Last updated Fri Mar 22 22:28 EDT 2019
Due-date: Wednesday 3/27/19
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/ 22 ("between" means "strictly between"), 23, 24ab, 25 (with 25c modified as indicated below), 26. Of these, hand in only 22 and 26. In your write-up of #26, you may assume the results of #25c.
- 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 #25 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!"
- Rosenlicht Chap. VII/ 3–5, 9, 10, 12. Of these, hand in only 3, 9, and 12. In these exercises, remember that for a sequence of functions, Rosenlicht's "converges" is my "converges pointwise" (and similarly for "convergent").
B. No non-book problems other than the ones in the Notes on Integration.
C. In the Notes on Integration, read Definition 6.113 and Remarks 6.104, 6.108, 6.114, 6.116, and 6.121. Do exercises 6.18–6.23 (located on pp. 60–61, 63–64, and 67). Of these, hand in only 6.19 and 6.21ab.
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