Homework Homework Assignment #5 Read Section 5.6 Page 341, Exercises: 1 – 19(Odd) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 341 1. Water flows into an empty reservoir at the rate of 3000 + 5t gal/hr. What is the quantity of water in the reservoir after 5 hrs? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 341 3. A population of insects increases at a rate of 200 + 10t + 0.25t2 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

Homework, Page 341 5. A factory produces bicycles at a rate of 95 + 0.1t2 – 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

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

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

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

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

Homework, Page 341 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 341 17. The traffic flow past a certain point on a highway is q(t) = 3,000 + 2,000t +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

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

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

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

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

Example, Page 349 Calculate du for the given function. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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

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

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

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

Example, Page 349 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 349 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 349 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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