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\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 Final Exam}
\\
{\bf Wednesday, May 10}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} Examination contains 8
 problems on 11 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      &  30  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  2      &  50  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  3      &  25  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  4      &  25  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  5      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  6      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  7      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  8      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline\hline
Total  & 175\mbox{\  \ } & \rule[-.3cm]{0cm}{0.9cm} \\ \hline
\end{tabular}
\end{center}

\vspace*{.5cm}
\begin{center}     {\bf GOOD LUCK}\end{center}
\begin{center} ...and best wishes for life after real analysis.
\end{center}

\newpage

\begin{problems}

\item Do all of the following four (10 points each).  
\begin{subproblem}
\item State the definition of uniform convergence $\{a_n\}_{n=1}^\infty$.
\vspace*{1.5in}
\item State the contraction mapping principle
\vspace*{1.5in}
\item State the definition of convergence for an infinite series.
\vspace*{1.5in}
\item State the inverse function theorem
\vspace*{1.5in}
\end{subproblem}
\newpage

\item One of the two statements in each of the following pairs is false.
Identify the false statement and give a counterexample.  In lieu of
a precise counterexample, you can get at least partial credit in some
cases by drawing an appropriate picture.  Do five of
six parts. (10 points each)
\begin{subproblem}
\item 
\begin{itemize}
\item
If $f_n:[0,1]\to\R$ converges uniformly to $f:[0,1]\to\R$, then $f_n$ 
converges to $f$ pointwise.
\item
If $f_n:[0,1]\to\R$ converges pointwise to $f:[0,1]\to\R$, then
$f_n$ converges to $f$ uniformly.
\end{itemize}
\vspace*{2.5in}

\item 
\begin{itemize}
\item 
If $\{a_n\}$ is a sequence converging to zero, then 
$\sum_{n=0}^\infty a_n$ converges.
\item
If $\sum_{n=0}^\infty a_n$ converges, then $\{a_n\}$ is a sequence 
converging to zero.
\end{itemize}
\vspace*{2.5in}

\item
\begin{itemize}
\item 
If all the partial derivatives of $f:\mathbf{R^n}\to\mathbf{R^m}$ 
exist at $\mathbf{0}$, then $f$ is differentiable at $\mathbf{0}$.
\item
If$f:\mathbf{R^n}\to\mathbf{R^m}$ is differentiable at $\mathbf{0}$, then
all the partial derivatives of $f$ exist at $\mathbf{0}$. 
\end{itemize}
\newpage

\item 
\begin{itemize}
\item If $f:\R^n\to\R^n$ is a $C^1$ function with a local inverse
$g$ defined near $f(\mathbf{x})$, then $g$ is $C^1$ and 
$Dg_{f(\mathbf{x})} = (Df_{\mathbf{x}})^{-1}$ 
\item
If $f:\R^n\to\R^n$ is a $C^1$ function and $Df_{\mathbf{x}}$ is
invertible, then there is a local inverse $g$ for $f$ defined
near $f(\mathbf{x})$.
\end{itemize}
\vspace*{2in}

\item 
Let $\sum_{j=0}^\infty c_j x^j$ be a power series.
\begin{itemize}
\item If the the series converges at $x=R>0$, then it converges at
      $x=-R$.
\item If the series converges at $x=R>0$, then it converges uniformly
      on $(-r,r)$ for all $r<R$.
\end{itemize}
\vspace*{2in}

\item
Let $f_n:\R\to\R$ be $C^1$ functions such that $f_n(0) = 0$ for every
$n\in\N$.
\begin{itemize}
\item
If $f_n$ converges uniformly on $\R$, then $f'_n$ converges uniformly on $\R$.
\item
If $f'_n$ converges uniformly to $g$ on $\R$, then $f_n$ converges 
uniformly on $\R$.
\end{itemize}
\vspace*{1.5in}
\end{subproblem}
\newpage

\item Let $f:\R^2\to\R^2$ be given by $f(x,y) = (e^x - e^y, \ln(x+y))$.
Note that $f(1,1) = (0,\log 2)$.  
\begin{subproblem}
\item Compute the Jacobian matrix for $f$ (5 points).  
\vspace*{2in}
\item Explain why there exists a local inverse $g$ for $f$ defined near
$(0,\log 2)$. (5 points)
\vspace*{2in}
\item Give the linear approximation at $(0,\log 2)$ for $g$. (10 points)
\newpage
\item Using $(1,1)$ as a first guess, find a better guess at a point
$(x,y)$ such that $f(x,y) = (.1,\log 2)$  {\it Do {\bf not} simplify your 
arithmetic!} (5 points). 
\vspace*{3in}
\end{subproblem}

\item Consider the intial value problem
$$
y'(t) = (t+y(t))^2,\qquad y(0) = 2
$$
Let $r>0$ be given and $E$ be the set of all continous functions 
$y:(-r,r)\to [0,4]$.
\begin{subproblem}
\item  
As in the proof of the existence and uniqueness theorem for solutions
to first order ordinary differential equations,
write down a a function $T:E\to E$ whose fixed point will be a solution
of the initial value problem. 
(8 points)
\newpage
\item  
Given a first guess of $y(t) = 2$, use your $T$ to find a better guess at a 
solution of the initial value problem.
(8 points)
\vspace*{3in}
\item 
Show that if $r$ (see statement of problem) is chosen small enough, then $T$ 
will be a contraction mapping (distance between functions is measured by the 
$\sup$ norm).  It is important here that functions in $E$ are bounded 
above and below by $4$ and $-4$, respectively.
(9 points)
\end{subproblem}
\newpage

\noindent{\bf Do 3 out of the following four problems. (15 points each)}

\item Let $f,g:\R^n\to\R^m$ be differentiable at $\mathbf{0}$.  Let
$a,b\in\R$ be constants and 
$h(\mathbf{x}) = a f(\mathbf{x}) + b g(\mathbf{x})$.  Show that
$h$ is differentiable at $\mathbf{0}$.
\newpage

\item Show that if $\sum_{n=0}^\infty a_n$ and
$\sum_{n=0}^\infty b_n$ converge absolutely, and $c_n = \max\{a_n,b_n\}$
for every $n$, then $\sum_{n=0}^\infty c_n$ converges.  
\newpage

\item Suppose that $a<b$ are numbers and $f_j:[a,b]\to\R$ are continuous
functions converging uniformly to $f:[a,b]\to\R$.  For each $x\in[a,b]$
and each $j\in\N$, let
$$
F_j(x) = \int_a^x f_j(t)\,dt.
$$
Show that $F_j(x)$ converges uniformly to 
$$
F(x) = \int_a^x f(t)\,dt.
$$
\newpage

\item Suppose that $f:\R\to\R$ is an infinitely differentiable function whose 
derivatives satisfy 
$$
\left|\frac{d^nf}{dx^n}\right| \leq n!
$$
for every $n\in\N$ and every $x\in\R$.  Let 
$\sum_{j=0}^\infty a_j x^j$ be the Taylor series centered at $0$ for $f$.
Give an interval $(-r,r)$ (i.e. find $r$) on which this Taylor series 
converges to $f$ and prove that your answer is correct.
\newpage

\end{problems}



\end{document}
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