Ch. 11 Additional Derivative Topics
11.1 Constant e, continuous compound interest
1 The Constant e
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) Find x to two decimal places.
x = 7,000e0.11
A) 7831.95
B) 8320.50
C) 7975.01
D) 7813.95
B) 2.9134
C) 2.6134
D) -2.8134
B) -70.1312
C) 44.321
D) -66.4815
2) Find t to four decimal places.
e-t = 0.06
A) 2.8134
3) Find t to four decimal places.
e -0.07t = 0.05
A) 42.7962
2 Continuous Compound Interest
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) What will the value of an account (to the nearest cent) be after 8 years if $100 is invested at 6.0% interest
compounded continuously?
A) $161.61
B) $849.47
C) $159.38
D) $175.32
2) If $5000 is invested at 5.25% compounded continuously, what is the amount in the account after 10 years?
A) $8452.29
B) $7420.65
C) $8442.52
D) $7625.00
3) How long will it take for the value of an account to be $890 if $350 is deposited at 11% interest compounded
continuously? Round your answer to the nearest hundredth.
A) 8.48 yr
B) 9.33 yr
C) 0.93 yr
D) 10.41 yr
4) How long will it take for $8400 to grow to $14.600 at an interest rate of 9.4% if the interest is compounded
continuously? Round the number of years to the nearest hundredth.
A) 5.88 yr
B) 0.59 yr
C) 0.06 yr
D) 58.81 yr
5) Suppose that $8000 is invested at an interest rate of 5.5% per year, compounded continuously. How long would
it take to double the investment?
A) 12.6 yr
B) 2 yr
C) 13.6 yr
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D) 11.6 yr
6) How long will it take money to double if it is invested at 5.25%, compounded continuously? Round your
answer to the nearest tenth.
A) 13.2 yr
B) 26.4 yr
C) 0.13 yr
D) 14 yr
7) An investor buys 100 shares of a stock for $20,000. After 5 years the stock is sold for $32,000. If interest is
compounded continuously, what annual nominal rate of interest did the original $20,000 investment earn?
(Represent the answer as a percent to three decimal places.)
A) 9.400%
B) 0.094%
C) 1.200%
D) 8.470%
8) Radioactive carbon-14 has a continuous compound rate of decay of r = -0.000124. Estimate the age of a skull
uncovered at an archaeological site if 6% of the original amount of carbon-14 is still present. (Compute answer
to the nearest year.)
A) 22,689 yr
B) 470 yr
C) 20,032 yr
D) 124,027 yr
11.2 Derivatives of Exponential, Logarithmic Functions
1 Derivative of e^x
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Differentiate.
1) Find fʹ(x) for f(x) = 3e-7x
A) fʹ(x) = -21e-7x
2) Find fʹ(x) for f(x) = e8x2 + x
A) fʹ(x) = 16xe8x2 + 1
B) fʹ(x) = 3e-7x
C) fʹ(x) = 21e-7x
D) fʹ(x) = -7e-7x
B) fʹ(x) = 16xex2 + 1
C) fʹ(x) = 16xe + 1
D) fʹ(x) = 16xe2x + 1
3) Find fʹ(x) for f(x) = y = ex4 - 2x + 1.
A) fʹ(x) = (4x3 - 2) e x4 - 2x + 1
B) fʹ(x) = 4x3 - 2
D) fʺ(x) = ex4 - 2x + 1
C) fʹ(x) = (4x3 - 2) ex
4) Find fʹ(x) for f(x) = e- x2 + 11x.
A) fʹ(x) = - e- x2 + 11x (2x - 11)
B) fʹ(x) = - e- x2 + 11x (2x + 11)
D) fʹ(x) = e- x2 + 11x (2x + 11)
C) fʹ(x) = e- x2 + 11x (2x - 11)
4
5) Find fʹ(x) for f(x) = (ex3 + 3)
3
A) fʹ(x) = 12 x2 ex3 (ex3 - 3)
3
B) fʹ(x) = 4 (ex3 - 3)
3
D) fʹ(x) = 4 x2 ex3 - 1 (ex3 - 3)
3
C) fʹ(x) = 4 (3x2 ex3 )
6) Find fʹ(x) for f(x) = A) fʹ(x) = 3ex
2ex + 1
3ex
(2ex + 1)2
B) fʹ(x) = ex
(2ex + 1)2
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C) fʹ(x) = 3ex
(2ex + 1)
D) fʹ(x) = 3ex
(2ex + 1)3
7) Find the derivative of f(x) = 5ex - 4x8 and simplify.
A) fʹ(x) = 5ex - 32x7
B) fʹ(x) = 5e5x - 32x7
C) fʹ(x) = 5ex - 32x8
D) fʹ(x) = 5ex - 4x8
Graph the exponential function.
