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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

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

10. 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

11. 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

12. 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

13. 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

14. 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

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

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

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

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

19. 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

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

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

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

23. 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