copyright © by houghton mifflin company, all rights reserved. calculus concepts 2/e latorre,...

Copyright © by Houghton Mifflin Company, All rights reserved.Copyright © by Houghton Mifflin Company, All rights reserved.

Calculus Concepts 2/eCalculus Concepts 2/eLaTorre, Kenelly, Fetta, Harris, and CarpenterLaTorre, Kenelly, Fetta, Harris, and Carpenter

Chapter 7Chapter 7Analyzing Accumulated Change: Analyzing Accumulated Change:

Integrals in ActionIntegrals in Action


Chapter 7 Key ConceptsChapter 7 Key Concepts• Integrals Involving Differences Integrals Involving Differences

• Future and Present ValuesFuture and Present Values

• Using Integrals in EconomicsUsing Integrals in Economics

• Average ValuesAverage Values

• Probability Density FunctionsProbability Density Functions


Integrals Involving DifferencesIntegrals Involving Differences• Area Between Two CurvesArea Between Two Curves

– If the graph of f lies above the graph of g from a to If the graph of f lies above the graph of g from a to b, then the area of the region between the two b, then the area of the region between the two curves from a to b is given bycurves from a to b is given by

b

adx)x(g)x(f

b

adx)x(g)x(f

b

adx)x(g)x(f

b

adx)x(g)x(f

• Difference of Two Accumulated ChangesDifference of Two Accumulated Changes– If f and g are two continuous rate-of-change If f and g are two continuous rate-of-change

functions, then the difference between the functions, then the difference between the accumulated change of f from a to b and the accumulated change of f from a to b and the accumulated change of g from a to b isaccumulated change of g from a to b is


Integrals Involving Differences: ExampleIntegrals Involving Differences: ExampleThe rate of change of sales accumulated since The rate of change of sales accumulated since 1989 by a European car manufacturer is given by 1989 by a European car manufacturer is given by s(t) = 3.7(1.19376)s(t) = 3.7(1.19376)tt million dollars per year. million dollars per year.

The rate of change of sales accumulated since The rate of change of sales accumulated since 1989 by an American car manufacturer is given by 1989 by an American car manufacturer is given by a(t) = 0.04t a(t) = 0.04t33 - 0.54t - 0.54t22 + 2.5t + 4.47 million dollars + 2.5t + 4.47 million dollars per year. per year.

By how much did the amount of the accumulated By how much did the amount of the accumulated sales differ from the end of 1995 through 2003?sales differ from the end of 1995 through 2003?


Integrals Involving Differences: ExampleIntegrals Involving Differences: Examples(t) = 3.7(1.19376)s(t) = 3.7(1.19376)tt million dollars per year million dollars per yeara(t) = 0.04ta(t) = 0.04t33 - 0.54t - 0.54t22 + 2.5t + 4.47 million dollars + 2.5t + 4.47 million dollars per yearper year

14

6dt)t(a)t(sRofArea

14

6dt)t(a)t(sRofArea

14

6

234t

t47.4t25.1t18.0t01.019376.1ln

)19376.1(7.3

14

6

234t

t47.4t25.1t18.0t01.019376.1ln

)19376.1(7.3

dollarsmillion9660.36 dollarsmillion9660.36


Integrals Involving Differences: Exercise 7.1 #3Integrals Involving Differences: Exercise 7.1 #3Sketch the graphs of the functions f and g on the Sketch the graphs of the functions f and g on the same axes, shade the region between a and b, same axes, shade the region between a and b, and calculate the area of the shaded region.and calculate the area of the shaded region.f(x) = xf(x) = x22 - 4x + 10 - 4x + 10 a = 1a = 1g(x) = 2xg(x) = 2x22 - 12x + 14 - 12x + 14 b = 7b = 7