8) f(x) = ex - 5
y
6
4
2
-6
-4
-2
2
4
x
6
-2
-4
-6
A)
B)
y
-6
-4
y
6
6
4
4
2
2
-2
2
4
6
x
-6
-4
-2
-2
-2
-4
-4
-6
-6
C)
2
4
6
x
2
4
6
x
D)
y
-6
-4
y
6
6
4
4
2
2
-2
2
4
6
x
-6
-4
-2
-2
-2
-4
-4
-6
-6
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9) f(x) = ex + 4
y
6
4
2
-6
-4
-2
2
4
x
6
-2
-4
-6
A)
B)
y
-6
-4
y
6
6
4
4
2
2
-2
2
4
6
x
-6
-4
-2
-2
-2
-4
-4
-6
-6
C)
2
4
6
x
2
4
6
x
D)
y
-6
-4
y
6
6
4
4
2
2
-2
2
4
6
x
-6
-4
-2
-2
-2
-4
-4
-6
-6
2 Derivative of ln x
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the derivative.
1) Find yʹ for y = ln 6x2
2
A) yʹ = x
B) yʹ = 12
x
C) yʹ = Page 204
2x
x2 + 6
D) yʹ = 1
2x + 6
2) Find yʹ for y = ln (6x3 - x2 )
18x - 2
A) yʹ = 6x2 - x
B) yʹ = 18x - 2
6x2
C) yʹ = 6x - 2
6x2 - x
D) yʹ = 18x - 2
6x3 - x
1
3) Find yʹ for y = x4 ln x - x3
3
A) yʹ = x3 - x2 + 4x3 ln x
B) yʹ = x3 - x2
C) yʹ = x4 ln x - x2 + 4x3
D) yʹ = 5x3 - x2
4) Find yʹ for y = ln (9x3 - x2 ).
27x - 2
A) yʹ = 9x2 - x
5) Find yʹ for y = ln (ln 7x).
1
A) yʹ = x ln 7x
6) Find yʹ for y = ln(1 - t)-2 .
2
A) yʹ = 1 - t
7) Find fʹ(x) for f(x) = ln(3x - 2).
3
A) fʹ(x) = 3x - 2
B) yʹ = 27x - 2
9x2
C) yʹ = 9x - 2
9x2 - x
D) yʹ = 27x - 2
9x3 - x
B) yʹ = 1
ln 7x
C) yʹ = 1
x
D) yʹ = 1
7x
B) yʹ = -2
1 - t
C) yʹ = 2
ln(1 - t)
D) yʹ = -2
ln(1 - t)
B) fʹ(x) = 3x - 2
3
C) fʹ(x) = e3x-2
Graph the function.
8) f(x) = 1 - ln x
y
5
5
-5
x
-5
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D) fʹ(x) = 3
ln(3x - 2)
A)
B)
y
y
5
5
x
5
-5
-5
-5
5
x
5
x
-5
C)
D)
y
y
5
5
x
5
-5
-5
-5
-5
3 Other Logarithmic and Exponential Functions
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the derivative.
1) Find fʹ(x) for f(x) = 8ex + 4 ln(x3 ).
12
12
A) 8ex + B) 8ex + x
x2
2) f(x) = ln 9 + e11x
11e11x
A)
9 + e 11x
3) Find A)
1
C)
9 + e 11x
D) 8ex + 1
11x
e
D)
12
x3
e11x
1 + e 11x
dy
given y = ln(-2x5 + x4 + 2x2 ). Simplfy your answer.
dx
A) 4) Find B)
4
C) 8ex + x2
-10x3 + 4x2 + 4
-2x4 + x3 + 2x
B)
-10x4 + 4x3 + 4x
-2x5 + x4 + 2x2
C)
10x3 + 4x2 + 4
2x4 + x3 + 2x
D)
10x4 + 4x3 + 4x
2x5 + x4 + 2x2
B)
2
dy
= (1 - t)
dt
C)
dy
1
= dt
ln (1 - t)
D)
dy
1
= dt
ln (1- t)2
dy
for y = ln(1 - t)-2 .
dt
2
dy
= 1 - t
dt
Page 206
5) Find fʹ(x) for f(x) = e-2x ln x .
e-2x
A) fʹ(x) = - 2e-2x ln x
x
C) fʹ(x) = - 2e-2x ln x
B) fʹ(x) = e-2x
x
D) fʹ(x) = e-2x
-2xe-2x
6) Find fʹ(x) for f(x) = x2 ln 7x.
