Question 1

Find the indicated derivative and simplify. ​   for   ​

Accepted Answer

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

Question 2

The table gives the yearly U.S. federal budget deficit (as a negative value) or surplus (as a positive value) in billions of dollars from 1990 to 2004. ​
 Source: Budget of the United States Government
Assume the federal budget deficit (or surplus) can be modeled with the function   , where D is in billions of dollars and t is the number of years past 1980. Use the model to find and interpret the instantaneous rate of change of the U.S. federal deficit (or surplus) in 1995.
​

Accepted Answer

A)  This model predicts a budget deficit increase of $590.3 billion from 1995 to 1996. 
B)  This model predicts a budget deficit increase of $81.2 billion from 1995 to 1996. 
C)  This model predicts a budget deficit decrease of $811.6billion from 1995 to 1996. 
D)  This model predicts a budget deficit decrease of $81.2 billion from 1995 to 1996. 
E)  This model predicts a budget deficit decrease of $590.3 billion from 1995 to 1996. 
A)  This model predicts a budget deficit increase of $590.3 billion from 1995 to 1996. 
B)  This model predicts a budget deficit increase of $81.2 billion from 1995 to 1996. 
C)  This model predicts a budget deficit decrease of $811.6billion from 1995 to 1996. 
D)  This model predicts a budget deficit decrease of $81.2 billion from 1995 to 1996. 
E)  This model predicts a budget deficit decrease of $590.3 billion from 1995 to 1996.

Question 3

Find the coordinates of points where the graph of   has horizontal tangents. As a check, graph   and see whether the points you found look as though they have horizontal tangents. ​   ​

Accepted Answer

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

Question 4

Find the second derivative. ​   ​

Accepted Answer

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

Question 5

At the indicated point, find the instantaneous rate of change of the function. ​   ​

Accepted Answer

A)  86 
B)  50 
C)  270 
D)  23 
E)  32 
A)  86 
B)  50 
C)  270 
D)  23 
E)  32

Question 6

Use properties of limits and algebraic methods to find the limit, if it exists. ​   ​

Accepted Answer

A)  63 
B)  65 
C)  60 
D)  68 
E)  does not exist 
A)  63 
B)  65 
C)  60 
D)  68 
E)  does not exist

Question 7

Graph the function with a graphing utility and use it to predict the limit. Check your work either by using the table feature of the graphing utility or by finding the limit algebraically. ​   ​

Accepted Answer

A)  0 
B)  9 
C)    
D)    
E)  does not exist 
A)  0 
B)  9 
C)    
D)    
E)  does not exist

Question 8

The monthly charge in dollars for x kilowatt-hours (kWh) of electricity used by a residential consumer of Excelsior Electric Membership Corporation from November through June is given by the function   . What is the monthly charge if 1,700 kWh of electricity is consumed in a month? Round your answer to two decimal places.

Accepted Answer

A)  $1,700.00 
B)  $169.80 
C)  $141.30 
D)  $73.80 
E)  $118.80 
A)  $1,700.00 
B)  $169.80 
C)  $141.30 
D)  $73.80 
E)  $118.80

Question 9

Find the indicated derivative and simplify. ​   for   ​

Accepted Answer

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

Question 10

At the indicated point, find the slope of the tangent to the curve. ​   ​

Accepted Answer

A)  -25 
B)  133 
C)  10 
D)  17 
E)  -53 
A)  -25 
B)  133 
C)  10 
D)  17 
E)  -53

Question 11

For the function given, find   ​   ​

Accepted Answer

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

Question 12

Find the derivative of the given function. Simplify and express the answer using positive exponents only. ​   ​

Accepted Answer

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

Question 13

Let   . Use the definition of derivative and the Procedure/Example table in this section to verify that   . Then, find the instantaneous rate of change of f (x) at   . ​

Accepted Answer

A)  10 
B)  2 
C)  8 
D)  4 
E)  -8 
A)  10 
B)  2 
C)  8 
D)  4 
E)  -8

Question 14

The table gives the yearly U.S. federal budget deficit (as a negative value) or surplus (as a positive value) in billions of dollars from 1990 to 2004. ​
 
Source: Budget of the United States Government
Assume the federal budget deficit (or surplus) can be modeled with the function   , where D is in billions of dollars and t is the number of years past 1980. Use the data to find an average rate of change that approximates the instantaneous rate in 1996.
​

