Question 1

The graph of a function f is is shown below. ​   ​
Sketch the graph of the derivative   .
​

Accepted Answer

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

Question 2

Use differentials to approximate the value of   .Round your answer to four decimal places. ​

Accepted Answer

A) 3.0470 
B) 3.0270 
C) 3.0170 
D) 3.0370 
E) 3.0570 
A) 3.0470 
B) 3.0270 
C) 3.0170 
D) 3.0370 
E) 3.0570

Question 3

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)  
(b)increasing:
  ;decreasing:
 
(c)relative max:
  ;relative min:
 
​ 
B) (a)  
(b)decreasing:
  ;increasing:
 
(c)relative min:
  ;relative max:
 
​ 
C) (a)  
(b)increasing:
  ;decreasing:
 
(c)relative max:
  ;no relative min.​ 
D) (a)  
(b)increasing:
  ;decreasing:
 
​(c)relative max:
  ;relative min:
 
​​ 
E) (a)  
(b)decreasing:
  ;increasing:
 
(c)relative min:
  ;relative max:
  
A) (a)  
(b)increasing:
  ;decreasing:
 
(c)relative max:
  ;relative min:
 
​ 
B) (a)  
(b)decreasing:
  ;increasing:
 
(c)relative min:
  ;relative max:
 
​ 
C) (a)  
(b)increasing:
  ;decreasing:
 
(c)relative max:
  ;no relative min.​ 
D) (a)  
(b)increasing:
  ;decreasing:
 
​(c)relative max:
  ;relative min:
 
​​ 
E) (a)  
(b)decreasing:
  ;increasing:
 
(c)relative min:
  ;relative max:

Question 4

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.
​
Use a graphing utility to confirm your results.
​

Accepted Answer

A) (a)  
(b)increasing:
  ;decreasing:
 
(c)relative max:
 
​ 
B) (a)  
(b)increasing:
  ;decreasing:
 
(c)relative max:
 
​ 
C) (a)  
(b)decreasing:
  ​;increasing:
 
(c)relative min:
 
​ 
D) (a)  
(b)decreasing:
  ;increasing:
 
(c)relative min:
  ;relative max:
 
​ 
E) (a)  
(b)decreasing:
  ;increasing:
 
(c)relative min:
 
​ 
A) (a)  
(b)increasing:
  ;decreasing:
 
(c)relative max:
 
​ 
B) (a)  
(b)increasing:
  ;decreasing:
 
(c)relative max:
 
​ 
C) (a)  
(b)decreasing:
  ​;increasing:
 
(c)relative min:
 
​ 
D) (a)  
(b)decreasing:
  ;increasing:
 
(c)relative min:
  ;relative max:
 
​ 
E) (a)  
(b)decreasing:
  ;increasing:
 
(c)relative min:
 
​

Question 5

A sector with central angle   is cut from a circle of radius 4 inches,and the edges of the sector are brought together to form a cone.Find the magnitude of   such that the volume of the cone is a maximum. ​

Accepted Answer

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

Question 6

Find the length and width of a rectangle that has perimeter 40 meters and a maximum area. ​

Accepted Answer

A) 10 m;10 m. 
B) 14 m;7 m. 
C) 1 m;19 m. 
D) 11 m;9 m. 
E) 5 m;15 m. 
A) 10 m;10 m. 
B) 14 m;7 m. 
C) 1 m;19 m. 
D) 11 m;9 m. 
E) 5 m;15 m.

Question 7

A calculator may be used for the following question.​Let   be a function such that   .In the interval   ,the graph of   has a point of inflection nearest   . ​

Accepted Answer

A) ​0.37 
B) ​1.233 
C) ​2.141 
D) ​2.268 
A) ​0.37 
B) ​1.233 
C) ​2.141 
D) ​2.268

Question 8

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

Accepted Answer

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

Question 9

The radius of a spherical balloon is measured to be 7 inches,with a possible error of 0.03 inch.Use differentials to approximate the maximum possible error in calculating the surface area of the sphere.Round your answer to two decimal places.

Accepted Answer

A) ±6.6 square inches 
B) ±9.24 square inches 
C) ±27.71 square inches 
D) ±18.47 square inches 
E) ±5.28 square inches 
A) ±6.6 square inches 
B) ±9.24 square inches 
C) ±27.71 square inches 
D) ±18.47 square inches 
E) ±5.28 square inches

Question 10

The height of an object t seconds after it is dropped from a height of 450 meters is   .Find the average velocity of the object during the first 3 seconds.Round to two decimal places. ​

Accepted Answer

A) 14.70 m/sec 
B) -14.70 m/sec 
C) -9.00 m/sec 
D) 9.00 m/sec 
E) -31.00 m/sec 
A) 14.70 m/sec 
B) -14.70 m/sec 
C) -9.00 m/sec 
D) 9.00 m/sec 
E) -31.00 m/sec

Question 11

Find the relative minima of   on the interval   by applying the First Derivative Test.Round numerical values in your answer to three decimal places. ​

Accepted Answer

A) relative minima:   
B) relative minima:   
C) relative minima:   
D) relative minima:   
E) relative minima:   
A) relative minima:   
B) relative minima:   
C) relative minima:   
D) relative minima:   
E) relative minima:

Question 12

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.Find the average value of   during the first year. ​

