By the way, recall that the due date for Homework 8 is now April 26.
Midterm 2 made me realize that for most of you time is an issue when doing problems involving ideas that are entirely new to you. Therefore, it will be better to have a take-home final. I'm back from a conference on May 22, so here's a suggestion: I would email you the exam on May 22 in the evening and you would have until May 25. Let me know how that sounds.
Sec. 7.2, ex. 2, 19, 21, 22;
Sec. 7.3, ex. 8, 13, 14, 22;
Sec. 8.1, ex. 20, 21, 22, 24;
Sec. 8.2, ex. 8, 9, 14, 16.
Btw, you are free to use all the results proved in Chapters 7 and 8.
When you are done reviewing, try this sample midterm 2. (Warning: I intentionally made it challenging.)
Our final exam is officially scheduled for May 19. However, I will be out of town attending a conference that day so we need to move it. I think we are obliged to use the make-up day, May 25. I would like to have the final from 9:45 to noon that day. Let me know if this doesn't work for you.
Hope you had an enjoyable spring break!
Sec. 6.1, ex. 7, 12, 14;
Sec. 6.2, ex. 7, 8, 12, 17;
Sec. 6.3, ex. 1, 5, 6;
Sec. 6.4, ex. 2, 4, 9, 15.
You don't need to write detailed solutions or turn them in, but you should definitely try to solve them all. It's a good idea to review homework problems as well. I'll post solutions to Homework 3 soon.
Yesterday in class we showed that if a function f is twice differentiable, then
[f(a+h)-2f(a)+f(a-h)]/h^2 --> f''(a), as h --> 0.
However, after two applications of L'Hospital's rule, we needed f'' to be continuous. This is not necessary. If we apply L'Hospital's rule just once, we get the limit of
[f'(a+h) - f'(a-h)]/2h which equals [f'(a+h)-f'(a)]/2h + [f'(a-h) - f'(a)]/(-2h),
which goes to f''(a), as h --> 0, by definition.
Solutions to Homework 2 have also been posted.
If you are still struggling with problem 16.(c) from section 6.2, feel free to study the proof of one of L'Hospital's rules (sec. 6.3) and imitate it. (However, you're not allowed to use L'Hospital's rules.)
Also, note the newly posted homework revision policy.
Finally, I posted a little handout called Good proof, bad proof.
|
Required text:
R. G. Bartle and D. R. Sherbert: Introduction to Real Analysis, John Wiliey & Sons, 3rd edition, 2000 |
Recommended reading:
|
C. C. Pugh: Real Mathematical Analysis,
Springer-Verlag, UTM, 2002
This is a more advanced book but it's beautifully written and covers a lot of interesting topics. |
1 class late: 50% penalty; 2 classes late: 75% penalty; 3 classes late: no credit.
| # | Due date | Assignment | Solutions |
|---|---|---|---|
| 1 | 2/6 | Sec. 3.2, #14. Sec. 3.4, #11. Sec. 3.7, #8. Sec. 4.3, #9. Sec. 5.1, # 11, 12. Sec. 5.2, #7. | |
| 2 | 2/13 | Sec. 6.1, #4, 13, 16. Sec. 6.2, #4, 16, 20. | |
| 3 | 2/24 | Sec. 6.3, #5, 10, 11. Sec. 6.4, #3, 10, 18. | |
| 4 | 3/10 | Sec. 7.1, #9, 11, 12. Sec. 7.2, #8, 10, 17. | |
| 5 | 3/20 | Sec. 7.3, #10, 12, 16, 20, 21. | |
| 6 | 3/24 | Sec. 8.1, #4, 14, 19, 23. | |
| 7 | 4/7 | Sec. 8.2, #3, 4, 5, 11, 13, 20. | |
| 8 | 4/26 | Sec. 9.4, #1abc, 6abc, 7, 9, 11, 17. | |
| 9 | 5/10 | Homework 9 problems |
Last modified: Thu Dec 7 10:09:25 PST 2006