7

1dx)x(g)x(fArea

7

1dx)x(g)x(fArea

7

1

22 dx14x12x210x4x 7

1

22 dx14x12x210x4x

54x4x4xdx4x8x7

1

2331

7

1

2 54x4x4xdx4x8x7

1

2331

7

1

2


Future and Present ValuesFuture and Present Values• Future Value of a Continuous Income StreamFuture Value of a Continuous Income Stream

– Suppose that an income stream flows Suppose that an income stream flows continuously into an interest-bearing checking continuously into an interest-bearing checking account at the rate of R(t) dollars per year where t account at the rate of R(t) dollars per year where t is measured in years and the account earns is measured in years and the account earns interest at the annual rate of 100r% compounded interest at the annual rate of 100r% compounded continuously. The future value of the account at continuously. The future value of the account at the end of T years isthe end of T years is

dollarsdte)t(RValueFutureT

0

)tT(r dollarsdte)t(RValueFutureT

0

)tT(r


Future and Present ValuesFuture and Present Values• Present Value of a Continuous Income StreamPresent Value of a Continuous Income Stream

– Suppose that an income stream flows continuously Suppose that an income stream flows continuously into an interest-bearing checking account at the rate into an interest-bearing checking account at the rate of R(t) dollars per year where t is measured in years of R(t) dollars per year where t is measured in years and the account earns interest at the annual rate of and the account earns interest at the annual rate of 100r% compounded continuously. The present 100r% compounded continuously. The present value of the account isvalue of the account is

dollarsdte)t(RValueesentPrT

0

rt dollarsdte)t(RValueesentPrT

0

rt


Future and Present Values: ExampleFuture and Present Values: ExampleAn investor is investing $3.3 million a year in an An investor is investing $3.3 million a year in an account returning 9.4% APR. Assuming a account returning 9.4% APR. Assuming a continuous income stream and continuous continuous income stream and continuous compounding of interest, how much will these compounding of interest, how much will these investments be worth 10 years from now?investments be worth 10 years from now?

dollarsdte)t(RValueFutureT

0

)tT(r dollarsdte)t(RValueFutureT

0

)tT(r

10

0

)t10(094.0 dte3.3 10

0

)t10(094.0 dte3.3

)0(094.094.0

)10(094.094.0

e094.0

e3.3e

094.0

e3.3

)0(094.0

94.0)10(094.0

94.0

e094.0

e3.3e

094.0

e3.3

million8.54$ million8.54$


Future and Present Values: Exercise 7.1 #5Future and Present Values: Exercise 7.1 #5For the year ending December 31, 1998, the For the year ending December 31, 1998, the revenue for the Sara Lee Corporation was $20.011 revenue for the Sara Lee Corporation was $20.011 billion. Assuming that Sara Lee’s revenue will billion. Assuming that Sara Lee’s revenue will increase by 5% per year and that beginning on increase by 5% per year and that beginning on January 1, 1999, 12.5% of the revenue is invested January 1, 1999, 12.5% of the revenue is invested each year (continuously) at an APR of 9% each year (continuously) at an APR of 9% compounded continuously. What is the future value compounded continuously. What is the future value of the investment at the end of 2006?of the investment at the end of 2006?

dollarsbilliondte)125.0()05.1(011.20FV8

0

)t8(09.0t dollarsbilliondte)125.0()05.1(011.20FV8

0

)t8(09.0t

dollarsbilliondtee)05.1(501.28

0

t09.072.0t dollarsbilliondtee)05.1(501.28

0

t09.072.0t

dollarsbillion35dx)959628.0(139.58

0

x dollarsbillion35dx)959628.0(139.58

0

x


Using Integrals in EconomicsUsing Integrals in Economics• Consumer’s Willingness and Ability to SpendConsumer’s Willingness and Ability to Spend

– For a continuous demand function q = D(p), the For a continuous demand function q = D(p), the maximum amount that consumers are willing maximum amount that consumers are willing and able to spend for a certain quantity qand able to spend for a certain quantity q00 of of

goods or services is the area of the shaded goods or services is the area of the shaded region. region.


