-3 Eponential Functions 7 87. Business Markup Policy. A bookstore sells a book with a wholesale price of $6 for $9.9 and one with a wholesale price of $0 for $.9. (A) If the markup policy for the store is assumed to be linear, find a function r m(w) that epresses the retail price r as a function of the wholesale price w. (B) Describe the function m verbally. (C) Find w m (r). (D) Describe the function m verbally. 88. Flight Conditions. In stable air, the air temperature drops about F for each,000-foot rise in altitude. (A) If the temperature at sea level is 63 F and a pilot reports a temperature of F at,000 feet, find a linear function t d(a) that epresses the temperature t in terms of the altitude a (in thousands of feet). (B) Find a d (t). Section -3 Eponential Functions Eponential Functions Basic Eponential Graphs Additional Eponential Properties Applications In this section we define eponential functions, look at some of their important properties including graphs and consider several significant applications. Eponential Functions Let s start by noting that the functions f and g given by f() and g() FIGURE f(). are not the same function. Whether a variable appears as an eponent with a constant base or as a base with a constant eponent makes a big difference. The function g is a quadratic function, which we have already discussed. The function f is a new type of function called an eponential function. The values of the eponential function f() for an integer are easy to compute [Fig. (a)]. If m/n is a rational number, then f(m/n) n m, which can be evaluated on almost any calculator [Fig. (b)]. Finally, a graphing utility can graph the function f() [Fig. (c)] for any given interval of values. 0 (a) (b) (c)
7 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS The only catch is that we have not yet defined for all real numbers. For eample, what does mean? The question is not easy to answer at this time. In fact, a precise definition of must wait for more advanced courses, where we can show that, if b is a positive real number and is any real number, then b names a real number, and the graph of f () is as indicated in Figure. We also can show that for irrational, b can be approimated as closely as we like by using rational number approimations for. Since.3..., for eample, the sequence.,.,.,... approimates improves., and as we use more decimal places, the approimation DEFINITION EXPONENTIAL FUNCTION The equation f() b b 0, b defines an eponential function for each different constant b, called the base. The independent variable may assume any real value. Thus, the domain of f is the set of all real numbers, and it can be shown that the range of f is the set of all positive real numbers. We require the base b to be positive to avoid imaginary numbers such as ( ) /. Basic Eponential Graphs Eplore/Discuss Compare the graphs of f() 3 and g() by graphing both functions in the same viewing window. Find all points of intersection of the graphs. For which values of is the graph of f above the graph of g? Below the graph of g? Are the graphs of f and g close together as? As? Discuss. It is useful to compare the graphs of y and y ( ) by plotting both on the same coordinate system, as shown in Figure (a). The graph of f() b b Fig. (b)
-3 Eponential Functions 73 looks very much like the graph of the particular case y, and the graph of f() b 0 b Fig. (b) FIGURE Basic eponential graphs. looks very much like the graph of y a horizontal asymptote for the graph. ( ) y y. Note in both cases that the ais is 8 6 y y y b 0 b y b b DOMAIN (, ) RANGE (0, ) (a) (b) The graphs in Figure suggest the following important general properties of eponential functions, which we state without proof: BASIC PROPERTIES OF THE GRAPH OF f() b, b 0, b. All graphs pass through the point (0, ). b 0 for any permissible base b.. All graphs are continuous, with no holes or jumps. 3. The ais is a horizontal asymptote.. If b, then b increases as increases.. If 0 b, then b decreases as increases. 6. The function f is one-to-one. Property 6 implies that an eponential function has an inverse, called a logarithmic function, which we discuss in Section -. Graphing eponential functions on a graphing utility is routine, but interpreting the results requires an understanding of the preceding properties. EXAMPLE Solution Graphing Eponential Functions Let f() ( ). Construct a table of values (rounded to two decimal places) for f() using integer values from 3 to 3. Graph f on a graphing utility and then sketch a graph by hand. Set the graphing utility in two-decimal-place mode, construct the table [Fig. 3(a)], and graph the function [Fig. 3(b)]. The points on the graph of f() for 0 are indistinguishable from the ais in Figure 3(b). However, from the properties of an eponential function, we know that f() 0 for all real numbers and that
7 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS f() 0 as. The hand sketch in Figure 3(c) illustrates the behavior for 0 more clearly. Of course, zooming in on the graphing utility will also illustrate this behavior. 0 y 30 0 0 0 3 FIGURE 3 (a) (b) (c) MATCHED PROBLEM ( ) Repeat Eample for y ( ). Additional Eponential Properties Eponential functions whose domains include irrational numbers obey the familiar laws of eponents for rational eponents (see Appendi A). We summarize these eponent laws here and add two other important and useful properties. EXPONENTIAL FUNCTION PROPERTIES For a and b positive, a, b, and and y real:. Eponent laws: a a y a y (a ) y a y (ab) a b a b a b a y a y a 7 7. a a y if and only if y. If 6 6, then, and. 3. For 0, a b if and only if a b. If a 3, then a 3.
