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\begin{document}







\begiincenter{Math 336, Test 3, Spring  1995}\endcenter

The test will be 50 minutes 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 (texbooks, notes) during the test.
What you write should be neat, grammatical, clear and concise. Good
luck.


%\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.}}\ }
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\define\R{\bold R}

\prob1 (20 pts)
Let $f(x)=\sum_{n=0}^\infty a_nx^n$ have radius of convergence $R>0$. Prove 
that $f$ is differentiable on $(-R,R)$ and that
$$f'(x)=\sum_{n=1}^\infty na_nx^{n-1}\quad\quad\text{for }\vert
x\vert<R.$$
 

\prob2 (20 pts) Give a careful statement of the Weierstrass
Approximation Theorem.


\prob3 (20 pts) Consider the following assertion: {\sl if $f$ is a
continuous function on $\R$, then there exists a sequence $(p_n)$ of
polynomial functions such that $(p_n)$ converges uniformly {\bf on
$\R$} to~$f$.} Show by example that this assertion is {\bf
false}. Give careful arguments to show that your example is a relevant
one. 




\prob4 (a) (10 pts) Give a definition of the log function $L(y)$ in
terms of a definite integral. (b) (10~pts) Use this definition to
prove that for any two positive numbers $y$ and $z$,
$L(yz)=L(y)+L(z)$.

\prob5 Let $f(x)=x^{\sin x}$ for $x>0$. (a) (10 pts) Write $f(x)$ in
terms of the exponential function $E$ and the log function $L$. (b)
(10 pts) Calculate $f'(x)$.


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