Using Integrals in EconomicsUsing Integrals in Economics• Consumer’s Willingness and Ability to SpendConsumer’s Willingness and Ability to Spend

P

p000

dp)p(Dqp P

p000

dp)p(Dqp

– pp0 0 is the market price at which qis the market price at which q00 units are in units are in

demand and P is the price above which demand and P is the price above which consumers will purchase none of the goods or consumers will purchase none of the goods or services. This area is calculated asservices. This area is calculated as


Using Integrals in EconomicsUsing Integrals in Economics• Producer's Willingness and Ability to ReceiveProducer's Willingness and Ability to Receive

– For a continuous supply function q = S(p), the For a continuous supply function q = S(p), the minimum amount that producers are willing and minimum amount that producers are willing and able to received for a certain quantity qable to received for a certain quantity q00 of of

goods or services is the area of the shaded goods or services is the area of the shaded region below.region below.


Using Integrals in EconomicsUsing Integrals in Economics• Producer's Willingness and Ability to ReceiveProducer's Willingness and Ability to Receive

– pp00 is the market price at which q is the market price at which q00 units are units are

supplied and psupplied and p11 is the shutdown price. (If there is the shutdown price. (If there

is no shutdown price, then pis no shutdown price, then p11 = 0.) This area is = 0.) This area is

calculated as calculated as 0

1

p

p 001 dp)p(Sqqp 0

1

p

p 001 dp)p(Sqqp


Using Integrals in Economics: ExampleUsing Integrals in Economics: ExampleSuppose the function for the average weekly supply Suppose the function for the average weekly supply of a certain brand of cellular phone can be modeled of a certain brand of cellular phone can be modeled by the following equation where p is the market price by the following equation where p is the market price in dollars per phone.in dollars per phone.

What is the least amount that producers are willing What is the least amount that producers are willing and able to receive for the quantity of phones that and able to receive for the quantity of phones that corresponds to a market price of $45.95? corresponds to a market price of $45.95?

15pwhen150p38.9p047.0

15pwhenphones0)p(S

2

15pwhen150p38.9p047.0

15pwhenphones0)p(S

2

95.45

15dp)p(S)95.45(S)95.45(S15Minimum

95.45

15dp)p(S)95.45(S)95.45(S15Minimum

53.300,16$dp)p(S247.680)247.680(1595.45

15 53.300,16$dp)p(S247.680)247.680(15

95.45

15


Integrals in Economics: Exercise 7.3 #9Integrals in Economics: Exercise 7.3 #9The willingness of answering machine producers to The willingness of answering machine producers to supply can be modeled by the following function supply can be modeled by the following function where S(p) is in thousands of machines:where S(p) is in thousands of machines:

How many machines will the producers supply if the How many machines will the producers supply if the market price is $40?market price is $40?

20pif60p2p024.0

20pif0)p(S

2

20pif60p2p024.0

20pif0)p(S

2

40

20dp)p(S)40(S)40(S20Minimum

40

20dp)p(S)40(S)40(S20Minimum

40

20

2 dp)60p2p024.0(40.18)40.18(20 40

20

2 dp)60p2p024.0(40.18)40.18(20

40

20

2 dp)60.41p2p024.0368 40

20

2 dp)60.41p2p024.0368

machinesthousand288 machinesthousand288


Average ValuesAverage Values• If y = f(x) is a smooth, continuous function If y = f(x) is a smooth, continuous function

from a to b, then the average value of f(x) from a to b, then the average value of f(x) from a to b is from a to b is

