Question 1

​Based on the graph of  pictured above,how many points of inflection exist for the twice differentiable function  on the interval  ? ​
​  ​

Accepted Answer

A) ​4 
B) ​3 
C) ​2 
D) ​1

Question 2

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  ;concave up on
 
D) inflection point at  ;concave up on
 ;concave down on
 
E) inflection point at  ;concave down on

Question 3

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

Question 4

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

Question 5

Analyze and sketch a graph of the function  . ​

Accepted Answer

A) ​
 
B) ​
 
C) ​
 
D) ​
 
E) ​

Question 6

A plane begins its takeoff at 2:00 P.M.on a 2100-mile flight.After 12.5 hours,the plane arrives at its destination.Explain why there are at least two times during the flight when the speed of the plane is 100 miles per hour. ​

Accepted Answer

A) By the Mean Value Theorem,there is a time when the speed of the plane must equal the average speed of 290 mi/hr.The speed was 100 mi/hr when the plane was accelerating to 290 mi/hr and decelerating from 290 mi/hr. 
B) By the Mean Value Theorem,there is a time when the speed of the plane must equal the average speed of 145 mi/hr.The speed was 100 mi/hr when the plane was accelerating to 145 mi/hr and decelerating from 145 mi/hr.​ 
C) By the Mean Value Theorem,there is a time when the speed of the plane must equal the average speed of 84 mi/hr.The speed was 100 mi/hr when the plane was accelerating to 84 mi/hr and decelerating from 84 mi/hr.​ 
D) By the Mean Value Theorem,there is a time when the speed of the plane must equal the average speed of 112 mi/hr.The speed was 100 mi/hr when the plane was accelerating to 112 mi/hr and decelerating from 112 mi/hr. 
E) By the Mean Value Theorem,there is a time when the speed of the plane must equal the average speed of 168 mi/hr.The speed was 100 mi/hr when the plane was accelerating to 168 mi/hr and decelerating from 168 mi/hr.​

Question 7

Sketch a graph of the function  over the interval  . ​

Accepted Answer

A) ​
 
B) ​
 
C) ​
 
D) ​
 
E) ​

Question 8

Analyze and sketch a graph of the function  . ​

Accepted Answer

A) ​
 
B) ​ ​
 
C) ​
 
D) ​
 
E) ​

Question 9

The measurement of the side of a square floor tile is 22 inches,with a possible error of  inch.Use differentials to approximate the possible propagated error in computing the area of the square. ​

Accepted Answer

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

Question 10

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

Question 11

​The graph of f ′(x)is given below for  .On which interval(s)is the function f(x)both increasing and concave up? ​  ​

Accepted Answer

A) ​  
B) ​  
C) ​  
D) ​

Question 12

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

Accepted Answer

A) absolute max: (0,1);absolute min:  
B) absolute max:  ;absolute min:
 
C) absolute max:  ;no absolute min 
D) no absolute max;absolute min:  
E) absolute max:  ;no absolute min

Question 13

The total stopping distance T of a vehicle is  where T is in feet and x is the speed in miles per hour.Use differentials to approximate the percent change in total stopping distance as speed changes from x = 22 to x = 27 miles per hour.Round your answer to one decimal place. ​

Accepted Answer

A) 51.8% 
B) 22.1% 
C) 42.7% 
D) 78.0% 
E) 122.7%

Question 14

A ball bearing is placed on an inclined plane and begins to roll.The angle of elevation of the plane is  radians.The distance (in meters)the ball bearing rolls in t seconds is  .Determine the value of  after one second.Round numerical values in your answer to one decimal place. ​

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 15

Use a graphing utility to graph the function  and locate the absolute extrema of the function on the interval  . ​

Accepted Answer

A) absolute minimum: 
No absolute maximum 
B) absolute minimum: 
No absolute maximum 
C) absolute minimum: 
No absolute maximum 
D) absolute minimum: 
Absolute maximum:
 
E) absolute maximum:  ,

Absolute minimum:
 ,

Question 16

Find the value o the derivative (if it exists)of the function  at the extremum point  . ​
​  ​

Accepted Answer

A)   
B)   
C) -1 
D) 0 
E) 1

Question 17

Find all values of x for which  is concave down.Explain your reasoning.

Accepted Answer

The answer of Find all values of x for which...

Question 18

Find two positive numbers such that the sum of the first and twice the second is 96 and whose product is a maximum. ​

Accepted Answer

A) 48 and 48 
B) 48 and 24 
C) 42 and 27 
D) 54 and 21 
E) 44 and 26

Question 19

Which of the following functions passes through the point  and satisfies  ? ​
​

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 20

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

Exam 4: Applications of Differentiation

Based on the graph of pictured above,how many points of inflection exist for the twice differentiable function on the interval ?

