Question 1

Use integration by parts to evaluate   . ​

Accepted Answer

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

Question 2

Find the producer's surplus for a product if its demand function is   and its supply function is   where p is in millions of dollars and x is the number of thousands of units. Round your answer to the nearest million dollars. ​

Accepted Answer

A)  53 million dollars 
B)  44 million dollars 
C)  40 million dollars 
D)  36 million dollars 
E)  47 million dollars 
A)  53 million dollars 
B)  44 million dollars 
C)  40 million dollars 
D)  36 million dollars 
E)  47 million dollars

Question 3

Equations are given whose graphs enclose a region. Find the area of the region. ​

Accepted Answer

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

Question 4

Suppose that the output of the machinery in a factory can be considered as a continuous income stream with an annual rate of flow at time t given by   in thousands of dollars per year. If the annual interest rate is 4% compounded continuously, find the capital value of the machinery. Round your answer to the nearest dollar. ​

Accepted Answer

A)  $4,406,250 
B)  $1,175,000 
C)  $1,958,333 
D)  $6,462,500 
E)  $2,937,500 
A)  $4,406,250 
B)  $1,175,000 
C)  $1,958,333 
D)  $6,462,500 
E)  $2,937,500

Question 5

The demand function for a product is   where p is in millions of dollars and x is the number of thousands of units. If the equilibrium price is $6 million, what is the consumer's surplus? Round your answer to the nearest million dollars. ​

Accepted Answer

A)  114 million dollars 
B)  149 million dollars 
C)  83 million dollars 
D)  88 million dollars 
E)  174 million dollars 
A)  114 million dollars 
B)  149 million dollars 
C)  83 million dollars 
D)  88 million dollars 
E)  174 million dollars

Question 6

If the marginal cost function for x units of a product is   dollars per unit, and if the fixed cost is $120, find the total cost function. ​

Accepted Answer

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

Question 7

Suppose that the Carter Car Service franchise finds that the income generated by its stores can be modeled by assuming that the income is a continuous stream with a monthly rate of flow at time t given by   (dollars per month). Find the total income from a Carter Car Service store for years 3 through 5 of operation and round to the nearest penny. ​

Accepted Answer

A)  $973,587.71 
B)  $2,307,500.23 
C)  $1,059,312.41 
D)  $1,684,306.74 
E)  $1,186,131.50 
A)  $973,587.71 
B)  $2,307,500.23 
C)  $1,059,312.41 
D)  $1,684,306.74 
E)  $1,186,131.50

Question 8

Suppose the Lorenz curve for the distribution of income of a certain country is given by   Find the Gini coefficient of income. Round your answer to three decimal places. ​

Accepted Answer

A)  2.632 
B)  2.896 
C)  4.264 
D)  5.632 
E)  0.264 
A)  2.632 
B)  2.896 
C)  4.264 
D)  5.632 
E)  0.264

Question 9

Find the average value of the given function over the given interval. ​

Accepted Answer

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

Question 10

Evaluate the integral   . ​

Accepted Answer

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

Question 11

The manufacturer of a medicine wants to test how a new 300-milligram capsule is released into the bloodstream. After a volunteer is given a capsule, blood samples are drawn every half-hour, and the number of milligrams of the drug in the bloodstream is calculated. The results obtained are as follows.  
Use the trapezoidal rule to approximate the average number of milligrams in the bloodstream during the first   hours and round your answer to two decimal places.
​

Accepted Answer

A)  433.60 mg 
B)  197.09 mg 
C)  123.18 mg 
D)  256.22 mg 
E)  335.05 mg 
A)  433.60 mg 
B)  197.09 mg 
C)  123.18 mg 
D)  256.22 mg 
E)  335.05 mg

Question 12

A continuous income stream has an annual rate of flow at time t given by   (dollars per year). Find the present value of this income stream for the next 9 years, if the money is worth 4% compounded continuously. Round to the nearest dollar. ​

Accepted Answer

A)  $47,664 
B)  $107,767 
C)  $178,262 
D)  $147,471 
E)  $81,028 
A)  $47,664 
B)  $107,767 
C)  $178,262 
D)  $147,471 
E)  $81,028

Question 13

Use the function   from   to   and n equal subintervals with the function evaluated at the left-hand endpoints of each subinterval. Find   by using the formula for the sum of the areas of the n rectangles (call this S). ​

Accepted Answer

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

Question 14

For what value of c does ?   ​

Accepted Answer

A)  16 
B)  384 
C)  ​-8 
D)  8 
E)  -16 
A)  16 
B)  384 
C)  ​-8 
D)  8 
E)  -16

Question 15

Suppose that the income from a slot machine in a casino flows continuously at a rate of   , where t is the time in hours since the casino opened. The total income during the first 8 hours is given by   . Find the average income over the first 8 hours. ​

Accepted Answer

A)  $289.02 
B)  $313.24 
C)  $728.90 
D)  $560.69 
E)  $953.18 
A)  $289.02 
B)  $313.24 
C)  $728.90 
D)  $560.69 
E)  $953.18

Question 16

Find the area of the shaded region. ​   ​

Accepted Answer

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

Question 17

The Carter Car Service franchise has a continuous income stream with a monthly rate of flow modeled by   (dollars per month). Find the average flow of income over years 4 to 5. Round your answer to the nearest cent. ​

Accepted Answer

A)  $165,371.54 
B)  $142,489.95 
C)  $391,947.09 
D)  $226,559.01 
E)  $159,548.60 
A)  $165,371.54 
B)  $142,489.95 
C)  $391,947.09 
D)  $226,559.01 
E)  $159,548.60

Question 18

Find the area of the shaded region. ​
​

Accepted Answer

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

Question 19

Evaluate the integral   by integration. Round your answer to two decimal places. ​

Accepted Answer

A)  4.00 
B)  9.79 
C)  0.71 
D)  7.34 
E)  3.77 
A)  4.00 
B)  9.79 
C)  0.71 
D)  7.34 
E)  3.77

Question 20

Suppose that the rate at which a nuclear power plant produces radioactive waste is proportional to the number of years it has been operating, according to   in pounds per year. Suppose also that the waste decays exponentially at a rate of 9% per year. Then the amount of radioactive waste that will accumulate in b years is given by   . Evaluate this integral. ​

Accepted Answer

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

Use integration by parts to evaluate .

