Question 1

Test for relative maximum and minimum. ​   ​

Accepted Answer

A)  relative maxima when x = 7, y = -8 and x = 7, y = 8; relative minima when x = -7, y = 8 and x = -7, y = -8 
B)  relative maximum when x = 7, y = -8; relative minimum when x = -7, y = 8 
C)  saddle point when x = -7, y = -8; relative minima when x = 0, y = 0 and x = -7, y = -8 
D)  saddle points when x = 7, y = -8 and x = -7, y = 8; relative maximum when x = -7, y = -8; relative minimum when x = 7, y = 8 
E)  saddle points when x = 7, y = 8 and x = -7, y = -8; relative maxima when x = 7, y = -8 and x = -7, y = 8 
A)  relative maxima when x = 7, y = -8 and x = 7, y = 8; relative minima when x = -7, y = 8 and x = -7, y = -8 
B)  relative maximum when x = 7, y = -8; relative minimum when x = -7, y = 8 
C)  saddle point when x = -7, y = -8; relative minima when x = 0, y = 0 and x = -7, y = -8 
D)  saddle points when x = 7, y = -8 and x = -7, y = 8; relative maximum when x = -7, y = -8; relative minimum when x = 7, y = 8 
E)  saddle points when x = 7, y = 8 and x = -7, y = -8; relative maxima when x = 7, y = -8 and x = -7, y = 8

Question 2

The following table shows the monthly mortgage payment R for a given interest rate 5% as a function of the amount financed A (in thousands of dollars) and the duration of the loan n in years. If   , use the table to find   . Round your answer to the nearest cent. ​

Accepted Answer

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

Question 3

Suppose that the number of crates of an agricultural product is given by   , where x is the number of hours of and y is the number of acres of the crop. Find the marginal productivity of the number of hours of labor when   and   . ​

Accepted Answer

A)  ​1,090.15 
B)  25.39 
C)  ​845.45 
D)  ​19.04 
E)  ​-30.97 
A)  ​1,090.15 
B)  25.39 
C)  ​845.45 
D)  ​19.04 
E)  ​-30.97

Question 4

Suppose that the total cost of producing 1 unit of a product is given by   dollars, where x represents the cost per pound of raw materials and y represents the hourly rate for labor. The present cost for raw materials is $11 per pound and the present hourly rate for labor is $24. How will an increase of $1 per hour in labor costs affect the total cost? Round your answer to the nearest cent. ​

Accepted Answer

A)  The total cost will increase approximately $75.28 if labor costs increase to $25 per hour. 
B)  The total cost will increase approximately $77.41 if labor costs increase to $25 per hour. 
C)  The total cost will increase approximately $133.36 if labor costs increase to $25 per hour. 
D)  The total cost will increase approximately $25.78 if labor costs increase to $25 per hour. 
E)  The total cost will increase approximately $27.95 if labor costs increase to $25 per hour. 
A)  The total cost will increase approximately $75.28 if labor costs increase to $25 per hour. 
B)  The total cost will increase approximately $77.41 if labor costs increase to $25 per hour. 
C)  The total cost will increase approximately $133.36 if labor costs increase to $25 per hour. 
D)  The total cost will increase approximately $25.78 if labor costs increase to $25 per hour. 
E)  The total cost will increase approximately $27.95 if labor costs increase to $25 per hour.

Question 5

There are different models for measuring the effects of high temperature and humidity. One of these is the Summer Simmer Index (S), given by   , where T is the air temperature (in degrees Fahrenheit) and H is the relative humidity (expressed as a decimal). ​
In a certain city, the measured temperature and humidity on a given day are:
Maximum:   with 47% humidity
Minimum:   with 78% humidity
​
Calculate the Summer Simmer Index S for both the maximum and minimum temperatures.
​

Accepted Answer

A)  maximum,   ;
Minimum,   
B)  maximum,   ;
Minimum,   
C)  ​maximum,   ;
Minimum,   ​ 
D)  maximum,   ;
Minimum,   
E)  ​maximum,   ;
Minimum,   
A)  maximum,   ;
Minimum,   
B)  maximum,   ;
Minimum,   
C)  ​maximum,   ;
Minimum,   ​ 
D)  maximum,   ;
Minimum,   
E)  ​maximum,   ;
Minimum,

Question 6

The total cost (in dollars) of producing 1 unit of a product is given by   , where x represents the cost per pound of raw materials and y represents the hourly rate for labor. The present cost for raw materials is $15 per pound and the present hourly rate for labor is $11. Indicate how to use the cost function C to determine how an increase of $1 per pound for raw materials will affect the total cost. Round your answer to two decimal places. ​

Accepted Answer

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

Question 7

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

Accepted Answer

A)  ​96,768 
B)  ​4,032 
C)  ​169,344 
D)  ​12,160 
E)  ​24,192 
A)  ​96,768 
B)  ​4,032 
C)  ​169,344 
D)  ​12,160 
E)  ​24,192

Question 8

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)    
A)    
B)    
C)    
D)    
E)

Question 9

The cost per day to society of an epidemic is   , where C is in dollars, x is the number of people infected on a given day, and y is the number of people who die on a given day. If 18,500 people are infected and 10 people die on a given day, what is the cost to society? Round your answer to the nearest dollar. ​

Accepted Answer

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

Question 10

Give the domain of the function   . ​

Accepted Answer

A)  domain: all reals p1 and p2 with   
B)  domain: all reals p1 and p2 with   
C)  domain: all reals p1 and p2 with   
D)  domain: all reals p1 and p2 with   
E)  domain: all reals p1 and p2 with   
A)  domain: all reals p1 and p2 with   
B)  domain: all reals p1 and p2 with   
C)  domain: all reals p1 and p2 with   
D)  domain: all reals p1 and p2 with   
E)  domain: all reals p1 and p2 with