Accepted Answer

A)  The budget deficit increased by an average rate of $85.5 billion from 1996 to 1997. 
B)  The budget deficit increased by an average rate of $831.3 billion from 1996 to 1997. 
C)  The budget deficit decreased by an average rate of $85.5 billion from 1996 to 1997. 
D)  The budget deficit decreased by an average rate of $83.1 billion from 1996 to 1997. 
E)  The budget deficit decreased by an average rate of $469.9 billion from 1996 to 1997. 
A)  The budget deficit increased by an average rate of $85.5 billion from 1996 to 1997. 
B)  The budget deficit increased by an average rate of $831.3 billion from 1996 to 1997. 
C)  The budget deficit decreased by an average rate of $85.5 billion from 1996 to 1997. 
D)  The budget deficit decreased by an average rate of $83.1 billion from 1996 to 1997. 
E)  The budget deficit decreased by an average rate of $469.9 billion from 1996 to 1997.

Question 15

For the function in this problem, approximate   near the point   by graphing the function on a graphing utility. Then, zoom in near the point until the graph appears straight, pick two points, and find the slope of the line you see. Round your answer to three decimal places.

Accepted Answer

A)  48.050 
B)  -9.000 
C)  -8.950 
D)  50.970 
E)  -8.999 
A)  48.050 
B)  -9.000 
C)  -8.950 
D)  50.970 
E)  -8.999

Question 16

Complete the table and use it to predict the limit, if it exists. ​   ​

Accepted Answer

A)  -5 
B)  5 
C)  -4 
D)  4 
E)  does not exist 
A)  -5 
B)  5 
C)  -4 
D)  4 
E)  does not exist

Question 17

For the given function, find the instantaneous rate of change of the function at   . ​   ​

Accepted Answer

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

Question 18

If the total revenue function for a toy is   dollars and the total cost function is   dollars. What is the instantaneous rate of change of profit if 10 units are produced and sold? Explain its meaning. ​

Accepted Answer

A)  0.20. Selling an additional unit when 10 have been sold increases profit by $0.20. 
B)  0.60. Selling an additional unit when 10 have been sold increases profit by $0.60. 
C)  6.80. Selling an additional unit when 10 have been sold increases profit by $6.80. 
D)  7.00. Selling an additional unit when 10 have been sold increases profit by $7.00. 
E)  234.00. Selling an additional unit when 10 have been sold increases profit by $234.00. 
A)  0.20. Selling an additional unit when 10 have been sold increases profit by $0.20. 
B)  0.60. Selling an additional unit when 10 have been sold increases profit by $0.60. 
C)  6.80. Selling an additional unit when 10 have been sold increases profit by $6.80. 
D)  7.00. Selling an additional unit when 10 have been sold increases profit by $7.00. 
E)  234.00. Selling an additional unit when 10 have been sold increases profit by $234.00.

Question 19

Use the numerical derivative feature of a graphing utility to approximate the given second derivative to two decimal places. Round your answer to two decimal places. ​   ​

Accepted Answer

A)  0.53 
B)  0.33 
C)  1.12 
D)  0.18 
E)  0.35 
A)  0.53 
B)  0.33 
C)  1.12 
D)  0.18 
E)  0.35

Question 20

In the figure given in this problem, at points A and E draw an approximate tangent line and then use it to answer the following question. Is   greater at point A or at point E? ​   ​

Accepted Answer

A)  point A 
B)  point E 
A)  point A 
B)  point E

Find the indicated derivative and simplify. for

Find the coordinates of points where the graph of has horizontal tangents. As a check, graph and see whether the points you found look as though they have horizontal tangents.

Find the second derivative.

At the indicated point, find the instantaneous rate of change of the function.

Use properties of limits and algebraic methods to find the limit, if it exists.

Graph the function with a graphing utility and use it to predict the limit. Check your work either by using the table feature of the graphing utility or by finding the limit algebraically.

Find the indicated derivative and simplify. for

At the indicated point, find the slope of the tangent to the curve.

For the function given, find

Find the derivative of the given function. Simplify and express the answer using positive exponents only.

Let . Use the definition of derivative and the Procedure/Example table in this section to verify that . Then, find the instantaneous rate of change of f (x) at .

For the function in this problem, approximate near the point by graphing the function on a graphing utility. Then, zoom in near the point until the graph appears straight, pick two points, and find the slope of the line you see. Round your answer to three decimal places.