-3 Eponential Functions 7 EXAMPLE Solution Using Eponential Function Properties Solve 3 8 for. Epress both sides in terms of the same base, and use property to equate eponents. 3 8 ( ) 3 3 6 3 6 3 9 Epress and 8 as powers of. (a ) y a y Property 9 Check (9/) 3 3/ ( ) 3 3 8 MATCHED PROBLEM Solve 7 9 for. Applications We now consider three applications that utilize eponential functions in their analysis: population growth, radioactive decay, and compound interest. Population growth and compound interest are eamples of eponential growth, while radioactive decay is an eample of negative eponential growth. Our first eample involves the growth of populations, such as people, animals, insects, and bacteria. Populations tend to grow eponentially and at different rates. A convenient and easily understood measure of growth rate is the doubling time that is, the time it takes for a population to double. Over short periods of time the doubling time growth model is often used to model population growth: P P 0 t/d where P Population at time t P 0 Population at time t 0 d Doubling time Note that when t d, P P 0 d/d P 0 and the population is double the original, as it should be. We use this model to solve a population growth problem in Eample 3.
76 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS EXAMPLE 3 Solutions Population Growth Meico has a population of around 00 million people, and it is estimated that the population will double in years. If population growth continues at the same rate, what will be the population: (A) years from now? (B) 30 years from now? Calculate answers to 3 significant digits. We use the doubling time growth model: P P 0 t/d FIGURE P 00( t/ ). P (millions) Substituting P 0 00 and d, we obtain P 00( t/ ) See Figure. 00 00 300 (A) Find P when t years: P 00( / ) 6 million people 00 00 0 0 30 0 Years 0 t (B) Find P when t 30 years: P 00( 30/ ) 69 million people MATCHED PROBLEM 3 The bacterium Escherichia coli (E. coli) is found naturally in the intestines of many mammals. In a particular laboratory eperiment, the doubling time for E. coli is found to be minutes. If the eperiment starts with a population of,000 E. coli and there is no change in the doubling time, how many bacteria will be present: (A) In 0 minutes? (B) In hours? Write answers to three significant digits. Eplore/Discuss The doubling time growth model would not be epected to give accurate results over long periods of time. According to the doubling time growth model of Eample 3, what was the population of Meico 00 years ago at the height of Aztec civilization? What will the population of Meico be 00 years from now? Eplain why these results are unrealistic. Discuss factors that affect human populations which are not taken into account by the doubling time growth model.
