\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} Investigate the family of curves
defined by the parametric equations $x=t^2$, $y=t^3 - ct$, where $c$
is a positive constant.

\medskip

\noindent a) Graph the curves for $c={1\over 4}$, $c= 1$ and $c=4$. What
features do all the curves have in common? (You may need to adjust the
graphing window as you change $c$.)  How does the shape change as $c$
increases? Find the $x$- and $y$-coordinates of all points where the
tangent line is horizontal or vertical.

\medskip
 
\noindent b) Verify that the point $(c,0)$ is on the curve for any
$c>0$. How many tangent lines does the curve have at the point $(c,
0)$? What are their slopes?

\noindent Check your answer numerically (for $c={1\over 4}$, $c= 1$
and $c=4$) by drawing the tangent lines on your graphing
calculator. 

\medskip

\noindent c) Consider the curve corresponding to $c={1\over 3}$. Part
of this curve is a loop. Find the length of that loop.










\vfil\eject\end

