Last updated Fri Jan 31 17:31 EST 2020
Due-date: Wednesday, 2/5/20
You are required to do all of the problems below. You will not be required to hand them all in. I will announce later 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 Chapter V/ 1, 2, 5, 10, 11. Of these, hand in only 2b, 10, 11 (just the "maximum" case). See notes below. Some notes that are specific to your write-ups of 2b and 10 have been added below, in boldface.
In #1, "Discuss the differentiability of the function" means "State, with proof, at which points the function is differentiable, at which of these points the derivative is continuous, and at which points the function not differentiable." Among the general properties of the sine function you may assume are that it is differentiable on \(\bf R\) and that its derivative is cosine.) Not covered in Chapter V, but which you may also assume for doing 1(c), is the Calc-1 rule for differentiating the square-root function on \((0,\infty)\).
In #2, do not use l'Hôpital's Rule, which is not proven within the text of Chapter V (its proof is a later exercise for Chapter V), and which would defeat the purpose of the problem. If you do this problem the intended way, you will likely find yourself using a special case of the lemma in non-book problem B2. In your write-up, you may assume the result of non-book problem B2.
Before doing #5, do non-book problem B4 (whose results you may then use, including the [correct] conclusion in part (b)).
In #10, "function of a function" means "composition of two composable real-valued functions". Hints for #10, with appropriate domains and codomains implicitly assumed: (i) You are not being asked to produce an explicit formula for the \(n^{\rm th}\) derivative of a composition for arbitrary \(n\). (ii) Use the Chain Rule Theorem to reduce the problem to showing that, for all \( m> 0\), the product of two \(m\)-times differentiable functions is \(m\) times differentiable. For a specific \(n,\) the relevant \(m\) here is not the same as \(n\). (Note: "Reduce proving statement A to proving statement B" doesn't mean that statements A and B are equivalent; it means that statement B implies statement A.) (iii) Use induction to show that every finite sum of \(m\)-times differentiable functions is \(m\) times differentiable, and produce the usual formula for the \(m^{\rm th}\) derivative of the sum in terms of the summands. When writing up this problem for hand-in, you may assume that these "finite sum" facts have already been proven. (iv) Use induction and (iii) to show that the product of two \(m\)-times differentiable functions is \(m\) times differentiable. This can be done without producing an explicit formula for the \(m\)th derivative of the product of two functions. (For the product, it's not hard to produce an explicit formula, with the exact coefficent of every term that arises; it's just unnecessary for this problem. It's quite a bit harder [and unnecessary] to produce to an explicit formula for the \(n\)th derivative of a composition. However, to get a feel for what happens, it's worthwhile to do the computation for \(n=2,3,4\).)
- B. Click here for non-book problems. Note: Taylor's Theorem, which is mentioned in a warning in B6, is not needed to do that problem. Of these, hand in only B3bc, B4a, B6ab, B7a.
- C. Continue reading the handout Some notes on normed vector spaces. Aim to finish it by Friday, Jan. 31, whether or not that ends up being the due-date for the rest of this assignment.
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