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

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

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

Analyze and sketch a graph of the function .

A plane begins its takeoff at 2:00 P.M.on a 2100-mile flight.After 12.5 hours,the plane arrives at its destination.Explain why there are at least two times during the flight when the speed of the plane is 100 miles per hour.

Sketch a graph of the function over the interval .

Analyze and sketch a graph of the function .

The measurement of the side of a square floor tile is 22 inches,with a possible error of inch.Use differentials to approximate the possible propagated error in computing the area of the square.

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

The graph of f ′(x) is given below for .On which interval(s) is the function f(x) both increasing and concave up?

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

The total stopping distance T of a vehicle is where T is in feet and x is the speed in miles per hour.Use differentials to approximate the percent change in total stopping distance as speed changes from x = 22 to x = 27 miles per hour.Round your answer to one decimal place.

A ball bearing is placed on an inclined plane and begins to roll.The angle of elevation of the plane is radians.The distance (in meters) the ball bearing rolls in t seconds is .Determine the value of after one second.Round numerical values in your answer to one decimal place.

Use a graphing utility to graph the function and locate the absolute extrema of the function on the interval .

Find the value o the derivative (if it exists) of the function at the extremum point .

Find all values of x for which is concave down.Explain your reasoning.

Find two positive numbers such that the sum of the first and twice the second is 96 and whose product is a maximum.

Which of the following functions passes through the point and satisfies ?

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 .

Exam 4: Applications of Differentiation

​Based on the graph of pictured above,how many points of inflection exist for the twice differentiable function on the interval ? ​ ​ ​

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

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

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

Analyze and sketch a graph of the function . ​

A plane begins its takeoff at 2:00 P.M.on a 2100-mile flight.After 12.5 hours,the plane arrives at its destination.Explain why there are at least two times during the flight when the speed of the plane is 100 miles per hour. ​

Sketch a graph of the function over the interval . ​

Analyze and sketch a graph of the function . ​

The measurement of the side of a square floor tile is 22 inches,with a possible error of inch.Use differentials to approximate the possible propagated error in computing the area of the square. ​

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

​The graph of f ′(x) is given below for .On which interval(s) is the function f(x) both increasing and concave up? ​ ​

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

The total stopping distance T of a vehicle is where T is in feet and x is the speed in miles per hour.Use differentials to approximate the percent change in total stopping distance as speed changes from x = 22 to x = 27 miles per hour.Round your answer to one decimal place. ​

A ball bearing is placed on an inclined plane and begins to roll.The angle of elevation of the plane is radians.The distance (in meters) the ball bearing rolls in t seconds is .Determine the value of after one second.Round numerical values in your answer to one decimal place. ​

Use a graphing utility to graph the function and locate the absolute extrema of the function on the interval . ​

Find the value o the derivative (if it exists) of the function at the extremum point . ​ ​ ​

Find all values of x for which is concave down.Explain your reasoning.

Find two positive numbers such that the sum of the first and twice the second is 96 and whose product is a maximum. ​

Which of the following functions passes through the point and satisfies ? ​ ​

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 . ​

Based on the graph of pictured above,how many points of inflection exist for the twice differentiable function on the interval ?

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

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

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

Analyze and sketch a graph of the function .

A plane begins its takeoff at 2:00 P.M.on a 2100-mile flight.After 12.5 hours,the plane arrives at its destination.Explain why there are at least two times during the flight when the speed of the plane is 100 miles per hour.

Sketch a graph of the function over the interval .

Analyze and sketch a graph of the function .

The measurement of the side of a square floor tile is 22 inches,with a possible error of inch.Use differentials to approximate the possible propagated error in computing the area of the square.

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

The graph of f ′(x) is given below for .On which interval(s) is the function f(x) both increasing and concave up?

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

The total stopping distance T of a vehicle is where T is in feet and x is the speed in miles per hour.Use differentials to approximate the percent change in total stopping distance as speed changes from x = 22 to x = 27 miles per hour.Round your answer to one decimal place.

A ball bearing is placed on an inclined plane and begins to roll.The angle of elevation of the plane is radians.The distance (in meters) the ball bearing rolls in t seconds is .Determine the value of after one second.Round numerical values in your answer to one decimal place.

Use a graphing utility to graph the function and locate the absolute extrema of the function on the interval .

Find the value o the derivative (if it exists) of the function at the extremum point .

Find two positive numbers such that the sum of the first and twice the second is 96 and whose product is a maximum.

Which of the following functions passes through the point and satisfies ?

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 .