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\begin{document}








\begincenter{Math 336, Test 1, Spring 1996}\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 (textbooks, 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}}
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\def\ourstrut{\vrule width 0.0pt depth 0.0pt height 15.0pt\relax}
\define\R{\Bbb R}
\define\Q{\Bbb Q} 
\define\Limit{\Cal L}

\prob1 (a) (20 pts) State and prove the Intermediate Value Theorem.
(b) (10~pts) Does there exist a continuous  function whose
domain is
the closed interval $[0,1]$  and whose range is the entire set
$[0,1]\cup[5,6]$? Justify
your answer. (c) (10~pts) Does there exist a continuous
function whose domain is the closed interval $[0,1]$ and whose range
is the entire real line? Justify your answer.

\prob2 (20 pts) 
(a) (10 pts) State the Mean Value Theorem. (b) (10~pts) Show that for
all real numbers $a$, $b$ there is an inequality
$$ \vert \sin a -\sin b\vert \le \vert a-b\vert\,\,.$$

\prob3 (a) (10 pts) Suppose that $f$ is a function defined on a set
$S$ of real numbers. Define what it means for $f$ to be {\sl uniformly
continuous} on~$S$. (b) (10~pts) Give an example of a function~$f$
which is continuous on $(0,1)$ but {\bf not} uniformly continuous on
this interval. Justify your answer carefully.

\prob4 (a) (10 pts) Suppose that $f$ is a function on $[a,b]$ with the
property that $f^{(n)}$ is continuous on $[a,b]$ and differentiable on
$(a,b)$. State the Taylor Mean Value Theorem for this function~$f$.
(b) (10~pts) In the situation above, take $a=0$, $b=1/2$,
$f(x)=\cos x$ and $n=2$.  Write down the expression provided by
Taylor's Theorem for $f(b)$.

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