Question 1

​Find the marginal profit for producing x units.(The profit is measured in dollars.) ​   ​

Accepted Answer

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

Question 2

​Find the indicated derivative. ​
Find   if   ​

Accepted Answer

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

Question 3

Find the slope of the tangent line to the graph of the function below at the given point. ​   ,   ​

Accepted Answer

A)  ​-12 
B)  ​-2 
C)  ​10 
D)  ​-18 
E)  ​none of the above 
A)  ​-12 
B)  ​-2 
C)  ​10 
D)  ​-18 
E)  ​none of the above A

Question 4

​Identify a function f(x)that has the given characteristics and then sketch the function. ​   ​;   ;   ​

Accepted Answer

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

Question 5

​Find the second derivative for the function   and solve the equation   ​

Accepted Answer

A)  ​-2 
B)  ​1 
C)  ​0 
D)  ​26 
E)  ​33 
A)  ​-2 
B)  ​1 
C)  ​0 
D)  ​26 
E)  ​33

Question 6

​Find the marginal cost for producing x units.(The cost is measured in dollars.) ​   ​

Accepted Answer

A)  ​$9,800 
B)  ​​$9,850 
C)  ​​$8,800 
D)  ​​$8,850 
E)  ​​​$9,750 
A)  ​$9,800 
B)  ​​$9,850 
C)  ​​$8,800 
D)  ​​$8,850 
E)  ​​​$9,750

Question 7

​Find   implicitly for   ​

Accepted Answer

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

Question 8

The graph shows the number of visitors V to a national park in hundreds of thousands during a one-year period,where   represents January.Estimate the rate of change of V over the interval   Round your answer to the nearest hundred thousand visitors per year. ​   ​

Accepted Answer

A)  187.5 hundred thousand visitors per year 
B)  281.25 hundred thousand visitors per year 
C)  225 hundred thousand visitors per year 
D)  562.5 hundred thousand visitors per year 
E)  1125 hundred thousand visitors per year 
A)  187.5 hundred thousand visitors per year 
B)  281.25 hundred thousand visitors per year 
C)  225 hundred thousand visitors per year 
D)  562.5 hundred thousand visitors per year 
E)  1125 hundred thousand visitors per year

Question 9

​Find the derivative of the following function using the limiting process. ​
​   ​

Accepted Answer

A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  ​none of the above 
A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  ​none of the above

Question 10

​A retail sporting goods store estimates that weekly sales and weekly advertising costs are related by the equation   The current weekly advertising costs are $1,500,and these costs are increasing at a rate of $135 per week.Find the current rate of change of weekly sales. ​

Accepted Answer

A)  ​156,600 dollars per week 
B)  ​158,880 dollars per week 
C)  ​85,980 dollars per week 
D)  ​83,700 dollars per week 
E)  ​823,080 dollars per week 
A)  ​156,600 dollars per week 
B)  ​158,880 dollars per week 
C)  ​85,980 dollars per week 
D)  ​83,700 dollars per week 
E)  ​823,080 dollars per week

Question 11

​A population of bacteria is introduced into a culture.The number of bacteria P can be modeled by   where t is the time (in hours).Find the rate of change of the population when   ​

Accepted Answer

A)  21.95 bacteria/hr 
B)  ​3.53 bacteria/hr 
C)  ​10.6 bacteria/hr 
D)  ​2.07 bacteria/hr 
E)  ​7.32 bacteria/hr 
A)  21.95 bacteria/hr 
B)  ​3.53 bacteria/hr 
C)  ​10.6 bacteria/hr 
D)  ​2.07 bacteria/hr 
E)  ​7.32 bacteria/hr

Question 12

​Assume that x and y are differentiable functions of t.Find   given that   ,   ,and   ​   ​

Accepted Answer

A)  ​-1.33 
B)  ​1.5 
C)  ​6 
D)  ​-8 
E)  ​2.67 
A)  ​-1.33 
B)  ​1.5 
C)  ​6 
D)  ​-8 
E)  ​2.67

Question 13

​Determine the point(s),(if any),at which the graph of the function has a horizontal tangent. ​
​   ​

Accepted Answer

A)  ​0 
B)  ​0 and 3 
C)  ​0 and -3 
D)  ​3 
E)  ​There are no points at which the graph has a horizontal tangent. 
A)  ​0 
B)  ​0 and 3 
C)  ​0 and -3 
D)  ​3 
E)  ​There are no points at which the graph has a horizontal tangent.