A) fʹ(x) = x(1 + 2 ln 7x )
C) fʹ(x) = B) fʹ(x) = (1 + 2 ln 7x )
x
1 + 2 ln 7x
D) fʹ(x) = x + 2 ln 7x
7) Find fʹ(x) for f(x) = 6e5x + 8 ln(8x + 1).
64
A) fʹ(x) = 30e5x + 8x + 1
C) fʹ(x) = 30e5x + 8
8x + 1
B) fʹ(x) = 6e5x + 8
8x + 1
D) fʹ(x) = 6e5x + 64
8x + 1
4 Exponential and Logarithmic Models
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) The salvage value S, in dollars, of a companyʹs mainframe computer after t years is estimated to be given by
S(t) = 700,000e-1.45t. What is the rate of depreciation in dollars per year after six years?
A) -$169 per year
B) -$1,015,000 per year
C) -$145 per year
D) -$210 per year
2) The sales in thousands of a new type of product are given by S(t) = 60 - 80e-.1t, where t represents time in
years. Find the rate of change of sales at the time when t = 3.
A) 5.9 thousand per year
B) 10.8 thousand per year
C) -5.9 thousand per year
D) -10.8 thousand per year
3) Suppose the price-demand equation for x units of a product is estimated to be p = 80e-0.02x , where x units are
sold per day at a price of p hundred dollars each. Find the production level and price that maximizes revenue.
A) x = 50 units; p = $2943.04
B) x = 45 units; p = $2943.04
C) x = 50 units; p = $29.43
D) x = 30 units; p = $2943.04
4) The percentage P of consumers who accept a new product is given by P(t) = 100(1 - e- 0.20t), where t is the
time in months. How many months will it take for 77% of the consumer to accept the new product?
A) 8 months
B) 7 months
C) 9 months
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D) 10 months
5) Suppose that the population of a town is given by P(t) = 8 ln 7t + 3 , where t is the time in years after 1990 and
P is the population of the town ,in thousands. Find Pʹ(t).
4
28
B) Pʹ(t) = A) Pʹ(t) = 7t + 3
7t + 3
C) Pʹ(t) = 8
7t + 3
D) Pʹ(t) = 28 ln 7t + 3
7t + 3
6) The market research department of a national food company chose a large city in the Midwest to test -market a
new cereal. They found that the weekly demand for the cereal is given approximately by p = 8 - 2 ln x, where x
is the number of boxes of cereal (in hundreds) sold each week and $p is the price of each box of cereal. If each
box of the cereal costs the company $1.15 to produce, how should the cereal be priced in order to maximize the
weekly profit?
A) $3.15
B) $7.96
C) $11.30
D) $3.50
11.3 Derivatives of Products, Quotients
1 Derivative of Products
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Differentiate.
1) Find fʹ(t) for f(x) = (4x - 3)(3x3 - x2 + 1)
A) fʹ(x) = 48x3 - 39x2 + 6x + 4
B) fʹ(x) = 12x3 + 13x2 - 39x + 4
C) fʹ(x) = 48x3 - 13x2 + 39x + 4
D) fʹ(x) = 36x3 + 39x2 - 13x + 4
2) Find fʹ(x) for f(x) = (2x - 4)(2x3 - x2 + 1).
A) fʹ(x) = 16x3 - 30x2 + 8x + 2
B) fʹ(x) = 16x3 - 10x2 + 30x + 2
C) fʹ(x) = 4x3 - 10x2 - 30x + 2
D) fʹ(x) = 12x3 + 30x2 - 10x + 2
3) Find fʹ(x) for f(x) = (5x3 + 4)(3x7 - 5).
A) fʹ(x) = 150x9 + 84x6 - 75x2
B) fʹ(x) = 150x9 + 84x6 - 75x
C) fʹ(x) = 20x9 + 84x6 - 75x2
D) fʹ(x) = 20x9 + 84x6 - 75x
4) Let f and g be functions that satisfy: f(4) = -1, g(4) = 3, fʹ(4) = 2, and gʹ(4) = -3. Find hʹ(4) for
h(x) = f(x)g(x) - 2f(x) + 7.
A) 5
B) 6
C) -5
D) -6
C) fʹ(t) = 6t2 + 4
D) fʹ(t) = 6t2 + 40
5) Find fʹ(t) if f(t) = 0.4t(5t2 + 1) and simplify.
A) fʹ(t) = 6t2 + 0.4
B) fʹ(t) = 6t2 - 0.4
Page 208
Provide an appropriate response.
6) One hour after x milligrams of a particular drug are given to a person, the change in body temperature T(x), in
degrees Celsius, is given approximately by:
5x2
x
160
T(x) = 1 - - ,
0 ≤ x ≤ 6
9
9
9
Find the sensitivity, Tʹ(x), of the body to a dosage of three milligrams.
