Question 1

The following figure shows the graph of revenue function   and cost function   . What is the relationship between marginal cost and marginal revenue when profit is at its maximum value? ​

Accepted Answer

A) > B) C) D) < E) no relationship exists A) > B) C) D) < E) no relationship exists

Question 2

Suppose that a chain of auto service stations, Quick-Oil, Inc., has found that its monthly sales volume y (in thousands of dollars) is related to the price p (in dollars) of an oil change by     . Is y increasing or decreasing for all values of   ​

Accepted Answer

A)  increasing 
B)  decreasing 
A)  increasing 
B)  decreasing

Question 3

For the given function, find the relative minima.

Accepted Answer

A)    
B)    
C)    
D)    
E)  no relative minima 
A)    
B)    
C)    
D)    
E)  no relative minima

Question 4

If the total cost function for a product is   dollars. Find the minimum average cost. Round your answer to two decimal places. ​

Accepted Answer

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

Question 5

A product can be produced at a total cost   dollars, where x is the number produced. If the total revenue is given by   dollars, determine the level of production x that will maximize the profit. ​

Accepted Answer

A)  21 units 
B)  56 units 
C)  66 units 
D)  33 units 
E)  44 units 
A)  21 units 
B)  56 units 
C)  66 units 
D)  33 units 
E)  44 units

Question 6

The base of a rectangular box is to be twice as long as it is wide. The volume of the box is 750 cubic inches. The material for the top costs $0.21 per square inch and the material for the sides and bottom costs $0.06 per square inch. Find the dimensions that will make the cost a minimum. ​​

Accepted Answer

A)  Dimensions are 15" by 10" by 4" (high). 
B)  Dimensions are 22" by 44" by 4" (high). 
C)  Dimensions are 5" by 10" by 15" (high). 
D)  Dimensions are 15" by 30" by 5" (high). 
E)  Dimensions are 4" by 8" by 22" (high). 
A)  Dimensions are 15" by 10" by 4" (high). 
B)  Dimensions are 22" by 44" by 4" (high). 
C)  Dimensions are 5" by 10" by 15" (high). 
D)  Dimensions are 15" by 30" by 5" (high). 
E)  Dimensions are 4" by 8" by 22" (high).

Question 7

For the given function find the relative minima, and sketch the graph. You may check your graph with a graphing utility.

Accepted Answer

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

Question 8

A time study showed that, on average, the productivity of a worker after t hours on the job can be modeled by   ,   , where   is the number of units produced per hour. After how many hours will productivity be maximized? Round your answer to two decimal places. ​​

Accepted Answer

A)  5.61 hours 
B)  8.00 hours 
C)  6.77 hours 
D)  2.34 hours 
E)  5.41 hours 
A)  5.61 hours 
B)  8.00 hours 
C)  6.77 hours 
D)  2.34 hours 
E)  5.41 hours

Question 9

An entrepreneur starts new companies and sells them when their growth is maximized. Suppose that the annual profit for a new company is given by   , where P is in thousands of dollars and x is the number of years after the company is formed. If she wants to sell the company before profits begin to decline, after how many years should she sell it? ​

Accepted Answer

A)  Sell the company after 4 years. 
B)  Sell the company after 5 years. 
C)  Sell the company after 6 years. 
D)  Sell the company after 7 years. 
E)  Sell the company after 8 years. 
A)  Sell the company after 4 years. 
B)  Sell the company after 5 years. 
C)  Sell the company after 6 years. 
D)  Sell the company after 7 years. 
E)  Sell the company after 8 years.

Question 10

A function and its graph are given. Use the second derivative to determine intervals on which the function is concave up. Check these results against the graph shown. ​   ​   ​

Accepted Answer

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

Question 11

For the given function, find the critical points.

Accepted Answer

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

Question 12

Use the indicated x-values on the graph of   to find all the intervals where   .   ​

Accepted Answer

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

Question 13

Find any vertical asymptotes for the given function.

