Deck 13: Section 6: Functions of Several Variables

Question 1

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)

Answer 1

Question 2

Evaluate at using the appropriate Chain Rule for where and . Round your answer to two decimal places.

A) 22.26
B) 22.34
C) 29.22
D) 26.94
E) 24.42

Answer 2

26.94

Question 3

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

Answer 3

1,210.14

Question 4

Differentiate implicitly to find .

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B)
C)
D)
E)

Question 5

Find using the appropriate Chain Rule for where , and .

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B)
C)
D)
E)

Question 6

Differentiate implicitly to find , given .

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B)
C)
D)
E)

Question 7

Find using the appropriate Chain Rule for where , and .

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B)
C)
D)
E)

Question 8

Find using the appropriate Chain Rule for where and .

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B)
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D)
E)

Question 9

Find using the appropriate Chain Rule for where and .

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B)
C)
D)
E)

Question 10

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

Question 11

Differentiate implicitly to find , given .

A)
B)
C)
D)
E)

Question 12

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?

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B)
C)
D)
E)

Question 13

Let be the angle between equal sides of an isosceles triangle and let x be the length of these sides. x is increasing at meter per hour and is increasing at radian per hour. Find the rate of increase of the area when and . Round your answer to three decimal places.

A)
B)
C)
D)
E)

Question 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.

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B)
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D)
E)

Question 15

Let , where and . Find .

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D)
E)

Question 16

Let , where , , and . Find .

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B)
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D)
E)

Question 17

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)

Question 18

Find using the appropriate Chain Rule for where .

A) 116t
B) 58t
C) 104t
D) 32t
E) 174t

Question 19

Let , where and . Find .

A)
B)
C)
D)
E)

Question 20

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