Question 1

Find the points of inflection and discuss the concavity of the function   on the interval   .

Accepted Answer

A)  no inflection points. concave up on   
B)  concave upward on   ; concave downward on   ; inflection point at   
C)  no inflection points. concave down on   
D)  concave downward on   ; concave upward on   ; inflection point at   
E)  none of the above 
A)  no inflection points. concave up on   
B)  concave upward on   ; concave downward on   ; inflection point at   
C)  no inflection points. concave down on   
D)  concave downward on   ; concave upward on   ; inflection point at   
E)  none of the above E

Question 2

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

Accepted Answer

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

Question 3

Find all points of inflection on the graph of the function   .

Accepted Answer

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

Question 4

Find all points of inflection, if any exist, of the graph of the function   . Round your answers to two decimal places.

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

Find the points of inflection and discuss the concavity of the function.

Accepted Answer

A)  inflection point at   ; concave upward on   ; concave downward on   
B)  inflection point at   ; concare downward on   ; concave upward on   
C)  inflection point at   ; concave downward on   ; concave upward on   
D)  inflection point at   ; concave downward on   ; concave upward on   
E)  inflection point at   ; concave upward on   ; concave downward on   
A)  inflection point at   ; concave upward on   ; concave downward on   
B)  inflection point at   ; concare downward on   ; concave upward on   
C)  inflection point at   ; concave downward on   ; concave upward on   
D)  inflection point at   ; concave downward on   ; concave upward on   
E)  inflection point at   ; concave upward on   ; concave downward on

Question 7

Find the point of inflection of the graph of the function   on the interval   .

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

Find all relative extrema of the function   . Use the Second Derivative Test where applicable.

Accepted Answer

A)  relative max: (1, -2); no relative min 
B)  no relative max or min 
C)  relative min: (0, -3); no relative max 
D)  relative max: (1, -2); relative min: (0, -3) 
E)  relative max: (0, -3); no relative min 
A)  relative max: (1, -2); no relative min 
B)  no relative max or min 
C)  relative min: (0, -3); no relative max 
D)  relative max: (1, -2); relative min: (0, -3) 
E)  relative max: (0, -3); no relative min

Question 10

Find all relative extrema of the function   . Use the Second Derivative Test where applicable.

Accepted Answer

A)  relative max:   ; no relative min 
B)  no relative max; no relative min 
C)  relative min:   ; relative max:   
D)  relative min:   ; no relative max 
E)  relative min:   ; relative max:   
A)  relative max:   ; no relative min 
B)  no relative max; no relative min 
C)  relative min:   ; relative max:   
D)  relative min:   ; no relative max 
E)  relative min:   ; relative max:

Question 11

Locate any relative extrema and inflection points of the function   . Use a graphing utility to confirm your results.

Accepted Answer

A)  relative minimum value:   ; no inflection points 
B)  relative minimum value:   inflection point:   
C)  relative maximum value:   inflection point:   
D)  relative minimum value:   no inflection points 
E)  relative maximum value:   ; no inflection points 
A)  relative minimum value:   ; no inflection points 
B)  relative minimum value:   inflection point:   
C)  relative maximum value:   inflection point:   
D)  relative minimum value:   no inflection points 
E)  relative maximum value:   ; no inflection points

Question 12

Determine the x-coordinate(s) of any relative extrema and inflection points of the function   .

Accepted Answer

A)  relative minimum:   ; inflection point:   
B)  no relative extrema; inflection point:   
C)  relative minimum:   ; no inflection points 
D)  no relative extrema; inflection point:   
E)  relative maximum:   inflection point:   
A)  relative minimum:   ; inflection point:   
B)  no relative extrema; inflection point:   
C)  relative minimum:   ; no inflection points 
D)  no relative extrema; inflection point:   
E)  relative maximum:   inflection point:

Question 13

Find the cubic function of the form   where   and the coefficients   are real numbers, which satisfies the conditions given below. Relative maximum:   Relative minimum:   Inflection point:

Accepted Answer

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

Question 14

Determine the x-coordinate(s) of any relative extrema and inflection points of the function   .

Accepted Answer

A)  relative minimum:   ; no inflection points 
B)  relative maximum:   ; inflection point:   
C)  no relative maximum or minimum; inflection point:   
D)  no relative extrema or inflection points. 
E)  relative minimum:   ; inflection point:   
A)  relative minimum:   ; no inflection points 
B)  relative maximum:   ; inflection point:   
C)  no relative maximum or minimum; inflection point:   
D)  no relative extrema or inflection points. 
E)  relative minimum:   ; inflection point:

Question 15

The graph of f is shown. Graph f, f' and f'' on the same set of coordinate axes.

