Due: December 1st, 2017
Math 142A Assignment 7
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The $\LaTeX$ file used to produce them is here.
 For each of the following functions, compute the derivative where it exists, and state the points at which it does not exist.
 $\frac{x^21}{x^2+2x+1}$
 $\frac{x^22x+1}{1x^2}$
 $\sin\left(\cos\left(2x^2\right)\right)$
 $\frac{x4}{\sqrt{x+1}}$

Let $a > 0$, and suppose that $f : \mathbb{R}\to\mathbb{R}$ is defined by $f(x) = a^x$.
Show that if $f$ is differentiable at $0$, then $f$ is differentiable everywhere, and $f'(x) = a^xf'(0)$.

Let the function $f$ be defined as follows:
$$\begin{align}f : \mathbb{R} &\longrightarrow \mathbb{R} \\ x &\longmapsto \begin{cases}x^2\sin\left(\frac1{x}\right)&\text{ if } x \neq 0\\0 & \text{ if } x = 0\end{cases}.\end{align}$$
Show that $f$ is differentiable everywhere, but its derivative is discontinuous at $0$.
(Hint: a lot of this problem is very similar to a problem on assignment 5.)

Suppose that $a < b$, and $g : [a, b] \to \mathbb{R}$ is continuous on its domain, and invertible.
Note that, as a consequence, $g([a,b])$ is a closed interval.

Suppose that $g$ is differentiable at $x_0 \in (a, b)$, and that $g'(x_0) \neq 0$.
Show that $g^{1}$ is differentiable at $g(x_0)$, and find its derivative there.

Suppose that $g$ is differentiable at $x_0 \in (a, b)$, and that $g'(x_0) = 0$.
Show that $g^{1}$ is not differentiable at $g(x_0)$.


Suppose that $a < b$, and that $f : [a, b] \to \mathbb{R}$ is strictly increasing and differentiable at $x_0 \in (a, b)$.
Show that $f'(x_0) \geq 0$.

Give an example of a strictly increasing differentiable function $f$ such that $f'(x) = 0$ for some $x$ in its domain.

Bonus:
Recall the definition of uniform convergence and of pointwise convergence from homework 5:
The supremum norm of a function $f : X \to \mathbb{R}$ is defined to be the quantity $$\f\_\infty = \sup\{f(x) : x \in X\}$$ if the supremum exists, and infinity otherwise.
A sequence of functions $(f_n)$, with $f_n : X \to \mathbb{R}$, is said to converge uniformly to a function $f : X \to \mathbb{R}$ if for every $\epsilon > 0$ there is $N \in \mathbb{N}$ such that for all $n > N$, $$\f_n  f\_\infty < \epsilon.$$
The sequence is said to converge to $f$ pointwise if for every $x \in X$, the sequence of real numbers $(f_n(x))$ converges to $f(x)$.
One of the results from that assignment was that if a sequence of continuous functions converges uniformly, then its limit is a continuous function.
In this problem we will demonstrate a function which is continuous everywhere, but differentiable nowhere.
For each $n \in \mathbb{N}$, let $f_n$ be the function defined as follows:
$$\begin{align}f_n : \mathbb{R} &\longrightarrow \mathbb{R} \\ x &\longmapsto \sum_{j=0}^n \left(\frac{3}{4}\right)^{\color{red} j}\sin\left(9^{\color{red} j}\pi x\right).\end{align}$$
 Show that the sequence $(f_n)$ converges pointwise to some function $f$.
 Show that the convergence is uniform. (It follows, then, that $f$ must be continuous.)
 Show that for any $x_0 \in \mathbb{R}$, there are points $y, z \in \mathbb{R}$ so that $x_0  y, x_0z \leq \frac1{9^n}$, but $f(y)  f(z) \geq \left(\frac34\right)^n$.
 Conclude that $f$ is not differentiable at $x_0$.