13.3 Partial Derivatives
788
____
6.
Find the partial derivative
for the function
.
a.
b.
c.
d.
e.
____
7.
Find the partial derivative
for the function
.
a.
b.
c.
d.
e.
789
Chapter 13: Functions of Several Variables
____
Find the partial derivative
for the function
.
a.
b.
c.
d.
e.
____ 9. For , evaluate at the point .
a.
b.
c.
d.
e.
____ 10. For , evaluate at the point .
a.
b.
c.
d.
e.
13.3 Partial Derivatives
790
____ 11. For , evaluate at the point .
a.
b.
c.
d.
e.
____ 12. For , evaluate at the point .
a.
b.
c.
d.
e.
791
Chapter 13: Functions of Several Variables
____
13.
Find the first partial derivative for the function
with
respect to z.
a.
b.
c.
d.
e.
____
14.
For
, evaluate
at the point
.
a.
b.
c.
d.
e.
____
15.
Find the second partial derivative for the function
with respect to
x.
a.
b.
13.3 Partial Derivatives
792
c.
d.
e.
____
16.
For
find all values of x and y such that
and
simultaneously.
a.
b.
c.
d.
e.
____
17.
For
find all values of x and y such that
and
simultaneously. a.
b.
c.
d.
e.
793
Chapter 13: Functions of Several Variables
____
18.
For
, find all values of x and y such that
and
simultaneously.
a.
b.
c.
d.
e.
____
19.
Suppose a pharmaceutical corporation has two plants that produce the same
over-the-counter medicine. If
and
are the numbers of units produced at plant 1 and plant 2,
respectively, then the total revenue for the product is given by
. When
and
find the marginal revenue
for plant 1.
a.
174
b.
172
c.
80
d.
124
e.
93
____ 20. Suppose a company manufactures two types of wood-burning stoves: a freestanding model
and a fireplace-insert model. The cost function for producing x freestanding and y
fireplace-insert stoves is . Find the marginal costs when
and . Round your answer to the nearest integer.
192
234
210
402
403
13.3 Partial Derivatives
794
____ 21. Suppose the temperature at any point in a steel plate is
where x and y are measured in meters. At the point find the rate of
change of the temperature with respect to the distance moved along the plate in the direction of the
x-axis. Round your answer to one decimal place.
a.
b.
c.
d.
e.
____ 22. Suppose the utility function is a measure of the utility (or satisfaction) derived
by a person from the consumption of two products x and y. Determine the marginal utility of
product x if the utility function is .
a.
b.
c.
d.
e.
____ 23. Suppose the utility function is a measure of the utility (or satisfaction) derived
by a person from the consumption of two products x and y. Determine the marginal utility of
product if the utility function is .
a.
b.
c.
d.
e.
795 Chapter 13: Functions of Several Variables
13.3 Partial Derivatives
Answer Section
13.3 Partial Derivatives
796
797 Chapter 13: Functions of Several Variables
13.4 Differentials
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1.
Find the total differential of the function
.
a.
b.
c.
d.
e.
____
2.
Find the total differential of the function
.
a.
b.
c.
d.
e.
13.4 Differentials
798
____
3.
Find the total differential for the function
.
a.
b.
c.
d.
e.
____
4.
Find the total differential for the function
.
a.
b.
c.
d.
e.
____ 5. Given , calculate by evaluating and .
Round your answer to four decimal places.
–7.9441
–45.9441
–19.9441
0.1118
–4.8882
____ 6. Given , use the total differential to approximate at towards
. Round your answer to two decimal places.
10.96
–6.04
49.96
31.92
33.92
799
Chapter 13: Functions of Several Variables
____
Given
, calculate
by evaluating
and
.
Round your answer to four decimal places.
7.3037
5.6074
–16.6963
–2.6963
–21.3926
____ 8. Given , use the total differential to approximate at towards
. Round your answer to four decimal places.
–35.7266
30.5500
17.2734
15.5468
–4.7266
____ 9. Find and use the total differential to approximate the quantity
. Round your answer to two decimal places.
5.80
2.80
4.40
2.40
0.40
____ 10. The radius r and height h of a right circular cylinder are measured with possible errors of
5% and 1%, respectively. Approximate the maximum possible percent error in measuring the
volume.
18 %
11 %
24 %
12 %
7 %
13.4 Differentials
800
____ 11. Suppose the formula for wind chill C (in degrees Fahrenheit) is given by
where v is the wind speed in miles per hour and T is
the temperature in degrees Fahrenheit. The wind speed is miles per hour and the temperature is
Fahrenheit. Use to estimate the maximum possible propagated error in calculating the
wind chill. Round your answer to four decimal places.
a.
b.
c.
d.
e.
