Question 1

If  , find  . ​

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 2

Evaluate the function  to find  . ​

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 3

If  , find  . ​

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 4

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

Question 5

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) ​

Question 6

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

Question 7

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

Question 8

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

Question 9

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)

Question 10

Suppose that the following table shows the average hourly earnings for full-time production workers in various industries for selected years. The linear regression equation for hourly earnings as a function of time is given by  (where x is the number of years after 1970 and y is the average hourly wage). What does this model predict for the average hourly earnings in 2024? Round your answer to two decimal places. ​

Accepted Answer

A) $42.51 
B) $327.10 
C) $43.41 
D) $326.20 
E) $325.80

Question 11

Suppose wind and cold temperatures combine to make the air temperature feel colder than it actually is. This combination is reported as wind chill. The following table shows that wind chill temperatures, WC, are a function of wind speed,  , and air temperature,  . If  , use the table to find  and interpret the answer. ​
Air Temperature   ​

Accepted Answer

A)  
For a wind speed of 5 mph and air temperature of 
, the wind-chill temperature is 
) 
B)  
For a wind speed of 5 mph and air temperature of 
, the wind-chill temperature is 
) 
C)  
For a wind speed of 5 mph and air temperature of 
, the wind-chill temperature is 
) 
D)  
For a wind speed of 5 mph and air temperature of 
, the wind-chill temperature is 
) 
E)  
For a wind speed of 5 mph and air temperature of 
, the wind-chill temperature is 
)

Question 12

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)

Question 13

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)

Question 14

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

Question 15

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) ​

Question 16

If  , find  . ​

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 17

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

Question 18

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

Question 19

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)

Exam 15: Functions of Two or More Variables

If , find . ​

Evaluate the function to find . ​

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. ​

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? ​

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

Test for relative maximum and minimum. ​ ​

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. ​

Test for relative maximum and minimum. ​

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. ​

If , find . ​

Test for relative maximum and minimum. ​ ​

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. ​

If , find .

Evaluate the function to find .

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.

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?

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

Test for relative maximum and minimum.

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.

Test for relative maximum and minimum.

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.

If , find .

Test for relative maximum and minimum.

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.