\noindent 2) (10 points) The following table gives the windchill $W$
 at various temperatures  $T$, ($^{\circ}F$) and wind speeds $v$
(mph).\vskip .2in
\moveright .25in
\vbox{\offinterlineskip
\halign{\strut $#$ \quad & \vrule \hfil\quad $#$ \quad \vrule & \hfil \quad $#$
 \quad \vrule & \hfil \quad $#$ \quad \vrule & \hfil \quad $#$
  \quad \vrule & \hfil \quad $#$ \quad \vrule & \hfil \quad $#$
  \quad \vrule & \hfil \quad $#$ \hfil \cr
  \noalign{\hrule}
   & T =15 & T =10 &T = 5 &T = 0 \cr
  \noalign{\hrule}
  v= 15& -11 & -18 & -25 &-31 \cr\noalign{\hrule}
   v= 20& -17 & -24 & -31 & -39 \cr\noalign{\hrule}
 v= 25& -22 & -29 & -36 & -44 \cr
  \noalign{\hrule}}}\par\vskip .2in

a) Use the table to find approximate values for $\part Wv $, $\part WT $
when $T=0$ and $v=25$.\par\vskip 2in\noindent
2a) continued \vskip 1in
b) Using the table and your answers to a), give (an approximation of) the
local linearization for $W$
around $T=0$ and $v=25$.\par\vskip 1in
c) Use the local linearization to approximate $W$  when $T=-5$ and
$v=25$.

\noindent 6.) (14 points) Suppose you are given two infinite
wires which are parallel to the $z$-axis, passing through the points
$(1,0)$ and $(-1,0)$ in the $xy$ plane which are homogeneously and
oppositely charged.
The potenial energy $P$ of a unit charge in the $xy$ plane
is given by
$$P=(1/2)\ln((x-1)^2+y^2)-(1/2)\ln((x+1)^2+y^2)$$
a) What is the gradient of the potential energy at $(3,3)$?\vskip 1.5in

b) What is the rate of change of $P$ in the direction $\vec u=1/2\vec
i-\sqrt 3/2\vec j$ at $(3,3)$? \vskip 1in

c) An {\it equipotential line} is a curve  along which the electrical potential
 is constant. What is the direction of the equipotential line
passing through  $(3,3)$? \vskip 1.25in

d) Another particle with a unit charge is moving up the $y$ axis at a rate
of $5m/sec$.
 What is the rate of change of the potential energy of this
particle?\vfill\eject