1. Homework Homework Assignment #7 Read Section 2.8 Page 106, Exercises: 1 – 25 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

2. Homework, Page 106 1. Use the IVT to show that f (x) = x 3 + x takes on the value of 9 for some x in [1, 2].. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

3. Homework, Page 106 5. Show that cos x = x has a solution in the interval [0, 1]. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

4. Homework, Page 106 Use the IVT to prove each of the following statements. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

5. Homework, Page 106 Use the IVT to prove each of the following statements. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

6. Homework, Page 106 17. Carry out three steps of the Bisection Method for f (x) =2 x – x 3 as follows: (a) Show that f (x) has a zero in [1, 1.5]. (b) Show that f (x) has a zero in [1.25, 1.5]. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

7. Homework, Page 106 17. (c) Determine whether [1.25, 1.375] or [1.375, 1.5] contains a zero. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

8. Homework, Page 106 Draw the graph of a function f (x) on [0, 4] with the given properties. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

9. Homework, Page 106 25. Corollary 2 is not foolproof. Let f (x) = x 2 – 1 and explain why the corollary fails to detect the roots at x = ± 1 if [a, b] contains [1, –1]. 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 2: Limits Section 2.8: The Formal Definition of a Limit

11. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In Figure 1(A), we see that the value of y approaches 1 as x approaches 0. In Figure 1(B), we see that the value of |f (x) – 1| < 0.2, if – 1 < x < 1. In Figure 1(C), we see that the value of |f (x) – 1| < 0.004, if – 0.15 < x < 0.15. As we continue to narrow the gap around 0, we lessen the value of |f (x) – 1|, leading to the formal definition of a limit.

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

13. The formal definition of the limit of a function Courtesy of Tom Reardon: tom@tomreardon.com

15. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company For y = 8x + 3, the formal definition gives us the limit as x approaches 3 as follows: and as illustrated in Figure 2 below.

16. Example, Page 113 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

17. Example, Page 113 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

18. Example, Page 113 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

19. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company For the parabola y = x 2, we have a similar result.

20. Example, Page 113 12. Based on the information conveyed in the graph, find values of c, L, ε, and δ > 0 such that the following statement holds: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

21. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 4 illustrates how a limit does not exist at a jump discontinuity.

22. Example, Page 113 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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

24. Homework Homework Assignment #8 Review Section 2.8 Page 113, Exercises: 1 – 13 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company