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\begin{document}

\begin{flushright}
  A. Miller\\
  M521 Exam 2\\
  April 14, 1994
\end{flushright}

\bigskip

\begin{center}
  Keep this sheet.  Use your own paper for the answers.\\
\end{center}

\bigskip

\begin{center}
  Results proved in class
\end{center}

\xprob{10} Suppose $f:[a,b]\arrow\reals$ and $g:[a,b]\arrow\reals$
are continuous on $[a,b]$ and differentiable on $(a,b)$.
Suppose also that $g(x)\not=0$ for every $x$ in $(a,b)$.
Prove that there exists a $c\in (a,b)$ such that
$${{f^\prime(c)}\over {g^\prime(c)}}=
{{f(b)-f(a)}\over {g(b)-g(a)}}.$$

\xprob{10} Prove that if $f_n:D\arrow \reals^m$ converges uniformly
to a $f:D\arrow \reals^m$
and each $f_n$ is continuous, then $f$ is continuous.  Begin by defining
uniform convergence.

\xprob{10} Prove the following theorem.
If $f$ is $C^\prime$, then $f$ is differentiable.
Begin by defining  $C^\prime$ and differentiable.

\xprob{10} Prove the following.
Suppose $g$ is $C^{(n+1)}$ on the interval $(-r,r)$, $x\in (-r,r)$,
and $R$ is defined by:
$$g(x)=g(0)+g^\prime(0)x+{{g^{(2)}(0)}\over2}x^2\cdots+
                         {{g^{(n)}(0)}\over {n!}}x^n + R.$$
Then for some $\zeta$ strictly between $x$ and $0$
$$R={{g^{(n+1)}(\zeta)}\over {(n+1)!}}x^{n+1}.$$


\newpage


\begin{center}
  Exercises
\end{center}

\xprob{10} Prove that the derivative of a function $f:\reals\arrow\reals$
has the intermediate value property, i.e., for any $a<b$ and $v$ if
$v$ is between $f^\prime(a)$ and $f^\prime(b)$, then for some $c\in (a,b)$
we have $f^\prime(c)=v$.


\xprob{10} Find and classify the critical points of
$f(x,y)=x^2+y^2+xy^2$.

\xprob{10} (a) Give an example of a function $f:\reals^2\arrow\reals$ such
that ${{\partial f}\over{\partial x}}$ and  ${{\partial f}\over{\partial y}}$
exist for every point $p\in\reals^2$, but $f$ is not differentiable
at $(0,0)$.

(b) Give an example of a function $f:\reals\arrow\reals$ which is
differentiable, but $f^\prime$ is not continuous.

Be sure and prove that your examples have the required properties.

\bigskip

\begin{center}
  New problems
\end{center}

\xprob{10} Give an example of a sequence $f_n:\reals\arrow\reals$ of
differentiable functions such that $f_n$ converges uniformly to
a function $f$ which is not differentiable.

\xprob{10}
Suppose $f:\reals^n\arrow\reals^n$ and its inverse function
$f^{-1}:\reals^n\arrow\reals^n$ are differentiable, and
$f(p)=q$.  Prove that the matrix $Df^{-1}(q)$ is the inverse of the matrix
of $Df(p)$.

\xprob{10} Suppose $f:\reals^2\arrow\reals$ and for every
$(x,y)$ we have $|f(x,y)|\leq |xy|$.  Prove that $f$ is differentiable
at $(0,0)$.

\end{document}