Question 1

Given   , find the average rate of change of f (x) over the following pair of intervals using a calculator if desired: ​   and   ​

Accepted Answer

A) 17.9 and 17.99 
B) 20.9 and 20.99 
C) 19.90 and 19.99 
D) 19.90 and 20.99 
E) 17.9 and 19.99 
A) 17.9 and 17.99 
B) 20.9 and 20.99 
C) 19.90 and 19.99 
D) 19.90 and 20.99 
E) 17.9 and 19.99

Question 2

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 3

For the function in this problem, find the instantaneous rate of change of the function at the given value. ​   ;   ​

Accepted Answer

A) 28 
B) 8 
C) 16 
D) 20 
E) 45 
A) 28 
B) 8 
C) 16 
D) 20 
E) 45

Question 4

If the instantaneous rate of change of g(x) at   is   , write the equation of the line tangent to the graph of g(x) at   ​

Accepted Answer

A) x = -  
Y +   
B) x =  
Y +   
C) x =  
Y -   
D) y =  
X -   
E) y =  
X +   
A) x = -  
Y +   
B) x =  
Y +   
C) x =  
Y -   
D) y =  
X -   
E) y =  
X +

Question 5

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 6

Find the indicated derivative. If   , find   .
​

Accepted Answer

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

Question 7

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 8

If the revenue for a product (in dollars) is   , and the average revenue per unit is   ,   , find   . ​

Accepted Answer

A) $500 per unit 
B) $0.8 per unit 
C) $50 per unit 
D) $0 per unit 
E) does not exist 
A) $500 per unit 
B) $0.8 per unit 
C) $50 per unit 
D) $0 per unit 
E) does not exist

Question 9

For the function given, find all x-values where the graph of   is falling. ​   ​

Accepted Answer

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

Question 10

The tangent line to the graph of   at   is shown. On the tangent line, P is the point of tangency and A is another point on the line. Find the instantaneous rate of change of   at P. ​   ​

Accepted Answer

A) -3 
B) 3 
C) -4 
D) -5 
E) does not exist 
A) -3 
B) 3 
C) -4 
D) -5 
E) does not exist

Question 11

Suppose that the cost C in dollars of obtaining water that contains p percent impurities is given by   . Find   , if it exists. ​

Accepted Answer

A) $0 
B) $20 
C) $200 
D) $2,000 
E) does not exist 
A) $0 
B) $20 
C) $200 
D) $2,000 
E) does not exist

Question 12

Find the indicated derivative and simplify. ​   for   ​

Accepted Answer

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

Question 13

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 14

Total revenue is in dollars and x is the number of units. Suppose that the total revenue function for a commodity is   . Find   , and tell what it represents. ​

Accepted Answer

A) $19.00. This represents the revenue from the sale of 50 units. 
B) $1,049.00. This represents the marginal revenue from the sale of 50 units. 
C) $1,050.00. This represents the marginal revenue from the sale of 50 units. 
D) $1,000.00. This represents the revenue from the sale of 50 units. 
E) $1.00. This represents the revenue from the sale of 50 units. 
A) $19.00. This represents the revenue from the sale of 50 units. 
B) $1,049.00. This represents the marginal revenue from the sale of 50 units. 
C) $1,050.00. This represents the marginal revenue from the sale of 50 units. 
D) $1,000.00. This represents the revenue from the sale of 50 units. 
E) $1.00. This represents the revenue from the sale of 50 units.

Question 15

Given the graph of   in the figure, determine all x-values from F, G, H, or I for which the function is differentiable. ​   ​

Accepted Answer

A) F G H I 
B) F H 
C) G H I 
D) F G 
E) F H I 
A) F G H I 
B) F H 
C) G H I 
D) F G 
E) F H I

Question 16

Total revenue is in dollars and x is the number of units. Suppose that the total revenue function for a commodity is   Find the marginal revenue function. ​

Accepted Answer

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

Question 17

Write the equation of the tangent line to each curve at the indicated point. As a check, graph both the function and the tangent line. ​   ​

Accepted Answer

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

Question 18

The total revenue function (in dollars) for a blender is   , where x is the number of units sold. What is the average rate of change in revenue R(x) as x increases from 20 to 30 units? ​

Accepted Answer

A) $254.60 per unit 
B) $59.80 per unit 
C) $48.20 per unit 
D) $49.80 per unit 
E) $49.00 per unit 
A) $254.60 per unit 
B) $59.80 per unit 
C) $48.20 per unit 
D) $49.80 per unit 
E) $49.00 per unit

Question 19

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 20

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)

Given , find the average rate of change of f (x) over the following pair of intervals using a calculator if desired: and

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

For the function in this problem, find the instantaneous rate of change of the function at the given value. ;

If the instantaneous rate of change of g(x) at is , write the equation of the line tangent to the graph of g(x) at

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

Find the indicated derivative. If , find .