5
5
A) degrees per mg
B) - degrees per mg
3
3
C) - 10
degrees per mg
9
D)
10
degrees per mg
3
2 Derivative of Quotients
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Differentiate.
1) Find fʹ(t) for f(x) = A) - 5
(7x - 5)2
2) Find fʹ(t) for f(x) = A)
4) Find B)
14x - 5
(7x - 5)2
C) - 5
7x - 5
D) - 5x
(7x - 5)2
B)
17
(2x - 7)2
C) - 17
(2x - 7)2
D) - 17
(3x - 2)2
B)
3x3 - 6x2 + 18x
(9 - 3x)2
C)
-9x2 + 18x
(9 - 3x)2
2x - 7
.
3x - 2
17
(3x - 2)2
3) Find yʹ for y = A)
x
7x - 5
x2
9 - 3x
-3x2 + 18x
(9 - 3x)2
dy -20x2 + 72x + 10
= dx
(4x2 + 2)2
B)
dy -20x2 + 62x + 28
= dx
(4x2 + 2)2
C)
dy 20x3 - 40x2 + 82x
= dx
(4x2 + 2)2
D)
dy 60x2 - 72x + 10
= dx
(4x2 + 2)2
C)
dy 2x3 + 3x2
= dx
(x - 1)2
A)
9x
(9 - 3x)2
dy
5x - 9
for y = dx
4x2 + 2
A)
5) Find D)
dy
x3
for y = .
dx
x - 1
dy 2x3 - 3x2
= dx
(x - 1)2
B)
dy - 2x3 + 3x2
= dx
(x - 1)2
Page 209
D)
dy -2x3 - 3x2
= dx
(x - 1)2
6) Find dy
x2 - 3x + 2
for y = .
dx
x7 - 2
A)
dy - 5x8 + 18x7 - 14x6 - 4x + 6
= dx
(x7 - 2)2
B)
dy - 5x8 + 19x7 - 14x6 - 4x + 6
= dx
(x7 - 2)2
C)
dy - 5x8 + 18x7 - 13x6 - 4x + 6
= dx
(x7 - 2)2
D)
dy - 5x8 + 18x7 - 14x6 - 3x + 6
= dx
(x7 - 2)2
Provide an appropriate response.
7) Find the derivative of the function f(x) = A)
8) Find 17
16
B)
2x - 7
at x = 2.
3x - 2
17
4
C) - 17
4
D) - 17
16
dy
-5x3 - 5x2 + 3
for y = . Do not simplify.
dx
-5x4 + 2
A)
(-5x4 + 2)(-15x2 - 10x) - (-5x3 - 5x2 + 3)(-20x3 )
(-5x4 + 2)2
B)
(-5x3 - 5x2 + 3)(-20x3 ) - (-5x4 + 2)(-15x2 - 10x)
(-5x4 + 2)2
C)
(-5x4 + 2)(-15x2 - 10x) - (-5x3 - 5x2 + 3)(-20x3 )
(-5x3 - 5x2 + 3)2
D)
(-5x3 - 5x2 + 3)(-20x3 ) - (-5x4 + 2)(-15x2 - 10x)
(-5x3 - 5x2 + 3)2
9) Find fʹx for f(x) = (3x + 4)2
. Do not simplify.
x3 - x2 + 3x
A)
6(x3 - x2 + 3x)(3x + 4) - (3x + 4)2 (3x2 - 2x + 3)
(x3 - x2 + 3x)2
B)
(3x + 4)2 (3x2 - 2x + 3) - 6(x3 - x2 + 3x)(3x + 4)
(x3 - x2 + 3x)2
C)
(3x + 4)2 (3x2 - 2x + 3) - 6(x3 - x2 + 3x)(3x + 4)
(3x + 4)4
D)
6(x3 - x2 + 3x)(3x + 4) - (3x + 4)2 (3x2 - 2x + 3)
(3x + 4)4
10) A publishing company has published a new magazine for young adults. The monthly sales S (in thousands) is
800t
given by S(t) = , where t is the number of months since the first issue was published. Find S(3) and Sʹ(3)
t + 2
and interpret the results.
A) At three months, the monthly sales are $480,000 and increasing at 64,000 magazines per month.
B) At three months, the monthly sales are $480,000 and decreasing at 64,000 magazines per month.
C) At three months, the monthly sales are $2,400,000 and increasing at 800,000 magazines per month.
D) At three months, the monthly sales are $2, 400,000 and increasing at 64,000 magazines per month.
Page 210
11) Find the values of x where the tangent line is horizontal for the graph of f(x) = A) x = 0, x = -4
B) x = -2
C) x = -2, x = 0, x = -4
D) x = 0, x = -2
4x2
.
x + 2
11.4 Chain Rule
1 Composite Functions
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) Write composite function y = (2x4 + 3x + 1)3 in the form y = f(u) and u = g(x).
