1. Homework Homework Assignment #30 Read Section 4.9 Page 282, Exercises: 1 – 13(Odd) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

2. Homework, Page 282 Use Newton’s Method with the given function and initial value x o to calculate x 1, x 2, and x 3. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

3. Homework, Page 282 Use Newton’s Method with the given function and initial value x o to calculate x 1, x 2, and x 3. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

4. Homework, Page 282 5. Use Figure 6 to choose an initial guess x o to the unique real root of x 3 + 2x +5 = 0. Then compute the first three iterates of Newton’s Method. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

5. Homework, Page 282 7. Use Newton’s Method to find two solutions of e x = 5x to three decimal places (Figure 7). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

6. Homework, Page 282 7. Use Newton’s Method to find two solutions of e x = 5x to three decimal places (Figure 7). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

7. Homework, Page 282 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

8. Homework, Page 282 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

9. Homework, Page 282 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

10. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Jon Rogawski Calculus, ET First Edition Chapter 4: Applications of the Derivative Section 4.9: Antiderivatives

11. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In some cases, we may know the derivative and wish to find the function itself. A function F (x) whose derivative is f (x) is called an antiderivative of f (x). Theorem 1 gives us a method of finding the general antiderivative of a function, namely: Theorem 1 is proven as follows:

12. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 1 graphically demonstrates how two functions (F (x) and F (x) + C) differing only by a constant C, have the same slope for a given value of x and, thus, the same derivative.

13. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The process of antidifferentiation is denoted by the Leibniz symbol ∫ which specifies finding the indefinite integral of a function. Just as we have rules for differentiation, we also have rules for integration, of which the first is:

14. Example, Page 292 Find the general antiderivative of f (x) and check your answer by differentiating. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

15. Example, Page 292 Find the general antiderivative of f (x) and check your answer by differentiating. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

16. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In Figure 2, we see that the slope of the graph of y = ln |x| is x –1 for all x. This leads to Theorem 3, which covers the exception to Theorem 2.

17. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The linearity rules for differentiation have their counterparts for integration as noted in Theorem 4.

18. Example, Page 292 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

19. Example, Page 292 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

20. Example, Page 292 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

21. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Recalling derivatives of the basic trigonometric functions, we can recognize the basic trigonometric integrals.

22. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Recalling the Chain Rule, we have the following trigonometric integrals.

23. Example, Page 292 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

24. Example, Page 292 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

25. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Remembering that the exponential function is its own derivative, we see how the indefinite integral of e x is e x + C. Applying the Chain rule in reverse to the exponential function, we obtain:

26. Example, Page 292 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

27. Homework Homework Assignment #31 Review Section 4.9 Page 292, Exercises: 1 – 41(EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company