Last updated Fri Mar 15 12:45 EDT 2019
Due-date: Monday, 3/18/19.
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/ 7 (see note below), 13, 15–20. Of these, hand in only 7, 13, 15, 16, 20.
- Note on #7: A function \(f:[a,b]\to {\bf R}\) (or, more generally, \(f:[a,b]\to X\), where \(X\) is any metric space) is called piecewise continuous if there exists a partition \( \{x_0, \dots, x_N\} \) of \([a,b]\) such that (i) the restriction of \(f\) to the open interval \( (x_{j-1},x_j) \) is continuous, \(1\leq j\leq N\); (ii) \(\lim_{x\to x_j-}f(x)\) and \(\lim_{x\to x_j+}f(x)\) exist, \(1\leq j\leq N-1\); and (iii) \( \lim_{x\to a+} f(x)\) and \( \lim_{x\to b-} f(x)\) exist. The function \(f\) in problem 7 is not assumed piecewise-continuous; condition (i) is assumed, but not conditions (ii) and (iii). For example, the function \(f:[0,1]\to {\bf R}\) defined by \(f(x)=\sin \frac{1}{x}\) for \(x\neq 0\), and \(f(0)=0\), is continuous except at the single point \(0\), but is not piecewise continuous (since \(\lim_{x\to 0+} \sin\frac{1}{x}\) does not exist). If you're able to do this problem only under the stronger assumption that \(f\) is piecewise continuous, that's still worth something, but recognize that you're adding an assumption that's not stated in the problem.
- Note on #13: To do this problem, 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. We haven't proven it yet, but it's a quick consequence of something we'll prove soon, and can also be proven using Rosenlicht homework problem II.11 from last semester.
Hint: 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 13, for the relevant sequence \(A_n\) and number \(M\), and a given \(\epsilon>0\), the way that you show \(A_n > M-\epsilon\) for all \(n\) sufficiently large will be quite different from the (easier) way that you show \(A_n < M+\epsilon\) for all \(n\) sufficiently large. Continuity of \(f\) is critical to the "\(A_n > M-\epsilon\)" argument, but irrelevant to the "\(A_n < M+\epsilon\)" argument.- Note on #16: It's implicit in the notation "\(C([a,b])\)" that the uniform metric is intended.
- Note on #18: No trig functions or their inverses are allowed. The point of the problem is to show directly that the two integrals are equal to each other, not that they are equal because they both yield \(\tan^{-1} x\).
- Note on #19: You may (and should) use the Lemma on p. 106 of Rosenlicht. Had I taken time in class to prove the version of Taylor's Theorem that I stated (the one in Rosenlicht section V.4), I'd have proceeded as Rosenlicht does, first proving this lemma. The version of Taylor's Theorem in problem VI.19 is actually more important than the one in Chapter V, because even though the Taylor remainder is now expressed as an integral, it's still an exact formula for the remainder; there are no unknown quantities like the \(c\) in Lagrange's form of the remainder (the one on p. 107).
- Note on #20: This problem is not easy. If you have what you think is a quick proof, you are probably overlooking something, making an implicit assumption, etc.
B. Click here for a non-book problem. Hand in this problem.
C. In the Notes on Integration, read the portions of Section 6.9 not covered in class (the second half of the proof of Proposition 6.89, and everything from Proposition 6.92 through the end of Section 6.9 except for Remark 6.94). Do exercises 6.7–6.9, 6.12–6.16 (located on pp. 31, 33, 35, 44, and 45). Of these, hand in only 6.12 and 6.15.
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