Last updated Thu Feb 21 18:15 EST 2019
"Virtual" due-date: Wednesday 2/20/19, the date of the first midterm. ("Virtual" means that you should have all the problems done before the first midterm, but there is nothing to hand in before the midterm.)
Due-date for hand-in problem: Monday 2/25/19
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.
A. Rosenlicht Chap. VI/ 1–4, 8, 10, 12. Do not hand in any of these problems. For these problems, you're allowed to use anything in sections 6.1–6.3 and 6.5 of my notes. You will probably find #s 2, 3, and 10 easier now that we've covered the "Step-Function Lemma", Proposition 6.45 in my Notes on Integration. However, all of these problems can be done without the "Step-Function Lemma".
Before you start working on #2, read the Q&A near the bottom of this page.
B. Click here for a non-book problem. This problem logically belongs with the "Exercises on the Mean Value Theorem" that were part of Assignment 2, but was postponed because Assignment 2 was long enough already. Do not hand in this problem.
C. In the Notes on Integration, do exercises 6.1–6.6 (located on pp. 6–7, 11, 12, 21, 23, and 28). You may need to clear your browser's recent cache to load the current version of the notes; the current exercises 6.4–6.6 are the former exercises 6.4–6.6 are the former exercises 6.5–6.7. Of these, hand in only exercise 6.1.
Reminder: Whenever I make a revision that causes a numbering-change or significant pagination-change in material we've already covered, I keep the previous versions of the notes posted on the Miscellaneous Handouts page, in order to help you find items that you remember by item-number or location but whose item-numbers or locations may have changed. I don't post new versions for minor revisions (such as fixing typos), or for revisions that are further along than wherever we are in the notes. However, I usually update the version-date on p. 1 whenever I make even minor changes, so don't be surprised or concerned if that date looks more recent than the last one you remember, and don't waste your time combing through the previously-covered sections of the notes to see what little things I might have changed. If I've changed anything important, I'll let you know, and will usually post the prior version.Q&A. Below are a question and answer from a previous time that I taught this class. A similar question can occur in many other problems. The answer is always the same (except, of course, for specific reference to the particular exercise).
Question. "For exercise 2 in Rosenlicht Chapter VI, may we assume that the integral exists and use that assumption to show that it is equal to 0, or do we have to prove existence first and then prove that it equals 0?"
Answer. You absolutely may not assume that the integral exists. Any time you're asked to prove "this = that", and "this" does not automatically exist, it's implicit that what you're being asked to prove is "this exists AND this = that." Another example of this sort of thing is "Prove that a certain limit = 1". It should be obvious that if you were asked this, you would not be allowed to assume from the start that the limit exists. If the writer of the exercise wanted you to make such an assumption, he/she would say so explicitly.
However, often in these cases, you end up exhibiting the value of "this" (or the limit, in my other example) at the same time that you prove existence. For example, you often prove that a limit exists by intelligently guessing the value of the limit (before trying to write a proof), then showing that your function, sequence, or whatever, approaches that value. Very often, in such a context, "intelligent guesswork" does involve a step of the type "Hmm, suppose the limit (or whatever) existed. What would it have to be?" That's perfectly good thinking when you're trying to figure out how you're going to prove something. Of course, this will not be part of the proof itself; you're going through this step to guide you to a proof.
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