Exploratory Problems for Chapter 9 331 (a) If the population has been increasing linearly, was the population in 1980 equal to 150,000, greater than 150,000, or less than 150,000? Explain your reasoning. (b) If the population has been increasing exponentially, was the population in 1980 equal to 150,000, greater than 150,000, or less than 150,000? Explain your rea- soning. Note: Your answers to parts (a) and (b) should be different! 5. Let D(t), H(t),and J(t)represent the annual salaries (in dollars) of David, Henry, and Jennifer, and suppose that these functions are given by the following formulas, where t is in years. t = 0 corresponds to this year’s salary, t = 1 to the salary one year from now, and so on. The domain of each function is t = 0, 1, 2, up to retirement. D(t) = 40,000 + 2500t H(t) =50,000(0.97) t J(t)=40,000(1.05) t (a) Describe in words how each employee’s salary is changing. (b) Suppose you are just four years away from retirement—you’ll collect a salary for four years, including the present year. Which person’s situation would you prefer to be your own? (c) If you are in your early twenties and looking forward to a long future with the company, which would you prefer? 6. In Anton Chekov’s play “Three Sisters,” Lieutenant-Colonel Vershinin says the fol- lowing in reply to Masha’s complaint that much of her knowledge is unnecessary. “I don’t think there can be a town so dull and dismal that intelligent and educated peo- ple are unnecessary in it. Let us suppose that of the hundred thousand people living in this town, which is, of course, uncultured and behind the times, there are only three of your sort Life will get the better of you, but you will not disappear without a trace. After you there may appear perhaps six like you, then twelve and so on until such as you form a majority. In two or three hundred years life on earth will be unimaginably beautiful, marvelous. Man needs such a life and, though he hasn’t it yet, he must have a presentiment of it, expect it, dream it, prepare for it; for that he must know more than his father and grandfather. And you complain about knowing a great deal that is unnecessary.” Let us assume that Vershinin means that this doubling occurs every generation and take a generation to be 25 years. Suppose that the total population of the town remains unchanged. (a) In approximately how many years will the people “such as [Masha] form a major- ity”? (b) What percentage of the town will be “intelligent and educated” in the 200 years that Vershinin mentions? (c) Now assume that the total population grows at a rate of 2% per year. Answer questions (a) and (b) with this new assumption. 7. Many trainers recommend that at the start of the season, a cyclist should increase his or her weekly mileage by not more than 15% each week. 332 CHAPTER 9 Exponential Functions (a) If a cyclist maintains a “base” of 50 miles per week during the winter, what is his or her maximum recommended weekly mileage for the fifth week of the season? (b) Find a formula for M(w), the maximum weekly recommended mileage w weeks into the season. Assume that initially the cyclist has a base of A miles per week. 8. Pasteurized milk is milk that has been heated enough to kill pathogenic bacteria. Pasteurization of milk is widespread because unpasteurized milk provides a good environment for bacterial growth. For example, tuberculosis can be transmitted from an infected cow to a human via unpasteurized milk. Mycobacterium tuberculosis has a doubling time of 12 to 16 hours. If a pail of milk contains 10 M. tuberculosis bacteria, after approximately how many hours should we expect there to be 1000 bacteria? Give a time interval. (Facts from The New Encylcopedia Britannica, 1993, volume 14, p. 581.) 