Question 1

Use analytic methods to find the limit as   for the given function. ​   ​

Accepted Answer

A)  -1500 
B)  1500 
C)  -620 
D)  620 
E)  does not exist 
A)  -1500 
B)  1500 
C)  -620 
D)  620 
E)  does not exist

Question 2

If an IRA is a variable-rate investment for 20 years at rate r percent per year, compounded monthly, then the future value S that accumulates from an initial investment of $1,000 is   . What is the rate of change of S with respect to r and what does it tell us if the interest rate is 7%? Round your answer to two decimal places. ​

Accepted Answer

A)  The future value increases by approximately $866.86 when the interest rate increases from 7% to 8%. 
B)  The future value increases by approximately $803.06 when the interest rate increases from 7% to 8%. 
C)  The future value increases by approximately $1634.43 when the interest rate increases from 7% to 8%. 
D)  The future value increases by approximately $781.46 when the interest rate increases from 7% to 8%. 
E)  The future value increases by approximately $1565.73 when the interest rate increases from 7% to 8%. 
A)  The future value increases by approximately $866.86 when the interest rate increases from 7% to 8%. 
B)  The future value increases by approximately $803.06 when the interest rate increases from 7% to 8%. 
C)  The future value increases by approximately $1634.43 when the interest rate increases from 7% to 8%. 
D)  The future value increases by approximately $781.46 when the interest rate increases from 7% to 8%. 
E)  The future value increases by approximately $1565.73 when the interest rate increases from 7% to 8%.

Question 3

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 4

Find the derivative of the function. ​   ​

Accepted Answer

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

Question 5

Write the equation of the line tangent to the graph of f (x) at   ​   ​

Accepted Answer

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

Question 6

Total revenue is in dollars and x is the number of units. Suppose that the total revenue function for a commodity is   Find   , and explain what this value represents. Round your answer to two decimal places. ​

Accepted Answer

A)  $17.90 is the marginal revenue from the 31st unit. 
B)  $15.80 is the marginal revenue from the 31st unit. 
C)  $537.00 is the marginal revenue from the 31st unit. 
D)  $552.73 is the actual revenue from the 31st unit. 
E)  $15.73 is the actual revenue from the 31st unit. 
A)  $17.90 is the marginal revenue from the 31st unit. 
B)  $15.80 is the marginal revenue from the 31st unit. 
C)  $537.00 is the marginal revenue from the 31st unit. 
D)  $552.73 is the actual revenue from the 31st unit. 
E)  $15.73 is the actual revenue from the 31st unit.

Question 7

Graph the marginal profit function for the profit function   where P(x) is in dollars and x is the number of units. At what level of production and sales will profit be at a maximum? ​

Accepted Answer

A)  0 
B)  100 
C)  184 
D)  16 
E)  200 
A)  0 
B)  100 
C)  184 
D)  16 
E)  200

Question 8

Determine whether the given function is continuous. If it is not, identify where it is discontinuous. You can verify your conclusions by graphing the function with a graphing utility, if one is available. ​   ​

Accepted Answer

A)  discontinuous at   
B)  discontinuous at   
C)  discontinuous at   
D)  discontinuous at   
E)  continuous everywhere 
A)  discontinuous at   
B)  discontinuous at   
C)  discontinuous at   
D)  discontinuous at   
E)  continuous everywhere

Question 9

A certain calling card costs 6.8 cents per minute to make a call, rounded up to the nearest minute. However, when the card is used at a pay phone, there is an additional 68 cent connection fee. If   is the charge from a pay phone for a call lasting t minutes, create a table of charges for calls lasting close to 1 minute and use it to find the following limit, if it exists. Round your answer to the nearest cent.   ​

Accepted Answer

A)  $0 
B)  $0.68 
C)  $0.75 
D)  $0.82 
E)  does not exist 
A)  $0 
B)  $0.68 
C)  $0.75 
D)  $0.82 
E)  does not exist

Question 10

Use analytic methods to evaluate the limit and determine which type of asymptote (if any) it represents. You can verify your conclusion by graphing the function with a graphing utility, if one is available. ​   ​

Accepted Answer

A)  limit = 4, vertical asymptote 
B)  limit = 8, horizontal asymptote 
C)  limit = 8, vertical asymptote 
D)  limit = 4, horizontal asymptote 
E)  does not exist 
A)  limit = 4, vertical asymptote 
B)  limit = 8, horizontal asymptote 
C)  limit = 8, vertical asymptote 
D)  limit = 4, horizontal asymptote 
E)  does not exist

Question 11

The graph of a company's profit function is shown. For the given problem, use the graph to answer the following questions about points A, B, and C. Rank from smallest to largest the marginal profit at these points. ​   ​

Accepted Answer

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

Question 12

Suppose that the revenue (in dollars) from the sale of a product is given by   , where x is the number of units sold. How fast is the marginal revenue   changing when   ​

Accepted Answer

A)  1.4 dollars per unit 
B)  -30 dollars per unit 
C)  30 dollars per unit 
D)  -0.1 dollars per unit 
E)  -1.4 dollars per unit 
A)  1.4 dollars per unit 
B)  -30 dollars per unit 
C)  30 dollars per unit 
D)  -0.1 dollars per unit 
E)  -1.4 dollars per unit

Question 13

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

Accepted Answer

A)  -45 
B)  54 
C)  45 
D)  -54 
E)  does not exist 
A)  -45 
B)  54 
C)  45 
D)  -54 
E)  does not exist

Question 14

Because the derivative of a function represents both the slope of the tangent to the curve and the instantaneous rate of change of the function, it is possible to use information about one to gain information about the other. In this problem, use the graph of the function   given in the figure.   Over what interval(s) (a) through (d) is the rate of change of f (x) negative?

