Section 4-4 The Exponential Function with Base e

Transcription

1 4-4 The Exponential Function with Base e Finance. A couple just had a new child. How much should they invest now at 8.% compounded daily in order to have $4, for the child s education 7 years from now? Compute the answer to the nearest dollar. 74. Finance. A person wishes to have $, cash for a new car years from now. How much should be placed in an account now if the account pays 9.7% compounded weekly? Compute the answer to the nearest dollar. 7. Finance. If $3, is deposited into an account earning 8% compounded daily and, at the same time, $, is deposited into an account earning % compounded daily, will the first account be worth more than the second? If so, when? 76. Finance. If $4, is deposited into an account earning 9% compounded weekly and, at the same time, $6, is deposited into an account earning 7% compounded weekly, will the first account be worth more than the second? If so, when? 77. Finance. Will an investment of $, at 8.9% compounded daily ever be worth more at the end of a quarter than an investment of $, at 9% compounded quarterly? Explain. 78. Finance. A sum of $, is invested at 3% compounded semiannually. Suppose that a second investment of $, is made at interest rate r compounded daily. For which values of r, to the nearest tenth of a percent, is the second investment better than the first? Discuss. Problems 79 and 8 require a graphing utility that can compute exponential regression equations of the form y ab x (consult your manual). 79. Depreciation. Table gives the market value of a minivan (in dollars) x years after its purchase. Find an exponential regression model of the form y ab x for this data set. Estimate the purchase price of the van. Estimate the value of the van years after its purchase. Round answers to the nearest dollar. x Depreciation. Table gives the market value of a luxury sedan (in dollars) x years after its purchase. Find an exponential regression model of the form y ab x for this data set. Estimate the purchase price of the sedan. Estimate the value of the sedan years after its purchase. Round answers to the nearest dollar. x T A B L E T A B L E Value ($),7 9,4 8, 6,84, 4,48 Value ($) 3, 9,,6,87 9,4 7, Section 4-4 The Exponential Function with Base e Base e Exponential Function Growth and Decay Applications Revisited Continuous Compound Interest A Comparison of Exponential Growth Phenomena Until now the number has probably been the most important irrational number you have encountered. In this section we will introduce another irrational number, e, that is just as important in mathematics and its applications.

2 84 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS Base e Exponential Function The following expression is important to the study of calculus and, as we will see later in this section, also is closely related to the compound interest formula discussed in the preceding section: x x () Explore/Discuss (A) Calculate the values of [ (/x)] x for x,, 3, 4, and. Are the values increasing or decreasing as x gets larger? (B) Graph y [ (/x)] x and discuss the behavior of the graph as x increases without bound. TABLE x x x , , , ,, Interestingly, by calculating the value of expression () for larger and larger values of x (see Table ), it appears that [ (/x)] x approaches a number close to.783. In a calculus course we can show that as x increases without bound, the value of [ (/x)] x approaches an irrational number that we call e. Just as irrational numbers such as and have unending, nonrepeating decimal representations (see Section A-), e also has an unending, nonrepeating decimal representation. To decimal places, e Exactly who discovered e is still being debated. It is named after the great Swiss mathematician Leonhard Euler (77 783), who computed e to 3 decimal places using [ (/x)] x. The constant e turns out to be an ideal base for an exponential function because in calculus and higher mathematics many operations take on their simplest form using this base. This is why you will see e used extensively in expressions and formulas that model real-world phenomena. e DEFINITION EXPONENTIAL FUNCTION WITH BASE e For x a real number, the equation f(x) e x defines the exponential function with base e.

