Last updated Tue Feb 5 13:35 EST 2019
Due-date: Friday, 2/8/19.
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 Chapter V/ 1, 2, 5, 7, 10, 11. Of these, hand in only 2b, 7, 10, 11 (just the "maximum" case). See notes below.
In #1, "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 \(\bf R\) 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.
You may find #5 easier after you do non-book problem B3.
In #7, whose result could be called "the intermediate value property of derivatives", assume that \(f'(a)\neq f'(b)\); otherwise, if you interpret "\(\gamma\) lies between \(f'(a)\) and \(f'(b)\)" to include the possibility "\(f'(a)=\gamma=f'(b)\)", the assertion becomes false. Note also that, for this problem, you can't just apply the "usual" Intermediate Value Theorem to \(f'\). (Here, by the " 'usual' Intermediate Value Theorem" I mean the result stated as a corollary at the bottom of p. 82 of Rosenlicht.) Why not?
The Intermediate Value Theorem referred to in #7 is a combination of (i) the theorem at the top of p. 82, and (ii) the fact that the only nonempty connected subsets of \(\bf R\) are intervals (the first Proposition on p. 60). In other words, the relevant "Intermediate Value Theorem" is: if \(E\) is a connected metric space and \(g:E\to {\bf R}\) is continuous, then \(g(E)\) is an interval. (By definition, an interval in \(\bf R\) is a nonempty set \(I\) with the property that if \(a\) and \(b\) are elements of \(I\), then every \(x\in {\bf R}\) that lies between \(a\) and \(b\) is an element of \(I\).) To use Rosenlicht's hint, you therefore have to know that the domain of the two-variable function in the hint is connected. For this, you may assume the result of Rosenlicht's problem IV.29(a) (a homework problem in my class last semester).
A consequence of #7 is that if \(f\) is a differentiable function on an open set \(U\), then \(f'\) cannot have any "jump discontinuities": there are no points \(x_0\) such that \(\lim_{x\to x_0+}f(x)\) and \(\lim_{x\to x_0-}f(x)\) both exist but are unequal.
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. (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 very term that arises; it's just unnecessay for this problem. It's quite a bit harder (and unneccsary) 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 B5, is not needed to do that problem. Of these, hand in only B2bc, B3a, B5ab, B6a.
- C. Continue reading the handout Some notes on normed vector spaces. Aim to finish it by Friday, Feb. 1, whether or not that ends up being the due-date for the rest of this assignment.
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