%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[12pt]{article}
\usepackage{amssymb}
\usepackage{epsfig}

\topmargin-2cm
\textheight 24cm
\textwidth18cm
\oddsidemargin-.7cm
\evensidemargin-.7cm
\parindent0cm
\parskip 2.5ex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\input{shortcuts.tex}
\newcommand{\re}{\rm Re}
\newcommand{\im}{\rm Im}
\newcommand{\Ds}{\displaystyle}
\newcommand{\dfrac}[2]{{\Ds \frac{#1}{#2}}}
\newcommand{\junk}[1]{}
\newcommand{\text}[1]{\mbox{\rm{#1}}}
\newcommand{\ignore}[1]{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcounter{prob}
\newcounter{subprob}
\newenvironment{problems}{
     \begin{list}{\bf{\arabic{prob}.}\hfill}{\usecounter{prob}
     \topsep-1.4ex \labelwidth.6cm \leftmargin.7cm \labelsep.1cm
     \rightmargin0cm \parsep2ex \itemsep2ex
     \partopsep1.4ex}}{\end{list}}
\newenvironment{subproblem}{
     \begin{list}{\bf{(\alph{subprob})}\hfill}{\usecounter{subprob}
     \topsep-1.4ex \labelwidth.6cm \leftmargin.8cm \labelsep.2cm
     \rightmargin0cm \parsep2ex \itemsep2ex
     \partopsep1.4ex}}{\end{list}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

{\bf Name:} \underline{\mbox{\hspace*{7cm}}}
\hfill{\bf Instructor:} Jeffrey Diller

\begin{center}
{\bf Math 336: Real Analysis II}
\\
{\bf Spring Semester 2000}
\\
{\bf Exam 2}
\\
{\bf Friday, April 14}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} Examination contains 6 problems on 7 sheets
of paper including the front cover.
Do all your work on the paper provided.
\\[1.5ex]
\hline
\end{tabular}

\begin{center} {\bf Scores}

\begin{tabular}{|c|c|c|}\hline
Question & Possible\quad &\ Actual\quad\rule[-.3cm]{0cm}{0.9cm} \\ \hline\hline
  1      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  2      &  40  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  3      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  4      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  5      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  6      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline\hline

Total  & 100\mbox{\  \ } & \rule[-.3cm]{0cm}{0.9cm} \\ \hline
\end{tabular}
\end{center}

\vspace*{.5cm}
\begin{center}     {\bf GOOD LUCK}\end{center}
\newpage

\begin{problems}

\item Do all three of the following (5 points each).  
\begin{subproblem}
\item Define what it means for a function $f:\R^n\to\R^m$ to be 
      differentiable at a point.
\vspace*{1.5in}
\item State the definition of an open subset of $\R^n$.
\vspace*{1.5in}
\item State the Weierstrass $M$-test.
\vspace*{1.5in}
\end{subproblem}
\newpage

\item Decide whether each of the following statements is true or false.  
If false, give a counterexample.  Do four out of five parts.  (10 points each)
\begin{subproblem}
\item $\sum_{j=1}^\infty a_j$ converges if and only if
      $\sum_{j=1}^\infty |a_j|$ converges.
\vspace*{1.3in}
\item Let $\sum_{j=1}^\infty a_j$ be a series whose terms are all
      greater than or equal to zero and whose partial sums are all less 
      than $\pi$.  Then the series converges.
\vspace*{1.3in}
\item Let $E_j\subset \mathbf{R}$ be closed for every $j\in\N$.  Then
      $\bigcup_{j=1}^\infty E_j$ is closed.
\vspace*{1.3in}
\item Let $f:\R^2\to\R$ be a function such that 
$\lim_{x\to 0} f(x,0) = \lim_{y\to 0} f(0,y) = 0$.  Then
$\lim_{(x,y)\to(0,0)} f(x,y) = 0$.
\vspace*{1.3in}
\item Suppose that $\sum_{j=0}^\infty a_j(x-3)^j$ converges at the
point $x = 1$.  Then it also converges at the point $x=4$.
\end{subproblem}

\newpage
\item Suppose that $f:\R^2\to\R^2$ is given by 
$f(x,y) = (\sin(x+y),\cos(x-y))$.
\begin{subproblem} 
\item 
Compute the Jacobian matrix of $f$ at an arbitrary point $(x,y)$ and explain 
why
the result implies that $f$ is differentiable everywhere in $\R^2$.
{\it Do {\bf not} try to use the definition of derivative in your explanation!}
\vspace*{3in}
\item
Suppose that $g:\R^2\to \R^3$ is a function such that 
$$
Dg_{(s,t)}(\mathbf{h}) = 
\left(
\begin{array}{cc}
s & t \\
t & s^2 \\
e^st & 1 
\end{array}
\right)
\left(
\begin{array}{c}
h_1 \\
h_2
\end{array}
\right).
$$
Compute the matrix for the derivative $D(g\circ f)_{(\pi/2,0)}$.
\end{subproblem}

\newpage
\noindent{\bf Do two out of the following three problems--15 points
          each}

\item Consider the series $\sum_{j=1}^\infty \frac{1}{(x+j)^4}$.
Show that this series converges uniformly for $x\in [0,1]$. 

\vspace*{5in} 
\noindent Explain why the limit function is actually $C^1$.



\newpage

\item The Taylor series centered at $0$ for $\cos x$ is 
$\sum_{j=0}^\infty \frac{x^{2j}}{(2j)!}$.
Show that this series converges to $\cos x$ for every $x\in\R$.
\newpage

\item Suppose that $f,g:\R^n\to\R^n$ are $C^1$ functions satisfying
$g\circ f(\mathbf{x}) = \mathbf{x}$ for every $\mathbf{x}\in\R$.  
Show that if $f$ has Jacobian matrix $A$ at $\mathbf{x}$, then $g$ has 
Jacobian matrix $A^{-1}$ at $f(\mathbf{x})$.  
\vspace*{6in}

\noindent Use the above assertion to show that if $f:\R^2\to\R^2$
is given by $f(x,y) = (x^3-2y^2,x\sin y)$, Then there exists no
$C^1$ function $g:\R^2\to\R^2$ satisfying $g\circ f(\mathbf{x}) = \mathbf{x}$
for every $\mathbf{x}\in\R^2$.
\newpage




\end{problems}



\end{document}
\end