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 .

If the revenue for a product (in dollars) is , and the average revenue per unit is , , find .

For the function given, find all x-values where the graph of is falling.

The tangent line to the graph of at is shown. On the tangent line, P is the point of tangency and A is another point on the line. Find the instantaneous rate of change of at P.

Suppose that the cost C in dollars of obtaining water that contains p percent impurities is given by . Find , if it exists.

Find the indicated derivative and simplify. for

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.

Total revenue is in dollars and x is the number of units. Suppose that the total revenue function for a commodity is . Find , and tell what it represents.

Given the graph of in the figure, determine all x-values from F, G, H, or I for which the function is differentiable.

Total revenue is in dollars and x is the number of units. Suppose that the total revenue function for a commodity is Find the marginal revenue function.

Write the equation of the tangent line to each curve at the indicated point. As a check, graph both the function and the tangent line.

The total revenue function (in dollars) for a blender is , where x is the number of units sold. What is the average rate of change in revenue R(x) as x increases from 20 to 30 units?

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.

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

Given , find the average rate of change of f (x) over the following pair of intervals using a calculator if desired: ​ and ​

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

For the function in this problem, find the instantaneous rate of change of the function at the given value. ​ ; ​

If the instantaneous rate of change of g(x) at is , write the equation of the line tangent to the graph of g(x) at ​

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

Find the indicated derivative. If , find . ​

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 . ​

If the revenue for a product (in dollars) is , and the average revenue per unit is , , find . ​

For the function given, find all x-values where the graph of is falling. ​ ​

The tangent line to the graph of at is shown. On the tangent line, P is the point of tangency and A is another point on the line. Find the instantaneous rate of change of at P. ​ ​

Suppose that the cost C in dollars of obtaining water that contains p percent impurities is given by . Find , if it exists. ​

Find the indicated derivative and simplify. ​ for ​

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. ​

Total revenue is in dollars and x is the number of units. Suppose that the total revenue function for a commodity is . Find , and tell what it represents. ​

Given the graph of in the figure, determine all x-values from F, G, H, or I for which the function is differentiable. ​ ​

Total revenue is in dollars and x is the number of units. Suppose that the total revenue function for a commodity is Find the marginal revenue function. ​

Write the equation of the tangent line to each curve at the indicated point. As a check, graph both the function and the tangent line. ​ ​

The total revenue function (in dollars) for a blender is , where x is the number of units sold. What is the average rate of change in revenue R(x) as x increases from 20 to 30 units? ​

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. ​ ​

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

Given , find the average rate of change of f (x) over the following pair of intervals using a calculator if desired: and

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

For the function in this problem, find the instantaneous rate of change of the function at the given value. ;

If the instantaneous rate of change of g(x) at is , write the equation of the line tangent to the graph of g(x) at

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

Find the indicated derivative. If , find .

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 .

If the revenue for a product (in dollars) is , and the average revenue per unit is , , find .

For the function given, find all x-values where the graph of is falling.

The tangent line to the graph of at is shown. On the tangent line, P is the point of tangency and A is another point on the line. Find the instantaneous rate of change of at P.

Suppose that the cost C in dollars of obtaining water that contains p percent impurities is given by . Find , if it exists.

Find the indicated derivative and simplify. for

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.

Total revenue is in dollars and x is the number of units. Suppose that the total revenue function for a commodity is . Find , and tell what it represents.

Given the graph of in the figure, determine all x-values from F, G, H, or I for which the function is differentiable.

Total revenue is in dollars and x is the number of units. Suppose that the total revenue function for a commodity is Find the marginal revenue function.

Write the equation of the tangent line to each curve at the indicated point. As a check, graph both the function and the tangent line.

The total revenue function (in dollars) for a blender is , where x is the number of units sold. What is the average rate of change in revenue R(x) as x increases from 20 to 30 units?

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.