Find the producer's surplus for a product if its demand function is and its supply function is where p is in millions of dollars and x is the number of thousands of units. Round your answer to the nearest million dollars.

Equations are given whose graphs enclose a region. Find the area of the region.

The demand function for a product is where p is in millions of dollars and x is the number of thousands of units. If the equilibrium price is $6 million, what is the consumer's surplus? Round your answer to the nearest million dollars.

If the marginal cost function for x units of a product is dollars per unit, and if the fixed cost is $120, find the total cost function.

Suppose the Lorenz curve for the distribution of income of a certain country is given by Find the Gini coefficient of income. Round your answer to three decimal places.

Find the average value of the given function over the given interval.

Evaluate the integral .

A continuous income stream has an annual rate of flow at time t given by (dollars per year). Find the present value of this income stream for the next 9 years, if the money is worth 4% compounded continuously. Round to the nearest dollar.

Use the function from to and n equal subintervals with the function evaluated at the left-hand endpoints of each subinterval. Find by using the formula for the sum of the areas of the n rectangles (call this S).

For what value of c does ?

Suppose that the income from a slot machine in a casino flows continuously at a rate of , where t is the time in hours since the casino opened. The total income during the first 8 hours is given by . Find the average income over the first 8 hours.

Find the area of the shaded region.

The Carter Car Service franchise has a continuous income stream with a monthly rate of flow modeled by (dollars per month). Find the average flow of income over years 4 to 5. Round your answer to the nearest cent.

Find the area of the shaded region.

Evaluate the integral by integration. Round your answer to two decimal places.

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Derivatives Continued

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Exam 13: Definite Integrals - Techniques

Use integration by parts to evaluate . ​

Find the producer's surplus for a product if its demand function is and its supply function is where p is in millions of dollars and x is the number of thousands of units. Round your answer to the nearest million dollars. ​

Equations are given whose graphs enclose a region. Find the area of the region. ​

The demand function for a product is where p is in millions of dollars and x is the number of thousands of units. If the equilibrium price is $6 million, what is the consumer's surplus? Round your answer to the nearest million dollars. ​

If the marginal cost function for x units of a product is dollars per unit, and if the fixed cost is $120, find the total cost function. ​

Suppose the Lorenz curve for the distribution of income of a certain country is given by Find the Gini coefficient of income. Round your answer to three decimal places. ​

Find the average value of the given function over the given interval. ​

Evaluate the integral . ​

A continuous income stream has an annual rate of flow at time t given by (dollars per year). Find the present value of this income stream for the next 9 years, if the money is worth 4% compounded continuously. Round to the nearest dollar. ​

Use the function from to and n equal subintervals with the function evaluated at the left-hand endpoints of each subinterval. Find by using the formula for the sum of the areas of the n rectangles (call this S). ​

For what value of c does ? ​

Suppose that the income from a slot machine in a casino flows continuously at a rate of , where t is the time in hours since the casino opened. The total income during the first 8 hours is given by . Find the average income over the first 8 hours. ​

Find the area of the shaded region. ​ ​

The Carter Car Service franchise has a continuous income stream with a monthly rate of flow modeled by (dollars per month). Find the average flow of income over years 4 to 5. Round your answer to the nearest cent. ​

Find the area of the shaded region. ​ ​

Evaluate the integral by integration. Round your answer to two decimal places. ​

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

Derivatives Continued

Indefinite Integrals

A: Definite Integrals - Techniques

Functions of Two or More Variables

Algebraic Concepts 2

Algebraic Concepts 3

Algebraic Concepts 4

Algebraic Concepts 5

Filters

Use integration by parts to evaluate .

Find the producer's surplus for a product if its demand function is and its supply function is where p is in millions of dollars and x is the number of thousands of units. Round your answer to the nearest million dollars.

Equations are given whose graphs enclose a region. Find the area of the region.

The demand function for a product is where p is in millions of dollars and x is the number of thousands of units. If the equilibrium price is $6 million, what is the consumer's surplus? Round your answer to the nearest million dollars.

If the marginal cost function for x units of a product is dollars per unit, and if the fixed cost is $120, find the total cost function.

Suppose the Lorenz curve for the distribution of income of a certain country is given by Find the Gini coefficient of income. Round your answer to three decimal places.

Find the average value of the given function over the given interval.

Evaluate the integral .

A continuous income stream has an annual rate of flow at time t given by (dollars per year). Find the present value of this income stream for the next 9 years, if the money is worth 4% compounded continuously. Round to the nearest dollar.

Use the function from to and n equal subintervals with the function evaluated at the left-hand endpoints of each subinterval. Find by using the formula for the sum of the areas of the n rectangles (call this S).

For what value of c does ?

Suppose that the income from a slot machine in a casino flows continuously at a rate of , where t is the time in hours since the casino opened. The total income during the first 8 hours is given by . Find the average income over the first 8 hours.

Find the area of the shaded region.

The Carter Car Service franchise has a continuous income stream with a monthly rate of flow modeled by (dollars per month). Find the average flow of income over years 4 to 5. Round your answer to the nearest cent.

Find the area of the shaded region.

Evaluate the integral by integration. Round your answer to two decimal places.