Complete the table and use it to predict the limit, if it exists.

For the given function, find the instantaneous rate of change of the function at .

If the total revenue function for a toy is dollars and the total cost function is dollars. What is the instantaneous rate of change of profit if 10 units are produced and sold? Explain its meaning.

Use the numerical derivative feature of a graphing utility to approximate the given second derivative to two decimal places. Round your answer to two decimal places.

In the figure given in this problem, at points A and E draw an approximate tangent line and then use it to answer the following question. Is greater at point A or at point E?

Linear Equations and Functions

Quadratic and Other Special Functions

Matrices

Inequalities and Linear Programming

Exponential and Logarithmic Functions

Mathematics of Finance

Introduction to Probability

Further Topics in Probability and Data Description

Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals - Techniques

A: Definite Integrals - Techniques

Functions of Two or More Variables

Algebraic Concepts 2

Algebraic Concepts 3

Algebraic Concepts 4

Algebraic Concepts 5

Filters

Exam 9: Derivatives

Find the indicated derivative and simplify. ​ for ​

Find the coordinates of points where the graph of has horizontal tangents. As a check, graph and see whether the points you found look as though they have horizontal tangents. ​ ​

Find the second derivative. ​ ​

At the indicated point, find the instantaneous rate of change of the function. ​ ​

Use properties of limits and algebraic methods to find the limit, if it exists. ​ ​

Graph the function with a graphing utility and use it to predict the limit. Check your work either by using the table feature of the graphing utility or by finding the limit algebraically. ​ ​

Find the indicated derivative and simplify. ​ for ​

At the indicated point, find the slope of the tangent to the curve. ​ ​

For the function given, find ​ ​

Find the derivative of the given function. Simplify and express the answer using positive exponents only. ​ ​

Let . Use the definition of derivative and the Procedure/Example table in this section to verify that . Then, find the instantaneous rate of change of f (x) at . ​

For the function in this problem, approximate near the point by graphing the function on a graphing utility. Then, zoom in near the point until the graph appears straight, pick two points, and find the slope of the line you see. Round your answer to three decimal places.

Complete the table and use it to predict the limit, if it exists. ​ ​

For the given function, find the instantaneous rate of change of the function at . ​ ​

If the total revenue function for a toy is dollars and the total cost function is dollars. What is the instantaneous rate of change of profit if 10 units are produced and sold? Explain its meaning. ​

Use the numerical derivative feature of a graphing utility to approximate the given second derivative to two decimal places. Round your answer to two decimal places. ​ ​

In the figure given in this problem, at points A and E draw an approximate tangent line and then use it to answer the following question. Is greater at point A or at point E? ​ ​

Linear Equations and Functions

Quadratic and Other Special Functions

Matrices

Inequalities and Linear Programming

Exponential and Logarithmic Functions

Mathematics of Finance

Introduction to Probability

Further Topics in Probability and Data Description

Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals - Techniques

A: Definite Integrals - Techniques

Functions of Two or More Variables

Algebraic Concepts 2

Algebraic Concepts 3

Algebraic Concepts 4

Algebraic Concepts 5

Filters

Find the indicated derivative and simplify. for

Find the coordinates of points where the graph of has horizontal tangents. As a check, graph and see whether the points you found look as though they have horizontal tangents.

Find the second derivative.

At the indicated point, find the instantaneous rate of change of the function.

Use properties of limits and algebraic methods to find the limit, if it exists.

Graph the function with a graphing utility and use it to predict the limit. Check your work either by using the table feature of the graphing utility or by finding the limit algebraically.

Find the indicated derivative and simplify. for

At the indicated point, find the slope of the tangent to the curve.

For the function given, find

Find the derivative of the given function. Simplify and express the answer using positive exponents only.

Let . Use the definition of derivative and the Procedure/Example table in this section to verify that . Then, find the instantaneous rate of change of f (x) at .

Complete the table and use it to predict the limit, if it exists.

For the given function, find the instantaneous rate of change of the function at .

If the total revenue function for a toy is dollars and the total cost function is dollars. What is the instantaneous rate of change of profit if 10 units are produced and sold? Explain its meaning.

Use the numerical derivative feature of a graphing utility to approximate the given second derivative to two decimal places. Round your answer to two decimal places.

In the figure given in this problem, at points A and E draw an approximate tangent line and then use it to answer the following question. Is greater at point A or at point E?