3 4-4 The Exponential Function with Base e 8 The exponential function with base e is used so frequently that it is often referred to as the exponential function. The graphs of y e x and y e x are shown in Figure. FIGURE Exponential functions with base e. y y e x y e x x Explore/Discuss (A) Graph y e x, y e.x, and y 3 e x in the same viewing window. How do these graphs compare with the graph of y b x for b? (B) Graph y e x, y e.x, and y 3 e x in the same viewing window. How do these graphs compare with the graph of y b x for b? (C) Use the properties of exponential functions to show that all of these functions are exponential functions. EXAMPLE Solution FIGURE f(x) 4 e x/. Analyzing a Graph Describe the graph of f(x) 4 e x/, including x and y intercepts, increasing and decreasing properties, and horizontal asymptotes. Round any approximate values to two decimal places. The graph of f is shown in Figure (a). The y intercept is f() 4 3 and the x intercept is.77 (to two decimal places). The graph shows that f is decreasing for all x. Since the exponential function e x/ as x, it follows that f(x) 4 e x/ 4 as x. The table in Figure (b) confirms this. Thus, the line y 4 is a horizontal asymptote for the graph. (a) (b)

4 86 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS MATCHED PROBLEM Describe the graph of f(x) e x/, including x and y intercepts, increasing and decreasing properties, and horizontal asymptotes. Round any approximate values to two decimal places. Growth and Decay Applications Revisited Most exponential growth and decay problems are modeled using base e exponential functions. We present two applications here and many more in Exercise 4-4. EXAMPLE Medicine Bacteria Growth Cholera, an intestinal disease, is caused by a cholera bacterium that multiplies exponentially by cell division as modeled by N N e.386t Solutions (A) Use N and t : where N is the number of bacteria present after t hours and N is the number of bacteria present at t. If we start with bacterium, how many bacteria will be present in (A) hours? (B) hours? Compute the answers to three significant digits. N N e.386t e.386(), (B) Use N and t : N N e.386t e.386() 6,7, MATCHED PROBLEM Repeat example if N N e.783t and all other information remains the same. EXAMPLE 3 Carbon 4 Dating Cosmic-ray bombardment of the atmosphere produces neutrons, which in turn react with nitrogen to produce radioactive carbon 4. Radioactive carbon 4 enters all living tissues through carbon dioxide, which is first absorbed by

5 , A Solutions FIGURE The Exponential Function with Base e 87 plants. As long as a plant or animal is alive, carbon 4 is maintained in the living organism at a constant level. Once the organism dies, however, carbon 4 decays according to the equation A A e.4t where A is the amount of carbon 4 present after t years and A is the amount present at time t. If, milligrams of carbon 4 are present at the start, how many milligrams will be present in (A), years? (B), years? Compute answers to three significant digits. Substituting A, in the decay equation, we have A,e.4t See Figure 3. (A) Solve for A when t,: A,e.4(,) 89 milligrams (B) Solve for A when t,:, t A,e.4(,).3 milligrams More will be said about carbon 4 dating in Exercise 4-4, where we will be interested in solving for t after being given information about A and A. MATCHED PROBLEM 3 Referring to Example 3, how many milligrams of carbon 4 would have to be present at the beginning in order to have milligrams present after, years? Compute the answer to four significant digits. EXAMPLE 4 Limited Growth in an Epidemic A community of, individuals is assumed to be homogeneously mixed. One individual who has just returned from another community has influenza. Assume the community has not had influenza shots and all are susceptible. The spread of the disease in the community is predicted to be given by the logistic curve N(t), 999e.3t where N is the number of people who have contracted influenza after t days. (A) How many people have contracted influenza after days? After days? Round answers to the nearest integer.

6 88 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS (B) How many days will it take until half the community has contracted influenza? Round answer to the nearest integer. (C) Does N approach a limiting value as t increases without bound? Explain. Solutions FIGURE 4 Logistic growth. (A) The table in Figure 4(a) shows that N() individuals and N() 88 individuals., (a) (b) (B) Figure 4(b) shows that the graph of N(t) intersects the line y after approximately 3 days. (C) The values in Figure 4(a) and the graph in Figure 4(b) both indicate that N approaches, as t increases without bound. We can confirm this algebraically by noting that since 999e.3t as t increases without bound, N(t),,,.3t 999e Thus, the upper limit on the growth of N is,, the total number of people in the community. MATCHED PROBLEM 4 A group of 4 parents, relatives, and friends are waiting anxiously at Kennedy Airport for a charter flight returning students after a year in Europe. It is stormy and the plane is late. A particular parent thought he had heard that the plane s radio had gone out and related this news to some friends, who in turn passed it on to others. The propagation of this rumor is predicted to be given by N(t) 4 399e.4t where N is the number of people who have heard the rumor after t minutes. (A) How many people have heard the rumor after minutes? After minutes? Round answers to the nearest integer. (B) How many minutes will it take until half the group has heard the rumor? Round answer to the nearest integer. (C) Does N approach a limiting value as t increases without bound? Explain. Continuous Compound Interest The constant e occurs naturally in the study of compound interest. Returning to the compound interest formula discussed in Section 4-3,

