Homework 15: The Derivative

Instructions. Assignments should be stapled and written clearly and legibly. For problems 1 - 4, I recommend against using \(\epsilon-\delta\) arguments. Instead use limit laws where appropriate.

1. Section 6.1, #9.
2. Suppose that \(f:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}\to \mathbb{R}\) are continuous functions satisfying (i) \(f(0)=0\), (ii) \(f'(0)=3\), and (iii) \(g(0)=2\). Prove that \(fg\) is differentiable at \(0\), and find \((fg)'(0)\).
   Note: The product rule for derivatives cannot be used for this problem, since \(g\) may not be differentiable at \(0\). You must use the definition of derivative.
3. Let \(f:(-1,1)\to \mathbb{R}\) be a bounded function. In Problem 5 of Homework 12, you proved that the function \(g:(-1,1)\to \mathbb{R}\) defined by \(g(x)=xf(x)\) is continuous at \(x=0\). Use this result to prove that the function \(h:(-1,1)\to \mathbb{R}\) defined by \(h(x)=x^2f(x)\) is differentiable at \(0\), and find \(h'(0)\).
   Note: The product rule for derivatives cannot be applied here either.
4. Suppose that \(f:\mathbb{R}\to \mathbb{R}\) and \( \lim\limits_{x\to 0}\frac{f(x)}{x}\) exists.
   1. Prove that \(\lim\limits_{x\to 0}f(x)\) exists and find its value.
   2. Prove that if \(f(0)=0\), then \(f\) is differentiable at \(0\).
5. Determine whether the following function is differentiable at 0. Prove your answer. \[f(x) = \begin{cases} x^2, & \text{if \(x\) is irrational}\\ 0, & \text{if \(x\) is rational} \end{cases} \]