Accepted Answer

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

Question 13

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 14

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 15

The graph of a function f is is shown below.Sketch the graph of the derivative   . ​   ​

Accepted Answer

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

Question 16

Determine whether the Mean Value Theorem can be applied to the function   on the closed interval   .If the Mean Value Theorem can be applied,find all numbers c in the open interval   such that   . ​

Accepted Answer

A) MVT applies;   
B) MVT applies;   
C) MVT applies;   
D) MVT applies;   
E) MVT applies;   
A) MVT applies;   
B) MVT applies;   
C) MVT applies;   
D) MVT applies;   
E) MVT applies;

Question 17

Find the relative extremum of   by applying the First Derivative Test. ​

Accepted Answer

A) relative minimum:   
B) relative maximum:   
C) relative minimum:   
D) relative minimum:   
E) relative maximum:   
A) relative minimum:   
B) relative maximum:   
C) relative minimum:   
D) relative minimum:   
E) relative maximum:

Question 18

Find the relative maxima of   on the interval   by applying the First Derivative Test.Round numerical values in your answer to three decimal places. ​

Accepted Answer

A) relative maxima:   
B) relative maxima:   
C) relative maxima:   
D) relative maxima:   
E) relative maxima:   
A) relative maxima:   
B) relative maxima:   
C) relative maxima:   
D) relative maxima:   
E) relative maxima:

Question 19

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:

Question 20

Find the critical number of the function   . ​

Accepted Answer

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

The graph of a function f is is shown below. Sketch the graph of the derivative .

Use differentials to approximate the value of .Round your answer to four decimal places.

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.

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. Use a graphing utility to confirm your results.

A sector with central angle is cut from a circle of radius 4 inches,and the edges of the sector are brought together to form a cone.Find the magnitude of such that the volume of the cone is a maximum.

Find the length and width of a rectangle that has perimeter 40 meters and a maximum area.

A calculator may be used for the following question.Let be a function such that .In the interval ,the graph of has a point of inflection nearest .

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

The radius of a spherical balloon is measured to be 7 inches,with a possible error of 0.03 inch.Use differentials to approximate the maximum possible error in calculating the surface area of the sphere.Round your answer to two decimal places.

The height of an object t seconds after it is dropped from a height of 450 meters is .Find the average velocity of the object during the first 3 seconds.Round to two decimal places.

Find the relative minima of on the interval by applying the First Derivative Test.Round numerical values in your answer to three decimal places.

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.Find the average value of during the first year.

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

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

The graph of a function f is is shown below.Sketch the graph of the derivative .

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

Find the relative extremum of by applying the First Derivative Test.

Find the relative maxima of on the interval by applying the First Derivative Test.Round numerical values in your answer to three decimal places.

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

Find the critical number of the function .

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

The graph of a function f is is shown below. ​ ​ Sketch the graph of the derivative . ​

Use differentials to approximate the value of .Round your answer to four decimal places. ​

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. ​

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. ​ Use a graphing utility to confirm your results. ​

A sector with central angle is cut from a circle of radius 4 inches,and the edges of the sector are brought together to form a cone.Find the magnitude of such that the volume of the cone is a maximum. ​

Find the length and width of a rectangle that has perimeter 40 meters and a maximum area. ​

A calculator may be used for the following question.​Let be a function such that .In the interval ,the graph of has a point of inflection nearest . ​

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

The radius of a spherical balloon is measured to be 7 inches,with a possible error of 0.03 inch.Use differentials to approximate the maximum possible error in calculating the surface area of the sphere.Round your answer to two decimal places.

The height of an object t seconds after it is dropped from a height of 450 meters is .Find the average velocity of the object during the first 3 seconds.Round to two decimal places. ​

Find the relative minima of on the interval by applying the First Derivative Test.Round numerical values in your answer to three decimal places. ​

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.Find the average value of during the first year. ​

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

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

The graph of a function f is is shown below.Sketch the graph of the derivative . ​ ​

Determine whether the Mean Value Theorem can be applied to the function on the closed interval .If the Mean Value Theorem can be applied,find all numbers c in the open interval such that . ​

Find the relative extremum of by applying the First Derivative Test. ​

Find the relative maxima of on the interval by applying the First Derivative Test.Round numerical values in your answer to three decimal places. ​

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

Find the critical number of the function . ​

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

The graph of a function f is is shown below. Sketch the graph of the derivative .

Use differentials to approximate the value of .Round your answer to four decimal places.

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.

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. Use a graphing utility to confirm your results.

A sector with central angle is cut from a circle of radius 4 inches,and the edges of the sector are brought together to form a cone.Find the magnitude of such that the volume of the cone is a maximum.

Find the length and width of a rectangle that has perimeter 40 meters and a maximum area.

A calculator may be used for the following question.Let be a function such that .In the interval ,the graph of has a point of inflection nearest .

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

The height of an object t seconds after it is dropped from a height of 450 meters is .Find the average velocity of the object during the first 3 seconds.Round to two decimal places.

Find the relative minima of on the interval by applying the First Derivative Test.Round numerical values in your answer to three decimal places.

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.Find the average value of during the first year.

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

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

The graph of a function f is is shown below.Sketch the graph of the derivative .

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

Find the relative extremum of by applying the First Derivative Test.

Find the relative maxima of on the interval by applying the First Derivative Test.Round numerical values in your answer to three decimal places.

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

Find the critical number of the function .