Last updated Mon Feb 20 19:50 EST 2017
Due-date: Monday 2/27/17
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.
At all times, you are EXPRESSLY FORBIDDEN from using online sources (other than materials I post for the class) to help you with your homework problems in any way. You are also EXPRESSLY FORBIDDEN from using any other sources, except the Rosenlicht textbook, to help you with homework, unless I give you specific permission. See the current homework rules (revised 2/6/17).
- A.
- Rosenlicht Chap. V/ 11, 12. Do not hand either of these in.
In #12, convex function is, essentially, what is often called "function whose graph is concave-up" in Calculus 1. The only difference is that "convex function" is more general: as you'll see in the problem, the definition of "convex function" makes sense whether or not the function is differentiable, let alone twice differentiable. Problem 12 is somewhat long, since there are two "if and only if"s to prove, and the "if"s can't be done just by reversing the steps in the "only if"s (or vice-versa), as far as I know. (Plus, you have to figure out how to state the indicated properties precisely in terms of inequalities.) There is a third equivalence that you could have been asked to show in this problem: that a differentiable function is convex if and only if its derivative is an increasing function (not necessarily strictly increasing). Don't assume this third equivalence is true unless you prove it (which you're not being required to do); I'm just stating it in case believing that it's true helps you figure out the proofs that you're being asked for.
- Rosenlicht Chap. VI/ 1–4, 8–10. Of these, hand in only 2, 3, 8, 9. You are permitted to use the result of B3 (which is also a hand-in problem) in #9, OR to use the result of #9 in B3, BUT NOT BOTH (i.e. circular reasoning is not allowed).
Before you start working on #2, read the Q&A near the bottom of this page.
It goes without saying that in a yes/no question such as #3, you are expected to prove your answer. You will probably find #3 rather challenging. A proof of the correct answer can be written in half a page, once you figure out the key idea, but finding the key idea is not so easy. I doubt you'll be able to do #3 until you've figured out a way to do #2, which itself is not so easy. Neither of these is a "just turn the crank" problem.
- B. Click here for non-book problems. Of these, hand in only B1 and B3.
Q&A. Below is a question and answer from a previous time that I taught this class. A similar question can occur in many other problems. The answer is always the same (except, of course, for specific reference to the particular exercise).
Question. "For exercise 2 in Rosenlicht Chapter VI, may we assume that the integral exists and use that assumption to show that it is equal to 0, or do we have to prove existence first and then prove that it equals 0?"
Answer. You absolutely may not assume that the integral exists. Any time you're asked to prove "this = that", and "this" does not automatically exist, it's implicit that what you're being asked to prove is "this exists AND this = that." Another example of this sort of thing is "Prove that a certain limit = 1". It should be obvious that if you were asked this, you would not be allowed to assume from the start that the limit exists. If the writer of the exercise wanted you to make such an assumption, he/she would say so explicitly.
However, often in these cases, you end up exhibiting the value of "this" (or the limit, in my other example) at the same time that you prove existence. For example, you usually prove that a limit exists by intelligently guessing the value of the limit (before trying to write a proof), then showing that your function, sequence, or whatever, approaches that value. Very often, in such a context, "intelligent guesswork" does involve a step of the type "Hmm, suppose the limit (or whatever) existed. What would it have to be?" That's perfectly good thinking when you're trying to figure out how you're going to prove something. Of course, this will not be part of the proof itself; you're going through this step to guide you to a proof.
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