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\begin{document}

\begin{center}
{\Large FINAL

Math 2210}
\end{center}

\begin{itemize}
\item  You have 2 hours to complete this exam.
\item  Show all your work (and make it neat!).  
  If I can't follow your work or it is missing you WILL NOT get credit.
\item  You can use a sheet of notes.
\item  If you run out of space, use the back of these sheets.
\item  Each problem is worth 10 points.
\end{itemize}



Name: \hrulefill


\begin{enumerate}


\item  Find the extrema of the function
\[
f(x,y) = x^2y-6y^2-3x^2
\]

\item  Evaluate the surface integral:
\[
\iint_G \mathbf{F} \dotprod \mathbf{n} \; dS
\]
where $G$ is the surface $z=\sqrt{9-x^2-y^2}$ and $z=0$ 
(so the surface is a hemisphere on top and a disc in the $xy$-plane on the bottom),
$\mathbf{n}$ is the outward pointing normal
and 
\[
\mathbf{F}=(\sin x, (1-\cos x)y, 4z)
\]
(Hint: think about different ways you might be able to compute this!)


\item  Consider the line integral
\[
\oint_C xy \; dx +(x^2+y^2)\;dy
\]
where $C$ is the unit circle traversed in the counter-clockwise direction.
\begin{enumerate}
\item  Compute the line integral directly (as a line integral!).
\item  Compute the line integral by appying Green's Theorem.
\end{enumerate}



\item  Let $R$ be the region in $\R^3$ defined by the following inequalities:
\[
z\leq 4- y^2 \qquad 
y \geq 0 \qquad 
z \geq 0 \qquad
x \geq z \qquad
x+z \leq 9
\]
Write the volume of $R$ as an iterated triple integral.
\\
(Don't compute the integral.)


\item  Consider the function
\[
f(x,y) = x^2y-xy^2+y^3
\]
From the point $(-2,1)$ and in the direction $(3,2)$, is $f$ increasing or decreasing and at what rate?


\item  Consider the double integral below:
\[
\iint_R y-x \;dA
\]
where $R$ is the triangle in the $xy$-plane with vertices $(0,0)$, $(0,-1)$, $(4,2)$.
Make the change of varibles below:
\begin{eqnarray*}
x & = & 2v \\
y & = & -u+v
\end{eqnarray*}
\begin{enumerate}
\item  Compute the Jacobian of this transformation.
\item  The transformation transforms the triangle $R$ into a triangle in the $uv$-plane (you don't have to show this).
	What are the vertices of this triangle in the $uv$-plane?
\item  Write the double integral as a double integral over the triangle in the $uv$-plane.
\item  Compute the integral.
\end{enumerate}



\end{enumerate}

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