Last updated Sat Nov 21 08:45 EST 2020
Due-date/time for handing in problems: Tuesday Nov. 24, 1:55 p.m.Due-date for doing the problems: Thursday Nov. 19, prior to that evening's midterm exam, so that you have a chance to ask me about them in Tuesday's office hour and still have time to absorb what you learn.
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 problems 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 are to 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: Bartle & Sherbert exercises:
- Section 6.1/ 2 (see note below), 4, 8ac, 9, 10, 12 (see notes below)
- Section 6.2/ 1cd, 2bc, 4 (if you're a statistics major, this should look very familiar to you!), 9, 15, 17
- Section 7.1/ 2ab, 15
Notes on some of these problems:
- 6.1/2: The entire purpose of this problem is to have you prove that this function is not differentiable at 0. Proofs are based on definitions and previously proven results, not on facts or ideas pulled out of thin air. For example, the definition of \(f'(0)\) is NOT \(\lim_{x\to 0}f'(x)\)—and, in fact, it is NOT true that the derivative of every differentiable function is continuous. Nor does non-book problem B1 apply, because the hypotheses are not met. I suggest that you also review the section "Some pitfalls to avoid when doing proofs" (especially the ones on p. 5) in the "What is a proof?" handout that you were assigned to read in Assignment 0.
- 6.1/12: (i) You may assume that the sine and cosine functions have their familiar properties. (ii) Change "\(f:{\bf R}\to {\bf R}\)" to "\(f:[0,\infty)\to {\bf R}\)", since for some rational exponents\(r>0\), we define \(x^r\) only for \(x\geq 0\)). (iii) "Determine", in this problem, includes proving your answer.
Of the B& S problems, hand in only these: 6.1/ 2 and 6.2/ 15
- B: Click for non-book problems. Of these, hand in only B3a, B4ab. For purposes of applying B3a to B4, you may assume that the result of B3a holds also with \(b\) replaced by \(\infty\). (I should have worded B3a to include this case; the same argument works in B3a whether or not the interval is bounded above.)
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