\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} a) Suppose that $z=f(x,y)$, that
$x=g(t)$ and $y=h(t)$, and that the functions $f$, $g$, and $h$ are
twice differentiable.  Use the Chain Rule to find an expression for
${{\textstyle d^2z}\over {\textstyle dt^2}}$.

\medskip
 

\noindent b) An insect crawls on a metal plate in the plane. At time
$t=1$ its position vector is ${\bf i}+2{\bf j}$, its velocity is
$2{\bf i}-{\bf j}$, and its acceleration is $3{\bf i}+4{\bf j}$.
Suppose that the temperature of the plate at the point $x,y$ is a
certain function $T(x,y)$ satisfying
 
$$\matrix{T(1,2) = 2,\ & T_x(1,2) = -1, \ & T_y(1,2) = 3, \cr
\noalign{\smallskip}
T_{xx}(1,2) = 0, \ & T_{xy}(1,2) = 1, \ & T_{yy}(1,2) = -2.\cr}$$


\noindent If $T(t)$ is the temperature experienced by the insect at
time $t$, find ${{\textstyle dT}\over {\textstyle dt}}$ and
${{\textstyle d^2 T}\over {\textstyle dt^2}}$ at time $t=1$.








\vfil\eject\end