A) y = f(u) = u3 and u = g(x) = 2x4 + 3x + 1.
C) y = f(u) = (2x4 + 3x + 1 )3 and u = g(x) = u
B) y = f(u) = 2x4 + 3x + 1 and u = g(x) = u3
D) y = f(u) = u and u = g(x) = (2x4 + 3x + 1)3
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
2) Write composite function y = e3x2 + x - 1 in the form y = f(u) and u = g(x).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
3) Find the composition f[g(x)] if f(u) = u5 and g(x) = 2 - 3x2 .
A) (2 - 3x2 )5
B) (2 - 3x2 )2
C) (10 - 15x2 )5
D) 2 - 3u10
4) Find the composite g[f(-k)] if f(x) = 8x2 - 5x and g(x) = 7x + 9.
B) 392k2 + 973k + 603
A) 56k2 + 35k + 9
5) Consider the function: f(x) = C) 56k2 - 35k + 9
D) 392k2 - 973k + 603
9 - x2
. Choose the answer choice that includes all of the pair(s) of functions
x2
from the list so that f(x) can be written as a composition: f(x) = g(h(x)).
A) g(x) = x ; h(x) = 9 - x2
x2
C) g(x) = x2 ;h(x) = 3-x
x
6) Consider the function: f(x) = B) g(x) = 9-x
;h(x)=x2
x
D) Both A and B
1
11 - x2
. Choose the answer choice that includes all of the pair(s) of functions
from the list so that f(x) can be written as a composition: f(x) = g(h(x)).
A) g(x) = 1
; h(x) = 11 - x2
x
C) g(x) = 1
; h(x) = x2
11 - x
B) g(x) = 11 - x ; h(x) = D) Both A and C
Page 211
1
x2
2 General Power Rule
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the derivative.
d
4
1) Find dω (ω2 + 3)5
A) - 40ω
(ω2 + 3)6
2) Find fʹ(x) for f(x) = (8x - 9)-4 .
32
A) - (8x - 9)5
3) Find: A)
B)
- 40
(ω2 + 3)6
B) - C)
32
40ω
(ω2 + 3)6
C) - (8x - 9)3
4
(8x - 9)5
D)
- 40ω
(ω2 + 3)5
D) - 4
(8x - 9)3
dy 8 7
8x - 10
dx
7x6
(8x7 - 10)7/8
4) y = (x-2 + x)-3
dy 3x5 (2 - x3 )
= A)
dx
(1 + x3 )4
B) 8
B)
7
8x7 - 10
C) 448x6
dy 3x4 (2 - x3 )
= dx
(1 + x3 )4
C)
7
8x7 - 10
dy 3x5 (2 - x3 )
= dx
(1 + x3 )3
D) D)
56x6
(8x7 - 10)7/8
dy 3x4 (2 - x3 )
= dx
(1 + x3 )3
5) Find fʹ(x) for f(x) = (4x2 + 3x)2 .
A) fʹ(x) = 64x3 + 72x2 + 18x
B) fʹ(x) = 32x3 + 36x2 + 18x
C) fʹ(x) = 32x3 + 36x2 + 9x
D) fʹ(x) = 64x3 + 36x2 + 18x
Provide an appropriate response.
6) If $2000 is invested at an annual interest rate r compounded monthly, the amount in the account after 5 years is
1 60
given by A = 2,000(1 + r) . Find the rate of change of the amount A with respect to the interest rate r.
12
1 59
1 59
1 59
1 59
B) 120,000(1 + r)
C) 12,000(1 + r)
D) 1000(1 + r)
A) 10,000(1 + r)
12
12
12
12
3 Chain Rule
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) Find fʹ(x) for f(x) = (x2 + 2)3 .
A) fʹ(x) = 6x5 + 24x3 + 24x
B) fʹ(x) = 3x5 + 24x3 + 24x
C) fʹ(x) = 6x5 + 20x3 + 24x
D) fʹ(x) = 6x5 + 12x3 + 12x
2) Find fʹ(x) for f(x) = (x2 + 2)3 .
A) fʹ(x) = 6x5 + 24x3 + 24x
B) fʹ(x) = 3x5 + 24x3 + 24x
C) fʹ(x) = 6x5 + 20x3 + 24x
D) fʹ(x) = 6x5 + 12x3 + 12x
Page 212
3) Find dy
for y = (5t2 - 4t)2 .
dt
A) 2(5t2 - 4t)(10t - 4)
B) (5t2 - 4t)(10t - 4)
C) 2(10t - 4)
D) 2(5t2 - 4t) + (10t - 4)
4) Find fʹ(x) for f(x) = (4x2 + 3x)2 .
