Question 1

Compare   and   for   at   with   Give your answers to four decimal places. ​

Accepted Answer

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

Question 2

Analyze and sketch a graph of the function   . ​

Accepted Answer

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

Question 3

​What value of c in the open interval (0,16)satisfies the Mean Value Theorem for   ? ​

Accepted Answer

A) ​0 
B) ​   
C) ​   
D) ​4 
A) ​0 
B) ​   
C) ​   
D) ​4

Question 4

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

Question 5

Identify the open intervals on which the function   is increasing or decreasing. ​

Accepted Answer

A) increasing on   ;decreasing on
  and
  
B) increasing on   and
  ;decreasing on
  
C) increasing on   and
  ;decreasing on
  
D) increasing on   and
  ;decreasing on
  
E) increasing on   and
  decreasing on
  
A) increasing on   ;decreasing on
  and
  
B) increasing on   and
  ;decreasing on
  
C) increasing on   and
  ;decreasing on
  
D) increasing on   and
  ;decreasing on
  
E) increasing on   and
  decreasing 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   ;concave 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   ;concave 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

​The points of inflection for   are at   and   .Which of the following is (are)true? ​
I.The points of inflection for   are at   and   .
II.The points of inflection for   are at   and   .
III.The points of inflection for   are at   and   .
​

Accepted Answer

A) ​I only 
B) ​II only 
C) ​I and II only 
D) ​I and III only 
A) ​I only 
B) ​II only 
C) ​I and II only 
D) ​I and III only

Question 8

Find the open interval(s)on which   is increasing or decreasing. ​

Accepted Answer

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

Question 9

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 10

Find the open interval(s)on which the function   is increasing in the interval   .Round numerical values in your answer to three decimal places. ​

Accepted Answer

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

Question 11

Find the equation of the tangent line T to the graph of   at the given point   . ​

Accepted Answer

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

Question 12

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder.The total volume of the solid is 28 cubic centimeters.Find the radius,r,of the cylinder that produces the minimum surface area.Round your answer to two decimal places. ​

Accepted Answer

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

Question 13

Find two positive numbers whose product is 186 and whose sum is a minimum. ​

Accepted Answer

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

Question 14

Find a function f that has derivative   and with graph passing through the point   . ​

Accepted Answer

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

Question 15

​A farmer has 225 yards of fencing to form two identical rectangular pens and a third pen that is twice as long as the other two pens,as shown in the diagram below.All three pens have the same width,x.Which value of y produces the maximum total fenced area? ​
​   ​
​

Accepted Answer

A) ​   
B) ​15 
C) ​   
D) ​   
A) ​   
B) ​15 
C) ​   
D) ​

Question 16

Find the differential   of the function   .

Accepted Answer

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

Question 17

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

Question 18

Locate the absolute extrema of the function   on the closed interval   . ​

Accepted Answer

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

Question 19

​The position of an object moving along a straight line for   is given by   ,and the position of a second object moving along the same line is given by   .If both objects begin at   ,at what time is the distance between the objects a minimum? ​

Accepted Answer

A) ​2 
B) ​   
C) ​   
D) ​0 
A) ​2 
B) ​   
C) ​   
D) ​0

Question 20

Suppose a manufacturer has determined that the total cost C of operating a factory is   where x is the number of units produced.At what level of production will the average cost per unit be minimized? (The average cost per unit is   . ) ​

Accepted Answer

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

Compare and for at with Give your answers to four decimal places.

Analyze and sketch a graph of the function .

What value of c in the open interval (0,16)satisfies the Mean Value Theorem for ?

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

Identify the open intervals on which the function is increasing or decreasing.

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

The points of inflection for are at and .Which of the following is (are)true? I.The points of inflection for are at and . II.The points of inflection for are at and . III.The points of inflection for are at and .

Find the open interval(s)on which is increasing or decreasing.

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

Find the open interval(s)on which the function is increasing in the interval .Round numerical values in your answer to three decimal places.

Find the equation of the tangent line T to the graph of at the given point .

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder.The total volume of the solid is 28 cubic centimeters.Find the radius,r,of the cylinder that produces the minimum surface area.Round your answer to two decimal places.

Find two positive numbers whose product is 186 and whose sum is a minimum.

Find a function f that has derivative and with graph passing through the point .

