Question 1

All edges of a cube are expanding at a rate of 9 centimeters per second.How fast is the volume changing when each edge is 2 centimeters? ​

Accepted Answer

A) 486   
B) 72   
C) 36   
D) 108   
E) 162   
A) 486   
B) 72   
C) 36   
D) 108   
E) 162

Question 2

Suppose the position function for a free-falling object on a certain planet is given by   .A silver coin is dropped from the top of a building that is 1372 feet tall.Find the instantaneous velocity of the coin when   . ​

Accepted Answer

A) -90   
B) -42   
C) -27   
D) -135   
E) -45   
A) -90   
B) -42   
C) -27   
D) -135   
E) -45

Question 3

A buoy oscillates in simple harmonic motion   as waves move past it.The buoy moves a total of 12.5 feet (vertically)between its low point and its high point.It returns to its high point every 8 seconds.Write an equation describing the motion of the buoy if it is at its high point at   . ​

Accepted Answer

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

Question 4

The volume of a cube with sides of length s is given by   .Find the rate of change of volume with respect to s when   centimeters. ​

Accepted Answer

A) 10125 cm2 
B) 3375 cm2 
C) 225 cm2 
D) 675 cm2 
E) 450 cm2 
A) 10125 cm2 
B) 3375 cm2 
C) 225 cm2 
D) 675 cm2 
E) 450 cm2

Question 5

The radius r of a sphere is increasing at a rate of 9 inches per minute.Find the rate of change of the volume when r = 11 inches. ​

Accepted Answer

A)    in3/min 
B)    in3/min 
C)    in3/min 
D)    in3/min 
E)    in3/min 
A)    in3/min 
B)    in3/min 
C)    in3/min 
D)    in3/min 
E)    in3/min

Question 6

​For what values of   and   is the piecewise function   differentiable? ​

Accepted Answer

A) ​   and
  
B) ​   and
  
C) ​   and​
  
D) ​   and
  
A) ​   and
  
B) ​   and
  
C) ​   and​
  
D) ​   and

Question 7

A man 6 feet tall walks at a rate of 10 feet per second away from a light that is 15 feet above the ground (see figure).When he is 13 feet from the base of the light,at what rate is the tip of his shadow moving? ​   ​​

Accepted Answer

A)      
B) 50   
C)      
D)      
E)      
A)      
B) 50   
C)      
D)      
E)

Question 8

Use the alternative form of the derivative to find the derivative of the function   at   . ​​

Accepted Answer

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

Question 9

Find   by implicit differentiation. ​   ​

Accepted Answer

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

Question 10

Find the derivative of the function   . ​

Accepted Answer

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

Question 11

An airplane flies at an altitude of ​h​ = 14 miles toward a point directly over an observer.Consider θ and x as shown in the following figure.Write θ as a function of x. ​   ​

Accepted Answer

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

Question 12

In a free-fall experiment,an object is dropped from a height of ​a = 144 feet.A camera on the ground 500 feet from the point of impact records the fall of the object as shown in the figure.Assuming the object is released at time t = 0.Find the rate of change of the angle of elevation of the camera when t = 1.Round your answer to four decimal places. ​​   ​​

Accepted Answer

A) -0.6006   
B) -0.0581   
C) 0.0601   
D) 0.0581   
E) -0.0601   
A) -0.6006   
B) -0.0581   
C) 0.0601   
D) 0.0581   
E) -0.0601

Question 13

​If   ,then   ​

Accepted Answer

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

Question 14

​If the nth derivative of   is denoted as   and   ,then   is the same as ​

Accepted Answer

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

Question 15

A ball is thrown straight down from the top of a 280-ft building with an initial velocity of -24 ft per second.The position function is   .What is the velocity of the ball after 4 seconds? ​

Accepted Answer

A) The velocity after 4 seconds is -88 ft per second. 
B) The velocity after 4 seconds is -104 ft per second. 
C) The velocity after 4 seconds is -152 ft per second. 
D) The velocity after 4 seconds is -40 ft per second. 
E) The velocity after 4 seconds is -304 ft per second. 
A) The velocity after 4 seconds is -88 ft per second. 
B) The velocity after 4 seconds is -104 ft per second. 
C) The velocity after 4 seconds is -152 ft per second. 
D) The velocity after 4 seconds is -40 ft per second. 
E) The velocity after 4 seconds is -304 ft per second.

