♦ssí✈❡❧ ❛ssí♥t♦t❛ ✈❡rt✐❝❛❧✱ x = −5❀ ♦ ❝á❧❝✉❧♦ ❞♦ ❧✐♠✐t❡ lim x→−5 x = −5 é ❛ssí♥t♦t❛ ✈❡rt✐❝❛❧✳ 5x2 − 8x + 3 = ±∞✱ ❧♦❣♦ x+5 5x2 − 8x + 3 = ±∞✱ ❧♦❣♦ ♥ã♦ t❡♠ ❛ssí♥t♦t❛ ❤♦r✐③♦♥t❛❧✳ x→±∞ x+5 g(x) 5x2 − 8x + 3 P♦r ♦✉tr♦ ❧❛❞♦✱ lim = lim = 5 ❛❧é♠ ❞✐ss♦✱ x→+∞ x x→+∞ x(x + 5) ❖❜s❡r✈❡✱ lim lim [g(x) − 5x] = lim x→+∞ x→+∞   5x2 − 8x + 3 − 5x = −33 x+5 ❆ss✐♠ y = 5x − 33 é ❛ssí♥t♦t❛ ♦❜❧íq✉❛✳ g(x) 5x2 − 8x + 3 = lim = 5 t❛♠❜é♠ x→−∞ x x→−∞ x(x + 5) P❛r❛ ♦ ❝❛s♦ x → −∞✱ t❡♠♦s lim lim [g(x) − 5x = lim x→−∞ x→−∞   5x2 − 8x + 3 − 5x = −33 x+5 P♦rt❛♥t♦✱ y = 5x − 33 é ❛ ú♥✐❝❛ ❛ssí♥t♦t❛ ♦❜❧íq✉❛✳ ❊①❡♠♣❧♦ ✻✳✷✹✳ ✸✷✼ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❉❡t❡r♠✐♥❡ ❛s ❛ssí♥t♦t❛s ❞❛ ❝✉r✈❛ y = ❙♦❧✉çã♦✳ 2x2 − 7x + 1 ✱ ❡ tr❛ç❛r ♦s r❡s♣❡❝t✐✈♦s ❣rá✜❝♦s✳ x−2 ❖ ❞♦♠í♥✐♦ D(f ) = R − {2}✳ ■♥t❡rs❡❝çõ❡s ❝♦♠ ♦s ❡✐①♦s✳ ❛✮ ❈♦♠ ♦ ❡✐①♦✲y ✿ ❜✮ ❈♦♠ ♦ ❡✐①♦✲x✿ ❡ C( 7− √ 41 4 1 1 ❀ é ♦ ♣♦♥t♦ A(0, − ) 2 2 √ √ 7 ± 41 7 + 41 y = 0 ❡♥tã♦ x = ❀ sã♦ ♦s ♣♦♥t♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s B( , 0) 4 4 x = 0 ❡♥tã♦ f (0) = − , 0) ❈á❧❝✉❧♦ ❞❡ ❛ssí♥t♦t❛s ✿ ❛✮ ❱❡rt✐❝❛✐s✿ −5 2x2 − 7x + 1 lim+ = + = +∞ x→2 x−2 0 −5 2x2 − 7x + 1 = − = −∞ lim x→2− x−2 0 ▲♦❣♦ x = 2 é ❛ssí♥t♦t❛ ✈❡rt✐❝❛❧✳ ❜✮ ❍♦r✐③♦♥t❛✐s✿ ❚❡♠♦s lim .f (x) = ±∞✳ x→±∞ ▲♦❣♦ ♥ã♦ t❡♠ ❛ssí♥t♦t❛s ❤♦r✐③♦♥t❛✐s✳ ❝✮ ❖❜❧íq✉❛s✿ f (x) 2x2 − 7x + 1 ❋✐❣✉r❛ ✻✳✶✷✿ = lim = 2✳ x→±∞ x x→±∞ x(x − 2) 2x2 − 7x + 1 b = lim [f (x) − mx] = lim [ − 2x] = x→±∞ x→±∞ x−2 −3x + 1 = −3 lim x→±∞ x − 2 P♦r t❛♥t♦ ❛ r❡t❛ y = 2x − 3 é ✉♠❛ ❛ssí♥t♦t❛ à ❞✐r❡✐t❛ ❡ ❡sq✉❡r❞❛ ❞❛ ❝✉r✈❛ y = f (x)✳ lim ❖ ❣rá✜❝♦ ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✻✳✶✷✮✳ ❊①❡♠♣❧♦ ✻✳✷✺✳ ❉❡t❡r♠✐♥❡ ❛s ❛ssí♥t♦t❛s ❞❛ ❝✉r✈❛ ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ g(x) = ▼♦str❡ s❡✉ r❡s♣❡❝t✐✈♦ ❣rá✜❝♦✳ ❙♦❧✉çã♦✳ √ 3 x3 − 3x2 − 9x + 27✳ ❖ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ é D(g) = R✳ ❖❜s❡r✈❡✱ ♥ã♦ t❡♠♦s ❛ssí♥t♦t❛s ✈❡rt✐❝❛✐s❀ ♣♦✐s ♥ã♦ ❡①✐st❡ ♥ú♠❡r♦ a t❛❧ q✉❡ ♦ ❧✐♠✐t❡ lim .g(x) = ±∞ ✐st♦ é ♥ã♦ ❡①✐st❡ ✈❛❧♦r r❡❛❧ q✉❡ ❢❛③ ③❡r♦ ♦ ❞❡♥♦♠✐♥❛❞♦r✳ x→a ◆ã♦ t❡♠♦s ❛ssí♥t♦t❛s ❤♦r✐③♦♥t❛✐s❀ ♥ã♦ ❡①✐st❡ ♥ú♠❡r♦ c t❛❧ q✉❡ ♦ ❧✐♠✐t❡ lim .g(x) = c✳ x→±∞ ✸✷✽ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈á❧❝✉❧♦ ❞❡ ❛ssí♥t♦t❛s ♦❜❧íq✉❛s✿ √ 3 g(x) x3 − 3x2 − 9x + 27 = lim . =1 m = lim . x→±∞ x→±∞ x x √ 3 b = lim [ x3 − 3x2 − 9x + 27 − 1 · x] = x→±∞ lim x→±∞   −3x2 − 9x + 27 √ √ = −1 ( 3 x3 − 3x2 − 9x + 27)2 + x( 3 x3 − 3x2 − 9x + 27) + x2 ❆ r❡t❛ y = x − 1 é ❛ssí♥t♦t❛ ❞✐r❡✐t❛ ❡ ❡sq✉❡r❞❛✳ ❈á❧❝✉❧♦ ❞❡ ❡①tr❡♠♦s r❡❧❛t✐✈♦s✿ (x + 1)(x + 3) g ′ (x) = p ❡♥tã♦ x = −1, x = 3 ❡ x = −3 sã♦ ♣♦♥t♦s ❝rít✐❝♦s✳ ( 3 (x − 3)2 (x + 3))2 ❖❜s❡r✈❡✱ ♣❛r❛ h ♣♦s✐t✐✈♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ t❡♠♦s g ′ (−1+h) < 0 ❡ g ′ (−1−h) > √ 0✱ ❧♦❣♦ t❡♠♦s ♠á①✐♠♦ r❡❧❛t✐✈♦ ♥♦ ♣♦♥t♦ A(−1, 3 32)❀ ♣♦r ♦✉tr♦ ❧❛❞♦✱ g ′ (3 − h) < 0 ❡ g ′ (3 + h) > 0✱ ❧♦❣♦ ❡♠ B(3, 0) t❡♠♦s ♠í♥✐♠♦ r❡❧❛t✐✈♦✳ ❖ ❣rá✜❝♦ ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✻✳✶✸✮✳ ❊①❡♠♣❧♦ ✻✳✷✻✳ ❚r❛ç❛r ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f (x) = ❙♦❧✉çã♦✳ √ 4 x4 − 5x3 − 4x2 + 20x ♠♦str❛♥❞♦ ❛s ❛ssí♥t♦t❛s✳ ❖ ❞♦♠í♥✐♦ ❞❡ ❞❡✜♥✐çã♦ é✱ D(f ) = { x ∈ R /. x4 − 5x3 − 4x2 + 20x ≥ 0 }✱ ✐st♦ é D(f ) = (−∞, −2] ∪ [0, 2] ∪ [5, +∞)✳ ■♥t❡rs❡❝çõ❡s ❝♦♠ ❡✐①♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s sã♦ ♦s ♣♦♥t♦s (−2, 0), (0, 0), (2, 0) ❡ (5, 0)✳ ◆ã♦ tê♠ ❛ssí♥t♦t❛s ✈❡rt✐❝❛✐s ♥❡♠ ❤♦r✐③♦♥t❛✐s✳ ❋✐❣✉r❛ ✻✳✶✸✿ ❋✐❣✉r❛ ✻✳✶✹✿ ❈á❧❝✉❧♦ ❞❡ ❛ssí♥t♦t❛s ♦❜❧íq✉❛s✿ f (x) m = lim = lim x→+∞ x x→+∞ √ 4 x4 − 5x3 − 4x2 + 20x =1 x ✸✷✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R √ 5 4 b = lim [f (x) − 1 · x = lim [ x4 − 5x3 − 4x2 + 20x − x] = − x→+∞ x→+∞ 4 5 ❆ r❡t❛ y = x − é ❛ssí♥t♦t❛ ♦❜❧íq✉❛ à ❞✐r❡✐t❛✳ 4 # "r 20 4 |x| 4 f (x) 5 = lim P♦r ♦✉tr♦ ❧❛❞♦✿ m = lim 1 − − 2 + 3 = −1✳ x→−∞ x x→−∞ x x x x √ 5 4 b = lim [f (x) − 1 · x] = lim [ x4 − 5x3 − 4x2 + 20x − (−1)x] = x→−∞ x→−∞ 4 5 ❆ r❡t❛ y = −x + é ❛ssí♥t♦t❛ ♦❜❧íq✉❛ à ❡sq✉❡r❞❛✳ 4 ❖ ❣rá✜❝♦ ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✻✳✶✹✮✳ ❊①❡♠♣❧♦ ✻✳✷✼✳ ❈♦♥str✉✐r ♦ ❣rá✜❝♦ ❞❛ ❝✉r✈❛ y = g(x)✱ ♠♦str❛♥❞♦ s✉❛s ❛ssí♥t♦t❛s✳ ❙♦❧✉çã♦✳  r x+3   , s❡✱ x > 0    x 3 x −x g(x) = , s❡✱ − 3 < x ≤ 0   (x√+ 1)(x + 4)    − 1 + x2 , s❡✱ x ≤ −3 ❖ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ D(g) = R − {−1}✳ ❈á❧❝✉❧♦ ❞❡ ❛ssí♥t♦t❛s ❤♦r✐③♦♥t❛✐s✿ lim .g(x) = lim x→+∞ x→+∞ r x+3 =1 x ❡ √ lim .g(x) = lim (− 1 + x2 ) = −∞ x→−∞ x→−∞ ❆ ú♥✐❝❛ ❛ssí♥t♦t❛ ❤♦r✐③♦♥t❛❧ é y = 1✳ ❈á❧❝✉❧♦ ❞❡ ❛ssí♥t♦t❛s ✈❡rt✐❝❛✐s✿ ❆s ♣♦ssí✈❡✐s ❛ssí♥t♦t❛s ✈❡rt✐❝❛✐s sã♦ ♦s ✈❛❧♦r❡s ❞❡ x ♣❛r❛ ♦s q✉❛✐s ♦ ❞❡♥♦♠✐♥❛❞♦r é ③❡r♦ ❡ ❡st❡s ✈❛❧♦r❡s sã♦✿ x = 0, x = −1, ❡ x = 4✳ ❖s ❧✐♠✐t❡s lim+ x→0 r x+3 = +∞; x 2 x3 − x = x→−1 (x + 1)(x + 4) 3 lim ❡♠ x = −4 ♥ã♦ t❡♠ s❡♥t✐❞♦ ❝❛❧❝✉❧❛r ♣❡❧♦ ❢❛t♦ ❡st❛r ❞❡✜♥✐❞❛ g(x) ♥♦ ✐♥t❡r✈❛❧♦ r❡❛❧ (−3, 0]✳ ▲♦❣♦ ❛ ú♥✐❝❛ ❛ssí♥t♦t❛ ✈❡rt✐❝❛❧ é x = 0✳ ❆ssí♥t♦t❛s ♦❜❧íq✉❛s✿ ◆ã♦ ❡①✐st❡ ❛ssí♥t♦t❛ ♦❜❧íq✉❛ à ❞✐r❡✐t❛✱ ♣♦✐s ❥á ❡①✐st❡ ✉♠❛ ❛ssí♥t♦t❛ ❤♦r✐③♦♥t❛❧✳ √ − 1 + x2 g(x) = lim = lim m = lim x→−∞ x→−∞ x→−∞ x x ✸✸✵ −|x| q x 1 x2 +1 =1 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ √ b = lim [g(x) − mx] = lim [− 1 + x2 − x] = lim √ x→−∞ x→−∞ x→−∞ ❆ ú♥✐❝❛ ❛ssí♥t♦t❛ ♦❜❧íq✉❛ é y = x✳ ❖ ❣rá✜❝♦ ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✻✳✶✺✮ −1 =0 1 + x2 − x ❋✐❣✉r❛ ✻✳✶✺✿ ❖❜s❡r✈❛çã♦ ✻✳✺✳ ❙❡ ❛ ❡q✉❛çã♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❡s❝r❡✈❡✲s❡ ♥❛ ❢♦r♠❛ x = g(y)✱ ♣❛r❛ ♦❜t❡r ❛ssí♥t♦t❛s ✉t✐❧✐③❛✲ ♠♦s ♦s r❡s✉❧t❛❞♦s ❞❛s Pr♦♣r✐❡❞❛❞❡s ✭✻✳✺✮ ✲ ✭✻✳✼✮ ♠♦❞✐✜❝❛♥❞♦ ❛s ✈❛r✐á✈❡✐s ❝♦rr❡s♣♦♥❞❡♥t❡s✳ ❉❡st❡ ♠♦❞♦✿ ✐✮ ❙❡ lim .g(y) = k ♦✉ s❡ lim .g(y) = k ❡♥tã♦ ❛ r❡t❛ x = k é ✉♠❛ ❛ssí♥t♦t❛ ✈❡rt✐❝❛❧✳ ✐✐✮ y→+∞ y→−∞ ❙❡ ❡①✐st❡ a ∈ R t❛❧ q✉❡ lim .g(y) = ±∞, lim+ .g(y) = ±∞ ♦✉ lim− .g(y) = ±∞✱ ❡♥tã♦ y→a y→a y→a ❛ r❡t❛ y = a é ✉♠❛ ❛ssí♥t♦t❛ ❤♦r✐③♦♥t❛❧✳ ✐✐✐✮ ❆ r❡t❛ x = ky + b é ✉♠❛ ❛ssí♥t♦t❛ ♦❜❧íq✉❛ s❡✿ lim . g(y) =k ❡ y y→+∞ lim . g(y) =k ❡ y y→−∞ y→+∞ y→−∞ lim [g(y) − ky] = b ♦✉ lim [g(y) − ky] = b ❊①❡♠♣❧♦ ✻✳✷✽✳ ❚r❛ç❛r ♦ ❣rá✜❝♦ ❞❛ ❝✉r✈❛ y 3 − y 2 x + y 2 + x = 0✱ ♠♦str❛♥❞♦ s✉❛s ❛ssí♥t♦t❛s✳ ❙♦❧✉çã♦✳ y 2 (y + 1) (y + 1)(y − 1) ❆ ✈❛r✐á✈❡❧ y ✭✐♠❛❣❡♠ ❞❛ ❢✉♥çã♦ y = f −1 (x) ♣❡rt❡♥❝❡ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s R − { −1, 1 }✳ ❉❛ ❡q✉❛çã♦ ❛ ❝✉r✈❛ t❡♠♦s✱ x = f (y) = ✸✸✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❆ssí♥t♦t❛s ✈❡rt✐❝❛✐s✿ ❖❜s❡r✈❡ ♦ ❧✐♠✐t❡✱ lim .f (y) = −∞❀ ❧♦❣♦ ♥ã♦ ❡①✐st❡ ❛ssí♥t♦t❛s ✈❡rt✐❝❛✐s✳ y→−∞ ❆ssí♥t♦t❛s ❤♦r✐③♦♥t❛✐s✿ ❙ã♦ ♣♦ssí✈❡✐s ❛ssí♥t♦t❛s ❤♦r✐③♦♥t❛✐s y = −1 ❡ y = 1✳ 1 y 2 (y + 1) =− y→−1 (y + 1)(y − 1) 2 lim y 2 (y + 1) lim = +∞ y→1+ (y + 1)(y − 1) ❡ ❡♥tã♦ ❛ ú♥✐❝❛ ❛ssí♥t♦t❛ ❤♦r✐③♦♥t❛❧ é y = 1✳ y 2 (y + 1) lim = −∞ y→1− (y + 1)(y − 1) ❋✐❣✉r❛ ✻✳✶✻✿ ❆ssí♥t♦t❛s ♦❜❧íq✉❛s✿ y 2 (y + 1) g(y) = lim =1 y→±∞ y(y + 1)(y − 1) y→±∞ y k = lim y 2 (y + 1) − y] = 1 y→±∞ (y + 1)(y − 1) b = lim [g(y) − ky] = lim y→±∞ ❧♦❣♦ ❛ ú♥✐❝❛ ❛ssí♥t♦t❛ é x = y + 1✳ ❖ ❣rá✜❝♦ ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✻✳✶✻✮ ❊①❡♠♣❧♦ ✻✳✷✾✳ ❉❡t❡r♠✐♥❡ ❛s ❝♦♥st❛♥t❡s L = lim x→+∞  m ❡ n q✉❡ ❝✉♠♣r❡♠ ❛ ❝♦♥❞✐çã♦✿  √ 15x3 + 7x + 4 √ 2 3 3 2 − x + 4x − 8x + 12x + 1 + 2mx − 3n = 0 3x2 + 4 ❙♦❧✉çã♦✳ ❚❡♠♦s L = lim x→+∞   √ 15x3 + 7x + 4 √ 2 3 3 2 − x + 4x − 8x + 12x + 1 + 2mx − 3n = 0 3x2 + 4 ✸✸✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❡♥tã♦ L = lim x→+∞   15x3 + 7x + 4 + (2m − 3)x − 3n − 3x2 + 4 − lim x→+∞ L = lim x→+∞  h√ i √ 3 ( x2 + 4x − x) + ( 8x3 + 12x2 + 1 − 2x) = 0  x3 (6m + 6) − 9nx2 + x(8m − 5) + (4 − 12n) − 3x2 + 4 # " 4x 12x2 + 1 =0 − lim √ + √ x→+∞ x2 + 4x + x ( 3 8x3 . . .)2 + (2x)() + (2x)2 ◗✉❛♥❞♦ m = −1✱ ❧♦❣♦ ❝❛❧❝✉❧❛♥❞♦ ♦ ❧✐♠✐t❡ ❞❛ s❡❣✉♥❞❛ ♣❛r❝❡❧❛ L = lim x→+∞     −9nx2 − 13x + (4 − 12n) 4 12 − =0 + 3x2 + 4 2 12 ◗✉❛♥❞♦ n = −1✱ ❧♦❣♦ ❛♦ ❝❛❧❝✉❧❛r ♦ ❧✐♠✐t❡ ❞❛ ♣r✐♠❡✐r❛ ♣❛r❝❡❧❛ L = lim x→+∞   9x2 − 13x + 16 −3=3−3=0 3x2 + 4 P♦rt❛♥t♦✱ ♦s ♥ú♠❡r♦s sã♦✿ m = −1 ❡ n = −1✳ ❊①❡♠♣❧♦ ✻✳✸✵✳ ❉❡t❡r♠✐♥❡ ♦ ❣rá✜❝♦ ❞❛ ❝✉r✈❛ y 3 x2 − y 2 + y + 2 = 0✱ ♠♦str❛♥❞♦ s✉❛s ❛ssí♥t♦t❛s✳ ❙♦❧✉çã♦✳ s 2 y − y − 2 y2 − y − 2 ❖❜s❡r✈❡✱ x2 = ✱ ❞❡ ♦♥❞❡ x = ± ✳ y3 y3 ❆♦ s✉❜st✐t✉✐r x ♣♦r −x ♥❛ ❡q✉❛çã♦ ♦r✐❣✐♥❛❧✱ s ❛ ♠❡s♠❛ ♥ã♦ ✈❛r✐❛✱ ❧♦❣♦ é s✐♠étr✐❝❛ y2 − y − 2 r❡s♣❡✐t♦ ❞♦ ❡✐①♦✲y ❀ ❡♥tã♦ é s✉✜❝✐❡♥t❡ ❛♥❛❧✐s❛r x = ✳ y3 2y 3 x ❉❡r✐✈❛♥❞♦ ✐♠♣❧í❝✐t❛♠❡♥t❡✱ 3y 2 x2 y ′ + 2y 3 x − 2yy ′ + y ′ = 0✱ ❧♦❣♦ y ′ = 1 + 2y − 3y 2 x2 ❡♥tã♦ x = 0 é ✉♠ ♣♦♥t♦✳ ◗✉❛♥❞♦ x = 0, y = 2 ♦✉ y = −1❀ ❡♠ (0, 2) t❡♠♦s ♠á①✐♠♦ r❡❧❛t✐✈♦ ❡✱ ❡♠ (0, −1) t❡♠♦s ♠í♥✐♠♦ r❡❧❛t✐✈♦✳s ❈♦♥s✐❞❡r❛♥❞♦ x = S y2 ✐♥t❡r✈❛❧♦ [−1, 0) [2, +∞)✳ ❖ ❧✐♠✐t❡ lim y→0 s −y−2 = y3 s (y − 2)(y + 1) ❡♥tã♦ ❛ ✐♠❛❣❡♠ y ♣❡rt❡♥❝❡ ❛♦ y3 y2 − y − 2 = +∞❀ ❧♦❣♦ y = 0 é ❛ssí♥t♦t❛ ❤♦r✐③♦♥t❛❧✳ y3 ✸✸✸ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ P♦r ♦✉tr♦ ❧❛❞♦✱ lim y→+∞ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ s R y2 − y − 2 = 0 ❡♥tã♦ x = 0 é ❛ ú♥✐❝❛ ❛ssí♥t♦t❛ ❤♦r✐③♦♥t❛❧✳ y3 ◆ã♦ t❡♠ ❛ssí♥t♦t❛s ♦❜❧íq✉❛s✱ s❡✉ ❣rá✜❝♦ ♠♦str❛✲s❡ ♥❛ ❋✐❣✉r❛ ✭✻✳✶✼✮✳ ❋✐❣✉r❛ ✻✳✶✼✿ ❖❜s❡r✈❛çã♦ ✻✳✻✳ P❛r❛ ♦ ❣rá✜❝♦ ❞❡ ❝✉r✈❛s ♣♦❞❡♠♦s ✉t✐❧✐③❛r r❡❝✉rs♦s ❛❞✐❝✐♦♥❛✐s ❞❡ ♣♦♥t♦s ❝rít✐❝♦s ❡✱ ♦✉ ❝r✐tér✐♦s ❞❛ ❞❡r✐✈❛❞❛✳ ✸✸✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✻✲✷ ✶✳ ❉❡t❡r♠✐♥❛r ♦s ✐♥t❡r✈❛❧♦s ❞❡ ❝r❡s❝✐♠❡♥t♦✱ ♦s ❡①tr❡♠♦s r❡❧❛t✐✈♦s ❡ ❡s❜♦ç❛r ♦ ❣rá✜❝♦ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✳ f (x) = x3 + 2x2 − 4x + 2 √ √ 3 3 f (x) = 5 x2 − x5 x 5. f (x) = 2 x +1 x2 + 2 7. g(x) = 2 x − 4x 9. g(x) = 3x5 − 125x3 + 2160x p p 11. h(x) = 3 (x + 2)2 − 3 (x − 2)2 x2 + 2x − 33 13. f (x) = x−4 1 − x + x2 15. f (x) = 1 + x + x2 1. 3. 2. 4. 6. 8. 10. 12. 14. 