• If y = f '(x) is a smooth, continuous rate-of-If y = f '(x) is a smooth, continuous rate-of-change function from a to b, then the average change function from a to b, then the average value of f '(x) from a to b is value of f '(x) from a to b is

ab

dx)x(fb

a

ab

dx)x(fb

a

ab

)a(f)b(f

ab

dx)x(fb

a

ab

)a(f)b(f

ab

dx)x(fb

a


Average Values: ExampleAverage Values: Example

yearperpeoplethousand4.32205210

dt)t(p210

205

yearperpeoplethousand4.32205210

dt)t(p210

205

The South Carolina population growth rate can be The South Carolina population growth rate can be modeled as p'(t) = 0.1552t + 0.223 thousand people modeled as p'(t) = 0.1552t + 0.223 thousand people per year where t is the number of years since 1790. per year where t is the number of years since 1790. The population in 1990 was 3486 thousand people. The population in 1990 was 3486 thousand people. What was the average rate of change in the What was the average rate of change in the population and what was the average population population and what was the average population from 1995 through 2000?from 1995 through 2000?

peoplethousand3725205210

dt)t(p210

205

peoplethousand3725205210

dt)t(p210

205


Average Values: Exercise 7.4 #7Average Values: Exercise 7.4 #7

22

1

22

1

21

5572.3967x6118.100

122

dx)x(n

22

1

22

1

21

5572.3967x6118.100

122

dx)x(n

The number of general-aviation aircraft accidents The number of general-aviation aircraft accidents from 1975 through 1997 can be modeled by from 1975 through 1997 can be modeled by

n(x) = -100.6118x + 3967.5572 accidentsn(x) = -100.6118x + 3967.5572 accidentswhere x is the number of years since 1975. where x is the number of years since 1975. Calculate the average rate of change in the yearly Calculate the average rate of change in the yearly number of accidents from 1976 through 1997.number of accidents from 1976 through 1997.

yearperaccidents6118.10021

85.2112

yearperaccidents6118.100

21

85.2112


Probability Density FunctionsProbability Density Functions• A probability density function y = f(x) for a A probability density function y = f(x) for a

random variable x is a continuous function or random variable x is a continuous function or piecewise continuous function such thatpiecewise continuous function such that

1dx)x(f.2

and,xnumberrealeachfor0)x(f.1

1dx)x(f.2

and,xnumberrealeachfor0)x(f.1

• The probability that a value of x lies in an The probability that a value of x lies in an interval with endpoints a and b, where a interval with endpoints a and b, where a b, b, is given byis given by

b

adx)x(f)bxa(P

b

adx)x(f)bxa(P


There is a 16.75% chance the recovery will occur There is a 16.75% chance the recovery will occur between 42 and 48 minutes.between 42 and 48 minutes.

Probability Density Functions: ExampleProbability Density Functions: ExampleThe proportion of patients who recover from mild The proportion of patients who recover from mild dehydration x hours after receiving treatment is dehydration x hours after receiving treatment is given bygiven by

1xor0xif0

1x0ifx12x12)x(f

32

1xor0xif0

1x0ifx12x12)x(f

32

Given that f is a probability density function, what is Given that f is a probability density function, what is the probability that the recovery time is between 42 the probability that the recovery time is between 42 and 48 minutes?and 48 minutes?

8.0

7.0

32 dxx12x12)8.0x7.0(P 8.0

7.0

32 dxx12x12)8.0x7.0(P

1675.0)x3x4(8.0

7.0

43 1675.0)x3x4(8.0

7.0

43


Probability Density: Exercise 7.5 #11Probability Density: Exercise 7.5 #11The manufacturer of a board game believes that the The manufacturer of a board game believes that the time it takes a 8 - 10 year-old child to learn the rules time it takes a 8 - 10 year-old child to learn the rules of its game has the probability density functionof its game has the probability density function

4tif0

4t0if)tt4()t(P

2323

4tif0

4t0if)tt4()t(P

2323

where t is time measured in minutes. Find and where t is time measured in minutes. Find and interpret P(0 interpret P(0 t t 1.5).1.5).

There is a 31.64% chance it will take 1.5 minutes There is a 31.64% chance it will take 1.5 minutes or less.or less.

5.1

0

2323 dt)tt4()5.1t0(P

5.1

0

2323 dt)tt4()5.1t0(P

3164.0)tt5.1

0

33212

163 3164.0)tt

5.1

0

33212

163

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