Question 1

A function and its graph are given. Use the graph to estimate the locations of any vertical asymptotes. ​   ​

Accepted Answer

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

Question 2

A rectangular box with a square base is to be formed from a square piece of metal with 30-inch sides. If a square piece with side x is cut from each corner of the metal and the sides are folded up to form an open box, the volume of the box is   . What value of x will maximize the volume of the box? ​

Accepted Answer

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

Question 3

The following figure shows the graph of revenue function   and cost function   . At which of the four x-values shown is the profit largest? ​

Accepted Answer

A)  A 
B)  C 
C)  B 
D)  D 
E)  Profit is largest at more than one x-value. 
A)  A 
B)  C 
C)  B 
D)  D 
E)  Profit is largest at more than one x-value.

Question 4

Use the graph shown in the figure and identify points from A through I that satisfy the given condition. ​   and

Accepted Answer

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

Question 5

The running yard for a dog kennel must contain at least 400 square feet. If a 15-foot side of the kennel is used as part of one side of a rectangular yard with 400 square feet, what dimensions will require the least amount of fencing? Round your answer to the nearest feet. ​​

Accepted Answer

A)  40 ft by 10 ft 
B)  400 ft by 1 ft 
C)  20 ft by 20 ft 
D)  40 ft by 5 ft 
E)  80 ft by 5 ft 
A)  40 ft by 10 ft 
B)  400 ft by 1 ft 
C)  20 ft by 20 ft 
D)  40 ft by 5 ft 
E)  80 ft by 5 ft

Question 6

Suppose that in an election year, the proportion p of voters who recognize a certain candidate's name t months after the campaign started is given by   . After how many months is the proportion maximized? ​

Accepted Answer

A)  3 months 
B)  7 months 
C)  2 months 
D)  9 months 
E)  6 months 
A)  3 months 
B)  7 months 
C)  2 months 
D)  9 months 
E)  6 months

Question 7

An inferior product with a large advertising budget sells well when it is introduced, but sales fall as people discontinue use of the product. Suppose that the weekly sales S are given by   where S is in millions of dollars and t is in weeks. After how many weeks will sales be maximized? ​

Accepted Answer

A)  two weeks 
B)  one week 
C)  ​three weeks 
D)  ​four weeks 
E)  ​five weeks 
A)  two weeks 
B)  one week 
C)  ​three weeks 
D)  ​four weeks 
E)  ​five weeks

Question 8

A function and its graph are given. Use the graph to estimate the locations of any horizontal asymptotes. ​

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by   ,   , where x is the number of machines used. Find the critical values of   that lie in the domain of the problem. ​
​

Accepted Answer

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

Question 11

A function and its graph are given. Use the second derivative to locate all x-values of points of inflection on the graph of   . Check these results against the graph shown. ​   ​   ​

Accepted Answer

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

Question 12

For the given function, find all intervals of x-values where the function is decreasing.

Accepted Answer

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

Question 13

For the revenue function given by   find the maximum average revenue. ​

Accepted Answer

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

Question 14

Use the sign diagram for   to determine the largest interval on which f (x) decreases, where   and   .

Accepted Answer

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

Question 15

For the cost function   , show that average costs are minimized at the x-value where   by finding this minimum. Round your answer to two decimal places. (​Note:   ) ​​

Accepted Answer

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

Question 16

Analytically determine any relative minima. Round your answer to two decimal places.

Accepted Answer

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

Question 17

The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by   ,   . Find the t-value at which x is maximum. Round your answer to two decimal places. ​​

Accepted Answer

A)  0 hours 
B)  1.41 hours 
C)  1767.77 hours 
D)  3.41 hours 
E)  3.45 hours 
A)  0 hours 
B)  1.41 hours 
C)  1767.77 hours 
D)  3.41 hours 
E)  3.45 hours

Question 18

From a tract of land, a developer plans to fence a rectangular region and then divide it into two identical rectangular lots by putting a fence down the middle. Suppose that the fence for the outside boundary costs $6 per foot and the fence for the middle costs $4 per foot. If each lot contains 4,600 square feet, find the dimensions of each lot that yield the minimum cost for the fence. Round your answer to two decimal places. ​​

Accepted Answer

A)  Dimensions are 42.19 ft for the side parallel to the divider and 109.03 ft for the other side. 
B)  Dimensions are 109.03 ft for the side parallel to the divider and 42.19 ft for the other side. 
C)  Dimensions are 67.82 ft for the side parallel to the divider and 67.82 ft for the other side. 
D)  Dimensions are 58.74 ft for the side parallel to the divider and 78.32 ft for the other side. 
E)  Dimensions are 78.32 ft for the side parallel to the divider and 58.74 ft for the other side. 
A)  Dimensions are 42.19 ft for the side parallel to the divider and 109.03 ft for the other side. 
B)  Dimensions are 109.03 ft for the side parallel to the divider and 42.19 ft for the other side. 
C)  Dimensions are 67.82 ft for the side parallel to the divider and 67.82 ft for the other side. 
D)  Dimensions are 58.74 ft for the side parallel to the divider and 78.32 ft for the other side. 
E)  Dimensions are 78.32 ft for the side parallel to the divider and 58.74 ft for the other side.

Question 19

Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by   ,   , where x is the number of machines used. Over what interval in the domain do average costs decrease? ​

Accepted Answer

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

Question 20

Use the sign diagram for   to determine x-values at which relative maxima occur, where   and   .

