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\newtheorem{theorem}{Theorem}
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\center/bf{ Math 336, Test 3, Spring (1994)}


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}}
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\prob1
Let $f$ be a continuous function on $[a,b]$. Prove
that $f$ is integrable on $[a,b]$.

\prob2 Let $f$ be a continuous function on $[a,b]$ which is
differentiable on $(a,b)$. Suppose that $f'$ is integrable on $[a,b]$.
Prove that $\int^b_af'=f(b)-f(a)$.

\prob3 Let $f$ be a continuous function on $[a,b]$. The {\bf Mean
Value Theorem for integrals} describes a particular property of $f$
with respect to integration.
\roster
\ritem a State the Mean Value Theorem for integrals.
\ritem b Give an example of a discontinuous function $f$ on a closed
interval $[a,b]$ for which the conclusion of the Mean Value Theorem
for integrals is {\bf false}. Justify your answer. (Hint: consider a step
function.)
\endroster


\prob4 
Calculate $$\lim_{x\to 0}\frac 1x\int_0^xe^{\sin t}dt\,\,.$$


\prob5 
Let $f$ be the function with $f(x)=e^x$ if $x$ is a rational number and
$f(x)=0$ if $x$ is not a rational number. Determine whether or not $f$
is integrable on $[0,1]$. Justify your answer.

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