\documentclass[fleqn, 12pt]{article}
%\pagestyle{empty}
\setlength{\topmargin}{-.5in}
\setlength{\oddsidemargin}{-.3in}
\setlength{\textheight}{9in}
\setlength{\textwidth}{7.2in}


\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{enumerate}

\newcommand{\adj}{\mathrm{Adj}}
\newcommand{\mbf}{\mathbf}
\newcommand{\R}{\mathbb{R}}
\newcommand{\ra}{\rightarrow}
\newcommand{\proj}{\mathrm{proj}}
\newcommand{\dotprod}{\bullet}

\newcommand{\sol}{{\textbf{Solution: }}}
\newcommand{\inch}{\vspace{1in}}

\newcommand{\lecture}{{\Large\textbf{Lecture Problems}}}

\begin{document}

\begin{center}
{\Large Warm-Up Problems and Lecture Problems

April 23, 2003}
\end{center}

\begin{enumerate}

\item  \label{one}
	 Find the Taylor series for $e^{-x}$ centered at $0$.

\inch

\item  We know the series for $e^x$:
	\[
	e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!} + \cdots
	\]
	How is this series related to your answer in question~\ref{one}?

\newpage

\lecture
\item  Using a Taylor polynomial for $f(x)=e^x$, find $e$, accurate to within $0.01$:
	\begin{enumerate}
	\item  For any $n$, what is the $n$-th derivative of $f(x)$?  $f^{(n)}(x)=$
\inch

	\item  Find an integer $M$ so that $|f^{(n)}(x)|\leq M$ for all $x$ in the interval $[-1,1]$.
		(This will be your value for $M$.)
\inch

	\item  Find the smallest integer $N$ so that $\frac{M(1^{N+1})}{(N+1)!} < 0.01$
\inch

	\item  For the value of $N$ that you found, write down the $N$-th Taylor polynomial $T_N(x)$:
\inch

	\item  For your value of $N$, compute $T_N(1)$.
\inch

	\item  Compare $T_N(1)$ with $e$ on your calculator.

	\end{enumerate}


\end{enumerate}


\end{document}