Question 1

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 2

The following figure shows the growth of a population as a function of time.   Which of A, B, and/or C correspond(s) to the upper limit of population?

Accepted Answer

A) B 
B) C 
C) A 
D) B and A 
E) none of the above 
A) B 
B) C 
C) A 
D) B and A 
E) none of the above

Question 3

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

Accepted Answer

A) 0 
B) 8 
C) 7 
D) 5 
E) does not exist 
A) 0 
B) 8 
C) 7 
D) 5 
E) does not exist

Question 4

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 5

Suppose that a chain of auto service stations, Quick-Oil, Inc., has found that its monthly sales volume y (in thousands of dollars) is related to the price p (in dollars) of an oil change by     . Is y increasing or decreasing for all values of   ​

Accepted Answer

A) increasing 
B) decreasing 
A) increasing 
B) decreasing

Question 6

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

Accepted Answer

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

Question 7

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 8

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 9

The owner of an orange grove must decide when to pick one variety of oranges. She can sell them for $14 a bushel if she sells them now, with each tree yielding an average of 5 bushels. The yield increases by half a bushel per week for the next 10 weeks, but the price per bushel decreases by $0.80 per bushel each week. When should the oranges be picked for maximum return?

Accepted Answer

A) 9 weeks from now 
B) 2 weeks from now 
C) 7 weeks from now 
D) 3 weeks from now 
E) 4 weeks from now 
A) 9 weeks from now 
B) 2 weeks from now 
C) 7 weeks from now 
D) 3 weeks from now 
E) 4 weeks from now

Question 10

A function and its first and second derivatives are given. Use these to find the relative maxima.

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

A product can be produced at a total cost   dollars, where x is the number produced. If the total revenue is given by   dollars, determine the level of production x that will maximize the profit. ​

Accepted Answer

A) 21 units 
B) 56 units 
C) 66 units 
D) 33 units 
E) 44 units 
A) 21 units 
B) 56 units 
C) 66 units 
D) 33 units 
E) 44 units

Question 13

Make a sign diagram for the function and determine all x-values at which relative minima occur.   ​

Accepted Answer

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

Question 14

For the given function and graph, estimate the coordinates of the relative maxima by observing the graph, where   . ​   ​   ​

Accepted Answer

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

Question 15

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 16

The following figure shows the growth of a population as a function of time.   If P represents the population and t represents the time, write a mathematical symbol that represents the rate of change (growth rate) of the population with respect to time.

Accepted Answer

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

Question 17

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

Accepted Answer

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

Question 18

Sketch the graph of the function   .

Accepted Answer

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

Question 19

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 20

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

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

The following figure shows the growth of a population as a function of time. Which of A, B, and/or C correspond(s) to the upper limit of population?

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

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

Suppose that a chain of auto service stations, Quick-Oil, Inc., has found that its monthly sales volume y (in thousands of dollars) is related to the price p (in dollars) of an oil change by . Is y increasing or decreasing for all values of

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

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

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: )

A function and its first and second derivatives are given. Use these to find the relative maxima.

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

A product can be produced at a total cost dollars, where x is the number produced. If the total revenue is given by dollars, determine the level of production x that will maximize the profit.

Make a sign diagram for the function and determine all x-values at which relative minima occur.

For the given function and graph, estimate the coordinates of the relative maxima by observing the graph, where .

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

The following figure shows the growth of a population as a function of time. If P represents the population and t represents the time, write a mathematical symbol that represents the rate of change (growth rate) of the population with respect to time.

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

Sketch the graph of the function .

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 find , if it exists. Confirm your results analytically.

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

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

The following figure shows the growth of a population as a function of time. Which of A, B, and/or C correspond(s) to the upper limit of population?

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

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

Suppose that a chain of auto service stations, Quick-Oil, Inc., has found that its monthly sales volume y (in thousands of dollars) is related to the price p (in dollars) of an oil change by . Is y increasing or decreasing for all values of ​

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

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

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: ) ​​

A function and its first and second derivatives are given. Use these to find the relative maxima.

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

A product can be produced at a total cost dollars, where x is the number produced. If the total revenue is given by dollars, determine the level of production x that will maximize the profit. ​

Make a sign diagram for the function and determine all x-values at which relative minima occur. ​

For the given function and graph, estimate the coordinates of the relative maxima by observing the graph, where . ​ ​ ​

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

The following figure shows the growth of a population as a function of time. If P represents the population and t represents the time, write a mathematical symbol that represents the rate of change (growth rate) of the population with respect to time.

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

Sketch the graph of the function .

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

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

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

Suppose that a chain of auto service stations, Quick-Oil, Inc., has found that its monthly sales volume y (in thousands of dollars) is related to the price p (in dollars) of an oil change by . Is y increasing or decreasing for all values of

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

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

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: )

A product can be produced at a total cost dollars, where x is the number produced. If the total revenue is given by dollars, determine the level of production x that will maximize the profit.

Make a sign diagram for the function and determine all x-values at which relative minima occur.

For the given function and graph, estimate the coordinates of the relative maxima by observing the graph, where .

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

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 find , if it exists. Confirm your results analytically.