\documentclass[11pt]{article}
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}

\textwidth = 6.5 in
\textheight = 9 in
\oddsidemargin = 0.0 in
\evensidemargin = 0.0 in
\topmargin = 0.0 in
\headheight = 0.0 in
\headsep = 0.0 in
\parskip = 0.2in
\parindent = 0.0in

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}{Definition}


\center{Math 336, Final Exam, Spring  1995}


The exam will be two hours in length.  Please write your name on the
cover of your blue book and write the solutions inside; start the
solution to each problem on a new page.  Each problem has the point
value indicated.  This test is being administered under the provisions
of the Honor Code. Your work should be your own, and you should not
make use of any outside material (textbooks, notes) during the test.
What you write should be neat, grammatical, clear and concise. Good
luck.

{\bf Please write your name at the top of this question sheet and
return the sheet inside
your exam booklet.}



%\define\prob#1{\par\medskip\noindent{\bf#1.}\ }
\define\prob#1{\par\medskip\noindent\hbox to 3ex{\bf#1.\hfil}}
%\define\fakeprob#1{\par\noindent\phantom{{\bf #1.}}\ }
\define\fakeprob#1{\par\noindent\hbox to 3ex{\hfil}}
\define\ritem#1{\item"(#1)"}
\define\dom{\operatorname{dom}}
\def\ourstrut{\vrule width 0.0pt depth 0.0pt height 15.0pt\relax}
\define\R{\bold R}

\prob1 (a) (10 pts) State Abel's Theorem. (b) (15~pts) Consider the
power series
$$ x -\frac {x^2}2 + \frac{x^3}3 -\frac{x^4}4 +\cdots =
\sum_{n=1}^\infty(-1)^{n+1}\frac{x^n}n\,\,.$$ 
Take for granted that this power series converges to $\log(1+x)$ for
$x$ in the {\bf open interval} $(-1,1)$. Use this fact, together with
Abel's theorem, to establish the identity
$$\log(2) = \sum_{n=1}^\infty\frac{(-1)^{n+1}}n\,\,.$$
Be sure to give a careful argument. Your argument is almost certainly
incomplete if it does not use the fact (which you may also take for
granted) that the function $\log(1+x)$ is continuous at $x=1$.


 

\prob2 (a) (10 pts) Let $(f_n)$ be a sequence of functions on a
set~$S$. Define what it means for the sequence $(f_n)$ to {\bf converge
uniformly} on $S$ to a function $f$. (b) (15~pts) Suppose that $(f_n)$
is a sequence of functions on $S$ which converges uniformly on $S$ to
a function $f$, and that $g$ is a bounded function on~$S$. (Here
``bounded'' means that there exists some $M>0$ such that $\vert
g(x)\vert<M$ for all $x\in S$). Using the definition from part (a),
show that the sequence $(gf_n)$ converges uniformly on~$S$ to the
product function $gf$. (c) (10 pts {\bf extra credit}) Show by example that
$(gf_n)$  does not necessarily converge uniformly to $gf$  if
$g$ is not bounded on~$S$.


\prob3 (a) (25 pts) Let $f$ be the function on $[-1,1]$ given by the
formula
$$f(x) = \cases -1 & -1\le x<0\\
                 0 & x=0\\
                 1 & 0<x\le 1
         \endcases
$$
Consider the question of whether there exists a sequence $(p_n)$ of
polynomial functions on $[-1,1]$ such that $(p_n)$ converges {\bf
uniformly} on $[-1,1]$ to~$f$. Either show that such a sequence exists
(perhaps by invoking an appropriate approximation theorem) or show
that such a sequence cannot exist. (b) (10 pts {\bf extra credit}) Answer
the same question (carefully!) with ``uniform convergence'' replaced
by ``pointwise convergence''.

\prob4 Let $f$ be the function defined as follows:
$$ f(x)=\cases x^2 \sin(1/x)& x\ne 0\\
               0 & x=0
        \endcases
$$
(a) (10 pts) Using the ordinary rules for differentiation, show that
$f$ is differentiable at $x$ for each $x\ne0$ and calculate $f'(x)$
for $x\ne0$.
(b) (10 pts) Use the {\bf definition} of the derivative to show that $f$ is
differentiable at $x=0$ and to calculate that $f'(0)=0$.
(c) (5 pts) Show that the function $f'$ is not continuous at $x=0$.

\prob5 (a) (10 pts) State the Mean Value Theorem. (b) (15 pts) Suppose
that $f$ is a function which is differentiable on $(a,b)$ and assume
that $f'(x)<0$ for all $x\in(a,b)$. Use the Mean Value Theorem to show
that  $f$ is strictly decreasing on $(a,b)$.

\prob6 Let  $f$ be a differentiable function on $(-\infty,\infty)$ and
suppose that
$$\lim_{x\to\infty} [f(x)+f'(x)] = L\,\,,$$
where $L$ is some number (i.e., $L$ is {\bf not} $\pm\infty$). (a) (15
pts) Show that $\lim_{x\to\infty} f(x)=L$. (Hint: You might want to make
the peculiar observation
$$\lim_{x\to\infty}f(x) = \lim_{x\to\infty}\frac{f(x)e^x}{e^x}$$
and calculate the limit on the right with L'Hospital's rule. If you
decide to do this, make sure that you explain why it is that
L'Hospital's rule applies to this situation.)  (b) (10 pts) Show that
$\lim_{x\to\infty} f'(x)=0$.

\prob7 (25 pts) Give an example of a function $f$ on $[0,1]$ which is
{\bf not} Darboux integrable. Explain thoroughly why your example is 
correct.

\prob8 (25 pts) Give an example of a function $f$ on $[0,1]$ such that
\roster
\item"$\bullet$" $f$ is Darboux integrable, and
\item"$\bullet$" the function $F(x)$ given by 
$$F(x)=\int_0^x f(t)\,dt$$
is {\bf not} differentiable on $(0,1)$. 
\endroster
Justify your answer carefully. It might be helpful to illustrate your
argument by drawing the graphs of $f$ and $F$.

\enddocument



% Local Variables: 
% mode: AmS-TeX
% TeX-master: t
% TeX-master: t
% End: 
