Question 1

For the given function, find   . ​   ​

Accepted Answer

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

Question 2

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

Accepted Answer

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

Question 3

A function and its first and second derivatives are given. Use these to find any relative minima.

Accepted Answer

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

Question 4

Find all 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 5

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

Accepted Answer

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

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. What is the minimum average cost? Round your answer to the nearest dollar. ​

Accepted Answer

A) $0 
B) $35 
C) $490 
D) $245 
E) $8,582 
A) $0 
B) $35 
C) $490 
D) $245 
E) $8,582

Question 7

The following figure shows the growth of a population as a function of time.   Which of A, B, and C correspond(s) to the point(s) at which the growth rate attains its maximum? ​

Accepted Answer

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

Question 8

Suppose that a company needs 800 items during a year and that preparation for each production run costs $60. Suppose further that it costs $8 to produce each item and $0.45 to store an item for one year. Use the inventory cost model to find the number of items in each production run that will minimize the total costs of production and storage. ​

Accepted Answer

A) 449 items 
B) 462 items 
C) 491 items 
D) 480 items 
E) 469 items 
A) 449 items 
B) 462 items 
C) 491 items 
D) 480 items 
E) 469 items

Question 9

Suppose that the oxygen level P (for purity) in a body of water t months after an oil spill is given by   . Find how long it will be before the oxygen level reaches its minimum. ​​

Accepted Answer

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

Question 10

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 11

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 12

A time study showed that, on average, the productivity of a worker after t hours on the job can be modeled by   ,   , where   is the number of units produced per hour. After how many hours will productivity be maximized? Round your answer to two decimal places. ​​

Accepted Answer

A) 5.61 hours 
B) 8.00 hours 
C) 6.77 hours 
D) 2.34 hours 
E) 5.41 hours 
A) 5.61 hours 
B) 8.00 hours 
C) 6.77 hours 
D) 2.34 hours 
E) 5.41 hours

Question 13

For the given function, classify the critical points as relative maxima, relative minima, or points of inflection. In each case, you may check your conclusions with a graphing utility.   ​

Accepted Answer

A)   
Is a relative minimum and  
Is point of inflection. 
B)   
Is a point of inflection and  
Is a relative minimum. 
C)   
Is a point of inflection and  
Is a relative maximum. 
D)   
Is a relative minimum and  
Is a relative maximum. 
E)   
Is a relative maximum and  
Is a relative minimum. 
A)   
Is a relative minimum and  
Is point of inflection. 
B)   
Is a point of inflection and  
Is a relative minimum. 
C)   
Is a point of inflection and  
Is a relative maximum. 
D)   
Is a relative minimum and  
Is a relative maximum. 
E)   
Is a relative maximum and  
Is a relative minimum.

Question 14

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 15

A product can be produced at a total cost   dollars, where x is the number produced. If the total revenue is given by   dollars, find the maximum profit. Round your answer to the nearest dollar. ​​

Accepted Answer

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

Question 16

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 17

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 18

For the given function, find the critical points.

Accepted Answer

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

Question 19

A small business has weekly average costs (in dollars) of   , where   is the number of units produced each week. The competitive market price for this business's product is $41 per unit. If production is limited to 277 units per week, find the level of production that yields maximum profit.

Accepted Answer

A) 88 units 
B) 277 units 
C) 140 units 
D) 110 units 
E) 125 units 
A) 88 units 
B) 277 units 
C) 140 units 
D) 110 units 
E) 125 units

Question 20

Both a function and its derivative are given. Use them to find all critical values.

Accepted Answer

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

For the given function, find .

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

A function and its first and second derivatives are given. Use these to find any relative minima.

Find all vertical asymptotes for the given function.

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

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. What is the minimum average cost? Round your answer to the nearest dollar.

The following figure shows the growth of a population as a function of time. Which of A, B, and C correspond(s) to the point(s) at which the growth rate attains its maximum?

Suppose that the oxygen level P (for purity) in a body of water t months after an oil spill is given by . Find how long it will be before the oxygen level reaches its minimum.

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

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

A time study showed that, on average, the productivity of a worker after t hours on the job can be modeled by , , where is the number of units produced per hour. After how many hours will productivity be maximized? Round your answer to two decimal places.

For the given function, classify the critical points as relative maxima, relative minima, or points of inflection. In each case, you may check your conclusions with a graphing utility.

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

A product can be produced at a total cost dollars, where x is the number produced. If the total revenue is given by dollars, find the maximum profit. Round your answer to the nearest dollar.

For the given function, find all critical values.

Find any vertical asymptotes for the given function.

For the given function, find the critical points.

A small business has weekly average costs (in dollars) of , where is the number of units produced each week. The competitive market price for this business's product is $41 per unit. If production is limited to 277 units per week, find the level of production that yields maximum profit.

Both a function and its derivative are given. Use them to find all critical values.

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

For the given function, find . ​ ​

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

A function and its first and second derivatives are given. Use these to find any relative minima.

Find all vertical asymptotes for the given function.

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

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. What is the minimum average cost? Round your answer to the nearest dollar. ​

The following figure shows the growth of a population as a function of time. Which of A, B, and C correspond(s) to the point(s) at which the growth rate attains its maximum? ​

Suppose that the oxygen level P (for purity) in a body of water t months after an oil spill is given by . Find how long it will be before the oxygen level reaches its minimum. ​​

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

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

A time study showed that, on average, the productivity of a worker after t hours on the job can be modeled by , , where is the number of units produced per hour. After how many hours will productivity be maximized? Round your answer to two decimal places. ​​

For the given function, classify the critical points as relative maxima, relative minima, or points of inflection. In each case, you may check your conclusions with a graphing utility. ​

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

A product can be produced at a total cost dollars, where x is the number produced. If the total revenue is given by dollars, find the maximum profit. Round your answer to the nearest dollar. ​​

For the given function, find all critical values.

Find any vertical asymptotes for the given function.

For the given function, find the critical points.

A small business has weekly average costs (in dollars) of , where is the number of units produced each week. The competitive market price for this business's product is $41 per unit. If production is limited to 277 units per week, find the level of production that yields maximum profit.

Both a function and its derivative are given. Use them to find all critical values.

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

For the given function, find .

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

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. What is the minimum average cost? Round your answer to the nearest dollar.

The following figure shows the growth of a population as a function of time. Which of A, B, and C correspond(s) to the point(s) at which the growth rate attains its maximum?

Suppose that the oxygen level P (for purity) in a body of water t months after an oil spill is given by . Find how long it will be before the oxygen level reaches its minimum.

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

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

A time study showed that, on average, the productivity of a worker after t hours on the job can be modeled by , , where is the number of units produced per hour. After how many hours will productivity be maximized? Round your answer to two decimal places.

For the given function, classify the critical points as relative maxima, relative minima, or points of inflection. In each case, you may check your conclusions with a graphing utility.

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

A product can be produced at a total cost dollars, where x is the number produced. If the total revenue is given by dollars, find the maximum profit. Round your answer to the nearest dollar.