Due date: Monday, 2/3/14
Last updated Jan 31 05:34 EDT 2014
You are required to do all of the problems below. You will not be required to hand them all in. I will announce which ones you do have to hand in.
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 pp. 108-110/ 1, 2, 4, 5, 7, 10, 11, 12. Of these, hand in only 2b, 7, 10, 11. See notes on #1, #7, and #12 below.
In #1, “discuss the differentiability of f” means “Discuss the differentiability of the function” means “State at which points the function is differentiable, at which points it is not differentiable, and prove your answers.” (Among the general properties of the sine function you may assume are that it is differentiable on R and that its derivative is cosine.)
In #7, which could be called an “intermediate value theorem for derivatives”, assume that f'(a)≠f'(b); otherwise the assertion is false. Note that, for this problem, you can’t just apply the (usual) Intermediate Value Theorem to f’. Why not? Include your answer in what you hand in for #7.
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. And please don’t go searching the web for solutions; that would defeat the purpose of my assigning the problem. - B. Click here for non-book problems. Of these, hand in only B1, B3ab, B5bc.
- C. Students who weren’t in my class last semester (or were in it but didn’t do that semester’s Assignment 1) should read (at least) the portion of my mathematical grammar handout that starts with the last paragraph on p. 1, and ends with “‘Gibberish’ version of the same argument” on p. 2. This handout was written originally for an MAS 4105 class, but all the general principles apply to all mathematical writing. Note that the handout is not suggesting that you should use mathematical symbols in place of words; it is trying to show or remind you that, among other things, in any mathematical argument you need words (or their symbolic equivalents) that logically connect your statements–including equations/inequalities–and is telling you how to use various symbols correctly, if you choose to use them.