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

\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 29\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 29} 
\bigskip


\item{1.}   An object's position at time $t$ is given by the path
$\sigma(t)$.  
\item{}  Suppose that $\sigma'(2)=(1,-3,7)$,
$\sigma'(2.5)=(2,0,4)$, $\sigma''(2)=(2,3,1)$, and
$\sigma''(2.5)=(0,1,-1)$.

\medskip
\itemitem{a)}  At time $t=2$, is the object's velocity changing?

\medskip
\itemitem{b)}  At time $t=2$, is the object's speed changing?  If so,
is it increasing or decreasing?

\medskip
\itemitem{c)}  At time $t=2.5$, is the object's velocity changing?

\medskip
\itemitem{d)}  At time $t=2.5$, is the object's speed changing?  If so,
is it increasing or decreasing?


\medskip 
\item{2.}  Suppose that an object's path is given by $\gamma(t)=
(t^2,\sin t)$.

\medskip
\itemitem{a)}  Sketch the path from time $t=-\pi$ to time $t=\pi$.

\medskip
\itemitem{b)}  Compute the velocity vector at times $t=0$ and times
$t=1$ and draw them on your graph based at the object's position at
each of those times. 

\medskip
\itemitem{c)}  Set up an integral that indicates the distance traveled by
the object from time $t=0$ to time $t=1$.

\medskip
\itemitem{d)}  Give a parameterization of this path in polar
coordinates. 

\medskip
\item{3.}  Recall from Worksheet 27, the sad predicament that our
good friends, the Idiots, faced:

\medskip
{\leftskip=1in\rightskip=1in\srm\noindent In the Odd Galaxy, there is
a planet called Id.  The Idiots, the inhabitants of Id, have noticed
that their planet goes in a large slow circular orbit given by the
path $\scriptstyle {\bf I}(t)=(\sin t,\cos t,0)$.  The unit of time is
one million Earth years, and the unit of length is one Idian
Astronomical Unit (IAU).  Their astronomers notice a very large
asteroid with path given by $\scriptstyle {\bf R}(t)=(\csc t,0,\cot
t)$ for $\scriptstyle 0