-3 Eponential Functions 77 Our second application involves radioactive decay, which is often referred to as negative growth. Radioactive materials are used etensively in medical diagnosis and therapy, as power sources in satellites, and as power sources in many countries. If we start with an amount A 0 of a particular radioactive isotope, the amount declines eponentially in time. The rate of decay varies from isotope to isotope. A convenient and easily understood measure of the rate of decay is the half-life of the isotope that is, the time it takes for half of a particular material to decay. In this section we use the following half-life decay model: A A 0 ( )t/h A 0 t/h where A Amount at time t A 0 Amount at time t 0 h Half-life Note that when t h, A A 0 h/h A 0 A 0 and the amount of isotope is half the original amount, as it should be. EXAMPLE Solutions Radioactive Decay The radioactive isotope gallium 67 ( 67 Ga), used in the diagnosis of malignant tumors, has a biological half-life of 6. hours. If we start with 00 milligrams of the isotope, how many milligrams will be left after (A) hours? (B) week? Compute answers to three significant digits. We use the half-life decay model: A A 0 ( )t/h A 0 t/h FIGURE A 00( t/6. ). A (milligrams) Using A 0 00 and h 6., we obtain A 00( t/6. ) See Figure. 00 0 (A) Find A when t hours: A 00( /6. ) 69.9 milligrams 00 Hours 00 t (B) Find A when t 68 hours ( week 68 hours): A 00( 68/6. ) 8.7 milligrams
78 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS MATCHED PROBLEM Radioactive gold 98 ( 98 Au), used in imaging the structure of the liver, has a half-life of.67 days. If we start with 0 milligrams of the isotope, how many milligrams will be left after: (A) day? (B) week? Compute answers to three significant digits. Our third application deals with the growth of money at compound interest. This topic is important to most people and is fundamental to many topics in the mathematics of finance. The fee paid to use another s money is called interest. It is usually computed as a percentage, called the interest rate, of the principal over a given period of time. If, at the end of a payment period, the interest due is reinvested at the same rate, then the interest earned as well as the principal will earn interest during the net payment period. Interest paid on interest reinvested is called compound interest. Suppose you deposit $,000 in a savings and loan that pays 8% compounded semiannually. How much will the savings and loan owe you at the end of years? Compounded semiannually means that interest is paid to your account at the end of each 6-month period, and the interest will in turn earn interest. The interest rate per period is the annual rate, 8% 0.08, divided by the number of compounding periods per year,. If we let A, A, A 3, and A represent the new amounts due at the end of the first, second, third, and fourth periods, respectively, then A $,000 $,000 0.08 $,000( 0.0) A A ( 0.0) [$,000( 0.0)]( 0.0) $,000( 0.0) A 3 A ( 0.0) [$,000( 0.0) ]( 0.0) $,000( 0.0) 3 A A 3 ( 0.0) [$,000( 0.0) 3 ]( 0.0) $,000( 0.0) What do you think the savings and loan will owe you at the end of 6 years? If you guessed A $,000( 0.0) P r n P r n P r n 3 P r n you have observed a pattern that is generalized in the following compound interest formula:
-3 Eponential Functions 79 COMPOUND INTEREST If a principal P is invested at an annual rate r compounded m times a year, then the amount A in the account at the end of n compounding periods is given by A P r m n The annual rate r is epressed in decimal form. EXAMPLE Solution Compound Interest If you deposit $,000 in an account paying 9% compounded daily, how much will you have in the account in years? Compute the answer to the nearest cent. We use the compound interest formula with P,000, r 0.09, m 36, and n (36),8: A P r m n FIGURE 6,000,000 0.09 36 8 $7,8.3 The graph of 0 3,60 A,000 0.09 36 0 is shown in Figure 6. MATCHED PROBLEM If $,000 is invested in an account paying 0% compounded monthly, how much will be in the account at the end of 0 years? Compute the answer to the nearest cent. EXAMPLE 6 Comparing Investments If $,000 is deposited into an account earning 0% compounded monthly and, at the same time, $,000 is deposited into an account earning % compounded monthly, will the first account ever be worth more than the second? If so, when?
