Deck 13: Functions of Several Variables

Question 1

Determine the continuity of the composite function where and . ​

A) continuous except at
​
B) continuous for all
​
C) continuous for all
​
D) continuous except at
​
E) continuous everywhere
​

Answer 1

B

Question 2

Discuss the continuity of the function.

A) f is continuous everywhere except at those points such that .
B) f is continuous everywhere except at those points such that .
C) f is continuous at those points such that .
D) f is continuous only when .
E) f is continuous everywhere in the xy-plane.

Answer 2

E

Question 3

Find and simplify for the given function . ​

A)
B)
C)
D)
E)

Answer 3

C

Question 4

Sketch the surface given by the function .

A)
B)
C)
D)
E)

Question 5

Find an equation of the tangent plane to the surface at the point .

A)
B)
C)
D)
E)

Question 6

A cargo container (in the shape of a rectangular solid) must have a volume of 520 cubic feet. The bottom will cost $7 per square foot to construct and the sides and the top will cost $3 per square foot to construct. Use Lagrange multipliers to find the dimensions of the container of this volume that has minimum cost.

A)
B)
C)
D)
E)

Question 7

According to the Ideal Gas Law, , where P is pressure, V is volume, T is temperature (in Kelvins), and k is a constant of proportionality. A tank contains 2,500 cubic inches of nitrogen at a pressure of 30 pounds per square inch and a temperature of 700 K. Determine k. ​

A)
B)
C)
D)
E)

Question 8

According to the Ideal Gas Law, , where P is pressure, V is volume, T is temperature (in Kelvins), and k is a constant of proportionality. A tank contains 2500 cubic inches of nitrogen at a pressure of 36 pounds per square inch and a temperature of 700 K. Write P as a function of V and T after evaluating k.

A)
B)
C)
D)
E)

Question 9

Find by using the limits and .

A)
B)
C)
D)
E)

Question 10

Find the path of a heat-seeking particle placed at point on a metal plate with a temperature field .

A)
B)
C)
D)
E)

Question 11

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 16 centimeters per minute (see figure). Find the rate at which the surface area is changing when the two radii are 19 centimeters and 24 centimeters, and the height is 8 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 12

Find the minimum cost of producing 60,000 units of a product , where x is the number of units of labor (at $80 per unit) and y is the number of units of capital (at $40 per unit). Round your answer to the nearest cent.

A) $190,278.22
B) $62,746.50
C) $37,020.43
D) $112,264.15
E) $55,724.88

Question 13

Sketch the level curves for the function for the given c-values .

A)
B)
C)
D)
E)

Question 14

Use Lagrange multipliers to find the maximum value of where and subject to the constraint . ​

A) maxima: ; minima:
​
B) maxima: ; minima:
​
C) maxima: ; minima:
​
D) maxima: ; minima:
​
E) maxima: ; minima:
​

Question 15

Find symmetric equations of the normal line to the surface at the point .

A)
B)
C)
D)
E)

Question 16

Describe the domain of the function .

A)
B)
C)
D)
E)

Question 17

Determine the continuity of the function .

A) continuous except at
B) continuous except at
C) continuous except at
D) continuous except at
E) continuous everywhere

Question 18

Sketch the surface given by the function.

A)
B)
C)
D)
E) none of these

Question 19

Find the least squares regression line for the points shown in the graph.

A)
B)
C)
D)
E)

Question 20

Find the second partial derivative for the function with respect to x.

A)
B)
C)
D)
E)

Question 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)

Question 22

Examine the function for relative extrema and saddle points.

A) saddle point: ; relative minimum:
B) saddle point: ; relative minimum:
C) relative minimum: ; relative maximum:
D) relative minimum: ; relative maximum:
E) saddle points: ,

Question 23

Find symmetric equations of the tangent line to the curve of intersection of the surfaces , at the point .

A)
B)
C)
D)
E)

Question 24

Find symmetric equations of the normal line to the surface at the point .

A)
B)
C)
D)
E)

Question 25

Find the absolute extrema of the function over the triangular region in the xy-plane with vertices and .

