\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} Do the following for each of the parametric curves below:

\medskip \noindent 
i) Compute the curvature. 

\smallskip \noindent ii) Understand and explain the result
independently of the curvature computation, for example by graphing
the parametric curve, recognizing it geometrically, and finding the
curvature from the geometric information.

\smallskip \noindent 
iii) Verify the geometric assertion in ii) by direct algebraic
manipulation.

\medskip

\def\vsp{\vphantom{a_{J_{J_J}}}}

\hskip 1.in\vtop{\hsize=2.5in \noindent {\bf Curve A}

\medskip

\noindent $\cases{\displaystyle x(t)={{1-t^2}\over{\vsp 1+t^2}}\cr
\displaystyle y(t)={{2t}\over{1+t^2}}\cr}$.

}

\vskip -.87in

\hskip 2.5in
\vtop{\hsize=2.in \noindent {\bf Curve B}

\medskip

\noindent $\cases{\displaystyle x(t)={{2-t^2}\over{\vsp 1+t^2}}\cr 
\displaystyle y(t)={{2t^2-2}\over{\vsp 1+t^2}}\cr
\displaystyle z(t)={{3t^2-2}\over{1+t^2}}\cr}$.

}

\medskip

\noindent {\bf Comment/hint} While these computations and graphs can
{\it possibly}\/ be done ``by hand'', certainly working with a
computational environment like {\tt Maple} will make investigation
more practical. But the direct computations must still be explained as
requested.

\vfil\eject\end