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\begin{document}

\begincenter{ Math 336, Test 3, 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}}
\define\ritem#1{\item"(#1)"}
\define\dom{\operatorname{dom}}
\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 (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 (a) (10 pts) State the Fundamental Theorem of Calculus.
(b) (10~pts) Let $f$ be a continuous function on $[-2,\infty)$, and
$F$ the function given by 
$F(x)=\int_{-2}^xf(t)\,dt$. Define a function $G$ on
$(-\infty,\infty)$ by $G(y)=F(\sin y)$, so that in more explicit
terms
$G(y)=\int_{-2}^{\sin y}f(t)\,dt$. Use the Fundamental Theorem of
Calculus and the Chain Rule to compute $G'(y)$.

\prob3 In the first three parts of this problem, 
$f$ denotes a continuous function on $[a,b]$ with the property that
$f(x)\ge0$ for all $x\in[a,b]$. We assume in addition
that there is a number
$c$ in $(a,b)$ such that $f(c)>0$. Let $y=f(c)$. 
\roster
\ritem a (5~pts) Show
that there exists a number $\delta>0$ such that
$I=(c-\delta,c+\delta)$ is contained in $(a,b)$ and such that $f(x)>y/2$ for all
$x\in I$. 
\ritem b (5~pts) Let $g$ be the function obtained by setting
$g(x)=y/2$ for $x\in I$ and $g(x)=0$ otherwise. Observe that $f(x)\ge
g(x)$ for all $x\in[a,b]$. 
\ritem c (5~pts) Conclude that $\int_a^bf\ge (y/2)\delta$; in particular,
conclude that $\int_a^bf$  is greater than zero.
\ritem d (5~pts) Give an example of an interval $[a,b]$ and a function
$f$ on $[a,b]$ such that $f$ is integrable, $f(x)\ge0$ for all
$x\in[a,b]$, $f(c)>0$ for some $c\in[a,b]$, but $\int_a^bf=0$. Is it
possible to find an example like this in which $f$ is continuous?
\endroster

\prob4 (a) (5 pts) Define the {\sl radius of convergence} of a power
series $\sum c_nx^n$.

For each of the following power series, compute both the
radius of convergence and the interval of convergence (5 points for
each series).
\def\ritem#1{\text{(#1) }}
$$\ritem b \sum_{n=1}^{\infty} x/\sqrt n \quad\quad
\ritem c \sum_{n=0}^\infty x^n/3^n        \quad\quad
\ritem d \sum_{n=1}^{\infty} n^2x^n
$$

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