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\begin{document}

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

\begin{center}
{\bf Math 366: Honors Real Analysis II}
\\
{\bf Spring Semester 2004}
\\
{\bf Exam 1}
\\
{\bf Wednesday, March 3}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} examination contains 5 problems.  Counting
the front cover and blank pages, the exam consists of 6 sheets
of paper.  
\\[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      &  25  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  2      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  3      &  30  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  4      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  5      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \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 each of the following (5 points each).  
\begin{subproblem}
\item State Weierstrass' approximation theorem.
\vspace*{2in}

\item Define what it means for a function $f:\R^n\to\R^m$ to
be \emph{differentiable} at a point $\mathbf{x}\in\R^n$.
\vspace*{1.5in}

\item Define the \emph{norm} of a linear transformation $T:\R^n \to \R^m$.
\vspace*{1.5in}

\item State the inverse function theorem.
\newpage

\item State the \emph{chain rule} (this semester's version!).
\end{subproblem}
\vspace*{2in}

\item Do one of the following two (use the back of the page if 
necessary, 15 points).
\begin{subproblem}
\item Consider $f:\R^2\to\R$ given by $f(0,0) = 0$ and 
$$
f(x,y) = \frac{xy}{x^2+y^2}
$$
otherwise.  Show that both partial derivatives of $f$ exist at every
point in $\R^2$ but that $f$ is not continuous at $0$.

\item Consider the vector space $\mathcal{C}([0,1],\R)$ of continuous
functions $f:[0,1]\to\R$ with the norm
$$
\norm{f} = \int_0^1 |f(x)|\,dx.
$$
Show that the function $T:\mathcal{C}([0,1],\R) \to \R$ given by
$T(f) = f(1)$ is a linear transformation with infinite norm.
\end{subproblem}

\newpage
\item Do two of the following three (15 points each).
\begin{subproblem}
\item Compute the norm of the linear transformation $T:\R^2\to\R^2$
given by $T(x,y) = (x+2y,y)$.
\item
Consider the system of equations
\begin{eqnarray*}
x^2 + y^3 + y + 2z & = & 0 \\ 
2x^5 + 3x + y^2 - z & = & 0.
\end{eqnarray*}
Clearly $x=0,y=0,z=0$ is a solution for the system.  Find (in a reasonably 
systematic way) `decent' values for $x$ and $y$ that, together with $z=.1$, 
approximate another solution to this system.
Indicate when you're finished what portion of your computation gives good
reason to believe that there actually \emph{is} a point $(x,y,.1)$ near
$(0,0,0)$ solving the system.
\item
Consider the set
$$
M = \{(x,y,z)\in\R^3:x+y+2yz = 1, zy^3 - 2x = -2\}.
$$
At what points might $M$ fail to be a submanifold of $\R^3$?  
{\bf Also,} describe the tangent space to $M$ at the point $(1,0,0)$.
\end{subproblem}

\newpage
\item Do one of the following two.  Feel free to use any results you
like from homework, the book, or lecture (15 points).
\begin{subproblem}
\item Let $f:\R^2\to\R$ be a differentiable function satisfying
$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y}$ at 
every point.  Show that $f(x,y) = f(x+y,0)$ for all $(x,y)\in\R^2$.

\item Suppose that $a > 0$ is a given real number and that 
$f:[-a,a] \to\R$ is a continuous function such that 
$$
\int_{-a}^a f(x)x^{2n}\,dx = 0
$$
for every $n\geq\N$.  Prove that $f$ must be an odd function (i.e. 
$f(-x) = -f(x)$ for every $x\in\R$).  {\it Hint:  use the fact that
$$
f(x) = g(x) + h(x)
$$
where $g(x) := \frac{f(x)+f(-x)}{2}$ is an even function of $x$ and
$h(x) := \frac{f(x)-f(-x)}{2}$ is an odd function of $x$.}
\end{subproblem}

\newpage
\item Do one of the problems you skipped (your choice, 15 points).

\end{problems}
\end{document}
\end
