Question 1

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

Accepted Answer

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

Question 2

Analytically determine the location(s) of any horizontal asymptote(s). Round your answer to two decimal places. ​   ​

Accepted Answer

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

Question 3

A function and its first and second derivatives are given. Use these to find all critical values.

Accepted Answer

A)    
B)  ​   
C)  ​   
D)  ​   
E)  no critical values 
A)    
B)  ​   
C)  ​   
D)  ​   
E)  no critical values

Question 4

The manufacturer of GRIPPER tires modeled its return to sales from television advertising expenditures in two regions, as follows: ​     ​
Where   and   are the sales revenue in millions of dollars, and   and   are millions of dollars of expenditures for television advertising. How much money will be needed to maximize sales revenue in both districts? Round your answer to the nearest million dollars.

Accepted Answer

A)  $16 million 
B)  ​$26 million 
C)  ​$9 million 
D)  ​$11 million 
E)  ​$12 million 
A)  $16 million 
B)  ​$26 million 
C)  ​$9 million 
D)  ​$11 million 
E)  ​$12 million

Question 5

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

Accepted Answer

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

Question 6

Both a function and its derivative are given. Use them to find the relative maxima.

Accepted Answer

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

Question 7

For the given function, find all critical values.

Accepted Answer

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

Question 8

Use the sign diagram for   to determine all critical values of f (x), where   and   .

Accepted Answer

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

Question 9

If the profit function for a commodity is   dollars, determine the number of units x that must be sold to result in maximum profit. ​​

Accepted Answer

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

Question 10

For the given function, find the horizontal points of inflection.

Accepted Answer

A)    
B)    
C)    
D)    
E)  no horizontal points of inflection 
A)    
B)    
C)    
D)    
E)  no horizontal points of inflection

Question 11

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

Accepted Answer

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

Question 12

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 13

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 14

Both a function and its derivative are given. Use them to find the relative minima.

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

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 production is maximized. ​

Accepted Answer

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

Question 17

Both a function and its derivative are given. Use them to find intervals on which the function is increasing.

Accepted Answer

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

Question 18

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 19

A firm has total revenue given by   dollars for x units of a product. Find the maximum revenue from sales of that product. ​

Accepted Answer

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

Question 20

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

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

Analytically determine the location(s) of any horizontal asymptote(s). Round your answer to two decimal places.

A function and its first and second derivatives are given. Use these to find all critical values.

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

Both a function and its derivative are given. Use them to find the relative maxima.

For the given function, find all critical values.

Use the sign diagram for to determine all critical values of f (x), where and .

If the profit function for a commodity is dollars, determine the number of units x that must be sold to result in maximum profit.

For the given function, find the horizontal points of inflection.

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

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

Find any horizontal asymptotes for the given function.

Both a function and its derivative are given. Use them to find the relative minima.

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

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 production is maximized.

Both a function and its derivative are given. Use them to find intervals on which the function is increasing.

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 firm has total revenue given by dollars for x units of a product. Find the maximum revenue from sales of that product.

For the given function, find the relative maxima.

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 second derivative to determine intervals on which the function is concave down. Check these results against the graph shown. ​ ​

Analytically determine the location(s) of any horizontal asymptote(s). Round your answer to two decimal places. ​ ​

A function and its first and second derivatives are given. Use these to find all critical values.

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

Both a function and its derivative are given. Use them to find the relative maxima.

For the given function, find all critical values.

Use the sign diagram for to determine all critical values of f (x), where and .

If the profit function for a commodity is dollars, determine the number of units x that must be sold to result in maximum profit. ​​

For the given function, find the horizontal points of inflection.

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

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

Find any horizontal asymptotes for the given function.

Both a function and its derivative are given. Use them to find the relative minima.

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

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 production is maximized. ​

Both a function and its derivative are given. Use them to find intervals on which the function is increasing.

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 firm has total revenue given by dollars for x units of a product. Find the maximum revenue from sales of that product. ​

For the given function, find the relative maxima.

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 second derivative to determine intervals on which the function is concave down. Check these results against the graph shown.

Analytically determine the location(s) of any horizontal asymptote(s). Round your answer to two decimal places.

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

If the profit function for a commodity is dollars, determine the number of units x that must be sold to result in maximum profit.

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

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

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

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 production is maximized.

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 firm has total revenue given by dollars for x units of a product. Find the maximum revenue from sales of that product.