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\newcommand{\myCourseNumber}{Math 142B}
\newcommand{\myAssignmentDueDate}{11:00pm Thursday, February 1, 2024}
\newcommand{\myAssignmentTitle}{Homework Assignment 2}
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\begin{questions}

%% Ross 29.14
\question Suppose $f$ is differentiable on $\mathbb{R}$, \ $1 \leq f'(x) \leq 2$ \ for all $x \in \mathbb{R}$, \ and \ $f(0) = 0$. \ Show that \ $x \leq f(x) \leq 2 x$ \ for all $x \geq 0$.

%% Ross 29.16
\question Define \ $f : \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \to \mathbb{R}$ by $f(x) = \tan(x)$ and let $g : \mathbb{R} \to \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ be the the inverse of $f$.  Determine a formula for $g'(y)$ for $y \in \mathbb{R}$.  (See Exercise 29.16 in your text.)

%% Ross 29.18
\question Let $f$ be differentiable on $\mathbb{R}$ with \ $d = \sup \left\{ \left| f'(x) \right| \ \mid \ x \in \mathbb{R} \right\} < 1$.
\begin{parts}
\part Select $s_0 \in \mathbb{R}$ and define \ $s_n = f(s_{n-1})$ \ for $n \geq 1$. \ Show that $(s_n)$ is a convergent sequence.  (See Exercise 29.18 in your text.)

[Hint: Show $(s_n)$ is Cauchy by first showing that \ $|s_{n+1} - s_{n}| \leq d \, |s_{n} - s_{n-1}|$ \ for $n \geq 1$.]

\part Show that $f$ has a \emph{fixed point}.  That is, show that \ $f(x_0) = x_0$ \ for some $x_0 \in \mathbb{R}$.
\end{parts}

%% Ross 30.4
\question Let $f$ be a function defined on some open interval $(0,a)$, and define $g(y) = f\left(\frac1{y}\right)$ for $y \in \left(\frac1{a}, \infty\right)$ with $\frac1{a} := 0$ when $a = \infty$.

Show that $\displaystyle \lim_{x \to 0^+} f(x)$ exists if and only if $\displaystyle \lim_{y \to \infty} g(y)$ exists, in which case the limits are equal.

%% Ross 30.6
\question Let $f$ be differentiable on some interval $(c, \infty)$ such that $\displaystyle \lim_{x \to \infty} \left[ f(x) + f'(x) \right] = L$, with $L$ finite.

Show that $\displaystyle \lim_{x \to \infty} f(x) = L$ and $\displaystyle \lim_{x \to \infty} f'(x) = 0$.  \Big[Hint: Write $f(x) = \frac{f(x) \ e^x}{e^x}$.\Big]

%% Ross 30.7
\question For $x \in \mathbb{R}$, let
\begin{displaymath}
f(x) = x + \cos(x)\sin(x) \quad \text{and} \quad g(x) = e^{\sin(x)} \left[x + \cos(x) \sin(x)\right].
\end{displaymath}
\begin{parts}
\part Show that \ $\displaystyle \lim_{x \to \infty} f(x) = \lim_{x \to \infty} g(x) = +\infty$.
\part Show that \ $\displaystyle f'(x) = 2 \, [\cos(x)]^2$ \ and \ $\displaystyle g'(x) = e^{\sin(x)} \cos(x) \, [2 \cos(x) + f(x)]$.
\part Show that \ $\displaystyle \frac{f'(x)}{g'(x)} = \frac{2 \, e^{-\sin(x)} \cos(x)}{2 \cos(x) + f(x)}$ if $\cos(x) \not= 0$ and $x > 3$.
\part Show that \ $\displaystyle \lim_{x \to \infty} \frac{2 e^{-\sin(x)} \cos(x)}{2 \cos(x) + f(x)} = 0$ and yet $\displaystyle \lim_{x \to \infty} \frac{f(x)}{g(x)}$ is \emph{not} defined.
\end{parts}

%% Ross 31.1
\question Find the Taylor series for $\cos(x)$ and indicate why it converges to $\cos(x)$ for all $x \in \mathbb{R}$.

%% Ross 31.5
\question Let $g(x) = \begin{cases} 0 & \text{if} \enspace x = 0, \\ e^{-1/x^2}  & \text{otherwise}. \end{cases}$
\begin{parts}
\part Show that \ $g^{(n)}(0)$ \ for all $n \in \mathbb{N}$.
\part Show that the Taylor series for $g$ about $0$ agrees with $g$ only at $x = 0$.
\end{parts}

%% Fitzpatrick 8.2.5
\question Prove that \enspace $\displaystyle \left| \sin(x + h) - \left(\sin(x) + h \cos(x)\right) \right| \leq \frac{h^2}{2}$ \enspace for every pair of real numbers $x$ and $h$.

%% Ross 31.11
\question Suppose $f$ is differentiable on $(a,b)$, \ $f'$ is bounded on $(a,b)$, \ $f'$ never vanishes on $(a,b)$, \ and the sequence $(x_n)$ in $(a,b)$ converges to $\bar{x} \in (a,b)$.

Show that if \ $\displaystyle x_n = x_{n-1} - \frac{f(x_{n-1})}{f'(x_{n-1})}$ \ for all $n \geq 0$, \ then \ $f(\bar{x}) = 0$.

\end{questions}

\end{document}
