1. Homework Homework Assignment #8 Read Section 5.8 Page 355, Exercises: 1 – 69(EOO) Quiz next time Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

2. Example, Page 355 Evaluate the definite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

3. Example, Page 355 Evaluate the definite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

4. Example, Page 355 Evaluate the definite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

5. Example, Page 355 Evaluate the definite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

6. Example, Page 355 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

7. Example, Page 355 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

8. Example, Page 355 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

9. Example, Page 355 Evaluate the definite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

10. Example, Page 355 Evaluate the integral using methods covered so far. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

11. Example, Page 355 Evaluate the integral using methods covered so far. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

12. Example, Page 355 Evaluate the integral using methods covered so far. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

13. Example, Page 355 Evaluate the integral using methods covered so far. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

14. Example, Page 355 Evaluate the integral using methods covered so far. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

15. Example, Page 355 Evaluate the integral using methods covered so far. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

16. Example, Page 355 Evaluate the integral using methods covered so far. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

17. Example, Page 355 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

18. Example, Page 355 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

19. Example, Page 355 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

20. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Chapter 5: The Integral Section 5.8: Exponential Growth and Decay Jon Rogawski Calculus, ET First Edition

21. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company There are many examples in biology, physics, and economics where “populations “ grow or decay at exponential rates. The general formula describing this growth or decay is: If k > 0, then the population is growing and if k < 0, the population is decaying.

22. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The e. coli bacteria shown in Figure 1 multiply exponentially. That is why, within hours of ingesting contaminated food or drink, a person becomes severely ill from this bacteria. As can be extrapolated from the graph in Figure 2, in less than 24 hours, an initial population of 1,000 bacteria would have grown to over 1,600,000, more than enough to affect the healthiest person.

23. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If a quantity is changing in proportion to the amount currently present, then it is increasing/decreasing exponentially

24. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The rate a which penicillin leaves the bloodstream is proportional to the amount present. If an initial amount of 450 mg is administered and 50 mg remains after seven hours, what is the decay constant? At what time after the injection, did 200 mg remain?

25. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company A frequently asked question is: “When will this process produce twice the original amount?” This question can be answered by solving 2 = 1 e kt for t.

26. Example, Page 366 2.A quantity P obeys exponential growth law P = e 5t (t in years). (a) At what time t is P = 10? (b) At what time t is P = 20? (c) What is the doubling time for P? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

27. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 4 shows a graph of the number of bachelor’s degrees awarded in physics each year from 1955 to about 1978. The figure is marked to emphasize the doubling that took place about every seven years.

28. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If the number of microscopic plants shown in Figure 5 doubles every 30 hours, what is the k value we should use?

29. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In the case of exponential decay, the half-life of the substance is calculated by using: Archeologists use the half-life of carbon-14 to date organic materials. The process was developed following World War II.