\documentclass[12pt]{amsart}


\begin{document}
\centerline{\bf MAT 205 SPRING 2000}
\centerline{\bf MIDTERM II}

\noindent
Name:
\\[.5cm]
SSN:
\\[.5cm]
\centerline{\bf THERE ARE FOUR PROBLEMS PLUS ONE EXTRA CREDIT PROBLEM}
\centerline{\bf SHOW YOUR WORK!!!}
\\[.5cm]
1.  Suppose 
\begin{align}
&z=2x^2y+xe^y \notag \\
&x(u,v)=v \sin{u} \notag \\
&y(u,v)=uv \notag 
\end{align}
Compute $\frac{\partial z}{\partial u}$ and $\frac{\partial z}{\partial
v}$.

\newpage
\noindent
\\[.5cm]
2.  A box has dimensions $l$, $w$, and $h$ which vary in the following
way:
\begin{align}
&l(t)=t^2+2 \notag \\
&h(t)=e^{2t}+1 \notag \\
&w(t)=4t^4+2t^2+2 \notag
\end{align}
Let $V = V(l,w,h)=l \times w \times h$ be the volume function of the box.
Find the rate
of change with respect to time of the volume of the box at $t=0$ seconds.	

\newpage
\noindent
\\[.5cm]
3.  The region $D$ in $\mathbb{R}^2$ shown on the blackboard is
bounded by
$x=1$, $y=e^x$, and $y=1-x^2$.  Compute
$$
\int \int _D xdA
$$

\newpage
\noindent
\\[.5cm]
4.  Let $z=f(x,y)=xy^3-x^2y$.  \\
a.  Find the gradient of $f$ at the point $(x,y)=(3,2)$.
\\[5cm]
b.  Find the equation of the tangent plane to the graph of the surface at
the point $(3,2,6)$.  
\\[5cm]
c.  Let ${\bf u}$ be the unit vector pointing in the direction given by
$\theta  
= - \frac{\pi}{3}$.  Find the derivative of $f$  in the direction ${\bf
u}$ at
the point $(x,y)=(3,2)$.

\newpage
\noindent
\\[.5cm]
Extra Credit:\\
Let $f(x,y)$ be a differentiable function on $\mathbb{R}^2$.  Two unit
vectors $\bf{u}$ and $\bf{v}$ are given by 
\begin{align}
&\bf{u}= \frac{\sqrt{2}}{2} \bf{i}+ \frac{\sqrt{2}}{2} \bf{j} \notag \\
&\bf{v}= \frac{\sqrt{2}}{2} \bf{i} - \frac{\sqrt{2}}{2} \bf{j} \notag 
\end{align}
Assume that $D_{\bf{u}}f(0,0) = \frac{3 \sqrt{2}}{2}$ and
$D_{\bf{v}}f(0,0) = \frac{\sqrt{2}}{2}$. Find $\nabla f(0,0)$. 





\end{document}

