Functions and Models

1. Functions and Models Figure 1 Often a graph is the best way to represent a function because it conveys so much information at a glance. Shown is a graph of the vertical ground acceleration created by the 2011 earthquake near Tohoku, Japan. The earthquake had a magnitude of 9.0 on the Richter scale and was so powerful that it moved northern Japan 8 feet closer Figure 1. A function f is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set E. We usually consider functions for which the sets D and E are sets of real numbers. The set D is called the domain of the function. The number f(x) is the value of f at x and is read “ f of x.” The range of f is the set of all possible values of f(x) as x varies throughout the domain. A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. A symbol that represents a number in the range of f is called a dependent variable. In Example A, for instance, r is the independent variable and A is the dependent variable. Figure 2 It’s helpful to think of a function as a machine (Figure 2). If x is in the domain of the function f, then when x enters the machine, it’s accepted as an input and the machine produces an output f(x) according to the rule of the function. Thus we can think of the domain as the set of all possible inputs and the range as the set of all possible outputs. The preprogrammed functions in a calculator are good examples of a function as a machine. For example, the square root key on your calculator computes such a function. You press the key labeled √ (or √𝑥 ) and enter the input x. If x < 0, then x is not in the domain of this function; that is, x is not an acceptable input, and the calculator will indicate an error. If x >= 0, then an approximation to √𝑥 will appear in the display. Thus the √𝑥 key on your calculator is not quite the same as the exact mathematical function f defined by f(x)= √𝑥 . Representations of Functions There are four possible ways to represent a function: ● Verbally(by a description in words) ● Numerically(by a table of values) ● Visually(by a graph) ● Algebraically(by an explicit formula) t (years 0 since 1900) 10 20 30 40 50 60 70 80 90 100 110 Population 1650 1750 1860 2070 2300 2560 3040 3710 4450 5280 6080 6870 (millions) We are given a description of the function in words: P(t) is the human population of the world at time t. Let’s measure t so that t = 0 corresponds to the year 1900. The table of values of world population provides a convenient representation of this function. If we plot these values, we get the graph (called a scatter plot) in Figure 4. It too is a useful representation; the graph allows us to absorb all the data at once. What about a formula? Of course, it’s impossible to devise an explicit formula that gives the exact human population P(t) at any time t. But it is possible to find an expression for a function that approximates P(t). In fact, using methods explained in Section mathematical model, we obtain the approximation 𝑃(𝑡) ≈ 𝑓(𝑡) = (1.43653 × 109 ). (1.01395)𝑡 Figure 4 shows that it is a reasonably good “fit.” The function f is called a mathematical model for population growth. In other words, it is a function with an explicit formula that approximates the behaviour of our given function. We will see, however, that the ideas of calculus can be applied to a table of values; an explicit formula is not necessary. Example: When you turn on a hot water faucet, the temperature T of the water depends on how long the water has been running. Draw a rough graph of T as a function of the time t that has elapsed since the faucet was turned on. Solution The initial temperature of the running water is close to room temperature because the water has been sitting in the pipes. When the water from the hot water tank starts flowing from the faucet, T increases quickly. In the next phase, T is constant at the temperature of the heated water in the tank. When the tank is drained, T decreases to the temperature of the water supply. This enables us to make the rough sketch of T as a function of t in Figure 5. Figure 5 Example: A function f is defined by Solution: Remember that a function is a rule. For this particular function the rule is the following: First look at the value of the input x. If it happens that x≤-1, then the value of f (x) is 1 - x. On the other hand, if x > -1, then the value of f (x) is x2. Figure 6 Symmetry If a function f satisfies f(-x) = f(x) for every number x in its domain, then f is called an even function. For instance, the function f(x) = x2 is even because f(-x) = (-x)2 = x2 = f(x) The geometric significance of an even function is that its graph is symmetric with respect to the y-axis (see Figure 7). This means that if we have plotted the graph of f for x >= 0, we obtain the entire graph simply by reflecting this portion about the y-axis. If f satisfies f(-x) = - f(x)sxd for every number x in its domain, then f is called an odd function. For example, the function f(-x) =x3 is odd because f(-x) = (-x)3 = - x3 = - f(x) Figure 7 Example: Determine whether each of the following functions is even, odd, or neither even nor odd. Solution: Increasing and Decreasing Functions 8 2. Mathematical model A mathematical model is a mathematical description (often by means of a function or an equation) of a real world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. Figure 9 below illustrates the process of mathematical modeling. Figure below illustrates the process of mathematical modeling. First task is to formulate a mathematical model by identifying and naming the independent and dependent variables and making assumptions that simplify the phenomenon enough to make it mathematically tractable. Figure 9 The second stage is to apply the mathematics that we know (the calculus) to the mathematical model that we have formulated in order to derive mathematical conclusions. Then, in the third stage, we take those mathematical conclusions and interpret them as information about the original real world phenomenon by way of offering explanations or making predictions. The final step is to test our predictions by checking against new real data. If the predictions don’t compare well with reality, we need to refine our model or to formulate a new model and start the cycle again. 