Accepted Answer

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

Question 16

The graph of f is shown. Graph f, f' and f'' on the same set of coordinate axes.

Accepted Answer

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

Question 17

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

Accepted Answer

A)  concave downward on   ; concave upward 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   ; concave upward 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

Question 18

Find the points of inflection and discuss the concavity of the function   .

Accepted Answer

A)  no inflection points; concave up on   
B)  no inflection points; concave down on   
C)  inflection point at x = 16; concave up on   
D)  inflection point at x = 0; concave up on   ; concave down on   
E)  inflection point at x = 16; concave down on   
A)  no inflection points; concave up on   
B)  no inflection points; concave down on   
C)  inflection point at x = 16; concave up on   
D)  inflection point at x = 0; concave up on   ; concave down on   
E)  inflection point at x = 16; concave down on

Question 19

Suppose the deflection D of a beam of length L is   where x is the distance from one end of the beam. Find the value of x that yields the maximum deflection.

Accepted Answer

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

Question 20

Find all relative extrema of the function   . Use the Second Derivative Test where applicable.

Accepted Answer

A)  relative max:   ; no relative min 
B)  relative min:   ; no relative max 
C)  relative min:   ; no relative max 
D)  relative max:   ; no relative min 
E)  no relative max or min 
A)  relative max:   ; no relative min 
B)  relative min:   ; no relative max 
C)  relative min:   ; no relative max 
D)  relative max:   ; no relative min 
E)  no relative max or min

Exam 4: Section 4: Applications of Differentiation

Find the points of inflection and discuss the concavity of the function on the interval .

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

Find all points of inflection on the graph of the function .

Find all points of inflection, if any exist, of the graph of the function . Round your answers to two decimal places.

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

Find the points of inflection and discuss the concavity of the function.

Find the point of inflection of the graph of the function on the interval .

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

Find all relative extrema of the function . Use the Second Derivative Test where applicable.

Find all relative extrema of the function . Use the Second Derivative Test where applicable.

Locate any relative extrema and inflection points of the function . Use a graphing utility to confirm your results.

Determine the x-coordinate(s) of any relative extrema and inflection points of the function .

Find the cubic function of the form where and the coefficients are real numbers, which satisfies the conditions given below. Relative maximum: Relative minimum: Inflection point:

Determine the x-coordinate(s) of any relative extrema and inflection points of the function .

The graph of f is shown. Graph f, f' and f'' on the same set of coordinate axes.

The graph of f is shown. Graph f, f' and f'' on the same set of coordinate axes.

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

Find the points of inflection and discuss the concavity of the function .

Suppose the deflection D of a beam of length L is where x is the distance from one end of the beam. Find the value of x that yields the maximum deflection.

Find all relative extrema of the function . Use the Second Derivative Test where applicable.

Section 1: Preparation for Calculus

Section 2: Preparation for Calculus

Section 3: Preparation for Calculus

Section 4: Preparation for Calculus

Section 5: Preparation for Calculus

Section 6: Preparation for Calculus

Section 1: Limits and Their Properties

Section 2: Limits and Their Properties

Section 3: Limits and Their Properties

Section 4: Limits and Their Properties

Section 5 : Limits and Their Properties

Section 1 : Differentiation

Section 2: Differentiation

Section 3: Differentiation

Section 4 : Differentiation

Section 5: Differentiation

Section 6: Differentiation

Section 7: Differentiation

Section 8: Differentiation

Section 1 : Applications of Differentiation

Section 2: Applications of Differentiation

Section 3: Applications of Differentiation

Section 5: Applications of Differentiation

Section 6: Applications of Differentiation

Section 7: Applications of Differentiation

Section 8: Applications of Differentiation

Section 1: Integration

Section 2: Integration

Section 3: Integration

Section 4: Integration

Section 5: Integration

Section 6: Integration

Section 7: Integration

Section 8: Integration

Section 9: Integration

Section 1: Differential Equations

Section 2: Differential Equations

Section 3: Differential Equations

Section 4: Differential Equations

Section 5: Differential Equations

Section 1: Applications of Integration

Section 2: Applications of Integration

Section 3: Applications of Integration

Section 4: Applications of Integration

Section 5: Applications of Integration

Section 6: Applications of Integration

Section 7: Applications of Integration

Section 1: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 2: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 3: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 4: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 5: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 6: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 7: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 8: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 1: Infinite Series

Section 2: Infinite Series

Section 3: Infinite Series

Section 4: Infinite Series