Question 1

Find three positive numbers x, y, and z whose sum is 21 and product is a maximum.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 2

Find and simplify the function   at the given value   .

Accepted Answer

A)  -102 
B)  -72 
C)  -60 
D)  -54 
E)  19 
A)  -102 
B)  -72 
C)  -60 
D)  -54 
E)  19

Question 3

Find the least squares regression line for the points .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 4

Identify any extrema of the function   by recognizing its given form or its form after completing the square.

Accepted Answer

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

Question 5

Find the partial derivative   for the function   . ​

Accepted Answer

A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​ 
A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​

Question 6

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

Accepted Answer

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

Question 7

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.

Accepted Answer

A)  18 % 
B)  11 % 
C)  24 % 
D)  12 % 
E)  7 % 
A)  18 % 
B)  11 % 
C)  24 % 
D)  12 % 
E)  7 %

Question 8

Examine the function   for relative extrema.

Accepted Answer

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

Question 9

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 120 volts are applied to a 5000-ohm resistor and the possible percent errors in measuring E and R are 8% and 2%, respectively. ​

Accepted Answer

A)  23% 
B)  43% 
C)  46% 
D)  38% 
E)  18% 
A)  23% 
B)  43% 
C)  46% 
D)  38% 
E)  18%

Question 10

Find   for function   .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 11

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   . ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 12

Find the critical points of the function   , and from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

Accepted Answer

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

Question 13

Find a unit normal vector to the surface   at the point   .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 14

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 15

Find the maximum production level   if the total cost of labor (at $70 per unit) and capital (at $48 per unit) is limited to $250,000, where x is the number of units of labor and y is the number of units of capital. Round your answer to the nearest integer.

Accepted Answer

A)  35,164 units produced 
B)  12,055 units produced 
C)  24,390 units produced 
D)  243,897 units produced 
E)  15,229 units produced 
A)  35,164 units produced 
B)  12,055 units produced 
C)  24,390 units produced 
D)  243,897 units produced 
E)  15,229 units produced

Question 16

For   find all values of x and y such that   and   simultaneously. ​

Accepted Answer

A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​ 
A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​

Question 17

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 y if the utility function is   . ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 18

Find the unit normal vector to the surface   at the point   . ​

Accepted Answer

A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​ 
A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​

Question 19

Identify any extrema of the function   by recognizing its given form or its form after completing the square.

Accepted Answer

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

Question 20

For   find all values of x and y such that   and   simultaneously. ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Find three positive numbers x, y, and z whose sum is 21 and product is a maximum.

Find and simplify the function at the given value .

Find the least squares regression line for the points .

Identify any extrema of the function by recognizing its given form or its form after completing the square.

Find the partial derivative for the function .

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

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.

Examine the function for relative extrema.

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 120 volts are applied to a 5000-ohm resistor and the possible percent errors in measuring E and R are 8% and 2%, respectively.

Find for function .

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 .

Find the critical points of the function , and from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

Find a unit normal vector to the surface at the point .

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

Find the maximum production level if the total cost of labor (at $70 per unit) and capital (at $48 per unit) is limited to $250,000, where x is the number of units of labor and y is the number of units of capital. Round your answer to the nearest integer.

For find all values of x and y such that and simultaneously.

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 y if the utility function is .

Find the unit normal vector to the surface at the point .

Identify any extrema of the function by recognizing its given form or its form after completing the square.

For find all values of x and y such that and simultaneously.

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Multiple Integration

Vector Anal

Filters

Exam 13: Functions of Several Variables

Find three positive numbers x, y, and z whose sum is 21 and product is a maximum.

Find and simplify the function at the given value .

Find the least squares regression line for the points .

Identify any extrema of the function by recognizing its given form or its form after completing the square.

Find the partial derivative for the function . ​

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

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.

Examine the function for relative extrema.

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 120 volts are applied to a 5000-ohm resistor and the possible percent errors in measuring E and R are 8% and 2%, respectively. ​

Find for function .

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 . ​

Find the critical points of the function , and from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

Find a unit normal vector to the surface at the point .

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

Find the maximum production level if the total cost of labor (at $70 per unit) and capital (at $48 per unit) is limited to $250,000, where x is the number of units of labor and y is the number of units of capital. Round your answer to the nearest integer.

For find all values of x and y such that and simultaneously. ​

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 y if the utility function is . ​

Find the unit normal vector to the surface at the point . ​

Identify any extrema of the function by recognizing its given form or its form after completing the square.

For find all values of x and y such that and simultaneously. ​

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Multiple Integration

Vector Anal

Filters

Find the partial derivative for the function .

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

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 120 volts are applied to a 5000-ohm resistor and the possible percent errors in measuring E and R are 8% and 2%, respectively.

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 .

For find all values of x and y such that and simultaneously.

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 y if the utility function is .

Find the unit normal vector to the surface at the point .

For find all values of x and y such that and simultaneously.