Download presentation

Presentation is loading. Please wait.

Published byBritton Gilbert Modified over 4 years ago

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

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google