\documentclass[oneside]{amsart}

% Boilerplate to put your name on every page
% This makes grading easier
\usepackage{authoraftertitle}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhead{}
\fancyfoot{}
\lhead{\MyAuthor}
\rhead{\thepage}

% Use letters in enumerated lists
\usepackage{enumitem}
\setenumerate[0]{label=(\alph*)}

% You can define abbreviations here
\newcommand{\problem}[1]{\par\noindent\textbf{Problem~#1}}
\newcommand{\answer}[1]{{\par\medskip\sffamily#1\par\medskip}}
\newcommand{\cl}{\mathop{\mathrm{cl}}\nolimits}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}

\begin{document}

% Put your title, name, and the date here
\title{Homework 9}
\author{Your Name Here}
\date{April 13, 2020}

\maketitle


\problem{2.4.2.} If $f$ and $g$ are functions on $(a,b)$ such that each has derivatives of order $k$ on $(a,b)$ for $1\leq k\leq n$, show that $fg$ has a derivative of order $n$ on $(a,b)$ and that
\[(fg)^{(n)} =\sum_{k=0}^n \frac{n!}{(n-k)! \, k!} f^{(k)}g^{(n-k)}.\]

\answer{Type your answer here.

A prime can be gotten by a simple apostrophe (you don't need the \texttt{\char`\\prime} command):
$f'$ \texttt{\$f'\$}

You get binary coefficients (``$n$ choose $k$'') with \texttt{\char`\$\char`\\binom\char`\{n\char`\}\char`\{k\char`\}\char`\$} $\binom{n}{k}$.

A multiline display can be done neatly as follows:\\
\begin{align*}
(x+y)^2 
    & = (x+y)(x+y) \\
    & = x^2+2xy+y^2.
\end{align*}
}

\clearpage

\problem{2.5.1.} Is Theorem 2.5.2 valid if $L=\infty$? Either prove Theorem 2.5.2 under these circumstances or find $f$ and $g$ that satisfy the conditions of Theorem 2.5.2  with $L=\infty$ but such that $\lim_{x\to c} f(x)/g(x)\neq \infty$.

\answer{Type answer here.}


\clearpage

\problem{(AB 7)} Let $a<c<b$. Suppose that $f \in C^{(n)}(a,b)$ and $f^{(n)}$ is differentiable on $(a,b)\setminus \{c\}$. Let $P_c(x)=\sum_{k=0}^n \frac{f^{(k)}(c)}{k!} (x-c)^k$ be the Taylor polynomial for $f$  of degree $n$ expanded about~$c$. Suppose that $G$ is continuous on $(a,b)$, and that $G$ is differentiable on $(c,b)$ with $G'(z)>0$ for all $z\in (c,b)$.
\begin{enumerate}
    \item Show that for every $x \in (c,b)$ there is a $y$ with $c<y<x$ and with $f (x)=P_c(x)+\frac{f^{(n+1)}(y)}{n!} (x-y)^n \frac{G(x)-G(c)}{G'(y)}$.

    \item What value of $G$ yields the formula $f (x)=P_c(x)+\frac{f^{(n+1)}(y)}{(n+1)!} (x-c)^{n+1}$ proven in class? 
\end{enumerate}


\answer{
\begin{enumerate}
    \item Answer to part (a)
    \item Answer to part (b)
\end{enumerate}
}


\end{document}