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774 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 .
a. 116t
b. 58t
c. 104t
d. 32t
e. 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 775
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.
a. 0.86
b. 1.54
c. 0.58
d. –1.73
e. –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.
a. –8.83
b. –5.89
c. –6.62
d. 2.72
e. –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.
a. 1,209.99
b. 806.71
c. 1,210.14
d. 806.69
e. 1,209.20
776 Chapter 13: Functions of Several Variables
____ 7. 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 777
____ 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.
778 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.
13.5 Chain Rules for Functions of Several Variables 779
____ 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.
a.
b.
c.
d.
e.
780 Chapter 13: Functions of Several Variables
13.5 Chain Rules for Functions of Several Variables
Answer Section
MULTIPLE CHOICE
