Question 1

Find the maximum value of the function   subject to the condition   . ​

Accepted Answer

A)  -10,650 
B)  ​-8,520 
C)  ​-6,390 
D)  ​-4,260 
E)  ​-2,130 
A)  -10,650 
B)  ​-8,520 
C)  ​-6,390 
D)  ​-4,260 
E)  ​-2,130

Question 2

Test for relative maximum and minimum. ​   ​

Accepted Answer

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

Question 3

If   , find   . ​

Accepted Answer

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

Question 4

Suppose that a manufacturer produces two brands of a product, brand 1 and brand 2. Suppose the demand for brand 1 is   thousand units and the demand for brand 2 is   thousand units, where p1 and p2 prices in dollars. If the joint cost function is   , in thousands of dollars, how many of each brand should be produced to maximize profit? ​

Accepted Answer

A)  74 units of brand 1
82 units of brand 2 
B)  9 units of brand 1
52 units of brand 2 
C)  74 units of brand 1
52 units of brand 2 
D)  14 units of brand 1
38 units of brand 2 
E)  27 units of brand 1
77 units of brand 2 
A)  74 units of brand 1
82 units of brand 2 
B)  9 units of brand 1
52 units of brand 2 
C)  74 units of brand 1
52 units of brand 2 
D)  14 units of brand 1
38 units of brand 2 
E)  27 units of brand 1
77 units of brand 2

Question 5

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

Accepted Answer

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

Question 6

In economics, the most economical quantity Q of goods (TVs, dress, 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 costs for each item (the cost of storage space, utilities, taxes, security, etc.). Calculate   for   ,   , and   . Round your answer to two decimal places.

Accepted Answer

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

Question 7

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 8

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

Accepted Answer

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

Question 9

Suppose that a company's production for Q units of its product is given by the Cobb-Douglas production function   , where K is dollars of capital investment and L is labor hours. Find Q if K = $8,600 and L = 640 hours. Round your answer to the nearest whole number. ​

Accepted Answer

A)  94,327 units 
B)  1,225 units 
C)  1,017 units 
D)  25,732 units 
E)  13,440 units 
A)  94,327 units 
B)  1,225 units 
C)  1,017 units 
D)  25,732 units 
E)  13,440 units

Question 10

If   , find   . ​

Accepted Answer

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

Question 11

If   , find   and   . ​

Accepted Answer

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

Question 12

Suppose that the cost of producing x units at plant X is   dollars and that the cost of producing y units of the same product at plant Y is   dollars. If the firm that owns the plants has an order for 240 units, how many should it produce at each plant to fill this order and minimize its cost of production? ​

Accepted Answer

A)  The production of 96 units at plant X and 144 units at plant Y minimizes the production costs. 
B)  The production of 144 units at plant X and 96 units at plant Y minimizes the production costs. 
C)  The production of 86 units at plant X and 154 units at plant Y minimizes the production costs. 
D)  The production of 111 units at plant X and 129 units at plant Y minimizes the production costs. 
E)  The production of 154 units at plant X and 86 units at plant Y minimizes the production costs. 
A)  The production of 96 units at plant X and 144 units at plant Y minimizes the production costs. 
B)  The production of 144 units at plant X and 96 units at plant Y minimizes the production costs. 
C)  The production of 86 units at plant X and 154 units at plant Y minimizes the production costs. 
D)  The production of 111 units at plant X and 129 units at plant Y minimizes the production costs. 
E)  The production of 154 units at plant X and 86 units at plant Y minimizes the production costs.

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

If   , find   . ​

Accepted Answer

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

Question 15

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 16

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 17

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

Accepted Answer

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

Question 18

Suppose an area of land is to be sprayed with two brands of pesticide: x liters of brand 1 and y liters of brand 2. If the number of thousands of insects killed is given by   . How many insects would be killed if 60 liters of brand 1 and 130 liters of brand 2 were used? Round your answer to the nearest integer. ​

Accepted Answer

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

Question 19

Test for relative maximum and minimum. ​   ​

Accepted Answer

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

Question 20

If   , find   . ​

Accepted Answer

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

Find the maximum value of the function subject to the condition .

Test for relative maximum and minimum.

If , find .

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

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.

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

Suppose that a company's production for Q units of its product is given by the Cobb-Douglas production function , where K is dollars of capital investment and L is labor hours. Find Q if K = $8,600 and L = 640 hours. Round your answer to the nearest whole number.

If , find .

If , find and .

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.

If , find .

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.

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.

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

Test for relative maximum and minimum.

If , find .

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

Find the maximum value of the function subject to the condition . ​

Test for relative maximum and minimum. ​ ​

If , find . ​

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

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. ​

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

Suppose that a company's production for Q units of its product is given by the Cobb-Douglas production function , where K is dollars of capital investment and L is labor hours. Find Q if K = $8,600 and L = 640 hours. Round your answer to the nearest whole number. ​

If , find . ​

If , find and . ​

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. ​

If , find . ​

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.

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. ​

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

Test for relative maximum and minimum. ​ ​

If , find . ​

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

Find the maximum value of the function subject to the condition .

Test for relative maximum and minimum.

If , find .

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

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.

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

Suppose that a company's production for Q units of its product is given by the Cobb-Douglas production function , where K is dollars of capital investment and L is labor hours. Find Q if K = $8,600 and L = 640 hours. Round your answer to the nearest whole number.

If , find .

If , find and .

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.

If , 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.

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

Test for relative maximum and minimum.

If , find .