Question 1

Identify the open intervals where the function   is increasing or decreasing.

Accepted Answer

A) decreasing:   ; increasing:   
B) increasing:   ; decreasing:   
C) increasing:   ; decreasing:   
D) increasing:   ; decreasing:   
E) decreasing on   
A) decreasing:   ; increasing:   
B) increasing:   ; decreasing:   
C) increasing:   ; decreasing:   
D) increasing:   ; decreasing:   
E) decreasing on   B

Question 2

Determine the open intervals on which the graph of   is concave downward or concave upward.

Accepted Answer

A) concave downward on   
B) concave downward on   ; concave upward on   
C) concave upward on   ; concave downward on   
D) concave downward on   ; concave upward on   
E) concave upward on   ; concave downward on   
A) concave downward on   
B) concave downward on   ; concave upward on   
C) concave upward on   ; concave downward on   
D) concave downward on   ; concave upward on   
E) concave upward on   ; concave downward on   E

Question 3

The resistance R of a certain type of resistor is   where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra system to find

Accepted Answer

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

Question 4

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

Accepted Answer

A)    
B) The derivative of f does not exist. 
C)    
D)    
E)    
A)    
B) The derivative of f does not exist. 
C)    
D)    
E)

Question 5

Determine whether Rolle's Theorem can be applied to   on the closed interval   If Rolle's Theorem can be applied, find all values of c in the open interval   such that

Accepted Answer

A) c = 3, c = 8 
B) c = 8 
C) c = 7, c = 3 
D) c = 7 
E) Rolle's Theorem does not apply 
A) c = 3, c = 8 
B) c = 8 
C) c = 7, c = 3 
D) c = 7 
E) Rolle's Theorem does not apply

Question 6

The resistance R of a certain type of resistor is   where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra system to determine the minimum resistance for this type of resistor. Round your answer to two decimal places.

Accepted Answer

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

Question 7

Find the limit.

Accepted Answer

A)    
B)    
C) 1 
D) 0 
E) does not exist 
A)    
B)    
C) 1 
D) 0 
E) does not exist

Question 8

Determine the slant asymptote of the graph of   .

Accepted Answer

A)    
B)    
C)    
D)    
E) no slant asymptote 
A)    
B)    
C)    
D)    
E) no slant asymptote

Question 9

Determine whether Rolle's Theorem can be applied to the function   on the closed interval [-1,5]. If Rolle's Theorem can be applied, find all values of c in the open interval (-1,5) such that   .

Accepted Answer

A) Rolle's Theorem applies; c = -2 
B) Rolle's Theorem applies; c = 0.5 
C) Rolle's Theorem does not apply 
D) Rolle's Theorem applies; c = 2 
E) Rolle's Theorem applies; c = 3.5 
A) Rolle's Theorem applies; c = -2 
B) Rolle's Theorem applies; c = 0.5 
C) Rolle's Theorem does not apply 
D) Rolle's Theorem applies; c = 2 
E) Rolle's Theorem applies; c = 3.5

Question 10

Determine whether Rolle's Theorem can be applied to the function   on the closed interval   . If Rolle's Theorem can be applied, find all numbers c in the open interval   such that   .

Accepted Answer

A) Rolle's Theorem applies; c =   
B) Rolle's Theorem applies; c =   ,   
C) Rolle's Theorem applies;   
D) Rolle's Theorem does not apply 
E) Rolle's Theorem does apply;   
A) Rolle's Theorem applies; c =   
B) Rolle's Theorem applies; c =   ,   
C) Rolle's Theorem applies;   
D) Rolle's Theorem does not apply 
E) Rolle's Theorem does apply;

Question 11

A business has a cost of   for producing x units. The average cost per unit is   Find the limit of   as x approaches infinity.

Accepted Answer

A) 250.0 
B) 300.0 
C) 0 
D) 1.0 
E) 1.2 
A) 250.0 
B) 300.0 
C) 0 
D) 1.0 
E) 1.2

Question 12

The graph of f is shown below. For which values of x is   zero?

Accepted Answer

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

Question 13

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

Accepted Answer

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

Question 14

Find the critical number of the function

Accepted Answer

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

Question 15

Analyze and sketch a graph of the function   .

Accepted Answer

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

Question 16

Analyze and sketch a graph of the function   .

Accepted Answer

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

Question 17

A company introduces a new product for which the number of units sold S is   where t is the time in months since the product was introduced. During what month does   equal the average value of S(t) during the first year?

Accepted Answer

A) March 
B) June 
C) April 
D) May 
E) September 
A) March 
B) June 
C) April 
D) May 
E) September

Question 18

Analyze and sketch a graph of the function   .

Accepted Answer

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

Question 19

The measurement of the radius of the end of a log is found to be 14 inches, with a possible error of   inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log.