A) fʹ(x) = 64x3 + 72x2 + 18x
B) fʹ(x) = 32x3 + 36x2 + 18x
C) fʹ(x) = 32x3 + 36x2 + 9x
D) fʹ(x) = 64x3 + 36x2 + 18x
5) Find dy
for y = (5t2 - 4t)2 .
dt
A) 2(5t2 - 4t)(10t - 4)
B) (5t2 - 4t)(10t - 4)
C) 2(10t - 4)
D) 2(5t2 - 4t) + (10t - 4)
6) Find A)
dy
for y = ln (3x3 - x2 )
dx
9x - 2
3x2 - x
B)
9x - 2
3x2
7) Find fʹ(x) for f(x) = (ln x)8
8 ln7 x
B) 8 ln7 x
A)
x
8) Find C)
C)
1
x8
D)
D)
9x - 2
3x3 - x
1
E)
(ln x)8
dy
for y = 3 x-1
dx
A) 3 x-1 ln(3)
C) 3 x-1 ln(x)
B) 3 ln(3)
9) Find fʹ(x) for f(x) = log7 (x6 + 1)
6x5
A)
(ln 7)(x6 + 1)
C)
3x - 2
3x2 - x
B)
6x5
x6 + 1
D)
D) 3 x-1 ln(3 x-1 )
6x5 (ln 7)
x6 + 1
1
(ln 7)(x6 + 1)
+ 6x5
Find the equation of the tangent line to the graph of the given function at the given value of x.
10) f(x) = (x2 + 4)2/3 ; x = 2
4
4
A) y = x + 3
3
4
B) y = x
3
4
20
C) y = x + 3
3
2
4
D) y = x + 3
3
Find all values of x for the given function where the tangent line is horizontal.
x
11) f(x) = (x2 + 3)3
A) ± 15
5
B) ± 3
5
C) 0, ± Page 213
15
5
D) 0
Solve the problem.
12) The concentration of a certain drug in the bloodstream t minutes after swallowing a pill containing the drug
1
can be approximated using the equation C(t) = 3t + 1 -1/2, where C(t) is the concentration in arbitrary units
4
and t is in minutes. Find the rate of change of concentration with respect to time at t = 5 minutes.
3
1
1
3
A) - units/min
B) - units/min
C) - units/min
D) - units/min
512
512
16
32
11.5 Implicit Differentiation
1 Special Function Notation
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) An equation that defines y as a function of x is given. Solve for y in terms of x, and replace y with the function
notation f(x).
5x - 6y = 5
5
5
5 - 5x
B) f(x) = -5x - C) f(x) = 5 - 5x
D) f(x) = 5 - x
A) f(x) = 6
6
-6
2) An equation that defines y as a function of x is given. Solve for y in terms of x, and replace y with the function
notation f(x).
9x2 + 7y = 6
A) f(x) = 6 - 9x2
7
B) f(x) = - 9x2 + 6
7
C) f(x) = 6 - 9x2
D) f(x) = 6 + 9x2
7
2 Implicit Differentiation
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) Find yʹ for y = y(x) defined implicitly by 5y2 - 8x4 + 3 = 0, and evaluate yʹ at (x, y) = (1, 1).
16x3
16x2
16
16
A) yʹ = ; yʹ (1, 1) = B) yʹ = ; yʹ (1, 1) = 5y
2
5
5
5y
C) yʹ = 11x3
11
; yʹ (1, 1) = 5
5y
D) yʹ = 2) Find yʹ for y = y(x) defined implicitly by 3xy - x2 - 4 = 0.
3x -2y
2x - 3y
B) yʹ = A) yʹ = 4
3x
4
d
3 - e(2x2 + x)
dx
3
A) 4(3 - e(2x2 + x)) e(2x2 + x)(-4x - 1)
3
C) 2(3 - e(2x2 + x)) e(2x2 + x)(-4x - 1)
11x2
11
; yʹ (1, 1) = 2
5
5y
2
C) yʹ = x
3
D) yʹ = 3y - 2x
3x
3) Find: 3
B) 4(3 - e(2x2 + x)) e(2x2 + x)(-4x2 - 1)
3
D) 4(3 - e(2x2 + x)) e(2x2 + x)(-4x+ 1)
Page 214
4) Find xʹ for x = x(t) defined implicitly by 3t + 4tx = 3e4x and evaluate xʹ at (t, x) = (1, 0).