____ 12. Suppose the formula for wind chill C (in degrees Fahrenheit) is given by
where v is the wind speed in miles per hour and T is
the temperature in degrees Fahrenheit. The wind speed is miles per hour and the temperature is
Fahrenheit. Use to estimate the relative error in calculating the wind chill. Round your
answer to two decimal places.
a.
b.
c.
d.
e.
____ 13. Suppose the centripetal acceleration of a particle moving in a circle is , where
v is the velocity and r is the radius of the circle. Approximate the maximum percent error in
measuring the acceleration due to errors of 4% in v and 2% in r.
11%
16%
10%
21%
13%
801 Chapter 13: Functions of Several Variables
____ 14. Suppose electrical power P is given by , where E is voltage and R is
resistance. Approximate the maximum percent error in calculating the power if 140 volts are applied
to a 3000-ohm resistor and the possible percent errors in measuring E and R are 5% and 8%,
respectively.
24%
42%
38%
43%
18%
____ 15.
Suppose the period T of a pendulum of length L is
where g is the
acceleration due to gravity. A pendulum is moved from the Canal Zone, where
feet per
second per second, to Greenland, where
feet per second per second. Because of the change
in temperature, the length of the pendulum changes from 2.6 feet to 2.45 feet. Approximate
the change in the period of the pendulum. Round your answer to four decimal places.
5.8508 seconds
–0.0746 second
0.9254 second
7.9254 seconds
–7.1492 seconds
13.4 Differentials
802
13.4 Differentials
Answer Section
803 Chapter 13: Functions of Several Variables
13.5 Chain Rules for Functions of Several Variables
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find using the appropriate Chain Rule for where .
116t
58t
104t
32t
174t
____ 2. Let , where and . Find .
a.
b.
c.
d.
e.
____ 3. Let , where , , and . Find .
a.
b.
c.
13.5 Chain Rules for Functions of Several Variables
804
d.
e.
____ 4. The parametric equations for the paths of two projectiles are given below. At what rate is
the distance between the two objects changing at ? Round your answer to two decimal
places.
0.86
1.54
0.58
–1.73
–0.65
____ 5. Find using the appropriate Chain Rule for where and
, and evaluate the partial derivative at and . Round your answer to two decimal
places.
–8.83
–5.89
–6.62
2.72
–8.09
____ 6. Find using the appropriate Chain Rule for where and
, and evaluate the partial derivative at and . Round your answer to two decimal
places.
1,209.99
806.71
1,210.14
806.69
1,209.20
805
Chapter 13: Functions of Several Variables
____
Find
using the appropriate Chain Rule for
where
, and
.
a.
b.
c.
d.
e.
____ 8. Find using the appropriate Chain Rule for where
, and .
a.
b.
c.
d.
e.
____ 9. Differentiate implicitly to find .
a.
b.
c.
d.
e.
13.5 Chain Rules for Functions of Several Variables
806
____
10.
Differentiate implicitly to find
, given
.
a.
b.
c.
d.
e.
____
11.
Differentiate implicitly to find
, given
.
a.
b.
c.
d.
e.
____ 12. The radius of a right circular cylinder is increasing at a rate of 6 inches per minute, and the
height is decreasing at a rate of 4 inches per minute. What is the rate of change of the volume when
the radius is 14 inches and the height is 32 inches?
a.
b.
c.
d.
e.
807
Chapter 13: Functions of Several Variables
____
13.
The radius of a right circular cylinder is increasing at a rate of 9 inches per minute,
and the height is decreasing at a rate of 7 inches per minute. What is the rate of change of the
surface area when the radius is 12 inches and the height is 32 inches?
a.
b.
c.
d.
e.
____ 14. The two radii of the frustum of a right circular cone are increasing at a rate of 4 centimeters
per minute, and the height is increasing at a rate of 12 centimeters per minute (see figure). Find the
rate at which the volume is changing when the two radii are 15 centimeters and 30 centimeters, and
the height is 10 centimeters.
a.
b.
c.
d.
e.
____ 15. The two radii of the frustum of a right circular cone are increasing at a rate of 7 centimeters
per minute, and the height is increasing at a rate of 11 centimeters per minute (see figure). Find the
rate at which the surface area is changing when the two radii are 16 centimeters and 26 centimeters,
and the height is 13 centimeters. [Note: The surface area does not include the top and bottom circles.]
Round your answer to two decimal places.