Question 1

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

Question 2

Test for relative maximum and minimum. ​  ​

Accepted Answer

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

Question 3

Give the domain of the function  . ​

Accepted Answer

A) domain: all reals 
And 
With  
B) domain: all reals 
And 
With  
C) domain: all reals 
And 
With  
D) domain: all reals 
And 
With  
E) domain: all reals 
And 
With

Question 4

Find the minimum value of  subject to the condition  . ​

Accepted Answer

A)   
B) ​  
C) ​  
D) ​  
E) ​

Question 5

Find the minimum value of  subject to the constraint  . ​

Accepted Answer

A)   
B) ​  
C) ​  
D) ​  
E) ​

Question 6

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

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 7

The following table shows the median age of automobiles and trucks on the road for a certain country. Use linear regression to find the linear equation that is the best fit for the data, with x equal to the number of years past 1980. ​

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 8

Give the domain of the function  . ​

Accepted Answer

A) domain: all reals with 
And  
B) domain: all reals with  
C) domain: all reals with  
D) domain: all reals with  
E) domain: all reals with 
And

Question 9

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)

Question 10

The total cost (in dollars) of producing 1 unit of a product is  , where x is the cost per pound of raw materials and y is the cost per hour of labor. If labor costs are held constant, at what rate will the total cost increase for each increase of $1 per pound in material cost? If material costs are held constant, at what rate will the total cost increase for each $1 per hour increase in labor costs? ​

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 11

The future value S of an investment earning 7% compounded continuously is a function of the principal P and the length of time t that the principal been invested. It is given by  . Find the future value of $42,000 invested for 30 years. ​

Accepted Answer

A) $45,045.34 
B) $343,326.42 
C) $300,979.14 
D) $8.17 
E) $342,979.14

Question 12

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

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 13

If  , find  and  . ​

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 14

If  , find  .

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 15

If  , find  and  . ​

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 16

Dr. Paul Siple conducted studies testing the effect of wind on the formation of ice at various temperatures and developed the concept of wind chill, which we hear reported during winter weather reports. One form of the formula that meteorologists use to calculate wind chill temperature is  , where s is wind speed and t is the actual air temperature. Find  when the temperature is  and the wind speed is 40 mph. Round your answer to two decimal places. ​

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 17

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
​

Question 18

Use the function  to find  . ​

Accepted Answer

A)    
B)    
C)    
D)    
E)

Question 19

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

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 20

If  , find  . ​

Accepted Answer

A)   
B)   
C)   
D)   
E)

Exam 15: Functions of Two or More Variables

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

Test for relative maximum and minimum.

Give the domain of the function .

Find the minimum value of subject to the condition .

Find the minimum value of subject to the constraint .

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

The following table shows the median age of automobiles and trucks on the road for a certain country. Use linear regression to find the linear equation that is the best fit for the data, with x equal to the number of years past 1980.

Give the domain of the function .

The future value S of an investment earning 7% compounded continuously is a function of the principal P and the length of time t that the principal been invested. It is given by . Find the future value of $42,000 invested for 30 years.

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

If , find and .

If , find .

If , find and .

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 .

Use the function to find .

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

If , find .

Exam 15: Functions of Two or More Variables

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

Test for relative maximum and minimum. ​ ​

Give the domain of the function . ​

Find the minimum value of subject to the condition . ​

Find the minimum value of subject to the constraint . ​

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

The following table shows the median age of automobiles and trucks on the road for a certain country. Use linear regression to find the linear equation that is the best fit for the data, with x equal to the number of years past 1980. ​

Give the domain of the function . ​

The future value S of an investment earning 7% compounded continuously is a function of the principal P and the length of time t that the principal been invested. It is given by . Find the future value of $42,000 invested for 30 years. ​

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

If , find and . ​

If , find .

If , find and . ​

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 .

Use the function to find . ​

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

If , find . ​

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

Test for relative maximum and minimum.

Give the domain of the function .

Find the minimum value of subject to the condition .

Find the minimum value of subject to the constraint .

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

The following table shows the median age of automobiles and trucks on the road for a certain country. Use linear regression to find the linear equation that is the best fit for the data, with x equal to the number of years past 1980.

Give the domain of the function .

The future value S of an investment earning 7% compounded continuously is a function of the principal P and the length of time t that the principal been invested. It is given by . Find the future value of $42,000 invested for 30 years.

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

If , find and .

If , find and .

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 .

Use the function to find .

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

If , find .