Question 14

​Determine whether the statement is true or false.If it is false,explain why or give an example that shows it is false. ​
If   ,then   ​

Accepted Answer

A) True 
 B)False

Question 15

Find the derivative of the following function using the limiting process. ​   ​

Accepted Answer

A)  ​2 
B)  ​   
C)  ​   
D)  ​   
E)  ​none of the above 
A)  ​2 
B)  ​   
C)  ​   
D)  ​   
E)  ​none of the above

Question 16

​For the function given,find   ​ ​   ​

Accepted Answer

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

Question 17

Find the derivative of the function. ​   ​

Accepted Answer

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

Question 18

​A brick becomes dislodged from the Empire State Building (at a height of 1,000 feet)and falls to the sidewalk below.Write the position s(t),velocity v(t),and acceleration a(t)as functions of time. ​

Accepted Answer

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

Question 19

​Profit.Suppose that the monthly revenue and cost (in dollars)for x units of a product are   and   At what rate per month is the profit changing if the number of units produced and sold is 100 and is increasing at a rate of 10 units per month? ​

Accepted Answer

A)  ​11,960 per month 
B)  ​1,160 per month 
C)  ​1,960 per month 
D)  760​ per month 
E)  19,960​ per month 
A)  ​11,960 per month 
B)  ​1,160 per month 
C)  ​1,960 per month 
D)  760​ per month 
E)  19,960​ per month

Question 20

​When the price of a glass of lemonade at a lemonade stand was $1.75,400 glasses were sold.When the price was lowered to $1.50,500 glasses were sold.Assume that the demand function is linear and that the marginal and fixed costs are $0.10 and $25,respectively.Find the profit P as a function of x,the number of glasses of lemonade sold. ​

Accepted Answer

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

Find the marginal profit for producing x units.(The profit is measured in dollars.)

Find the indicated derivative. Find if

Find the slope of the tangent line to the graph of the function below at the given point. ,

Identify a function f(x)that has the given characteristics and then sketch the function. ; ;

Find the second derivative for the function and solve the equation

Find the marginal cost for producing x units.(The cost is measured in dollars.)

Find implicitly for

The graph shows the number of visitors V to a national park in hundreds of thousands during a one-year period,where represents January.Estimate the rate of change of V over the interval Round your answer to the nearest hundred thousand visitors per year.

Find the derivative of the following function using the limiting process.

A retail sporting goods store estimates that weekly sales and weekly advertising costs are related by the equation The current weekly advertising costs are $1,500,and these costs are increasing at a rate of $135 per week.Find the current rate of change of weekly sales.

A population of bacteria is introduced into a culture.The number of bacteria P can be modeled by where t is the time (in hours).Find the rate of change of the population when

Assume that x and y are differentiable functions of t.Find given that , ,and

Determine the point(s),(if any),at which the graph of the function has a horizontal tangent.

Determine whether the statement is true or false.If it is false,explain why or give an example that shows it is false. If ,then

Find the derivative of the following function using the limiting process.

For the function given,find

Find the derivative of the function.

A brick becomes dislodged from the Empire State Building (at a height of 1,000 feet)and falls to the sidewalk below.Write the position s(t),velocity v(t),and acceleration a(t)as functions of time.

Profit.Suppose that the monthly revenue and cost (in dollars)for x units of a product are and At what rate per month is the profit changing if the number of units produced and sold is 100 and is increasing at a rate of 10 units per month?