3. Linear Models When we say that y is a linear function of x, we mean that the graph of the function is a line, so we can use the slope intercept form of the equation of a line to write a formula for the function as: where m is the slope of the line and b is the y intercept. 10 Figure 10 Example: (a) As dry air moves upward, it expands and cools. If the ground temperature is 20°C and the temperature at a height of 1 km is 10°C, express the temperature T (in °C) as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b) Draw the graph of the function in part (a). What does the slope represent? (c) What is the temperature at a height of 2.5 km? Solution: (a) Because we are assuming that T is a linear function of h, we can write T = mh + b We are given that T = 20 when h = 0, so 20 = m . 0 + b = b In other words, the y-intercept is b = 20. We are also given that T = 10 when h = 1, so 10 = m . 1 + 20 The slope of the line is therefore m = 10 - 20 = -10 and the required linear function is T = -10 h + 20 (b) The graph is sketched in Figure. The slope is m = - 10°C/km, and this represents the rate of change of temperature with respect to height. (c) At a height of h = 2.5 km, the temperature is T = - 10(2.5) + 20 = - 5°C 4. Polynomials A function P is called a polynomial if Figure 11 A polynomial of degree 3 is of the form and is called a cubic function. Figure 8 shows the graph of a cubic function in part (a) and graphs of polynomials of degrees 4 and 5 in parts (b) and (c). We will see later why the graphs have these shapes. Figure 12 Power Functions A function of the form f (x) = xa, where a is a constant, is called a power function. We consider several cases. (i) a = n, where n is a positive integer The graphs of f (x) = xn for n = 1, 2, 3, 4, and 5 are shown in Figure 12. (These are polynomials with only one term.) We already know the shape of the graphs of y = x (a line through the origin with slope 1) and y = x2. Figure 12 Graphs of f (x) = xn for n = 1, 2, 3, 4, 5 The general shape of the graph of f (x) = xn depends on whether n is even or odd. If n is even, then f (x) = xn is an even function and its graph is similar to the parabola y = x2. If n is odd, then f (x) = xn is an odd function and its graph is similar to that of y = x3. Notice from Figure 13, however, that as n increases, the graph of y = xn becomes flatter near 0 and steeper when | x| >= 1. Figure 13 (ii) a = 1/n, where n is a positive integer 𝑛 The function f (x) = x1/n = √𝑥 is a root function. For n = 2 it is the square root function f (x) =√𝑥, whose domain is [0,∞) and whose graph is the upper half of the parabola 𝑛 x = y2. [See Figure 14 (a).] For other even values of n, the graph of y = √𝑥 is similar to that 3 of y =√𝑥. For n = 3 we have the cube root function f(x) = √𝑥 whose domain is R (recall that every real number has a cube root) and whose graph is shown in Figure 14 (b). The graph of 𝑛 3 y = √𝑥 for n odd n > 3) is similar to that of y = √𝑥. Figure 14 Graphs of root functions (iii) a = -1 The graph of the reciprocal function f(x) = x-1 = 1/x is shown in Figure 15. Its graph has the equation y = 1/x, or xy = 1, and is a hyperbola with the coordinate axes as its asymptotes. This function arises in physics and chemistry in connection with Boyle’s Law, which says that, when the temperature is constant, the volume V of a gas is inversely proportional to the pressure P: V = C/P where C is a constant. Thus the graph of V as a function of P (see Figure 16) has the same general shape as the right half of Figure. Figure 15 The reciprocal function Figure 16 Volume as a function of pressure at constant temperature Rational Functions A rational function f is a ratio of two polynomials: f(x) = 𝑃(𝑥) 𝑄(𝑥) where P and Q are polynomials. The domain consists of all values of x such that 𝑄(𝑥) ≠ 0. A simple example of a rational function is the function𝑓(𝑥) = 1/𝑥, whose domain is {𝑥|𝑥 ≠ 𝑜}; this is the reciprocal function graphed in Figure 15. The function 𝑓(𝑥) = 2𝑥 4 − 𝑥 2 + 1 𝑥2 − 4 is a rational function with domain {𝑥 | 𝑥 ≠ ±2}. Its graph is shown in Figure 17. Figure 17 Algebraic Functions A function f is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is automatically an algebraic function. Here are two more examples: Figure 18 Figure 18 Trigonometric Functions In calculus the convention is that radian measure is always used (except when otherwise indicated). For example, when we use the function 𝑓(𝑥) = sin 𝑥, it is understood that sin x means the sine of the angle whose radian measure is x. Thus the graphs of the sine and cosine functions are as shown in Figure 19. Figure 19 An important property of the sine and cosine functions is that they are periodic functions and have period 2𝜋. This means that, for all values of x, The periodic nature of these functions makes them suitable for modeling repetitive phenomena such as tides, vibrating springs, and sound waves. For instance, in Example 1.3.4 we will see that a reasonable model for the number of hours of daylight in Philadelphia t days after January 1 is given by the function Exponential Functions The exponential functions are the functions of the form 𝑓(𝑥) = 𝑏 𝑥 , where the base b is a positive constant. The graphs of 𝑦 = 2𝑥 and 𝑦 = (0.5)𝑥 are shown in Figure 20. In both cases the domain is (−∞, ∞) and the range is (0, ∞). Exponential functions will be studied in detail, and we will see that they are useful for modeling many natural phenomena, such as population growth and radioactive decay. Figure 20 Logarithmic Functions The logarithmic functions 𝑓(𝑥) = 𝑙𝑜𝑔𝑏 𝑥, where the base b is a positive constant, are the inverse functions of the exponential functions. Figure 21 shows the graphs of four logarithmic functions with various bases. In each case the domain is (0, ∞) the range is (−∞, ∞), and the function increases slowly when x >1. Example: Classify the following functions as one of the types of functions that we have discussed. Solution: Figure 21 Reference: James Stewart, Calculus Early Transcendentals, (2015), Eighth Edition, Student Edition, McMaster University and University of Toronto.