7 4-4 The Exponential Function with Base e 89 A P r m n Compound interest recall that P is the principal invested at an annual rate compounded m times a year and A is the amount in the account after n compounding periods. For the purposes of the discussion here, it is convenient to let n mt, where t is time in years, so that A P r m mt is the amount in the account after t years. Suppose P, r, and t are held fixed, and m, the number of compounding periods in year, is increased without bound. Will the amount A increase without bound or will it tend to some limiting value? Let s examine a specific case numerically before we attack the general problem. If P $, r.8, and t years, then A.8 m m The amount A is computed for several values of m in Table. Notice that the largest gain appears in going from annually to semiannually. Then, the gains slow down as m increases. In fact, it appears that A might be tending to something close to $7.3 as m gets larger and larger. TABLE Effect of Compounding Frequency Compounding Frequency Annually $6.64 Semiannually Quarterly Weekly Daily Hourly 8, n A.8 m m We now return to the general problem to see if we can determine what happens to A P[ (r/m)] mt as m increases without bound. A little algebraic manipulation of the compound interest formula will lead to an answer and a significant result in the mathematics of finance: A P r m mt P m/r (m/r)rt x x rt Change algebraically. Let x m/r.

8 9 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS The expression within the square brackets should look familiar. Recall from the first part of this section that x x e as x Since r is fixed, x m/r as m. Thus, P r m mt Pe rt as m and we have arrived at the continuous compound interest formula, a very important and widely used formula in business, banking, and economics. CONTINUOUS COMPOUND INTEREST FORMULA If a principal P is invested at an annual rate r compounded continuously, then the amount A in the account at the end of t years is given by A Pe rt The annual rate r is expressed as a decimal. EXAMPLE Continuous Compound Interest If $ is invested at an annual rate of 8% compounded continuously, what amount, to the nearest cent, will be in the account after years? Solution Use the continuous compound interest formula to find A when P $, r.8, and t : A Pe rt $e (.8)() 8% is equivalent to r.8. $7.3 Compare this result with the values calculated in Table. MATCHED PROBLEM What amount will an account have after years if $ is invested at an annual rate of % compounded annually? Quarterly? Continuously? Compute answers to the nearest cent. The continuous compound interest formula may also be used to model short-term population growth. If a population P is assumed to grow continuously at an annual rate r, then the population A at the end of t years is given by A Pe rt.

9 4-4 The Exponential Function with Base e 9 A Comparison of Exponential Growth Phenomena The equations and graphs given in Table 3 compare several widely used growth models. These are divided basically into two groups: unlimited growth and limited growth. Following each equation and graph is a short, incomplete list of areas in which the models are used. We have only touched on a subject that has been extensively developed and that you are likely to study in greater depth in the future. TABLE 3 Exponential Growth and Decay Description Equation Graph Uses Unlimited growth y ce kt c, k y Short-term population growth (people, bacteria, etc.); growth of money at continuous compound interest c t Exponential decay y ce kt c, k c y Radioactive decay; light absorption in water, glass, and the like; atmospheric pressure; electric circuits t Limited growth y c( e kt ) c, k c y Learning skills; sales fads; company growth; electric circuits t Logistic growth M y ce kt c, k, M M y Long-term population growth; epidemics; sales of new products; company growth t Answers to Matched Problems. y intercept: 3; x intercept:.83; increasing for all x; horizontal asymptote: y. (A) bacteria (B), bacteria mg 4. (A) 48 individuals; 33 individuals (B) minutes (C) N approaches an upper limit of 4, the number of people in the entire group.. Annually: $76.3; quarterly: $8.6; continuously: $8.

