Question 1

Test for relative maximum and minimum. ​   ​

Accepted Answer

A) relative maxima when x = 7, y = -8 and x = 7, y = 8; relative minima when x = -7, y = 8 and x = -7, y = -8 
B) relative maximum when x = 7, y = -8; relative minimum when x = -7, y = 8 
C) saddle point when x = -7, y = -8; relative minima when x = 0, y = 0 and x = -7, y = -8 
D) saddle points when x = 7, y = -8 and x = -7, y = 8; relative maximum when x = -7, y = -8; relative minimum when x = 7, y = 8 
E) saddle points when x = 7, y = 8 and x = -7, y = -8; relative maxima when x = 7, y = -8 and x = -7, y = 8 
A) relative maxima when x = 7, y = -8 and x = 7, y = 8; relative minima when x = -7, y = 8 and x = -7, y = -8 
B) relative maximum when x = 7, y = -8; relative minimum when x = -7, y = 8 
C) saddle point when x = -7, y = -8; relative minima when x = 0, y = 0 and x = -7, y = -8 
D) saddle points when x = 7, y = -8 and x = -7, y = 8; relative maximum when x = -7, y = -8; relative minimum when x = 7, y = 8 
E) saddle points when x = 7, y = 8 and x = -7, y = -8; relative maxima when x = 7, y = -8 and x = -7, y = 8

Question 2

Suppose that x units of one input and y units of a second input result in   units of a product. What is the maximum production? ​

Accepted Answer

A) 1,675 
B) ​3,081 
C) ​1,152 
D) ​1,027 
E) ​1,425 
A) 1,675 
B) ​3,081 
C) ​1,152 
D) ​1,027 
E) ​1,425

Question 3

Suppose that the production function for a product is   , where x represents the number of work hours and y represents the available capital (per week). Find the marginal productivity of y. ​

Accepted Answer

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

Question 4

If   , find   . ​

Accepted Answer

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

Question 5

Give the domain of the function   . ​

Accepted Answer

A) domain: all reals x and y with   
B) domain: all reals x and y with   
C) domain: all reals x and y with   
D) domain: all reals x and y with  
And   
E) domain: all reals x and y with   
A) domain: all reals x and y with   
B) domain: all reals x and y with   
C) domain: all reals x and y with   
D) domain: all reals x and y with  
And   
E) domain: all reals x and y with

Question 6

Use the function   to find   . ​

Accepted Answer

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

Question 7

The cost per day to society of an epidemic is   , where C is in dollars, x is the number of people infected on a given day, and y is the number of people who die on a given day. If 18,500 people are infected and 10 people die on a given day, what is the cost to society? Round your answer to the nearest dollar. ​

Accepted Answer

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

Question 8

A firm has two plants, X and Y. Suppose that the cost of producing x units at plant X is   dollars and the cost of producing y units of the same product at plant Y is given by   dollars. If the firm has an order for 3,000 units, how many should it produce at each plant to fill this order and minimize the cost of production? ​

Accepted Answer

A) 1,500 at plant X; 1,500 at plant Y 
B) 2,500 at plant X; 500 at plant Y 
C) 1,000 at plant X; 2,000 at plant Y 
D) 3,000 at plant X; 0 at plant Y 
E) 2,000 at plant X; 1,000 at plant Y 
A) 1,500 at plant X; 1,500 at plant Y 
B) 2,500 at plant X; 500 at plant Y 
C) 1,000 at plant X; 2,000 at plant Y 
D) 3,000 at plant X; 0 at plant Y 
E) 2,000 at plant X; 1,000 at plant Y

Question 9

If   , find   . ​

Accepted Answer

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

Question 10

Find the maximum value of   subject to   ,   ,   . Round your answer to the nearest integer. ​

Accepted Answer

A) ​908 
B) ​384 
C) ​586 
D) ​1,089 
E) ​1,099 
A) ​908 
B) ​384 
C) ​586 
D) ​1,089 
E) ​1,099

Question 11

Suppose that the number of crates of an agricultural product is given by   , where x is the number of hours of and y is the number of acres of the crop. Find the marginal productivity of the number of hours of labor when   and   . ​

Accepted Answer

A) ​1,090.15 
B) 25.39 
C) ​845.45 
D) ​19.04 
E) ​-30.97 
A) ​1,090.15 
B) 25.39 
C) ​845.45 
D) ​19.04 
E) ​-30.97

Question 12

Find the slope of the tangent in the positive x-direction to the surface   at the point   .

Accepted Answer

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

Question 13

Use the function   to find   . ​

Accepted Answer

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

Question 14

The joint cost (in dollars) for two products is given by   , where x represents the quantity of product X produced and y represents the quantity of product Y produced. Find the marginal cost with respect to x if 2 units of product X and 9 units of product Y are produced. ​

Accepted Answer

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

Question 15

Test   for relative maximum and minimum. ​

Accepted Answer

A) The point  
Is a saddle point.A relative maximum occurs at  
) 
B) The point  
Is a saddle point.A relative maximum occurs at  
) 
C) The point  
Is a saddle point.A relative maximum occurs at  
) 
D) The point  
Is a saddle point.A relative minimum occurs at  
) 
E) The point  
Is a saddle point.A relative minimum occurs at  
) 
A) The point  
Is a saddle point.A relative maximum occurs at  
) 
B) The point  
Is a saddle point.A relative maximum occurs at  
) 
C) The point  
Is a saddle point.A relative maximum occurs at  
) 
D) The point  
Is a saddle point.A relative minimum occurs at  
) 
E) The point  
Is a saddle point.A relative minimum occurs at  
)