80 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS Solution Let y and y represent the amounts in the first and second accounts, respectively, then y,000( 0.0/) y,000( 0.0/) where is the number of compounding periods (months). Eamining the graphs of y and y [Fig. 7(a)], we see that the graphs intersect at 39.38 months. Since compound interest is paid at the end of each compounding period, we compare the amount in the accounts after 39 months and after 0 months [Fig. 7(b)]. Thus, the first account is worth more than the second for 0 months or years and 8 months. FIGURE 7,000 0 0 0 (a) (b) MATCHED PROBLEM 6 If $,000 is deposited into an account earning 0% compounded quarterly and, at the same time, $,000 is deposited into an account earning 6% compounded quarterly, when will the first account be worth more than the second? Answers to Matched Problems. y ( ) y 3 3.00 8.00.00 0 0.0 0.3 0.03 3 0.0 y 0 30 0 0 3. 3. (A),30 (B),00,000.0 0 6 3. (A) 3.9 mg (B) 8. mg. $,707.0 6. After 3 quarters
-3 Eponential Functions 8 EXERCISE -3 A. Match each equation with the graph of f, g, m, or n in the figure. (A) y (0.) (B) y (C) y ( 3 ) (D) y f g 6 m n In Problems 9, simplify. 9. 0 3 0 0. ( 3 ) y.. 3.. ( 3 y ) z B 3 y 3z In Problems 36, solve for. 3 3. 3 6. 0 3 0 6 7. 7 7 3 8. 6 9. ( ) ( ) 30. 3 ( ) 3 3. 3. 9 3. Match each equation with the graph of f, g, m, or n in the figure. (A) y (B) y (0.) (C) y 3 f 0 g 6 m n 0 (D) y In Problems 3 8, compute answers to four significant digits. 3. 3. 3. 6. 3 7. 3 3 8. ( ) 33. 3. 00,000 3. 9 3 3 36. 3 37. Find all real numbers a such that a a. Eplain why this does not violate the second eponential function property in the bo on page 7. 38. Find real numbers a and b such that a b but a b. Eplain why this does not violate the third eponential function property in the bo on page 7. 39. Eamine the graph of y on a graphing utility and eplain why cannot be the base for an eponential function. 0. Eamine the graph of y 0 on a graphing utility and eplain why 0 cannot be the base for an eponential function. [Hint: Turn the aes off before graphing.] Graph each function in Problems 8 using the graph of f shown in the figure. f() Before graphing the functions in Problems 9 8, classify each function as increasing or decreasing, find the y intercept, and identify any asymptotes. Then eamine the graph to check your answers. 9. y 3 0. y. y (. y ( ) 3 ) 3 3. g() 3. f(). h() (3 ) 6. f() ( ) 7. y 3 3 8. y. y f(). y f() 3. y f( ). y f( ). y f() 6. y 3 f() 7. y 3f( ) 8. y f( )
8 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS In Problems 9 6, sketch the graph of a function of the form f() a(b ) c that satisfies the given conditions. 9. f is an increasing function, asymptotic to the ais, and satisfying f( ) 0., f(0), and f(0) 6. 0. f is an increasing function, asymptotic to the ais, and satisfying f( ) 0., f(0), and f(3) 8.. f is a decreasing function, asymptotic to the ais, and satisfying f( ), f(0) 6, and f() 0.37.. f is a decreasing function, asymptotic to the ais, and satisfying f( ) 3, f(0), and f() 0.. 3. f is a decreasing function, asymptotic to the line y 3, and satisfying f( 0).7, f(0), and f(0).. f is an increasing function, asymptotic to the line y, and satisfying f( 0) 0, f(0) 3, and f(0) 3.7.. f is an increasing function, asymptotic to the line y, and satisfying f( ).7, f(0), and f(). 6. f is a decreasing function, asymptotic to the line y, and satisfying f( ), f(0), and f().7. C In Problems 7 60, simplify. 7. (6 6 )(6 6 ) 8. (3 3 )(3 3 ) 9. (6 6 ) (6 6 ) 60. (3 3 ) (3 3 ) In Problems 6 6, use a graphing utility to approimate local etrema and intercepts to two decimal places. Investigate the behavior as and as and identify any horizontal asymptotes. 6. m() (3 ) 6. h() 3( ) 63. 