Question 1

Evaluate the function   to find   . ​

Accepted Answer

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

Question 2

Find the values for each of the dimensions of a closed-top box of length x, width y, and height z (in inches) if the volume equals 32,768 cubic inches and the box requires the least amount of material to make. ​

Accepted Answer

A)  x = 46 inches, y = 46 inches, z = 46 inches 
B)  x = 32 inches, y = 32 inches, z = 32 inches 
C)  x = 41 inches, y = 41 inches, z = 41 inches 
D)  x = 45 inches, y = 45 inches, z = 45 inches 
E)  x = 34 inches, y = 34 inches, z = 34 inches 
A)  x = 46 inches, y = 46 inches, z = 46 inches 
B)  x = 32 inches, y = 32 inches, z = 32 inches 
C)  x = 41 inches, y = 41 inches, z = 41 inches 
D)  x = 45 inches, y = 45 inches, z = 45 inches 
E)  x = 34 inches, y = 34 inches, z = 34 inches

Question 3

Suppose that the utility function for two commodities is given by   and that the budget constraint is   . What values of x and y will maximize utility? ​

Accepted Answer

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

Question 4

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 5

Use the points given in the table to write the equation of the line that is the best fit for the points. Round numerical values in your answer to two decimal places. ​

Accepted Answer

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

Question 6

Suppose the production function for a product is   , where x is the capital expenditures and y is the number of work-hours. Find the marginal productivity of x. ​

Accepted Answer

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

Question 7

A company manufactures two products, A and B. If x is the number of thousands of units of A and y is the number of thousands of units of B, then the cost and revenue in thousands of dollars are ​   ​
Find the number of each type of product that should be manufactured to maximize profit.
​

Accepted Answer

A)  3,000 units of product A
48,000 units of product B 
B)  6,000 units of product A
24,000 units of product B 
C)  12,000 units of product A
12,000 units of product B 
D)  3,000 units of product A
12,000 units of product B 
E)  12,000 units of product A
48,000 units of product B 
A)  3,000 units of product A
48,000 units of product B 
B)  6,000 units of product A
24,000 units of product B 
C)  12,000 units of product A
12,000 units of product B 
D)  3,000 units of product A
12,000 units of product B 
E)  12,000 units of product A
48,000 units of product B

Question 8

The demand functions for two products are given by ​   ​
Where p1 and p2 are in dollars and q1 and q2 are numbers of units.
Find the demand for each of the products if the price of the first is   and the price of the second is   .

Accepted Answer

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

Question 9

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 x. ​

Accepted Answer

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

Question 10

Find the minimum value of   subject to the constraint   . ​

Accepted Answer

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

Question 11

Test   for relative maximum and minimum. ​

Accepted Answer

A)  A relative minimum occurs at (1,0,13) 
B)  A relative maximum occurs at (0,0,13) 
C)  A relative maximum occurs at (1,0,7) 
D)  A relative minimum occurs at (0,0,7) 
E)  A relative maximum occurs at (13,0,0) 
A)  A relative minimum occurs at (1,0,13) 
B)  A relative maximum occurs at (0,0,13) 
C)  A relative maximum occurs at (1,0,7) 
D)  A relative minimum occurs at (0,0,7) 
E)  A relative maximum occurs at (13,0,0)

Question 12

Find x and y such that   subject to the condition   attains a minimum value. ​

Accepted Answer

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

Question 13

Find the maximum value of   subject to the constraint   ,   ,   , and   . ​

Accepted Answer

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

Question 14

Suppose the production function for a product is   , where x is the capital expenditures and y is the number of work-hours. Find the marginal productivity of y. ​

Accepted Answer

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

Question 15

If   , find   . ​

Accepted Answer

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

Question 16

Find the maximum value of   subject to the constraint   . ​

Accepted Answer

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

Question 17

Evaluate the function at the given values of the independent variables. ​   ,   ,   ​

Accepted Answer

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

Question 18

Suppose that the number of thousands of insects killed by two brands of pesticide is given by   , where x is the number of liters of Brand X and y is the number of liters of Brand Y. What is the rate of change of insect deaths with respect to the number of liters of Brand X if 100 liters of each brand are currently being used? Round your answer to two decimal places.

Accepted Answer

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

Question 19

Suppose that the utility function for two commodities is given by   , and the budget constraint is   . Find the values of x and y that maximize utility. ​

Accepted Answer

A)  The maximum is at   . 
B)  The maximum is at   . 
C)  The maximum is at   . 
D)  The maximum is at   . 
E)  The maximum is at   . 
A)  The maximum is at   . 
B)  The maximum is at   . 
C)  The maximum is at   . 
D)  The maximum is at   . 
E)  The maximum is at   .

Question 20

The cost (in dollars) of manufacturing one item is given by   , where x is the cost of 1 hour of labor and y is the cost of 1 pound of material. If the hourly cost of labor is $5, and the material costs $2 per pound, what is the cost of manufacturing one of these items? ​

Accepted Answer

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

Evaluate the function to find .

Find the values for each of the dimensions of a closed-top box of length x, width y, and height z (in inches) if the volume equals 32,768 cubic inches and the box requires the least amount of material to make.

Suppose that the utility function for two commodities is given by and that the budget constraint is . What values of x and y will maximize utility?

Test for relative maximum and minimum.

