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, #, and%representtheannualsalaries(indollars)ofDavid, Henry, and Jennifer, and suppose that these functions are given by the following formulas, where is in years. 0 corresponds to this yearÕs salary, 1 to the salary one year from now, and so on. The domain of each function is 0, 1, 2, up to retirement.
(a) Describe in words how each employeeÕs salary is changing.
(b) Suppose you are just four years away from retirementÑyouÕll collect a salary for four years, including the present year. Which personÕs 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 ChekovÕs play ÒThree Sisters,Ó Lieutenant-Colonel Vershinin says the fol-lowing in reply to MashaÕs complaint that much of her knowledge is unnecessary. ÒI donÕt 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 hasnÕt 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.Ó
LetusassumethatVershininmeansthatthisdoublingoccurseverygenerationand 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 Òsuch as [Masha] form a major-ityÓ?
(b) What percentage of the town will be Òintelligent and educatedÓ 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 ÒbaseÓ of 50 miles per week during the winter, what is his or her maximum recommended weekly mileage for the Þfth week of the season?
(b) Find a formula for ", the maximum weekly recommended mileage weeks into the season. Assume that initially the cyclist has a base of 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 Þre ofÞcials, a 1996 Þre on the Warm Springs Reservation in central Oregon tripled in size to 65,000 acres in one day. A Þre in Upper Lake, California, quintupled in size to 10,200 acres in one day.
(a) Assuming exponential growth, determine the doubling time for each Þre. (b) What was the hourly percentage growth of each Þre?
10. Duringthedecadefrom1985to1995,HarvardÕsaveragereturnonÞnancialinvestments in its endowment was 11.1% per year. Over the same period, YaleÕs total return on its investments was 287.3%. (Boston Globe, July 26, 1996.) LetÕs assume both Harvard and YaleÕs endowments are growing exponentially.
(a) What was HarvardÕs total return over this ten-year period? (b) What was YaleÕs average annual rate of return?
(c) Which school got the higher return on its investments?
(d) What was the doubling time for each schoolÕs investments?
(e) In 1995, HarvardÕs 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 101023, 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 & 31621.065, 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 or 2?
(a) Using what you know about these two functions and experimenting numerically
and graphically, guess the following limits:
2 2 i. lim 2 ii. lim2 iii. lim 2 iv. lim 2
(b) For large, which function is dominant, 2 or 2? Would you have answered differently if we looked at 3 and 3 instead?
Exploratory Problems for Chapter 9 333
13. Devaluation: Due to inßation, 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 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 0) there were 100 beavers. In June 1994 ( 1) there were 130 beavers.
(a) Write a function that gives the number of beavers at time .
(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 ßying school, a school for training pilots. Our benchmark time, 0, is one year after the school opened.
When 0, the number of people who have graduated 25. When 3, the number of people who have graduated 127.
Find the cumulative number of people who have graduated at time 5 if
(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 OfÞce, during the 14-year period betweentheschoolyear1980—1981andtheschoolyear1994—1995,theaveragetuition at four-year public colleges increased by 234%. During the same period, average household income increased by 82%, and the Labor DepartmentÕs 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 familyÕs totalincome.Assumingthattheirincomeincreasedatthesamepaceastheaverage 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 tickets is to subtract from 100% the probability that you will lose on all 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 , the percentage chance of winning at least once if you play the game times.
(c) How many tickets must you buy in order to have a 25% chance of winning? A 50% chance?
(e) Sketchagraphof.Use[0,100]astherangeofthegraph.Explainthepractical signiÞcance of any asymptotes.
9.4 THE DERIVATIVE OF AN EXPONENTIAL FUNCTION
Asanexploratoryproblemyouinvestigatedthederivativesof2,3,and10 bynumerically approximating the derivatives at various points. Below are tables of values of approxima-tions to the derivatives of 2, 3, and 10.13
2 3 10
0 1 0.693 0 1 2 1.386 1 2 4 2.774 2 3 8 5.547 3
4 16 11.094 4
1 1.099 3 3.298 9 9.893
1 10 2 100 3 1000
23.052 230.524 2305.238
Based on the data gathered, we can conjecture that
0.6932, 1.0993, 2.30510.
Ineachcasethederivativeoftheexponentialfunctionatanypointappearstobeproportional to the value of the function at that point. Furthermore, the proportionality constant appears to be the derivative of the function at 0.
0 2 0 3 0 10
13 The approximations to were made with for 0.001. In each case the following was used:
The second term was calculated, multiplied by the Þrst, 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 , then 0 .
If , then
(the slope of the tangent line to at 0) .
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. LetÕs go back to the limit deÞnition of derivative and see if we can prove our conjecture.
Proof of Conjecture
Let , where is a positive constant. Consider for any real number .
lim 1 . 0
As goes to zero is unaffected, so
lim 1 . 0
0 1 lim 0 lim 0.
This is precisely the deÞnition of 0, the derivative of at 0. How delightful! We have now proven our conjecture.
If , then 0 0 .
We can approximate the derivatives at zero numerically, as done in the exploratory problem. Our results stand as
2 0.6932, 3 1.0983, 10 2.30210.