10 9 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS EXERCISE 4-4 A. Match each equation with the graph of f, g, m, or n in the figure. (A) y e.x (B) y e.6x (C) y e.4x f g 6 m (D) y e.x n B, and (/x) is greater than, so [ (/x)] x approaches infinity as x. (B) Which number does [ (/x)] x approach as x approaches? Before graphing the functions in Problems 7 6, classify each function as increasing or decreasing, find the x and y intercepts, and identify any asymptotes. Round any approximate values to two decimal places. Examine the graph to check your answers y e x 8. y e x 9. y e.x. y e.x. Match each equation with the graph of f, g, m, or n in the figure. (A) y e.x (B) y e.7x (C) y e.4x (D) y e.3x f 4 g 6 m n In Problems 3 8, compute answers to four significant digits. 3. e e 4. e e. e 6. e 7. e e 8. e e In Problems 9 4, simplify. 9. e x e 3x. (e x ) 4. (e x ) 3 e x e 4 3x. e 4x e 6x e x e x. (A) Explain what is wrong with the following reasoning about the expression [ (/x)] x : As x gets large, (/x) approaches because /x approaches, and raised to any power is, so [ /x] x approaches. (B) Which number does [ (/x)] x approach as x approaches? 6. (A) Explain what is wrong with the following reasoning about the expression [ (/x)] x : If b, then the exponential function b x approaches as x approaches 4. f(t) e.t. g(t) e.t 3. F(x) e x 4. G(x) e x 3. m(t) e 3t 6. n(t) 3 e t In Problems 7 3, describe the transformations that can be used to obtain the graph of g from the graph of f(x) e x (see Section -). Check your answers by graphing f and g in the same viewing window. 7. g(x) e x 8. g(x) e x 3 9. g(x) e x 3. g(x) e x 3. g(x) e (x ) 3. g(x).e (x ) In Problems 33 38, simplify. x 3 e x 3x e x x 4 e x 4x 3 e x (e x e x ) (e x e x ) 36. e x (e x ) e x (e x ) x 6 e x (e x e x ) e x (e x e x ) e x e x (e x e x ) (e x e x )e x e x In Problems 39 4, solve each equation. [Remember: e x for any real number x.] 39. xe x 4. (x 3)e x 4. x e x xe x 4. 3xe x x e x In Problems 43, use a graphing utility to find local extrema, y intercepts, and x intercepts. Investigate the behavior as x and as x and identify any horizontal x 8