6. f() g() 3 3 APPLICATIONS 6. Gaming. A person bets on red and black on a roulette wheel using a Martingale strategy. That is, a $ bet is placed on red, and the bet is doubled each time until a win occurs. The process is then repeated. If black occurs n times in a row, then L n dollars is lost on the nth bet. Graph this function for n 0. Even though the function is defined only for positive integers, points on this type of graph are usually joined with a smooth curve as a visual aid. 66. Bacterial Growth. If bacteria in a certain culture double every hour, write an equation that gives the number of bacteria N in the culture after t hours, assuming the culture has 00 bacteria at the start. Graph the equation for 0 t. 67. Population Growth. Because of its short life span and frequent breeding, the fruit fly Drosophila is used in some genetic studies. Raymond Pearl of Johns Hopkins University, for eample, studied 300 successive generations of descendants of a single pair of Drosophila flies. In a laboratory situation with ample food supply and space, the doubling time for a particular population is. days. If we start with male and female flies, how many flies should we epect to have in (A) week? (B) weeks? 68. Population Growth. If Kenya has a population of about 30,000,000 people and a doubling time of 9 years and if the growth continues at the same rate, find the population in (A) 0 years (B) 30 years Compute answers to two significant digits. 69. Insecticides. The use of the insecticide DDT is no longer allowed in many countries because of its long-term adverse effects. If a farmer uses pounds of active DDT, assuming its half-life is years, how much will still be active after (A) years? (B) 0 years? Compute answers to two significant digits. 70. Radioactive Tracers. The radioactive isotope technetium 99m ( 99m Tc) is used in imaging the brain. The isotope has a half-life of 6 hours. If milligrams are used, how much will be present after (A) 3 hours? (B) hours? Compute answers to three significant digits. 7. Finance. Suppose $,000 is invested at % compounded weekly. How much money will be in the account in (A) year? (B) 0 years? Compute answers to the nearest cent. 7. Finance. Suppose $,00 is invested at 7% compounded quarterly. How much money will be in the account in 3 (A) year? (B) years? Compute answers to the nearest cent.
- The Eponential Function with Base e 83 73. Finance. A couple just had a new child. How much should they invest now at 8.% compounded daily in order to have $0,000 for the child s education 7 years from now? Compute the answer to the nearest dollar. 7. Finance. A person wishes to have $,000 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 $3,000 is deposited into an account earning 8% compounded daily and, at the same time, $,000 is deposited into an account earning % compounded daily, will the first account be worth more than the second? If so, when? 76. Finance. If $,000 is deposited into an account earning 9% compounded weekly and, at the same time, $6,000 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 $0,000 at 8.9% compounded daily ever be worth more at the end of a quarter than an investment of $0,000 at 9% compounded quarterly? Eplain. 78. Finance. A sum of $,000 is invested at 3% compounded semiannually. Suppose that a second investment of $,000 is made at interest rate r compounded daily. For which values of r, to the nearest tenth of a percent, is the second investment better than the first? Discuss. Problems 79 and 80 require a graphing utility that can compute eponential regression equations of the form y ab (consult your manual). 79. Depreciation. Table gives the market value of a minivan (in dollars) years after its purchase. Find an eponential regression model of the form y ab for this data set. Estimate the purchase price of the van. Estimate the value of the van 0 years after its purchase. Round answers to the nearest dollar. 3 6 80. Depreciation. Table gives the market value of a luury sedan (in dollars) years after its purchase. Find an eponential regression model of the form y ab for this data set. Estimate the purchase price of the sedan. Estimate the value of the sedan 0 years after its purchase. Round answers to the nearest dollar. 3 6 T A B L E T A B L E Value ($),7 9, 8, 6,8,,8 Value ($) 3, 9,00,6,87 9,0 7, Section - The Eponential Function with Base e Base e Eponential Function Growth and Decay Applications Revisited Continuous Compound Interest A Comparison of Eponential 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.