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\newtheorem{theorem}{Theorem}
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\begin{document}






\center{ Math 336, Test 3, Spring (1993)}\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.  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}}
\def\ourstrut{\vrule width 0.0pt depth 0.0pt height 25.0pt\relax}

\prob1
Suppose that $g$ is a continuous function on $[a,b]$ which is
differentiable on $(a,b)$, and assume that $g'$ is integrable on $[a,b]$.
Prove that
$$\int^b_ag'=g(b)-g(a)\quad.$$


\prob2 (a) List the properties satisfied by the distance function~$d$
in a metric space. 
\fakeprob2 (b) Let $B$ be the set of all bounded sequences
$\bold x=(x_1,x_2,\ldots)$ and define $$d(\bold x, \bold
y)=\operatorname{sup}\{\vert x_j-y_j\vert : j=1,2,\ldots\}\quad.$$ Prove
that $d$~satisfies the triangle inequality.


\prob3 (a) State the Weierstrass Approximation Theorem as it applies
to functions $f$ on a closed interval $[a,b]$.  \fakeprob3 (b) Give an example to
show that the conclusion of the theorem is false if the closed
interval $[a,b]$ is replaced in the statement by $(-\infty,\infty)$.
Justify your answer.


\prob4 (a) Use the identity
$\sum_{n=0}^\infty(-1)^nx^{2n}=1/(1+x^2)$, $x\in(-1,1)$, to find
a power series formula for $\operatorname{arctan}x$ that is valid for
$x\in(-1,1)$.  \fakeprob4 (b) Combine this formula with the calculation
$\operatorname{tan}(\pi/6)=1/\sqrt 3$ to obtain an explicit series formula
for~$\pi$.


\prob5  Differentiate the following two functions.
\roster
\ritem a $$f(x)=\int_0^{x^2} \sin t \,dt$$
\ritem b $$g(x)=x^{\cos x}$$
\endroster

\enddocument



