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\centerline{\bf Math 336, Final Exam, Spring  1996 }

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.}\ }
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\define\dom{\operatorname{dom}}
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\define\R{\bold R}

\prob1 (20 pts) Suppose that $f$ is a function which is continuous on
the closed interval $[a,b]$. Prove that $f$~is uniformly continuous on
$[a,b]$.

\prob2 (20 pts) Let $f$ be a function which is continuous on the
closed interval $[a,b]$ and differentiable on the open interval
$(a,b)$. Assume that $f(a)=f(b)$. Prove that there exists a number
$c\in(a,b)$ with $f'(c)=0$. (This is {\sl Rolle's Theorem}).

\prob3 (20 pts) Show that there exists a number~$x$ between $0$ and~$1$
such that $2^x=10x$.

\prob4 (20 pts) Show that if $f'(x)$ exists, then
$$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}\,\,.$$
Bonus (10 extra credit pts): give an example of a function $f$ and a
number~$x$ such that $f$ is defined in a neighborhood of $x$ and the
above limit exists, but $f'(x)$ does {\bf not} exist.

\prob5 (a) (10 pts)  State the Taylor Mean Value Theorem. Write out
the degree~$n$ Taylor expansion of the function $f(x)=e^x$, including
the remainder term. (b) (10 pts) Set $x=1$ in the above Taylor
expansion of $e^x$ and show that the remainder term approaches~$0$ as
$n$ approaches infinity. Conclude that $e=\sum_{n=0}^\infty 1/n!$.


\prob6 (a) (10 pts) Describe the {\sl integral test} for determining
whether or not a series converges. (b) (10 pts) Use the integral test
or any other method to determine whether or not the series $\sum
_{n=2}^\infty 1/n\log n$ converges.

\prob7 (10 pts) Consider the series
$$
\sum_{n=1}^\infty \left(\frac{(-1)^n}{\sqrt n} +\frac 1{n^2} +\frac
{4^n}{(n!)^2}\right)\,\,.$$
Determine whether or not the series converges. Give reasons for your
answer. (Hint: you may have to break the series up and apply more than
one test.)

\prob8(20 pts)  Give an example of a bounded function $f$ on the
closed interval $[0,1]$ which is {\bf not} integrable. Produce a
careful argument to justify the conclusion that $\int_0^1f$ does not
exist. At the very least, you will have to refer to the definition of
the integral.


 


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