Question 11

The following table gives the actual or projected population in millions for a certain city for selected years from 2000 to 2050. 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 2000. ​

Accepted Answer

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

Question 12

Use the function   to find   . ​

Accepted Answer

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

Question 13

If   , find   and   . ​

Accepted Answer

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

Question 14

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 15

If   , find   . ​

Accepted Answer

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

Question 16

If   , find   . ​

Accepted Answer

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

Question 17

Use the function   to find   . ​

Accepted Answer

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

Question 18

Suppose that the data in the table below shows the average earnings of year-round full-time workers by gender for several different levels of educational attainment. Find and interpret the slope of the linear regression line. Round the slope to three decimal places. ​

Accepted Answer

A)  The slope is 0.910. The average annual earnings of females is decreasing at a rate of $0.910 per dollar increase in the annual of males. 
B)  The slope is 0.807. The average annual earnings of females is decreasing at a rate of $0.807 per dollar decrease in the average annual earnings of males. 
C)  The slope is 0.759. The average annual earnings of females is decreasing at a rate of $0.759 per dollar decrease in the average annual earnings of males. 
D)  The slope is 0.900. The average annual earnings of females is increasing at a rate of $0.900 per dollar increase in the average annual earnings of males. 
E)  The slope is 0.749. The average annual earnings of females is increasing at a rate of $0.749 per dollar increase in the average annual earnings of males. 
A)  The slope is 0.910. The average annual earnings of females is decreasing at a rate of $0.910 per dollar increase in the annual of males. 
B)  The slope is 0.807. The average annual earnings of females is decreasing at a rate of $0.807 per dollar decrease in the average annual earnings of males. 
C)  The slope is 0.759. The average annual earnings of females is decreasing at a rate of $0.759 per dollar decrease in the average annual earnings of males. 
D)  The slope is 0.900. The average annual earnings of females is increasing at a rate of $0.900 per dollar increase in the average annual earnings of males. 
E)  The slope is 0.749. The average annual earnings of females is increasing at a rate of $0.749 per dollar increase in the average annual earnings of males.

Question 19

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 20

Suppose that the utility function for two goods X and Y is given by   , and a consumer purchases 2 units of X and 6 units of Y. If the consumer purchases 2 units of Y, how many units of X must be purchased to retain the same level of utility? ​

Accepted Answer

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

Test for relative maximum and minimum.

The following table shows the monthly mortgage payment R for a given interest rate 5% as a function of the amount financed A (in thousands of dollars) and the duration of the loan n in years. If , use the table to find . Round your answer to the nearest cent.

Suppose that the number of crates of an agricultural product is given by , where x is the number of hours of and y is the number of acres of the crop. Find the marginal productivity of the number of hours of labor when and .

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

Give the domain of the function .

The following table gives the actual or projected population in millions for a certain city for selected years from 2000 to 2050. 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 2000.

Use the function to find .

If , find and .

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.

If , find .

If , find .

Use the function to find .

Suppose that the data in the table below shows the average earnings of year-round full-time workers by gender for several different levels of educational attainment. Find and interpret the slope of the linear regression line. Round the slope to three decimal places.

Suppose that the utility function for two goods X and Y is given by , and a consumer purchases 2 units of X and 6 units of Y. If the consumer purchases 2 units of Y, how many units of X must be purchased to retain the same level of utility?

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. ​ ​

The following table shows the monthly mortgage payment R for a given interest rate 5% as a function of the amount financed A (in thousands of dollars) and the duration of the loan n in years. If , use the table to find . Round your answer to the nearest cent. ​

Suppose that the number of crates of an agricultural product is given by , where x is the number of hours of and y is the number of acres of the crop. Find the marginal productivity of the number of hours of labor when and . ​

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

Give the domain of the function . ​

The following table gives the actual or projected population in millions for a certain city for selected years from 2000 to 2050. 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 2000. ​

Use the function to find . ​

If , find and . ​

If , find . ​

If , find . ​

Use the function to find . ​

Suppose that the data in the table below shows the average earnings of year-round full-time workers by gender for several different levels of educational attainment. Find and interpret the slope of the linear regression line. Round the slope to three decimal places. ​

Suppose that the utility function for two goods X and Y is given by , and a consumer purchases 2 units of X and 6 units of Y. If the consumer purchases 2 units of Y, how many units of X must be purchased to retain the same level of utility? ​

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.

The following table shows the monthly mortgage payment R for a given interest rate 5% as a function of the amount financed A (in thousands of dollars) and the duration of the loan n in years. If , use the table to find . Round your answer to the nearest cent.

Suppose that the number of crates of an agricultural product is given by , where x is the number of hours of and y is the number of acres of the crop. Find the marginal productivity of the number of hours of labor when and .

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

Give the domain of the function .

The following table gives the actual or projected population in millions for a certain city for selected years from 2000 to 2050. 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 2000.

Use the function to find .

If , find and .

If , find .

If , find .

Use the function to find .

Suppose that the data in the table below shows the average earnings of year-round full-time workers by gender for several different levels of educational attainment. Find and interpret the slope of the linear regression line. Round the slope to three decimal places.

Suppose that the utility function for two goods X and Y is given by , and a consumer purchases 2 units of X and 6 units of Y. If the consumer purchases 2 units of Y, how many units of X must be purchased to retain the same level of utility?