Last updated Sat Dec 11 16:43 EST 2021
Due-dates: various; see below.
You are required to do all of the problems AND READING below. (I've listed the reading first, but still called it "part C" for consistency with other assignments.) For this assignment, you will not be handing in anything in.
Although nothing on this assignment is to be handed in, In most years I'd be giving (at least) assignments 5 and 6, with assignment 5 collected on or about the last day of class, and assignment 6 based on material covered after the cutoff for assignment. This year's assignment 5 may be more like a double-assignment, corresponding to most years' assignments 5 and 6.
- C (reading):
- In Abbott, read Example 4.3.6, Theorem 4.3.9, and Section 4.4. (Due-date Mon. Nov. 29.) Note: In Theorem 4.4.2, a hypothesis is omitted: the compact set \(K\) should be assumed to be non-empty.
For each of the theorems in Section 4.4, you should do the following (due-date Wed. Dec. 1.):
- Read the statement of the theorem, but not the proof. Copy the statement down onto a piece of paper (or type it into a file, if you prefer).
Close the book, and try to prove theorem on your own.
After you either think you've succeeded in proving the theorem, or have given up trying, read the proof in the book. Your proof-attempts don't have to be done all at once. If you hit a snag, it's okay to move onto another one of the theorems, do some exercises from this assignment, or just take a break and do something unrelated to this course (as long as you get back to trying seriously to prove the theorem, leaving yourself enough time for the attempt[s].)
- If you did write what you thought was a proof, Compare it to the book's, and decide (as best you can) whether your proof is correct. ("Correct" does not mean "identical to what Abbott wrote." There is never just one correct argument that proves something, and never just one correct way to write the same argument. Plus, Abbott's proofs are not always perfect, even given that they are intended not just to prove whatever, but to teach the reader some principle(s) and/or method(s) at the same time—something that your own proofs need not do, and generally should not do.)
- If you decide that your proof was not correct, wait a little while—long enough that you're not going to remember large portions of Abbott's proof nearly verbatim—and then try again.
- Repeat steps 2–5 as many times as necessary until you can write what you are pretty sure is a correct proof that is not a (nearly) verbatim repetition of Abbott's proof. If the best you can do is to parrot Abbott's proof, afraid to change a single word, then you don't understand the proof.
Steps 1 and 2 above should be part of the way you read any theorems and proofs in a textbook!
- Read the files maa4211_f20_lect29.pdf (due-date Mon. Dec. 6) and maa4211_f20_lect30.pdf (due-date Wed. Dec. 8)that were emailed to the class on 12/3/21, and that are now posted in Canvas as well. (All you really need to read are the parts I didn't cover in class, but it won't hurt you to read the entire files.) As mentioned in the 12/3 email, these are lecture notes that I wrote for last year's class, and "[you should] ignore references to the textbook (which was different last year). Also ignore references to 'good linear approximation' (at least for now), which I have not yet covered this year, and may not get to at all. There may also be 'hand typos'."
- A: Abbott exercises. See notes and additional instructions below this list before starting these problems.
- 4.2/ 1, 5d, 6, 7, 8ab, 9–11 (due-date Mon. Nov. 29).
- 4.3/ 3, 5, 8, 9, 10a, 13 (due-date Mon. Nov. 29).
- 4.4/ 1, 7, 9, 11, 13 (due-date Wed. Dec. 1).
- 5.3/ 4, 5a (due-date Thurs. Dec. 9). (This due-date is after the last class, so it's really just my strong recommendatation that you try to get these problems done by this date. The same goes for the non-book problems with this due-date. In the last few days before the final exam, you're going to have plenty to review and study; don't saddle yourself with homework problems you've never looked at before.)
Notes and additional instructions on some of these problems:
- Some of the exercises from Sections 4.3 and 4.4 relate to the assigned reading from these sections, so for each of these sections, do the assigned reading first.
- 4.3.5: I did this in class briefly, but you should do it yourself anyway.
- 4.3.13(c): "linear function through the origin" means "function whose graph is a straight line passing through the origin."
- 4.4.9: (i) In the definition of "Lipschitz", it's more customary (and convenient) to write "\(|f(x)-f(y)|\leq M|x-y|\)" instead of "\(\big|\frac{f(x)-f(y)}{x-y}\big|\leq M\)" or "\( \frac{|f(x)-f(y)|}{|x-y|}\leq M\)," so that the inequality is valid for all \(x,y\in A\) without the restriction \(x\neq y\). (ii) To do part (b), show that the square-root function from \([0,\infty)\to{\bf R} \), which you showed to be uniformly continuous in Exercise 4.4.7, is not Lipschitz. (There's only one "problem point" in the domain, namely 0. For any $a>0$, the function \(x\mapsto\sqrt{x}\) is Lipschitz on the domain \( [a, \infty) \).)
- 4.4.13. Correct the notation and wording as follows. (i) Replace "\( (x_n) \subseteq A\) is a Cauchy sequence" by "\( (x_n) \) is a Cauchy sequence in \(A\). (Optionally, replace "\( (x_n) \)" by "\( (x_n)_{n=1}^\infty.\)") (ii) Replace "\(f(x_n)\)" by "\( (f(x_n)) \)" or by "\( (f(x_n))_{n=1}^\infty.\)") In both cases, Abbott is being sloppy, something not to be encouraged; I would grade this writing as wrong. The first correction is needed because a sequence is not a subset of its codomain; it is a function from the naturals to that codomain. The second correction is needed because a sequence is not the same thing as an arbitrary term of that sequence. Do not copy Abbott's bad habits.
- 5.3.4(b). I didn't notice definitions of "twice differentiable" or \(f''\) in Abbott, but they're just what you learned in Calculus 1.
- Read Theorem 5.3.5 first, of course.
- B: Click for non-book problems. Due-dates: 12/1 for B1, B2, B4, B5; 12/2 for B3, B6, B7; 12/8 for B8, B9, B10; 12/9 for B11, B12, B13.
General homework page
Class home page