1. 1 Copyright © Cengage Learning. All rights reserved. 5. Inverse, Exponential and Logarithmic Functions 5.3 The natural Exponential Function

2. 2 The Natural Exponential Function The compound interest formula is where P is the principal invested, r is the annual interest rate (expressed as a decimal), n is the number of interest periods per year, and t is the number of years that the principal is invested. The next example illustrates what happens if the rate and total time invested are fixed, but the interest period is varied.

3. 3 Example 1 – Using the compound interest formula Suppose $1000 is invested at a compound interest rate of 9%. Find the new amount of principal after one year if the interest is compounded quarterly, monthly, weekly, daily, hourly, and each minute. Solution: If we let P = $1000, t = 1, and r = 0.09 in the compound interest formula, then for n interest periods per year.

4. 4 Example 1 – Solution The values of n we wish to consider are listed in the following table, where we have assumed that there are 365 days in a year and hence (365)(24) = 8760 hours and (8760)(60) = 525,600 minutes. (In many business transactions an investment year is considered to be only 360 days.) cont’d

5. 5 Example 1 – Solution Using the compound interest formula (and a calculator), we obtain the amounts given in the following table. cont’d

6. 6 The Natural Exponential Function Note that, in the preceding example, after we reach an interest period of one hour, the number of interest periods per year has no effect on the final amount. If interest had been compounded each second, the result would still be $1094.17. (Some decimal places beyond the first two do change.) Thus, the amount approaches a fixed value as n increases. Interest is said to be compounded continuously if the number n of time periods per year increases without bound.

7. 7 The Natural Exponential Function If we let P = 1, r = 1, and t = 1 in the compound interest formula, we obtain The expression on the right-hand side of the equation is important in calculus. In Example 1 we considered a similar situation: as n increased, A approached a limiting value.

8. 8 The Natural Exponential Function The same phenomenon occurs for this formula, as illustrated by the following table.

9. 9 The Natural Exponential Function In calculus it is shown that as n increases without bound, the value of the expression [1 + (1/n)] n approaches a certain irrational number, denoted by e. The number e arises in the investigation of many physical phenomena. An approximation is e  2.71828. We denote the fact as follows.

10. 10 The Natural Exponential Function In the following definition we use e as a base for an important exponential function. The natural exponential function is one of the most useful functions in advanced mathematics and applications.

11. 11 The Natural Exponential Function Since 2 < e < 3, the graph of y = e x lies between the graphs of y = 2 x and y = 3 x, as shown in Figure 1. Figure 1

12. 12 Application: Continuously Compounded Interest The compound interest formula is If we let 1/k = r/n, then k = n/r, n = kr, and nt = krt, and we may rewrite the formula as The Natural Exponential Function

13. 13 For continuously compounded interest we let n (the number of interest periods per year) increase without bound, denoted by n → or, equivalently, by k →. Using the fact that → e as k →, we see that → P [e] rt = Pe rt as k →. The Natural Exponential Function

14. 14 This result gives us the following formula. The next example illustrate the use of this formula. The Natural Exponential Function

15. 15 Example 2 – Using the continuously compounded interest formula Suppose $20,000 is deposited in a money market account that pays interest at a rate of 6% per year compounded continuously. Determine the balance in the account after 5 years. Solution: Applying the formula for continuously compounded interest with P = 20,000, r = 0.06, and t = 5, we have A = Pe rt = 20,000e 0.06(5) = 20,000e 0.3 Using a calculator, we find that A = $26,997.18.

16. 16 The continuously compounded interest formula is just one specific case of the following law. The Natural Exponential Function

17. 17 Example 5 – Using the law of decay formula The isotope plutonium-238 is used in powering spacecraft and decays at a rate of about 0.79% per year. To the nearest tenth of a gram, how much of a 100-gram sample will remain in 88 years? Solution: We apply the decay formula q = q 0 e rt with initial quantity q 0 = 100, rate of decay r = –0.0079, and time t = 88 years. The amount remaining after 88 years is 100e –0.0079(88) = 100e –0.6952  49.9.

18. 18 Example 5 – Solution Since 49.9 is close to one half the original amount, we know that the half-life of 238Pu is about 88 years. cont’d

19. 19 The function f in the next example is important in advanced applications of mathematics. The Natural Exponential Function

20. 20 Example 6 – Sketching a graph involving two exponential functions Sketch the graph of f if Solution: Note that f is an even function, because Thus, the graph is symmetric with respect to the y-axis.

21. 21 Example 6 – Solution Using a calculator, we obtain the following approximations of f(x). Plotting points and using symmetry with respect to the y-axis gives us the sketch in Figure 3. The graph appears to be a parabola; however, this is not actually the case. cont’d Figure 3

22. 22 Application: Flexible Cables The function f of Example 6 occurs in applied mathematics and engineering, where it is called the hyperbolic cosine function. This function can be used to describe the shape of a uniform flexible cable or chain whose ends are supported from the same height, such as a telephone or power line cable (see Figure 4). Figure 4 The Natural Exponential Function

23. 23 If we introduce a coordinate system, as indicated in the figure, then it can be shown that an equation that corresponds to the shape of the cable is where a is a real number. The graph is called a catenary, after the Latin word for chain. The function in Example 6 is the special case in which a = 1. The Natural Exponential Function

24. 24 Example 8 – Sketching a Gompertz growth curve In biology, the Gompertz growth function G, given by G(t) = ke (–Ae –Bt ) where k, A, and B are positive constants, is used to estimate the size of certain quantities at time t. The graph of G is called a Gompertz growth curve. The function is always positive and increasing, and as t increases without bound, G(t) levels off and approaches the value k. Graph G on the interval [0, 5] for k = 1.1, A = 3.2, and B = 1.1, and estimate the time t at which G(t) = 1.

25. 25 Example 8 – Solution We begin by assigning 1.1e (–3.2e –1.1t ) to Y 1. Since we wish to graph G on the interval [0, 5], we choose Xmin = 0 and Xmax = 5. Because G(t) is always positive and does not exceed the value k = 1.1, we choose Ymin = 0 and Ymax = 2. Hence, the viewing rectangle dimensions are [0, 5] by [0, 2].

26. 26 Example 8 – Solution Graphing G gives us a display similar to Figure 6. The endpoint values of the graph are approximately (0, 0.045) and (5, 1.086). To determine the time when y = G(t) = 1, we use an intersect feature, with Y 2 = 1, to obtain x = t  3.194. Figure 6 [0, 5] by [0, 2] cont’d