Last updated Wed Apr 17 22:23 EDT 2019
"Virtual" due-date: Wednesday 4/10/19, the date of the second midterm. ("Virtual" means that you should have all the problems done before the midterm, but there is nothing to hand in before the midterm.)
Due-date for hand-in problems: Monday 4/22/19
You are required to do all of the problems below. You will not be required to hand them all in. I have indicated below which ones you do have to hand in on the due-date. Don't make the mistake of thinking that I'm collecting only the problems I think are important.
The "due date" above is the date that your written-up problems should be handed in, but don't wait to get started on the assignment. You should always get started on problems as soon as we cover the relevant material in class.
- C. (This will help with several exercises in this assignment.) In the "Improper Integrals" handout on the Miscellaneous Handouts webpage, read Lemma 1 and its proof. This reading is self-contained; you don't need to have read anything else in the handout.
- A.
- Rosenlicht Chap. VI/ 27, 28. Of these, hand in only 28. In your writeup you may assume the analog, for "real-valued limits at infinity", of Lemma 1 in the Improper Integrals handout. Specifically, this analog is: For any \(a\in {\bf R}\) and function \(f:[a,\infty)\to{\bf R}\) (or \(f:(a,\infty)\to{\bf R}\)),
\(\lim_{x\to\infty} f(x)\) exists if and only if for all \(\epsilon > 0\), there exists \(b\in{\bf R}\) such that for all \(x_1,x_2 > b\) we have \(|f(x_1)-f(x_2)| < \epsilon.\)
The Improper Integrals handout on the Miscellaneous Handouts webpage covers a lot more than is in these two problems, at least some of which I think you'd find surprising (e.g. the Remark on p. 6, and exercise 18 on p. 8, if you're interested). But I'm giving you a break, just having you read Lemma 1, and assigning these two Rosenlicht problems, instead of having you read the whole handout and do all the exercises.
- Rosenlicht Chap. VII/ 25 (do non-book problem B2 before this, in order to make sense of the sum), 26 (Problem 15, referred to here, is B2(d)), 27. Of these, hand in only 27.
Problems 35 and 36, previously posted here, have been moved to the next assignment.- B. Click here for non-book problems. Of these, hand in only B2abcd.
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