Question 1

If the total cost function for a product is   dollars, determine how many units x should be produced to minimize the average cost per unit? ​​

Accepted Answer

A) 90 units 
B) 200 units 
C) 44 units 
D) 63 units 
E) 51 units 
A) 90 units 
B) 200 units 
C) 44 units 
D) 63 units 
E) 51 units

Question 2

The consumer price data can be modeled by the function   , where   represents 1945 and   is the consumer price index (CPI) in year   . During what year does the model predict that the rate of change of the CPI reached its maximum? ​

Accepted Answer

A) 1955 
B) 1981 
C) 1996 
D) 1977 
E) 1947 
A) 1955 
B) 1981 
C) 1996 
D) 1977 
E) 1947

Question 3

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 4

For the given function, find the critical points.

Accepted Answer

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

Question 5

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) 0 
B) 1 
C) 8 
D) 6 
E) 4 
A) 0 
B) 1 
C) 8 
D) 6 
E) 4

Question 6

For the revenue function given by   find the maximum average revenue. ​

Accepted Answer

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

Question 7

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 8

The following figure shows the graph of revenue function   and cost function   . At which of the four   -values shown is the profit largest?   ​

Accepted Answer

A) B 
B) C 
C) D 
D) A 
E) Profit is largest at more than one x-value. 
A) B 
B) C 
C) D 
D) A 
E) Profit is largest at more than one x-value.

Question 9

From a tract of land, a developer plans to fence a rectangular region and then divide it into two identical rectangular lots by putting a fence down the middle. Suppose that the fence for the outside boundary costs $6 per foot and the fence for the middle costs $4 per foot. If each lot contains 4,600 square feet, find the dimensions of each lot that yield the minimum cost for the fence. Round your answer to two decimal places. ​​

Accepted Answer

A) Dimensions are 42.19 ft for the side parallel to the divider and 109.03 ft for the other side. 
B) Dimensions are 109.03 ft for the side parallel to the divider and 42.19 ft for the other side. 
C) Dimensions are 67.82 ft for the side parallel to the divider and 67.82 ft for the other side. 
D) Dimensions are 58.74 ft for the side parallel to the divider and 78.32 ft for the other side. 
E) Dimensions are 78.32 ft for the side parallel to the divider and 58.74 ft for the other side. 
A) Dimensions are 42.19 ft for the side parallel to the divider and 109.03 ft for the other side. 
B) Dimensions are 109.03 ft for the side parallel to the divider and 42.19 ft for the other side. 
C) Dimensions are 67.82 ft for the side parallel to the divider and 67.82 ft for the other side. 
D) Dimensions are 58.74 ft for the side parallel to the divider and 78.32 ft for the other side. 
E) Dimensions are 78.32 ft for the side parallel to the divider and 58.74 ft for the other side.

Question 10

A firm can produce 100 units per week. If its total cost function is   dollars, and its total revenue function is   dollars, how many units x should it produce to maximize its profit? ​​

Accepted Answer

A) 1,850 units 
B) 950 units 
C) 96 units 
D) 50 units 
E) 100 units 
A) 1,850 units 
B) 950 units 
C) 96 units 
D) 50 units 
E) 100 units

Question 11

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 12

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

Accepted Answer

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

Question 13

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 14

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

Accepted Answer

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

Question 15

A function and its first and second derivatives are given. Use these to find all points of inflection.

Accepted Answer

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

Question 16

Use the sign diagram for   to determine x-values at which relative maxima occur, where   and   .

Accepted Answer

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

Question 17

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 18

The profit from a grove of orange trees is given by   dollars, where x is the number of orange trees per acre. How many trees per acre will maximize the profit? ​

Accepted Answer

A) 300 
B) 80 
C) 140 
D) 75 
E) 150 
A) 300 
B) 80 
C) 140 
D) 75 
E) 150

Question 19

Use the sign diagram for   to determine x-values at which relative minima occur, where   and   .   ​

Accepted Answer

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

Question 20

The percent   of impurities that can be removed from the waste water of a manufacturing process at a cost of   dollars is given by   . Find   values for which p is increasing.

Accepted Answer

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

If the total cost function for a product is dollars, determine how many units x should be produced to minimize the average cost per unit?

The consumer price data can be modeled by the function , where represents 1945 and is the consumer price index (CPI) in year . During what year does the model predict that the rate of change of the CPI reached its maximum?

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

For the given function, find the critical points.

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

For the revenue function given by find the maximum average revenue.

Find any vertical asymptotes for the given function.

The following figure shows the graph of revenue function and cost function . At which of the four -values shown is the profit largest?

A firm can produce 100 units per week. If its total cost function is dollars, and its total revenue function is dollars, how many units x should it produce to maximize its profit?

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

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

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

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

A function and its first and second derivatives are given. Use these to find all points of inflection.

Use the sign diagram for to determine x-values at which relative maxima occur, where and .

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.

The profit from a grove of orange trees is given by dollars, where x is the number of orange trees per acre. How many trees per acre will maximize the profit?

Use the sign diagram for to determine x-values at which relative minima occur, where and .

The percent of impurities that can be removed from the waste water of a manufacturing process at a cost of dollars is given by . Find values for which p is increasing.

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

If the total cost function for a product is dollars, determine how many units x should be produced to minimize the average cost per unit? ​​

The consumer price data can be modeled by the function , where represents 1945 and is the consumer price index (CPI) in year . During what year does the model predict that the rate of change of the CPI reached its maximum? ​

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

For the given function, find the critical points.

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

For the revenue function given by find the maximum average revenue. ​

Find any vertical asymptotes for the given function.

The following figure shows the graph of revenue function and cost function . At which of the four -values shown is the profit largest? ​

A firm can produce 100 units per week. If its total cost function is dollars, and its total revenue function is dollars, how many units x should it produce to maximize its profit? ​​

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

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

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

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

A function and its first and second derivatives are given. Use these to find all points of inflection.

Use the sign diagram for to determine x-values at which relative maxima occur, where and .

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. ​ ​

The profit from a grove of orange trees is given by dollars, where x is the number of orange trees per acre. How many trees per acre will maximize the profit? ​

Use the sign diagram for to determine x-values at which relative minima occur, where and . ​

The percent of impurities that can be removed from the waste water of a manufacturing process at a cost of dollars is given by . Find values for which p is increasing.

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

If the total cost function for a product is dollars, determine how many units x should be produced to minimize the average cost per unit?

The consumer price data can be modeled by the function , where represents 1945 and is the consumer price index (CPI) in year . During what year does the model predict that the rate of change of the CPI reached its maximum?

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

For the revenue function given by find the maximum average revenue.

The following figure shows the graph of revenue function and cost function . At which of the four -values shown is the profit largest?

A firm can produce 100 units per week. If its total cost function is dollars, and its total revenue function is dollars, how many units x should it produce to maximize its profit?

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

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

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

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.

The profit from a grove of orange trees is given by dollars, where x is the number of orange trees per acre. How many trees per acre will maximize the profit?

Use the sign diagram for to determine x-values at which relative minima occur, where and .