Accepted Answer

A)    
B)    
C)    
D)    
E)  no vertical asymptotes 
A)    
B)    
C)    
D)    
E)  no vertical asymptotes

Question 14

A function and its graph are given. Use the graph to find the horizontal asymptotes, if they exist. Confirm your results analytically. ​   ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)  no horizontal asymptotes 
A)    
B)    
C)    
D)    
E)  no horizontal asymptotes

Question 15

Determine whether the given function is concave up or concave down at the indicated point.   at

Accepted Answer

A)  concave down 
B)  concave up 
A)  concave down 
B)  concave up

Question 16

Suppose that the oxygen level P (for purity) in a body of water t months after an oil spill is given by   . Find how long it will be before the rate of change of P is maximized. That is, find the point of diminishing returns. ​

Accepted Answer

A)    
B)    
C)    
D)    
E)  none of the above 
A)    
B)    
C)    
D)    
E)  none of the above

Question 17

A function and its first and second derivatives are given. Use these to find any relative minima.

Accepted Answer

A)    
B)    
C)    
D)    
E)  no relative minima 
A)    
B)    
C)    
D)    
E)  no relative minima

Question 18

A function and its graph are given. From the graph estimate where   has a relative minimum. ​   ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)  no relative minima 
A)    
B)    
C)    
D)    
E)  no relative minima

Question 19

Analytically determine the location of any vertical asymptotes. Round your answer to two decimal places.

Accepted Answer

A)    
B)    
C)    
D)    
E)  no vertical asymptotes 
A)    
B)    
C)    
D)    
E)  no vertical asymptotes

Question 20

In this problem,   and its graph are given. Use the graph of   to determine where   is concave down. ​

Accepted Answer

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

The following figure shows the graph of revenue function and cost function . What is the relationship between marginal cost and marginal revenue when profit is at its maximum value?

Suppose that a chain of auto service stations, Quick-Oil, Inc., has found that its monthly sales volume y (in thousands of dollars) is related to the price p (in dollars) of an oil change by . Is y increasing or decreasing for all values of

For the given function, find the relative minima.

If the total cost function for a product is dollars. Find the minimum average cost. Round your answer to two decimal places.

A product can be produced at a total cost dollars, where x is the number produced. If the total revenue is given by dollars, determine the level of production x that will maximize the profit.

The base of a rectangular box is to be twice as long as it is wide. The volume of the box is 750 cubic inches. The material for the top costs $0.21 per square inch and the material for the sides and bottom costs $0.06 per square inch. Find the dimensions that will make the cost a minimum.

For the given function find the relative minima, and sketch the graph. You may check your graph with a graphing utility.

A time study showed that, on average, the productivity of a worker after t hours on the job can be modeled by , , where is the number of units produced per hour. After how many hours will productivity be maximized? Round your answer to two decimal places.

A function and its graph are given. Use the second derivative to determine intervals on which the function is concave up. Check these results against the graph shown.

For the given function, find the critical points.

Use the indicated x-values on the graph of to find all the intervals where .

Find any vertical asymptotes for the given function.

A function and its graph are given. Use the graph to find the horizontal asymptotes, if they exist. Confirm your results analytically.

Determine whether the given function is concave up or concave down at the indicated point. at

Suppose that the oxygen level P (for purity) in a body of water t months after an oil spill is given by . Find how long it will be before the rate of change of P is maximized. That is, find the point of diminishing returns.

A function and its first and second derivatives are given. Use these to find any relative minima.

A function and its graph are given. From the graph estimate where has a relative minimum.

Analytically determine the location of any vertical asymptotes. Round your answer to two decimal places.

In this problem, and its graph are given. Use the graph of to determine where is concave down.