Use the points given in the table to write the equation of the line that is the best fit for the points. Round numerical values in your answer to two decimal places.

Suppose the production function for a product is , where x is the capital expenditures and y is the number of work-hours. Find the marginal productivity of x.

A company manufactures two products, A and B. If x is the number of thousands of units of A and y is the number of thousands of units of B, then the cost and revenue in thousands of dollars are Find the number of each type of product that should be manufactured to maximize profit.

The demand functions for two products are given by Where p₁ and p₂ are in dollars and q₁ and q₂ are numbers of units. Find the demand for each of the products if the price of the first is and the price of the second is .

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 x.

Find the minimum value of subject to the constraint .

Test for relative maximum and minimum.

Find x and y such that subject to the condition attains a minimum value.

Find the maximum value of subject to the constraint , , , and .

Suppose the production function for a product is , where x is the capital expenditures and y is the number of work-hours. Find the marginal productivity of y.

If , find .

Find the maximum value of subject to the constraint .

Evaluate the function at the given values of the independent variables. , ,

Suppose that the utility function for two commodities is given by , and the budget constraint is . Find the values of x and y that maximize utility.

The cost (in dollars) of manufacturing one item is given by , where x is the cost of 1 hour of labor and y is the cost of 1 pound of material. If the hourly cost of labor is $5, and the material costs $2 per pound, what is the cost of manufacturing one of these items?

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 and Data Description

Derivatives

Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals - Techniques

A: Definite Integrals - Techniques

Algebraic Concepts 2

Algebraic Concepts 3

Algebraic Concepts 4

Algebraic Concepts 5

Filters

Exam 14: Functions of Two or More Variables

Evaluate the function to find . ​

Find the values for each of the dimensions of a closed-top box of length x, width y, and height z (in inches) if the volume equals 32,768 cubic inches and the box requires the least amount of material to make. ​

Suppose that the utility function for two commodities is given by and that the budget constraint is . What values of x and y will maximize utility? ​

Test for relative maximum and minimum. ​

Use the points given in the table to write the equation of the line that is the best fit for the points. Round numerical values in your answer to two decimal places. ​

Suppose the production function for a product is , where x is the capital expenditures and y is the number of work-hours. Find the marginal productivity of x. ​

A company manufactures two products, A and B. If x is the number of thousands of units of A and y is the number of thousands of units of B, then the cost and revenue in thousands of dollars are ​ ​ Find the number of each type of product that should be manufactured to maximize profit. ​

The demand functions for two products are given by ​ ​ Where p1 and p2 are in dollars and q1 and q2 are numbers of units. Find the demand for each of the products if the price of the first is and the price of the second is .

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 x. ​

Find the minimum value of subject to the constraint . ​

Test for relative maximum and minimum. ​

Find x and y such that subject to the condition attains a minimum value. ​

Find the maximum value of subject to the constraint , , , and . ​

Suppose the production function for a product is , where x is the capital expenditures and y is the number of work-hours. Find the marginal productivity of y. ​

If , find . ​

Find the maximum value of subject to the constraint . ​

Evaluate the function at the given values of the independent variables. ​ , , ​

Suppose that the utility function for two commodities is given by , and the budget constraint is . Find the values of x and y that maximize utility. ​

The cost (in dollars) of manufacturing one item is given by , where x is the cost of 1 hour of labor and y is the cost of 1 pound of material. If the hourly cost of labor is $5, and the material costs $2 per pound, what is the cost of manufacturing one of these items? ​

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 and Data Description

Derivatives

Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals - Techniques

A: Definite Integrals - Techniques

Algebraic Concepts 2

Algebraic Concepts 3

Algebraic Concepts 4

Algebraic Concepts 5

Filters

Evaluate the function to find .

Find the values for each of the dimensions of a closed-top box of length x, width y, and height z (in inches) if the volume equals 32,768 cubic inches and the box requires the least amount of material to make.

Suppose that the utility function for two commodities is given by and that the budget constraint is . What values of x and y will maximize utility?

Test for relative maximum and minimum.

Use the points given in the table to write the equation of the line that is the best fit for the points. Round numerical values in your answer to two decimal places.

Suppose the production function for a product is , where x is the capital expenditures and y is the number of work-hours. Find the marginal productivity of x.

A company manufactures two products, A and B. If x is the number of thousands of units of A and y is the number of thousands of units of B, then the cost and revenue in thousands of dollars are Find the number of each type of product that should be manufactured to maximize profit.

The demand functions for two products are given by Where p₁ and p₂ are in dollars and q₁ and q₂ are numbers of units. Find the demand for each of the products if the price of the first is and the price of the second is .

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 x.

Find the minimum value of subject to the constraint .

Test for relative maximum and minimum.

Find x and y such that subject to the condition attains a minimum value.

Find the maximum value of subject to the constraint , , , and .

Suppose the production function for a product is , where x is the capital expenditures and y is the number of work-hours. Find the marginal productivity of y.

If , find .

Find the maximum value of subject to the constraint .

Evaluate the function at the given values of the independent variables. , ,

Suppose that the utility function for two commodities is given by , and the budget constraint is . Find the values of x and y that maximize utility.

The cost (in dollars) of manufacturing one item is given by , where x is the cost of 1 hour of labor and y is the cost of 1 pound of material. If the hourly cost of labor is $5, and the material costs $2 per pound, what is the cost of manufacturing one of these items?