Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 36 potx

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.