Accepted Answer

A)  a 
B)  d 
C)  b d 
D)  c 
E)  a b c 
A)  a 
B)  d 
C)  b d 
D)  c 
E)  a b c

Question 15

The following table shows the numbers of U.S. cellular subscriberships (in millions) from 1985 to 2002. ​
 
Source: The CTIA Semi-Annual Wireless Industry Survey
Assume that the number of U.S. cellular subscriberships, in millions, can be modeled by the function   , where t is the number of years past 1980. Find the function that models the instantaneous rate of change of the U.S. cellular subscriberships.
​

Accepted Answer

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

Question 16

For the given function, find the slope of the tangent line at   . ​   ​

Accepted Answer

A)  145 
B)  91 
C)  457 
D)  137 
E)  does not exist 
A)  145 
B)  91 
C)  457 
D)  137 
E)  does not exist

Question 17

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 18

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

Accepted Answer

A)  44.0 
B)  -22.0 
C)  22.0 
D)  -0.5 
E)  does not exist 
A)  44.0 
B)  -22.0 
C)  22.0 
D)  -0.5 
E)  does not exist

Question 19

A graph of   is shown and a c-value is given. For this problem, use the graph to find   . ​   ​   ​

Accepted Answer

A)  5 
B)  0 
C)  -15 
D)  15 
E)  does not exist 
A)  5 
B)  0 
C)  -15 
D)  15 
E)  does not exist

Question 20

Find the third derivative. ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)  0 
A)    
B)    
C)    
D)    
E)  0

Use analytic methods to find the limit as for the given function.

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

Find the derivative of the function.

Write the equation of the line tangent to the graph of f (x) at

Total revenue is in dollars and x is the number of units. Suppose that the total revenue function for a commodity is Find , and explain what this value represents. Round your answer to two decimal places.

Graph the marginal profit function for the profit function where P(x) is in dollars and x is the number of units. At what level of production and sales will profit be at a maximum?

Determine whether the given function is continuous. If it is not, identify where it is discontinuous. You can verify your conclusions by graphing the function with a graphing utility, if one is available.

Use analytic methods to evaluate the limit and determine which type of asymptote (if any) it represents. You can verify your conclusion by graphing the function with a graphing utility, if one is available.

The graph of a company's profit function is shown. For the given problem, use the graph to answer the following questions about points A, B, and C. Rank from smallest to largest the marginal profit at these points.

Suppose that the revenue (in dollars) from the sale of a product is given by , where x is the number of units sold. How fast is the marginal revenue changing when

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

For the given function, find the slope of the tangent line at .

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

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

A graph of is shown and a c-value is given. For this problem, use the graph to find .

Find the third derivative.

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

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Exam 9: Derivatives

Use analytic methods to find the limit as for the given function. ​ ​

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

Find the derivative of the function. ​ ​

Write the equation of the line tangent to the graph of f (x) at ​ ​

Total revenue is in dollars and x is the number of units. Suppose that the total revenue function for a commodity is Find , and explain what this value represents. Round your answer to two decimal places. ​

Graph the marginal profit function for the profit function where P(x) is in dollars and x is the number of units. At what level of production and sales will profit be at a maximum? ​

Determine whether the given function is continuous. If it is not, identify where it is discontinuous. You can verify your conclusions by graphing the function with a graphing utility, if one is available. ​ ​

Use analytic methods to evaluate the limit and determine which type of asymptote (if any) it represents. You can verify your conclusion by graphing the function with a graphing utility, if one is available. ​ ​

The graph of a company's profit function is shown. For the given problem, use the graph to answer the following questions about points A, B, and C. Rank from smallest to largest the marginal profit at these points. ​ ​

Suppose that the revenue (in dollars) from the sale of a product is given by , where x is the number of units sold. How fast is the marginal revenue changing when ​

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

For the given function, find the slope of the tangent line at . ​ ​

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

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

A graph of is shown and a c-value is given. For this problem, use the graph to find . ​ ​ ​

Find the third derivative. ​ ​

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

Use analytic methods to find the limit as for the given function.

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

Find the derivative of the function.

Write the equation of the line tangent to the graph of f (x) at

Total revenue is in dollars and x is the number of units. Suppose that the total revenue function for a commodity is Find , and explain what this value represents. Round your answer to two decimal places.

Graph the marginal profit function for the profit function where P(x) is in dollars and x is the number of units. At what level of production and sales will profit be at a maximum?

Determine whether the given function is continuous. If it is not, identify where it is discontinuous. You can verify your conclusions by graphing the function with a graphing utility, if one is available.

Use analytic methods to evaluate the limit and determine which type of asymptote (if any) it represents. You can verify your conclusion by graphing the function with a graphing utility, if one is available.

The graph of a company's profit function is shown. For the given problem, use the graph to answer the following questions about points A, B, and C. Rank from smallest to largest the marginal profit at these points.

Suppose that the revenue (in dollars) from the sale of a product is given by , where x is the number of units sold. How fast is the marginal revenue changing when

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

For the given function, find the slope of the tangent line at .

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

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

A graph of is shown and a c-value is given. For this problem, use the graph to find .

Find the third derivative.