Question 16

Find the derivative of the function   . ​

Accepted Answer

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

Question 17

The displacement from equilibrium of an object in harmonic motion on the end of a spring is   where y is measured in feet and t is the time in seconds.Determine the velocity of the object when   .Round your answer to two decimal places. ​

Accepted Answer

A) 5.46 ft/sec 
B) 2.40 ft/sec 
C) 3.46 ft/sec 
D) 4.46 ft/sec 
E) 0.87 ft/sec 
A) 5.46 ft/sec 
B) 2.40 ft/sec 
C) 3.46 ft/sec 
D) 4.46 ft/sec 
E) 0.87 ft/sec

Question 18

Determine all values of x, (if any),at which the graph of the function has a horizontal tangent. ​   ​

Accepted Answer

A)    
B)    and   
C)    and   
D)    
E) The graph has no horizontal tangents. 
A)    
B)    and   
C)    and   
D)    
E) The graph has no horizontal tangents.

Question 19

The length of a rectangle is   and its height is   ,where t is time in seconds and the dimensions are in inches.Find the rate of change of area,A,with respect to time. ​

Accepted Answer

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

Question 20

A manufacturer of digital audio players estimates that the profit for selling a particular model is   where P is the profit in dollars and x is the advertising expense in 10,000's of dollars (see figure).Find the smaller of two advertising amounts that yield a profit P of $1,900,000.Round your answer to the nearest dollar. ​   ​

Accepted Answer

A) $532,519 
B) $291,308 
C) $271,308 
D) $460,892 
E) $315,369 
A) $532,519 
B) $291,308 
C) $271,308 
D) $460,892 
E) $315,369

All edges of a cube are expanding at a rate of 9 centimeters per second.How fast is the volume changing when each edge is 2 centimeters?

Suppose the position function for a free-falling object on a certain planet is given by .A silver coin is dropped from the top of a building that is 1372 feet tall.Find the instantaneous velocity of the coin when .

A buoy oscillates in simple harmonic motion as waves move past it.The buoy moves a total of 12.5 feet (vertically)between its low point and its high point.It returns to its high point every 8 seconds.Write an equation describing the motion of the buoy if it is at its high point at .

The volume of a cube with sides of length s is given by .Find the rate of change of volume with respect to s when centimeters.

The radius r of a sphere is increasing at a rate of 9 inches per minute.Find the rate of change of the volume when r = 11 inches.

For what values of and is the piecewise function differentiable?

A man 6 feet tall walks at a rate of 10 feet per second away from a light that is 15 feet above the ground (see figure).When he is 13 feet from the base of the light,at what rate is the tip of his shadow moving?

Use the alternative form of the derivative to find the derivative of the function at .

Find by implicit differentiation.

Find the derivative of the function .

An airplane flies at an altitude of h = 14 miles toward a point directly over an observer.Consider θ and x as shown in the following figure.Write θ as a function of x.

If ,then

If the nth derivative of is denoted as and ,then is the same as

A ball is thrown straight down from the top of a 280-ft building with an initial velocity of -24 ft per second.The position function is .What is the velocity of the ball after 4 seconds?

Find the derivative of the function .

The displacement from equilibrium of an object in harmonic motion on the end of a spring is where y is measured in feet and t is the time in seconds.Determine the velocity of the object when .Round your answer to two decimal places.

Determine all values of x, (if any),at which the graph of the function has a horizontal tangent.

The length of a rectangle is and its height is ,where t is time in seconds and the dimensions are in inches.Find the rate of change of area,A,with respect to time.

