Question 1

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

Accepted Answer

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

Question 2

Use the derivative to locate all critical points. Use a graphing utility if desired. Round your answers to two decimal places.   ​

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

The owner of an orange grove must decide when to pick one variety of oranges. She can sell them for $18 a bushel if she sells them now, with each tree yielding an average of 7 bushels. The yield increases by half a bushel per week for the next 10 weeks, but the price per bushel decreases by $0.70 per bushel each week. When should the oranges be picked for maximum return? ​​

Accepted Answer

A)  10 weeks from now 
B)  2 weeks from now 
C)  7 weeks from now 
D)  5 weeks from now 
E)  6 weeks from now 
A)  10 weeks from now 
B)  2 weeks from now 
C)  7 weeks from now 
D)  5 weeks from now 
E)  6 weeks from now

Question 5

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 6

A firm can produce 100 units per week. If its total cost function is   dollars, and its total revenue function is   dollars, find the maximum profit. ​​

Accepted Answer

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

Question 7

The following figure shows the graph of revenue function   and cost function   . At which of the four x-values shown is the slope of the tangent to the revenue curve equal to the slope of the tangent to the cost curve? ​

Accepted Answer

A)  ​D 
B)  B 
C)  C 
D)  A 
E)  profit largest at more than one x-value 
A)  ​D 
B)  B 
C)  C 
D)  A 
E)  profit largest at more than one x-value

Question 8

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 9

A function and its graph are given. Use the graph to find   , if it exists, where   . Confirm your results analytically. ​     ​

Accepted Answer

A)    
B)    
C)    
D)    
E)  does not exist 
A)    
B)    
C)    
D)    
E)  does not exist

Question 10

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

Accepted Answer

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

Question 11

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 12

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

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

Use the derivative to locate all critical points. Use a graphing utility if desired. Round your answers to two decimal places.

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

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

A firm can produce 100 units per week. If its total cost function is dollars, and its total revenue function is dollars, find the maximum profit.

The following figure shows the graph of revenue function and cost function . At which of the four x-values shown is the slope of the tangent to the revenue curve equal to the slope of the tangent to the cost curve?

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 , if it exists, where . Confirm your results analytically.

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

Find any vertical asymptotes for the given function.

Find any horizontal asymptotes for the given function.

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

Exam 10: Derivatives

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

Use the derivative to locate all critical points. Use a graphing utility if desired. Round your answers to two decimal places. ​

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

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

A firm can produce 100 units per week. If its total cost function is dollars, and its total revenue function is dollars, find the maximum profit. ​​

The following figure shows the graph of revenue function and cost function . At which of the four x-values shown is the slope of the tangent to the revenue curve equal to the slope of the tangent to the cost curve? ​

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 , if it exists, where . Confirm your results analytically. ​ ​

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

Find any vertical asymptotes for the given function.

Find any horizontal asymptotes for the given function.

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

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

Use the derivative to locate all critical points. Use a graphing utility if desired. Round your answers to two decimal places.

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

A firm can produce 100 units per week. If its total cost function is dollars, and its total revenue function is dollars, find the maximum profit.

The following figure shows the graph of revenue function and cost function . At which of the four x-values shown is the slope of the tangent to the revenue curve equal to the slope of the tangent to the cost curve?

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 , if it exists, where . Confirm your results analytically.