Question 1

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

Accepted Answer

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

Question 2

If   find   and   . ​

Accepted Answer

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

Question 3

The demand functions for qA and qB units of two related products, A and B, are given. ​   ​
Where pA and pB are in dollars.
Find the marginal demand of qA with respect to pB and the marginal demand of qB with respect to pA. Are the two goods competitive or complementary?
​

Accepted Answer

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

Question 4

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

Accepted Answer

A) 79 
B) ​45 
C) ​56 
D) ​50 
E) ​157 
A) 79 
B) ​45 
C) ​56 
D) ​50 
E) ​157

Question 5

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

Accepted Answer

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

Question 6

If   , find   . ​

Accepted Answer

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

Question 7

Suppose that the joint cost function for two products is   dollars. Find the marginal cost with respect to x. ​

Accepted Answer

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

Question 8

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 9

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 a pair of prices p1 and p2 such that the demands for the products will be equal. Round your answers to the nearest cent.
​

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

Suppose that 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 and interpret the marginal cost with respect to y if 6 units of product X and 10 units of product Y are produced. ​

Accepted Answer

A) If x remains at 6, the expected change in cost for an 11th unit of y is $32. 
B) If x remains at 6, the expected change in cost for an 11th unit of y is $15. 
C) If x remains at 6, the expected change in cost for an 11th unit of y is $20. 
D) If x remains at 6, the expected change in cost for an 11th unit of y is $36. 
E) If x remains at 6, the expected change in cost for an 11th unit of y is $51. 
A) If x remains at 6, the expected change in cost for an 11th unit of y is $32. 
B) If x remains at 6, the expected change in cost for an 11th unit of y is $15. 
C) If x remains at 6, the expected change in cost for an 11th unit of y is $20. 
D) If x remains at 6, the expected change in cost for an 11th unit of y is $36. 
E) If x remains at 6, the expected change in cost for an 11th unit of y is $51.

Question 12

Suppose the Cobb-Douglas production function for a company is given by   , where x is the company's capital investment and y is the size of the labor force (in work-hours). Find the marginal productivity of x. Then, find the marginal productivity of y. ​

Accepted Answer

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

Question 13

Suppose that if $100,000 is borrowed to purchase a home, then the monthly payment R is a function of the interest rate i (expressed as a percent) and the number of years n before the mortgage is paid, and is given by   . Find   and interpret your answer. Round your answer to two decimal places. ​

Accepted Answer

A)   
The monthly payment for $100,000 borrowed for 50 years at 2.25% interest is $141.51. 
B)   
The monthly payment for $100,000 borrowed for 50 years at 2.25% interest is $4,265.02. 
C)   
The monthly payment for $100,000 borrowed for 50 years at 2.25% interest is $277.78. 
D)   
The monthly payment for $100,000 borrowed for 50 years at 2.25% interest is $3,801.72. 
E)   
The monthly payment for $100,000 borrowed for 50 years at 2.25% interest is $278.78. 
A)   
The monthly payment for $100,000 borrowed for 50 years at 2.25% interest is $141.51. 
B)   
The monthly payment for $100,000 borrowed for 50 years at 2.25% interest is $4,265.02. 
C)   
The monthly payment for $100,000 borrowed for 50 years at 2.25% interest is $277.78. 
D)   
The monthly payment for $100,000 borrowed for 50 years at 2.25% interest is $3,801.72. 
E)   
The monthly payment for $100,000 borrowed for 50 years at 2.25% interest is $278.78.

Question 14

A new food is designed to add weight to mature beef cattle. The weight in pounds is given by   , where x is the number of units of the first ingredient and y is the number of units of the second ingredient. How many units of each ingredient will maximize the weight? ​

Accepted Answer

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

Question 15

In economics, the most economical quantity Q of goods (TVs, dresses, gallons of paint, etc.) for a store to order is given by Wilson's lot size formula   , where K is the cost of placing the order, M is the number of items sold per week, and h is the weekly holding cost for each item (the cost of storage space, utilities, taxes, security, etc.). Find   . ​

Accepted Answer

A) 205 
B) 14 
C) 580 
D) 42,000 
E) 1,640 
A) 205 
B) 14 
C) 580 
D) 42,000 
E) 1,640

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

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 y.

