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\begin{document}

\title{Math 18.100B/C, Spring 2012 - Problem Set \# 7}

%\email{jspeck@math.mit.edu}


\maketitle



\fbox{
\begin{minipage}{1 \textwidth}
\textbf{Professors:} Roman Bezrukavnikov and Jared Speck  \ 
\\
\begin{center} 
{\large \textbf{Due: by 3pm on 4-20-12} (in the Undergraduate Math Office: Room 2-108)}
\end{center}
\end{minipage}
}

\bigskip

\textbf{Instructions:} Write your name and whether you 
are registered for 18.100B or 18.100C. 
Students registered for 18.100C are expected to typeset all of their solutions using LaTex.

Collaboration is permitted but you are advised to first attempt
solving the problems yourself, and you must write up the solutions
independently and mention your collaborators and other sources
in your paper. 

As always in this course (unless stated otherwise) an answer must be
accompanied by a proof.
%\renewcommand{\labelenumi}{\textbf{\Roman{enumi}}.}
%\noindent \textbf{Part I:}

\bigskip
\begin{enumerate}

\item Problem 2 on pg. 114.

\item Problem 6 on pg. 114.

\item Problem 9 on pg. 115.

%\item Problem 15 on pg. 115.

%\item Problem 16 on pg. 116.

%\item Problem 21 on pg. 117.

\item Problem 22 on pg. 117.

\item Problem 26 on pg. 119.

\item Problem 27 on pg. 119.

\item In this problem, you can use the following four facts regarding the function 
$g(x) = e^x;$ we will prove these facts later in the course: \textbf{i)} $g: \mathbb{R} \to (0, \infty)$ 
is an infinitely differentiable function on $\mathbb{R};$ \textbf{ii)} $g' = g;$ \textbf{iii)} $g(-x) = \frac{1}{g(x)};$ 
\textbf{iv)} $\lim_{x \to \infty} g(x) = \infty.$ Now consider the function

	\begin{align*}
				f(x) = \begin{cases}
								0, & \mbox{if} \ x \leq 0, \\
								e^{-1/x^2}, & \mbox{if} \ x > 0.
								\end{cases}
			\end{align*}
Show that $f$ is an infinitely differentiable function on $\mathbb{R}.$ In particular, show that
$f^{(n)}(0) = 0$ for $n = 0,1,2,\cdots.$ Note that this shows that the Taylor polynomials 
centered at $x_0=0$ are not very useful; to the right of $x=0,$ they approximate $f(x)$ with 
$100 \%$ relative error.

\end{enumerate}	


\end{document}