Question 16

There are different models for measuring the effects of high temperature and humidity. One of these is the Summer Simmer Index (S), given by   , where T is the air temperature (in degrees Fahrenheit) and H is the relative humidity (expressed as a decimal). ​
In a certain city, the measured temperature and humidity on a given day are:
Maximum:   with 47% humidity
Minimum:   with 78% humidity
​
Calculate the Summer Simmer Index S for both the maximum and minimum temperatures.
​

Accepted Answer

A) maximum,  
;
Minimum,   
B) maximum,  
;
Minimum,   
C) ​maximum,  
;
Minimum,  
​ 
D) maximum,  
;
Minimum,   
E) ​maximum,  
;
Minimum,   
A) maximum,  
;
Minimum,   
B) maximum,  
;
Minimum,   
C) ​maximum,  
;
Minimum,  
​ 
D) maximum,  
;
Minimum,   
E) ​maximum,  
;
Minimum,

Question 17

Suppose that the production function for a product is   , where x represents the number of work hours per month and y is the number of available machines. Determine the marginal productivity of x. ​

Accepted Answer

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

Question 18

Suppose that   tons is the production function for a product with x units of one input and y units of a second input. Find the values of x and y that will maximize production. ​

Accepted Answer

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

Question 19

Find x and y such that   subject to   ,   ,   attains a maximum value. ​

Accepted Answer

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

Question 20

Find the slope of the tangent in the positive   -direction to the surface   at   . ​

Accepted Answer

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

Test for relative maximum and minimum.

Suppose that x units of one input and y units of a second input result in units of a product. What is the maximum production?

Suppose that the production function for a product is , where x represents the number of work hours and y represents the available capital (per week). Find the marginal productivity of y.

If , find .

Give the domain of the function .

Use the function to find .

If , find .

Find the maximum value of subject to , , . Round your answer to the nearest integer.

Suppose that the number of crates of an agricultural product is given by , where x is the number of hours of and y is the number of acres of the crop. Find the marginal productivity of the number of hours of labor when and .

Find the slope of the tangent in the positive x-direction to the surface at the point .

Use the function to find .

The joint cost (in dollars) for two products is given by , where x represents the quantity of product X produced and y represents the quantity of product Y produced. Find the marginal cost with respect to x if 2 units of product X and 9 units of product Y are produced.

Test for relative maximum and minimum.

Suppose that the production function for a product is , where x represents the number of work hours per month and y is the number of available machines. Determine the marginal productivity of x.

Suppose that tons is the production function for a product with x units of one input and y units of a second input. Find the values of x and y that will maximize production.

Find x and y such that subject to , , attains a maximum value.

Find the slope of the tangent in the positive -direction to the surface at .

Algebraic Concepts

Linear Equations and Functions

Quadratic and Other Special Functions

Matrices

Inequalities and Linear Programming

Exponential and Logarithmic Functions

Mathematics of Finance

Introduction to Probability

Further Topics in Probability; Data Description

Derivatives

Applications of Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals: Techniques of Integration

Filters

Exam 15: Functions of Two or More Variables

Test for relative maximum and minimum. ​ ​

Suppose that x units of one input and y units of a second input result in units of a product. What is the maximum production? ​

Suppose that the production function for a product is , where x represents the number of work hours and y represents the available capital (per week). Find the marginal productivity of y. ​

If , find . ​

Give the domain of the function . ​

Use the function to find . ​

If , find . ​

Find the maximum value of subject to , , . Round your answer to the nearest integer. ​

Suppose that the number of crates of an agricultural product is given by , where x is the number of hours of and y is the number of acres of the crop. Find the marginal productivity of the number of hours of labor when and . ​

Find the slope of the tangent in the positive x-direction to the surface at the point .

Use the function to find . ​

The joint cost (in dollars) for two products is given by , where x represents the quantity of product X produced and y represents the quantity of product Y produced. Find the marginal cost with respect to x if 2 units of product X and 9 units of product Y are produced. ​

Test for relative maximum and minimum. ​

Suppose that the production function for a product is , where x represents the number of work hours per month and y is the number of available machines. Determine the marginal productivity of x. ​

Suppose that tons is the production function for a product with x units of one input and y units of a second input. Find the values of x and y that will maximize production. ​

Find x and y such that subject to , , attains a maximum value. ​

Find the slope of the tangent in the positive -direction to the surface at . ​

Algebraic Concepts

Linear Equations and Functions

Quadratic and Other Special Functions

Matrices

Inequalities and Linear Programming

Exponential and Logarithmic Functions

Mathematics of Finance

Introduction to Probability

Further Topics in Probability; Data Description

Derivatives

Applications of Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals: Techniques of Integration

Filters

Test for relative maximum and minimum.

Suppose that x units of one input and y units of a second input result in units of a product. What is the maximum production?

Suppose that the production function for a product is , where x represents the number of work hours and y represents the available capital (per week). Find the marginal productivity of y.

If , find .

Give the domain of the function .

Use the function to find .

If , find .

Find the maximum value of subject to , , . Round your answer to the nearest integer.

Suppose that the number of crates of an agricultural product is given by , where x is the number of hours of and y is the number of acres of the crop. Find the marginal productivity of the number of hours of labor when and .

Use the function to find .

The joint cost (in dollars) for two products is given by , where x represents the quantity of product X produced and y represents the quantity of product Y produced. Find the marginal cost with respect to x if 2 units of product X and 9 units of product Y are produced.

Test for relative maximum and minimum.

Suppose that the production function for a product is , where x represents the number of work hours per month and y is the number of available machines. Determine the marginal productivity of x.

Suppose that tons is the production function for a product with x units of one input and y units of a second input. Find the values of x and y that will maximize production.

Find x and y such that subject to , , attains a maximum value.

Find the slope of the tangent in the positive -direction to the surface at .