Question 1

Find all points of inflection of the given function. Round your answer to two decimal places.   ​

Accepted Answer

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

Question 2

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   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 6 years. 
B) Sell the company after 7 years. 
C) Sell the company after 8 years. 
D) Sell the company after 9 years. 
E) Sell the company after 10 years. 
A) Sell the company after 6 years. 
B) Sell the company after 7 years. 
C) Sell the company after 8 years. 
D) Sell the company after 9 years. 
E) Sell the company after 10 years.

Question 3

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 increase? ​

Accepted Answer

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

Question 4

The graph shows a total cost function. Determine the level of production at which average cost is minimized.

Accepted Answer

A) 200 units 
B) 800 units 
C) 400 units 
D) 0 units 
E) 600 units 
A) 200 units 
B) 800 units 
C) 400 units 
D) 0 units 
E) 600 units

Question 5

A graph of   is given. Use the graph to determine where the graph of   has a relative maximum. ​   ​   ​

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

For the given function and graph, determine all critical value(s), where   .

Accepted Answer

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

Question 8

A box with an open top and a square base is to be constructed to contain 6,912 cubic inches. Find the dimensions that will require the minimum amount of material to construct the box. ​​

Accepted Answer

A) The dimensions of the box using minimum material are 24 in. by 24 in. by 12 in. 
B) The dimensions of the box using minimum material are 21 in. by 21 in. by 17 in. 
C) The dimensions of the box using minimum material are 12 in. by 12 in. by 24 in. 
D) The dimensions of the box using minimum material are 17 in. by 17 in. by 21 in. 
E) The dimensions of the box using minimum material are 21 in. by 21 in. by 12 in. 
A) The dimensions of the box using minimum material are 24 in. by 24 in. by 12 in. 
B) The dimensions of the box using minimum material are 21 in. by 21 in. by 17 in. 
C) The dimensions of the box using minimum material are 12 in. by 12 in. by 24 in. 
D) The dimensions of the box using minimum material are 17 in. by 17 in. by 21 in. 
E) The dimensions of the box using minimum material are 21 in. by 21 in. by 12 in.

Question 9

Suppose that the total number of units produced by a worker in   hours of an 8-hour shift can be modeled by the production function   . Find the number of hours before production is maximized.

Accepted Answer

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

Question 10

Find any horizontal asymptotes for the given function.

Accepted Answer

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

Question 11

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)   
   
E) no relationship exists 
A)   
   
B)   
   
E) no relationship exists

Question 12

Find the x-value at which the absolute minimum of f (x) occurs on the interval [a, b].   ,

Accepted Answer

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

Question 13

If the total revenue function for a blender is   , find the maximum 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 given function, find the relative maxima.

Accepted Answer

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

Question 16

A function and its first and second derivatives are given. Use these to find any horizontal asymptotes.

Accepted Answer

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

Question 17

Suppose that the total number of units produced by a worker in t hours of an 8-hour shift can be modeled by the production function   . Find the number of hours before the rate of production is maximized. That is, find the point of diminishing returns. ​​

Accepted Answer

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

Question 18

Find any horizontal asymptotes for the given function.

Accepted Answer

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

Question 19

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 20

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

Find all points of inflection of the given function. 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 increase?

The graph shows a total cost function. Determine the level of production at which average cost is minimized.

A graph of is given. Use the graph to determine where the graph of has a relative maximum.

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

For the given function and graph, determine all critical value(s), where .

A box with an open top and a square base is to be constructed to contain 6,912 cubic inches. Find the dimensions that will require the minimum amount of material to construct the box.

Suppose that the total number of units produced by a worker in hours of an 8-hour shift can be modeled by the production function . Find the number of hours before production is maximized.

Find any horizontal asymptotes for the given function.

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?

Find the x-value at which the absolute minimum of f (x) occurs on the interval [a, b]. ,

If the total revenue function for a blender is , find the maximum revenue.

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

For the given function, find the relative maxima.

A function and its first and second derivatives are given. Use these to find any horizontal asymptotes.

Suppose that the total number of units produced by a worker in t hours of an 8-hour shift can be modeled by the production function . Find the number of hours before the rate of production is maximized. That is, find the point of diminishing returns.

Find any horizontal asymptotes for the given function.

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?

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.

Algebraic Concepts

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; Data Description

Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals: Techniques of Integration

Functions of Two or More Variables

Filters

Exam 11: Applications of Derivatives

Find all points of inflection of the given function. 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 increase? ​

The graph shows a total cost function. Determine the level of production at which average cost is minimized.

A graph of is given. Use the graph to determine where the graph of has a relative maximum. ​ ​ ​

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

For the given function and graph, determine all critical value(s), where .

A box with an open top and a square base is to be constructed to contain 6,912 cubic inches. Find the dimensions that will require the minimum amount of material to construct the box. ​​

Suppose that the total number of units produced by a worker in hours of an 8-hour shift can be modeled by the production function . Find the number of hours before production is maximized.

Find any horizontal asymptotes for the given function.

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?

Find the x-value at which the absolute minimum of f (x) occurs on the interval [a, b]. ,

If the total revenue function for a blender is , find the maximum revenue. ​

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

For the given function, find the relative maxima.

A function and its first and second derivatives are given. Use these to find any horizontal asymptotes.

Suppose that the total number of units produced by a worker in t hours of an 8-hour shift can be modeled by the production function . Find the number of hours before the rate of production is maximized. That is, find the point of diminishing returns. ​​

Find any horizontal asymptotes for the given function.

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? ​

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. ​​

Algebraic Concepts

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; Data Description

Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals: Techniques of Integration

Functions of Two or More Variables

Filters

Find all points of inflection of the given function. 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 increase?

A graph of is given. Use the graph to determine where the graph of has a relative maximum.

A box with an open top and a square base is to be constructed to contain 6,912 cubic inches. Find the dimensions that will require the minimum amount of material to construct the box.

If the total revenue function for a blender is , find the maximum revenue.

Suppose that the total number of units produced by a worker in t hours of an 8-hour shift can be modeled by the production function . Find the number of hours before the rate of production is maximized. That is, find the point of diminishing returns.

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?

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.