Question 1

For the given function, find the relative minima.

Accepted Answer

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

Question 2

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 3

For the given function, find   ​   ​

Accepted Answer

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

Question 4

Analytically determine any relative maxima. Round your answer to two decimal places.

Accepted Answer

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

Question 5

Find all relative minima of the given function.

Accepted Answer

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

Question 6

The base of a rectangular box is to be twice as long as it is wide. The volume of the box is 450 cubic inches. The material for the top costs $0.17 per square inch and the material for the sides and bottom costs $0.1 per square inch. Find the dimensions that will make the cost a minimum.

Accepted Answer

A) Dimensions are 9" by 10" by 12" (high). 
B) Dimensions are 11" by 22" by 12" (high). 
C) Dimensions are 5" by 10" by 9" (high). 
D) Dimensions are 9" by 18" by 5" (high). 
E) Dimensions are 12" by 24" by 11" (high). 
A) Dimensions are 9" by 10" by 12" (high). 
B) Dimensions are 11" by 22" by 12" (high). 
C) Dimensions are 5" by 10" by 9" (high). 
D) Dimensions are 9" by 18" by 5" (high). 
E) Dimensions are 12" by 24" by 11" (high).

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

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 9

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

Accepted Answer

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

Question 10

Determine whether the given function is concave up or concave down at the indicated point.   at

Accepted Answer

A) concave down 
B) concave up 
A) concave down 
B) concave up

Question 11

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 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

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. How many machines give minimum average costs? ​

Accepted Answer

A) Using 25 machines gives the minimum average costs. 
B) Using zero machines gives the minimum average costs. 
C) Using 35 machines gives the minimum average costs. 
D) Using 50 machines gives the minimum average costs. 
E) Using 55 machines gives the minimum average costs. 
A) Using 25 machines gives the minimum average costs. 
B) Using zero machines gives the minimum average costs. 
C) Using 35 machines gives the minimum average costs. 
D) Using 50 machines gives the minimum average costs. 
E) Using 55 machines gives the minimum average costs.

Question 15

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 16

Use the graph of   to identify at which of the indicated points the derivative   changes from negative to positive. ​

Accepted Answer

A) (-1, 2), (2, 4) 
B) (4, 1) 
C) (-1, 2), (2, 4), (5, 6) 
D) (2, 4) 
E) (2, 4), (5, 6) 
A) (-1, 2), (2, 4) 
B) (4, 1) 
C) (-1, 2), (2, 4), (5, 6) 
D) (2, 4) 
E) (2, 4), (5, 6)

Question 17

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 18

The monthly demand function for x units of a product sold by a monopoly is   dollars per unit, and its average cost is   dollars. If production is limited to 111 units, find the number of units that maximizes profit. (​Note: ​   ) ​​

Accepted Answer

A) 40 units 
B) 111 units 
C) 107 units 
D) 11 units 
E) 54 units 
A) 40 units 
B) 111 units 
C) 107 units 
D) 11 units 
E) 54 units

Question 19

The percent p of impurities that can be removed from the waste water of a manufacturing process at a cost of C dollars is given by   . Can 100% of the pollution be removed? ​

Accepted Answer

A) ​Yes 
B) ​No 
A) ​Yes 
B) ​No

Question 20

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

For the given function, find the relative minima.

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

For the given function, find

Analytically determine any relative maxima. Round your answer to two decimal places.

Find all relative minima of the given function.

The base of a rectangular box is to be twice as long as it is wide. The volume of the box is 450 cubic inches. The material for the top costs $0.17 per square inch and the material for the sides and bottom costs $0.1 per square inch. Find the dimensions that will make the cost a minimum.

Find any vertical asymptotes for the given function.

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

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

Determine whether the given function is concave up or concave down at the indicated point. at

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

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 estimate the locations of any horizontal asymptotes.

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. How many machines give minimum average costs?

Find any horizontal asymptotes for the given function.

Use the graph of to identify at which of the indicated points the derivative changes from negative to positive.

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

The monthly demand function for x units of a product sold by a monopoly is dollars per unit, and its average cost is dollars. If production is limited to 111 units, find the number of units that maximizes profit. (Note: )

The percent p of impurities that can be removed from the waste water of a manufacturing process at a cost of C dollars is given by . Can 100% of the pollution be removed?

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

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 the relative minima.

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

For the given function, find ​ ​

Analytically determine any relative maxima. Round your answer to two decimal places.

Find all relative minima of the given function.

The base of a rectangular box is to be twice as long as it is wide. The volume of the box is 450 cubic inches. The material for the top costs $0.17 per square inch and the material for the sides and bottom costs $0.1 per square inch. Find the dimensions that will make the cost a minimum.

Find any vertical asymptotes for the given function.

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

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

Determine whether the given function is concave up or concave down at the indicated point. at

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

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 estimate the locations of any horizontal asymptotes. ​

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. How many machines give minimum average costs? ​

Find any horizontal asymptotes for the given function.

Use the graph of to identify at which of the indicated points the derivative changes from negative to positive. ​

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

The monthly demand function for x units of a product sold by a monopoly is dollars per unit, and its average cost is dollars. If production is limited to 111 units, find the number of units that maximizes profit. (​Note: ​ ) ​​

The percent p of impurities that can be removed from the waste water of a manufacturing process at a cost of C dollars is given by . Can 100% of the pollution be removed? ​

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

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 and graph, determine all critical point(s), where .

For the given function, find

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

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

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. How many machines give minimum average costs?

Use the graph of to identify at which of the indicated points the derivative changes from negative to positive.

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

The monthly demand function for x units of a product sold by a monopoly is dollars per unit, and its average cost is dollars. If production is limited to 111 units, find the number of units that maximizes profit. (Note: )

The percent p of impurities that can be removed from the waste water of a manufacturing process at a cost of C dollars is given by . Can 100% of the pollution be removed?

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