Accepted Answer

A)    
B)  ​   
C)    and   
D)    
E)  no relative maxima 
A)    
B)  ​   
C)    and   
D)    
E)  no relative maxima

A function and its graph are given. Use the graph to estimate the locations of any vertical asymptotes.

A rectangular box with a square base is to be formed from a square piece of metal with 30-inch sides. If a square piece with side x is cut from each corner of the metal and the sides are folded up to form an open box, the volume of the box is . What value of x will maximize the volume of the box?

The following figure shows the graph of revenue function and cost function . At which of the four x-values shown is the profit largest?

Use the graph shown in the figure and identify points from A through I that satisfy the given condition. and

The running yard for a dog kennel must contain at least 400 square feet. If a 15-foot side of the kennel is used as part of one side of a rectangular yard with 400 square feet, what dimensions will require the least amount of fencing? Round your answer to the nearest feet.

Suppose that in an election year, the proportion p of voters who recognize a certain candidate's name t months after the campaign started is given by . After how many months is the proportion maximized?

An inferior product with a large advertising budget sells well when it is introduced, but sales fall as people discontinue use of the product. Suppose that the weekly sales S are given by where S is in millions of dollars and t is in weeks. After how many weeks will sales be maximized?

A function and its graph are given. Use the graph to estimate the locations of any horizontal asymptotes.

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

Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. Find the critical values of that lie in the domain of the problem.

A function and its graph are given. Use the second derivative to locate all x-values of points of inflection on the graph of . Check these results against the graph shown.

For the given function, find all intervals of x-values where the function is decreasing.

For the revenue function given by find the maximum average revenue.

Use the sign diagram for to determine the largest interval on which f (x) decreases, where and .

For the cost function , show that average costs are minimized at the x-value where by finding this minimum. Round your answer to two decimal places. (Note: )

Analytically determine any relative minima. Round your answer to two decimal places.

The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by , . Find the t-value at which x is maximum. Round your answer to two decimal places.

Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. Over what interval in the domain do average costs decrease?

Use the sign diagram for to determine x-values at which relative maxima occur, where and .

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Exponential and Logarithmic Functions

Mathematics of Finance

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Further Topics in Probability and Data Description

Derivatives

Derivatives Continued

Indefinite Integrals

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A: Definite Integrals - Techniques

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Exam 10: Derivatives

A function and its graph are given. Use the graph to estimate the locations of any vertical asymptotes. ​ ​

The following figure shows the graph of revenue function and cost function . At which of the four x-values shown is the profit largest? ​

Use the graph shown in the figure and identify points from A through I that satisfy the given condition. ​ and

The running yard for a dog kennel must contain at least 400 square feet. If a 15-foot side of the kennel is used as part of one side of a rectangular yard with 400 square feet, what dimensions will require the least amount of fencing? Round your answer to the nearest feet. ​​

Suppose that in an election year, the proportion p of voters who recognize a certain candidate's name t months after the campaign started is given by . After how many months is the proportion maximized? ​

An inferior product with a large advertising budget sells well when it is introduced, but sales fall as people discontinue use of the product. Suppose that the weekly sales S are given by where S is in millions of dollars and t is in weeks. After how many weeks will sales be maximized? ​

A function and its graph are given. Use the graph to estimate the locations of any horizontal asymptotes. ​

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

Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. Find the critical values of that lie in the domain of the problem. ​ ​

A function and its graph are given. Use the second derivative to locate all x-values of points of inflection on the graph of . Check these results against the graph shown. ​ ​ ​

For the given function, find all intervals of x-values where the function is decreasing.

For the revenue function given by find the maximum average revenue. ​

Use the sign diagram for to determine the largest interval on which f (x) decreases, where and .

For the cost function , show that average costs are minimized at the x-value where by finding this minimum. Round your answer to two decimal places. (​Note: ) ​​

Analytically determine any relative minima. Round your answer to two decimal places.

The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by , . Find the t-value at which x is maximum. Round your answer to two decimal places. ​​

Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. Over what interval in the domain do average costs decrease? ​

Use the sign diagram for to determine x-values at which relative maxima occur, where and .

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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A function and its graph are given. Use the graph to estimate the locations of any vertical asymptotes.

The following figure shows the graph of revenue function and cost function . At which of the four x-values shown is the profit largest?

Use the graph shown in the figure and identify points from A through I that satisfy the given condition. and

The running yard for a dog kennel must contain at least 400 square feet. If a 15-foot side of the kennel is used as part of one side of a rectangular yard with 400 square feet, what dimensions will require the least amount of fencing? Round your answer to the nearest feet.

Suppose that in an election year, the proportion p of voters who recognize a certain candidate's name t months after the campaign started is given by . After how many months is the proportion maximized?

An inferior product with a large advertising budget sells well when it is introduced, but sales fall as people discontinue use of the product. Suppose that the weekly sales S are given by where S is in millions of dollars and t is in weeks. After how many weeks will sales be maximized?

A function and its graph are given. Use the graph to estimate the locations of any horizontal asymptotes.

Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. Find the critical values of that lie in the domain of the problem.

A function and its graph are given. Use the second derivative to locate all x-values of points of inflection on the graph of . Check these results against the graph shown.

For the revenue function given by find the maximum average revenue.

For the cost function , show that average costs are minimized at the x-value where by finding this minimum. Round your answer to two decimal places. (Note: )

The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by , . Find the t-value at which x is maximum. Round your answer to two decimal places.

Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. Over what interval in the domain do average costs decrease?