Question 1

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

Accepted Answer

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

Question 2

Find all critical numbers of the function   ,   .​ ​

Accepted Answer

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

Question 3

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;4 
C) MVT applies;   
D) MVT applies;8 
E) MVT does not apply 
A) MVT applies;   
B) MVT applies;4 
C) MVT applies;   
D) MVT applies;8 
E) MVT does not apply

Question 4

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 speed of the ball bearing after t seconds.Round to one decimal place. ​

Accepted Answer

A) speed:   meters per second 
B) speed:   meters per second 
C) speed:   meters per second 
D) speed:   meters per second 
E) speed:   meters per second 
A) speed:   meters per second 
B) speed:   meters per second 
C) speed:   meters per second 
D) speed:   meters per second 
E) speed:   meters per second

Question 5

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

Accepted Answer

A) The absolute maximum is   ,and it occurs at the critical number
  .The absolute minimum is
  ,and it occurs at the left endpoint
  . 
B) The absolute maximum is   ,and it occurs at either endpoint
  .The absolute minimum is 0,and it occurs at the critical number
  . 
C) The absolute maximum is   ,and it occurs only at the left endpoint
  .The absolute minimum is 0,and it occurs at the critical number
  . 
D) The absolute maximum is   ,and it occurs at the critical number
  .The absolute minimum is
  ,and it occurs at the right endpoint
  . 
E) The absolute maximum is   ,and it occurs only at the right endpoint
  .The absolute minimum is 0,and it occurs at the critical number
  . 
A) The absolute maximum is   ,and it occurs at the critical number
  .The absolute minimum is
  ,and it occurs at the left endpoint
  . 
B) The absolute maximum is   ,and it occurs at either endpoint
  .The absolute minimum is 0,and it occurs at the critical number
  . 
C) The absolute maximum is   ,and it occurs only at the left endpoint
  .The absolute minimum is 0,and it occurs at the critical number
  . 
D) The absolute maximum is   ,and it occurs at the critical number
  .The absolute minimum is
  ,and it occurs at the right endpoint
  . 
E) The absolute maximum is   ,and it occurs only at the right endpoint
  .The absolute minimum is 0,and it occurs at the critical number
  .

Question 6

For what values of x is   not defined? Explain your reasoning.

Accepted Answer

Since @#IMG-DLM& ,then @#IMG-DLM& .
​ @#IMG-DLM& i

Question 7

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

Question 8

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

Accepted Answer

A) left endpoint:   absolute minimum
Right endpoint:
  absolute maximum
Critical points: none 
B) left endpoint:   absolute minimum
Right endpoint:
  absolute maximum
Critical points: none 
C) left endpoint:   absolute minimum
Right endpoint:
 
Critical points:
  absolute maximum 
D) left endpoint:   absolute maximum
Right endpoint:
  absolute minimum
Critical points: none 
E) left endpoint:   absolute minimum
Right endpoint:
  absolute maximum
Critical points:
  
A) left endpoint:   absolute minimum
Right endpoint:
  absolute maximum
Critical points: none 
B) left endpoint:   absolute minimum
Right endpoint:
  absolute maximum
Critical points: none 
C) left endpoint:   absolute minimum
Right endpoint:
 
Critical points:
  absolute maximum 
D) left endpoint:   absolute maximum
Right endpoint:
  absolute minimum
Critical points: none 
E) left endpoint:   absolute minimum
Right endpoint:
  absolute maximum
Critical points:

Question 9

For time   ,a particle moves along the x-axis with position given by   .During what time intervals is the speed of the particle increasing? ​

Accepted Answer

A) ​   < t ≤ 10 only 
B) ​0 ≤ t < 1 and   < t <
  
C) ​0 ≤ t < 1 and   < t ≤ 10 
D) ​​1 < t <   and
  < t ≤ 10 
A) ​   < t ≤ 10 only 
B) ​0 ≤ t < 1 and   < t <
  
C) ​0 ≤ t < 1 and   < t ≤ 10 
D) ​​1 < t <   and
  < t ≤ 10

Question 10

Find the point on the graph of the function   that is closest to the point   .Round all numerical values in your answer to four decimal places. ​

Accepted Answer

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

Question 11

The range R of a projectile is   where ν0 is the initial velocity in feet per second and θ is the angle of elevation.If ν0 = 2300 feet per second and θ is changed from 13° to 14° use differentials to approximate the change in the range.Round your answer to the nearest integer. ​

Accepted Answer

A) 2593 ft 
B) 4568 ft 
C) 5623 ft 
D) 2811 ft 
E) 5186 ft 
A) 2593 ft 
B) 4568 ft 
C) 5623 ft 
D) 2811 ft 
E) 5186 ft

Question 12

The radius of a spherical balloon is measured to be 9 inches,with a possible error of 0.02 inch.Use differentials to approximate the maximum possible error in calculating the volume of the sphere.Round your answer to two decimal places. ​

Accepted Answer

A) ±10.18 cubic inches 
B) ±5.65 cubic inches 
C) ±30.54 cubic inches 
D) ±20.36 cubic inches 
E) ±4.52 cubic inches 
A) ±10.18 cubic inches 
B) ±5.65 cubic inches 
C) ±30.54 cubic inches 
D) ±20.36 cubic inches 
E) ±4.52 cubic inches

Question 13

Find the differential   of the function   . ​

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

Find the differential   of the function   . ​

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

Let   be defined as   in the open interval (-3,3).Determine the coordinate(s)of the relative maxima of   in the open interval.Explain your reasoning.

