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\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 30\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}


\centerline
{\bf WORKSHEET 30} 
\bigskip

\item{1.}  Imagine swinging in a circular motion a weight tied to a
string.  What can be said about this situation?
\smallskip
\item{}  This is a very open ended question.  Play with it.  How can
you use vector functions and their derivatives to talk about this
situation?  What interesting questions can you ask?  Can you answer
those questions?

\medskip 
\item{2.}  Consider the paths 
$$\eqalign{
\gamma(t)&=(\cos t,t,t^2) \cr
\hbox{and}\qquad
\sigma(t)&=(t^2+1,-t,\sin t). \cr}
$$
Find the equation of a plane which is tangent to both of these curves
at a point of their intersection.


\medskip
\item{3.}  Sketch the graph of the parametric curve $x(t)=\sqrt{1-t}$,
$y(t)=\sqrt{t}$.  Find the value(s) of $t$ (if any) where the slope is
$0$, $\pm1$, $\pm\infty$.


\medskip
\item{4.}  Parametrize the unit circle and use this to find an
equation for the tangent line to the circle at the point
$(1/2,\sqrt3/2)$. What is the equation of the tangent line at 
arbitrary point?


\medskip
\item{5.}  Suppose that $(x,y,z)$ is a point on the helix with radius
$a$ and height $h$.  What is the equation of the tangent line to the
helix at this point?


\medskip
\item{6.}  Suppose you are constructing a smokestack that is 30 feet
tall and six feet in diameter.  In order to lend support to the
structure, you decide to add a {\it strake} which is a support
cut to spiral up the outside of the smokestack exactly once.  You
decide to use pieces of metal for the strake which are cut as a
portion of a circle of some diameter.  What diameter should you use to
best fit the strake onto the smokestack?

\vfill




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