3.4 The Chain Rule
149
____
21.
Find the derivative of the function
.
a.
b.
c.
d.
e.
____
22.
Find the derivative of the function
.
a.
b.
c.
d.
e.
____
23.
Find the derivative of the function
.
a.
b.
c.
d.
e.
150 Chapter 3: Differentiation
____ 24. Find the second derivative of the function.
a.
b.
c.
d.
e.
____ 25. Find the second derivative of the function .
a.
b.
c.
d.
e.
____ 26. Find if .
a.
b.
c.
d.
e.
3.4 The Chain Rule
151
____
27.
Evaluate the derivative of the function
.
a.
b.
c.
d.
e.
____
28.
Evaluate the derivative of the function at the given point.
,
a.
b.
c.
d.
e.
152 Chapter 3: Differentiation
____ 29. Evaluate the derivative of the function at the point .
a.
b.
c.
d.
e.
____ 30. Evaluate the derivative of the function at the point .
a.
b.
c.
d.
e.
____ 31. Find an equation to the tangent line for the graph of f at the given point.
,
a.
b.
c.
d.
e.
3.4 The Chain Rule
153
____
32.
Find an equation to the tangent line to the graph of the function
at the point
. The coefficients below are given to two decimal places.
a.
b.
c.
d.
e.
____
33.
Find an equation of the tangent line to the graph of
at the point (1,0).
a.
b.
c.
d.
e.
____
34.
Find an equation of the tangent line to the graph of
at the point
.
a.
b.
c.
d.
e.
____
35.
Find
if
.
a.
b.
c.
d.
e.
154
Chapter 3: Differentiation
____
36.
Find
if
.
a.
b.
c.
d.
e.
____
37.
Find
if
.
a.
b.
c.
d.
e.
155
a. 8.00 ft/sec b.
4.68 ft/sec c.
6.00 ft/sec d.
1.00 ft/sec e.
5.00 ft/sec
____ 40. Suppose a 15-centimeter pendulum moves according to the equation where is the
angular displacement from the vertical in radians and t is the time in seconds. Determine the
rate of change of when t = 7 seconds. Round your answer to four decimal places.
a. 2.5034 radians per second b.
3.6185 radians per second c.
0.3129 radians per second d.
3.1535 radians per second e.
4.1724 radians per second
seconds. Determine the velocity of the object when
decimal places.
. Round your answer to two
where y is measured in feet and t is the time in
end of a spring is
____ 39.
The displacement from equilibrium of an object in harmonic motion on the
seconds. Determine the position of the object when
decimal places.
a. 0.12 feet b.
0.21 feet
c.
17.53 feet d.
0.46 feet
e.
0.11 feet
. Round your answer to two
where y is measured in feet and t is the time in
end of a spring is
____ 38.
3.4 The Chain Rule
The displacement from equilibrium of an object in harmonic motion on the
156 Chapter 3: Differentiation
____ 41. A buoy oscillates in simple harmonic motion as waves move past it. The
buoy moves a total of 3.5 feet (vertically) between its low point and its high point. It returns
to its high point every 14 seconds. Write an equation describing the motion of the buoy if it
is at its high point at t = 0.
a.
b.
c.
d.
e.
____ 42. A buoy oscillates in simple harmonic motion as waves move past it. The
buoy moves a total of 13.5 feet (vertically) between its low point and its high point. It returns
to its high point every 10 seconds. Determine the velocity of the buoy as a function of t.
a.
b.
c.
d.
e.
____ 43. Suppose the table below shows the temperature at which a given liquid boils at
selected pressures p (pounds per square inch). A model that approximates the data is
. Find the rate of change of T with respect to p when P = 20.
Round your answer to two decimal places.
P
5
10
14.696
20
30
40
60
80
100
(1 atm)
T
234.68
288.88
320.55
346.89
383.19
410.28
450.7
481.26
506.22
a.
b.
c.
d.
e.
3.4 The Chain Rule
157
____
44.
If the annual rate of inflation averages
over the next
years, the approximate
cost
of goods or services during any year in that decade is
where
is the time in
years and
is the present cost. The price of an oil change for your car is presently
. Estimate
the price
years from now. Round your answer to two decimal places.
$26.30
$30.67
$78.89
$25.61
$94.31
____ 45. If the annual rate of inflation averages over the next years, the approximate
cost of goods or services during any year in that decade is where is the time in
years and is the present cost. The price of an oil change for your car is presently . Find the
rate of change of with respect to when . Round your answer to three decimal places.
32.176 dollars per year
1.966 dollars per year
2.293 dollars per year
5.962 dollars per year
32.730 dollars per year
158 Chapter 3: Differentiation
3.4 The Chain Rule
Answer Section
3.4 The Chain Rule 159
160
Chapter 3: Differentiation
3.5 Implicit Differentiation
161
3.5 Implicit Differentiation
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find by implicit differentiation.
a.
b.
c.
d.
e.
____ 2. Find by implicit differentiation.
a.
b.
c.
d.
162 Chapter 3: Differentiation e.
____ 3. Find by implicit differentiation given that .
a.
b.
c.
d.
e.
____ 4. Find by implicit differentiation.
a.
b.
c.
d.
e.
3.5 Implicit Differentiation
163
____ 5. Find by implicit differentiation given that .
a.
b.
c.
d.
e.
____ 6. Use implicit differentiation to find
a.
b.
c.
d.
e.
164
Chapter 3: Differentiation
____
7.
Find
by implicit differentiation.
a.
b.
c.
d.
e.
____
8.
Find
by implicit differentiation.
a.
b.
c.
d.
e.
3.5 Implicit Differentiation
165
____ 9. Find by implicit differentiation.
a.
b.
c.
d.
e.
____ 10. Find by implicit differentiation given that . Use the original equation
to simplify your answer.
a.
b.
c.
d.
e.
166 Chapter 3: Differentiation
____ 11. Use implicit differentiation to find .
a.
b.
c.
d.
e.
____ 12. Use implicit differentiation to find .
a.
b.
c.
d.
e.
3.5 Implicit Differentiation
167
____
13.
Evaluate
for the equation
at the given point
. Round your
answer to two decimal places.
a.
b.
c.
d.
e.
____
14.
Evaluate
for the equation
at the given point
. Round
your answer to two decimal
places. a.
b.
c.
d.
e.
____ 15. Evaluate for the equation at the given point . Round
your answer to two decimal places.
a.
b.
c.
d.
e.
168
Chapter 3: Differentiation
____
16.
Use implicit differentiation to find
at the point
.
a.
b.
c.
d.
e.
____
17.
Find the slope of the tangent line
at the given point
. Round
your answer to two decimal places.
a.
0.67
b.
2.00
c.
1.00
d.
1.67
e.
3.00
____ 18. Find an equation of the tangent line to the graph of the function given below at the
given point.
(The coefficients below are given to two decimal places.)
a.
b.
c.
d.
e.