\input amssym.def
\def\d{\displaystyle}
\input epsf
\def\d{\displaystyle}
\nopagenumbers
\magnification= \magstep1

\parindent=0pt
\font\largerbold = cmbx10 scaled \magstep2

\centerline {\largerbold Problems, Week 10}


%finishing up the semester with a few problems on polar coordinates
%and parametric curves. please give some ``plain vanilla'' problems in
%addition to these if you have time. 

\bigskip


1.~If $A$ is a positive number, the parametric curve 
$\d { \cases{x = t^3-t\cr y= {\textstyle A\vphantom{'}\over
{\textstyle{1+t^{2\vphantom{'}}}}}\cr}}$ looks like:

\medskip

\centerline{\epsfbox{pic10-1.eps}} 

\medskip

There's one value of the number $A$ so that the ``self-intersection''
of the curve is perpendicular and the picture looks like:

\medskip
\centerline{\epsfbox{pic10-2.eps}}
\medskip
Find that value of $A$.

\bigskip

2.~A window is in the shape of a $2'$ by $3'$ rectangle surmounted by
a semicircle\footnote*{\sevenrm It is traditional for a calculus
course to have a problem discussing this sort of window, including the
rather uncommon word ``surmount'' as part of the description. We honor
tradition here!}. Describe the boundary of the window as $r=f(\theta)$
in a polar coordinate system whose location and orientation you
specify. Note that $f(\theta)$ will need to be defined ``piecewise''.

\medskip

\centerline{\epsfbox{pic10-3.eps}}

\vfil\eject

3.~A point $P$ travels on the parabola $y=x^2$.

\smallskip

a) Give parametric formulas for the location of $P$ where the
parameter is the first coordinate of the point $P$. 

\medskip

b) Give parametric formulas for the location of $P$ where the
parameter is the second coordinate of the point $P$. The parametric
formulas will have to be given ``piecewise''.

\medskip

c) Give parametric formulas for the location of $P$ where the
parameter is the angle that the ray from the origin to
$P$ makes with the positive $x$-axis.

\medskip

d) Give parametric formulas for the location of $P$ where the
parameter is the angle that the ray from the point $(0,1)$ to
$P$ makes with the ray from $(0,1)$ to the origin. 

\medskip

e) Give parametric formulas for the location of $P$ where the
parameter is the distance from the origin to $P$. The
parametric formulas again will have to be given ``piecewise''. 


\bigskip

4.~Find the Cartesian coordinates of $\underline{\rm all}$ points of
intersection of the curves with polar equations $r^2= 4\sin \theta$
and $r= 1-\sin \theta$.  (Warning: $(0,{\theta}_1)$ and
$(0,{\theta}_2)$ represent the same point {\it always}; also, so do
$(r,\theta)$ and $(-r, \theta \pm \pi)$.)

\vfil\eject\end




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