3 + 4x
3
7
3 + 4x
A) xʹ = ; xʹ (1, 0) = B) xʹ = ; xʹ (1, 0) = 8
8
4x
4x
12e - 4t
12e - 4t
C) xʹ = 3 + 4x
1
; xʹ (1, 0) = 4
12e4x - 4t
D) xʹ = 4 + 4x
1
; xʹ (1, 0) = 2
12e4x - 4t
5) Find dy/dx by implicit differentiation.
x3 + y3 = 5
A)
dy
x2
= - dx
y2
B)
dy x2
= dx y2
C)
dy
y2
= - dx
x2
D)
dy y2
= dx x2
C)
dy
x
= dx y - x
D)
dy
x
= dx x - y
C)
dy
x2 + 3xy
= - dx
x2 + y 2
D)
dy x2 + 3xy
= dx
x2 + y 2
D)
x
1
; - 4
3
3xy - x
6) Find dy/dx by implicit differentiation.
2xy - y2 = 1
A)
dy
y
= dx y - x
B)
dy
y
= dx x - y
7) Find dy/dx by implicit differentiation.
x3 + 3x 2 y + y 3 = 8
A)
x2 + 2xy
dy
= - dx
x2 + y 2
B)
dy x2 + 2xy
= dx
x2 + y 2
8) Find the equation(s) of the tangent line(s) to the graph of y2 - xy + 3 = 0 at x = -4.
1
3
3
B) y = x - 3
A) y = x + 3 and y = - x - 3
2
2
2
3
1
D) y = x + 2
2
3
C) y = - x - 3
2
9) Find yʹ and the slope of the tangent line to the graph of ln (xy) = y3 + 1 at (1, -1).
1
y
1
y
1
y
; B)
; - C)
; - A)
4
4
4
3
3
3
3xy - x
3xy
3xy - x
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
10) Find xʹ for x = x(t) defined implicitly by t3 - 5x2 = ln t and evaluate xʹ at (t, x) = (0, -1).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
11) The demand equation for a certain product is 8p2 + q 2 = 1200, where p is the price per unit in dollars and q is
the number of units demanded. Find dq/dp.
A) dq/dp = -8p/q
B) dq/dp = -8q/p
Page 215
C) dq/dp = -q/8p
D) dq/dp = -p/8q
12) The position of a particle at time t is given by s, where s 3 + 4st + 4t3 - 12t = 0. Find the velocity ds/dt.
12 - 4s - 12t 2
12 + 4s - 12t 2
A) ds/dt = B) ds/dt = 3s2 + 4t
3s2 + 4t
C) ds/dt = 12 + 4s - 12t 2
3s2 - 4t
D) ds/dt = 12 - 4s - 12t 2
3s2 - 4t
11.6 Related Rates
1 Related Rates
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) Assume x = x(t) and y = y(t). Find A) -4
B) -6
2) Assume x = x(t) and y = y(t). Find A) - 13
30
dx
dy
if x2 + y2 = 25 and = 3 when x = 3 and y = 4.
dt
dt
B)
C) 4
D) 6
dy
dx
if x2 (y - 6) = 12y + 3 and = 2 when x = 5 and y = 12.
dt
dt
13
20
C) - 20
13
D)
20
13
3) Evaluate dy/dt for the function at the point.
x3 + y 3 = 9; dx/dt = -3, x = 1, y = 2
A)
3
4
B) - 3
4
C)
4
3
D) - 4
3
C)
1
12
D) - 1
12
4) Evaluate dy/dt for the function at the point.
x + y
= x2 + y 2 ; dx/dt = 12, x = 1, y = 0
x - y
A) 12
B) - 12
5) A point is moving on the graph of xy = 24. When the point is at (4, 6), its x coordinate is increasing at the rate of
9 units per second. How fast is the y coordinate changing at that moment?
27
B) decreasing at 9 units per second
A) decreasing at units per second
2
C) increasing at 27
units per second
2
D) increasing at 9 units per second
6) Suppose two automobiles leave from the same point at the same time. If one travels north at 60 miles per hour
and the other travels east at 45 miles per hour, how fast will the distance between them be changing after three
hours?
A) 75 mph
B) 150 mph
C) 125 mph
Page 216
D) 50 mph
7) A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second,
at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feet
away from the wall?
A) 4.8 ft/sec
B) 9.6 ft/sec
C) 2.4 ft/sec
D) 5.2 ft/sec
8) A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 3 feet per second,
at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 9 feet
away from the wall?
A) 8.1 ft/sec
C) -8.1 ft/sec
B) 4.9 ft/sec
D) 5.4 ft/sec
9) A man 6 ft tall walks at a rate of 5 ft/sec away from a lamppost that is 13 ft high. At what rate is the length of
his shadow changing when he is 65 ft away from the lamppost?