A) absolute maximum: at absolute minimum: at and along the line
B) absolute maximum: at absolute minimum: at and along the line
C) absolute maximum: at absolute minimum: at and along the line
D) absolute maximum: at absolute minimum: at
E) absolute maximum: at absolute minimum: at

Question 26

Find the partial derivative for the function .

A)
B)
C)
D)
E)

Question 27

Find and simplify the function at the given value .

A) -45
B) -18
C) 45
D) 18
E) 0

Question 28

For function , find the maximum value of the directional derivative at (3,2). ​

A) /9 ​
B) /36 ​
C) /144 ​
D) /72 ​
E) /108 ​

Question 29

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 7% in v and 5% in r. ​

A) 23%
B) 25%
C) 19%
D) 40%
E) 24%

Question 30

Differentiate implicitly to find , given . ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 31

Determine the continuity of the function .

A) continuous except at
B) continuous except at
C) continuous for all
D) continuous for all
E) continuous everywhere

Question 32

Find three positive numbers x, y, and z whose sum is 21 and the sum of the squares is a minimum.

A)
B)
C)
D)
E)

Question 33

A company manufactures two types of sneakers: running shoes and basketball shoes. The total revenue from units of running shoes and units of basketball shoes is: ​ ,
​
Where and are in thousands of units. Find and so as to maximize the revenue.
​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 34

Find and simplify the function at the given value . ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 35

The temperature at the point on a metal plate is modeled by , . Find the direction of greatest increase in heat from the point . ​

A) The greatest increase is in the direction of the gradient .
​
B) The greatest increase is in the direction of the gradient .
​
C) The greatest increase is in the direction of the gradient .
​
D) The greatest increase is in the direction of the gradient .
​
E) The greatest increase is in the direction of the gradient .
​

Question 36

Find the directional derivative of the function at P in the direction of .

A) 30/
B) -10/
C) 6/
D) 46/
E) -46/

Question 37

Examine the function for relative extrema and saddle points.

A) saddle point:
B) relative minimum:
C) relative minimum:
D) saddle point:
E) saddle point:

Question 38

Find the directional derivative of the function at in the direction of . Round your answer to two decimal places. ​

A) -8.02
B) -1.71
C) -5.67
D) -1.03
E) -5.14

Question 39

Find the total differential for the function .

A)
B)
C)
D)
E)

Question 40

Find an equation of the tangent plane to the surface at the point . ​

A)
B)
C)
D)
E)

Question 41

Determine the continuity of the function .

A) continuous except at
B) continuous except at
C) continuous except at
D) continuous except at
E) continuous everywhere

Question 42

The material for constructing the base of an open box costs 1.5 times as much per unit area as the material for constructing the sides. For a fixed amount of money $400.00, find the dimensions of the box of largest volume that can be made.

A)
B)
C)
D)
E)

Question 43

The temperature at the point on a metal plate is modeled by , . Find the directions of no change in heat on the plate from the point .

A) There will be no change in directions perpendicular to the gradient .
B) There will be no change in directions parallel to the gradient .
C) There will be no change in directions parallel to the gradient .
D) There will be no change in directions perpendicular to the gradient .
E) There will be no change in directions perpendicular to the gradient .

Question 44

The radius of a right circular cylinder is increasing at a rate of 8 inches per minute, and the height is decreasing at a rate of 6 inches per minute. What is the rate of change of the surface area when the radius is 16 inches and the height is 37 inches?

A)
B)
C)
D)
E)

Question 45

Use Lagrange multipliers to minimize the function subject to the following constraint. ​ ​
Assume that x, y, and z are positive.
​

A)
B)
C)
D)
E)

Question 46

Find the partial derivative for the function . ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 47

Find the limit . ​ ​

A) ​
B) ​
C) ​
D) ​
E) 0
​

Question 48

Given , calculate by evaluating and . Round your answer to four decimal places. ​

A) -4.4276
B) -38.4276
C) -14.4276
D) 0.1448
E) 5.1448

Question 49

For , evaluate at the point .

A)
B)
C)
D)
E)

Question 50

Examine the function for relative extrema.

