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\begin{center}
{\Large TEST 2

Math 2210 - 2}
\end{center}

Instructions:
\begin{itemize}
\item  You have 50 minutes to complete this exam.
\item  Show all your work.  If I can't follow your work or it is missing you WILL NOT get credit.  
	(This holds for graphing too!)
\item  Use the back of these sheets if you run out of room.  Make sure I can find your work.
\item  No calculators or notes.
\item  The test is out of 60 points -- each problem is worth 10 points.
\end{itemize}

\hrule

\begin{enumerate}

\item  
\begin{enumerate}
\item  Let $(x,y,z)=(-2,-2,-3)$, find the coordinates of this point in cylindrical coordinates.
\item  Let $(\rho,\theta,\phi)=(4,5\pi/3,3\pi/4)$, find the Cartesian coordinates of this point.
\end{enumerate}


\item  Consider the function:
\[
f(x,y)=xe^y + \cos(xy)
\]
\begin{enumerate}
\item  Find all first partial derivatives ($f_x, f_y$)
\item  Find all second partial derivatives ($f_{xx}, f_{xy}, f_{yx}, f_{yy}$)
\end{enumerate}


\item  Graph the level curves $z=k$ for $k=-2,0,2$:
\[
z=-x^2-4y^2+1
\]


\item  Graph the following function:
\[
z = 1-x^2
\]


\item  Consider the limit:
\[
\lim_{(x,y) \ra (0,0)} \frac{x^2}{x^2-y^2}
\]
Consider the paths to the origin along the $x$-axis and then along the $y$-axis.
Show that the limit does not exist.




\item  Describe the following surfaces where ``$\mathrm{constant}$'' could be any constant number. 
	(You might want to graph them to help me understand your description).
\begin{enumerate}
\item  $\theta=\mbox{constant}$, in cylindrical coordinates.
\item  $\theta=\mbox{constant}$, in spherical coordinates.
\item  $\phi=\mbox{constant}$, in spherical coordinates.
\item  $r=\mbox{constant}$, in cylindrical coordinates.
\item  $\rho=\mbox{constant}$, in spherical coordinates.
\end{enumerate}



\end{enumerate}

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