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Copyright © by Houghton Mifflin Company, All rights reserved. Chapter 7 Key Concepts Integrals Involving DifferencesIntegrals Involving DifferencesIntegrals Involving DifferencesIntegrals Involving Differences Future and Present ValuesFuture and Present ValuesFuture and Present ValuesFuture and Present Values Using Integrals in EconomicsUsing Integrals in EconomicsUsing Integrals in EconomicsUsing Integrals in Economics Average ValuesAverage ValuesAverage ValuesAverage Values Probability Density FunctionsProbability Density FunctionsProbability Density FunctionsProbability Density Functions

Copyright © by Houghton Mifflin Company, All rights reserved. Integrals Involving Differences Area Between Two CurvesArea Between Two Curves –If the graph of f lies above the graph of g from a to b, then the area of the region between the two curves from a to b is given by Difference of Two Accumulated ChangesDifference of Two Accumulated Changes –If f and g are two continuous rate-of-change functions, then the difference between the accumulated change of f from a to b and the accumulated change of g from a to b is

Copyright © by Houghton Mifflin Company, All rights reserved. Integrals Involving Differences: Example The rate of change of sales accumulated since 1989 by a European car manufacturer is given by s(t) = 3.7(1.19376) t million dollars per year. The rate of change of sales accumulated since 1989 by an American car manufacturer is given by a(t) = 0.04t 3 - 0.54t 2 + 2.5t + 4.47 million dollars per year. By how much did the amount of the accumulated sales differ from the end of 1995 through 2003?

Copyright © by Houghton Mifflin Company, All rights reserved. Integrals Involving Differences: Example s(t) = 3.7(1.19376) t million dollars per year a(t) = 0.04t 3 - 0.54t 2 + 2.5t + 4.47 million dollars per year

Copyright © by Houghton Mifflin Company, All rights reserved. Integrals Involving Differences: Exercise 7.1 #3 Sketch the graphs of the functions f and g on the same axes, shade the region between a and b, and calculate the area of the shaded region. f(x) = x 2 - 4x + 10a = 1 g(x) = 2x 2 - 12x + 14b = 7

Copyright © by Houghton Mifflin Company, All rights reserved. Future and Present Values Future Value of a Continuous Income StreamFuture Value of a Continuous Income Stream –Suppose that an income stream flows continuously into an interest-bearing checking account at the rate of R(t) dollars per year where t is measured in years and the account earns interest at the annual rate of 100r% compounded continuously. The future value of the account at the end of T years is

Copyright © by Houghton Mifflin Company, All rights reserved. Future and Present Values Present Value of a Continuous Income StreamPresent Value of a Continuous Income Stream –Suppose that an income stream flows continuously into an interest-bearing checking account at the rate of R(t) dollars per year where t is measured in years and the account earns interest at the annual rate of 100r% compounded continuously. The present value of the account is

Copyright © by Houghton Mifflin Company, All rights reserved. Future and Present Values: Example An investor is investing \$3.3 million a year in an account returning 9.4% APR. Assuming a continuous income stream and continuous compounding of interest, how much will these investments be worth 10 years from now?

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

Copyright © by Houghton Mifflin Company, All rights reserved. Using 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 maximum amount that consumers are willing and able to spend for a certain quantity q 0 of goods or services is the area of the shaded region.

Copyright © by Houghton Mifflin Company, All rights reserved. Using Integrals in Economics Consumer’s Willingness and Ability to SpendConsumer’s Willingness and Ability to Spend –p 0 is the market price at which q 0 units are in demand and P is the price above which consumers will purchase none of the goods or services. This area is calculated as

Copyright © by Houghton Mifflin Company, All rights reserved. Using 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 minimum amount that producers are willing and able to received for a certain quantity q 0 of goods or services is the area of the shaded region below.

Copyright © by Houghton Mifflin Company, All rights reserved. Using Integrals in Economics Producer's Willingness and Ability to ReceiveProducer's Willingness and Ability to Receive –p 0 is the market price at which q 0 units are supplied and p 1 is the shutdown price. (If there is no shutdown price, then p 1 = 0.) This area is calculated as

Copyright © by Houghton Mifflin Company, All rights reserved. Using Integrals in Economics: Example Suppose the function for the average weekly supply of a certain brand of cellular phone can be modeled by the following equation where p is the market price in dollars per phone. What is the least amount that producers are willing and able to receive for the quantity of phones that corresponds to a market price of \$45.95?

Copyright © by Houghton Mifflin Company, All rights reserved. Integrals in Economics: Exercise 7.3 #9 The willingness of answering machine producers to supply can be modeled by the following function where S(p) is in thousands of machines: How many machines will the producers supply if the market price is \$40?

Copyright © by Houghton Mifflin Company, All rights reserved. Average Values If y = f(x) is a smooth, continuous function from a to b, then the average value of f(x) from a to b isIf y = f(x) is a smooth, continuous function from a to b, then the average value of f(x) from a to b is If y = f '(x) is a smooth, continuous rate-of- change function from a to b, then the average value of f '(x) from a to b isIf y = f '(x) is a smooth, continuous rate-of- change function from a to b, then the average value of f '(x) from a to b is

Copyright © by Houghton Mifflin Company, All rights reserved. Average Values: Example The South Carolina population growth rate can be modeled as p'(t) = 0.1552t + 0.223 thousand people per year where t is the number of years since 1790. The population in 1990 was 3486 thousand people. What was the average rate of change in the population and what was the average population from 1995 through 2000?

Copyright © by Houghton Mifflin Company, All rights reserved. Average Values: Exercise 7.4 #7 The number of general-aviation aircraft accidents from 1975 through 1997 can be modeled by n(x) = -100.6118x + 3967.5572 accidents where x is the number of years since 1975. Calculate the average rate of change in the yearly number of accidents from 1976 through 1997.

Copyright © by Houghton Mifflin Company, All rights reserved. Probability Density Functions A probability density function y = f(x) for a random variable x is a continuous function or piecewise continuous function such thatA probability density function y = f(x) for a random variable x is a continuous function or piecewise continuous function such that The probability that a value of x lies in an interval with endpoints a and b, where a  b, is given byThe probability that a value of x lies in an interval with endpoints a and b, where a  b, is given by

Copyright © by Houghton Mifflin Company, All rights reserved. There is a 16.75% chance the recovery will occur between 42 and 48 minutes. Probability Density Functions: Example The proportion of patients who recover from mild dehydration x hours after receiving treatment is given by Given that f is a probability density function, what is the probability that the recovery time is between 42 and 48 minutes?

Copyright © by Houghton Mifflin Company, All rights reserved. Probability Density: Exercise 7.5 #11 The manufacturer of a board game believes that the time it takes a 8 - 10 year-old child to learn the rules of its game has the probability density function where t is time measured in minutes. Find and interpret P(0  t  1.5). There is a 31.64% chance it will take 1.5 minutes or less.