$$ \newcommand{\cis}{\operatorname{cis}} \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\paren}[1]{\left(#1\right)} \newcommand{\sq}[1]{\left[#1\right]} \newcommand{\abs}[1]{\left\lvert#1\right\rvert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\ang}[1]{\left\langle#1\right\rangle} \newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} \newcommand{\ceil}[1]{\left\lceil#1\right\rceil} \newcommand{\C}{\mathbb{C}} \newcommand{\D}{\mathbb{D}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\F}{\mathbb{F}} \newcommand{\T}{\mathbb{T}} \renewcommand{\S}{\mathbb{S}} \newcommand{\intr}{{\large\circ}} \newcommand{\limni}[1][n]{\lim_{#1\to\infty}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\cA}{\mathcal{A}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cC}{\mathcal{C}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cG}{\mathcal{G}} \newcommand{\cH}{\mathcal{H}} \newcommand{\cI}{\mathcal{I}} \newcommand{\cJ}{\mathcal{J}} \newcommand{\cK}{\mathcal{K}} \newcommand{\cL}{\mathcal{L}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cN}{\mathcal{N}} \newcommand{\cO}{\mathcal{O}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cQ}{\mathcal{Q}} \newcommand{\cR}{\mathcal{R}} \newcommand{\cS}{\mathcal{S}} \newcommand{\cT}{\mathcal{T}} \newcommand{\cU}{\mathcal{U}} \newcommand{\cV}{\mathcal{V}} \newcommand{\cW}{\mathcal{W}} \newcommand{\cX}{\mathcal{X}} \newcommand{\cY}{\mathcal{Y}} \newcommand{\cZ}{\mathcal{Z}} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bI}{\mathbb{I}} \newcommand{\bJ}{\mathbb{J}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bL}{\mathbb{L}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bS}{\mathbb{S}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bU}{\mathbb{U}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bW}{\mathbb{W}} \newcommand{\bX}{\mathbb{X}} \newcommand{\bY}{\mathbb{Y}} \newcommand{\bZ}{\mathbb{Z}} $$
Due: November 22nd, 2019

Math 104 Assignment 10

  1. A criterion for differentiability

    Suppose $f : (a, b) \to \R$ and $x \in (a, b)$. Show that $f$ is differentiable at $x$ if and only if there is a function $E : (a, b) \to \R$ continuous at $x$ and a constant $C \in \R$ so that $E(x) = 0$ and for all $t \in (a, b)$, \[f(t) = f(x) + C(t-x) + E(t)(t-x).\] Moreover, show that in this case $C = f'(x)$.

  2. Some counterexamples of converses

    Give an example of continuous functions $f, g : \R \to \R$ so that:
    1. $f+g$ is differentiable but $f$ is not.
    2. $fg$ is differentiable, $f$ is not, and $g(x) \gt 0$ for all $x \in \R$.
    3. $f\circ g$ and $g$ are differentiable but $f$ is not.
    4. $f\circ g$ and $f$ are differentiable but $g$ is not.

  3. More derivatives

    Recall that $\sin$ and $\cos$ are differntiable functions $\R\to[-1,1]$ with the properties that $\sin'(x) = \cos(x)$, $\cos'(x) = -\sin(x)$, $\sin(x) \gt 0$ for $x \in (0, \pi)$, $\cos(x) = \sin(x + \frac\pi2)$ (from which the other translations follow, e.g. $\sin(x) = \sin(x+2\pi)$, $\cos(x) = \cos(x+2\pi)$, ...), $\sin(0) = 0$, and $\cos(0) = 1$. (In fact, $\sin$ is the only differentiable function on $\R$ with $\sin'' = -\sin$, $\sin(0) = 0$, and $\sin'(0) = 1$, but showing this is difficult, and defining $\sin$ as the unique function with these properties makes it rather difficult to show basic properties such as periodicity.)

    Let \begin{align*} f : \R &\longrightarrow \R \\ t &\longmapsto \begin{cases} 0 & \text{ if } t = 0 \\ t^2\sin\paren{\frac1t} & \text{ if } t \neq 0.\end{cases} \end{align*}

    1. Show that $f$ is differentiable on $\R$, and find its derivative.
    2. Show that $f'$ is discontinuous at $0$.

  4. Inverses

    Suppose that $f : (a, b) \to \R$ is differentiable with $f'(x) \gt 0$ for all $x \in (a, b)$.

    1. Prove that $f$ is strictly monotonically increasing, i.e., if $a \lt x \lt y \lt b$ then $f(x) \lt f(y)$.
    2. Note that $f$ is one-to-one, and so has an inverse function $g : f((a, b)) \to (a, b)$. Show that $g$ is continuous.
    3. Show that for all $x \in (a, b)$, \[g'(f(x)) = \frac1{f'(x)}.\]

  5. Positive derivative at a point

    Show that there is a differentiable function $f : \R \to \R$ with $f'(0) \gt 0$, but so that there is no $\delta\gt0$ with $f$ monotonically increasing on $(-\delta, \delta)$. (I.e., there is no $\delta \gt 0$ so that for all $x, y\in\R$ with $-\delta \lt x \lt y \lt \delta$ we have $f(x) \leq f(y)$.)

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.