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Indefinite Integrals

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A: Definite Integrals - Techniques

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Exam 10: Derivatives

The following figure shows the graph of revenue function and cost function . What is the relationship between marginal cost and marginal revenue when profit is at its maximum value? ​

Suppose that a chain of auto service stations, Quick-Oil, Inc., has found that its monthly sales volume y (in thousands of dollars) is related to the price p (in dollars) of an oil change by . Is y increasing or decreasing for all values of ​

For the given function, find the relative minima.

If the total cost function for a product is dollars. Find the minimum average cost. Round your answer to two decimal places. ​

A product can be produced at a total cost dollars, where x is the number produced. If the total revenue is given by dollars, determine the level of production x that will maximize the profit. ​

The base of a rectangular box is to be twice as long as it is wide. The volume of the box is 750 cubic inches. The material for the top costs $0.21 per square inch and the material for the sides and bottom costs $0.06 per square inch. Find the dimensions that will make the cost a minimum. ​​

For the given function find the relative minima, and sketch the graph. You may check your graph with a graphing utility.

A time study showed that, on average, the productivity of a worker after t hours on the job can be modeled by , , where is the number of units produced per hour. After how many hours will productivity be maximized? Round your answer to two decimal places. ​​

A function and its graph are given. Use the second derivative to determine intervals on which the function is concave up. Check these results against the graph shown. ​ ​ ​

For the given function, find the critical points.

Use the indicated x-values on the graph of to find all the intervals where . ​

Find any vertical asymptotes for the given function.

A function and its graph are given. Use the graph to find the horizontal asymptotes, if they exist. Confirm your results analytically. ​ ​ ​

Determine whether the given function is concave up or concave down at the indicated point. at

Suppose that the oxygen level P (for purity) in a body of water t months after an oil spill is given by . Find how long it will be before the rate of change of P is maximized. That is, find the point of diminishing returns. ​

A function and its first and second derivatives are given. Use these to find any relative minima.

A function and its graph are given. From the graph estimate where has a relative minimum. ​ ​ ​

Analytically determine the location of any vertical asymptotes. Round your answer to two decimal places.

In this problem, and its graph are given. Use the graph of to determine where is concave down. ​

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 Continued

Indefinite Integrals

Definite Integrals - Techniques

A: Definite Integrals - Techniques

Functions of Two or More Variables

Algebraic Concepts 2

Algebraic Concepts 3

Algebraic Concepts 4

Algebraic Concepts 5

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The following figure shows the graph of revenue function and cost function . What is the relationship between marginal cost and marginal revenue when profit is at its maximum value?

Suppose that a chain of auto service stations, Quick-Oil, Inc., has found that its monthly sales volume y (in thousands of dollars) is related to the price p (in dollars) of an oil change by . Is y increasing or decreasing for all values of

If the total cost function for a product is dollars. Find the minimum average cost. Round your answer to two decimal places.

A product can be produced at a total cost dollars, where x is the number produced. If the total revenue is given by dollars, determine the level of production x that will maximize the profit.

The base of a rectangular box is to be twice as long as it is wide. The volume of the box is 750 cubic inches. The material for the top costs $0.21 per square inch and the material for the sides and bottom costs $0.06 per square inch. Find the dimensions that will make the cost a minimum.

A time study showed that, on average, the productivity of a worker after t hours on the job can be modeled by , , where is the number of units produced per hour. After how many hours will productivity be maximized? Round your answer to two decimal places.

A function and its graph are given. Use the second derivative to determine intervals on which the function is concave up. Check these results against the graph shown.

Use the indicated x-values on the graph of to find all the intervals where .

A function and its graph are given. Use the graph to find the horizontal asymptotes, if they exist. Confirm your results analytically.

Suppose that the oxygen level P (for purity) in a body of water t months after an oil spill is given by . Find how long it will be before the rate of change of P is maximized. That is, find the point of diminishing returns.

A function and its graph are given. From the graph estimate where has a relative minimum.

In this problem, and its graph are given. Use the graph of to determine where is concave down.