Question 1

Test for relative maximum and minimum. ​   ​

Accepted Answer

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

Question 2

If   , find   . ​

Accepted Answer

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

Question 3

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 4

If   , find   . ​

Accepted Answer

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

Question 5

Test for relative maximum and minimum. ​   ​

Accepted Answer

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

Question 6

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 7

If   , find   . ​

Accepted Answer

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

Question 8

Find the dimensions (in centimeters) of the box with square base, open top, and volume 1,098,500 cubic centimeters that requires the least materials to make. ​

Accepted Answer

A)  base = 130 cm; height = 65 cm 
B)  base =   cm; height =   cm 
C)  base =   cm; height = 130 cm 
D)  base = 65 cm; height = 260 cm 
E)  base = 10,985 cm; height = 100 cm 
A)  base = 130 cm; height = 65 cm 
B)  base =   cm; height =   cm 
C)  base =   cm; height = 130 cm 
D)  base = 65 cm; height = 260 cm 
E)  base = 10,985 cm; height = 100 cm

Question 9

Find the values for each of the dimensions of an open-top box of length x, width y, and height   (in inches) such that the box requires the least amount of material to make. ​

Accepted Answer

A)  length = 8 inches
Width = 8 inches
Height = 4 inches 
B)  length = 4 inches
Width = 8 inches
Height = 8 inches 
C)  length = 8 inches
Width = 4 inches
Height = 8 inches 
D)  length = 4 inches
Width = 4 inches
Height = 16 inches 
E)  length = 16 inches
Width = 4 inches
Height = 4 inches 
A)  length = 8 inches
Width = 8 inches
Height = 4 inches 
B)  length = 4 inches
Width = 8 inches
Height = 8 inches 
C)  length = 8 inches
Width = 4 inches
Height = 8 inches 
D)  length = 4 inches
Width = 4 inches
Height = 16 inches 
E)  length = 16 inches
Width = 4 inches
Height = 4 inches

Question 10

Test   for relative maximum and minimum. ​

Accepted Answer

A)  A relative minimum at (0,0,8) 
B)  A relative maximum at (0,0,3) 
C)  A relative maximum at (1,0,8) 
D)  A relative minimum at (0,0,3) 
E)  A relative maximum at (3,0,0) 
A)  A relative minimum at (0,0,8) 
B)  A relative maximum at (0,0,3) 
C)  A relative maximum at (1,0,8) 
D)  A relative minimum at (0,0,3) 
E)  A relative maximum at (3,0,0)

Question 11

A company has the Cobb-Douglas production function   , where x is the number of units of labor, y is the number of units of capital, and z is the units of production. Suppose labor costs $200 per unit, capital costs $150 per unit, and the total cost of labor and capital is limited to $300,000. Find the number of units of labor and the number of units of capital that maximize production. ​

Accepted Answer

A)  1,500 units of labor; 2,000 units of capital 
B)  1,200 units of labor; 2,000 units of capital 
C)  1,500 units of labor; 400 units of capital 
D)  1,200 units of labor; 400 units of capital 
E)  12,000 units of labor; 4,000 units of capital 
A)  1,500 units of labor; 2,000 units of capital 
B)  1,200 units of labor; 2,000 units of capital 
C)  1,500 units of labor; 400 units of capital 
D)  1,200 units of labor; 400 units of capital 
E)  12,000 units of labor; 4,000 units of capital

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

On the basis of past experience, a company has determined that its sales revenue (in dollars) is related to its advertising according to the formula   , where x is the amount spent on radio advertising and y is the amount spent on television advertising. If the company plans to spend $30,000 on these two means of advertising, how much should it spend on each method to maximize its sales revenue? ​

Accepted Answer

A)  $8,401.47 on radio; $21,598.53 on television 
B)  $10,801.89 on radio; $19,198.11 on television 
C)  $12,002.10 on radio; $17,997.90 on television 
D)  $14,402.52 on radio; $15,597.48 on television 
E)  $15,602.73 on radio; $14,397.27 on television 
A)  $8,401.47 on radio; $21,598.53 on television 
B)  $10,801.89 on radio; $19,198.11 on television 
C)  $12,002.10 on radio; $17,997.90 on television 
D)  $14,402.52 on radio; $15,597.48 on television 
E)  $15,602.73 on radio; $14,397.27 on television

Question 14

If   , find   . ​

Accepted Answer

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

Question 15

Suppose that an indifference curve for two goods, X and Y, has the equation   . If 30 units of X are purchased, how many units of Y must be purchased to remain on this indifference curve? ​

Accepted Answer

A)    units of Y. 
B)  57,600 units of Y. 
C)  64 units of Y. 
D)  87 units of Y. 
E)  101 units of Y. 
A)    units of Y. 
B)  57,600 units of Y. 
C)  64 units of Y. 
D)  87 units of Y. 
E)  101 units of Y.

