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\center Math 336, Test 2, Spring (1994) \endtitle


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.  
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.}\ }
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%\define\fakeprob#1{\par\noindent\phantom{{\bf #1.}}\ }
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\prob1 (40 pts) Prove the following theorem.
\par\smallskip\par\noindent {{\bf Abel's Theorem}.}
Let $f(x)=\sum_{n=0}^\infty$ be a power series with
finite positive radius of convergence~$R$. If the series
converges at $x=R$, then $f$ is continuous at $x=R$. If the
series converges at $x=-R$, then $f$ is continuous at $x=-R$.

\prob2 (20 pts)
Let $f$ be a continuous function on $[a,b]$ that is differentiable on
$(a,b)$ and satisfies $f(a)=f(b)$. Show that there exists at least one
$x$ in $(a,b)$ such that $f'(x)=0$. (This is {\bf Rolle's Theorem}.)


\prob3 (20 pts) {\bf Weierstrass's Approximation Theorem} states that
if $f$ is a continuous function on a closed interval $[a,b]$, then
there exists a sequence $(p_n)$ of polynomials such that $p_n\to f$
uniformly on $[a,b]$. Show that the conclusion of the theorem fails
without the assumption that the interval involved is closed and
bounded, by showing that there is no sequence of polynomials which
converges uniformly on ${\Bbb R}$ to $f(x)=\cos x$.


\prob4 (20 pts) 
Let $f(x)$ be the function with domain ${\Bbb R}$ and range
$(-\pi/2,\pi/2)$ given by $f(x)=\operatorname{arctan}x$. Use the Mean
Value Theorem to show that 
$$ |f(x)-f(y)|\le|x-y|$$
for all $x$ and $y$ in $\Bbb R$.

\enddocument



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