9. According to fire officials, a 1996 fire on the Warm Springs Reservation in central Oregon tripled in size to 65,000 acres in one day. A fire in Upper Lake, California, quintupled in size to 10,200 acres in one day. (a) Assuming exponential growth, determine the doubling time for each fire. (b) What was the hourly percentage growth of each fire? 10. During the decade from1985 to 1995, Harvard’s average return on financial investments in its endowment was 11.1% per year. Over the same period, Yale’s total return on its investments was 287.3%. (Boston Globe, July 26, 1996.) Let’s assume both Harvard and Yale’s endowments are growing exponentially. (a) What was Harvard’s total return over this ten-year period? (b) What was Yale’s average annual rate of return? (c) Which school got the higher return on its investments? (d) What was the doubling time for each school’s investments? (e) In 1995, Harvard’s endowment was approximately $8 billion. What was its in- stantaneous rate of growth (from investment only, ignoring new contributions)? Include units in your answer. 11. (a) If rabbits grow according to R(t) = 1010(2) t/3 , t in years, after how many years does the rabbit population double? What is the percent increase in growth each year? (b) If the sheep population in Otrahonga, New Zealand, is growing according to S(t) = 3162(1.065) t , t in years, after approximately how many years does the sheep population double? What is the percent increase in growth each year? 12. Exploratory: Which grows faster, 2 x or x 2 ? (a) Using what you know about these two functions and experimenting numerically and graphically, guess the following limits: i. lim x→∞ x2 −x ii. lim x→−∞ x 2 2 x iii. lim x→∞ x 2 2 x iv. lim x→∞ 2 x x 2 (b) For |x| large, which function is dominant, 2 x or x 2 ? Would you have answered differently if we looked at 3 x and x 3 instead? Exploratory Problems for Chapter 9 333 13. Devaluation: Due to inflation, a dollar loses its purchasing power with time. Suppose that the dollar loses its purchasing power at a rate of 2% per year. (a) Find a formula that gives us the purchasing power of $1 t years from now. (b) Use your calculator to approximate the number of years it will take for the pur- chasing power of the dollar to be cut in half. 14. A population of beavers is growing exponentially. In June 1993 (our benchmark year when t = 0) there were 100 beavers. In June 1994 (t = 1) there were 130 beavers. (a) Write a function B(t) that gives the number of beavers at time t. (b) What is the percent increase in the beaver population from one year to the next? 15. We are given two data points for the cumulative number of people who have graduated from a newly established flying school, a school for training pilots. Our benchmark time, t = 0, is one year after the school opened. When t = 0, the number of people who have graduated = 25. When t = 3, the number of people who have graduated = 127. Find the cumulative number of people who have graduated at time t = 5if (a) the cumulative number of people who have graduated is a linear function of time. (b) the cumulative number of people who have graduated is an exponential function of time. 16. According to a report from the General Accounting Office, during the 14-year period between the school year 1980–1981 and the school year 1994–1995, the average tuition at four-year public colleges increased by 234%. During the same period, average household income increased by 82%, and the Labor Department’s Consumer Price Index (CPI) increased by 74%. (Boston Globe, August 16, 1996.) (a) Assuming exponential growth, determine the annual percentage increase for each of these three measures. (b) The average cost of tuition in 1994–1995 was $2865 for in-state students. What was it in 1980–1981? (c) Starting with an initial value of one unit for each of the three quantities, average tuition at four-year public colleges, average household income, and the Consumer Price Index, sketch on a single set of axes the graphs of the three functions over this 14-year period. (d) Suppose that a family has two children born 14 years apart. In 1980–1981, the tuition cost of sending the elder child to college represented 15% of the family’s total income. Assuming that their income increased at the same pace as the average household, what percent of their income was needed to send the younger child to college in 1994–1995? 