30
15
325
30
ft/sec
B)
ft/sec
C)
ft/sec
D)
ft/sec
A)
19
19
6
7
11.7 Elasticity of Demand
1 Relative Rate of Changes
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
For the given demand function, find the value(s) of p for which total revenue is maximized.
1) x = D(p) = 700 - p
A) 350
B) 700
C) 1400
D) 280
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
2) Find the relative rate of change of f(x) = 150x - 0.08x 2 .
3) Find the relative rate of change of f(x) = 15x + 4x ln x
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
4) A company is manufacturing a new digital watch and can sell all it manufactures. The cost (in dollars) is given
by C(x) = 5000 + 2x, where the production output in one day is x watches. If production is increasing at
5 watches per day when production is 375 watches per day, find the rate of increase in cost.
A) $10 per day
B) $5 per day
C) $175 per day
D) $75 per day
5) A company is manufacturing a new digital watch and can sell all it manufactures. The revenue (in dollars) is
x2
given by R(x) = 50x - , where the production output in one day is x watches. If production is increasing at
50
5 watches per day when production is 375 watches per day, find the rate of increase in revenue.
A) $175 per day
B) $250 per day
C) $150 per day
Page 217
D) $75 per day
6) Given the revenue and cost functions R = 26x - 0.3x2 and C = 3x + 10, where x is the daily production, find the
rate of change of profit with respect to time when 20 units are produced and the rate of change of production is
7units per day per day.
A) $77.00 per day
B) $156.80 per day
C) $98.00 per day
D) $149.00 per day
2 Elasticity of Demand
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the elasticity of the demand function as a function of p.
1) x = D(p) = 800 - p
p
A) E(p) = 800 - p
2) x = D(p) = 900 - p
p
A) E(p) = 1800 - 2p
3) x = D(p) = B) E(p) = p
p - 800
C) E(p) = B) E(p) = p
2p - 1800
C) E(p) = B) E(p) = 2
p + 6
C) E(p) = 1
800 - p
p
900 - p
D) E(p) = p(800 - p)
D) E(p) = 1
1800 - 2p
700
(p + 6)2
A) E(p) = 2p
p + 6
1400p
(p + 6)3
D) E(p) = 1400p(p + 6)
Solve the problem.
4) A beverage company works out a demand function for its sale of soda and finds it to be
x = D(p) = 3100 - 24p
where x = the quantity of sodas sold when the price per can, in cents, is p. At what prices, p, is the elasticity of
demand inelastic?
A) For p < 65 cents
B) For p < 129 cents
C) For p > 37,200 cents
D) For p > 258 cents
Page 218
Ch. 11 Additional Derivative Topics
Answer Key
11.1 Constant e, continuous compound interest
1 The Constant e
1) A
2) A
3) A
2 Continuous Compound Interest
1) A
2) A
3) A
4) A
5) A
6) A
7) A
8) A
11.2 Derivatives of Exponential, Logarithmic Functions
1 Derivative of e^x
1) A
2) A
3) A
4) A
5) A
6) A
7) A
8) A
9) A
2 Derivative of ln x
1) A
2) A
3) A
4) A
5) A
6) A
7) A
8) A
3 Other Logarithmic and Exponential Functions
1) A
2) A
3) A
4) A
5) A
6) A
7) A
4 Exponential and Logarithmic Models
1) A
2) A
3) A
4) A
5) A
6) A
Page 219
11.3 Derivatives of Products, Quotients
1 Derivative of Products
1) A
2) A
3) A
4) A
5) A
6) A
2 Derivative of Quotients
1) A
2) A
3) A
4) A
5) A
6) A
7) A
8) A
9) A
10) A
11) A
11.4 Chain Rule
1 Composite Functions
1) A
2) y = f(u) = ex and u = g(x) = 3x2 + x - 1.
3) A
4) A
5) D
6) D
2 General Power Rule
1) A
2) A
3) A
4) A
5) A
6) A
3 Chain Rule
1) A
2) A
3) A
4) A
5) A
6) A
7) A
8) A
9) A
10) A
11) A
12) A
11.5 Implicit Differentiation
1 Special Function Notation
1) A
2) A
2 Implicit Differentiation
1) A
Page 220
2)
3)
4)
5)
6)
7)
8)
9)
A
A
A
A
A
A
A
A
10) xʹ = 3t3 - 1 1
; 10x
10
11) A
12) A
11.6 Related Rates
1 Related Rates
1) A
2) A
3) A
4) A
5) A
6) A
7) A
8) A
9) A
11.7 Elasticity of Demand
1 Relative Rate of Changes
1) A
150 - 0.16x
2)
150x - 0.08x 2
3)
19 + 4ln x
15x + 4x ln x
4) A
5) A
6) A
2 Elasticity of Demand
1) A
2) A
3) A
4) A
Page 221