\def\wsn{33}
\input worksheet.tex

\item{1.}  Compute the following integrals
$$
\pt a \int_0^{1\over\sqrt2}{1\over\sqrt{2-x^2}}\,dx\hskip1in
\pt b \int{dx\over7+3x^2}\hskip1in
\pt c \int{x\,dx\over\sqrt{1-4x^4}}
$$


\hbox{\vbox{\hsize=5in
\itemitem{2.\hskip12pt a)}  Sketch a graph of the function
$\displaystyle{f(x)={1\over1+x^2}{\atop.}}$   

\medskip
\itemitem{b)}  Find the area between the graph and the $x$-axis from
$x=-1$ to $x=1$. 

\medskip
\itemitem{c)}  Find the area between the graph and the $x$-axis from
$x=-100$ to $x=100$.}
\hskip.3in
\epsfxsize1.5in\epsfbox{/home/oehrtman/m210/cartmansimon.ps}}

\medskip
\itemitem{d)}  Let $M$ be a very large number.  Find the area between
the graph and the $x$-axis from $x=-M$ to $x=M$.  

\medskip
\itemitem{e)}  What happens to the area as $M\to\infty$?  What is the
meaning of this in terms of your graph?

\vskip.5in
\item{3.}  A billboard $k$ feet wide is perpendicular to a straight
road and is $s$ feet from the road.  At what point on the road would a
motorist have the best view of the billboard; that is, at what point
on the road is the angle subtended by the billboard a maximum?

\vskip.5in
\item{4.}    Find the fallacy in the following argument that $0=1$.

\medskip
\hbox{\item{}\epsfxsize1in\epsfbox{/home/oehrtman/m210/pinkscarf.ps}
\hskip.3in\vbox{\hsize=4.5in
$$
{\eqalign{u&={1\over x}\cr du&=-{1\over x^2}dx\cr}}\hskip1in
{\eqalign{dv&=dx\cr v&=x\cr}}
$$}}

\item{}
So integration by parts applied to $\int 1/x\,dx$ yields:
$$
\eqalign{
0+\int{dx\over x}
&=\left({1\over x}\right)x-\int\left({-1\over x^2}\right)x\,dx\cr
&=1+\int{dx\over x}{\atop.}}
$$
Thus $0=1$.


\hbox{\vbox{\hsize=4.5in
\item{5.}  One of the most important functions in analysis is the
{\it gamma function}, 
$$
\Gamma(x)=\int_0^\infty e^{-t}t^{x-1}\,dt,\qquad x>0.
$$
\medskip
\itemitem{a)}  Use integration by parts to prove that
$\Gamma(x+1)=x\Gamma(x)$.}
\hskip.3in
\epsfxsize2in\epsfbox{/home/oehrtman/m210/turkeyrush2.ps}}

\medskip
\itemitem{b)}  Show that $\Gamma(1)=1$.  Conclude that
$\Gamma(n)=(n-1)!$ for all natural numbers $n$.

\medskip
\item{}  The gamma function provides a simple example of a
continuous function which {\bf interpolates} the values of $n!$ for
natural numbers $n$.



\bye