\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} a) Suppose $f(x,y)=x^y$. Rewrite $f$
as a composition of standard functions, and then find the domain of
$f$ and the first partial derivatives of $f$ in its domain. Surely
$2^3 = 8$. If $x$ is increased by $.01$ (so $(x,y)$ changes from
$(2,3)$ to $(2.01,3)$), approximate the change in $f$ using linear
approximation. If $y$ is increased by $.01$ (so $(x,y)$ changes from
$(2,3)$ to $(2,3.01)$), approximate the change in $f$ using linear
approximation.  Compare the ``exact answers'' to the linearization
answers.

\medskip

\noindent b) Suppose $g(x,y,z)= x^{\left( y^z \right)}$. Rewrite $g$
as a composition of standard functions, and then find the domain of
$g$ and the first partial derivatives of $g$ in its domain.  Surely
$2^{\left( 3^4 \right)} \approx 2.417 \cdot
10^{24}$ (it is {\it exactly}\/ $24178\ 51639\
22925\ 83494\ 12352$.). If one of the variables in $g$ is increased by
$.01$, which variable will likely make the biggest change in the value
of $g$?  Support your assertion by an argument using linear
approximation based on the derivatives which have been calculated. The
exact values of $g$ are not requested, but only a decision based on
the linearized approximations to the perturbed function values.

\vfil\eject\end