Accepted Answer

A)    square inches 
B)    square inches 
C)    square inches 
D)    square inches 
E)    square inches 
A)    square inches 
B)    square inches 
C)    square inches 
D)    square inches 
E)    square inches

Question 20

For the function   : (a) Find the critical numbers of f (if any);
(b) Find the open intervals where the function is increasing or decreasing; and
(c) Apply the First Derivative Test to identify all relative extrema.
Then use a graphing utility to confirm your results.

Accepted Answer

A) (a) x = 0 , 4 (b) increasing:   ; decreasing:   (c) relative max:   ; relative min:   
B) (a) x = 0 , 4 (b) decreasing:   ; increasing:   (c) relative min:   ; relative max:   
C) (a) x = 0 , 1 (b) increasing:   ; decreasing:   (c) relative max:   ; relative min:   
D) (a) x = 0 , 1 (b) decreasing:   ; increasing:   (c) relative min:   ; relative max:   
E) (a) x = 0 , 1 (b) increasing:   ; decreasing:   (c) relative max:   ; no relative min. 
A) (a) x = 0 , 4 (b) increasing:   ; decreasing:   (c) relative max:   ; relative min:   
B) (a) x = 0 , 4 (b) decreasing:   ; increasing:   (c) relative min:   ; relative max:   
C) (a) x = 0 , 1 (b) increasing:   ; decreasing:   (c) relative max:   ; relative min:   
D) (a) x = 0 , 1 (b) decreasing:   ; increasing:   (c) relative min:   ; relative max:   
E) (a) x = 0 , 1 (b) increasing:   ; decreasing:   (c) relative max:   ; no relative min.

Identify the open intervals where the function is increasing or decreasing.

Determine the open intervals on which the graph of is concave downward or concave upward.

The resistance R of a certain type of resistor is where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra system to find

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

Determine whether Rolle's Theorem can be applied to on the closed interval If Rolle's Theorem can be applied, find all values of c in the open interval such that

The resistance R of a certain type of resistor is where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra system to determine the minimum resistance for this type of resistor. Round your answer to two decimal places.

Find the limit.

Determine the slant asymptote of the graph of .

Determine whether Rolle's Theorem can be applied to the function on the closed interval [-1,5]. If Rolle's Theorem can be applied, find all values of c in the open interval (-1,5) such that .

Determine whether Rolle's Theorem can be applied to the function on the closed interval . If Rolle's Theorem can be applied, find all numbers c in the open interval such that .

A business has a cost of for producing x units. The average cost per unit is Find the limit of as x approaches infinity.

The graph of f is shown below. For which values of x is zero?

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

Find the critical number of the function

Analyze and sketch a graph of the function .

Analyze and sketch a graph of the function .

A company introduces a new product for which the number of units sold S is where t is the time in months since the product was introduced. During what month does equal the average value of S(t) during the first year?

Analyze and sketch a graph of the function .

The measurement of the radius of the end of a log is found to be 14 inches, with a possible error of inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log.

For the function : (a) Find the critical numbers of f (if any); (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. Then use a graphing utility to confirm your results.

Prerequisites

Functions and Their Graphs

Polynomial and Rational Functions

Limits and Their Properties

Differentiation

Integration

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions and Calculus

Trigonometric Functions

Analytic Trigonometry

Trigonometric Functions and Calculus

Topics in Analytic Geometry

Additional Topics in Trigonometry

Filters

Exam 6: Applications of Differentiation

Identify the open intervals where the function is increasing or decreasing.

Determine the open intervals on which the graph of is concave downward or concave upward.

The resistance R of a certain type of resistor is where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra system to find

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

Determine whether Rolle's Theorem can be applied to on the closed interval If Rolle's Theorem can be applied, find all values of c in the open interval such that

The resistance R of a certain type of resistor is where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra system to determine the minimum resistance for this type of resistor. Round your answer to two decimal places.

Find the limit.

Determine the slant asymptote of the graph of .

Determine whether Rolle's Theorem can be applied to the function on the closed interval [-1,5]. If Rolle's Theorem can be applied, find all values of c in the open interval (-1,5) such that .

Determine whether Rolle's Theorem can be applied to the function on the closed interval . If Rolle's Theorem can be applied, find all numbers c in the open interval such that .

A business has a cost of for producing x units. The average cost per unit is Find the limit of as x approaches infinity.

The graph of f is shown below. For which values of x is zero?

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

Find the critical number of the function

Analyze and sketch a graph of the function .

Analyze and sketch a graph of the function .

A company introduces a new product for which the number of units sold S is where t is the time in months since the product was introduced. During what month does equal the average value of S(t) during the first year?

Analyze and sketch a graph of the function .

The measurement of the radius of the end of a log is found to be 14 inches, with a possible error of inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log.

For the function : (a) Find the critical numbers of f (if any); (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. Then use a graphing utility to confirm your results.

Prerequisites

Functions and Their Graphs

Polynomial and Rational Functions

Limits and Their Properties

Differentiation

Integration

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions and Calculus

Trigonometric Functions

Analytic Trigonometry

Trigonometric Functions and Calculus

Topics in Analytic Geometry

Additional Topics in Trigonometry

Filters