Question 1

Suppose that the demand function for q units of an appliance priced at $p per unit is given by   find the rate of change of price with respect to the number of appliances. ​

Accepted Answer

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

Question 2

Differentiate the function   . ​

Accepted Answer

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

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

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

Accepted Answer

A) 10 
B) -15 
C) 0 
D) 15 
E) does not exist 
A) 10 
B) -15 
C) 0 
D) 15 
E) does not exist

Question 5

Find   ​   ​

Accepted Answer

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

Question 6

Use the graph of   and the given c-value to find   . ​   ​   ​

Accepted Answer

A) -6 
B) 0 
C) 8 
D) 2 
E) does not exist 
A) -6 
B) 0 
C) 8 
D) 2 
E) does not exist

Question 7

Determine whether the given function is continuous. If it is not, identify where it is discontinuous. ​   ​

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 8

For the function in this problem, find the slope of the tangent line at the given value. ​   ;   ​

Accepted Answer

A) 5 
B) 19 
C) 13 
D) -1 
E) 21 
A) 5 
B) 19 
C) 13 
D) -1 
E) 21

Question 9

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 10

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

Accepted Answer

A)    
B) -3 
C)    
D)    
E) does not exist 
A)    
B) -3 
C)    
D)    
E) does not exist

Question 11

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 12

If the cost function for a commodity is   dollars, find the marginal cost at   units, and tell what this predicts about the cost of producing 2 additional units. Round your answer to two decimal places. ​

Accepted Answer

A) $18.31. The cost of producing 2 additional units is approximately $36.63. 
B) $67.88. The cost of producing 2 additional units is approximately $67.88. 
C) $33.94. The cost of producing 2 additional units is approximately $33.94. 
D) $33.94. The cost of producing 2 additional units is approximately $67.88. 
E) $6.38. The cost of producing 2 additional units is approximately $12.75. 
A) $18.31. The cost of producing 2 additional units is approximately $36.63. 
B) $67.88. The cost of producing 2 additional units is approximately $67.88. 
C) $33.94. The cost of producing 2 additional units is approximately $33.94. 
D) $33.94. The cost of producing 2 additional units is approximately $67.88. 
E) $6.38. The cost of producing 2 additional units is approximately $12.75.

Question 13

In this problem, cost, revenue, and profit are in dollars and x is the number of units. Suppose that the total revenue function is given by   , and that the total cost function is given by   . Find   at   , and explain what it predicts. Round your answer to two decimal places. ​

Accepted Answer

A) 130.00. The profit will increase by approximately $130.00 from the sale of the 101st item. 
B) 40.00. The profit will increase by approximately $40.00 from the sale of the 101st item. 
C) 10.00. The profit will increase by approximately $10.00 from the sale of the 101st item. 
D) 60.00. The profit will decrease by approximately $60.00 from the sale of the 101st item. 
E) 90.00. The profit will decrease by approximately $90.00 from the sale of the 101st item. 
A) 130.00. The profit will increase by approximately $130.00 from the sale of the 101st item. 
B) 40.00. The profit will increase by approximately $40.00 from the sale of the 101st item. 
C) 10.00. The profit will increase by approximately $10.00 from the sale of the 101st item. 
D) 60.00. The profit will decrease by approximately $60.00 from the sale of the 101st item. 
E) 90.00. The profit will decrease by approximately $90.00 from the sale of the 101st item.

Question 14

Determine whether the function is continuous or discontinuous at the given x-value. ​   ​

Accepted Answer

A) continuous 
B) discontinuous 
A) continuous 
B) discontinuous

Question 15

Find the second derivative of   . ​

Accepted Answer

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

Question 16

The graph of a company's total cost function is shown. Does this total cost function represent a manufacturing process that is getting more efficient or less efficient as the production level increases? ​   ​