11 4-4 The Exponential Function with Base e 93 asymptotes. Round any approximate values to two decimal places. 43. f(x) e x 44. g(x) 3 e x 4. m(x) e x 46. n(x) e x 47. s(x) e x 48. r(x) e x 49. F(x). G(x) 3e x e x C. Use a graphing utility to investigate the behavior of f(x) ( x) /x as x approaches.. Use a graphing utility to investigate the behavior of f(x) ( x) /x as x approaches. It is common practice in many applications of mathematics to approximate nonpolynomial functions with appropriately selected polynomials. For example, the polynomials in Problems 3 6, called Taylor polynomials, can be used to approximate the exponential function f(x) e x. To illustrate this approximation graphically, in each problem graph f(x) e x and the indicated polynomial in the same viewing window, 4 x 4 and y P (x) x x P (x) x x 6 x3 P 3 (x) x x 6 x3 4 x4 6. P 4 (x) x x 6 x3 4 x4 x 7. Investigate the behavior of the functions f (x) x/e x, f (x) x /e x, and f 3 (x) x 3 /e x as x and as x, and find any horizontal asymptotes. Generalize to functions of the form f n (x) x n /e x, where n is any positive integer. 8. Investigate the behavior of the functions g (x) xe x, g (x) x e x, and g 3 (x) x 3 e x as x and as x, and find any horizontal asymptotes. Generalize to functions of the form g n (x) x n e x, where n is any positive integer. APPLICATIONS 9. Population Growth. If the world population is about 6 billion people now and if the population grows continuously at an annual rate of.7%, what will the population be in years? Compute the answer to two significant digits. 6. Population Growth. If the population in Mexico is around million people now and if the population grows continuously at an annual rate of.3%, what will the population be in 8 years? Compute the answer to two significant digits. 6. Population Growth. In 996 the population of Russia was 48 million and the population of Nigeria was 4 million. If the populations of Russia and Nigeria grow continuously at annual rates of.6% and 3.%, respectively, when will Nigeria have a greater population than Russia? 6. Population Growth. In 996 the population of Germany was 84 million and the population of Egypt was 64 million. If the populations of Germany and Egypt grow continuously at annual rates of.% and.9%, respectively, when will Egypt have a greater population than Germany? 63. Space Science. Radioactive isotopes, as well as solar cells, are used to supply power to space vehicles. The isotopes gradually lose power because of radioactive decay. On a particular space vehicle the nuclear energy source has a power output of P watts after t days of use as given by P 7e.3t Graph this function for t. 64. Earth Science. The atmospheric pressure P, in pounds per square inch, decreases exponentially with altitude h, in miles above sea level, as given by P 4.7e.h Graph this function for h. 6. Marine Biology. Marine life is dependent upon the microscopic plant life that exists in the photic zone, a zone that goes to a depth where about % of the surface light still remains. Light intensity I relative to depth d, in feet, for one of the clearest bodies of water in the world, the Sargasso Sea in the West Indies, can be approximated by I I e.94d where I is the intensity of light at the surface. What percentage of the surface light will reach a depth of (A) feet? (B) feet? 66. Marine Biology. Refer to Problem 6. In some waters with a great deal of sediment, the photic zone may go down only to feet. In some murky harbors, the intensity of light d feet below the surface is given approximately by I I e.3d What percentage of the surface light will reach a depth of (A) feet? (B) feet? 67. Money Growth. If you invest $, in an account paying.38% compounded continuously, how much money will be in the account at the end of (A) 6. years? (B) 7 years? 68. Money Growth. If you invest $7, in an account paying 8.3% compounded continuously, how much money will be in the account at the end of (A). years? (B) years?

