Beauty: Analysis Midterm

Thursday, March 06, 2008

Analysis Midterm

Well I had my undergraduate analysis midterm today. I think it is pretty safe to assume that I dominated that test. When I first saw the test I began to worry, but as I started to work on the problems thing began to settle down. The test only covered material from the chapters on differentiation and integration.
Here are the problem statements as I remember them, I might post my solutions later on.

1:
If $|f'|<M<\infty$ for some interval $I$ , show that $f$ is uniformly continuous on this interval. (Note that $I$ need not be closed, or bounded)

2:
part 1: Suppose that $g$ is continuous at $x=0$ . Prove that $f=xg(x)$ is differentiable at $x=0$ .
part 2: Define a function $f:\mathbb{R}\mapsto\mathbb{R}$ by $f(x)=x^2$ if $x\in\mathbb{Q}$ and $f(x)=0$ if $x\notin\mathbb{Q}$ . Find the value of $f'(0)$ and prove your result.

3:
Define $f(x)=x^a\sin\left(\frac{1}{x^2}\right)$ when $x\neq{0}$ and 0 at the origin. What is the minimum value of $a$ that causes $f'$ to be bounded. Also for this $a$ is $f'$ continuous at 0? Prove your result. Finally characterize the values of $a$ such that $f'$ is continuous on $\mathbb{R}$ .

4:
Prove that $f\in{R(\alpha)}$ if and only if for every $\epsilon>0$ there exist a partition $P$ such that $U(P,f,\alpha)-L(P,f,\alpha)<\epsilon$ .

5:
Suppose that $f$ is continuous on $[0,1]$ , and $f(0)=C$ . Prove that,
$\displaystyle\lim_{x\to 0}\frac{1}{x}\int_0^xf(t)dt=C$ .

6:
Let $f$ be differentiable on the closed interval $[a,b]$ such that $f'(a)<\lambda<f'(b)$ . Prove that there exist a $c\in{(a,b)$ such that $f'(c)=\lambda$ .

As you can see, nothing really difficult about the exam. I don't know why I panicked at the beginning, maybe I was anxious or something.