Functions, graphs, and Limits

Applications of the Derivative

Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions

Probability and Calculus

Series and Taylor Polynomials

Differential Equations

Filters

Exam 2: Differentiation

​Find the marginal profit for producing x units.(The profit is measured in dollars.) ​ ​

​Find the indicated derivative. ​ Find if ​

Find the slope of the tangent line to the graph of the function below at the given point. ​ , ​

​Identify a function f(x)that has the given characteristics and then sketch the function. ​ ​; ; ​

​Find the second derivative for the function and solve the equation ​

​Find the marginal cost for producing x units.(The cost is measured in dollars.) ​ ​

​Find implicitly for ​

The graph shows the number of visitors V to a national park in hundreds of thousands during a one-year period,where represents January.Estimate the rate of change of V over the interval Round your answer to the nearest hundred thousand visitors per year. ​ ​

​Find the derivative of the following function using the limiting process. ​ ​ ​

​A retail sporting goods store estimates that weekly sales and weekly advertising costs are related by the equation The current weekly advertising costs are $1,500,and these costs are increasing at a rate of $135 per week.Find the current rate of change of weekly sales. ​

​A population of bacteria is introduced into a culture.The number of bacteria P can be modeled by where t is the time (in hours).Find the rate of change of the population when ​

​Assume that x and y are differentiable functions of t.Find given that , ,and ​ ​

​Determine the point(s),(if any),at which the graph of the function has a horizontal tangent. ​ ​ ​

​Determine whether the statement is true or false.If it is false,explain why or give an example that shows it is false. ​ If ,then ​

Find the derivative of the following function using the limiting process. ​ ​

​For the function given,find ​ ​ ​

Find the derivative of the function. ​ ​

​A brick becomes dislodged from the Empire State Building (at a height of 1,000 feet)and falls to the sidewalk below.Write the position s(t),velocity v(t),and acceleration a(t)as functions of time. ​

​Profit.Suppose that the monthly revenue and cost (in dollars)for x units of a product are and At what rate per month is the profit changing if the number of units produced and sold is 100 and is increasing at a rate of 10 units per month? ​

Functions, graphs, and Limits

Applications of the Derivative

Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions

Probability and Calculus

Series and Taylor Polynomials

Differential Equations

Filters

Find the marginal profit for producing x units.(The profit is measured in dollars.)

Find the indicated derivative. Find if

Find the slope of the tangent line to the graph of the function below at the given point. ,

Identify a function f(x)that has the given characteristics and then sketch the function. ; ;

Find the second derivative for the function and solve the equation

Find the marginal cost for producing x units.(The cost is measured in dollars.)

Find implicitly for

The graph shows the number of visitors V to a national park in hundreds of thousands during a one-year period,where represents January.Estimate the rate of change of V over the interval Round your answer to the nearest hundred thousand visitors per year.

Find the derivative of the following function using the limiting process.

A retail sporting goods store estimates that weekly sales and weekly advertising costs are related by the equation The current weekly advertising costs are $1,500,and these costs are increasing at a rate of $135 per week.Find the current rate of change of weekly sales.

A population of bacteria is introduced into a culture.The number of bacteria P can be modeled by where t is the time (in hours).Find the rate of change of the population when

Assume that x and y are differentiable functions of t.Find given that , ,and

Determine the point(s),(if any),at which the graph of the function has a horizontal tangent.

Determine whether the statement is true or false.If it is false,explain why or give an example that shows it is false. If ,then

Find the derivative of the following function using the limiting process.

For the function given,find

Find the derivative of the function.

A brick becomes dislodged from the Empire State Building (at a height of 1,000 feet)and falls to the sidewalk below.Write the position s(t),velocity v(t),and acceleration a(t)as functions of time.

Profit.Suppose that the monthly revenue and cost (in dollars)for x units of a product are and At what rate per month is the profit changing if the number of units produced and sold is 100 and is increasing at a rate of 10 units per month?