A farmer has 225 yards of fencing to form two identical rectangular pens and a third pen that is twice as long as the other two pens,as shown in the diagram below.All three pens have the same width,x.Which value of y produces the maximum total fenced area?

Find the differential of the function .

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 .

Locate the absolute extrema of the function on the closed interval .

The position of an object moving along a straight line for is given by ,and the position of a second object moving along the same line is given by .If both objects begin at ,at what time is the distance between the objects a minimum?

Suppose a manufacturer has determined that the total cost C of operating a factory is where x is the number of units produced.At what level of production will the average cost per unit be minimized? (The average cost per unit is . )

Preparation for Calculus

Limits and Their Properties

Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques, L'Hôpital's Rule, and Improper Integrals

Infinite Series

Parametric Equations, Polar Coordinates, and Vectors

Mechanical Ventilation and Sedation: Assessment, Medications, and Complications

Filters

Exam 4: Applications of Differentiation

Compare and for at with Give your answers to four decimal places. ​

Analyze and sketch a graph of the function . ​

​What value of c in the open interval (0,16)satisfies the Mean Value Theorem for ? ​

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

Identify the open intervals on which the function is increasing or decreasing. ​

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

​The points of inflection for are at and .Which of the following is (are)true? ​ I.The points of inflection for are at and . II.The points of inflection for are at and . III.The points of inflection for are at and . ​

Find the open interval(s)on which is increasing or decreasing. ​

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

Find the open interval(s)on which the function is increasing in the interval .Round numerical values in your answer to three decimal places. ​

Find the equation of the tangent line T to the graph of at the given point . ​

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder.The total volume of the solid is 28 cubic centimeters.Find the radius,r,of the cylinder that produces the minimum surface area.Round your answer to two decimal places. ​

Find two positive numbers whose product is 186 and whose sum is a minimum. ​

Find a function f that has derivative and with graph passing through the point . ​

​A farmer has 225 yards of fencing to form two identical rectangular pens and a third pen that is twice as long as the other two pens,as shown in the diagram below.All three pens have the same width,x.Which value of y produces the maximum total fenced area? ​ ​ ​ ​

Find the differential of the function .

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 . ​

Locate the absolute extrema of the function on the closed interval . ​

​The position of an object moving along a straight line for is given by ,and the position of a second object moving along the same line is given by .If both objects begin at ,at what time is the distance between the objects a minimum? ​

Suppose a manufacturer has determined that the total cost C of operating a factory is where x is the number of units produced.At what level of production will the average cost per unit be minimized? (The average cost per unit is . ) ​

Preparation for Calculus

Limits and Their Properties

Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques, L'Hôpital's Rule, and Improper Integrals

Infinite Series

Parametric Equations, Polar Coordinates, and Vectors

Mechanical Ventilation and Sedation: Assessment, Medications, and Complications

Filters

Compare and for at with Give your answers to four decimal places.

Analyze and sketch a graph of the function .

What value of c in the open interval (0,16)satisfies the Mean Value Theorem for ?

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

Identify the open intervals on which the function is increasing or decreasing.

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

The points of inflection for are at and .Which of the following is (are)true? I.The points of inflection for are at and . II.The points of inflection for are at and . III.The points of inflection for are at and .

Find the open interval(s)on which is increasing or decreasing.

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

Find the open interval(s)on which the function is increasing in the interval .Round numerical values in your answer to three decimal places.

Find the equation of the tangent line T to the graph of at the given point .

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder.The total volume of the solid is 28 cubic centimeters.Find the radius,r,of the cylinder that produces the minimum surface area.Round your answer to two decimal places.

Find two positive numbers whose product is 186 and whose sum is a minimum.

Find a function f that has derivative and with graph passing through the point .

A farmer has 225 yards of fencing to form two identical rectangular pens and a third pen that is twice as long as the other two pens,as shown in the diagram below.All three pens have the same width,x.Which value of y produces the maximum total fenced area?

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 .

Locate the absolute extrema of the function on the closed interval .

The position of an object moving along a straight line for is given by ,and the position of a second object moving along the same line is given by .If both objects begin at ,at what time is the distance between the objects a minimum?

Suppose a manufacturer has determined that the total cost C of operating a factory is where x is the number of units produced.At what level of production will the average cost per unit be minimized? (The average cost per unit is . )