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**Homework Homework Assignment #5 Read Section 5.6**

Page 341, Exercises: 1 – 19(Odd) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 341 1. Water flows into an empty reservoir at the rate of t gal/hr. What is the quantity of water in the reservoir after 5 hrs? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 341 3. A population of insects increases at a rate of t t2 insects/day. Find the insect population after 3 days, assuming that there are 35 insects at t = 0. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 341 5. A factory produces bicycles at a rate of t2 – t bicycles per week (t in weeks). How many bicycles were produced from day 8 to day 21? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 341 7. A cat falls from a tree (with zero initial velocity) at time t = 0. How far does the cat fall between t = 0.5 and t = 1 s? Use Galileo’s formula v(t) = –32t ft/s. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 341 Assume that a particle moves in a straight line with given velocity. Find the total displacement and total distance traveled over the time interval, and draw a motion diagram, with distance and time labels. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 341 Assume that a particle moves in a straight line with given velocity. Find the total displacement and total distance traveled over the time interval, and draw a motion diagram, with distance and time labels. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 341 13. The rate (in liters per minute) at which water drains from a tank is recorded at half-minute intervals. Use the average of the left- and right endpoint approximation to estimate the amount of water drained in the first 3 min. t 0.5 1.0 1.5 2.0 2.5 3.0 l/min 50 48 46 44 42 40 38 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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**Homework, Page 341 Rogawski Calculus**

Copyright © 2008 W. H. Freeman and Company

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Homework, Page 341 17. The traffic flow past a certain point on a highway is q(t) = 3, ,000t +300t2, where t is in hours and t = 0 is 8 AM. How many cars pass by during the time interval from 8 to 10 AM? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 341 19. To encourage manufacturers to reduce pollution, a carbon tax on each ton of CO2 released into the atmosphere has been proposed. To model the effects of such a tax, policymakers study the marginal cost of abatement B(x), defined as the cost of increasing CO2 reduction from x to x + 1 tons (in units of 10,000 tons – Figure 4). Which quantity is represented by Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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**Calculus, ET First Edition**

Jon Rogawski Calculus, ET First Edition Chapter 5: The Integral Section 5.6: Substitution Method Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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When differentiating functions, we sometimes need to use the Chain Rule. We will now cover the Substitution Method of integration which is the Chain Rule “in reverse.” Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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**The Substitution Method is formally stated in Theorem 1.**

Breaking down the integral as follows: We see that the antiderivative of f (u) du is F(u) + C Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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**Example, Page 349 Calculate du for the given function.**

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 349 Write the integral in terms of u and du. Then evaluate. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 349 Write the integral in terms of u and du. Then evaluate. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 349 Write the integral in terms of u and du. Then evaluate. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 349 Show that the integral is equal to a multiple of sin(u(x)) + C for an appropriate choice of u(x). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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**Example, Page 349 Evaluate the indefinite integral. Rogawski Calculus**

Copyright © 2008 W. H. Freeman and Company

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**Example, Page 349 Evaluate the indefinite integral. Rogawski Calculus**

Copyright © 2008 W. H. Freeman and Company

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**Example, Page 349 Evaluate the indefinite integral. Rogawski Calculus**

Copyright © 2008 W. H. Freeman and Company

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**Homework Homework Assignment #6 Review Section 5.6**

Page 349, Exercises: 1 – 57(EOO) Quiz next time Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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