Exponential Functions PDF

1.4 Exponential Functions Copyright © Cengage Learning. All rights reserved. Exponential Functions The function f(x) = 2x is called an exponential function because the variable, x, is the exponent. It should not be confused with the power function g(x) = x2, in which the variable is the ba...

1.4 Exponential Functions Copyright © Cengage Learning. All rights reserved. Exponential Functions The function f(x) = 2x is called an exponential function because the variable, x, is the exponent. It should not be confused with the power function g(x) = x2, in which the variable is the base. In general, an exponential function is a function of the form f(x) = bx where b is a positive constant. Let’s recall what this means. If x = n, a positive integer, then 2 Exponential Functions If x = 0, then b0 = 1, and if x = –n, where n is a positive integer, then If x is a rational number, x = p /q, where p and q are integers and q > 0, then But what is the meaning of bx if x is an irrational number? For instance, what is meant by or 5 ? 3 Exponential Functions To help us answer this question we first look at the graph of the function y = 2x, where x is rational. A representation of this graph is shown in Figure 1. We want to enlarge the domain of y = 2x to include both rational and irrational numbers. There are holes in the graph in Figure 1 corresponding to irrational Representation of y = 2x, x rational values of x. Figure 1 We want to fill in the holes by defining f(x) = 2x, where x  , so that f is an increasing function. 4 Exponential Functions In particular, since the irrational number satisfies we must have and we know what 21.7 and 21.8 mean because 1.7 and 1.8 are rational numbers. 5 Exponential Functions Similarly, if we use better approximations for we obtain better approximations for 6 Exponential Functions It can be shown that there is exactly one number that is greater than all of the numbers 21.7, 21.73, 21.732, 21.7320, 21.73205, … and less than all of the numbers 21.8, 21.74, 21.733, 21.7321, 21.73206, … We define to be this number. Using the preceding approximation process we can compute it correct to six decimal places: 7 Exponential Functions Similarly, we can define 2x (or bx, if b > 0) where x is any irrational number. Figure 2 shows how all the holes in Figure 1 have been filled to complete the graph of the function f(x) = 2x, x  Representation of y = 2x, x rational y = 2x, x real Figure 1 Figure 2 8 Exponential Functions The graphs of members of the family of functions y = bx are shown in Figure 3 for various values of the base b. Figure 3 9 Exponential Functions Notice that all of these graphs pass through the same point (0, 1) because b0 = 1 for b  0. Notice also that as the base b gets larger, the exponential function grows more rapidly (for x > 0). You can see from Figure 3 that there are basically three kinds of exponential functions y = bx. If 0 < b < 1, the exponential function decreases; if b = 1, it is a constant; and if b > 1, it increases. 10 Exponential Functions These three cases are illustrated in Figure 4. (a) y = bx, 0 < b < 1 (b) y = 1x (c) y = bx, b > 1 Figure 4 11 Exponential Functions Observe that if b  1, then the exponential function y = bx has domain and range (0, ). Notice also that, since (1/b)x = 1/bx = b–x, the graph of y = (1/b)x is just the reflection of the graph of y = bx about the y-axis. 12 Exponential Functions One reason for the importance of the exponential function lies in the following properties. If x and y are rational numbers, then these laws are well known from elementary algebra. It can be proved that they remain true for arbitrary real numbers x and y. 13 Example 1 Sketch the graph of the function y = 3 – 2x and determine its domain and range. Solution: First we reflect the graph of y = 2x [shown in Figures 2 and 5(a)] about the x-axis to get the graph of y = –2x in Figure 5(b). Then we shift the graph of y = –2x upward 3 units to obtain the graph of y = 3 – 2x in Figure 5(c). The domain is and the range is ( , 3). (a) y = 2x (b) y = –2x (c) y = 3 – 2x 14 Figure 5 Applications of Exponential Functions The exponential function occurs very frequently in mathematical models of nature and society. Here we indicate briefly how it arises in the description of population growth and radioactive decay. First we consider a population of bacteria in a homogeneous nutrient medium. Suppose that by sampling the population at certain intervals it is determined that the population doubles every hour. 15 Applications of Exponential Functions If the number of bacteria at time t is p(t), where t is measured in hours, and the initial population is p(0) = 1000, then we have p(1) = 2p(0) = 2  1000 p(2) = 2p(1) = 22  1000 p(3) = 2p(2) = 23  1000 It seems from this pattern that, in general, p(t) = 2t  1000 = (1000)2t 16 Applications of Exponential Functions This population function is a constant multiple of the exponential function y = 2t, so it exhibits the rapid growth. Under ideal conditions (unlimited space and nutrition and absence of disease) this exponential growth is typical of what actually occurs in nature. 