12 94 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS 69. Money Growth. Barron s, a national business and financial weekly, published the following Top Savings Deposit Yields for -year certificate of deposit accounts: Gill Savings Richardson Savings and Loan USA Savings 8.3% (CC) 8.4% (CQ) 8.% (CD) where CC represents compounded continuously, CQ compounded quarterly, and CD compounded daily. Compute the value of $, invested in each account at the end of years. 7. Money Growth. Refer to Problem 69. In another issue of Barron s, -year certificate of deposit accounts included: Alamo Savings Lamar Savings 8.% (CQ) 8.% (CC) Compute the value of $, invested in each account at the end of year. 7. Present Value. A promissory note will pay $3, at maturity years from now. How much should you be willing to pay for the note now if the note gains value at a rate of 9% compounded continuously? 7. Present Value. A promissory note will pay $, at maturity years from now. How much should you be will- ing to pay for the note now if the note gains value at a rate of % compounded continuously? 73. AIDS Epidemic. In June 996 the World Health Organization estimated that 7.7 million cases of AIDS (acquired immunodeficiency syndrome) had occurred worldwide since the beginning of the epidemic. Assuming that the disease spreads continuously at an annual rate of 7%, estimate the total number of AIDS cases that have occurred by June of the year (A) (B) AIDS Epidemic. In June 996 the World Health Organization estimated that 8 million people worldwide had been infected with HIV (human immunodeficiency virus) since the beginning of the AIDS epidemic. Assuming that HIV infection spreads continuously at an annual rate of 9%, estimate the total number of people who have been infected with HIV by June of the year (A) (B) 4 7. Learning Curve. People assigned to assemble circuit boards for a computer manufacturing company undergo on-the-job training. From past experience, it was found that the learning curve for the average employee is given by N 4( e.t ) where N is the number of boards assembled per day after t days of training. (A) How many boards can an average employee produce after 3 days of training? After days of training? Round answers to the nearest integer. (B) How many days of training will it take until an average employee can assemble boards a day? Round answer to the nearest integer. (C) Does N approach a limiting value as t increases without bound? Explain. 76. Advertising. A company is trying to expose a new product to as many people as possible through television advertising in a large metropolitan area with million potential viewers. A model for the number of people N, in millions, who are aware of the product after t days of advertising was found to be N ( e.37t ) (A) How many viewers are aware of the product after days? After days? Express answers as integers, rounded to three significant digits. (B) How many days will it take until half of the potential viewers will become aware of the product? Round answer to the nearest integer. (C) Does N approach a limiting value as t increases without bound? Explain. 77. Newton s Law of Cooling. This law states that the rate at which an object cools is proportional to the difference in temperature between the object and its surrounding medium. The temperature T of the object t hours later is given by T T m (T T m )e kt where T m is the temperature of the surrounding medium and T is the temperature of the object at t. Suppose a bottle of wine at a room temperature of 7 F is placed in the refrigerator to cool before a dinner party. If the temperature in the refrigerator is kept at 4 F and k.4, find the temperature of the wine, to the nearest degree, after 3 hours. (In Exercise 4-7 we will find out how to determine k.) 78. Newton s Law of Cooling. Refer to Problem 77. What is the temperature, to the nearest degree, of the wine after hours in the refrigerator? 79. Photography. An electronic flash unit for a camera is activated when a capacitor is discharged through a filament of wire. After the flash is triggered, and the capacitor is discharged, the circuit (see the figure) is connected and the battery pack generates a current to recharge the capacitor. The time it takes for the capacitor to recharge is called the recycle time. For a particular flash unit using a -volt battery pack, the charge q, in coulombs, on the capacitor t seconds after recharging has started is given by q.9( e.t )

13 4- Logarithmic Functions 9 Find the value that q approaches as t increases without bound and interpret. I 8. Medicine. An electronic heart pacemaker utilizes the same type of circuit as the flash unit in Problem 79, but it is designed so that the capacitor discharges 7 times a minute. For a particular pacemaker, the charge on the capacitor t seconds after it starts recharging is given by q. 8( e t ) Find the value that q approaches as t increases without bound and interpret. 8. Wildlife Management. A herd of white-tailed deer is introduced to a coastal island where there had been no deer before. Their population is predicted to increase according to the logistic curve N R V C S 4e.4t where N is the number of deer expected in the herd after t years. (A) How many deer will be present after years? After 6 years? Round answers to the nearest integer. (B) How many years will it take for the herd to grow to deer? Round answer to the nearest integer. (C) Does N approach a limiting value as t increases without bound? Explain. 8. Training. A trainee is hired by a computer manufacturing company to learn to test a particular model of a personal computer after it comes off the assembly line. The learning curve for an average trainee is given by N 4 e.t (A) How many computers can an average trainee be expected to test after 3 days of training? After 6 days? Round answers to the nearest integer. (B) How many days will it take until an average trainee can test 3 computers per day? Round answer to the nearest integer. (C) Does N approach a limiting value as t increases without bound? Explain. Section 4- Logarithmic Functions Definition of Logarithmic Function From Logarithmic Form to Exponential Form, and Vice Versa Properties of Logarithmic Functions Definition of Logarithmic Function We now define a new class of functions, called logarithmic functions, as inverses of exponential functions. Since exponential functions are one-to-one, their inverses exist. Here you will see why we placed special emphasis on the general concept of inverse functions in Section 4-. If you know quite a bit about a function, then, based on a knowledge of inverses in general, you will automatically know quite a bit about its inverse. For example, the graph of f is the graph of f reflected across the line y x, and the domain and range of f are, respectively, the range and domain of f. If we start with the exponential function f: y x and interchange the variables x and y, we obtain the inverse of f: f : x y The graphs of f, f, and the line y x are shown in Figure. This new function is given the name logarithmic function with base. Since we cannot solve