Last updated Mon Jan 30 12:25 EST 2017
Due-date: Friday 2/3/17
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.
- A. Rosenlicht pp. 108–110/ 1, 2, 5, 7, 10. Of these, hand in only 2b, 7. See notes on #s 1, 2, and 7 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 you probably have never seen proven, which is not proven within the text of Chapter V (its proof is a later exercise for Chapter V, and will either be proven in class or assigned at the appropriate time), and which would defeat the purpose of the problem.
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 problem V.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). (I assigned this as a homework problem in my section last semester. If you have never done it, you should do it as part of this assignment.)
- B. Click here for non-book problems. Note: Taylor's Theorem, which we have not covered as of this writing (1/27/17) and which is mentioned in a warning in B4, is not necessary to do B4. Of these, hand in only B2, B4ab, B6bc.
- C. Read this handout on solving equations. It was written for a low-level class, but many students even in high-level classes have never seen, or thought about, the logic that's involved in manipulating or solving equations. The same logical principles in this handout apply when manipulating or solving inequalities; the only ideas in the handout that aren't relevant to inequalities are "one-to-one" and "onto".
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