A) relative maximum:
B) relative minimum:
C) relative maximum:
D) relative minimum:
E) no relative extrema

Question 51

Find the first partial derivative for the function with respect to z.

A)
B)
C)
D)
E)

Question 52

Find the partial derivative for the function . ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 53

Use the gradient to find a normal vector to the graph of the equation at the given point.

A)
B)
C)
D)
E)

Question 54

Find using the appropriate Chain Rule for where and . ​

A) 104t
B) 52t
C) 80t
D) 44t
E) 156t

Question 55

Let , where , , and . Find . ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 56

Find the maximum value of the directional derivative at the point of the function . Round your answer to two decimal places.

A) 0.32
B) 0.14
C) 0.33
D) 0.10
E) 0.04

Question 57

Suppose a corporation manufactures candles at two locations. The cost of producing units at location 1 is and the cost of producing units at location 2 is . The candles sell for $12 per unit. Find the quantity that should be produced at each location to maximize the profit . ​

A)
B)
C)
D)
E)

Question 58

A rectangular box with an open top has a length of x feet, a width of y feet, and a height of z feet. It costs $1.80 per square foot to build the base and $0.45 per square foot to build the sides. Write the cost C of constructing the box as a function of x, y, and z.

A)
B)
C)
D)
E)

Question 59

Describe the domain of the function .

A)
B)
C)
D)
E)

Question 60

Let represent the temperature at each point on the sphere . Find the maximum temperature on the curve formed by the intersection of the sphere and the plane .

A)
B)
C)
D)
E)

Question 61

Differentiate implicitly to find . ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 62

Find the limit for the function . ​

A) -4y
B) 0
C)
D) 4y
E) 4x

Question 63

Given , use the total differential to approximate at towards . Round your answer to four decimal places. ​

A) -32.5115
B) -0.0200
C) 25.4885
D) 2.9770
E) -9.5115

Question 64

For the function given by describe the level surface given by . ​

A) sphere of radius 8 centered at the origin
​
B) circle of radius 8 centered at the origin
​
C) elliptic cone centered at the origin
​
D) sphere of radius 64 centered at the origin
​
E) right circular cone centered at the origin
​

Question 65

Find the highest point on the curve of intersection of the following surfaces. Cone: , Plane:

A)
B)
C)
D)
E)

Question 66

Find the partial derivative for the function . ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 67

Find the limit.

A)
B)
C) 0
D) 1
E)

Question 68

Examine the function for relative extrema and saddle points. ​ ​

A) relative minimum:
B) relative minimum:
C) relative maximum:
D) saddle point:
E) saddle point:

Question 69

Sketch the level curves of the function for the given c-values .

A)
B)
C)
D)
E)

Question 70

Find using the appropriate Chain Rule for where and , and evaluate the partial derivative at and . Round your answer to two decimal places. ​

A) 1,210.01
B) 806.73
C) 1,210.16
D) 806.71
E) 1,209.34

Question 71

Determine the continuity of the function .

A) continuous except at
B) continuous for
C) continuous except at
D) continuous except at
E) continuous for

Question 72

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)

Question 73

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) -1.15
B) -1.94
C) -0.77
D) -4.95
E) -2.76

Question 74

Use the gradient to find the directional derivative of the function at P in the direction of Q.

A)
B)
C)
D)
E)

Question 75

Find the minimum distance from the point to the plane .

A) ​
B) ​
C) ​
D) ​
E) ​

Question 76

Use Lagrange multipliers to minimize the function subject to the following two constraints. Assume that x, y, and z are nonnegative.

A)
B)
C)
D)
E)

Question 77

Use Lagrange multipliers to find the maximum value of where and , subject to the constraint . ​

A)
B)
C)
D)
E)

Question 78

Use polar coordinates to find the limit.

Question 79

Given , use the total differential to approximate at towards . Round your answer to two decimal places. ​

A) 4.08
B) -10.92
C) 40.08
D) 22.16
E) 25.16

Question 80

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) 127
B) 125
C) 56
D) 104
E) 70