\nopagenumbers
\def\pt#1{\hbox{#1) }}
\input amssym.def
\input amssym.tex
\input epsf
\def\R{{\Bbb R}}
\font\srm=cmr8
\magnification=\magstep1

\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 25\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 25} 
\bigskip


\item{1.}  Below is a graph, in polar coordinates, of $r=f(\theta)$.
On each of the six axes provided, sketch a graph of the modified
function indicated.

\medskip
\centerline{
\epsfxsize=2in\epsfbox{/home/oehrtman/m210s96/polarblob.eps}}

\hbox{\vbox{\hsize=3.2in
\itemitem{a)} $r=f(\theta+\pi/2)$ 

\centerline{
\epsfxsize=1.5in\epsfbox{/home/oehrtman/m210s96/polar_axes2.eps}}

\itemitem{c)} $r=f(\theta)-1$ 

\centerline{
\epsfxsize=1.5in\epsfbox{/home/oehrtman/m210s96/polar_axes1.eps}}

\itemitem{e)} $r=f(-\theta)$ 

\centerline{
\epsfxsize=1.5in\epsfbox{/home/oehrtman/m210s96/polar_axes2.eps}}}

\vbox{\hsize=3.2in
\itemitem{b)} $r=f(\theta-\pi/4)$

\centerline{
\epsfxsize=1.5in\epsfbox{/home/oehrtman/m210s96/polar_axes2.eps}}

\itemitem{d)} $r=2f(\theta)$

\centerline{
\epsfxsize=1.5in\epsfbox{/home/oehrtman/m210s96/polar_axes4.eps}}

\itemitem{f)} $r=f(0)$

\centerline{
\epsfxsize=1.5in\epsfbox{/home/oehrtman/m210s96/polar_axes3.eps}}}}

\eject
\item{2.}  Graph each function below in two different ways: first plot
r on the vertical axis as a function of $\theta$ on the horizontal
axis, then plot $r$ at a radial distance from the origin as a function
of the angle $\theta$ from the positive $x$-axis.

\medskip
\itemitem{a)} $r=1+\cos\theta$ \hskip3in b) $r=\cos\theta-1$.

\centerline{
\hskip.2in
\epsfxsize=2.4in\epsfbox{/home/oehrtman/m210s96/cart_axes1.eps}
\epsfxsize=2.4in\epsfbox{/home/oehrtman/m210s96/cart_axes2.eps}}

\bigskip

\centerline{
\hskip.3in
\epsfxsize=2.3in\epsfbox{/home/oehrtman/m210s96/polar_axes5.eps}
\hskip.7in
\epsfxsize=2.3in\epsfbox{/home/oehrtman/m210s96/polar_axes5.eps}}

\medskip
\item{3.}  If two points have polar coordinates $(r_1,\theta_1)$ and
$(r_2,\theta_2)$, show that the distance $d$ between them is given by
$$
d^2=r_1^2+r_2^2-2r_1r_2\cos(\theta_1-\theta_2).
$$
What does this mean geometrically?


\medskip
\item{4.}    Write down the Maclaurin series for $\sin x$, $\cos x$, and
$e^x$.  Prove that $e^{i\theta}=\cos\theta+i\sin\theta$.  Using this
formula, express the given numbers in the form $re^{i\theta}$ for a
positive real number $r$ and argument $\theta$, $-\pi<\theta\leq\pi$:
$$
{\eqalign{
\pt a &{e^2\over\sqrt2}-{e^2\over\sqrt2}i\cr\cr
\pt b &3\left(\cos{\pi\over4}+i\sin{\pi\over4}\right)\cr}}\qquad
{\eqalign{
\pt c &5\left(\cos{\pi\over6}+i\sin{\pi\over6}\right)\cdot
3\left(\cos{\pi\over2}+i\sin{\pi\over2}\right)\cr\cr
\pt d &7\left(\cos{7\pi\over3}+i\sin{7\pi\over3}\right)\cr}}
$$


\medskip
\item{5.}  Find the intersection of the curve $r=1-\cos\theta$ and the
circle $r=\cos\theta$.



\bye