17 Applications of Exponential Functions What about the human population? Table 1 shows data for the population of the world in the 20th century and Figure 8 shows the corresponding scatter plot. Scatter plot for world population growth Figure 8 18 Applications of Exponential Functions The pattern of the data points in Figure 8 suggests exponential growth, so we use a graphing calculator with exponential regression capability to apply the method of least squares and obtain the exponential model P = (1436.53) (1.01395)t where t = 0 corresponds to 1900. Figure 9 shows the graph of this exponential function together with the original data points. Exponential model for population growth Figure 9 We see that the exponential curve fits the data reasonably well. The period of relatively slow population growth is explained by the two world wars and the Great Depression of the 1930s. 19 Example 3 The half-life of strontium-90, 90Sr, is 25 years. This means that half of any given quantity of 90Sr will disintegrate in 25 years. (a) If a sample of 90Sr has a mass of 24 mg, find an expression for the mass m(t) that remains after t years. (b) Find the mass remaining after 40 years, correct to the nearest milligram. (c) Use a graphing device to graph m(t) and use the graph to estimate the time required for the mass to be reduced to 5 mg. 20 Example 3 – Solution (a) The mass is initially 24 mg and is halved during each 25-year period, so 21 Example 3 – Solution cont’d From this pattern, it appears that the mass remaining after t years is This is an exponential function with base (b) The mass that remains after 40 years is 22 Example 3 – Solution cont’d (c) We use a graphing calculator or computer to graph the function in Figure 12. Figure 12 We also graph the line m = 5 and use the cursor to estimate that m(t) = 5 when t  57. So the mass of the sample will be reduced to 5 mg after about 57 years. 23 The Number e Of all possible bases for an exponential function, there is one that is most convenient for the purposes of calculus. The choice of a base b is influenced by the way the graph of y = bx crosses the y-axis. 24 The Number e Figures 13 and 14 show the tangent lines to the graphs of y = 2x and y = 3x at the point (0, 1). Figure 13 Figure 14 (For present purposes, you can think of the tangent line to an exponential graph at a point as the line that touches the graph only at that point.) If we measure the slopes of these tangent lines at (0, 1), we find that m  0.7 for y = 2x and m  1.1 for y = 3x. 25 The Number e It turns out, that some of the formulas of calculus will be greatly simplified if we choose the base b so that the slope of the tangent line to y = bx at (0, 1) is exactly 1. (See Figure 15.) The natural exponential function crosses the y-axis with a slope of 1. Figure 15 26 The Number e In fact, there is such a number and it is denoted by the letter e. (This notation was chosen by the Swiss mathematician Leonhard Euler in 1727, probably because it is the first letter of the word exponential.) 27 The Number e In view of Figures 13 and 14, it comes as no surprise that the number e lies between 2 and 3 and the graph of y = ex lies between the graphs of y = 2x and y = 3x. (See Figure 16) Figure 13 Figure 14 Figure 16 28 The Number e We will see that the value of e, correct to five decimal places, is e  2.71828 We call the function f(x) = ex the natural exponential function. 29 Example 4 Graph the function and state the domain and range. Solution: We start with the graph of y = ex from Figures 15 and 17(a) and reflect about the y-axis to get the graph of y = e–x in Figure 17(b). (Notice that the graph crosses the y-axis with a slope of –1). The natural exponential function crosses the y-axis with a slope of 1. (a) y = ex (b) y = e–x Figure 15 Figure 17 30 Example 4 – Solution cont’d Then we compress the graph vertically by a factor of 2 to obtain the graph of in Figure 17(c). Finally, we shift the graph downward one unit to get the desired graph in Figure 17(d). Figure 17 The domain is and the range is (–1, ). 31 Exercise 1.4 (p 53) 32 33 Exercises (version 9) 34 Exercises (version 9) 35 Exercises (version 9) 36 Exercises (version 9) 37