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\newtheorem{theorem}{Theorem}
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\begincenter{Math 336, Test 2, 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.}}\ }
\define\fakeprob#1{\par\noindent\hbox to 3ex{\hfil}}
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\define\dom{\operatorname{dom}}
\def\ourstrut{\vrule width 0.0pt depth 0.0pt height 15.0pt\relax}
\define\R{\bold R}

\prob1
(40 pts) Write an essay at least two pages in length on the
theory of integration. State the major definitions in the theory,
describe the major theorems, and give the proof of at least one of
these theorems.

\prob2 (20 pts) Let $(f_n)$ be a sequence of continuous functions on $[a,b]$,
and suppose that $f_n\to f$ uniformly on $[a,b]$. Prove that
$$\lim_{n\to\infty}\int_a^bf_n(x)\,dx=\int_a^bf(x)\,dx.$$

\prob3 (20 pts) Calculate
$$\lim_{n\to\infty}\int_1^2e^{-nx^2}\,dx.$$ It is probably not a good
idea to attempt to evaluate these definite integrals
directly. If you make use of any theorem in calculating the limit, 
give careful and complete arguments to show that the
assumptions of the theorem are satisfied.




\prob4 
Let $f$ be the following function on $[-1,1]$: $f(t)=0$ for $-1\le
t<0$; $f(t)=1$ for $0\le t<1/2$ or for $1/2<t\le1$; $f(1/2)=-1$.
Define $F(x)$ for $x\in[-1,1]$ by
$$F(x)=\int_{-1}^x f(t)\,dt.$$
\roster
\ritem a (5 pts) Sketch the graphs of $f(x)$ and of $F(x)$.
\ritem b (5 pts) Where is the function $F$ continuous?
\ritem c (5 pts) Where is the function $F$ differentiable?
Calculate $F'(x)$ for all $x$ at which $F$ is differentiable.
\ritem d (5 pts) Explain how your answer to (c) fits in with the
statement of
the second form of the fundamental theorem of calculus.
\endroster


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