17. Suppose that in a certain scratch-ticket lottery game, the probability of winning with the purchase of one card is 1 in 500, or 0.2%; hence, the probability of losing is 100% − 0.2% = 99.8%. But what if you buy more than one ticket? One way to cal- culate the probability that you will win at least once if you buy n tickets is to subtract from 100% the probability that you will lose on all n cards. This is an easy calcu- lation; the probability that you will lose two times in a row is (99.8%)(99.8%) = 334 CHAPTER 9 Exponential Functions 99.6004%, so the probability that you will win at least once if you play two times is 1 − (99.8%)(99.8%) = 0.3996%. (a) What is the probability that you will win at least once if you play three times? (b) Find a formula for P (n), the percentage chance of winning at least once if you play the game n times. (c) How many tickets must you buy in order to have a 25% chance of winning? A 50% chance? (d) Does doubling the number of tickets you buy also double your chances of winning? (e) Sketch a graph of P (n). Use [0, 100] as the range of the graph. Explain the practical significance of any asymptotes. 9.4 THE DERIVATIVE OF AN EXPONENTIAL FUNCTION As an exploratory problem you investigated the derivatives of 2 x ,3 x ,and 10 x by numerically approximating the derivatives at various points. Below are tables of values of approxima- tions to the derivatives of 2 x ,3 x ,and 10 x . 13 f(x)=2 x g(x) = 3 x h(x) = 10 x xf(x)f  (x) x g(x) g  (x) x h(x) h  (x) 0 1 0.693 0 1 1.099 0 1 2.305 1 2 1.386 1 3 3.298 1 10 23.052 2 4 2.774 2 9 9.893 2 100 230.524 3 8 5.547 3 27 29.679 3 1000 2305.238 4 16 11.094 4 81 89.036 4 10,000 23052.381 Based on the data gathered, we can conjecture that f  (x) ≈ (0.693)2 x , g  (x) ≈ (1.099)3 x , h  (x) ≈ (2.305)10 x . Observation In each case the derivative of the exponential function at any point appears to be proportional to the value of the function at that point. Furthermore, the proportionality constant appears to be the derivative of the function at x = 0. f  (x) ≈ f  (0) · 2 x g  (x) ≈ g  (0) · 3 x h  (x) ≈ h  (0) · 10 x 13 The approximations to f  (c) were made with f(c+h)−f(c) h for h = 0.001. In each case the following was used: b c+h − b c h = b c  b h − 1 h  . The second term was calculated, multiplied by the first, and rounded off after three decimal places. You may have done this slightly differently, but our answers ought to be in the same ballpark. 9.4 The Derivative of an Exponential Function 335 This leads us to conjecture that, more generally, if f(x)=b x ,then f  (x) = f  (0) · b x . Conjecture If f(x)=b x ,then f  (x) = (the slope of the tangent line to b x at x = 0) · b x . All this is simply conjecture. The data we have gathered are based purely on numerical approximation; we have done only a few test cases with a few bases. The conjecture agrees with what we know about exponential functions. For example, if a population is growing exponentially, then we expect its rate of growth (its derivative) to be proportional to the size of the population (the value of the function itself.) Conjectures are wonderful; mathematicians are continually looking for patterns and making conjectures. After experimenting and conjecturing, a mathematician is interested in trying to prove his or her conjectures. Let’s go back to the limit definition of derivative and see if we can prove our conjecture. Proof of Conjecture Let f(x)=b x ,where b is a positive constant. Consider f  (c) for any real number c. f  (c) = lim h→0 f(c+h) − f(c) h f  (c) = lim h→0 b c+h − b c h f  (c) = lim h→0 b c · b h − b c h = lim h→0 b c  b h − 1 h  . As h goes to zero b c is unaffected, so f  (c) = b c  lim h→0 (b h − 1) h  . Notice that lim h→0 (b h − 1) h = lim h→0 (b h − b 0 ) h = lim h→0 f (h) − f(0) h − 0 . This is precisely the definition of f  (0), the derivative of b x at x = 0. How delightful! We have now proven our conjecture. If f(x)=b x , then f  (x) = f  (0) · b x = f  (0) · f(x). We can approximate the derivatives at zero numerically, as done in the exploratory problem. Our results stand as d dx 2 x ≈ (0.693)2 x , d dx 3 x ≈ (1.098)3 x , d dx 10 x ≈ (2.302)10 x . 