16. f (x) = x4 − 14x2 − 24x + 1 f (x) =| x2 − 9 | x+1 f (x) = 2 x +x+1 x2 − 5x + 6 f (x) = 2 x − 4x√− 5 3 g(x) = (x − 1) x2 3 g(x) = x3 + x x2 + x + 1 f (x) = 1 − x + x2 1 − x + x2 f (x) = 1 + x − x2 ✷✳ ❙✉♣♦♥❤❛♠♦s ai > 0, i = 1, 2, . . . n✱ ♣❛r❛ ♦s s❡❣✉✐♥t❡s ❡①❡r❝í❝✐♦s ❞❡t❡r♠✐♥❡ ♦s ♣♦♥t♦s ❞❡ ♠á①✐♠♦ ♦✉ ♠í♥✐♠♦✱ ❝❛s♦ ❡①✐st✐r✳ ✶✳ f (x) = (a1 − x)2 + (a2 − x)2 ♣❛r❛ a 6= b✳ ✷✳ ✸✳ ✹✳ f (x) = (a1 − x)2 + (a2 − x)2 + (a3 − x)2 + · · · + (an − x)2 f (x) = (a1 − 2x2 )2 + (a2 − 2x2 )2 + (a3 − 2x2 )2 + · · · + (an − 2x2 )2 ✳ f (x) = (a1 − x)m + (a2 − x)m + (a3 − x)m + · · · + (an − x)m ✳ ✸✳ ❉❡t❡r♠✐♥❡ ♦s ✐♥t❡r✈❛❧♦s ❞❡✱ ❝r❡s❝✐♠❡♥t♦ ❡ ❞❡❝r❡s❝✐♠❡♥t♦ ♣❛r❛ ❛s ❢✉♥çõ❡s✿ ✶✳ y = x(1 + √ x) ✷✳ y = x − 2senx✱ s❡ 0 ≤ x ≤ 2π ✹✳ ❆♥❛❧✐s❛r ♦s ❡①tr❡♠♦s ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ ✶✳ y = (x − 5)ex ✷✳ ✸✳ y = (x − 1)4 ✹✳ ✺✳ ❉❡t❡r♠✐♥❛r ♦s ✈❛❧♦r❡s a, b ❡ c s❡✿ ✶✳ √ y = x 1 − x2 p y = 1 − 3 (x − 2)4 f (x) = 2x3 + ax2 + b t❡♠ ❡①tr❡♠♦ r❡❧❛t✐✈♦ ❡♠ (−1, 2)✳ g(x) = ax2 + bx + c t❡♠ ❡①tr❡♠♦ r❡❧❛t✐✈♦ ❡♠ (1, 7) ❡ ♦ ❣rá✜❝♦ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ (2, −2)✳ ✷✳ ✸✸✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✻✳ P❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✱ ❞❡t❡r♠✐♥❡ ♦ ♠á①✐♠♦ ♦✉ ♠í♥✐♠♦ ❛❜s♦❧✉t♦ ♥♦s ✐♥t❡r✈❛❧♦s ✐♥❞✐❝❛❞♦s✳ 1 1 1 1 ✷✳ f (x) = ❡♠ [− , 1] ✶✳ f (x) = 3x4 − 8x3 + 6x2 ❡♠ [− , ] ✸✳ 2 2 1 ❡♠ [−0, 5] f (x) = 5 x +x+1 ✹✳ 2 2 x+1 1 f (x) = 2 ❡♠ [−1, ] x +1 2 ✼✳ ❉❡t❡r♠✐♥❡ ♦ r❛✐♦ ❞❛ ❜❛s❡ ❡ ❛ ❛❧t✉r❛ h ❞❡ ✉♠ ❝✐❧✐♥❞r♦ r❡t♦ ❝♦♠ ✈♦❧✉♠❡ ❝♦♥st❛♥t❡ V ✱ ❞❡ ♠♦❞♦ q✉❡ s✉❛ ár❡❛ t♦t❛❧ s❡❥❛ ♠í♥✐♠❛✳ ✽✳ ❉❡t❡r♠✐♥❡ ❡①tr❡♠♦s r❡❧❛t✐✈♦s ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿   0 s❡✱ x é ✐rr❛❝✐♦♥❛❧ f (x) = 1 p  s❡✱ x = é ❢r❛çã♦ ✐rr❡❞✉tí✈❡❧, q q   1, s❡✱ x = 1 ♣❛r❛ ❛❧❣✉♠ n ∈ N f (x) = n  0, ♥♦s ❞❡♠❛✐s ❝❛s♦s ✶✳ ✷✳ p, q ∈ Z+ ✾✳ ❆❝❤❛r ♦s ❧❛❞♦s ❞♦ r❡tâ♥❣✉❧♦✱ ❞❡ ♠❛✐♦r ár❡❛ ♣♦ssí✈❡❧✱ ✐♥s❝r✐t♦ ♥❛ ❡❧✐♣s❡✿ a2 y 2 = a2 b2 ✳ b2 x 2 + ✶✵✳ ❙❡❥❛♠ ♦s ♣♦♥t♦s A(1, 4) ❡ B(3, 0) ❞❛ ❡❧✐♣s❡ 2x2 + y 2 = 18✳ ❆❝❤❛r ✉♠ t❡r❝❡✐r♦ ♣♦♥t♦ C ♥❛ ❡❧✐♣s❡ t❛❧ q✉❡✱ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ABC s❡❥❛ ❛ ♠❛✐♦r ♣♦ssí✈❡❧✳ ✶✶✳ ❊♥tr❡ ♦s r❡tâ♥❣✉❧♦s ❞❡ ♣❡rí♠❡tr♦ 10✱ q✉❛❧ ❞❡❧❡s é ❛q✉❡❧❡ q✉❡ t❡♠ ♠❛✐♦r ár❡❛❄ ✶✷✳ P❛r❛ ♦s s❡❣✉✐♥t❡s ❡①❡r❝í❝✐♦s✱ tr❛ç❛r ♦ ❣rá✜❝♦ ❞❛ ❝✉r✈❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ✐♥❞✐❝❛♥❞♦ s✉❛s ❛ssí♥t♦t❛s✳ 1. f (x) = √ 1 + x2 + 2x x−5 − 7x + 10 x2 + 9 5. f (x) = (x − 3)2 √ 7. f (x) = x + x2 − x r x4 − 5x2 + 4 9. f (x) = x2 + 2x − 24 x2 + 2x + 1 11 f (x) = r x 2 4 21 + 4x − x 13. f (x) = r x2 + 7x − 8 9x2 − 6x − 8 15. f (x) = r 16x2 + 4x − 6 16x2 + 4x − 6 17. f (x) = 9x2 − 6x − 8 3. f (x) = x2 19. y 3 − 6x2 + x3 = 0 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. ✸✸✻ 1 − x2 x2 − 4 r 7x2 − x3 + x − 7 6 f (x) = x3 − 9x2 − 9x + 81 √ f (x) = 3 x3 − 5x2 − 25x + 125 √ f (x) = 4 x4 − x3 − 9x2 + 9x √ 3x3 + 3x + 1 2 f (x) = 4 + x + 2 x +x−6 √ 5 + 4x4 − 6x5 4 f (x) = 36x + 5 + 3 x − 6x2 − 4x + 24 5 4 √ x − 5x + 1 3 f (x) = 4 − x3 + 1 2 x − 11x − 80 √ x2 − x3 + 1 f (x) = 4 + x2 + x2 + 1 r 6 4 2 4 x − 9x − x + 9 f (x) = +x r x2 − 25 6 4 2 4 x − 9x − x + 9 f (x) = x − x2 − 25 f (x) = 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✷✶✳ ✷✷✳ ✷✸✳ ✷✹✳ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R  x2   √ , s❡✱ | x |< 1 1 − x2 f (x) =   3x + 3x, s❡✱ | x |≥ 1 2x + 1    5x − 17   , s❡✱ x ≤ −3 x+   x+3  | 10x − 1 | +50x2 − 19 f (x) = , s❡✱ − 3 < x < 1   (x − 2)(x2 + 4x + 3)   √  4 8x − 8x6 − 5x2 , s❡✱ x ≥ 1    2   2+ , s❡✱ x ≤ −3    x  r  3 2x + 2 f (x) = , s❡✱ − 3 < x < 1  x − 1    (x − 1)3   , s❡✱ x ≥ 1  (x + 1)2  r  2+x   x· , s❡✱ | x |< 2 2−x f (x) = 2x2    2 , s❡✱ | x |≥ 2 x +x ✶✸✳ ❈♦♥str✉✐r ♦ ❣rá✜❝♦ ❞❛s s❡❣✉✐♥t❡s ❝✉r✈❛s✱ ♠♦str❛♥❞♦ s✉❛s ❛ssí♥t♦t❛s✳ 1. y 3 = (x − a)2 (x − c), a > 0, c > 0 3. x3 − 2y 2 − y 3 = 0 5. 4x3 = (a + 3x)(x2 + y 2 ), 2. y 2 (x − 2a) = x3 − a3 4. xy 2 + yx2 = a3 , a>0 7. x2 (x − y)2 = y 4 − 1 a>0 6. x2 (x − y)2 = a2 (x2 + y 2 ) ✶✹✳ P❛r❛ ❝❛❞❛ ❡①❡r❝í❝✐♦✱ ❞❡t❡r♠✐♥❡ ❛s ❝♦♥st❛♥t❡s m ❡ n q✉❡ ❝✉♠♣r❡♠ ❛ ❝♦♥❞✐çã♦✿ ✶✳ ✷✳ ✸✳ ✹✳ ✺✳ ✻✳ " # √ x2 − 3 3 x2 + 1 + 3 lim − mx − n = 0 x→+∞ x−3 # " √ x2 + 3 3 x3 + 1 + 5 − mx − n = 0 lim x→+∞ x+3 " # √ √ 5x3 − 4 x8 + 1 − 3 x6 + 1 + 1 lim − mx − n = 0 x→+∞ x2 − 4 # " √ √ 5x3 + 4 x8 + 1 + 3 x6 + 1 + 5 − mx − n = 0 lim x→+∞ x2 + 4 " # √ √ 6x4 + 4 4 x12 + 1 − x3 − 3 x9 + 1 + 7 lim − 3mx − 2n = 0 x→+∞ x3 − 8 # " √ √ 6x4 + 5 4 x12 + 1 − 7x3 − 3 x9 + 1 − 9 − 2mx − 3n = 0 lim x→+∞ x3 − 8 ✸✸✼ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✼✳ lim x→+∞ " ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ 20x3 + 15x2 + 6 √ 2 − 4x + 5x + 3x2 + 4 r 3 R # 8x5 + 3x + 1 + 4mx + 17n = 0 x2 + 1 ✶✺✳ P❛r❛ ❝❛❞❛ ✉♠ ❞♦s s❡❣✉✐♥t❡s ❡①❡r❝í❝✐♦s✱ ❝❛❧❝✉❧❛r ❛ssí♥t♦t❛s✱ ♣♦♥t♦s ❞❡ ♠á①✐♠♦ ♦✉ ♠í♥✐♠♦s ❡ ❞❡s❡♥❤❛r ❛ r❡❣✐ã♦ ✶✳ ✷✳ ✸✳ ✹✳ ✺✳ ✻✳ ✼✳ ✽✳ ✾✳ ✶✵✳ ✶✶✳ ✶✷✳ ✶✸✳ ✶✹✳ ✶✺✳ ✶✻✳ ✶✼✳ ✶✽✳ ✶✾✳ ✷✵✳ ✷✶✳ ✷✷✳ A✿ π π x=− , x= , y =0} 2 2 2 2 A = { (x, y) ∈ R /. y = x + 2x − 3, x = −2, x = 0, y = 0 } A = { (x, y) ∈ R2 /. y = cos x, A = { (x, y) ∈ R2 /. y = 9 − x2 , y = x2 + 1 } x2 − x , y = 0, x = −1, x = 2 } A = { (x, y) ∈ R2 /. y = 1 + x2 A = { (x, y) ∈ R2 /. y = 3x − x2 , y = x2 − x } 2 A = { (x, y) ∈ R2 /. y = tan x, x = 0, y = cos x } 3 A = { (x, y) ∈ R2 /. y = x3 + x, x = 0, y = 2, y = 0 } 3x A = { (x, y) ∈ R2 /. y = arctan x, y = arccos , y = 0 } 2 2 A = { (x, y) ∈ R /. y = arcsenx, y = arccos x, x = 1 } A = { (x, y) ∈ R2 /. y = x3 − 3x2 + 2x + 2, A = { (x, y) ∈ R2 /. y = 4 − Ln(x + 1), A = { (x, y) ∈ R2 /. y 2 − x = 0, y = 2x2 − 4x + 2 } y = Ln(x + 1), x = 0 } y − x3 = 0, x + y − 2 = 0 } A = { (x, y) ∈ R2 /. y(x2 + 4) = 4(2 − x), y = 0, x = 0 } A = { (x, y) ∈ R2 /. y = x3 + x − 4, A = { (x, y) ∈ R2 /. y = ex , y = x, y = 8 − x } y = e−x , x = 1 } A = { (x, y) ∈ R2 /. y = 2x + 2, x = y 2 + 1, x = 0, y = 0, x = 2 } A = { (x, y) ∈ R2 /. y =| x − 2 |, y + x2 = 0, x = 1, x = 3 } √ A = { (x, y) ∈ R2 /. y = x2 − 3, y =| x − 1 |, y = 0 } A = { (x, y) ∈ R2 /. y =| senx | x2 − 4 A = { (x, y) ∈ R2 /. y = 2 , x − 16 A = { (x, y) ∈ R2 /. y = arcsenx, A = { (x, y) ∈ R2 /. y = tan2 x, ♣❛r❛ x ∈ [0, 2π], y = −x, x = 2π } x = −3, x = 3, y = 0 } y = arccos x, x = 0 } π y = 0, x = , x = 0 } 3 ✶✻✳ ❈❛❧❝✉❧❡ ❛s ❞✐♠❡♥sõ❡s ❞♦ r❡tâ♥❣✉❧♦ ❞❡ ♣❡rí♠❡tr♦ ♠á①✐♠♦ q✉❡ ♣♦❞❡ s❡r ✐♥s❝rt♦ ❡♠ ✉♠❛ s❡♠✐❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ r✳ ✶✼✳ ❆❝✉♠✉❧❛✲s❡ ❛r❡✐❛ ❡♠ ❢♦r♠❛ ❝ô♥✐❝❛ ❛ r❛③ã♦ ❞❡ 10dm3 /min✳ ❙❡ ❛ ❛❧t✉r❛ ❞♦ ❝♦♥❡ é s❡♠♣r❡ ✐❣✉❛❧ ❛ ❞♦✐s ✈❡③❡s ♦ r❛✐♦ ❞❡ s✉❛ ❜❛s❡✱ ❛ q✉❡ r❛③ã♦ ❝r❡s❝❡ ❛ ❛❧t✉r❛ ❞♦ ❝♦♥❡ q✉❛♥❞♦ ❡st❛ é ✐❣✉❛❧ ❛ 8dm❄ ✸✸✽ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✻✳✸ ❋♦r♠❛s ✐♥❞❡t❡r♠✐♥❛❞❛s ❚r❛t❛r❡♠♦s ❞❛s r❡❣r❛s ❞❡ ▲✬❍♦s♣✐t❛❧ q✉❡ ♣❡r♠✐t❡♠ ❝❛❧❝✉❧❛r ❧✐♠✐t❡s ❞❛ ❢♦r♠❛✿ ∞ , ∞ 0 , 0 ❚❡♦r❡♠❛ ✻✳✶✳ 0 · ∞, 00 , ∞∞ , 1∞ ❞❡ ❈❛✉❝❤②✳ ❙❡❥❛♠ ❛s ❢✉♥çõ❡s r❡❛✐s f (x) ❡ ❛✮ ❙❡❥❛♠ ❝♦♥tí♥✉❛s ♥♦ ✐♥t❡r✈❛❧♦ ❜✮ ❙❡❥❛♠ ❞❡r✐✈á✈❡✐s ❡♠ ❝✮ g ′ (x) 6= 0 ∞ − ∞, g(x)✱ t❛✐s q✉❡✿ [a, b]✳ (a, b)✳ ∀ x ∈ (a, b) ❡♥tã♦✱ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ c ∈ (a, b) t❛❧ q✉❡✿ f (b) − f (a) f ′ (c) = ′ ✳ g(b) − g(a) g (c) ❉❡♠♦♥str❛çã♦✳ ❖❜s❡r✈❡✱ g(a) 6= g(b) ❘♦❧❧❡ ❀ ✐st♦ ✐♠♣❧✐❝❛r✐❛ q✉❡ ❡①✐st❡ ❙❡❥❛ k= f (b) − f (a) ✱ g(b) − g(a) g(a) = g(b) ❝✉♠♣r✐r✐❛ ❛s ❝♦♥❞✐çõ❡s ❞♦ ❚❡♦r❡♠❛ c ∈ (a, b) t❛❧ q✉❡ g ′ (c) = 0 ❝♦♥trár✐♦ à ❤✐♣ót❡s❡✳ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ❡♥tã♦ f (b) − f (a) = k(g(b) − g(a)) ✭✻✳✸✮ F (x) = f (x)−f (a)−k(g(x)−g(a)) ♣❛r❛ t♦❞♦ x ∈ [a, b]❀ ❡♥tã♦ F é ❝♦♥tí♥✉❛ ❡♠ [a, b], F é ❞❡r✐✈á✈❡❧ ❡♠ (a, b) ❡ F (b) = F (a) = 0❀ ❧♦❣♦ F ❝✉♠♣r❡♠ ′ ❛s ❝♦♥❞✐çõ❡s ❞♦ ❚❡♦r❡♠❛ ❞❡ ❘♦❧❧❡✱ ♣♦rt❛♥t♦ ❡①✐st❡ c ∈ (a, b) t❛❧ q✉❡ F (c) = 0✳ ❈♦♥s✐❞❡r❡✲s❡ ❛ ❢✉♥çã♦ ❛✉①✐❧✐❛r F ′ (x) = f ′ (x) − kg ′ (x)✱ ❡♥tã♦ F ′ (c) = f ′ (c) − kg ′ (c) = 0 f ′ (c) (a, b), k = ′ ✳ g (c) f ′ (c) f (b) − f (a) = ′ ✳ P♦rt❛♥t♦✱ ❡♠ ✭✻✳✸✮t❡♠♦s g(b) − g(a) g (c) ❙❡♥❞♦ Pr♦♣r✐❡❞❛❞❡ ✻✳✽✳ ❙❡ ❛s ❢✉♥çõ❡s lim+ x→a g ′ (c) 6= 0 ∀ c ∈ Pr✐♠❡✐r❛ r❡❣r❛ ❞❡ ▲✬❍♦s♣✐t❛❧✳ f, g : R −→ R ❝✉♠♣r❡♠✿ a) ❈♦♥tí♥✉❛s ♥♦ ✐♥t❡r✈❛❧♦ [a, a + h], h > 0 c) g ′ (x) 6= 0 ∀ x ∈ (a, a + h) f ′ (x) f ′ (x) e) lim+ ′ = L ♦✉ lim+ ′ = ±∞ x→a g (x) x→a g (x) ❡♥tã♦ ❡ ❝♦♠♦ f ′ (x) f (x) = lim+ ′ =L g(x) x→a g (x) ♦✉ ✸✸✾ b) ❞❡r✐✈á✈❡✐s ❡♠ (a, a + h) d) f (a) = g(a) = 0 ±∞✳ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❉❡♠♦♥str❛çã♦✳ ❖❜s❡r✈❡ q✉❡ g ′ (x) 6= 0 ∀ x ∈ (a, a + h)✳ ❆♣❧✐❝❛♥❞♦ ♦ ❚✳❱✳▼✳ à ❢✉♥çã♦ g ♥♦ ✐♥t❡r✈❛❧♦ [a, a + h] t❡♠♦s q✉❡ ❡①✐st❡ ♦♥❞❡ c ∈ (a, x) t❛❧ q✉❡ g(x) − g(a) = (x − a)g ′ (c)❀ ❛s ❤✐♣ót❡s❡s ❝✮ ❡ ❞✮ ✐♠♣❧✐❝❛♠ g(x) = (x − a)g ′ (c) 6= 0✳ f (x) ❡ ❝♦♠♦ ♣♦r ❤✐♣ót❡s❡ f (a) = g(a) = 0✱ g(x) f (x) − f (a) f ′ (x) f (x) = = ′ ♣❛r❛ a < d < x✳ ♦ t❡♦r❡♠❛ ❞❡ ❈❛✉❝❤② ♣❡r♠✐t❡ ❡s❝r❡✈❡r g(x) g(x) − g(a) g (x) ❖❜s❡r✈❛♥❞♦✱ q✉❛♥❞♦ x → a+ ✱ ❡♥tã♦ d → a+ ✱ ❞❛ ❤✐♣ót❡s❡ ❡✮ s❡❣✉❡✿ P❛r❛ x ∈ (a, a + h) ❝♦♥s✐❞❡r❡ ♦ q✉♦❝✐❡♥t❡ lim+ x→a f (x) f (x) − f (a) f ′ (x) f ′ (x) = lim+ = lim+ ′ = lim+ ′ = L ♦✉ d→a g (x) x→a g (x) g(x) x→a g(x) − g(a) ±∞ ❖❜s❡r✈❛çã♦ ✻✳✼✳ ❙❡ ❛s ❝♦♥❞✐çõ❡s ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭✻✳✽✮ sã♦ ✈❡r✐✜❝❛❞❛s ♥✉♠ ✐♥t❡r✈❛❧♦ [a − h, a] ♦✉ [a − h, a + h]✱ ❛ Pr♦♣r✐❡❞❛❞❡ ✭✻✳✽✮ é ✈❡r❞❛❞❡✐r❛ q✉❛♥❞♦ x → a− ♦✉ x → a Pr♦♣r✐❡❞❛❞❡ ✻✳✾✳ ❙❡ ❛s ❝♦♥❞✐çõ❡s ❛✮✱ ❜✮ ❡ ❝✮ ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭✻✳✽✮ sã♦ ✈❡r✐✜❝❛❞❛s ♥✉♠ ✐♥t❡r✈❛❧♦ 1 1 [ , +∞) ♦✉ (−∞, − ] ❡ lim .f (x) = lim .g(x) = 0 ♦✉ lim .f (x) = lim .g(x) = 0 ✱ x→+∞ x→+∞ x→−∞ x→−∞ h h ❡♥tã♦ f ′ (x) f (x) = lim ′ x→+∞ g (x) x→+∞ g(x) lim ♦✉ f (x) f ′ (x) = lim ′ x→−∞ g(x) x→−∞ g (x) lim s❡♠♣r❡ q✉❡ ♦ ❧✐♠✐t❡ ❞♦ s❡❣✉♥❞♦ ♠❡♠❜r♦ ❡①✐st❛✳ ❉❡♠♦♥str❛çã♦✳ 1 1 1 ❙❡ x → +∞✱ ❝♦♥s✐❞❡r❛♥❞♦ x = ❛s ❞✉❛s ❢✉♥çõ❡s f ( ) ❡ g( ) tê♠ ❧✐♠✐t❡ ③❡r♦ q✉❛♥❞♦ t t t t → 0+ ✳ 1 1 ❉❡✜♥✐♥❞♦ f ( ) = g( ) = 0 ♣❛r❛ t = 0✱ ♦❜tê♠✲s❡ ❞✉❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ♥♦ ✐♥t❡r✈❛❧♦ t t [0, h] q✉❡ ✈❡r✐✜❝❛♠ ❛s ❝♦♥❞✐çõ❡s ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭✻✳✽✮✳ ❆♣❧✐❝❛♥❞♦ ❡st❛ Pr♦♣r✐❡❞❛❞❡ ✭✻✳✽✮✿ − t12 · f ′ ( 1t ) f ( 1t ) [f ( 1t )]′ f ′ ( 1t ) f ′ (x) f (x) = lim+ 1 ′ 1 = lim+ ′ 1 = lim ′ = lim+ 1 = lim+ lim 1 x→+∞ g (x) x→+∞ g(x) t→0 − 2 · g ( ) t→0 g( ) t→0 [g( )]′ t→0 g ( ) t t t t t ❉❡ ♠♦❞♦ s✐♠✐❧❛r ♠♦str❛✲s❡ q✉❛♥❞♦ t → −∞✳ Pr♦♣r✐❡❞❛❞❡ ✻✳✶✵✳ ❙❡ f ′ (a) = g ′ (a) = 0 ❡ f ′ ❡ g ′ ❝✉♠♣r❡♠ ❛s ❝♦♥❞✐çõ❡s ❞❛ Pr♦♣r✐❡❞❛❞❡ ✭✻✳✽✮✱ ❡♥tã♦✿ lim+ x→a f (x) f ′ (x) f ′′ (x) = lim+ ′ = lim+ ′′ g(x) x→a g (x) x→a g (x) ❉❡♠♦♥str❛çã♦✳ ✸✹✵ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❆ ❞❡♠♦♥str❛çã♦ é ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❖❜s❡r✈❛çã♦ ✻✳✽✳ ❙❡ ♥ã♦ ❡①✐st❡ ♦ ❧✐♠✐t❡ ❞❡ f (x) g(x) ❡①✐st❡ ♦ ❧✐♠✐t❡ ❞❡ f ′ (x) g ′ (x) q✉❛♥❞♦ x → a✱ ❡♥tã♦ ♥ã♦ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♥ã♦ ❝♦♠♦ ♠♦str❛ ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✳ ❊①❡♠♣❧♦ ✻✳✸✶✳ ❙❡❥❛♠ ❛s ❢✉♥çõ❡s g(x) = senx ❈❛❧❝✉❧❛♥❞♦ ♦ ❧✐♠✐t❡✿ P♦r ♦✉tr♦ ❧❛❞♦✱ 2xsen( x1 ) − cos( x1 ) f ′ (x) = lim x→0 x→0 g ′ (x) cos x f, g : R −→ R x→a s❡✱ x=0 ✳ ♥ã♦ ❡①✐st❡✳ ❝✉♠♣r❡♠✿ h>0 b) d) e) f (a) = g(a) = 0 lim+ x 6= 0 ❙❡❣✉♥❞❛ r❡❣r❛ ❞❡ ▲✬❍♦s♣✐t❛❧✳ a) ❈♦♥tí♥✉❛s ♥♦ ✐♥t❡r✈❛❧♦ (a, a + h], c) g ′ (x) 6= 0 ∀ x ∈ (a, a + h) ❡♥tã♦ s❡✱   x2 · sen( x1 ) x 1 f (x) = lim = lim · x · sen = 0✳ lim x→0 x→0 senx x→0 g(x) senx x lim Pr♦♣r✐❡❞❛❞❡ ✻✳✶✶✳ ❙❡ ❛s ❢✉♥çõ❡s ❡   x2 · sen( 1 ) f (x) = x  0 f) f ′ (x) f (x) = lim+ ′ =L g(x) x→a g (x) ♦✉ ❞❡r✐✈á✈❡✐s ❡♠ (a, a + h) lim+ .f (x) = lim+ .g(x) = ∞ x→a x→a f ′ (x) lim+ ′ =L x→a g (x) ♦✉ ±∞ ±∞ ❉❡♠♦♥str❛çã♦✳ ❊①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❊①❡♠♣❧♦ ✻✳✸✷✳ ❈❛❧❝✉❧❛r ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ 1 ❛✮ ❙♦❧✉çã♦✳ x− 4 lim+ x→0 Lnx ❛✮ ➱ ❞❛ ❢♦r♠❛ ❙♦❧✉çã♦✳ ❜✮ ❜✮ ➱ ❞❛ ❢♦r♠❛ ∞ ✱ ∞ ∞ ✱ ∞ ❧♦❣♦ ❧♦❣♦ lim x→0 1 x 1 x + sen( x1 )   5 1 − 41 x− 4 1 −1 x− 4 = lim+ − x 4 = −∞✳ lim = lim+ x→0+ Lnx x→0 x→0 x−1 4 lim x→0 1 x ú❧t✐♠♦ ❧✐♠✐t❡ ♥ã♦ ❡①✐st❡✱ ♣♦ré♠ 1 x −x−2 1 = lim −2 = lim 1 1 x→0 x→0 x (1 + cos( x )) 1 + cos( x1 ) + sen( x ) 1 1 x lim = lim = 1✳ x→0 1 + sen( 1 ) x→0 1 + xsen( 1 ) x x x ✸✹✶ ❡st❡ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❖❜s❡r✈❛çã♦ ✻✳✾✳ ✐✮ ❆♣❧✐❝❛♥❞♦ ♦ ❚✳❱✳▼✳ ♠♦str❛✲s❡ q✉❡✱ s❡ lim .f (x) = ∞ ❡ lim .g(x) = ∞✱ ❡♥tã♦ x→a x→a lim .f ′ (x) = ∞ ❡ lim .g ′ (x) = ∞✱ ✐st♦ s✐❣♥✐✜❝❛ q✉❡ ♥✉♠ ❝❡rt♦ s✉❜❝♦♥❥✉♥t♦ ❞❡ x→a x→a f ′ (x) f ′ (x) q✉❛♥❞♦ x → a ✳ ◆ã♦ ♦❜st❛♥t❡ ♦ q✉♦❝✐❡♥t❡ ♣♦❞❡ g ′ (x) g ′ (x) f (x) ♥ã♦ ♣❡r♠✐t❡ ✭❊①❡♠♣❧♦ ✭✻✳✸✷✮ ❛✮ ✮ ♣❡r♠✐t✐r s✐♠♣❧✐✜❝❛çõ❡s q✉❡ g(x) D(f ) é ✐♥❞❡t❡r♠✐♥❛❞♦ ✐✐✮ ✐✐✐✮ P❛r❛ ❛ ❢♦r♠❛ ∞ 0 ✈❡r✐✜❝❛♠✲s❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❛♥á❧♦❣❛s ❛♦s ❞❛ ❢♦r♠❛ ✳ ∞ 0 f ′ (x) ∞ ♥ã♦ t❡♠ ❧✐♠✐t❡ q✉❛♥❞♦ x → a✱ ♥ã♦ é ♥❡❝❡ssár✐♦ ❝♦♥s✐❞❡r❛r q✉❡✱ s❡ ′ ∞ g (x) f (x) ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♥ã♦ t❡♥❤❛ ❧✐♠✐t❡ ✭❊①❡♠♣❧♦ ✭✻✳✸✷✮ ❜✮ ✮ g(x) ◆❛ ❢♦r♠❛ ❊①❡♠♣❧♦ ✻✳✸✸✳ ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳ ex−3 − e3−x x→3 sen(x − 3) lim 0 ✱ ❝♦♠♦ ❛s ❝♦♥❞✐çõ❡s ❞❛ r❡❣r❛ ❞❡ ▲✬❍♦s♣✐t❛❧ 0 1+1 ex−3 + e3−x ex−3 − e3−x = lim = = 2✳ ✭Pr♦♣r✐❡❞❛❞❡ ✭✻✳✶✶✮✮ ❝✉♠♣r❡♠✱ ❡♥tã♦✿ lim x→3 cos(x − 3) x→3 sen(x − 3) 1 ex−3 − e3−x = 2✳ P♦rt❛♥t♦✱ lim x→3 sen(x − 3) ◗✉❛♥❞♦ x → 3✱ ♦ ❧✐♠✐t❡ t❡♥❞❡ ♣❛r❛ ❊①❡♠♣❧♦ ✻✳✸✹✳ ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳ ex−2 + e2−x − 2 lim ✳ x→2 1 − cos(x − 2) ex−2 + e2−x − 2 = x→2 1 − cos(x − 2) 0 0 ❖ ❧✐♠✐t❡ é ❞❛ ❢♦r♠❛ ✱ ❛♣❧✐❝❛♥❞♦ ❛ r❡❣r❛ ❞❡ ▲✬❍♦s♣✐t❛❧ t❡♠♦s✿ lim ex−2 − e2−x ✳ x→2 sen(x − 2) lim ❊st❡ ú❧t✐♠♦ ❧✐♠✐t❡ é ❞❛ ❢♦r♠❛ ex−2 + e2−x = 2✳ x→2 cos(x − 2) 0 ✱ ❛♣❧✐❝❛♥❞♦ ♥♦✈❛♠❡♥t❡ ❛ 0 Pr♦♣r✐❡❞❛❞❡ ✭✻✳✽✮ t❡♠♦s✿ lim ex−2 + e2−x − 2 = 2✳ x→2 1 − cos(x − 2) P♦rt❛♥t♦✱ lim ❊①❡♠♣❧♦ ✻✳✸✺✳ ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳ 1 − cos x − 21 x2 ✳ lim x→0 x4 ✸✹✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ❊st❡ ❧✐♠✐t❡ é ❞❛ ❢♦r♠❛ 0 ✱ 0 R ❛♣❧✐❝❛♥❞♦ três ✈❡③❡s ❛ r❡❣r❛ ❞❡ ▲✬❍♦s♣✐t❛❧ t❡♠♦s✿ 1 − cos x − 21 x2 1 senx − x cos x − 1 lim = lim = lim = − ✳ x→0 x→0 x→0 x4 4x3 12x2 24 P♦rt❛♥t♦✱ 1 − cos x − 12 x2 1 =− lim 4 x→0 x 24 ❊①❡♠♣❧♦ ✻✳✸✻✳ 1 ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳ ❙❡ x→0 e− x lim x→0 x + ✱ ❡st❡ ❧✐♠✐t❡ é ❞❛ ❢♦r♠❛ 0 ✱ 0 1 ❡♥tã♦ 1 1 e− x x−2 · e− x e− x lim+ = lim+ = lim+ 2 x→0 x→0 x→0 x 1 x ♦❜s❡r✈❡✱ s❡ ❝♦♥t✐♥✉❛♠♦s ❝♦♠ ♦ ♣r♦❝❡ss♦ ♥ã♦ ♣♦❞❡r❡♠♦s ❡✈✐t❛r ❛ ✐♥❞❡t❡r♠✐♥❛çã♦✳ 1 1 e− x x P♦r ♦✉tr♦ ❧❛❞♦ ❡s❝r❡✈❡♥❞♦✱ lim = lim+ 1 ❡st❡ ú❧t✐♠♦ ❧✐♠✐t❡ é ❞❛ x→0+ x x→0 e x 1 1 e− x −x−2 x ❆♣❧✐❝❛♥❞♦ ❛ r❡❣r❛ ❞❡ ▲✬❍♦s♣✐t❛❧✱ lim = lim = lim 1 1 = 0✳ x→0+ x x→0+ e x x→0+ −x−2 e x 1 e− x 1 −1 − ❙❡ x → 0 ✱t❡♠♦s lim = lim · e x = (−∞)(+∞) = −∞✳ x→0− x x→0− x 1 e− x ♥ã♦ ❡①✐st❡✳ P♦rt❛♥t♦ ♦ ❧✐♠✐t❡ lim x→0 x ❢♦r♠❛ ∞ ✳ ∞ ❊①❡♠♣❧♦ ✻✳✸✼✳ 1 ❈❛❧❝✉❧❛r lim x→+∞ e − x2 − 1 1 x2 . ❙♦❧✉çã♦✳ ➱ ❞❛ ❢♦r♠❛ 0 ✱ 0 1 ❡♥tã♦ lim x→+∞ 1 P♦rt❛♥t♦ lim x→+∞ e − x2 − 1 1 x2 e − x2 − 1 1 x2 1 1 e − x2 e− x2 · 2x−3 = lim = −1✳ = lim x→+∞ −1 x→+∞ −2x−3 = −1✳ ❊①❡♠♣❧♦ ✻✳✸✽✳ ❈❛❧❝✉❧❛r lim+ x→0 Ln(senx) ✳ Ln(tan x) ❙♦❧✉çã♦✳ ∞ Ln(senx) cot x ✱ ❧♦❣♦ lim+ = lim+ sec2 x = lim+ cos2 x✳ x→0 Ln(tan x) x→0 x→0 ∞ tan x Ln(senx) = 1✳ lim+ x→0 Ln(tan x) ❖ ❧✐♠✐t❡ é ❞❛ ❢♦r♠❛ P♦rt❛♥t♦ ❊①❡♠♣❧♦ ✻✳✸✾✳ ❈❛❧❝✉❧❛r limπ x→ 2 tan x ✳ tan(3x) ❙♦❧✉çã♦✳ ✸✹✸ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ∞ ✱ ❞❛s ✐❞❡♥t✐❞❛❞❡s tr✐❣♦♥♦♠étr✐❝❛s s❡❣✉❡✲s❡ ✿ ∞   tan x senx · cos 3x = = limπ lim x→ 2 sen3x · cos x x→ π2 tan(3x)     h senx i cos 3x cos 3x = limπ · limπ = (−1) limπ ✳ x→ 2 sen3x x→ 2 x→ 2 cos x cos x ❖ ❧✐♠✐t❡ é ❞❛ ❢♦r♠❛ ❆ ❛♣❧✐❝❛♥❞♦ ❛ r❡❣r❛ ❞❡ ▲✬❍♦s♣✐t❛❧ ❛ ♦ ú❧t✐♠♦ ❧✐♠✐t❡✿     cos 3x −3sen3x = limπ = −3✳ lim x→ 2 x→ π2 cos x −senx lim P♦rt❛♥t♦ x→ π2 tan x = (−1)(−3) = 3✳ tan(3x) ❊①❡♠♣❧♦ ✻✳✹✵✳ ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳ xn , x→+∞ ex lim n ∈ N✳ ∞ n ∈ N ❡ ♦ ❧✐♠✐t❡ é ❞❛ ❢♦r♠❛ ✱ ❛♣❧✐❝❛♥❞♦ ❛ r❡❣r❛ ❞❡ ▲✬❍♦s♣✐t❛❧ s✉❝❡ss✐✈❛♠❡♥t❡ ∞ n! xn n ✈❡③❡s✱t❡♠♦s ✿ lim x = lim x = 0✳ x→+∞ e x→+∞ e xn P♦rt❛♥t♦✱ lim = 0✳ x→+∞ ex ❈♦♠♦ ❊①❡♠♣❧♦ ✻✳✹✶✳ ❉❡t❡r♠✐♥❡ ♦ s❡❣✉✐♥t❡ ❧✐♠✐t❡ ✿ ❙♦❧✉çã♦✳ ❖ ❧✐♠✐t❡ é ❞❛ ❢♦r♠❛ n − 1 < r < n✳ ∞ ✱ ∞ xr , x→+∞ ex ❝♦♠♦ lim r r ∈ Q − N, r > 0✳ é ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦✱ ❡①✐st❡ ❆♣❧✐❝❛♥❞♦ ❛ r❡❣r❛ ❞❡ ▲✬❍♦s♣✐t❛❧ s✉❝❡ss✐✈❛♠❡♥t❡ n n ∈ N t❛❧ q✉❡ ✈❡③❡s t❡♠♦s✿ r(r − 1)(r − 2)(r − 3) · · · (r − (n − 1))xr−n xr = lim =0 x→+∞ x→+∞ ex ex lim ♣♦✐s✱ ex (r − n) < 0✳ é ✐♥✜♥✐t♦ ❞❡ ✑ ♦r❞❡♠ ♠❛✐♦r q✉❡ q✉❛❧q✉❡r ♣♦tê♥❝✐❛ r P♦rt❛♥t♦✱ lim x → +∞ ❞❡ x✑✳ ❊st❡ r❡s✉❧t❛❞♦ ♠♦str❛ q✉❡✱ q✉❛♥❞♦ x→+∞ x = 0✳ ex ♦ ❧✐♠✐t❡ ❞❛ ❡①♣♦♥❡♥❝✐❛❧ ❊①❡♠♣❧♦ ✻✳✹✷✳ ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳ Lnx , x→+∞ xr lim r ∈ N, r > 0✳ ∞ ✱ ❛♣❧✐❝❛♥❞♦ ❛ r❡❣r❛ ❞❡ ▲✬❍♦s♣✐t❛❧ ∞ −1 x 1 Lnx = lim = lim = 0✳ lim x→+∞ r · xr−1 x→+∞ r · xr x→+∞ xr ❖ ❧✐♠✐t❡ é ❞❛ ❢♦r♠❛ ✸✹✹ t❡♠♦s✿ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❖ r❡s✉❧t❛❞♦ ♠♦str❛ q✉❡✱ q✉❛♥❞♦ ❞❡ x r ✑✱ ♣❛r❛ r > 0✳ Lnx é ✐♥✜♥✐t♦ ❞❡ ✏♦r❞❡♠ ✐♥❢❡r✐♦r Lnx = 0. x→+∞ xr ❋♦r♠❛s ✐♥❞❡t❡r♠✐♥❛❞❛s r❡❞✉tí✈❡✐s à ❢♦r♠❛ ❆s r❡❣r❛s ❞❡ ▲✬❍♦s♣✐t❛❧ ❛♣❧✐❝❛♠✲s❡ à ❢♦r♠❛ 0 · ∞, ❛ ❢✉♥çã♦ lim P♦rt❛♥t♦✱ ✻✳✸✳✶ x → +∞ ∞ − ∞, 00 , ∞0 ❡ 1∞ ❀ ♣♦❞❡♠ s❡r 0 0 ♦✉ ∞ ∞ ∞ ❀ ♣♦ré♠ ❛s ❢♦r♠❛s ✐♥❞❡t❡r♠✐♥❛❞❛s✿ ∞ 0 ∞ tr❛♥s❢♦r♠❛❞❛s ♣❛r❛ ❢♦r♠❛ ♦✉ ✳ 0 ∞ 0 0 ♦✉ ✻✳✸✳✶✳✶ ❆ ❢♦r♠❛ ✐♥❞❡t❡r♠✐♥❛❞❛ 0 · ∞ lim .f (x) = 0 ❡ lim .g(x) = ∞ ✭♣♦❞❡ s❡r ±∞✮✱ t❡♠♦s✿ [lim .f (x)]· x→a x→a [lim .g(x)] = lim .f (x)g(x) = 0 · ∞✳ ❊st❡ ❧✐♠✐t❡ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ✉t✐❧✐③❛♥❞♦ ❛ r❡❣r❛ ❞❡ ◗✉❛♥❞♦ ♣♦r ❡①❡♠♣❧♦ x→a x→a x→a ▲✬❍♦s♣✐t❛❧✱ s❡❣✉♥❞♦ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s tr❛♥s❢♦r♠❛çõ❡s✿ 1a f ·g = 2a f ·g = f 0 ✳ 0 g ∞ ✳ ∞ 1 ❡ r❡s✉❧t❛ ❞❛ ❢♦r♠❛ g 1 ❡ r❡s✉❧t❛ ❞❛ ❢♦r♠❛ f ❖❜s❡r✈❛çã♦ ✻✳✶✵✳ ✐✮ ◗✉❛♥❞♦ ✉♠ ❞♦s ❢❛t♦r❡s é ✉♠❛ ❢✉♥çã♦ tr❛♥s❝❡♥❞❡♥t❡ ❝♦♠ ❞❡r✐✈❛❞❛s ❛❧❣é❜r✐❝❛s✱ ❝♦♥✈é♠ ❝♦♥s✐❞❡r❛r ❡st❡ ❢❛t♦r ❝♦♠♦ ♦ ♥✉♠❡r❛❞♦r ❛♥t❡s ❞❡ ✉t✐❧✐③❛r ❛ r❡❣r❛ ❞❡ ▲✬❍♦s♣✐t❛❧✳ ✐✐✮ ◆ã♦ ❝♦♥❢✉♥❞✐r ❝♦♠ ❛ Pr♦♣r✐❡❞❛❞❡ ✭✻✳✶✶✮✲✭❞✮ ♦ ✈❛❧♦r ❞❡ ✉♠ ❞♦s ❧✐♠✐t❡s ♥ã♦ é ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ❊①❡♠♣❧♦ ✻✳✹✸✳ ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳ lim [tan x · Ln(senx)] x→0 0 · ∞✱ ❛♣❧✐❝❛♥❞♦ ❛ r❡❣r❛ ♣r❡❝❡❞❡♥t❡ Ln(senx) ∞ lim [tan x · Ln(senx)] = lim é ❞❛ ❢♦r♠❛ ✳ x→0 x→0 cot x ∞ ❖❜s❡r✈❡ q✉❡ ♦ ❧✐♠✐t❡ é ❞❛ ❢♦r♠❛ ✭✻✳✶✵✮t❡♠♦s✿ ▲♦❣♦✱ lim [tan x · Ln(senx)] = lim x→0 P♦rt❛♥t♦✱ x→0 ❡ ❞❛ ❖❜s❡r✈❛çã♦ Ln(senx) cot x = lim = − lim (cos x)(senx) = 0✳ x→0 x→0 cot x − csc2 x lim [tan x · Ln(senx)] = 0✳ x→0 ✸✹✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✻✳✸✳✶✳✷ ❆ ❢♦r♠❛ ✐♥❞❡t❡r♠✐♥❛❞❛ P♦r ❡①❡♠♣❧♦✱ s❡ R ∞−∞ lim .f (x) = ∞ ❡ lim .g(x) = ∞✱t❡♠♦s✿ lim .f (x)− lim .g(x) = ∞−∞✳ x→a x→a x→a x→a ❊st❡ ❧✐♠✐t❡ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ✉t✐❧✐③❛♥❞♦ ❛ r❡❣r❛ ❞❡ ▲✬❍♦s♣✐t❛❧✱ s❡❣✉♥❞♦ ❛ tr❛♥s❢♦r♠❛✲ çã♦✿ 1 1 f − g = f · g[ − ]✳ g f ❊①❡♠♣❧♦ ✻✳✹✹✳ ❈❛❧❝✉❧❛r ❙♦❧✉çã♦✳ 1 lim+ [ − csc x]✳ x→0 x −(x − senx) csc x 1 1 1 ][ − 1 ] = lim+ = lim+ [ − csc x] = lim+ [ x→0 x→0 x→0 x x csc x x · senx x cos x − 1 0 −senx = lim+ = lim+ = =0 x→0 senx + x · cos x x→0 cos x + cos x − x · senx 2 1 P♦rt❛♥t♦✱ lim [ − csc x] = 0✳ x→0+ x ❚❡♠♦s ✻✳✸✳✶✳✸ ❆s ❢♦r♠❛s ✐♥❞❡t❡r♠✐♥❛❞❛s 00 , ∞0 ❡ 1∞ 0 · ∞✱ s❡ ❛♦ = eg(x)·Ln[f (x)] ✳ ❚♦❞❛s ❡st❛s ❢♦r♠❛s sã♦ r❡❞✉tí✈❡✐s à ❢♦r♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❧♦❣❛r✐t♠♦ q✉❡ ❞✐③✿ f (x) g(x) ❝❛❧❝✉❧❛r ♦ ❧✐♠✐t❡ ✉t✐❧✐③❛♠♦s ❛ ❊①❡♠♣❧♦ ✻✳✹✺✳ ❈❛❧❝✉❧❛r lim [x + senx]tan x ✳ x→0+ ❙♦❧✉çã♦✳ ❖❜s❡r✈❡✿ lim tan x·Ln(x+senx) lim+ [x + senx]tan x = ex→0+ ✭✻✳✹✮ x→0 1+cos x P♦r ♦✉tr♦ ❧❛❞♦✱ Ln(x + senx) = lim+ x+senx = lim+ tan x · Ln(x + senx) = lim+ x→0 − csc2 x x→0 x→0 cot x lim+ (1 + cos x) · lim+ x→0 x→0 sen2 x 2senx cos x = (−2) lim+ = (−2)(0) = 0 x→0 x + senx 1 + cos x lim tan x·Ln(x+senx) ◆❛ ❡①♣r❡ssã♦ ✭✻✳✹✮ t❡♠♦s P♦rt❛♥t♦✱ lim+ [x + senx]tan x = ex→0+ x→0 tan x lim [x + senx] x→0+ = e 0 = 1✳ = 1✳ ❊①❡♠♣❧♦ ✻✳✹✻✳ π ❈❛❧❝✉❧❛r limπ [tan x] 2 −x ✳ x→ 2 ❙♦❧✉çã♦✳ ❚❡♠♦s✿ lim [tan x] x→ π2 π −x 2 =e lim ( π2 −x)Ln[tan x] x→ π 2 ✸✹✻ ✭✻✳✺✮ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ P♦r ♦✉tr♦ ❧❛❞♦✱ = limπ x→ 2 limπ ( x→ 2 2 ( π2 − x)2 −2( π2 − x) 0 = limπ = =0 = limπ 2 2 x→ 2 cos x − sen x x→ 2 senx · cos x −1 lim [tan x] x→ π2 π −x 2 lim [tan x] x→ π2 R Ln(tan x) π = − x)Ln[tan x] = limπ 1 x→ 2 2 ( π −x) sec2 x tan x 1 π ( 2 −x)2 ❊♠ ✭✻✳✺✮ s❡❣✉❡✲s❡✱ P♦rt❛♥t♦✱ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ π −x 2 =e lim ( π2 −x)Ln(tan x) x→ π 2 = e 0 = 1✳ = 1✳ ❊①❡♠♣❧♦ ✻✳✹✼✳ 1 lim [1 + x2 ] x·senx ✳ ❈❛❧❝✉❧❛r x→0 ❙♦❧✉çã♦✳ ❊st❡ ❧✐♠✐t❡ é ❞❛ ❢♦r♠❛ 1 ∞ 2 =e lim [1 + x ] ❡①♣♦❡♥t❡ ❞❡ e s❡❣✉❡✲s❡✿ ✱ ❡ t❡♠♦s✿ P❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ ❧✐♠✐t❡ ❞♦ 1 x·senx lim x→0 Ln(1+x2 ) x·senx x→0 ✳ 2x Ln(1 + x2 ) 1+x2 lim = lim = x→0 x · senx x→0 senx + x · cos x 1 2x 2 · lim = (1) lim =1 2 x→0 1 + x x→0 senx + x · cos x x→0 2 cos x − senx lim P♦rt❛♥t♦✱ 1 lim [1 + x2 ] x·senx = e1 = e✳ x→0 ❊①❡♠♣❧♦ ✻✳✹✽✳ ❈❛❧❝✉❧❛r lim xx ✳ x→0+ ❙♦❧✉çã♦✳ ❖ ❧✐♠✐t❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♥❛ ❢♦r♠❛✿ lim x·Lnx lim+ xx = ex→0+ ✭✻✳✻✮ x→0 ❚❡♠♦s✿ lim+ x · Lnx = lim+ x→0 x→0 P♦rt❛♥t♦✱ ❡♠ ✭✻✳✻✮ Lnx 1 x = lim+ x→0 1 x − x12 = lim+ (−x) = 0✳ x→0 lim xx = e0 = 1✳ x→0+ ❊①❡♠♣❧♦ ✻✳✹✾✳ ▼♦str❡ q✉❡ ❙♦❧✉çã♦✳ ◗✉❛♥❞♦ x − senx x→+∞ x + senx lim x → ∞✱ t❡♠♦s ❡①✐st❡✱ ♣♦ré♠ ♥ã♦ é ♥❡❝❡ssár✐♦ ❛♣❧✐❝❛r ❛ r❡❣r❛ ❞❡ ▲✬❍♦s♣✐t❛❧✳ y= 1 → 0✱ x ❧♦❣♦ x − senx lim = lim x→+∞ x + senx y→0 1 y 1 y − sen y1 + sen y1 ✸✹✼ = lim y→0 1 − ysen y1 1 + ysen y1 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ 1 |≤ 1 ✭ é ❧✐♠✐t❛❞❛✮✱ ❡♥tã♦ lim y · sen y1 = 0✱ y→0 y 1 1 − ysen y x − senx = 1✳ P♦rt❛♥t♦ lim = 1✳ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ lim 1 x→+∞ x + senx y→0 1 + ysen y ❈♦♠♦ ❝✉♠♣r❡✲s❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ | sen ◆ã♦ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ❛ r❡❣r❛ ❞❡ ▲✬❍♦s♣✐t❛❧ ♦❜s❡r✈❡✱ é ❞❛ ❢♦r♠❛ x − senx ∄ = x→+∞ x + senx ∄ lim ❊①❡♠♣❧♦ ✻✳✺✵✳ ❛✮ ❉❛r ✉♠ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ❢✉♥çã♦ f (x) ♣❛r❛ ♦ q✉❛❧ ❡①✐st❡ lim .f (x)✱ ♣♦ré♠ ♥ã♦ ❡①✐st❡ x→∞ lim .f ′ (x) x→∞ ❜✮ ▼♦str❡ q✉❡✱ s❡ ❡①✐st❡♠ lim .f (x) ❡ lim .f ′ (x)✱ ❡♥tã♦ lim .f ′ (x) = 0✳ ❝✮ ▼♦str❡ q✉❡ s❡ ❡①✐st❡ lim .f (x) ❡ ❡①✐st❡ lim .f ′′ (x) ❡ t❛♠❜é♠ lim .f ′′ (x) = 0✳ x→∞ x→∞ x→∞ ❙♦❧✉çã♦✳ x→∞ x→∞ x→∞ ❛✮ senx2 senx2 ✳ ❖❜s❡r✈❡✱ lim .f (x) = lim =0 x→∞ x→∞ x x senx2 2x2 cos x2 − senx2 2 = 2 cos x − ✳ ♣♦ré♠❀ f ′ (x) = x2  x2  senx2 ◆♦ ❧✐♠✐t❡ lim .f ′ (x) = lim 2 cos x2 − 2 = ∄ − 1 = ? x→∞ x→∞ x ➱ s✉✜❝✐❡♥t❡ ❝♦♥s✐❞❡r❛r ❛ ❢✉♥çã♦ f (x) = ❙♦❧✉çã♦✳ ❜✮ ❙✉♣♦♥❤❛♠♦s q✉❡ lim .f ′ (x) = L > 0✳ ❊♥tã♦ ❡①✐st❡ ❛❧❣✉♠ N t❛❧ q✉❡ | f ′ (x) − L |< x→∞ L L 2 ♣❛r❛ x > N ✱ ✐st♦ ✐♠♣❧✐❝❛ q✉❡ f ′ (x) > ✳ 2 P♦ré♠ s❡❣✉♥❞♦ ♦ t❡♦r❡♠❛ ❞♦ ✈❛❧♦r ♠é❞✐♦ ✐st♦ t❛♠❜é♠ ✐♠♣❧✐❝❛ q✉❡ f (x) > f (N ) + x−N |L| 2 ♣❛r❛ x > N ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ lim .f (x) ♥ã♦ ❡①✐st❡✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ ♠♦str❛✲s❡ q✉❡ ♥ã♦ ♣♦❞❡ x→∞ ❛❝♦♥t❡❝❡r lim .f ′ (x) = L < 0✳ x→∞ P♦rt❛♥t♦ lim .f ′ (x) = 0 ❙♦❧✉çã♦✳ ❝✮ x→∞ ❙❡❥❛ lim .f ′′ (x) = L > 0 ✱ ❡♥tã♦ ♦ ♠❡s♠♦ q✉❡ ♥❛ ♣❛rt❡ ❛✮ t❡rí❛♠♦s q✉❡ lim .f ′ (x) = x→∞ x to∞ ∞✳ ❆♣❧✐❝❛♥❞♦ ♥♦✈❛♠❡♥t❡ ♦ t❡♦r❡♠❛ ❞♦ ✈❛❧♦r ♠é❞✐♦ ♠♦str❛✲s❡ q✉❡ lim .f (x) = ∞✱ ✐st♦ é x→∞ ❝♦♥tr❛❞✐çã♦ ❝♦♠ ❛ ❤✐♣ót❡s❡✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ ♥ã♦ ♣♦❞❡ ❛❝♦♥t❡❝❡r lim .f ′′ (x) = L < 0✳ x→∞ P♦rt❛♥t♦ lim .f ′′ (x) = 0✳ x→∞ ❊♠ ❣❡r❛❧✱ s❡ ❡①✐st❡♠ lim .f (x) ❡ lim .f (k) (x)✱ ❡♥tã♦ lim .f ′ (x) = lim .f ′′ (x) = ′′′ lim .f (x) = · · · = lim .f x→∞ x→∞ x→∞ (k) (x) = 0 x→∞ k ∈ N✳ ✸✹✽ x→∞ x→∞ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡r❝í❝✐♦s ✻✲✸ ✶✳ ❈❛❧❝✉❧❛r ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ 1. 4. 7. 10. 13. 16. 19. 22. 25. 28. 31. ex − 1 x→0 Ln(x + 1) ex − 1 lim x→0 x(x + 1) s  tan x x lim x→0+ x lim aLnx − x x→1 Lnx √ lim x sec x lim 2. 5. 8. 11. 14. x→0 lim senx x 17. x→0 Lnx lim √ x→+∞ x tan x lim+ x→0 x √ 1 + x2 lim x→+∞ x √ x2 x2 + a2 lim x→∞ 5 lim+ x 1+2Lnx 20. 23. 26. 29. 32. x→0 x − tan x x − senx √ lim x ex + x lim 3. x→0 6. x→0  1 x lim − x→1 Lnx x−1 1 ] x→0 x2 x · enx − x lim x→0 1 − cos(nx) sen2 x − senx2 lim x→0 x · cos x lim+ [Lnx + 1 1 − 2] 2 x→0 sen x x √ 2 lim x 1 + x lim [ x→0 lim x · (Ln | x |) 2 x→0 1 x − ] x→1 x − 1 Lnx 1 1 lim [ − x ] x→0 x e −1 lim [  9. 12. 15. 18. 21. 24. 27. 30. 33. x + sen(πx) x − sen(πx) ax − (a + 1)x lim x→0 x lim x→0 x − arcsenx x→0 x lim ax − x · Lnx − cos x x→0 sen2 x 1 lim [ − cot x] x→0 x · cos x π limπ [ − x · tan x] x→ 2 2 cos x   5 seny + senx 1+2Lnx lim Ln x→0 seny − senx lim lim+ (1 − ex )Ln(senx) x→0 tan2 (x−1 ) lim x→+∞ Ln2 (1 + 4x−1 ) x · sen(senx) lim x→0 1 − cos(senx)  2  x + 3x + 5 5 lim − x→0 senx x ✷✳ ❱❡r✐✜❝❛r ❛ ✈❛❧✐❞❛❞❡ ❞❛s s❡❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s✿ ✶✳ ✷✳ √ 5 p cos7 (x + 1)] 15 Ln17 (x + 2) Ln2 p = lim ✳ √ √ 5 x→−1 [5 x+1 ] · tan3 ( 3 x + 1) · arcsen 9 (x + 1)14 Ln5 p p √ √ 15 (a − x)13 [cos 3 a − x − cos(sen3 3 a − x)]sen(2 3 (a − x)2 ) 1 p p = lim 3 3 x→a 6 [ea−x − 1] · sen3 (a − x)2 [1 − cos(sen4 (a − x)2 )] [2 (x+1)4 ][1 − p 9 ✸✳ ❖♥❞❡ s❡ ❡♥❝♦♥tr❛ ♦ ❡rr♦ ♥❛ ❛♣❧✐❝❛çã♦ ❞❛ r❡❣r❛ ❞❡ ▲✬❍♦s♣✐t❛❧❄ x3 + x − 2 = x→1 x2 − 3x + 2 lim 3x2 + 1 = x→1 2x − 3 lim 6 =3 x→1 2 lim ◆❛ ✈❡r❞❛❞❡ ♦ ❧✐♠✐t❡ é −4✳ ✸✹✾ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✹✳ ❉❡t❡r♠✐♥❡ ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿ cos2 x − 1 x→0 x2 ❛✮ ❜✮ lim ✺✳ ❉❡t❡r♠✐♥❡ f ′ (0) s❡✿ f (x) = g ′′ (0) = 17✳ x x→1 tan x lim g(x) s❡ x 6= 0, x f (0) = 0, g(0) = g ′ (0) = 0 ❡ ✻✳ ▼♦str❡ ❛s s❡❣✉✐♥t❡s r❡❣r❛s ❞❡ ▲✬❍♦s♣✐t❛❧✿ f (x) f ′ (x) = L ✱ ❡♥tã♦ lim = L, ✶✳ ❙❡ lim .f (x) = lim .g(x) = 0 ❡ lim ′ x→a+ g(x) x→a+ x→a+ x→a+ g (x) ✭❛♥á❧♦❣♦ ♣❛r❛ ❧✐♠✐t❡s á ❡sq✉❡r❞❛✮✳ f ′ (x) f (x) = 0✱ ❡♥tã♦ lim+ = 0✱ ′ x→a x→a x→a g (x) x→a g(x) ✭❛♥á❧♦❣♦ ♣❛r❛ −∞ ♦✉ s❡ x → a+ ♦✉ x → a− ✮ f ′ (x) f (x) ✸✳ ❙❡ lim .f (x) = lim .g(x) = 0 ❡ lim ′ = L✱ ❡♥tã♦ lim = L✳ x→∞ x→∞ x→∞ g (x) x to∞ g(x) f (x) f ′ (x) = ∞✱ ❡♥tã♦ lim = ∞✳ ✹✳ ❙❡ lim .f (x) = lim .g(x) = 0 ❡ lim ′ x→∞ g(x) x→∞ x→∞ x→∞ g (x) ✷✳ ❙❡ lim+ .f (x) = lim+ .g(x) = ∞ ❡ lim+ ✼✳ ▼♦str❡ q✉❡ lim x→0 ▲✬❍♦s♣✐t❛❧✳ x2 · sen x1 = 0✱ ♣♦ré♠ ♥ã♦ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ❛♣❧✐❝❛♥❞♦ ❛ r❡❣r❛ ❞❡ senx ✽✳ ❉❡t❡r♠✐♥❡ ♦s ❧✐♠✐t❡s ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ ✶✳ ✹✳ ✼✳ ✶✵✳ x lim sen x ✷✳ lim (x + 1)Lnx ✺✳ x→0 x→0 lim x→∞ lim x→0 √ x x + 2x q x2 tan x x ✽✳ ✶✶✳ 2 ✶✳ ✸✳ x→0 ✷✳ ✹✳ ✻✳ ✾✳ lim (π − 2x)cos x ✶✷✳ x→ π2 ✾✳ ❱❡r✐✜❝❛r ♦ ❝á❧❝✉❧♦ ❞♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿  1 lim x · cot πx = x→0 π √ x2 3 lim cos 2x = e−6 ✸✳ lim x · Lnx   Ln(x − 1) lim x→1 cot πx   1 1 lim − x→1 x − 1 Lnx x→0  1 2 2 lim 2 − cot x = x→0 x i 3 h 2−(ex +e−x ) cos x = lim x4 x→0  1 1 − x lim x→0 x e −1 i h Ln(x−a) lim Ln(ex −ea ) x→a   p q − lim x→1 1 − xp 1 − xq   π − 2 arctan x √ lim x 3 x→∞ e −1  1 3 ✶✵✳ ◗✉❛❧ ❞♦s tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s ❞❡ ♣❡rí♠❡tr♦ ❞❛❞♦ 2p✱ t❡♠ ♠❛✐♦r ár❡❛ ❄ ✶✶✳ ❉❡ ✉♠❛ ❢♦❧❤❛ ❝✐r❝✉❧❛r✱ t❡♠♦s q✉❡ ❝♦rt❛r ✉♠ s❡t♦r ❝✐r❝✉❧❛r ❞❡ ♠♦❞♦ q✉❡ ♣♦ss❛♠♦s ❝♦♥str✉✐r ✉♠ ❢✉♥✐❧ ❞❡ ♠❛✐♦r ❝❛♣❛❝✐❞❛❞❡ ♣♦ssí✈❡❧✳ ❉❡t❡r♠✐♥❡ ♦ â♥❣✉❧♦ ❝❡♥tr❛❧ α ❞♦ s❡t♦r ❝✐r❝✉❧❛r✳ ✶✷✳ ❖❜t❡r ✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s ❞❡ ár❡❛ ♠á①✐♠❛ ✐♥s❝r✐t♦ ♥✉♠ ❝ír❝✉❧♦ ❞❡ r❛✐♦ 12cm✳ ✸✺✵ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ✻✳✹ ❆♣❧✐❝❛çõ❡s ❞✐✈❡rs❛s ❆♣r❡s❡♥t❛✲s❡ ❛ s❡❣✉✐r ♣r♦❜❧❡♠❛s ❛♣❧✐❝❛❞♦s ❛ ❞✐✈❡rs♦s r❛♠♦s ❞❛s ❝✐ê♥❝✐❛s✱ t❛✐s ❝♦♠♦ ♣r♦❜❧❡♠❛s ❞❡ ❢ís✐❝❛✱ q✉í♠✐❝❛✱ ❜✐♦❧♦❣✐❛✱ ❡t❝✳ ❊①❡♠♣❧♦ ✻✳✺✶✳ ❉❡t❡r♠✐♥❡ ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ❞❡ ♠♦❞♦ q✉❡ s✉❛ s♦♠❛ s❡❥❛ 60 ❡ s❡✉ ♣r♦❞✉t♦ ♦ ♠❛✐♦r ♣♦ssí✈❡❧✳ ❙♦❧✉çã♦✳ 60 − x✱ ❡♥tã♦ ♦ ♣r♦❞✉t♦ P (x) = x(60 − x)✱ ❧♦❣♦ P ′ (x) = 60 − 2x q✉❛♥❞♦ P ′ (x) = 0 s❡❣✉❡ q✉❡ x = 30 ✭é ♣♦♥t♦ ❝rít✐❝♦ ❞❡ P (x)✮❀ t❛♠❜é♠ P ′′ (x) = −2 ❡ P ′′ (30) = −2 < 0✳ P❡❧♦ ❝r✐tér✐♦ ❞❛ ❞❡r✐✈❛❞❛ s❡❣✉♥❞❛ ❞❡ P (x)✱ ❡♠ x = 30 t❡♠♦s ♠á①✐♠♦ ♣❛r❛ P (x)✳ ▲♦❣♦ ♦s ♥ú♠❡r♦s sã♦ 30 ❡ 30✳ ❙❡❥❛♠ ♦s ♥ú♠❡r♦s x ❡ ❊①❡♠♣❧♦ ✻✳✺✷✳ ❉❛❞❛ ✉♠❛ ❢♦❧❤❛ ❞❡ ♣❛♣❡❧ã♦ q✉❛❞r❛❞❛ ❞❡ ❧❛❞♦ a✱ ❞❡s❡❥❛✲s❡ ❝♦♥str✉✐r ✉♠❛ ❝❛✐①❛ ❞❡ ❜❛s❡ q✉❛❞r❛❞❛ s❡♠ t❛♠♣❛ ❝♦rt❛♥❞♦ ❡♠ s✉❛s ❡sq✉✐♥❛s q✉❛❞r❛❞♦s ✐❣✉❛✐s ❡ ❞♦❜r❛♥❞♦ ❝♦♥✈❡♥✐❡♥✲ t❡♠❡♥t❡ ❛ ♣❛rt❡ r❡st❛♥t❡✳ ❉❡t❡r♠✐♥❛r ♦ ❧❛❞♦s ❞♦s q✉❛❞r❛❞♦s q✉❡ ❞❡✈❡♠ s❡r ❝♦rt❛❞♦s ❞❡ ♠♦❞♦ q✉❡ ♦ ✈♦❧✉♠❡ ❞❛ ❝❛✐①❛ s❡❥❛ ♠á①✐♠♦ ♣♦ssí✈❡❧✳ ❙♦❧✉çã♦✳ x ♦ ❧❛❞♦ ❞♦ q✉❛❞r❛❞♦ ❛ s❡r ❝♦rt❛❞♦ ❡♠ ❝❛❞❛ ❡sq✉✐♥❛✱ ♦ ✈♦❧✉♠❡ ❞❛ ❝❛✐①❛ é a V (x) = x(a − 2x)2 ♦♥❞❡ 0 < x < ✳ ❉❡r✐✈❛♥❞♦t❡♠♦s V ′ (x) = a2 − 8ax + 12x2 ✱ q✉❛♥❞♦ 2 a V ′ (x) = 0 t❡♠♦s q✉❡ ♦ ú♥✐❝♦ ♣♦♥t♦ ❝rít✐❝♦ q✉❡ ❝✉♠♣r❡ ❛ ❝♦♥❞✐çã♦ é ❀ ♣♦r ♦✉tr♦ ❧❛❞♦✱ 6 ′′ a ′′ V (x) = −8a + 24x ❡ V ( ) = −4a < 0✳ 6 ❙❡♥❞♦ P♦rt❛♥t♦✱ ♦ ✈♦❧✉♠❡ s❡rá ♠á①✐♠♦ q✉❛♥❞♦ ♦s ❝♦rt❡s ❞♦s q✉❛❞r❛❞♦s ♥❛s ❡sq✉✐♥❛s s❡❥❛♠ ✐❣✉❛✐s à s❡①t❛ ♣❛rt❡ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❧❛❞♦ a✳ ❊①❡♠♣❧♦ ✻✳✺✸✳ ❉❡s❡❥❛✲s❡ ❝♦♥str✉✐r ✉♠ ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r r❡t♦ ❝♦♠ t❛♠♣❛✱ ❝✉❥❛ ❜❛s❡ s❡❥❛ ✉♠❛ ❝✐r❝✉♥❢❡✲ rê♥❝✐❛✱ ❞❡ ♠♦❞♦ ❛ ❣❛st❛r ❛ ♠❡♥♦r q✉❛♥t✐❞❛❞❡ ❞❡ ♠❛t❡r✐❛❧✳ ◗✉❛❧ é ❛ r❡❧❛çã♦ ❡♥tr❡ ❛ ❛❧t✉r❛ ❡ ♦ r❛✐♦ ❞❛ ❜❛s❡ ♣❛r❛ ✐st♦ ❛❝♦♥t❡❝❡r ❄ ❙♦❧✉çã♦✳ ❉❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ♠❛t❡♠át✐❝♦✱ ♦ ♣r♦❜❧❡♠❛ ❛♣r❡s❡♥t❛ ❞♦✐s ❛s♣❡❝t♦s✳ ❛✮ ❉❡ t♦❞♦s ♦s ❝✐❧✐♥❞r♦s q✉❡ ♣♦ss✉❡♠ ár❡❛ t♦t❛❧ ✐❣✉❛❧✱ t❡rá ♠❡♥♦r ❣❛st♦ ❞❡ ♠❛t❡r✐❛❧ ❛q✉❡❧❡ q✉❡ t❡♥❤❛ ♠❛✐♦r ✈♦❧✉♠❡✳ ❜✮ ❉❡ t♦❞♦s ♦s ❝✐❧✐♥❞r♦s q✉❡ ♣♦ss✉❡♠ ♦ ♠❡s♠♦ ✈♦❧✉♠❡✱ t❡rá ♠❡♥♦r ❣❛st♦ ❞❡ ♠❛t❡r✐❛❧ ❛q✉❡❧❡ q✉❡ s✉❛ ár❡❛ s❡❥❛ ♠í♥✐♠❛✳ ✸✺✶ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈♦♥s✐❞❡r❡♠♦s ♦ ❝❛s♦ ❞❛ ♣❛rt❡ ❛✮✳ ❙✉♣♦♥❤❛ ✉♠ ❝✐❧✐♥❞r♦ ❞❡ ❛❧t✉r❛ h ❡ r❛✐♦ ❞❛ ❜❛s❡ r ❀ ❡♥tã♦ s✉❛ ár❡❛ t♦t❛❧ é A V = πr2 h✳ A − 2πr2 ✱ ✈❡♠✱ h = 2πr = 2πr2 +2πrh ✭A é ❝♦♥st❛♥t❡✮ ❡ s❡✉ ✈♦❧✉♠❡ ❉♦ ❞❛❞♦ ❞❛ ár❡❛ t♦t❛❧ s✉❜st✐t✉✐♥❞♦ ❡st❡ ✈❛❧♦r ❡♠ V t❡♠♦s Ar A A − 2πr2 )= − πr3 ⇒ V ′ (r) = − 3πr2 2πr 2 2 r A ♦t✐♠✐③❛♥❞♦ ❡st❛ ❢✉♥çã♦ ❡♥❝♦♥tr❛✲s❡ r0 = é ♣♦♥t♦ ❝rít✐❝♦✱ ❡ V ′′ (r0 ) = −6πr < 0✱ 6π ❛ss✐♠ ♦ ✈♦❧✉♠❡ é ♠á①✐♠♦✳ r A ⇒ A = 6πr2 ✱ s✉❜st✐t✉✐♥❞♦ ♥❛ ❛❧t✉r❛ h t❡♠♦s ❈♦♥s✐❞❡r❡ r = r0 = 6π V (r) = πr2 ( h= ▲♦❣♦ ❛ r❡❧❛çã♦ h:r é 2 : 1❀ A − 2πr2 6πr2 − 2πr2 = = 2r 2πr 2πr ✐st♦ é✱ ❛ ❛❧t✉r❛ é ♦ ❞♦❜r♦ ❞♦ r❛✐♦ ❞❛ ❜❛s❡✳ ❊①❡♠♣❧♦ ✻✳✺✹✳ ❯♠ ❛r❛♠❡ ❞❡ 80cm ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡✈❡ s❡r ❝♦rt❛❞♦ ❡♠ ❞♦✐s ♣❡❞❛ç♦s✳ ❞❡❧❡s ❞❡✈❡✲s❡ ❝♦♥str✉✐r ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❡ ❝♦♠ ♦ ♦✉tr♦ ✉♠ q✉❛❞r❛❞♦✳ ❈♦♠ ✉♠ ◗✉❛✐s sã♦ ❛s ❞✐♠❡♥sõ❡s ❞♦s ❛r❛♠❡s ❞❡ ♠♦❞♦ q✉❡ ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦ ❝ír❝✉❧♦ ❡ q✉❛❞r❛❞♦ s❡❥❛♠✿ ♠í♥✐♠❛❀ ❜✮ ❛✮ ♠á①✐♠❛✳ ❙♦❧✉çã♦✳ ❙✉♣✉♥❤❛ ❛ r❛✐♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ s❡❥❛ r✱ m❀ ❡ s❡❥❛♠ ♦s ❝♦♠♣r✐✲ 4m = 80 − x✳ ❆ s♦♠❛ ❞❛s ár❡❛s ❡ ♦ ❧❛❞♦ ❞♦ q✉❛❞r❛❞♦ xcm ❡ (80 − x)cm❀ ❡♥tã♦ 2πr = x ❡ h x i2  80 − x 2 2 2 ✳ + é✿ S = πr + m = π 2π 4     x (80 − x) 1 1 4+π ′ − =x + − 10 = x − 10❀ ♦ ú♥✐❝♦ ♣♦♥t♦ ▲♦❣♦✱ S (x) = 2π 8  8π  2π 8 8π ❝rít✐❝♦ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ x = 10 ≈ 35, 19✳ 4+π   1 1 ′′ ❆ ❞❡r✐✈❛❞❛ s❡❣✉♥❞❛ ❞❡ S(x) é✿ S (x) = > 0✳ + 2π 8 ❈♦♠♦ ❛ ❢✉♥çã♦ S(x) s♦♠❡♥t❡ t❡♠ ♠í♥✐♠♦ r❡❧❛t✐✈♦ ❡♠ x ≈ 35, 19✱ ❛ ár❡❛ ♠í♥✐♠❛  2  2 35, 19 80 − 35, 19 é S(35, 19) = π + = 224, 13cm2 ✱ ♣❡❧♦ ❢❛t♦ ♥ã♦ ♣♦ss✉✐r ♠❛✐s 2π 4 ♠❡♥t♦s ❞♦ ❛r❛♠❡ ♣♦♥t♦s ❝rít✐❝♦s✱ ❛ ár❡❛ ♠á①✐♠❛ ❞❡✈❡ ♦❝♦rr❡r ❡♠ ✉♠ ❞♦s ♣♦♥t♦s ❞♦ ❡①tr❡♠♦✳ ◗✉❛♥❞♦ x 80, S(80) = π  80 2π 2 = 509.75cm2 = t❡♠✲s❡ ár❡❛ ♠á①✐♠❛✳ ✸✺✷ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❊①❡♠♣❧♦ ✻✳✺✺✳ ●❡r❛❞♦r é ✉♠ ❛♣❛r❡❧❤♦ q✉❡ tr❛♥s❢♦r♠❛ q✉❛❧q✉❡r t✐♣♦ ❞❡ ❡♥❡r❣✐❛ ❡♠ ❡♥❡r❣✐❛ ❡❧étr✐❝❛✳ ❙❡ ❛ ♣♦tê♥❝✐❛ P ✱ ❡♠ watts✱ q✉❡ ✉♠ ❝❡rt♦ ❣❡r❛❞♦r ❧❛♥ç❛ ♥✉♠ ❝✐r❝✉✐t♦ ❡❧étr✐❝♦✱ é ❞❛❞♦ ♣♦r✿ P (i) = 20.i − 51i2 ✱ ♦♥❞❡ i é ❛ ✐♥t❡♥s✐❞❛❞❡ ❞❛ ❝♦rr❡♥t❡ ❡❧étr✐❝❛ q✉❡ ❛tr❛✈❡ss❛ ♦ ❣❡r❛❞♦r✱ ❡♠ ❛♠♣❡r❡s ✭amp✮✱ ♣❡❞❡✲s❡✿ ❛✮ P❛r❛ q✉❡ ✐♥t❡♥s✐❞❛❞❡ ❞❛ ❝♦rr❡♥t❡ ❡❧étr✐❝❛ ❡st❡ ❣❡r❛❞♦r ❧❛♥ç❛ ♥♦ ❝✐r❝✉✐t♦ ♣♦tê♥❝✐❛ ♠á①✐♠❛❄ ❜✮ P❛r❛ q✉❡ ✐♥t❡♥s✐❞❛❞❡ ❞❛ ❝♦rr❡♥t❡ ❡❧étr✐❝❛✱ ❡st❡ ❣❡r❛❞♦r ❧❛♥ç❛ ♥♦ ❝✐r❝✉✐t♦ ✉♠❛ ♣♦tê♥❝✐❛ ♠❛✐♦r q✉❡ 15W ❄ ❙♦❧✉çã♦✳ ❆ ♣♦tê♥❝✐❛ é ♠á①✐♠❛ q✉❛♥❞♦ ❡①✐st❡ i✱ ❞❡ ♠♦❞♦ q✉❡ s❡❥❛ ❛ ❢✉♥çã♦ P (i) ♠á①✐♠❛✳ ❉❡ P (i) = 20.i − 51i2 t❡♠♦s q✉❡ P ′ (i) = 20 − 102i ♦♥❞❡ i = 20 20 é ♦ ✈❛❧♦r ❝rít✐❝♦❀ 102 = 0.196amp✳ ♦❜s❡r✈❡ q✉❡ P ′′ (i) = −102 < 0✱ ❧♦❣♦ ❡♠ i = 102 P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❡❞❡✲s❡ ♦ ✈❛❧♦r ❞❡ i q✉❛♥❞♦ P (i) = 15❀ ✐st♦ é 15 = 20.i − 51i2 ❧♦❣♦ 51i2 − 20.i + 15 = 0 ⇒ i= 20 ± p 202 − 4(51)(15) s❡♥❞♦ ♦ ✐♥t❡r✐♦r ❞❛ ♣❛rt❡ r❛❞✐❝❛❧ 2(51) ♥❡❣❛t✐✈❛✱ ♥ã♦ ❡①✐st❡ i✳ P♦rt❛♥t♦ ❛ r❡s♣♦st❛ ♣❛r❛ ❛ ♣❛rt❡ ❛✮ é i = 0.196amp ❡ ♣❛r❛ ❛ ♣❛rt❡ ❜✮ ♥♦ ❡①✐st❡ s♦❧✉çã♦✳ ❊①❡♠♣❧♦ ✻✳✺✻✳ ❉♦✐s ♣♦st❡s ✈❡rt✐❝❛✐s ❞❡ 6 ❡ 8 ♠❡tr♦s ❡stã♦ ♣❧❛♥t❛❞♦s ♥✉♠ t❡rr❡♥♦ ♣❧❛♥♦✱ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ❞❡ ✶✵♠ ❡♥tr❡ s✉❛s ❜❛s❡s✳ ❈❛❧❝✉❧❛r ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ♠í♥✐♠♦ ❞❡ ✉♠ ✜♦ q✉❡ ♣❛rt✐♥❞♦ ❞♦ t♦♣♦ ❞❡ ✉♠ ❞❡st❡s ♣♦st❡s✱ t♦q✉❡ ♦ s♦❧♦ ♥❛ r❡t❛ q✉❡ ✉♥❡ ❛s ❜❛s❡s ❡✱ ❧♦❣♦ ♦ t♦♣♦ ❞♦ ♦✉tr♦ ♣♦st❡✳ ❙♦❧✉çã♦✳ ◆❛ s❡❣✉✐♥t❡ ❋✐❣✉r❛ ✭✻✳✶✽✮✱ s❡❥❛ AC = 10, AB = 6 ❡ CD = 8✱ ❡♥tã♦ ❛ ❤✐♣♦t❡♥✉s❛ p √ BM = 36 + x2 ❡ M D = 64 + (10 − x)2 ✳ ❆ ❢✉♥çã♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ✜♦ q✉❡ ♠♦❞❡❧❛ ♦ ♣r♦❜❧❡♠❛ é✿ √ x2 p + 64 + (10 − x)2 D 8m C ✉ ❅ ❅ ❅ ❅ ❅ ❅ 10 − x ✉B ❅ ❅ M❅ 6m x A 36 + 50 m ▲❡♠❜r❡ q✉❡ x ≥ 0❀ ❧♦❣♦ ♥♦ ❝á❧❝✉❧♦ ❞♦s ❋✐❣✉r❛ ✻✳✶✽✿ ♣♦♥t♦s ❝rít✐❝♦s ❞❡ f (x) t❡♠♦s✿ 10 − x x ✱ − p f ′ (x) = √ 36 + x2 64 + (10 − x)2 30 q✉❛♥❞♦ f ′ (x) = 0✱ t❡♠♦s x = é ♣♦♥t♦ ❝rít✐❝♦✳ 7 64 30 36 + > 0 ❡ f ′′ ( ) > 0✱ ❆ ❞❡r✐✈❛❞❛ s❡❣✉♥❞❛ ❞❡ f (x) é✱ f ′′ (x) = 2 2 36 + x 64 + (10 − x) 7 s  2 30 30 30 ❀ ❛ss✐♠ f ( ) = 36 + + ❧♦❣♦ t❡♠♦s ❝♦♠♣r✐♠❡♥t♦ ♠í♥✐♠♦ q✉❛♥❞♦ x = 7 7 7 f (x) = ✸✺✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ s  30 64 + 10 − 7 2 = 17.20m✳ ❊①❡♠♣❧♦ ✻✳✺✼✳ s(t) = 12t2 + 6t✳ ❛✮ ❯♠ ❛✉t♦♠ó✈❡❧ ❞❡s❝❡ ✉♠ ♣❧❛♥♦ ✐♥❝❧✐♥❛❞♦ s❡❣✉♥❞♦ ❛ ❡q✉❛çã♦ ❛ ✈❡❧♦❝✐❞❛❞❡ 3 s❡❣✉♥❞♦s ❞❡♣♦✐s ❞❛ ♣❛rt✐❞❛❀ ❜✮ ❆❝❤❛r ❞❡t❡r♠✐♥❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥✐❝✐❛❧✳ ❙♦❧✉çã♦✳ ❖ ❛✉t♦♠ó✈❡❧ q✉❡ ❡st❛✈❛ ❡♠ r❡♣♦✉s♦✱ ❞❡s❝r❡✈❡ ✉♠ ♠♦✈✐♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ t❡♠♣♦ s(t) = 12t2 + 6t❀ v(t) = s′ (t) = 24t + 6✳ ♠❡❞✐❛♥t❡ ❛ ❡q✉❛çã♦ tr❛❥❡tór✐❛ é s✉❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛ ❡♠ q✉❛❧q✉❡r ♣♦♥t♦ ❞❛ ❛✮ v(3) = 24(3) + 6 = 78m/sg ✳ ❜✮ ❆ ✈❡❧♦❝✐❞❛❞❡ ✐♥✐❝✐❛❧ q✉❛♥❞♦ t = 0✱ ❢♦✐ v(0) = 6m/sg ✳ ❊①❡♠♣❧♦ ✻✳✺✽✳ t 1o ❞❡ ❥❛♥❡✐r♦ ❞❡ 1.994 s❡❥❛ 4.000 ✳ ✭❛✮ ❯s❡ ❛ ❞❡r✐✈❛❞❛ ♣❛r❛ ❡st✐♠❛r ❛ ♠✉❞❛♥ç❛ ❡s♣❡r❛❞❛ ♥❛ ♣♦♣✉❧❛çã♦ f (t) = 10.000− t+1 o o ❞❡ 1 ❞❡ ❥❛♥❡✐r♦ ❞❡ 1998 ❛ 1 ❞❡ ❥❛♥❡✐r♦ ❞❡ 1999❀ ✭❜✮ ❆❝❤❡ ❛ ♠✉❞❛♥ç❛ ❡①❛t❛ ❡s♣❡r❛❞❛ ♥❛ o o ♣♦♣✉❧❛çã♦ 1 ❞❡ ❥❛♥❡✐r♦ ❞❡ 1998 ❛ 1 ❞❡ ❥❛♥❡✐r♦ ❞❡ 1999✳ ❊s♣❡r❛✲s❡ q✉❡ ❛ ♣♦♣✉❧❛çã♦ ❞❡ ✉♠❛ ❝❡rt❛ ❝✐❞❛❞❡ ❛♥♦s ❛♣ós ❙♦❧✉çã♦✳❛✮ ❚❡♠♦s ♦ 1 o 1o t = 0 ❡ f (0) = 6.000 1.998 t❡♠♦s t = 4✳ ❞❡ ❥❛♥❡✐r♦ ❞❡ ❥❛♥❡✐r♦ ❞❡ ❤❛❜✐t❛♥t❡s✳ ❈♦♠♦ t é ❞❛❞♦ ❡♠ ❛♥♦s✱ ❡♠ P♦r ♦✉tr♦ ❧❛❞♦✱ ❡♠ ❣❡r❛❧ f ′ (t) ≈ ❛ss✐♠ f (t + 1) − f (t) (t + 1) − t f ′ (4) = (10.000 − ❧♦❣♦ f ′ (4) ≈ f (4 + 1) − f (4) = f (5) − f (4) (4 + 1) − 4 4.000 4.000 ) − (10.000 − ) = 200 5 4 ❛ ♠✉❞❛♥ç❛ ❡s♣❡r❛❞❛ é ❞❡ 200 ❤❛❜✐t❛♥t❡s ❛ ♠❛✐s✳ ❙♦❧✉çã♦✳ ❜✮ ▲❡♠❜r❡✱ f ′ (x) = 4.000 ✱ (t + 1)2 ❧♦❣♦ ❛ ♠✉❞❛♥ç❛ ❡s♣❡r❛❞❛ ❡①❛t❛ é f ′ (4) = 4.000 = 160✳ (4 + 1)2 ❊①❡♠♣❧♦ ✻✳✺✾✳ ❯♠❛ ♣❡❞r❛ é ❧❛♥ç❛❞❛ ♣❛r❛ ❝✐♠❛ ✈❡rt✐❝❛❧♠❡♥t❡❀ s✉♣♦♥❤❛ ❛t✐♥❥❛ s✉❛ ❛❧t✉r❛ 10t ❡♠ ♠❡tr♦s ❞❡♣♦✐s ❞❡ t h(t) = −5t2 + s❡❣✉♥❞♦s ❞♦ ❧❛♥ç❛♠❡♥t♦✳ ◗✉❡ ❛❧t✉r❛ ♠á①✐♠❛ ❛t✐♥❣✐rá ❛ ♣❡❞r❛❄ ◗✉❛♥t♦s s❡❣✉♥❞♦s ❛♣ós t❡r s✐❞♦ ❧❛♥ç❛❞❛❄ ❙♦❧✉çã♦✳ h(t) = −5t2 + 10t✱ ❡♥tã♦ h′ (t) = h′′ (t) = −10 < 0 ❛ss✐♠ ❡♠ t = 1 t❡♠♦s ♠á①✐♠♦ ➱ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♠á①✐♠♦ r❡❧❛t✐✈♦✱ ♣♦r ❤✐♣ót❡s❡ −10t + 10 ⇒ t=1 é ♣♦♥t♦ ❝rít✐❝♦❀ ✸✺✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R r❡❧❛t✐✈♦ ✭t❛♠❜é♠ ❛❜s♦❧✉t♦✮ ♦♥❞❡ h(1) = −5(1)2 + 10(1) = 5 m ❡❧❛ ❛t✐♥❣❡ ♦ ♣♦♥t♦ ♠❛✐s ❛❧t♦ 1 s❡❣✉♥❞♦ ❛♣ós ❞❡ ❧❛♥ç❛❞❛ ♣❛r❛ ❛rr✐❜❛✳ ❊①❡♠♣❧♦ ✻✳✻✵✳ ❆❝❤❛r ♦s ✈❛❧♦r❡s ❞❡ x ❡ y ✱ ❛ ✜♠ ❞❡ q✉❡ ❛ ❡①♣r❡ssã♦ xn y m s❡❥❛ ♠á①✐♠❛✱ s❡♥❞♦ x+y = a✱ ♦♥❞❡ a é ❝♦♥st❛♥t❡✳ ❙♦❧✉çã♦✳ ❚❡♠♦s y = a − x✱ ♣♦r ♦✉tr♦ ❧❛❞♦ ❞❛ ❡①♣r❡ssã♦ xn y m ♣♦❞❡♠♦s ♦❜t❡r ❛ ❢✉♥çã♦ f (x) = xn (a − x)m ✳ an ❚❡♠♦s f ′ (x) = xn (a − x)m [na − x(n + m)]❀ sã♦ ♣♦♥t♦s ❝rít✐❝♦s x = a ❡ x = ✳ m+n ◗✉❛♥❞♦ x = a t❡♠♦s y = 0 ❡ x y ♥ã♦ é ✉♠❛ ❡①♣r❡ssã♦ ♠á①✐♠❛ ✭é ❝♦♥st❛♥t❡✮✳ n m an an ✱ ❡♥tã♦ f ′ (x1 ) < 0❀ ❡ s❡ x2 < ❡♥tã♦ f ′ (x2 ) > 0❀ ❛ss✐♠ f (x) t❡♠ m+n m+n an ♠á①✐♠♦ q✉❛♥❞♦ x = ✳ m+n n(x + y) ❈♦♠♦ x + y = a ❡♥tã♦ x = ⇒ x.m = n.y ❧♦❣♦ ❛ ❡①♣r❡ssã♦ xn y m s❡rá m+n n x : ✳ ♠á①✐♠❛ q✉❛♥❞♦ é ❝✉♠♣r❡ ❛ r❡❧❛çã♦ ✿ y m ❙❡❥❛ x1 > ❊①❡♠♣❧♦ ✻✳✻✶✳ ❉✉❛s ❧✐♥❤❛s ❢érr❡❛s s❡ ❝r✉③❛♠ ❡♠ â♥❣✉❧♦ r❡t♦✳ ❉✉❛s ❧♦❝♦♠♦t✐✈❛s✱ ❞❡ 20 m ❝❛❞❛ ✉♠❛✱ ❡♠ ❣r❛♥❞❡ ✈❡❧♦❝✐❞❛❞❡✱ ❛♣r♦①✐♠❛♠✲s❡ ❞♦ ❝r✉③❛♠❡♥t♦✱ s❡ ❞❡s❧♦❝❛♥❞♦ ❡♠ ❝❛❞❛ ✉♠❛ ❞❡ss❛s ❧✐♥❤❛s ❢❡rr❡❛s✳ ❆ ♣r✐♠❡✐r❛ ❞❡❧❛s✱ s❡ ❡♥❝♦♥tr❛ ❡♠ ✉♠❛ ❡st❛çã♦ A ❛ 65 km ❞♦ ❝r✉③❛♠❡♥t♦❀ ❛ ♦✉tr❛✱ s❡ ❡♥❝♦♥tr❛ ♥❛ ❡st❛çã♦ B ❛ 40 km✳ ❆ ♣r✐♠❡✐r❛ s❡ ❞❡s❧♦❝❛ ❛ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ 600 m/min✱ ❡♥q✉❛♥t♦ ♦ ♦✉tr❛ ✈✐❛❥❛ ❛ 800 m/min✳ ◗✉❛♥t♦s ♠✐♥✉t♦s t❡rã♦ tr❛♥s❝♦rr✐❞♦s ❞❡s❞❡ ❛ ♣❛rt✐❞❛ ❛té ♦ ✐♥st❛♥t❡ ❡♠ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❛s ❞✉❛s ❧♦❝♦♠♦t✐✈❛s s❡❥❛ ♠í♥✐♠❛❄ ◗✉❛❧ é ❡ss❛ ❞✐stâ♥❝✐❛❄ ❙♦❧✉çã♦✳ ❙✉♣♦♥❤❛♠♦s t❡rã♦ tr❛♥s❝♦rr✐❞♦ x min ❛té ❝❤❡❣❛r ❛♦ ❝r✉③❛♠❡♥t♦✳ ❆ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ♣r✐✲ ♠❡✐r❛ é 600m/min = 0, 6km/min ❡✱ ❞❛ s❡✲ ❣✉♥❞❛ 0, 8km/min✳ ❙❡❣✉♥❞♦ ♦ ❣rá✜❝♦ ❞❛ ❋✐❣✉r❛ ✭✻✳✶✾✮ ❚❡♠♦s AO = 65km ❡ OB = 40km❀ ❛♣ós tr❛♥s❝♦rr✐✲ ❞♦s x min t❡♠♦s ❛s ❞✐stâ♥❝✐❛s✿ OC = (65 − 0.6x), B ◗ ◗ D ◗◗ ◗ p(40 ◗ − 0, 8x)2 + (65 − 0, 6x)2 ◗ ◗ ◗ ◗ ◗ 40 km 40 − 0, 8x ◗ ◗ ◗ 65 − 0, 6x ◗◗ ◗ ◗ O −16, 8km ✑ E ✑ ✑ ✑ ✑ C ✑✑ A ❋✐❣✉r❛ ✻✳✶✾✿ OD = (40 − 0.8x) ❙❡❥❛ CD ❛ ❞✐stâ♥❝✐❛ q✉❡ s❡♣❛r❛ ❛s ❞✉❛s ❧♦❝♦♠♦t✐✈❛s✱ ❧♦❣♦✿ CD = p (65 − 0.6x)2 + (40 − 0.8x)2 ✳ ✸✺✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ❆ ❢✉♥çã♦ q✉❡ ❞❡s❝r❡✈❡ ❛ ❞✐stâ♥❝✐❛ f (x) = p CD R é✿ (65 − 0, 6x)2 + (40 − 0, 8x)2 ❈❛❧❝✉❧❡♠♦s ♦ ♠í♥✐♠♦ r❡❧❛t✐✈♦ ❞❛ ❢✉♥çã♦ f (x)✿ 71 − x 0, 6(65 − 0, 6x) + 0, 8(40 − 0, 8x) = −p f ′ (x) = − p (65 − 0, 6x)2 + (40 − 0, 8x)2 (65 − 0, 6x)2 + (40 − 0, 8x)2 q✉❛♥❞♦ x = 71 f ′ (x) = 0 t❡♠♦s x = 71❀ x1 > 71, f ′ (x1 ) > 0 s❡ ❡ s❡ x2 < 71, f ′ (x2 ) < 0✱ ❧♦❣♦ ❡♠ t❡♠♦s ♠í♥✐♠♦ r❡❧❛t✐✈♦✳ 40−0.8(71) = −16, 8 = OE, ❖❜s❡r✈❡ q✉❡✿ P♦rt❛♥t♦✱ t❡rã♦ tr❛♥s❝♦rr✐❞♦ 71 65−0, 6(71) = 22, 4 = OC ❡ f (71) = 28✳ ♠✐♥✉t♦s ❡ ❛ ❞✐stâ♥❝✐❛ ♠í♥✐♠❛ ❡♥tr❡ ❡❧❛s é ❞❡ 28km✳ ❊①❡♠♣❧♦ ✻✳✻✷✳ ❊♥❝❤❡✲s❡ ✉♠ ❜❛❧ã♦ ❡s❢ér✐❝♦✱ ❞❡ t❛❧ ♠♦❞♦ q✉❡ s❡✉ ✈♦❧✉♠❡ ❡stá ❝r❡s❝❡♥❞♦ à r❛③ã♦ ❞❡ 5 cm2 /min.✳ ❊♠ q✉❡ r❛③ã♦ ♦ ❞✐â♠❡tr♦ ❝r❡s❝❡ q✉❛♥❞♦ ♦ ❞✐â♠❡tr♦ é 12 cm ❄ ❙♦❧✉çã♦✳ ❖ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛ ❞❡ r❛✐♦ r é 3 V (x) = 4πx ✳ 3(8) 4πr3 V = ❀ 3 ❖ ❞✐❢❡r❡♥❝✐❛❧ ❞♦ ✈♦❧✉♠❡ ❡♠ r❡❧❛çã♦ ❛♦ ❞✐â♠❡tr♦ s❡♥❞♦ s❡✉ ❞✐â♠❡tr♦ x é✱ d(V ) = 2 ❞❛❞♦s✱ d(V ) = 5 cm min. ❡ x = 12 cm✱ x = 2r✱ 12πx2 dx❀ 24 t❡♠♦s s❡❣✉♥❞♦ ♦s ❧♦❣♦ 5 cm2 12π(12 cm)2 = dx min. 24 P♦r t❛♥t♦ ♦ ❞✐â♠❡tr♦ ❝r❡s❝❡ ♥❛ r❛③ã♦ ❞❡ ⇒ dx = 10 cm 144π min 10 cm ✳ 144π min ❊①❡♠♣❧♦ ✻✳✻✸✳ ◆✉♠ ❝✐r❝✉✐t♦ ❡❧étr✐❝♦✱ s❡ ❛♠♣❡r❡s é ❛ ❝♦rr❡♥t❡✱ ❛ ❧❡✐ ❝♦♥st❛♥t❡✱ ♠♦str❡ q✉❡ R E volts ❞❡ Ohm ❞❡❝r❡s❝❡ ❛ R ohns é ❛ r❡s✐stê♥❝✐❛✱ I ❡st❛❜❡❧❡❝❡ q✉❡ I · R = E ✳ ❙✉♣♦♥❤❛ q✉❡ E s❡❥❛ ✉♠❛ t❛①❛ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ✐♥✈❡rs♦ ❞♦ q✉❛❞r❛❞♦ ❞❡ I ✳ é ❛ ❢♦rç❛ ❡❧❡tr♦♠♦tr✐③✱ ❙♦❧✉çã♦✳ ➱ ✐♠❡❞✐❛t♦✱ ♣❡❧♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✱ t❡♠♦s q✉❡✱ s❡♥❞♦ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❡ P♦rt❛♥t♦✱ R R é ❞❛❞❛ ♣❡❧❛ ❡①♣r❡ssã♦ é ❞❡❝r❡s❝❡♥t❡ ✭dR q✉❛❞r❛❞♦ ❞❛ ❝♦rr❡♥t❡ < 0✮❀ E  ′ E dI dR = I ❝♦♥st❛♥t❡✱ ❡♥tã♦ ✐st♦ é✱ dR = − R(I) = E dI ✳ I2 E ❀ I ❞❡❝r❡s❝❡ ❛ ✉♠❛ t❛①❛ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ✐♥✈❡rs♦ ❞♦ I✳ ✸✺✻ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❊①❡♠♣❧♦ ✻✳✻✹✳ ❉❡t❡r♠✐♥❡ ❛s ❞✐♠❡♥sõ❡s ❞♦ ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r r❡t♦ ❞❡ ✈♦❧✉♠❡ ♠á①✐♠♦ q✉❡ ♣♦❞❡ s❡r ✐♥s❝r✐t♦ ❡♠ ✉♠❛ ❡s❢❡r❛ ❞❡ r❛✐♦ R = 12 m ❋✐❣✉r❛ ✭✻✳✷✵✮ ❙♦❧✉çã♦✳ ❖❜s❡r✈❡ ❛ ❙❡❥❛ r ❋✐❣✉r❛ 1 R = 12 = AB = OB = AO✳ 2 ❡♥tã♦ AC = 2r ❡ ❛ ❛❧t✉r❛ ❞♦ ❝✐❧✐♥❞r♦ ✭✻✳✷✵✮✱ s❛❜❡♠♦s ♦ r❛✐♦ ❞❛ ❜❛s❡ ❞♦ ❝✐❧✐♥❞r♦✱ BC = q √ 2 2 AB − AC = 2 122 − r2 ♦ ✈♦❧✉♠❡ ❞♦ ❝✐❧✐♥❞r♦ é ❞❛❞♦ ♣❡❧❛ ❢✉♥çã♦ ❚❡♠♦s ❛ ❞❡r✐✈❛❞❛ V ′ (r) = sã♦ ♣♦♥t♦s ❝rít✐❝♦s✳ é √ V (r) = 2πr2 122 − r2 2πr(288 − 3r2 ) √ 122 − r2 q✉❛♥❞♦ √ r=4 6 √ r=4 6 ❡ ❙♦♠❡♥t❡ t❡♠♦s ✈♦❧✉♠❡ ♠á①✐♠♦ q✉❛♥❞♦ P♦rt❛♥t♦✱ ♦ r❛✐♦ ❞❛ ❜❛s❡ ❞♦ ❝✐❧✐♥❞r♦ é √ V ′ (r) = 0 ❡♥tã♦ r = ±4 6 ❡ ❡ r=0 √ BC = 8 3✳ s✉❛ ❛❧t✉r❛ √ BC = 8 3✳ A ❙ ❙ 4 km ❋✐❣✉r❛ ✻✳✷✵✿ ❙ ❙ √ ❙ 4 2 + x2 ❙ ❙ ❙ (4 − x) km x km ❙❙ O C B ❋✐❣✉r❛ ✻✳✷✶✿ ❊①❡♠♣❧♦ ✻✳✻✺✳ A✱ ❛ 4 km ❞♦ ♣♦♥t♦ ♠❛✐s ♣❡rt♦ O ❞❡ ✉♠❛ ❝♦st❛ r❡t❛❀ ♥♦ ♣♦♥t♦ B t❛♠❜é♠ ❞❛ ❝♦st❛ ❡ ❛ 4 km ❞❡ O ❡①✐st❡ ✉♠❛ t❡♥❞❛✳ ❙❡ ♦ ❣✉❛r❞❛ ❞♦ ❢❛r♦❧ ♣♦❞❡ r❡♠❛r ❛ 4 km/hora ❡ ❝❛♠✐♥❤❛r 5 km/hora✱ q✉❛❧ ♦ ❝❛♠✐♥❤♦ q✉❡ ❞❡✈❡ s❡❣✉✐r ♣❛r❛ ❝❤❡❣❛r ❯♠ ❢❛r♦❧ ❡♥❝♦♥tr❛✲s❡ ♥✉♠ ♣♦♥t♦ ❞♦ ❢❛r♦❧ à t❡♥❞❛ ♥♦ ♠❡♥♦r t❡♠♣♦ ♣♦ssí✈❡❧ ❄ ❙♦❧✉çã♦✳ ❙✉♣♦♥❤❛♠♦s ❛❝♦♥t❡ç❛ ♦ ❞❡s❡♥❤♦ ❞❛ s✐t✉❛❞♦ ❡♥tr❡ ❙❡❥❛ T O ❡ B ❋✐❣✉r❛ ✭✻✳✷✶✮✱ ✐st♦ é✱ ❞❡✈❡ r❡♠❛r ❛té ♦ ♣♦♥t♦ C ❧♦❣♦ ❝❛♠✐♥❤❛r ♦ r❡st♦ ❞♦ ❝❛♠✐♥❤♦✳ ♦ t❡♠♣♦ ✉t✐❧✐③❛❞♦ ❞❡s❞❡ ♦ ♣♦♥t♦ A ❛té ❝❤❡❣❛r ❛♦ ♣♦♥t♦ B✳ ❊♥tã♦✱ ❝♦♠♦ ♦ t❡♠♣♦ é ❛ r❡❧❛çã♦ ❞♦ ❡s♣❛ç♦ ❞✐✈✐❞✐❞♦ ❡♥tr❡ ✈❡❧♦❝✐❞❛❞❡✱ t❡♠♦s q✉❡ ♦ ✸✺✼ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ | AC | | CB | + ✳ 4 5 √ ❖❜s❡r✈❡✱ | AC |= 42 + x2 ❡ | CB |= 4 − x✱ ❧♦❣♦ t❡♠♣♦ T = T (x) = √ 4 2 + x2 4 − x + 4 5 ⇒ 0≤x≤4 1 x − ✱ q✉❛♥❞♦ T ′ (x) = 0 ❡ ❝♦♥s✐❞❡r❛♥❞♦ 2 5 4 16 + x 16 ♦ ❞♦♠í♥✐♦ ❞❡ ❞❡✜♥✐çã♦ ❞❛ ❢✉♥çã♦✱ ♦❜t❡♠♦s ♦ ♣♦♥t♦ ❝rít✐❝♦ x = ± q✉❡ ♥ã♦ ♣❡rt❡♥❝❡ ❛♦ 3 ❞♦♠í♥✐♦✳ ❆ ❝♦♥❝❧✉sã♦ é q✉❡ ♥ã♦ ❡①✐st❡ ♠á①✐♠♦ ♦✉ ♠í♥✐♠♦ r❡❧❛t✐✈♦ q✉❛♥❞♦ 0 ≤ x ≤ 4✳ √ 9 9 P♦r ♦✉tr♦ ❧❛❞♦✱ T (0) = ❡ T (4) = 2 < = T (0)✳ 5 5 √ 9 ❈♦♠♦ T (4) = 2 < = T (0)✱ é ♠❛✐s rá♣✐❞♦ r❡♠❛r ❞✐r❡t❛♠❡♥t❡ ❛té B ❡ ♥ã♦ ❝❛♠✐♥❤❛r✳ 