Accepted Answer

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

Question 18

If   , find   . ​

Accepted Answer

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

Question 19

If   , find   and   . ​

Accepted Answer

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

Question 20

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) ​

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

If find and .

The demand functions for q_A and q_B units of two related products, A and B, are given. Where p_A and p_B are in dollars. Find the marginal demand of q_A with respect to p_B and the marginal demand of q_B with respect to p_A. Are the two goods competitive or complementary?

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

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

If , find .

Suppose that the joint cost function for two products is dollars. Find the marginal cost with respect to x.

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.

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 a pair of prices p₁ and p₂ such that the demands for the products will be equal. Round your answers to the nearest cent.

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

Suppose that 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 and interpret the marginal cost with respect to y if 6 units of product X and 10 units of product Y are produced.

Suppose the Cobb-Douglas production function for a company is given by , where x is the company's capital investment and y is the size of the labor force (in work-hours). Find the marginal productivity of x. Then, find the marginal productivity of y.

Suppose that if $100,000 is borrowed to purchase a home, then the monthly payment R is a function of the interest rate i (expressed as a percent) and the number of years n before the mortgage is paid, and is given by . Find and interpret your answer. Round your answer to two decimal places.

A new food is designed to add weight to mature beef cattle. The weight in pounds is given by , where x is the number of units of the first ingredient and y is the number of units of the second ingredient. How many units of each ingredient will maximize the weight?

Find the maximum value of subject to the constraint .

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 y.

If , find .

If , find 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.

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

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

If find and . ​

The demand functions for qA and qB units of two related products, A and B, are given. ​ ​ Where pA and pB are in dollars. Find the marginal demand of qA with respect to pB and the marginal demand of qB with respect to pA. Are the two goods competitive or complementary? ​

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

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

If , find . ​

Suppose that the joint cost function for two products is dollars. Find the marginal cost with respect to x. ​

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. ​

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 a pair of prices p1 and p2 such that the demands for the products will be equal. Round your answers to the nearest cent. ​

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

Suppose the Cobb-Douglas production function for a company is given by , where x is the company's capital investment and y is the size of the labor force (in work-hours). Find the marginal productivity of x. Then, find the marginal productivity of y. ​

Suppose that if $100,000 is borrowed to purchase a home, then the monthly payment R is a function of the interest rate i (expressed as a percent) and the number of years n before the mortgage is paid, and is given by . Find and interpret your answer. Round your answer to two decimal places. ​

A new food is designed to add weight to mature beef cattle. The weight in pounds is given by , where x is the number of units of the first ingredient and y is the number of units of the second ingredient. How many units of each ingredient will maximize the weight? ​

Find the maximum value of subject to the constraint . ​

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 y.

If , find . ​

If , find 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. ​

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

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

If find and .

The demand functions for q_A and q_B units of two related products, A and B, are given. Where p_A and p_B are in dollars. Find the marginal demand of q_A with respect to p_B and the marginal demand of q_B with respect to p_A. Are the two goods competitive or complementary?

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

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

If , find .

Suppose that the joint cost function for two products is dollars. Find the marginal cost with respect to x.

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.

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 a pair of prices p₁ and p₂ such that the demands for the products will be equal. Round your answers to the nearest cent.

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

Suppose the Cobb-Douglas production function for a company is given by , where x is the company's capital investment and y is the size of the labor force (in work-hours). Find the marginal productivity of x. Then, find the marginal productivity of y.

Suppose that if $100,000 is borrowed to purchase a home, then the monthly payment R is a function of the interest rate i (expressed as a percent) and the number of years n before the mortgage is paid, and is given by . Find and interpret your answer. Round your answer to two decimal places.

A new food is designed to add weight to mature beef cattle. The weight in pounds is given by , where x is the number of units of the first ingredient and y is the number of units of the second ingredient. How many units of each ingredient will maximize the weight?

Find the maximum value of subject to the constraint .

If , find .

If , find 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.