336 CHAPTER 9 Exponential Functions We have proven that d dx b x = αb x , where α = the slope of the tangent to the graph of b x at x = 0. It follows that d dx Cb x = αCb x . Regardless of whether we are looking at average or at instantaneous rates of change, we can state the following. Exponential functions grow at a rate proportional to themselves. Exponential functions have a constant percent change. Question: Is there a base “b” such that the derivative of b x is b x ? This question asks us to look for a function whose derivative is, itself, a function f such that the slope of the graph of f at any point (x, y) is given simply by the y-coordinate. If we can find a base “b” such that the slope of the tangent line to b x at x = 0 is 1, then we have found such a function. Since the slope of b x at x = 0 increases as b increases, based on the data we’ve collected we posit the existence of such a number. Graphically and numerically it seems reasonable to believe such a number exists. We begin to look for such a base between 2 and 3 because we know that d dx 2 x ≈ (0.693)2 x , and d dx 3 x = (1.099) · 3 x . Some numerical experimentation allows us to narrow in on a value between 2.71 and 2.72. Definition We define the number e to be the base such that the slope of the tangent line to e x at x = 0 is 1. In other words, we define e to be the number such that d dx e x = e x . The number e is between 2.71 and 2.72. Later we will pin down the size of e more closely. Given that d dx e x = e x , we can find the derivative of e 2x and e 3x using the Product Rule. We know that e 2x = e x · e x and e 3x = e x · e 2x .  e 2x   = e x · e x + e x · e x = 2e x · e x = 2e 2x  e 3x   = e x · 2e 2x + e x · e 2x = 3e x · e 2x = 3e 3x Do you notice a pattern? The mathematician in you wants to generalize. In fact, using induction, we can show that this pattern holds for any integer k: d dx e kx = ke kx . Partitioning the problem into cases facilitates this generalization. Proof that d dx e kx = ke kx for any positive integer k Our statement (e kx )  = ke kx holds for k = 1. We need to show that if d dx e nx = ne nx , then d dx e (n+1)x = (n + 1)e (n+1)x . d dx e (n+1)x = d dx  e nx · e x  = ne nx · e x + e nx · e x = (n + 1)e nx · e x = (n + 1)e (n+1)x 9.4 The Derivative of an Exponential Function 337 Therefore, our statement holds true for any positive integer and the proof is complete. Check on your own that this statement holds true for the case k =0. Using the Quotient Rule, we can show that this works for any negative integer as well. Consider the function f(x)=e −kx , where k is positive. We can rewrite the function as f(x)= 1 e kx and apply the Quotient Rule. d dx  1 e kx  = e kx ·0 −1 ·ke kx  e kx  2 = −ke kx e 2kx =−ke kx−2kx =−ke −kx Therefore, d dx e kx =ke kx for any integer k. Answers to Selected Exercises Exercise 9.3 Answers i. 3 √ −8 = 1 2 . Instead 3 √ −8 3 √ x −2 =−2·x −2/3 ,not 1 2 x −2/3 . ii. 1 2 −3 = 2 −3 . We should have B −6 C 3 2 −3 = 2 3 C 3 B 6 = 8C 3 B 6 . iii. √ 4  x 2 + 4y 2 = 2  x 2 + 4y 2 . This cannot be simplified.  x 2 + 4y 2 = x +2y. Try squaring the latter if you’re unconvinced. iv. (AD) n +(CB) n BD n is as far as we can go. (AD) n + (CB) n = (AD + CB) n (except for n = 1) v. 2  1 x + 1 y  −1 = 2  y xy + x xy  −1 = 2  y+x xy  −1 = 2xy y+x The error is that  1 x + 1 y  −1 =x + y. For instance,  1 2 + 1 2  −1 =1 −1 =1, not 2 + 2 = 4. Exercise 9.4 Answers i. 2 .5 x 1.5 2 −.5 x −.5 = 2 .5−(−.5) · x 1.5−(−.5) = 2 .5+.5 · x 1.5+.5 = 2x 2 ii. 4(9y −x ) 1/2 = 4  9 y x = 4·3 √ y x = 12 y x/2 or 12y −x/2 iii. b x+w −b x b x = b x ·b w −b x b x = b x (b w −1) b x = b w − 1. iv. ( 3 √ 64x 3 ) 1/2  1 2  −1 = (4x 3 ) 1/2 (2 −1 ) −2 = 2x 3/2 2 2 = x 3/2 2 v. Q 3 R +Q R +1 Q 2R = Q R+2R +Q R+1 Q R+R = Q R (Q 2R +Q) Q R Q R = Q 2R +Q Q R or Q 2R Q R + Q Q R = Q R + Q 1−R Exercise 9.6 Answers About $43.10 more if interest is compounded quarterly Exercise 9.8 Answers (a) C(t) =C 0 (0.5) t/5730 (b) ≈ 2948.5 years ago 338 CHAPTER 9 Exponential Functions Exercise 9.9 Answers i. C(t) = C 0 (0.8) t = C 0 0.8 t ii. C(t) = C 0 (0.9) 2t = C 0 0.81 t iii. C(t) = C 0 (0.951) 4t ≈ C 0 0.81794 t iv. C(t) = C 0 (0.4) t/4 ≈ C 0 0.795 t The most efficient system results in the smallest number of contaminants; 60% every 4 hours is therefore the most efficient. PROBLEMS FOR SECTION 9.4 1. Let g(t) = 3 5t . Show that g(t + h) − g(t) h = g(t) · g(h) − g(0) h . Some tips: (i) Write out the equation using the actual function. (ii) Now your job is to make the left and right sides look the same. Use the laws of exponents to do this. 2. Let f(t)=3 t . (a) Sketch a graph of f . (b) Approximate f  (1), the slope of the tangent line to the graph of f(t)=3 t at t = 1, by computing the slope of the secant line through (1, f(1)) and (1.0001, f (1.0001)). (c) Approximate f  (0), the slope of the tangent line to the graph of f(t)=3 t at t = 0, by computing the slope of the secant line through (0, f(0)) and (0.0001, f (0.0001)). (d) Sketch a rough graph of the slope function f  . 3. The Exploratory Problems indicated that exponential functions grow at a rate pro- portional to themselves, i.e., if f(x)=a x , then f  (x) = ka x , for some constant k. Approximate the appropriate constant if f(x)=7 x . 4. In the Exploratory Problems you approximated the derivatives of 2 x ,3 x ,and 10 x for various values of x, and, after looking at your results, you conjectured about the patterns. Now, using the definition of the derivative of f at x = a, we return to this, focusing on the function f(x)=5 x . (a) Using the definition of the derivative of f at x = a, f  (a) = lim h→0 f(a+h) − f(a) h , give an expression for f  (0), the slope of the tangent line to the graph of at x = 0. (b) Show that for the function f(x)=5 x ,the difference quotient, f(x+h)−f(x) h , is equal to f(x)· f(h)−f(0) h . (c) Using the definition of derivative, 9.4 The Derivative of an Exponential Function 339 f  (x) = lim h→0 f(x+h) − f(x) h , conclude that the derivative of f(x)=5 x is f  (0) · f(x). Notice that you have now proven that the derivative of 5 x is proportional to 5 x , with the proportionality constant being the slope of the tangent line to 5 x at x = 0. f  (x) = f  (0) · f(x) (d) Approximate the slope of the tangent line to 5 x at x = 0 numerically. For Problems 5 through 9, differentiate the function given. 5. f(x)=x 3 e x 6. f(x)= e 2x x 7. f(x)=3e −x 8. f(x)= x 2 +x e x +1 9. f(x)=e 2x (x 2 + 2x + 2) 10. Use the tangent line approximation of e x at x = 0 to approximate e −1 . Is your answer larger than e −1 or smaller? 11. Find the equation of the line tangent to f(x)=e x at x = 1. 12. Differentiate the following. (a) f(x)= x 2 e x 3 (b) f(x)= 5x 2 3e x (c) f(x)= 1 xe 5x 13. Double, double, toil and trouble; Fire burn and caldron bubble. Macbeth Act IV scene I. It is the eve of Halloween and the witches are emerging. As the evening progresses the number of witches grows exponentially with time. At the moment when the first star of the evening is sighted, there are 40 witches and the number of witches is growing at a rate of 10 witches per hour. Later, at the moment when there are 88 witches, at what rate is the number of witches increasing? Explain your reasoning. 14. Money in a bank account is growing exponentially. When there is $4000 in the account, the account is growing at a rate of $100 per year. How fast is the money growing when there is $5500 in the account? It is not necessary to find an equation for M(t)in order to solve this problem. (In fact, you have not been given enough information to find M(t).) 340 CHAPTER 9 Exponential Functions 15. Consider the function f(x)= x 2 e x . (a) Compute f  (x). (The Quotient Rule is unnecessary.) (b) For what values of x is f  (x) positive? For what values of x is f  (x) negative? (c) For what values of x is f(x)increasing? For which is it decreasing? Give exact answers. (d) What is the smallest value ever taken on by f(x)?Explain your reasoning. . chances of winning? (e) Sketch a graph of P (n). Use [0, 100] as the range of the graph. Explain the practical significance of any asymptotes. 9.4 THE DERIVATIVE OF AN EXPONENTIAL FUNCTION As an. the slope of the tangent to the graph of b x at x = 0. It follows that d dx Cb x = αCb x . Regardless of whether we are looking at average or at instantaneous rates of change, we can state the. 1980 equal to 150,000, greater than 150,000, or less than 150,000? Explain your rea- soning. Note: Your answers to parts (a) and (b) should be different! 5. Let D(t), H(t) ,and J(t)represent the annual