Preparation for Calculus

Limits and Their Properties

Applications of 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 3: Differentiation

All edges of a cube are expanding at a rate of 9 centimeters per second.How fast is the volume changing when each edge is 2 centimeters? ​

Suppose the position function for a free-falling object on a certain planet is given by .A silver coin is dropped from the top of a building that is 1372 feet tall.Find the instantaneous velocity of the coin when . ​

A buoy oscillates in simple harmonic motion as waves move past it.The buoy moves a total of 12.5 feet (vertically)between its low point and its high point.It returns to its high point every 8 seconds.Write an equation describing the motion of the buoy if it is at its high point at . ​

The volume of a cube with sides of length s is given by .Find the rate of change of volume with respect to s when centimeters. ​

The radius r of a sphere is increasing at a rate of 9 inches per minute.Find the rate of change of the volume when r = 11 inches. ​

​For what values of and is the piecewise function differentiable? ​

A man 6 feet tall walks at a rate of 10 feet per second away from a light that is 15 feet above the ground (see figure).When he is 13 feet from the base of the light,at what rate is the tip of his shadow moving? ​ ​​

Use the alternative form of the derivative to find the derivative of the function at . ​​

Find by implicit differentiation. ​ ​

Find the derivative of the function . ​

An airplane flies at an altitude of ​h​ = 14 miles toward a point directly over an observer.Consider θ and x as shown in the following figure.Write θ as a function of x. ​ ​

​If ,then ​

​If the nth derivative of is denoted as and ,then is the same as ​

A ball is thrown straight down from the top of a 280-ft building with an initial velocity of -24 ft per second.The position function is .What is the velocity of the ball after 4 seconds? ​

Find the derivative of the function . ​

The displacement from equilibrium of an object in harmonic motion on the end of a spring is where y is measured in feet and t is the time in seconds.Determine the velocity of the object when .Round your answer to two decimal places. ​

Determine all values of x, (if any),at which the graph of the function has a horizontal tangent. ​ ​

The length of a rectangle is and its height is ,where t is time in seconds and the dimensions are in inches.Find the rate of change of area,A,with respect to time. ​

Preparation for Calculus

Limits and Their Properties

Applications of 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

All edges of a cube are expanding at a rate of 9 centimeters per second.How fast is the volume changing when each edge is 2 centimeters?

Suppose the position function for a free-falling object on a certain planet is given by .A silver coin is dropped from the top of a building that is 1372 feet tall.Find the instantaneous velocity of the coin when .

A buoy oscillates in simple harmonic motion as waves move past it.The buoy moves a total of 12.5 feet (vertically)between its low point and its high point.It returns to its high point every 8 seconds.Write an equation describing the motion of the buoy if it is at its high point at .

The volume of a cube with sides of length s is given by .Find the rate of change of volume with respect to s when centimeters.

The radius r of a sphere is increasing at a rate of 9 inches per minute.Find the rate of change of the volume when r = 11 inches.

For what values of and is the piecewise function differentiable?

A man 6 feet tall walks at a rate of 10 feet per second away from a light that is 15 feet above the ground (see figure).When he is 13 feet from the base of the light,at what rate is the tip of his shadow moving?

Use the alternative form of the derivative to find the derivative of the function at .

Find by implicit differentiation.

Find the derivative of the function .

An airplane flies at an altitude of h = 14 miles toward a point directly over an observer.Consider θ and x as shown in the following figure.Write θ as a function of x.

If ,then

If the nth derivative of is denoted as and ,then is the same as

A ball is thrown straight down from the top of a 280-ft building with an initial velocity of -24 ft per second.The position function is .What is the velocity of the ball after 4 seconds?

Find the derivative of the function .

The displacement from equilibrium of an object in harmonic motion on the end of a spring is where y is measured in feet and t is the time in seconds.Determine the velocity of the object when .Round your answer to two decimal places.

Determine all values of x, (if any),at which the graph of the function has a horizontal tangent.

The length of a rectangle is and its height is ,where t is time in seconds and the dimensions are in inches.Find the rate of change of area,A,with respect to time.