Accepted Answer

A) less efficient 
B) more efficient 
A) less efficient 
B) more efficient

Question 17

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 18

Find the slope of the tangent to the graph of   at any point. ​   ​

Accepted Answer

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

Question 19

Find the second derivative. ​   ​

Accepted Answer

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

Question 20

The tangent line to a curve at a point closely approximates the curve near the point. In fact, for x-values close enough to the point of tangency, the function and its tangent line are virtually indistinguishable. This problem explores this relationship. Write the equation of the tangent line to the curve at the indicated point. To check your answer, use a graphing utility to graph both the function and its tangent line, and repeatedly zoom in on the point of tangency until the function and the tangent line cannot be distinguished. ​   ​

Accepted Answer

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

Suppose that the demand function for q units of an appliance priced at $p per unit is given by find the rate of change of price with respect to the number of appliances.

Differentiate the function .

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

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

Find

Use the graph of and the given c-value to find .

Determine whether the given function is continuous. If it is not, identify where it is discontinuous.

For the function in this problem, find the slope of the tangent line at the given value. ;

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.

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

If the cost function for a commodity is dollars, find the marginal cost at units, and tell what this predicts about the cost of producing 2 additional units. Round your answer to two decimal places.

In this problem, cost, revenue, and profit are in dollars and x is the number of units. Suppose that the total revenue function is given by , and that the total cost function is given by . Find at , and explain what it predicts. Round your answer to two decimal places.

Determine whether the function is continuous or discontinuous at the given x-value.

Find the second derivative of .

The graph of a company's total cost function is shown. Does this total cost function represent a manufacturing process that is getting more efficient or less efficient as the production level increases?

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?

Find the slope of the tangent to the graph of at any point.

Find the second derivative.

Algebraic Concepts

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; Data Description

Applications of Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals: Techniques of Integration

Functions of Two or More Variables

Filters

Exam 10: Derivatives

Suppose that the demand function for q units of an appliance priced at $p per unit is given by find the rate of change of price with respect to the number of appliances. ​

Differentiate the function . ​

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

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

Find ​ ​

Use the graph of and the given c-value to find . ​ ​ ​

Determine whether the given function is continuous. If it is not, identify where it is discontinuous. ​ ​

For the function in this problem, find the slope of the tangent line at the given value. ​ ; ​

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. ​ ​

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

If the cost function for a commodity is dollars, find the marginal cost at units, and tell what this predicts about the cost of producing 2 additional units. Round your answer to two decimal places. ​

In this problem, cost, revenue, and profit are in dollars and x is the number of units. Suppose that the total revenue function is given by , and that the total cost function is given by . Find at , and explain what it predicts. Round your answer to two decimal places. ​

Determine whether the function is continuous or discontinuous at the given x-value. ​ ​

Find the second derivative of . ​

The graph of a company's total cost function is shown. Does this total cost function represent a manufacturing process that is getting more efficient or less efficient as the production level increases? ​ ​

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? ​

Find the slope of the tangent to the graph of at any point. ​ ​

Find the second derivative. ​ ​

Algebraic Concepts

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; Data Description

Applications of Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals: Techniques of Integration

Functions of Two or More Variables

Filters

Suppose that the demand function for q units of an appliance priced at $p per unit is given by find the rate of change of price with respect to the number of appliances.

Differentiate the function .

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

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

Find

Use the graph of and the given c-value to find .

Determine whether the given function is continuous. If it is not, identify where it is discontinuous.

For the function in this problem, find the slope of the tangent line at the given value. ;

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.

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

If the cost function for a commodity is dollars, find the marginal cost at units, and tell what this predicts about the cost of producing 2 additional units. Round your answer to two decimal places.

In this problem, cost, revenue, and profit are in dollars and x is the number of units. Suppose that the total revenue function is given by , and that the total cost function is given by . Find at , and explain what it predicts. Round your answer to two decimal places.

Determine whether the function is continuous or discontinuous at the given x-value.

Find the second derivative of .

The graph of a company's total cost function is shown. Does this total cost function represent a manufacturing process that is getting more efficient or less efficient as the production level increases?

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?

Find the slope of the tangent to the graph of at any point.

Find the second derivative.