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


\center{Math 336, Test 1, Spring  1995}

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.  The problems are worth
20~points apiece; in the case of a problem with multiple sections, the
credit will be divided equally between the parts unless otherwise
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.}}\ }
\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
State and prove Taylor's theorem.

\prob2 Give examples to show that the following two statements are
{\bf FALSE}:
\roster
\ritem a A uniformly continuous function $f:\R\to\R$ is
bounded.
\ritem b Every sequence which has a convergent subsequence is
bounded above.
\endroster

\prob3 
(a) Carefully state the Weierstrass M-test for the uniform convergence
of a series of functions. (b) Show that $\sum_{n=1}^\infty(1/n^2)\cos
nx$ converges uniformly on $\R$ to a continuous function.


\prob4 
(a) Carefully state the Mean Value Theorem. (b) Suppose that $f$ is a
function which is continuous on $[0,\infty)$ and differentiable on
$(0,\infty)$. Assume that $f(0)=0$ and that $f'(x)<0$ for $x>0$. Use
the Mean Value Theorem to show that $f(x)<0$ for $x>0$.

\prob5 Let $(f_n)$ be a sequence of functions on a closed interval
$[a,b]$.
\roster
\ritem a (5 pts) Define what it means for $(f_n)$ to converge
{\bf pointwise} to a function~$f$.
\ritem b (5 pts) Define what it means for $(f_n)$ to converge
{\bf uniformly} to a function~$f$.
\ritem c (10 pts) Give an example of a sequence $(f_n)$ which converges pointwise but does not converge uniformly.
\endroster

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