Question 16

Test for relative maximum and minimum. ​   ​

Accepted Answer

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

Question 17

Use the function   to find   . ​

Accepted Answer

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

Question 18

If   , find   and   . ​

Accepted Answer

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

Question 19

A biologist has conjectured that the weight of a certain lizard in pounds as a function of age A in years and length L in inches is given by   . If a lizard were to remain 13 inches long as it aged from 4 to 5 years old, then determine how much the lizard's weight would be predicted to increase over that year. Round to three decimal places. ​

Accepted Answer

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

Question 20

Suppose that a gas satisfies the universal gas law,   with n equal to 40 moles of the gas and R, the universal gas constant, equal to 0.082054. What is V if   (kelvins, the units in which temperature is measured on the Kelvin scale) and   atmosphere? Round your answer to four decimal places. ​

Accepted Answer

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

Test for relative maximum and minimum.

If , find .

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

If , find .

Test for relative maximum and minimum.

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

If , find .

Find the dimensions (in centimeters) of the box with square base, open top, and volume 1,098,500 cubic centimeters that requires the least materials to make.

Find the values for each of the dimensions of an open-top box of length x, width y, and height (in inches) such that the box requires the least amount of material to make.

Test for relative maximum and minimum.

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

If , find .

Suppose that an indifference curve for two goods, X and Y, has the equation . If 30 units of X are purchased, how many units of Y must be purchased to remain on this indifference curve?

Test for relative maximum and minimum.

Use the function to find .

If , find and .

Suppose that a gas satisfies the universal gas law, with n equal to 40 moles of the gas and R, the universal gas constant, equal to 0.082054. What is V if (kelvins, the units in which temperature is measured on the Kelvin scale) and atmosphere? Round your answer to four decimal places.

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

Test for relative maximum and minimum. ​ ​

If , find . ​

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

If , find . ​

Test for relative maximum and minimum. ​ ​

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

If , find . ​

Find the dimensions (in centimeters) of the box with square base, open top, and volume 1,098,500 cubic centimeters that requires the least materials to make. ​

Find the values for each of the dimensions of an open-top box of length x, width y, and height (in inches) such that the box requires the least amount of material to make. ​

Test for relative maximum and minimum. ​

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

If , find . ​

Suppose that an indifference curve for two goods, X and Y, has the equation . If 30 units of X are purchased, how many units of Y must be purchased to remain on this indifference curve? ​

Test for relative maximum and minimum. ​ ​

Use the function to find . ​

If , find and . ​

Suppose that a gas satisfies the universal gas law, with n equal to 40 moles of the gas and R, the universal gas constant, equal to 0.082054. What is V if (kelvins, the units in which temperature is measured on the Kelvin scale) and atmosphere? Round your answer to four decimal places. ​

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

Test for relative maximum and minimum.

If , find .

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

If , find .

Test for relative maximum and minimum.

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

If , find .

Find the dimensions (in centimeters) of the box with square base, open top, and volume 1,098,500 cubic centimeters that requires the least materials to make.

Find the values for each of the dimensions of an open-top box of length x, width y, and height (in inches) such that the box requires the least amount of material to make.

Test for relative maximum and minimum.

If , find .

Suppose that an indifference curve for two goods, X and Y, has the equation . If 30 units of X are purchased, how many units of Y must be purchased to remain on this indifference curve?

Test for relative maximum and minimum.

Use the function to find .

If , find and .

Suppose that a gas satisfies the universal gas law, with n equal to 40 moles of the gas and R, the universal gas constant, equal to 0.082054. What is V if (kelvins, the units in which temperature is measured on the Kelvin scale) and atmosphere? Round your answer to four decimal places.