\documentclass[11pt]{article}
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}

\textwidth = 6.5 in
\textheight = 9 in
\oddsidemargin = 0.0 in
\evensidemargin = 0.0 in
\topmargin = 0.0 in
\headheight = 0.0 in
\headsep = 0.0 in
\parskip = 0.2in
\parindent = 0.0in

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

\center{Math 336, Test 2, Spring  (1993)}


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 first problem is worth 40
points and the others 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\ritem#1{\item"(#1)"}
\define\dom{\operatorname{dom}}
\def\ourstrut{\vrule width 0.0pt depth 0.0pt height 25.0pt\relax}

\prob1 [40 pts] Describe the construction of the integral $\int_a^bf$
and discuss the major properties of the integral.

\prob2 Prove that if $f$ is a function which is continuous on the
closed interval $[a,b]$ then $f$ is integrable on $[a,b]$.


\prob3 Determine which of the following functions are integrable on
the interval $[0,2]$. Justify your answers carefully.  For each function which is
integrable, compute the integral.
\roster
\ritem a [6 pts] $$f(x) = \cases1, &\text{for  $0\le x\le1$}\\
                                x, &\text{ for $1<x\le2$}.\endcases$$
\ritem b [7 pts] $$f(x)=   \cases0,&\text{ if $x$ is a rational
number}\\
                                 1,&\text{otherwise}.\endcases$$
\ritem c [7 pts] $$f(x)= \cases1/x^2,&\text{if $x\ne0$}\\
                                 0,&\text{if $x=0$}.\endcases$$
\endroster

\prob4 (a) Find the Taylor Series expansion of $f(x)=\sin x$. (b)
Indicate why this series converges to $\sin x$ for all $x\in\Bbb R$.

\enddocument


