Last updated Sun Dec 6 06:02 EST 2020
Due-date: Monday, 12/7/20You are required to do all of the problems and reading 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 7.2/ 8 (see note below), 9, 10, 18 (see note below), 19
- Section 7.3/ 6, 10, 18bc
- Section 8.1/ 2, 3, 4, 12, 13, 14, 21, 22, 23
- Section 8.2/ 1, 3–6, 8
Notes on some of these problems:
- Note on 7.2/ 8: Observe that this problem is equivalent to the following: if \(f:[a,b]\to{\bf R}\) is continuous, and \(f(x)\geq 0\) for all \(x\in [a,b]\), and \(f(c)>0\) for some \(c\in [a,b]\), then \(\int_a^b f > 0\). The result of problem 7.2/ 8 was stated in this form as a "We'll see later that ...", without proof, in class. This homework problem is that "We'll see later"!
- Note on 7.2/ 18: To do this problem (which is tricky enough to warrant my giving you a hint below), you'll need to use the fact that \(\lim_{n\to\infty} c^{1/n}=1\) for every real number \(c>0\). You may assume this fact for this problem. You are asked to prove it in non-book problem B1, so that your proof of result of problem 7.2/ 18 will then be complete.
Hint for #18: By definition, a sequence \( (a_n)_{n=1}^\infty\) in \({\bf R}\) converges to \(M\in {\bf R}\) if and only if given any \(\epsilon > 0\), we have \(a_n\in (M-\epsilon, M+\epsilon)\) for all \(n\) sufficiently large. In problem 18, for the sequence \(M_n\) and relevant number \(M\), and a given \(\epsilon>0\), the way that you show \(M_n > M-\epsilon\) for all \(n\) sufficiently large will be quite different from the (easier) way that you show \(M_n < M+\epsilon\) for all \(n\) sufficiently large. Continuity of \(f\) is critical to the "\(M_n > M-\epsilon\)" argument, but irrelevant to the "\(M_n < M+\epsilon\)" argument.Of the B& S problems, hand in only these:
7.2/ 8, 18
8.1/ 4, 14, 22
8.2/ 5- B: Click for non-book problems. Of these, hand in only B2.
- C: Read the handout Logarithms and Exponentiation . (Due-date: Tuesday 12/8/20.)
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