5 ❉❡r✐✈❛♥❞♦ T (x) ♦❜t❡♠♦s T ′ (x) = √ ❊①❡♠♣❧♦ ✻✳✻✻✳ ❆s ♠❛r❣❡♥s s✉♣❡r✐♦r ❡ ✐♥❢❡r✐♦r ❞❡ ✉♠❛ ♣á❣✐♥❛ sã♦ 3cm ❝❛❞❛ ✉♠❛ ❡ ❛s ♠❛r❣❡♥s ❧❛t❡r❛✐s ❞❡ 2, 5 cm ❝❛❞❛ ✉♠❛✳ ❙❡ ❛ ár❡❛ ❞♦ ♠❛t❡r✐❛❧ ✐♠♣r❡ss♦ ❞❡✈❡ s❡r ✜①❛ ❡ ✐❣✉❛❧ ❛ 623, 7 cm2 ✳ ◗✉❛✐s sã♦ ❛s ❞✐♠❡♥sõ❡s ❞❛ ♣á❣✐♥❛ ❞❛ ár❡❛ ♠í♥✐♠❛❄✳ ❙♦❧✉çã♦✳ ❙✉♣♦♥❤❛♠♦s t❡♠♦s ❛ ♣á❣✐♥❛ ❝♦♠♦ ♥❛ ❋✐❣✉r❛ ✭✻✳✷✷✮✳ ❆ ár❡❛ ❞❛ ♠❡s♠❛ é  623, 7 + 6 cm2 A(x) = (x + 5) x  623 cm x ( 623 + 6) cm x P❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ♣♦♥t♦s ❝rít✐❝♦s t❡♠♦s A′ (x) = 6 − x cm 3.118, 6 =0 x2 (x + 5) cm ❡♥tã♦ x = 22, 80 ✭❛♣r♦①✐♠❛❞❛♠❡♥t❡✮✳ ❙❡♥❞♦ ❛ ❞❡r✐✲ ✈❛❞❛ s❡❣✉♥❞❛ ♣♦s✐t✐✈❛✱ ❡♠ x = 22.8 t❡♠♦s ♠í♥✐♠♦ r❡❧❛t✐✈♦✳ P♦rt❛♥t♦ ❛ ♣á❣✐♥❛ ❞❡✈❡ t❡r 27, 80 cm ♣♦r 33, 32 cm✳ ❋✐❣✉r❛ ✻✳✷✷✿ ❊①❡♠♣❧♦ ✻✳✻✼✳ ❙✉♣♦♥❤❛ q✉❡ ✉♠❛ ♣❡ss♦❛ ♣♦s❛ ❛♣r❡♥❞❡r f (t) ♣❛❧❛✈r❛s s❡♠ s❡♥t✐❞♦ ❡♠ t ❤♦r❛s ❡ f (t) = 25 t2 ✱ ♦♥❞❡ 0 ≤ t ≤ 9✳ ❆❝❤❡ ❛ t❛①❛ ❞❡ ❛♣r❡♥❞✐③❛❞♦ ❞❛ ♣❡ss♦❛ ❛♣ós✿ ✭❛✮ 1 ❤♦r❛❀ ✭❜✮ 8 ❤♦r❛s ✳ ❙♦❧✉çã♦✳ ❛✮ √ 5 ✸✺✽ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❆ t❛①❛ ❞❡ ❛♣r❡♥❞✐③❛❞♦ ❞❡♣♦✐s ❞❛ ♣r✐♠❡✐r❛ ❤♦r❛ é ❙♦❧✉çã♦✳ ❜✮ ❆ t❛①❛ ❞❡ ❛♣r❡♥❞✐③❛❞♦ ❞❡♣♦✐s ❞❛s 2, 77 ♣❛❧❛✈r❛s ❛♣ós ❞❡ 8 ❤♦r❛s✳ f (1) − f (0) = f (1) = 25 9−8 8 ♣r✐♠❡✐r❛s ❤♦r❛s é ♣❛❧❛✈r❛s✳ √ √ f (9) − f (8) 5 5 = 25( 92 − 82 ) = 9−8  √  2 5 −3 t ✱ q✉❛♥❞♦ t = 8 t❡♠♦s f ′ (8) = 2, 87. f (t) = 25 3 ′ ❆ t❛①❛ ❞❡ ❛♣r❡♥❞✐③❛❞♦ ❡①❛t♦ é ❊①❡♠♣❧♦ ✻✳✻✽✳ ◗✉❛♥❞♦ t♦ss✐♠♦s ♦ r❛✐♦ ❞❡ ♥♦ss❛ tr❛q✉é✐❛ ❞✐♠✐♥✉✐✱ ❛❢❡t❛♥❞♦ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❛r q✉❡ ♣❛ss❛ ♥❡ss❡ ór❣ã♦✳ ❙❡♥❞♦ r0 ❡ r r❡s♣❡❝t✐✈❛♠❡♥t❡ ♦ r❛✐♦ ❞❛ tr❛q✉é✐❛ ♥❛ s✐t✉❛çã♦ ♥♦r♠❛❧ ❡ ♥♦ ♠♦♠❡♥t♦ ❞❛ t♦ss❡✱ ❛ r❡❧❛çã♦ ❡♥tr❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❢♦r♠❛ 2 V (r) = ar (r0 − r)✱ ❈❛❧❝✉❧❡ ♦ r❛✐♦ r ♦♥❞❡ a V ❡ r é ❞❛❞❛ ♣♦r ✉♠❛ ❢✉♥çã♦ ❞❛ é ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛✳ q✉❡ ♣❡r♠✐t❡ ❛ ♠❛✐♦r ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❛r✳ ❙♦❧✉çã♦✳ ❚❡r❡♠♦s q✉❡ ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞❡ r q✉❡ ♠❛①✐♠✐③❛ ❛ ❢✉♥çã♦ V (r)❀ ❝♦♠ ❡❢❡✐t♦ V ′ (r) = 2 2 3ar( r0 − r) ♦ ✈❛❧♦r ❝rít✐❝♦ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ r = r0 ✳ 3 3 2 2 ❙❡❥❛ r1 > r0 ✱ ❡♥tã♦ V ′ (r1 ) < 0❀ ❡ s❡ x2 < r0 ❡♥tã♦ V ′ (r2 ) > 0❀ ❛ss✐♠ V (r) 3 3 2 2 ♠á①✐♠♦ q✉❛♥❞♦ x = r0 ✳ ❖ r❛✐♦ r q✉❡ ♣❡r♠✐t❡ ❛ ♠❛✐♦r ✈❡❧♦❝✐❞❛❞❡ é r = r0 ✳ 3 3 t❡♠ ❊①❡♠♣❧♦ ✻✳✻✾✳ ❆ s♦♠❛ ❞❡ três ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s é ♠❛✐s ♦ q✉á❞r✉♣❧♦ ❞♦ t❡r❝❡✐r♦ s♦♠❛♠ 80✳ 40✱ ♦ ♣r✐♠❡✐r♦ ♠❛✐s ♦ tr✐♣❧♦ ❞♦ s❡❣✉♥❞♦ ❉❡t❡r♠✐♥❡ ♦s ♥ú♠❡r♦s ❞❡ ♠♦❞♦ q✉❡ s❡✉ ♣r♦❞✉t♦ s❡❥❛ ♦ ♠❛✐♦r ♣♦ssí✈❡❧✳ ❙♦❧✉çã♦✳ ❙❡❥❛♠ ♦s ♥ú♠❡r♦s r❡❛✐s a, b, c ✭♥❡ss❛ ♦r❞❡♠✮ ❡ s✉♣♦♥❤❛♠♦s q✉❡ a = 40 − (b + c)✱ [40 − (b + c)] + 3b + 4c = 80 ⇒ ❧♦❣♦ 2b = 40 − 3c ♦ ♣r♦❞✉t♦ é P = abc = [40 − ( 40 − 3c 40 − 3c 1 + c)]( )c = (40 + c)(40 − 3c)c 2 2 4 1 P ′ (c) = − (9c2 + 160c − 1600) ♦♥❞❡ c = 6, 22 é ♣♦♥t♦ 4 ❞❡ ♠á①✐♠♦ r❡❧❛t✐✈♦✳ ❖ ♥ú♠❡r♦ ♣r♦❝✉r❛❞♦ ♣ró①✐♠♦ ❞❡ 6, 22 é 6✳ P♦rt❛♥t♦ ♦s ♥ú♠❡r♦s q✉❡ ❝✉♠♣r❡♠ ♦ ♣r♦❜❧❡♠❛ sã♦ 23, 11 ❡ 6✳ ❉❡r✐✈❛♥❞♦ ❛ ❢✉♥çã♦ P ♦❜t❡♠♦s ❊①❡♠♣❧♦ ✻✳✼✵✳ ✸✺✾ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ 50cm3 ♣♦r ♠❡❞❡ 15cm❄ ❯♠❛ ❜♦❧❛ ❡s❢ér✐❝❛ ❞❡ ♥❡✈❡ ❡stá s❡ ❞❡rr❡t❡♥❞♦ à r❛③ã♦ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ❡stá ❞✐♠✐♥✉✐♥❞♦ ♦ r❛✐♦ ❞❛ ❜♦❧❛ q✉❛♥❞♦ ❡st❡ R ♠✐♥✉t♦✳ ❈♦♠ q✉❛❧ ❙♦❧✉çã♦✳ ❈♦♠♦ ♦ r❛✐♦ t ✐♥st❛♥t❡ r ❡stá ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦✱ ❧♦❣♦ ♠✐♥✉t♦s ❡stá ❞❛❞♦ ♣♦r 4 V (t) = π[r(t)]3 3 r = r(t) ❛ss✐♠ −50 = 4π[r(t)]2 · r′ (t) ◗✉❛♥❞♦ r(t) = 15cm ⇒ V (t) ❞❛ ❜♦❧❛ ♥♦ ❝❡♥tí♠❡tr♦s ❝ú❜✐❝♦s✳ ❆ r❛♣✐❞❡③ ❝♦♠ q✉❡ ❛ ❜♦❧❛ s❡ ❞❡rr❡t❡ é ❞❛❞♦ ♣♦r 4π[r(t)]2 · r′ (t) ❡ ♦ ✈♦❧✉♠❡ V ′ (t) = −50cm3 ✱ r′ (t) = − t❛♠❜é♠ V ′ (t) = 50 4π[r(t)]2 s❡❣✉❡ r′ (t) = − 50 1 =− cm/min 2 4π[15] 18π ❛ ❞❡r✐✈❛❞❛ é ♥❡❣❛t✐✈❛✱ ❡r❛ ❞❡ ❡s♣❡r❛r ♣❡❧♦ ❢❛t♦ ♦ r❛✐♦ ❡st❛r ❞✐♠✐♥✉✐♥❞♦✳ P♦rt❛♥t♦✱ q✉❛♥❞♦ ♦ r❛✐♦ ♠❡❞❡ 15cm ❡st❛ ❞✐♠✐♥✉✐❞♦ à r❛③ã♦ ❞❡ 1 cm/min✳ 18π ❊①❡♠♣❧♦ ✻✳✼✶✳ ◗✉❡r❡♠♦s ❢❛❜r✐❝❛r ✉♠❛ ❜♦✐❛ ❢♦r♠❛❞❛ ♣♦r ❞♦✐s ❝♦♥❡s r❡t♦s ❞❡ ❢❡rr♦ ✉♥✐❞♦s ♣❡❧❛s s✉❛s ❜❛s❡s✳ P❛r❛ s✉❛ ❝♦♥str✉çã♦ t❡♠♦s ♣❧❛❝❛s ❝✐r❝✉❧❛r❡s ❞❡ 3m ❞❡ r❛✐♦✳ ❉❡t❡r♠✐♥❡ ❛s ❞✐♠❡♥✲ sõ❡s ❞❛ ❜♦✐❛ ♣❛r❛ q✉❡ s❡✉ ✈♦❧✉♠❡ s❡❥❛ ♠á①✐♠♦✳ ❙♦❧✉çã♦✳ 1 V = 2( )πx2 y ✳ ❉♦ tr✐â♥✲ 3 t❡♠♦s x2 = 9 − y 2 ✱ ❛ss✐♠ ❖ ✈♦❧✉♠❡ ❞❛ ❜♦✐❛ ést❛ ❞❛❞♦ ♣♦r ❣✉❧♦ ♠♦str❛❞♦ ♥❛ ❋✐❣✉r❛ ✭✻✳✷✸✮ 2 1 V = 2( )πx2 y = ( )π(9 − y 2 )y 3 3 q✉❛♥❞♦ =0 s❡❣✉❡ q✉❡ y= √ 3 dV 2π = (9 − 3y 2 ) dy 3 ⇒ ❞❡ ♦♥❞❡ r= √ 6 ❆ ❞❡r✐✈❛❞❛ s❡❣✉♥❞❛ 2π d2 V = (−6y) 2 dy 3 ⇒ √ d2 V √ 2π ( (−6 3) = 3) < 0 dy 2 3 P♦rt❛♥t♦✱ ♦ ✈♦❧✉♠❡ ❞❛ ❜♦✐❛ s❡rá ♠á①✐♠♦ q✉❛♥❞♦ ♦ r❛✐♦ ❞❛ ❜❛s❡ ❞♦s ❝♦♥❡s s❡❥❛ r= √ 6 ❡ s✉❛ ❛❧t✉r❛ y= √ ✸✻✵ ❋✐❣✉r❛ ✻✳✷✸✿ 3 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❊①❡r❝í❝✐♦s ✻✲✹ ✶✳ ❯♠❛ ♣❡ss♦❛ ❛t✐r❛ ✈❡rt✐❝❛❧♠❡♥t❡ ♣❛r❛ ♦ ❝é✉ ✉♠❛ ❜♦❧❛ ❞♦ t♦♣♦ ❞❡ ✉♠ ❡❞✐❢í❝✐♦✳ ❉❡♣♦✐s ❞❡ 2 s❡❣✉♥❞♦s✱ ❛ ❜♦❧❛ ♣❛ss❛ ♣♦r ❡❧❡✱ ❝❤❡❣❛♥❞♦ ❛♦ s♦❧♦ 2 s❡❣✉♥❞♦s ❞❡♣♦✐s✳ ✶✳ ◗✉❛❧ ❡r❛ ❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥✐❝✐❛❧ ❞❛ ❜♦❧❛ ❄ ✷✳ ❈♦♠ q✉❡ ✈❡❧♦❝✐❞❛❞❡ ❛ ❜♦❧❛ ♣❛ss♦✉ ♣❡❧❛ ♣❡ss♦❛✱ q✉❛♥❞♦ ❝❛í❛ ❡♠ ❞✐r❡çã♦ ❛♦ s♦❧♦ ❄ ✸✳ ❈♦♠ q✉❡ ✈❡❧♦❝✐❞❛❞❡ ❛ ❜♦❧❛ ❝❤❡❣❛rá ❛♦ s♦❧♦ ❄ ✹✳ ◗✉❛❧ é ❛ ❛❧t✉r❛ ❞♦ ❡❞✐❢í❝✐♦ ❄ ✷✳ ❆s ❡q✉❛çõ❡s ❞♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠ ♣r♦❥❡t✐❧ ❡stã♦ ❞❛❞❛s ♣❡❧❛s ❡q✉❛çõ❡s x = t(v0 cos α) ❡ y = t(v0 senα) − 16t2 ✱ ♦♥❞❡ v0 é ❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥✐❝✐❛❧✱ α é ♦ â♥❣✉❧♦ ❞❡ ❡❧❡✈❛çã♦ ❞♦ ❝❛♥❤ã♦✱ t é ♦ t❡♠♣♦ ❡♠ s❡❣✉♥❞♦s✱ x ❡ y sã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣r♦❥❡t✐❧✳ ❉❡t❡r♠✐♥❡ ❛ ❛❧t✉r❛ ♠á①✐♠❛ q✉❡ ❛❧❝❛♥ç❛ ♦ ♣r♦❥❡t✐❧ ❡ ✈❡r✐✜q✉❡ q✉❡ ♦ ♠❛✐♦r ❛❧❝❛♥❝❡ s❡ ♦❜tê♠ π q✉❛♥❞♦ α = ✳ 4 ✸✳ ❆ ❧❡✐ ❞❡ ❇♦②❧❡ ♣❛r❛ ❛ ❡①♣❛♥sã♦ ❞❡ ✉♠ ❣ás é P V = C ✱ ♦♥❞❡ P é ♦ ♥ú♠❡r♦ ❞❡ q✉✐❧♦s ♣♦r ✉♥✐❞❛❞❡ q✉❛❞r❛❞❛ ❞❡ ♣r❡ssã♦✱ V é ♦ ♥ú♠❡r♦ ❞❡ ✉♥✐❞❛❞❡s ❝ú❜✐❝❛s ♥♦ ✈♦❧✉♠❡ ❞♦ ❣ás ❡ C é ✉♠❛ ❝♦♥st❛♥t❡ ✳ ❆❝❤❡ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ✐♥st❛♥tâ♥❡❛ ❞❡ V ❡♠ r❡❧❛çã♦ ❛ P q✉❛♥❞♦ P = 4 ❡ V = 8✳ ✹✳ ❊st❛ s❡♥❞♦ ❞r❡♥❛❞♦ á❣✉❛ ❞❡ ✉♠❛ ♣✐s❝✐♥❛ ❞❡ ✈♦❧✉♠❡ V ✱ ♦ ✈♦❧✉♠❡ ❞❡ á❣✉❛ ❛♣ós t ♠✐♥✉t♦s ❞♦ ✐♥✐❝✐♦ ❞❛ ❞r❡♥❛❣❡♠ é V = 250(1.600 − 80t + t2 )✳ ❆❝❤❛r✿ ✭❛✮ ❆ t❛①❛ ♠é❞✐❛ s❡❣✉♥❞♦ ❛ q✉❛❧ ❛ á❣✉❛ ❞❡✐①❛ ❛ ♣✐s❝✐♥❛ ❞✉r❛♥t❡ ♦s 5 ♣r✐♠❡✐r♦s ♠✐♥✉t♦s✳ ✭❜✮ ❆ ✈❡❧♦❝✐❞❛❞❡ ❛ q✉❛❧ ❛ á❣✉❛ ❡stá ✢✉✐♥❞♦ ❞❛ ♣✐s❝✐♥❛ 5 ♠✐♥✉t♦s ❛♣ós ♦ ❝♦♠❡ç♦ ❞❛ ❞r❡♥❛❣❡♠✳ ✺✳ ❙✉♣♦♥❤❛ q✉❡ ✉♠ ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r r❡t♦ t❡♥❤❛ ✉♠❛ ❛❧t✉r❛ ❝♦♥st❛♥t❡ ❞❡ 10 cm✳ ❙❡ V cm3 ❢♦✐ ♦ ✈♦❧✉♠❡ ❞♦ ❝✐❧✐♥❞r♦ ❡ r ♦ r❛✐♦ ❞❡ s✉❛ ❜❛s❡✱ ❛❝❤❡ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ♠é❞✐❛ ❞❡ V ❡♠ r❡❧❛çã♦ ❛ r q✉❛♥❞♦ r ✈❛r✐❛ ❞❡✿ ✭❛✮ 5, 00 ❛ 5, 40❀ ✭❜✮ 5, 00 ❛ 5, 10❀ ✭❝✮ 5, 00 ❛ 5, 01❀ ✭❞✮ ❛❝❤❡ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ✐♥st❛♥tâ♥❡❛ ❞❡ V ❡♠ r❡❧❛çã♦ ❛ r q✉❛♥❞♦ r é 5, 00✳ ❙✉❣❡stã♦✿ ❆ ❢ór♠✉❧❛ ♣❛r❛ ❡♥❝♦♥tr❛r ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r r❡t♦ é V = πr2 h✱ ♦♥❞❡ h cm é ❛❧t✉r❛ ❞♦ ❝✐❧✐♥❞r♦✳ ✻✳ ❯♠ tr♦♥❝♦ ❞❡ ár✈♦r❡ ♠❡❞❡ 20 m✱ t❡♠ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❝♦♥❡ tr✉♥❝❛❞♦✳ ❖s ❞✐â♠❡tr♦s ❞❡ s✉❛s ❜❛s❡s ♠❡❞❡♠ 2 m ❡ 1 m✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡✈❡✲s❡ ❝♦rt❛r ✉♠❛ ✈✐❣❛ ❞❡ s❡çã♦ tr❛♥s✈❡rs❛❧ q✉❛❞r❛❞❛ ❝✉❥♦ ❡✐①♦ ❝♦✐♥❝✐❞❛ ❝♦♠ ❛ ❞♦ tr♦♥❝♦ ❡ ❝✉❥♦ ✈♦❧✉♠❡ s❡❥❛ ♦ ♠❛✐♦r ♣♦ssí✈❡❧✳ ◗✉❡ ❞✐♠❡♥sõ❡s ❞❡✈❡ t❡r ❛ ✈✐❣❛❄ ✸✻✶ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ R1 ❡ R2 ❡stã♦ ✉♥✐❞❛s ❡♠ ♣❛r❛❧❡❧♦✱ ❛ r❡s✐stê♥❝✐❛ 1 1 1 t♦t❛❧ R ❡stá ❞❛❞❛ ♣♦r = + ✳ ❙❡ R1 ❡ R2 ❛✉♠❡♥t❛♠ ❛ r❛③ã♦ ❞❡ 0.01ohms/sg R R1 R2 ❡ 0.02 ohms/sg ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ◗✉❛❧ é ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❡ R ♥♦ ✐♥st❛♥t❡ ❡♠ q✉❡ R1 = 30 ohms ❡ R2 = 90 ohms ❄ ✼✳ ◗✉❛♥❞♦ ❞✉❛s r❡s✐stê♥❝✐❛s ❡❧étr✐❝❛s ✽✳ ❯♠ ❢♦❣✉❡t❡ é ❧❛♥ç❛❞♦ ✈❡rt✐❝❛❧♠❡♥t❡ ❡ s✉❛ tr❛❥❡tór✐❛ t❡♠ ❡q✉❛çã♦ ❤♦rár✐❛ 2 160t − 5t ✱ ♦ 2 s ❞❡♣♦✐s ❞♦ s = s❡♥t✐❞♦ ♣♦s✐t✐✈♦ ♣❛r❛ ♦ ❝é✉✳ ❉❡t❡r♠✐♥❡✿ ❛✮ ❆ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ❢♦❣✉❡t❡ ❧❛♥ç❛♠❡♥t♦✳ ❜✮ ❖ t❡♠♣♦ q✉❡ ❧❡✈❛ ♦ ❢♦❣✉❡t❡ ♣❛r❛ ❛❧❝❛♥ç❛r s✉❛ ❛❧t✉r❛ ♠á①✐♠❛✳ ✾✳ ❯♠❛ ♣❡❞r❛ é ❧❛♥ç❛❞❛ ❛ ✉♠❛ ❧❛❣♦❛ ❡ ♣r♦❞✉③ ✉♠❛ sér✐❡ ❞❡ ♦♥❞✉❧❛çõ❡s ❝♦♥❝ê♥tr✐❝❛s✳ ❙❡ ♦ r❛✐♦ r 1.8 m/s✱ ❞❡t❡r♠✐♥❡ ❛ ❛✮ ◗✉❛♥❞♦ r = 3 m✳ ❜✮ ◗✉❛♥❞♦ ❞❛ ♦♥❞❛ ❡①t❡r✐♦r ❝r❡s❝❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ❛ r❛③ã♦ ❞❡ t❛①❛ ❝♦♠ ❛ q✉❡ ❛ á❣✉❛ ♣❡rt✉r❜❛❞❛ ❡stá ❝r❡s❝❡♥❞♦ r = 6 m✳ ✶✵✳ ❯♠❛ ♣❡❞r❛ s❡ ❞❡✐①❛ ❝❛✐r ✭❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ✐♥✐❝✐❛❧ ③❡r♦✮ ❞♦ t♦♣♦ ❞❡ ✉♠ ❡❞✐❢í❝✐♦ ❞❡ 144 ♠❡tr♦s ❞❡ ❛❧t✉r❛✳ ❛✮ ❊♠ q✉❡ ♠♦♠❡♥t♦ ❛ ♣❡❞r❛ ❝❤❡❣❛rá ❛♦ s♦❧♦ ❄ ❜✮ ◗✉❛❧ s❡rá ❛ ✈❡❧♦❝✐❞❛❞❡ ❛♦ ❝❤❡❣❛r ❛♦ s♦❧♦ ❄✳ ❙✉❣❡stã♦✿ P❛r❛ ✉♠ ♦❜❥❡t♦ q✉❡ s❡ ❛t✐r❛ ♦✉ ❝❛✐ ✈❡rt✐❝❛❧♠❡♥t❡✱ ❛ ❛❧t✉r❛ q✉❡ r❡❝♦rr❡ ❞❡♣♦✐s ❞❡ ♦♥❞❡ V0 é ❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥✐❝✐❛❧ ❞♦ ♦❜❥❡t♦ ❡ A0 t s❡❣✉♥❞♦s é✿ A(t) = −16t2 + V0 + A0 ✱ é ❛ ❛❧t✉r❛ ✐♥✐❝✐❛❧✳ ✶✶✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ✉♠ ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r r❡t♦ ❡ ❢❡❝❤❛❞♦ t❡♥❤❛ ✉♠❛ ár❡❛ ❞❡ 100 cm2 ✳ ◗✉❡ ✈❛❧♦r❡s ❞❡✈❡♠ t❡r ♦ r❛✐♦ ❡ s✉❛ ❛❧t✉r❛ ♣❛r❛ q✉❡ s❡✉ ✈♦❧✉♠❡ s❡❥❛ ♠á①✐♠♦ ❄ ✶✷✳ ▼♦str❡ q✉❡ ♦ ❝✐❧✐♥❞r♦ r❡t♦ ❞❡ ♠❛✐♦r ✈♦❧✉♠❡ q✉❡ ♣♦❞❡ s❡r ✐♥s❝r✐t♦ ❡♠ ✉♠ ❝♦♥❡✱ é ❞♦ ✈♦❧✉♠❡ ❞♦ ❝♦♥❡✳ ✶✸✳ ❯♠ ❝♦♥❡ ❝✐r❝✉❧❛r r❡t♦ t❡♠ ✉♠ ✈♦❧✉♠❡ ❞❡ 120 cm3 4 9 ◗✉❛✐s sã♦ ❛s ❞✐♠❡♥sõ❡s q✉❡ ❞❡✈❡ t❡r ❡st❡ ❝♦♥❡ ♣❛r❛ q✉❡ s✉❛ ár❡❛ ❧❛t❡r❛❧ s❡❥❛ ♠í♥✐♠❛❄ ✶✹✳ ◆✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s ❛ ❡ss❡ ❧❛❞♦ ♠❡❞❡ h✳ ABC ♦ ❧❛❞♦ ❞❡s✐❣✉❛❧ ❉❡t❡r♠✐♥❡ ✉♠ ♣♦♥t♦ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s ❞❡ P P AC ♠❡❞❡ 2a ❡ ❛ ❛❧t✉r❛ ❝♦rr❡s♣♦♥❞❡♥t❡ s♦❜r❡ ❛ ❛❧t✉r❛ ♠❡♥❝✐♦♥❛❞❛ ♣❛r❛ q✉❡ ❛ ❛té ♦s três ✈ért✐❝❡s s❡❛ ♠í♥✐♠❛✳ 80cm ♣♦r 50cm✳ ❈♦rt❛♥❞♦ ❝♦♥✈❡♥✐❡♥t❡♠❡♥t❡ ❧❛❞♦ x q✉❡r❡♠♦s ❝♦♥str✉✐r ✉♠❛ ❝❛✐①❛✳ ❈❛❧❝✉❧❡ x ✶✺✳ ❚❡♠♦s ✉♠❛ ❢♦❧❤❛ ❞❡ ♣❛♣❡❧ã♦ ♠❡❞✐♥❞♦ ❡♠ ❝❛❞❛ ✈ért✐❝❡ ♥✉♠ q✉❛❞r❛❞♦ ❞❡ ♣❛r❛ q✉❡ ❛ r❡❢❡r✐❞❛ ❝❛✐①❛ t❡♥❤❛ ✉♠ ✈♦❧✉♠❡ ♠á①✐♠♦✳ ✶✻✳ ❚❡♠♦s ✉♠ ❛r❛♠❡ ❞❡ 1m ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❡ ❞❡s❡❥❛♠♦s ❞✐✈✐❞✐✲❧♦ ❡♠ ❞✉❛s ♣❛rt❡s ♣❛r❛ ❢♦r♠❛r ❝♦♠ ✉♠❛ ❞❡❧❛s ✉♠ ❝ír❝✉❧♦ ❡ ❝♦♠ ❛ ♦✉tr❛ ✉♠ q✉❛❞r❛❞♦✳ ❉❡t❡r♠✐♥❡ ♦ ❝♦♠♣r✐♠❡♥t♦ q✉❡ t❡♠ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ♣❡ç❛s ❞❡ ♠♦❞♦ q✉❡ ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦ ❝ír❝✉❧♦ ❡ q✉❛❞r❛❞♦ s❡❥❛ ♠í♥✐♠❛✳ ✸✻✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ▼✐s❝❡❧â♥❡❛ ✻✲✶   x2 · sen 1 , s❡✱ x 6= 0 ✶✳ ❊st✉❞❡♠♦s ❛ s❡❣✉✐♥t❡ ❢✉♥çã♦✿ f (x) = ✳ x  0, s❡✱ x = 0 P❡❧♦ ❚✳❱✳▼ ♥♦ ✐♥t❡r✈❛❧♦ [0, x] t❡♠♦s✿ f (x)−f (0) = x·f ′ (c) q✉❛♥❞♦ ✭0 < c < x)✳ 1 1 x c ◗✉❛♥❞♦ x t❡♥❞❡ ♣❛r❛ ③❡r♦✱ c 1 q✉❡✿ lim cos = 0✳ ❊①♣❧✐❝❛r ❡st❡ c→0 c 1 c ■st♦ é✿ x2 sen = x(2c · sen − cos )✱ ❞❡ ♦♥❞❡ cos 1 1 1 = 2csen − x · sen ✳ c c x t❛♠❜é♠ t❡♥❞❡ ♣❛r❛ ③❡r♦❀ ❞❡st❡ ♠♦❞♦ ❝♦♥❝❧✉í♠♦s r❡s✉❧t❛❞♦ ♣❛r❛❞♦①❛❧✳ ✷✳ P❛r❛ ✉♠❛ ❝♦♥st❛♥t❡ a > 0✱ ❞❡t❡r♠✐♥❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ✈❛❧♦r ♠á①✐♠♦ ❡ ♠í♥✐♠♦ 1 r❡❧❛t✐✈♦ ❞❛ ❢✉♥çã♦ g(x) = (a − − x)(4 − 3x2 )✳ a ✸✳ ❙❡❥❛♠ f ❡ g ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡♠ (a, b) t❛✐s q✉❡ f ′ (x) > g ′ (x) ∀ x ∈ (a, b)✳ ❙❡ ❡①✐st❡ c ∈ (a, b) t❛❧ q✉❡ f (c) = g(c)✱ ♠♦str❡ q✉❡ f (x) < g(x) ∀ x ∈ (a, c) ❡ g(x) < f (x) ∀ x ∈ (c, b)✳ ✹✳ ❙❡❥❛ f ❞❡r✐✈á✈❡❧ ❡♠ R ❡ g(x) = ❞❡ g ✱ ♠♦str❡ q✉❡✿ ✶✳ ✷✳ f (x) , x x 6= 0✳ ❙❡ c é ✉♠ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❧♦❝❛❧ c · f ′ (c) − f (c) = 0✳ ❆ r❡t❛ t❛♥❣❡♥t❡ ❛♦ ❣rá✜❝♦ ❞❡ f ♥♦ ♣♦♥t♦ (c, f (c)) ♣❛ss❛ ♣❡❧❛ ♦r✐❣❡♠✳ ✺✳ ❉❡t❡r♠✐♥❡ ♦s ✐♥t❡r✈❛❧♦s ❞❡ ❝r❡s❝✐♠❡♥t♦ ♦✉ ❞❡❝r❡s❝✐♠❡♥t♦ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿ 1. y = 2 − 3x + x3 2. y = x.e−x 3. y = p (x2 − 1)3 √ 4. y = (2 − x)(x + 1)2 5. y = x2 (1 − x x) 6. y = Ln(x2 + 1) p x 8. y = (2x − 1) 3 (x − 3)2 9. y = x − 2sen2 x 7. y = Lnx 10. y = e1,5senx ✻✳ ❖ ✈❛❧♦r ❞❡ ✉♠ ❞✐❛♠❛♥t❡ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ q✉❛❞r❛❞♦ ❞♦ s❡✉ ♣❡s♦✳ ❉✐✈✐❞❡ ✉♠ ❞✐❛✲ ♠❛♥t❡ ❞❡ 2 ❣r❛♠❛s ❡♠ ❞✉❛s ♣❛rt❡s ❞❡ t❛❧ ♠♦❞♦ q✉❡ ❛ s♦♠❛ ❞♦s ✈❛❧♦r❡s ❞♦s ❞✐❛♠❛♥t❡s ♦❜t✐❞♦s s❡❥❛ ♠í♥✐♠❛✳ ✼✳ ❉❡t❡r♠✐♥❡ ♦ ❝✐❧✐♥❞r♦ ❞❡ s✉♣❡r❢í❝✐❡ t♦t❛❧ S ✱ t❛❧ q✉❡ s❡✉ ✈♦❧✉♠❡ s❡❥❛ ♠á①✐♠♦✳ ✸✻✸ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✽✳ P❛r❛ ♦s s❡❣✉✐♥t❡s ❡①❡r❝í❝✐♦s✱ tr❛ç❛r ♦ ❣rá✜❝♦ ❞❛ ❝✉r✈❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ✐♥❞✐❝❛♥❞♦ s✉❛s ❛ssí♥t♦t❛s✳ ✶✳ ✷✳ ✸✳ ✹✳  √  x2 + x − x, s❡✱ 2 f (x) =  x − 81 , s❡✱ x2 − 9x  x2   √ , s❡✱ 4 − x2 f (x) =   3x + 6x, s❡✱ 2x + 1  r 8   5 x + 2x + 1     x3 + 8 x +1 f (x) = −   x+3    √ 3 6x2 − x3  r   3 x + 3  s❡✱   x−3   3|x+3| f (x) = s❡✱  x + 1      2   s❡✱  5+ x | x |≥ 9 | x |< 9 | x |> 2 | x |≤ 2 s❡✱ x ≤ −1 s❡✱ −1<x≤1 s❡✱ x > 1 x ≤ −3 −3<x≤2 x>2 ✾✳ ❯♠❛ ❡s❝❛❞❛ ❝♦♠ 6m ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❡stá ❛♣♦✐❛❞❛ ❡♠ ✉♠❛ ♣❛r❡❞❡ ✈❡rt✐❝❛❧✳ ❙❡ ❛ ❜❛s❡ ❞❛ ❡s❝❛❞❛ ❝♦♠❡ç❛ ❛ s❡ ❞❡s❧✐③❛r ❤♦r✐③♦♥t❛❧♠❡♥t❡✱ à r❛③ã♦ ❞❡ 0, 6m/s✱ ❝♦♠ q✉❡ ✈❡❧♦❝✐❞❛❞❡ ♦ t♦♣♦ ❞❛ ❡s❝❛❞❛ ❞❡s❝❡ ❛ ♣❛r❡❞❡✱ q✉❛♥❞♦ ❡stá ❛ 4m ❞♦ s♦❧♦❄ ✶✵✳ ❯♠ ❤♦♠❡♠ ❞❡ 1, 80m✱ ❝❛♠✐♥❤❛♥❞♦ à ✈❡❧♦❝✐❞❛❞❡ ❞❡ 1, 5m/s ✱ ❛❢❛st❛✲s❡ ❞❡ ✉♠❛ ❧â♠♣❛❞❛ s✐t✉❛❞❛ ❛ 5m ❛❝✐♠❛ ❞♦ ❝❤ã♦✳ ❈❛❧❝✉❧❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❝♦♠ q✉❡ s❡ ♠♦✈❡ ❛ s♦♠❜r❛ ❞♦ ❤♦♠❡♠ ❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❝♦♠ q✉❡ s❡ ♠♦✈❡ ❛ ❡①tr❡♠✐❞❛❞❡ ❞❡❧❛✳ ✶✶✳ ❖ ❣ás ❞❡ ✉♠ ❜❛❧ã♦ ❡s❢ér✐❝♦ ❡s❝❛♣❛ à r❛③ã♦ ❞❡ 2dm3 /min✳ ❊♥❝♦♥tr❡ ❛ r❛③ã♦ ❝♦♠ q✉❡ ❞✐♠✐♥✉✐ ❛ s✉♣❡r❢í❝✐❡ ❞♦ ❜❛❧ã♦ q✉❛♥❞♦ ♦ r❛✐♦ é ❞❡ 12dm✳ ✶✷✳ ❯♠ ❜❛❧ã♦ ❡s❢ér✐❝♦ ❡stá s❡♥❞♦ ✐♥✢❛❞♦ ❡ s❡✉ r❛✐♦ é R ♥♦ ✜♠ ❞❡ t s❡❣✉♥❞♦s✳ ❊♥✲ ❝♦♥tr❡ ♦ r❛✐♦ ♥♦ ✐♥st❛♥t❡ ❡♠ q✉❡ ❛s t❛①❛s ❞❡ ✈❛r✐❛çã♦ ❞❛ s✉♣❡r❢í❝✐❡ ❡ ❞♦ r❛✐♦ sã♦ ♥✉♠ér✐❝❛♠❡♥t❡ ✐❣✉❛✐s✳ ✶✸✳ ▼♦str❡ q✉❡ ❛ s✉❜t❛♥❣❡♥t❡ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ q✉❛❧q✉❡r ♣♦♥t♦ ❞❛ ♣❛rá❜♦❧❛ y = ax2 é ✐❣✉❛❧ à ♠❡t❛❞❡ ❞❛ ❛❜s❝✐ss❛ ❞♦ ♣♦♥t♦ ❞❡ t❛♥❣❡♥❝✐❛✳ ✶✹✳ ❈❛❧❝✉❧❡ ❛s ❞✐♠❡♥sõ❡s ❞♦ tr❛♣é③✐♦ r❡❣✉❧❛r ❞❡ ♣❡rí♠❡tr♦ ♠á①✐♠♦ q✉❡ ♣♦❞❡✲s❡ ✐♥s❝r❡✈❡r ❡♠ ✉♠❛ s❡♠✐❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ r s❡ ✉♠❛ ❜❛s❡ ❞♦ tr❛♣é③✐♦ ♦❝✉♣❛ t♦❞♦ ♦ ❞✐â♠❡tr♦ ❞❡ ❧❛ s❡♠✐❝✐r❝✉♥❢❡rê♥❝✐❛✳ ✸✻✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R R$5, 00 ❡ ❝❛❧❝✉❧❛ q✉❡✱ (20 − x) ✉♥✐❞❛❞❡s ♣♦r ❞✐❛✳ ✶✺✳ ❯♠ ❝♦♠❡r❝✐❛♥t❡ ♣r♦❞✉③ ❝❡rt♦ ♣r♦❞✉t♦ ❛♦ ❝✉st♦ ✉♥✐tár✐♦ ❞❡ s❡ ✈❡♥❞ê✲❧♦s ❛ x r❡❛✐s ❛ ✉♥✐❞❛❞❡✱ ♦s ❝❧✐❡♥t❡s ❝♦♠♣r❛rã♦ ❆ q✉❡ ♣r❡ç♦ ♦ ❢❛❜r✐❝❛♥t❡ ❞❡✈❡ ✈❡♥❞❡r s❡✉ ♣r♦❞✉t♦ ♣❛r❛ q✉❡ s❡❥❛ ♠á①✐♠♦ ♦ ❧✉❝r♦ ♦❜t✐❞♦ ❄ α é ❝❤❛♠❛❞♦ ✏r❛✐③ ❞✉♣❧❛✑ ❞❛ ❢✉♥çã♦ ♣♦❧✐♥ô♠✐❝❛ f ✱ s❡ f (x) = (x − α)2 g(x) ❛❧❣✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥ô♠✐❝❛ g(x) ✶✻✳ ❖ ♥ú♠❡r♦ ♣❛r❛ α ✶✳ ▼♦str❡ q✉❡ é r❛✐③ ❞✉♣❧❛ ❞❡ ✷✳ ❊♠ q✉❛✐s ❝♦♥❞✐çõ❡s ❛ ❢✉♥çã♦ f s❡✱ ❡ s♦♠❡♥t❡ s❡ t❛♠❜é♠ é r❛✐③ ❞❡ f (x) = ax2 + bx + c✱ ❝♦♠ ✶✼✳ ❖ ♥ú♠❡r♦ ❞❡ ❜❛❝tér✐❛s ❞❡ ❝❡rt♦ ❝✉❧t✐✈♦ ♥✉♠ ✐♥st❛♥t❡ 1000(25 + tet/20 ) ♣❛r❛ t❡♠ r❛✐③ ❞✉♣❧❛❄ é ❞❛❞♦ ♣❡❧❛ ❢ór♠✉❧❛ N = 0 ≤ t ≤ 100✳ ✶✳ ❊♠ q✉❡ ✐♥st❛♥t❡ ❞❡ss❡ ✐♥t❡r✈❛❧♦✱ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ❜❛❝tér✐❛s❄ ✷✳ t a 6= 0 f ′✳ 0 ≤ t ≤ 100✱ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ ♠á①✐♠♦ ❡ ✉♠ ❊♠ q✉❡ ✐♥st❛♥t❡ é ♠❛✐s ❧❡♥t♦ ♦ ❝r❡s❝✐♠❡♥t♦ ♦✉ ❞❡❝r❡s❝✐♠❡♥t♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❜❛❝tér✐❛s❄ ✶✽✳ ❆ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ✉♠ ♠ó✈✐❧ q✉❡ ♣❛rt❡ ❞❛ ♦r✐✲ ❣❡♠ ❡stá ❞❛❞❛ ❡♠ m/s ❡ ♣❡❧♦ ❣rá✜❝♦✿ ✶✳ ❈❛❧❝✉❧❛r ❛ ❢✉♥çã♦ ✏❡s♣❛ç♦ ♣❡r❝♦rr✐❞♦✑✳ ✷✳ ●r❛✜❝❛r ❛ ❢✉♥çã♦ ❡s♣❛ç♦ ♣❡r❝♦rr✐❞♦✲ t❡♠♣♦✳ ✸✳ Pr♦✈❡ q✉❡ ❛ ár❡❛ s♦❜ ❛ ❝✉r✈❛ q✉❡ ❞❛ ❛ ✈❡❧♦❝✐❞❛❞❡ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ❡s♣❛ç♦ t♦t❛❧ v ✻ 2 ✳ ✳ ✳ ✳✳ ✳❍ ✳ ❍ ✓✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳❍ ✳❍ ✳✳ 1 ✳✓ ✳ ✳ ❍❍ ✳ ✓ ✳ 0 1 2 3 4 5 t ✲ 6 ♣❡r❝♦rr✐❞♦✳ ✶✾✳ ❉❡t❡r♠✐♥❛r ♠á①✐♠♦s ❡ ♠í♥✐♠♦s ❞❛ ❢✉♥çã♦ ✷✵✳ ❙❡❥❛ ❙❡ d f (x) = 2senx + cos 2x✳ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s x ❡ y r❡s♣❡❝t✐✈❛♠❡♥t❡✳ x ❛✉♠❡♥t❛ ❝♦♠ ✉♠❛ r❛♣✐❞❡③ ❞❡ 0, 5m/s ❡ y ❞✐♠✐♥✉✐ ❝♦♠ ✉♠❛ r❛♣✐❞❡③ ❞❡ 0, 25m/s✳ ✶✳ ◗✉❛❧ ❛ r❛③ã♦ ❡♠ q✉❡ ❡st❛ ♠✉❞❛♥❞♦ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❞✐❛❣♦♥❛❧ q✉❛♥❞♦ ❡ x = 3m y = 4m✳ ✷✳ ❆ ❞✐❛❣♦♥❛❧ ❡stá ❛✉♠❡♥t❛♥❞♦ ♦✉ ❞✐♠✐♥✉í❞♦ ♥❡ss❡ ✐♥st❛♥t❡✳ 500 ❝❡♥tí♠❡tr♦s ❝ú❜✐❝♦s✱ t❡♠ ✉♠ r❛✐♦ ❞❡ 2cm q✉❡ ❞❡✈❡♠♦s ❛❝❡✐t❛r ❛♦ ♠❡❞✐r s✉❛ ❛❧t✉r❛ h ❞❛ ✷✶✳ ❯♠ r❡❝✐♣✐❡♥t❡ ❝✐❧í♥❞r✐❝♦ ❞❡ ❝❛♣❛❝✐❞❛❞❡ ❡ ❡stá ❝❤❡✐♦ ❞❡ á❣✉❛✳ ◗✉❛❧ ♦ ❡rr♦ á❣✉❛ ❞♦ r❡❝✐♣✐❡♥t❡ ♣❛r❛ ❛ss❡❣✉r❛r q✉❡ t❡r❡♠♦s ♠❡✐♦ ❧✐tr♦ ❞❡ á❣✉❛ ❝♦♠ ✉♠ ❡rr♦ ❞❡ ♠❡♥♦s 1%❄ ✸✻✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ✷✷✳ ◆✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s ABC ♦ ❧❛❞♦ ❞❡s✐❣✉❛❧ AC ♠❡❞❡ 2a ❡ ❛ ❛❧t✉r❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ❡ss❡ ❧❛❞♦ ♠❡❞❡ h✳ ❉❡t❡r♠✐♥❡ ✉♠ ♣♦♥t♦ P s♦❜r❡ ❛ ❛❧t✉r❛ ♠❡♥❝✐♦♥❛❞❛ ♣❛r❛ q✉❡ ❛ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s ❞❡ P ❛té ♦s três ✈ért✐❝❡s s❡❛ ♠í♥✐♠❛✳ ✷✸✳ ❯♠ r❛✐♦ ❞❡ ❧✉③ ✭❢ót♦♥✮ ♣❛rt❡ ❞❡ ✉♠ ♣♦♥t♦ A ♣❛r❛ ✉♠ ♣♦♥t♦ B s♦❜r❡ ✉♠ ❡s♣❡❧❤♦ ♣❧❛♥♦✱ s❡♥❞♦ r❡✢❡t✐❞♦ q✉❛♥❞♦ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ P ✳ ❊st❛❜❡❧❡❝❡r ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ♦ ❝❛♠✐♥❤♦ AP B s❡❥❛ ♦ ♠❛✐s ❝✉rt♦ ♣♦ssí✈❡❧✳ ✷✹✳ ◗✉❛❧ ❞♦s tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s ❞❡ ♣❡rí♠❡tr♦ ❞❛❞♦ 2p✱ t❡♠ ♠❛✐♦r ár❡❛ ❄ ✷✺✳ ❖ ❝✉st♦ ✈❛r✐á✈❡❧ ❞❛ ❢❛❜r✐❝❛çã♦ ❞❡ ✉♠ ❝♦♠♣♦♥❡♥t❡ ❡❧étr✐❝♦ é R$8, 05 ♣♦r ✉♥✐❞❛❞❡✱ ❡ ♦ ❝✉st♦ ✜①♦ R$500, 00✳ ❊s❝r❡✈❛ ♦ ❝✉st♦ C ❝♦♠♦ ❢✉♥çã♦ ❞❡ x✱ ♦ ♥ú♠❡r♦ ❞❡ ✉♥✐❞❛❞❡s ♣r♦❞✉③✐❞❛s✳ ▼♦str❡ q✉❡ ❛ ❞❡r✐✈❛❞❛ ❞❡ss❛ ❢✉♥çã♦ ❝✉st♦ é ❝♦♥st❛♥t❡ ❡ ✐❣✉❛❧ ❛♦ ❝✉st♦ ✈❛r✐á✈❡❧✳ ✷✻✳ ❉❡ t♦❞♦s ♦s tr✐â♥❣✉❧♦s ✐sós❝❡❧❡s ❞❡ 12m ❞❡ ♣❡rí♠❡tr♦✱ q✉❛✐s ❞❡❧❡s t❡♠ ár❡❛ ♠á①✐♠❛❄ ✷✼✳ Pr❡t❡♥❞❡✲s❡ ❢❛❜r✐❝❛r ✉♠❛ ❧❛t❛ ❝✐❧í♥❞r✐❝❛ ❞❡ ♠❡t❛❧ ❝♦♠ t❛♠♣❛ q✉❡ ❝♦♥t❡♥❤❛ ✉♠ ❧✐tr♦ ❞❡ ❝❛♣❛❝✐❞❛❞❡ ♣❛r❛ ❝♦♥s❡r✈❛r ♠❛♥t❡✐❣❛✳ ◗✉❛✐s s❡rã♦ ❛s ❞✐♠❡♥sõ❡s ♣❛r❛ q✉❡ s❡ ✉t✐❧✐③❡ ❛ ♠❡♥♦r q✉❛♥t✐❞❛❞❡ ❞❡ ♠❡t❛❧❄ ✷✽✳ ❯♠ s❡t♦r ❝✐r❝✉❧❛r t❡♠ ♣❡rí♠❡tr♦ ❞❡ 10m✳ ❉❡t❡r♠✐♥❡ ♦ r❛✐♦ ❡ ❛♠♣❧✐t✉❞❡ ❞♦ s❡t♦r ❞❡ ♠❛✐♦r ár❡❛ ❝♦♠ ❡ss❡ ♣❡rí♠❡tr♦✳ ✷✾✳ ❯♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s ❞❡ ♣❡rí♠❡tr♦ 30cm✱ ❣✐r❛ ❡♥t♦r♥♦ ❞❡ s✉❛ ❛❧t✉r❛ ❡♥❣❡♥❞r❛♥❞♦ ✉♠ ❝♦♥❡✳ ◗✉❛❧ ♦ ✈❛❧♦r ❛ ❞❛r ❛ ❜❛s❡ ♣❛r❛ q✉❡ ♦ ✈♦❧✉♠❡ s❡❥❛ ♠á①✐♠♦❄ ✸✵✳ ❉❡❝♦♠♣♦r ♦ ♥ú♠❡r♦ 44 ❡♠ ❞♦✐s s♦♠❛♥❞♦ ❞❡ ♠♦❞♦ q✉❡ ❛ s❡①t❛ ♣❛rt❡ ❞♦ q✉❛❞r❛❞♦ ❞♦ ♣r✐♠❡✐r♦ ♠❛✐s ❛ q✉✐♥t❛ ♣❛rt❡ ❞♦ q✉❛❞r❛❞♦ ❞♦ s❡❣✉♥❞♦ s❡❥❛ ♠í♥✐♠❛❄ ✸✶✳ ❯♠❛ ❢♦❧❤❛ ❞❡ ♣❛♣❡❧ ❞❡✈❡ t❡r ❞❡ 18cm2 ❞❡ t❡①t♦ ✐♠♣r❡ss♦✱ ❛s ♠❛r❣❡♥s s✉♣❡r✐♦r ❡ ✐♥❢❡r✐♦r ❞❡ 2cm ❞❡ ❛❧t✉r❛ ❡ ❛s ♠❛r❣❡♥s ❧❛t❡r❛✐s ❞❡ 1cm ❧❛r❣✉r❛✳ ❖❜t❡r r❛③♦❛✈❡❧♠❡♥t❡ ❛s ❞✐♠❡♥sõ❡s q✉❡ ♠✐♥✐♠✐③❛♠ ❛ s✉♣❡r❢í❝✐❡ ❞♦ ♣❛♣❡❧✳ ✳ ✸✻✻ 09/02/2021 ❘❡❢❡rê♥❝✐❛s ❬✶❪ ❆❜❡❧❧❛♥❛s P✳ & P❡r❡③ ❇❡❛t♦ ▼✳✲ ❈✉rs♦ ❞❡ ▼❛t❡♠át✐❝❛s ❡♠ ❋♦r♠❛ ❞❡ Pr♦❜❧❡✲ ♠❛s✳✲ ❙♦❝✐❡❞❛❞ ❆♥ó♥✐♠❛ ❊s♣❛ñ♦❧❛ ❞❡ ❚r❛❞✉❝t♦r❡s ② ❆✉t♦r❡s✳ ✶✾✻✵✳ ❈á❧❝✉❧♦ ■ ✲ ❉✐❢❡r❡♥❝✐❛❧ ✲ ❈♦❧❡❝❝✐ó♥ ❍❛r♣❡r✳ ❊❞✐t♦r ❚♦rr❡❧❛r❛ ❊s✲ ❬✷❪ ➪❧✈❛r♦ P✐♥③ó♥✳✲ ♣❛ñ❛ 1973✳ ❬✸❪ ❇❡r♠❛♥ ●✳ ◆✳✲ ▼♦s❝♦ú✳ Pr♦❜❧❡♠❛s ② ❊❥❡r❝í❝✐♦s ❞❡ ❆♥á❧✐s✐s ▼❛t❡♠át✐❝♦✲ ❊❞✐t♦r✐❛❧ ▼■❘ 1977✳ Pr♦❜❧❡♠❛s ② ❊❥❡r❝í❝✐♦s ❞❡ ❆♥á❧✐s✐s ▼❛t❡♠át✐❝♦✳✲ ❬✹❪ ❉❡♠✐♥♦✈✐❝❤ ❇✳✲ ▼■❘ ▼♦s❝♦ú✳ ❬✺❪ ▲❛♥❣ ❙❡r❣❡✳✲ 1971✳ ❈á❧❝✉❧♦ ■✳✲ ❋♦♥❞♦ ❊❞✉❛t✐✈♦ ■♥t❡r❛♠❡r✐❝❛♥♦ ❙✳ ❆✳ 1973✳ ❬✻❪ ▲❡✐t❤♦❧❞ ▲♦✉✐s✳✲ ❍❆❘❇❘❆ ❬✼❪ ❖✬❈♦♥♥♦r ❊❞✐t♦r✐❛❧ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ➪ ❊❝♦♥♦♠✐❛ ❡ ❆❞♠✐♥✐str❛çã♦✳✲ ❊❞✐t♦r❛ 1988✳ ❏✳ ❏✳ & ❘♦❜❡rts♦♥ ❊✳ ❋✳✲ ❍✐stór✐❛ ❞♦ ❈á❧❝✉❧♦✳ ❤tt♣✿✴✴✇✇✇✲ ❤✐st♦r②✳♠❝s✳st✲❛♥❞r❡✇s✳❛❝✳✉❦✳ ❬✽❪ ❑✉❞r✐á✈ts❡✈ ▲✳ ❉✳ ❛t ❡❧❧✳✲ ▼■❘ ▼♦s❝♦ú✳ ❬✾❪ ❘✐✈❛✉❞ ❏✳✲ Pr♦❜❧❡♠❛s ❞❡ ❆♥á❧✐s✐s ▼❛t❡♠át✐❝♦✳✲❱♦❧ ■ ✲ ❊❞✐t♦r✐❛❧ 1984✳ ❊①❡r❝✐❝❡s ❞✬❛♥❛❧②s❡✲ ▲✐✈r❛r✐❡ ✈✉✐❜❡rt P❛r✐s✳ ❚♦♠♦ ■ 1971✳ ❬✶✵❪ ❙t❡✇❛rt✳ ❏❛♠❡s✱✲ ❈á❧❝✉❧♦ ❞❡ ✉♥❛ ❱❛r✐❛❜❧❡✿ ❚r❛s❝❡♥❞❡♥t❡s t❡♠♣r❛♥❛s✳✲ ❙❡①t❛ ❊❞✐çã♦ ✲ ❈❊◆●❆●❊ ▲❡❛r♥✐♥❣✳ ❬✶✶❪ ❙♣✐✈❛❦ ▼✐❝❤❛❡❧✳✲ 1989✳ ❈❛❧❝✉❧✉s✿ ❈á❧❝✉❧♦ ■♥✜♥✐t❡s✐♠❛❧✳ ✲ ❊❞✐t♦r✐❛❧ ❘❡✈❡rt❡✳ 1983✳ ❬✶✷❪ ❙✇♦❦♦✇s❦✐✳ ❊❛r❧ ❲✳✲ ❈á❧❝✉❧♦ ❝♦♥ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✳✲ ❙❡❣✉♥❞❛ ❊❞✐çã♦ ✲ ●r✉♣♦ ❊❞✐t♦r✐❛❧ ■❜❡r♦❛♠ér✐❝❛✳ 2008✳ ❬✶✸❪ ❚✐❜✐r✐❝❛ ❉✳ ❆❧t❛♠✐r❛♥♦✳✲ ❈✉rs♦ ❞❡ ❈á❧❝✉❧♦ ■♥✜♥✐t❡s✐♠❛❧✳✲ ❚♦♠♦ ■✱ P✉❜❧✐❝❛çã♦ ❞❛ ❋✉♥❞❛çã♦ ●♦r❝✐❡①✳❖✉r♦ Pr❡t♦ 1962✳ ✸✻✼ ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ✸✻✽ R 09/02/2021 ❮♥❞✐❝❡ ❞❡ ♥ú♠❡r♦s ♣♦s✐t✐✈♦s✱ ✶✼ ❞❡ ♣❛rt✐❞❛✱ ✻✼ ✐♥❞✉t✐✈♦✱ ✹✽ ♥✉♠ér✐❝♦✱ ✹ s♦❧✉çã♦✱ ✷✾ ❈♦♥s❡r✈❛çã♦ ❞♦ s✐♥❛❧✱ ✶✼✷✱ ✶✽✶ ❈♦♥tí♥✉❛ ♣❡❧❛ ❞✐r❡✐t❛✱ ✷✷✾ ❡sq✉❡r❞❛✱ ✷✷✾ ❈♦♥t✐♥✉✐❞❛❞❡ ❡♠ ✐♥t❡r✈❛❧♦s✱ ✷✷✾ ♥✉♠ ❝♦♥❥✉♥t♦✱ ✷✷✶ ♥✉♠ ♣♦♥t♦✱ ✷✶✾ ❈♦♥tr❛❞♦♠í♥✐♦✱ ✻✽✱ ✼✼ ❈♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈✐❝❛✱ ✽✶ ❈♦rt❡s✱ ✸ ❈✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛✱ ✷✽✵ ❈✉st♦ ♠é❞✐♦✱ ✶✵✵ t♦t❛❧✱ ✶✵✵ ❮♥✜♠♦✱ ✹✼ ❆❝❡❧❡r❛çã♦ ✐♥st❛♥tâ♥❡❛✱ ✸✶✶ ❆❞✐çã♦✱ ✻ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✱ ✶✵ ❆ssí♥t♦t❛ ❤♦r✐③♦♥t❛❧✱ ✸✷✺ ♦❜❧íq✉❛✱ ✸✷✺ ✈❡rt✐❝❛❧✱ ✸✶✾ ❆①✐♦♠❛ ❞❡ ❡①✐stê♥❝✐❛✱ ✶✼ ❞♦ s✉♣r❡♠♦✱ ✺✱ ✹✼ ❇❤❛s❦❛r❛✱ ✶✼✼ ❈❛r❧ ❋✳ ●❛✉ss ✭1777 − 1855✮✱ ✸ ❈❛t❡♥ár✐❛ ✱ ✸✶✸ ❈❛✉❝❤②✱ ✸✱ ✻✸ ❆✳ ▲✳✱ ✷✹✽ ❈❤r✐st♦♣❤ ●✉❞❡r♠❛♥♥✱ ✷✶✼ ❈✐❧✐♥❞r♦✱ ✶✶ ❈♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r✱ ✾✸✱ ✷✺✸ ❈♦♠❜✐♥❛çã♦ ❧✐♥❡❛r✱ ✶✷✼ ❈♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s✱ ✶✵✽ ❈♦♠♣r✐♠❡♥t♦ ❞❛ ♥♦r♠❛❧✱ ✷✺✶ t❛♥❣❡♥t❡✱ ✷✺✶ ❈♦♥❥✉♥t♦ ✐♠❛❣❡♠✱ ✼✼ ❞❡ ❝❤❡❣❛❞❛✱ ✻✼ ❉❡❞❡❦✐♥❞ ❘✳✱ ✸ ❉❡♠❛♥❞❛✱ ✶✵✵ ❉❡♣❡♥❞ê♥❝✐❛ ❢✉♥❝✐♦♥❛❧✱ ✽✼ ❉❡r✐✈❛❞❛ á ❡sq✉❡r❞❛✱ ✷✺✸ à ❞✐r❡✐t❛✱ ✷✺✹ ❞❛ ❢✉♥çã♦ ✐♥✈❡rs❛✱ ✷✻✸ ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r✱ ✷✻✷ ✸✻✾ ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❛r❝♦ ❝♦t❛♥❣❡♥t❡✱ ✶✹✾ ❛r❝♦ s❡❝❛♥t❡✱ ✶✺✵ ❛r❝♦ s❡♥♦✱ ✶✹✽ ❛r❝♦ t❛♥❣❡♥t❡✱ ✶✹✾ ❇✐❥❡t✐✈❛✱ ✽✶ ❜✐✉♥í✈♦❝❛✱ ✽✶ ❝♦❧❝❤❡t❡✱ ✾✻ ❝♦♥st❛♥t❡✱ ✾✶ ❝♦♥tí♥✉❛✱ ✷✶✾ ❝♦ss❡♥♦✱ ✶✹✸ ❝♦t❛♥❣❡♥t❡✱ ✶✹✺ ❝✉st♦ ♠é❞✐♦✱ ✶✵✶ ❞❡ ❞❡♠❛♥❞❛✱ ✾✾ ❞❡ ❧✉❝r♦ t♦t❛❧✱ ✶✵✵ ❞❡ ♦❢❡rt❛✱ ✾✾ ❞❡ r❡❝❡✐t❛ t♦t❛❧✱ ✶✵✵ ❞❡r✐✈á✈❡❧✱ ✷✹✽ ❞❡r✐✈❛❞❛✱ ✷✹✽ ❞❡s❝♦♥tí♥✉❛✱ ✷✶✾ ❞♦ ❝✉st♦ t♦t❛❧✱ ✶✵✵ ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡✱ ✸✶✺ ❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡✱ ✸✶✺ ❡①♣♦♥❡♥❝✐❛❧✱ ✶✸✸ ❤♦♠♦❣rá✜❝❛✱ ✶✶✺✱ ✶✶✾ ✐❞❡♥t✐❞❛❞❡✱ ✾✷✱ ✶✶✷ ✐♠♣❛r✱ ✶✷✸ ✐♠♣❧í❝✐t❛✱ ✶✷✶ ✐♥❥❡t✐✈❛✱ ✽✵ ✐♥❥❡t♦r❛✱ ✽✶ ✐♥✈❡rs❛✱ ✶✶✶ ❧✐♠✐t❛❞❛✱ ✶✷✺✱ ✷✸✹ ❧✐♥❡❛r✱ ✾✷ ❧♦❣❛rít♠✐❝❛✱ ✶✸✺ ❧✉❝r♦✱ ✶✶✽ ♠❛✐♦r ✐♥t❡✐r♦✱ ✾✺ ♠❛♥t✐ss❛✱ ✶✷✷ ♠♦♥♦tô♥✐❝❛✱ ✶✷✹ ♥ã♦ ❝r❡s❝❡♥t❡✱ ✸✶✺ ✐♠♣❧í❝✐t❛✱ ✷✻✻ ❉❡s❝❛rt❡s✱ ✻✻ ❉❡s❝♦♥t✐♥✉✐❞❛❞❡ ❡ss❡♥❝✐❛❧✱ ✷✷✵ ❡✈✐tá✈❡❧✱ ✷✷✵✱ ✷✷✽ r❡♠♦✈í✈❡❧✱ ✷✷✵ ❉❡s✐❣✉❛❧❞❛❞❡✱ ✷✾ ❞❡ ❍♦❧❞❡r✱ ✸✷✷ tr✐❛♥❣✉❧❛r✱ ✹✶ ❉✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ✷✽✻ ❉✐✈✐s✐❜✐❧✐❞❛❞❡✱ ✺✹ ❉✐✈✐s♦r ❝♦♠✉♠✱ ✾✱ ✺✺ ❉♦♠í♥✐♦ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ✼✻ ❞❡ ✉♠❛ r❡❧❛çã♦✱ ✻✽ ❊q✉❛çã♦✱ ✶✾ ❞❛ r❡t❛✱ ✾✸ ❞❡ ❞❡♠❛♥❞❛✱ ✾✾ ❞✐❢❡r❡♥❝✐❛❧✱ ✷✽✶ ❊q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s✱ ✷✽✵ ❊q✉✐❧í❜r✐♦ ❞❡ ♠❡r❝❛❞♦✱ ✶✵✷ ❊rr♦ ♣❡r❝❡♥t✉❛❧✱ ✷✾✵ r❡❧❛t✐✈♦✱ ✷✾✵ ❊✉❧❡r✱ ✹✾ ❋ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛✱ ✶✾ ❞❡ ▲❡✐❜♥✐t③✱ ✷✻✷ ❋❡r♠❛t✱ ✹✾✱ ✻✻ ❋♦r♠❛s ✐♥❞❡t❡r♠✐♥❛❞❛s✱ ✶✼✺ ❋✉♥çã♦✱ ✼✺ ❛✜♠✱ ✾✶ ❛❧❣é❜r✐❝❛✱ ✶✷✷ ❛r❝♦ ❝♦ss❡❝❛♥t❡✱ ✶✺✵ ❛r❝♦ ❝♦ss❡♥♦✱ ✶✹✽ ✸✼✵ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ♥ã♦ ❞❡❝r❡s❝❡♥t❡✱ ✸✶✺ ♥ã♦ ❧✐♠✐t❛❞❛✱ ✶✷✺ ♣❛r✱ ✶✷✸ ♣❡r✐ó❞✐❝❛✱ ✶✷✶ ♣♦❧✐♥ô♠✐❝❛✱ ✷✷✸ ♣♦s✐çã♦✱ ✸✵✽ q✉❛❞rát✐❝❛✱ ✾✼ r❛❝✐♦♥❛❧✱ ✾✼ r❛✐③ q✉❛❞r❛❞❛✱ ✾✻ r❡❝❡✐t❛ ♠é❞✐❛✱ ✶✵✶ s❡❝❛♥t❡✱ ✶✹✻ s❡♥♦✱ ✶✹✸ s♦❜r❡❥❡t✐✈❛✱ ✽✶ s♦❜r❡❥❡t♦r❛✱ ✽✶ t❛♥❣❡♥t❡✱ ✶✹✹ ✉♠✲❛✲✉♠✱ ✽✶ ✉♥í✈♦❝❛✱ ✽✶ ✈❛❧♦r ❛❜s♦❧✉t♦✱ ✾✻ ❋✉♥çõ❡s ❡❧❡♠❡♥t❛r❡s✳✱ ✶✷✼ ❤✐♣❡r❜ó❧✐❝❛s✱ ✶✺✷ ✐❣✉❛✐s✱ ✶✵✼ ♣♦❧✐♥♦♠✐❛✐s✱ ✸✸ tr❛♥s❝❡♥❞❡♥t❡s✱ ✶✸✸ R ❏♦❤♥ ❱❡♥♥✱ ✹ ❏♦r❣❡ I ✱ ✸✵✼ ▲ó❣✐❝❛ ♠❛t❡♠át✐❝❛✱ ✶ ▲❛❣r❛♥❣❡ ❏✳ ▲✳✱ ✷✹✽ ▲❛♣❧❛❝❡✱ ✷✹✺ ▲❡✐ ❞❛s t❛♥❣❡♥t❡s✱ ✶✹✽ ❞❡ ❇♦②❧❡✱ ✸✻✶ ❞❡ ❖❤♠✱ ✸✺✻ ❞♦s ❝♦ss❡♥♦s✱ ✶✹✼ ❞♦s s❡♥♦s✱ ✶✹✼ ❤♦rár✐❛✱ ✷✹✻ ▲❡✐❜♥✐t③✱ ✷✹✼ ●✳ ❲✳✱ ✷✹✽ ▲❡♠❛ ❞❡ ❊✉❝❧✐❞❡s✱ ✺✺ ▲✐♠✐t❛çã♦ ❣❧♦❜❛❧✱ ✷✸✸ ▲✐♠✐t❡ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✱ ✷✵✹ ❧♦❣❛rít♠✐❝❛✱ ✷✵✹ ▲✐♠✐t❡ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ✶✻✹ ▲✐♠✐t❡s ❛♦ ✐♥✜♥✐t♦✱ ✶✽✺ ✐♥✜♥✐t♦s✱ ✶✾✺ ❧❛t❡r❛✐s✱ ✶✽✸ ▲✐♠✐t❡s ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s✱ ✷✵✵ tr✐❣♦♥♦♠étr✐❝❛s ✐♥✈❡rs❛s✱ ✷✵✷ ▲✉❝r♦ ♠é❞✐♦✱ ✶✶✽ ●❛②✲▲✉ss❛❝✱ ✾✵ ●♦❧❞❜❛❝❤✱ ✺✵ ●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ▲❡✐❜♥✐t③✱ ✸✵✼ ●rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ✼✺ ■♠❛❣❡♠ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ✼✻ ❞❡ ✉♠❛ r❡❧❛çã♦✱ ✻✽ ■♥❞✉çã♦ ♠❛t❡♠át✐❝❛✱ ✺✵ ■♥❡q✉❛çã♦✱ ✷✾ ■♥✜♠♦ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ✶✷✻ ■♥t❡r✈❛❧♦s✱ ✸✵ ▼á①✐♠♦✱ ✹✹✱ ✹✽ ❛❜s♦❧✉t♦✱ ✷✾✵ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ✶✷✻ ❞✐✈✐s♦r ❝♦♠✉♠✱ ✺✺ ❧♦❝❛❧✱ ✷✾✶ r❡❧❛t✐✈♦✱ ✷✾✶ ✸✼✶ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ ▼é❞✐❛ ❛r✐t♠ét✐❝❛✱ ✷✹✱ ✻✶ ❣❡♦♠étr✐❝❛✱ ✷✺✱ ✻✶ ▼í♥✐♠♦✱ ✹✹✱ ✹✽ ❛❜s♦❧✉t♦✱ ✷✾✵ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ✶✷✻ ❧♦❝❛❧✱ ✷✾✶ r❡❧❛t✐✈♦✱ ✷✾✶ ▼í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠✱ ✺✻ ▼❡♥♦r q✉❡✱ ✺ R Pr✐♠❡✐r❛ ❞❡r✐✈❛❞❛✱ ✷✹✽ Pr✐♥❝í♣✐♦ ❞❛ ❜♦❛ ♦r❞❡♠✱ ✹✽ ❞❡ ❆rq✉✐♠❡❞❡s✱ ✷✷ Pr♦❞✉t♦✱ ✻ Pr♦♣r✐❡❞❛❞❡s ❞♦s ❧✐♠✐t❡s✱ ✶✼✶ ◗✉❛♥t✐❞❛❞❡ ❞❛ ❞❡♠❛♥❞❛✱ ✶✵✵ ❞❡ ❡q✉✐❧í❜r✐♦✱ ✶✵✷ ◆ú♠❡r♦ ❝♦♠♣♦st♦✱ ✾✱ ✶✵✱ ✺✻ ✐rr❛❝✐♦♥❛❧✱ ✶✸ ♣❛r✱ ✶✸ ♣r✐♠♦✱ ✾✱ ✺✻ r❛❝✐♦♥❛❧✱ ✶✸ ◆ú♠❡r♦s ♣r✐♠♦s✱ ✺✵ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s✱ ✶✵ ◆❡✇t♦♥✱ ✷✹✼ ❘❛✐③ q✉❛❞r❛❞❛✱ ✶✾ ❘❡❝❡✐t❛ ♠é❞✐❛✱ ✶✵✵ t♦t❛❧✱ ✶✵✵ ❘❡❣r❛ ❞❛ ❝❛❞❡✐❛✱ ✷✻✺ ❞❡ ▲✬❍♦s♣✐t❛❧✱ ✸✸✾ ❘❡❣r❛s ❞❡ ❞❡r✐✈❛çã♦✱ ✷✺✼ ❘❡❧❛çã♦✱ ✻✼ ❞❡ ♦r❞❡♠✱ ✶✼ ♥✉❧❛✱ ✻✼ ❘❡sí❞✉♦✱ ✶✵ ❘❡s♦❧✈❡r ✉♠❛ ❡q✉❛çã♦✱ ✶✾ ❘❡str✐çã♦ ♣r✐♥❝✐♣❛❧✱ ✶✹✽ ❘❡t❛ ❛♠♣❧✐❛❞❛✱ ✸✵ ♥♦r♠❛❧✱ ✷✺✶ ♥✉♠ér✐❝❛✱ ✺ t❛♥❣❡♥t❡✱ ✷✹✻✱ ✷✺✶ ❖❢❡rt❛✱ ✶✵✵ ❖♣❡r❛çõ❡s ❝♦♠ ❢✉♥çõ❡s✱ ✶✵✼ ❖r❞❡♠ ♠❛✐♦r✱ ✸✹✹ P❛râ♠❡tr♦✱ ✹✵✱ ✶✶✷✱ ✷✹✼ P❛rt❡ ✐♥t❡✐r❛✱ ✷✶ P✐❡rr❡ ❋❡r♠❛t✱ ✷✹✻ P✐t❛❣ór✐❝♦s✱ ✻✻ P♦♥t♦ ❝rít✐❝♦✱ ✸✷✱ ✷✾✹ ❞❡ ❛❝✉♠✉❧❛çã♦✱ ✷✹✼✱ ✷✺✸✱ ✷✽✻ ❞❡ ❡q✉✐❧í❜r✐♦✱ ✶✵✷ ❞❡ ❡①tr❡♠♦✱ ✷✾✶ ❞❡ ✐♥✢❡①ã♦✱ ✸✶✺ ✜①♦✱ ✸✵✹ ❧✐♠✐t❡✱ ✷✹✼ s✐♥❣✉❧❛r✱ ✷✾✹ P♦s✐t✐✈✐❞❛❞❡✱ ✶✼ ❙❡çã♦ tr❛♥s✈❡rs❛❧✱ ✷✷✼ ❙✐st❡♠❛ ♥✉♠ér✐❝♦✱ ✸ ❙✉❜♥♦r♠❛❧✱ ✷✺✶ ❙✉❜t❛♥❣❡♥t❡✱ ✷✺✶ ❙✉❜tr❛çã♦✱ ✹ ❙✉♣r❡♠♦✱ ✹✼ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ✶✷✻ ✸✼✷ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❚r✐❝♦t♦♠✐❛✱ ✶✼ ❚❛①❛ ❞❡ ✈❛r✐❛çã♦✱ ✷✹✷✱ ✸✵✻ ❯♥✐❝✐❞❛❞❡ ❞♦ ❧✐♠✐t❡✱ ✶✼✶ ♠é❞✐❛✱ ✷✹✼ ♣♦st❛❧✱ ✷✶✾ ❱❛❧♦r ❛❜s♦❧✉t♦✱ ✹✶ ❚❡♦r❡♠❛ ❱❛❧♦r ❡①tr❡♠♦✱ ✷✾✶ ❞❡ ❇♦❧③❛♥♦✱ ✷✸✶ ❱❛r✐á✈❡❧ ❞❡ ❈❛✉❝❤②✱ ✸✸✾ ❞❡♣❡♥❞❡♥t❡✱ ✼✻ ❞❡ P✐tá❣♦r❛s✱ ✷✸✱ ✽✼ ✐♥❞❡♣❡♥❞❡♥t❡✱ ✼✻ ❞❡ ❘♦❧❧❡✱ ✷✾✺✱ ✸✸✾ ❱❡❧♦❝✐❞❛❞❡ ❞❡ ❲❡✐❡rstr❛ss✱ ✷✸✹✱ ✸✷✵ ✐♥st❛♥tâ♥❡❛✱ ✸✵✾ ❞♦ ❝♦♥❢r♦♥t♦✱ ✶✼✷ ♠é❞✐❛✱ ✸✵✽ ❞♦ s❛♥❞✉í❝❤❡✱ ✶✼✷ ❱✐③✐♥❤❛♥ç❛✱ ✶✻✷ ❞♦ ✈❛❧♦r ✐♥t❡r♠é❞✐♦✱ ✷✸✺ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ❛r✐t♠ét✐❝❛✱ ✺✻ ❲❡✐❡rstr❛ss✱ ❑❛r❧ ❚❤❡♦❞♦r ❲✐❧❤❡❧♠ ✱ ✷✶✼ ✸✼✸ 09/02/2021 ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❈❤r✐st✐❛♥ ❏♦s❡ ◗✉✐♥t❛♥❛ P✐♥❡❞♦ ♣♦ss✉✐ ❇❛❝❤❛r❡❧❛t♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛✱ ♣❡❧❛ ✉♥✐✈❡rs✐❞❛❞❡ ❞❡❝❛♥❛ ❞❛ ❆♠ér✐❝❛ ✲ ❯♥✐✈❡rs✐❞❛❞ ◆❛❝✐♦♥❛❧ ▼❛②♦r ❞❡ ❙❛♥ ▼❛r❝♦s ✲ ▲✐♠❛✴P❡r✉ ✭1980✮✱ ♠❡str❛❞♦ ✭1990✮ ❡ ❞♦✉t♦r❛❞♦ ✭1997✮ ❡♠ ❈✐ê♥❝✐❛s ▼❛t❡♠át✐❝❛s✱ ♣❡❧❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✳ ❈♦♠♦ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠át✐❝❛✱ ❞❡s❞❡ 1977✱ ❛t✉♦✉ ♥❛s ✉♥✐✲ ✈❡rs✐❞❛❞❡s✿ ✭✶✮ ◆❛❝✐♦♥❛❧ ▼❛②♦r ❞❡ ❙❛♥ ▼❛r❝♦s✱ ✭✷✮ ◆❛❝✐♦✲ ♥❛❧ ❞❡ ■♥❣❡♥✐❡r✐❛✱ ✭✸✮ ❚é❝♥✐❝❛ ❞❡❧ ❈❛❧❧❛♦✱ ✭✹✮ ❉❡ ▲✐♠❛✱ ✭✺✮ ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❙❛♥ ▼❛rt✐♥✱ ❡♠ ▲✐♠❛ ✲ P❡r✉✳ ◆♦ ❇r❛s✐❧✱ ❛t✉♦✉ ♥❛s ✉♥✐✈❡r✲ s✐❞❛❞❡s✿ ✭✶✮ ❯♥✐♦❡st❡ ✭❈❛s❝❛✈❡❧✮✱ ✭✷✮ ❚❡❝♥♦❧ó❣✐❝❛ ❋❡❞❡r❛❧ ❞♦ P❛r❛♥á ✭P❛t♦ ❇r❛♥❝♦✮ ❡ ✭✸✮ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❚♦❝❛♥t✐♥s ✲ ❯❋❚✳ ➱ ♣r♦❢❡ss♦r ❛ss♦❝✐❛❞♦ ❞❛ ❋✉♥❞❛çã♦ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❚♦❝❛♥t✐♥s ❡ ❈♦♦r❞❡♥❛❞♦r ❞♦ ❈✉rs♦ ❞❛ ▲✐❝❡♥❝✐❛t✉r❛ ❡♠ ▼❛t❡♠át✐❝❛ ❊❆❉✴❯❆❇✴❯❋❚✳ ❉❡s❞❡ 2005 ♣❡rt❡♥❝❡ ❛♦ ❇❛♥❝♦ ❞❡ ❛✈❛✲ ❧✐❛❞♦r❡s ❞♦ ■♥st✐t✉t♦ ◆❛❝✐♦♥❛❧ ❞❡ ❊st✉❞♦s ❡ P❡sq✉✐s❛s ❊❞✉❝❛❝✐♦♥❛✐s ❆♥ís✐♦ ❚❡✐①❡✐r❛ ✲ ■♥❡♣✳ ❚❡♠ ❡①♣❡r✐ê♥❝✐❛ ♥❛ ár❡❛ ❞❡ ❊❞✉❝❛çã♦✱ ❝♦♠ ê♥❢❛s❡ ❡♠ ❊❞✉❝❛çã♦ P❡r♠❛♥❡♥t❡✱ ❛t✉✲ ❛♥❞♦ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦s s❡❣✉✐♥t❡s t❡♠❛s✿ ❡❞✉❝❛çã♦ ♠❛t❡♠át✐❝❛✱ ♠❛t❡♠át✐❝❛✱ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛✱ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❡ ❡❞✉❝❛çã♦✳ ➱ ♠❡♠❜r♦ ❞♦ ❈♦♥s❡❧❤♦ ❊❞✐t♦r✐❛❧ ❞❛ ■❊❙ ❈❧❛r❡t✐❛♥♦✱ ❡♠ ❙ã♦ P❛✉❧♦✱ ❡ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❚♦❝❛♥t✐♥s ✲ ❯❋❚ ✭♣❡rí♦❞♦ 2012 − 2014✮✳ ❈❤r✐st✐❛♥ t❡♠ tr❛❜❛❧❤♦s ♣✉❜❧✐❝❛❞♦s ♥❛ ár❡❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❡♠ ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s✱ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛ ❡ ♦✉tr♦s❀ s✉❛s ❧✐♥❤❛s ❞❡ ♣❡sq✉✐s❛ sã♦✿ ❍✐stó✲ r✐❛ ❞❛ ▼❛t❡♠át✐❝❛✱ ❋✐❧♦s♦✜❛ ❞❛ ▼❛t❡♠át✐❝❛✱ ❊♣✐st❡♠♦❧♦❣✐❛ ❞❛ ▼❛t❡♠át✐❝❛ ❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❡♠ ❉❡r✐✈❛❞❛s P❛r❝✐❛✐s✳ ✸✼✹ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R ❉❖ ▼❊❙▼❖ ❆❯❚❖❘ ▲✐✈r♦s Pá❣✐♥❛s • ■♥tr♦❞✉çã♦ ❛s ❊str✉t✉r❛s ❆❧❣é❜r✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 278 • ■♥tr♦❞✉çã♦ à ▲ó❣✐❝❛ ▼❛t❡♠át✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 197 • ❋✉♥❞❛♠❡♥t♦s ❞❛ ▼❛t❡♠át✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 273 • ❈á❧❝✉❧♦ ■♥t❡❣r❛❧ ❡ ❋✉♥çõ❡s ❞❡ ❱ár✐❛s ❱❛r✐á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ • ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ • ❙ér✐❡s ❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ • ❚❡♦r✐❛ ❞❛ ❉❡♠♦♥str❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ • ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 365 • ■♥tr♦❞✉çã♦ à ❆♥á❧✐s❡ ❘❡❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 390 R✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 390 490 74 246 ◆♦t❛s ❞❡ ❆✉❧❛ 01 ❙✉♣❧❡♠❡♥t♦ ❞❡ ❈á❧❝✉❧♦ ■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 522 04 ❙✉♣❧❡♠❡♥t♦ ❞❡ ❈á❧❝✉❧♦ ■❱ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 676 05 ■♥tr♦❞✉çã♦ à ❊♣✐st❡♠♦❧♦❣✐❛ ❞❛ ▼❛t❡♠át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 200 06 ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 280 02 ❙✉♣❧❡♠❡♥t♦ ❞❡ ❈á❧❝✉❧♦ ■■✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 398 03 ❙✉♣❧❡♠❡♥t♦ ❞❡ ❈á❧❝✉❧♦ ■■■ ✭❡♠ ❡❞✐çã♦✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 07 ❈♦♠♣❧❡♠❡♥t♦ ❞❛ ▼❛t❡♠át✐❝❛ ■✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 194 08 ❈♦♠♣❧❡♠❡♥t♦ ❞❛ ▼❛t❡♠át✐❝❛ ■■✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 214 09 ❙✉♣❧❡♠❡♥t♦ ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 10 ❖ ❈á❧❝✉❧♦ ❝♦♠ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s 11 ❙✉♣❧❡♠❡♥t♦ ❞❡ ❆♥á❧✐s❡ ❘❡❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 160 12 ❖ ❈á❧❝✉❧♦ ❝♦♠ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s 13 ❈á❧❝✉❧♦ ❱❡t♦r✐❛❧ ❡ ❙ér✐❡s ◆✉♠ér✐❝❛s ✭❡♠ ❡❞✐çã♦✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 222 C C 410 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 100 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 100 ✸✼✺ 09/02/2021 ❈❤r✐st✐❛♥ ◗✳ P✐♥❡❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡♠ R 14 ▼❛♥✉❛❧ ❞♦ ❊st✉❞❛♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 15 ❊str✉t✉r❛çã♦ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛ ✲ Pró✲❈✐ê♥❝✐❛s ✲ ❱♦❧ ✶ ✲ ✶✾✾✾✳ ✳ ✳ ✳ ✳ ✳ 140 13 ❊str✉t✉r❛çã♦ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛ ✲ Pró✲❈✐ê♥❝✐❛s ✲ ❱♦❧ ✷ ✲ ✶✾✾✾✳ ✳ ✳ ✳ ✳ ✳ 236 16 ❊str✉t✉r❛çã♦ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 180 17 ❚ó♣✐❝♦s ❞❡ ❈á❧❝✉❧♦ ■✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 142 ❙✉❣❡stõ❡s✿ 50 ❝❤r✐st✐❛♥❥q♣❅②❛❤♦♦✳❝♦♠✳❜r ✸✼✻ 09/02/2021