Last updated Mon Mar 20 18:10 EDT 2017
Due-date: Friday 3/24/17
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), 11–13, 15, 16 (it's implicit in the notation "\(C[a,b])\)" that the uniform metric is intended here), 17–20, 22, 23, 24ab, 25 (with 25c modified as indicated below), 26. Hand in only 7, 10, 11, 15, 16, 20, 22. (#10 was actually part of the previous assignment, but I'm having you hand it in with this one.)
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). 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 #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 #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.
Hint for #22: \([1,n] = [1,2]\cup [2, 3] \cup \dots \cup [n-1,n].\)
Hint for #23: Divide one function by another.
Note on #25: At the very least, do parts (a)–(c) without using l'Hôpital's Rule, based on the "pre-(a)" part of the problem. You should really be able to do even the pre-(a) part of the problem without l'Hôpital's Rule. That's why Rosenlicht gave the pre-(a) part of the problem; if he'd wanted you to use l'Hôpital's Rule in (a)–(c), there would have been no point in stating the pre-(a) part of the problem.
Modification for #25c: Replace \( x^n/e^x\) by \(x^\alpha/e^{\epsilon x}\), where \(\alpha\) is an arbitrary real number and \(\epsilon\) is an arbitrary positive real number. Of course, replacing the book's \(n\in {\bf R}\) by my \(\alpha\) makes no difference, since Rosenlicht said that \(n\) could be any real number in this problem; I just didn't want you overlooking that \(n\) is not assumed to be an integer.A good way to think of (and remember) what #26 is showing is that "logs are weaker than powers, and powers are weaker than exponentials." Equivalently: "exponentials are stronger than powers, and powers are stronger than logs." In any battle between functions of two of these types—where "battle" means an (initially) indeterminate limit of type "\(\frac{\infty}{\infty}\)" or "\(0\times\infty\)"—the stronger function wins. For example, in part (b), a limit of type "\(0\times\infty\)", the factor \(x^{\alpha}\) wants the limit to be \(0\), while the factor \(\log x\) wants the limit to be \(-\infty\). "Puny log!" scoffs the power-function \(x^\alpha\). "You are helpless to stop me from making the limit \(0\)!" As \(x\to\infty\), exponential functions \(x\mapsto e^{\epsilon x}\) (with \(\epsilon>0\)) similarly laugh at power-functions \(x\mapsto x^\alpha\) , no matter how large \(\alpha\) is (say, \(\alpha=1,000,000,000\)) or how small \(\epsilon\) is (say, \(\epsilon=10^{-100}\)). "You tortoise!" says the exponential function to the power function. "In a race to infinity, you eat my dust!"
Die-hard fans of l'Hôpital's Rule should do the following exercise:(a) Use l'Hôpital's Rule alone to compute \(\lim_{x\to\infty}\frac{(x^3+e^{x/2})^{\sqrt{2}}}{(x^2+e^{x\sqrt{2}})^{1/2}}\). No stopping till you get an answer or you die, whichever comes first. (b) In your next life (or this one, if you disobeyed the last instruction in part (a)), ask yourself the question, "Which is better: knowing the algebraic and growth properties of common functions, or knowing l'Hôpital's Rule?"
- B.
- In the Notes on Integration, read Section 6.9 (Integration of vector-valued functions). This is assigned in lieu of my assigning Rosenlicht's problem VI.6, which I've always assigned in the past (for which I also usually had to concurrently assign problems IV.22 and IV.23). Based on my experience with past students, even though Section 6.9 is rather long, the time it will take you to read it is much less than the time it would take you to do problems IV.22, IV.23, and VI.6, if you could succeed in them at all. (My past students have not had much success with this suite of problems. The most important results in IV.22 and IV.23 are now proved in the handout "Some notes on normed vector spaces" linked to your Miscellaneous Handouts page.)
- Read this handout on improper integrals. Exercises in the handout should be read, since some results are stated in the exercises. However, doing these exercises is not part of this assignment (though some might be part of a future assignment)
Either or both of the reading-assignments above should make for good spring-break reading!
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