Accepted Answer

For @#IMG-DLM& ,the relative maximum will occur at

Question 18

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 19

​Let M represent the absolute maximum of f(x)in an interval. Let R represent a root of f(x)in the given interval.
Let m represent the absolute minimum of f(x)in the interval.
​
If   ,then which of the following is true over the closed interval   ?
​

Accepted Answer

A) ​M and R occur at a critical point and m occurs at an endpoint. 
B) ​M and m occur at critical points. 
C) ​M,m,and R occur at endpoints of the given interval. 
D) ​M occurs at an endpoint,whereas m and R occur at a critical point. 
A) ​M and R occur at a critical point and m occurs at an endpoint. 
B) ​M and m occur at critical points. 
C) ​M,m,and R occur at endpoints of the given interval. 
D) ​M occurs at an endpoint,whereas m and R occur at a critical point.

Question 20

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:

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

Find all critical numbers of the function , .

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 .

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 speed of the ball bearing after t seconds.Round to one decimal place.

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

For what values of x is not defined? Explain your reasoning.

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 .

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

For time ,a particle moves along the x-axis with position given by .During what time intervals is the speed of the particle increasing?

Find the point on the graph of the function that is closest to the point .Round all numerical values in your answer to four decimal places.

The range R of a projectile is where ν₀ is the initial velocity in feet per second and θ is the angle of elevation.If ν₀ = 2300 feet per second and θ is changed from 13° to 14° use differentials to approximate the change in the range.Round your answer to the nearest integer.

The radius of a spherical balloon is measured to be 9 inches,with a possible error of 0.02 inch.Use differentials to approximate the maximum possible error in calculating the volume of the sphere.Round your answer to two decimal places.

Find the differential of the function .

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

Find the differential of the function .

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

Let be defined as in the open interval (-3,3).Determine the coordinate(s)of the relative maxima of in the open interval.Explain your reasoning.

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:

Let M represent the absolute maximum of f(x)in an interval. Let R represent a root of f(x)in the given interval. Let m represent the absolute minimum of f(x)in the interval. If ,then which of the following is true over the closed interval ?

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

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 f is shown below.For which value of x is zero? ​ ​

Find all critical numbers of the function , .​ ​

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 . ​

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 speed of the ball bearing after t seconds.Round to one decimal place. ​

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

For what values of x is not defined? Explain your reasoning.

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 . ​

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

For time ,a particle moves along the x-axis with position given by .During what time intervals is the speed of the particle increasing? ​

Find the point on the graph of the function that is closest to the point .Round all numerical values in your answer to four decimal places. ​

The range R of a projectile is where ν0 is the initial velocity in feet per second and θ is the angle of elevation.If ν0 = 2300 feet per second and θ is changed from 13° to 14° use differentials to approximate the change in the range.Round your answer to the nearest integer. ​

The radius of a spherical balloon is measured to be 9 inches,with a possible error of 0.02 inch.Use differentials to approximate the maximum possible error in calculating the volume of the sphere.Round your answer to two decimal places. ​

Find the differential of the function . ​

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

Find the differential of the function . ​

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

Let be defined as in the open interval (-3,3).Determine the coordinate(s)of the relative maxima of in the open interval.Explain your reasoning.

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: ​

​Let M represent the absolute maximum of f(x)in an interval. Let R represent a root of f(x)in the given interval. Let m represent the absolute minimum of f(x)in the interval. ​ If ,then which of the following is true over the closed interval ? ​

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

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 f is shown below.For which value of x is zero?

Find all critical numbers of the function , .

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 .

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 speed of the ball bearing after t seconds.Round to one decimal place.

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

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 .

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

For time ,a particle moves along the x-axis with position given by .During what time intervals is the speed of the particle increasing?

Find the point on the graph of the function that is closest to the point .Round all numerical values in your answer to four decimal places.

The range R of a projectile is where ν₀ is the initial velocity in feet per second and θ is the angle of elevation.If ν₀ = 2300 feet per second and θ is changed from 13° to 14° use differentials to approximate the change in the range.Round your answer to the nearest integer.

The radius of a spherical balloon is measured to be 9 inches,with a possible error of 0.02 inch.Use differentials to approximate the maximum possible error in calculating the volume of the sphere.Round your answer to two decimal places.

Find the differential of the function .

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

Find the differential of the function .

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

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:

Let M represent the absolute maximum of f(x)in an interval. Let R represent a root of f(x)in the given interval. Let m represent the absolute minimum of f(x)in the interval. If ,then which of the following is true over the closed interval ?

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