Question 1

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

Accepted Answer

A)  ​I 
B)  F 
C)  C 
D)  D 
E)  H 
A)  ​I 
B)  F 
C)  C 
D)  D 
E)  H

Question 2

The percent p of impurities that can be removed from the waste water of a manufacturing process at a cost of C dollars is given by   . Find any horizontal asymptotes. ​

Accepted Answer

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

Question 3

For the revenue function given by   , find the x-value where   that is, where the average revenue equals the marginal revenue. (​Note:   ) ​​

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

For the given function, use the graph to identify x-values for which   . You may use the derivative to check your conclusion. ​

Accepted Answer

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

Question 6

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. How many machines give minimum average costs? ​

Accepted Answer

A)  Using 25 machines gives the minimum average costs. 
B)  Using zero machines gives the minimum average costs. 
C)  Using 35 machines gives the minimum average costs. 
D)  Using 50 machines gives the minimum average costs. 
E)  Using 55 machines gives the minimum average costs. 
A)  Using 25 machines gives the minimum average costs. 
B)  Using zero machines gives the minimum average costs. 
C)  Using 35 machines gives the minimum average costs. 
D)  Using 50 machines gives the minimum average costs. 
E)  Using 55 machines gives the minimum average costs.

Question 7

A graph of   is given. Use the graph to determine all critical values of   . ​

Accepted Answer

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

Question 8

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 9

Sketch the graph of the function   .

Accepted Answer

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

Question 10

A printer has a contract to print 62,000 posters for a political candidate. He can run the posters by using any number of plates from 1 to 30 on his press. If he uses x metal plates, they will produce x copies of the poster with each impression of the press. The metal plates cost $3.00 to prepare, and it costs $15.00 per hour to run the press. If the press can make 1000 impressions per hour, how many metal plates should the printer make to minimize costs? Round your answer to the nearest count of metal plates. ​

Accepted Answer

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

Question 11

Use the indicated x-values on the graph of   to find the x-coordinate of any horizontal point of inflection.

Accepted Answer

A)  a 
B)  c 
C)  f 
D)  e 
E)  d 
A)  a 
B)  c 
C)  f 
D)  e 
E)  d

Question 12

In this problem,   and its graph are given. Use the graph of   to determine where   has a point of inflection. ​     ​

Accepted Answer

A)    
B)    
C)    
D)    
E)  no point of inflection 
A)    
B)    
C)    
D)    
E)  no point of inflection

Question 13

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

Accepted Answer

A)  2.83 mg 
B)  530.33 mg 
C)  1166.73 mg 
D)  8.77 mg 
E)  1125.30 mg 
A)  2.83 mg 
B)  530.33 mg 
C)  1166.73 mg 
D)  8.77 mg 
E)  1125.30 mg

Question 14

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 15

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 16

Analytically determine the location(s) of any horizontal asymptote(s).

Accepted Answer

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

Question 17

Find all relative maxima of the given function.

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

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

Question 20

A function and its graph are given. From the graph, estimate where   . ​       ​

Accepted Answer

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

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

The percent p of impurities that can be removed from the waste water of a manufacturing process at a cost of C dollars is given by . Find any horizontal asymptotes.

For the revenue function given by , find the x-value where that is, where the average revenue equals the marginal revenue. (Note: )

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

For the given function, use the graph to identify x-values for which . You may use the derivative to check your conclusion.

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. How many machines give minimum average costs?

A graph of is given. Use the graph to determine all critical values of .

Find any vertical asymptotes for the given function.

Sketch the graph of the function .

Use the indicated x-values on the graph of to find the x-coordinate of any horizontal point of inflection.

In this problem, and its graph are given. Use the graph of to determine where has a point of inflection.

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

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 vertical asymptotes for the given function.

Analytically determine the location(s) of any horizontal asymptote(s).

Find all relative maxima of the given function.

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

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

A function and its graph are given. From the graph, estimate where .

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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

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Exam 10: Derivatives

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

The percent p of impurities that can be removed from the waste water of a manufacturing process at a cost of C dollars is given by . Find any horizontal asymptotes. ​

For the revenue function given by , find the x-value where that is, where the average revenue equals the marginal revenue. (​Note: ) ​​

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

For the given function, use the graph to identify x-values for which . You may use the derivative to check your conclusion. ​

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. How many machines give minimum average costs? ​

A graph of is given. Use the graph to determine all critical values of . ​

Find any vertical asymptotes for the given function.

Sketch the graph of the function .

Use the indicated x-values on the graph of to find the x-coordinate of any horizontal point of inflection.

In this problem, and its graph are given. Use the graph of to determine where has a point of inflection. ​ ​

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

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 vertical asymptotes for the given function.

Analytically determine the location(s) of any horizontal asymptote(s).

Find all relative maxima of the given function.

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

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

A function and its graph are given. From the graph, estimate where . ​ ​

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

Filters

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

The percent p of impurities that can be removed from the waste water of a manufacturing process at a cost of C dollars is given by . Find any horizontal asymptotes.

For the revenue function given by , find the x-value where that is, where the average revenue equals the marginal revenue. (Note: )

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

For the given function, use the graph to identify x-values for which . You may use the derivative to check your conclusion.

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. How many machines give minimum average costs?

A graph of is given. Use the graph to determine all critical values of .

In this problem, and its graph are given. Use the graph of to determine where has a point of inflection.

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

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.

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

A function and its graph are given. From the graph, estimate where .