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\begin{center}{ \Large{Worksheets}  \\\large{9/21/2017} }\end{center}
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\begin{enumerate}
\item Let $f(x,y,z)=xy^2z^3$. Compute the following partial derivatives\[\frac{\partial f}{\partial x},~\frac{\partial f}{\partial y},~\frac{\partial f}{\partial z},~\frac{{{\partial }^{2}}f}{\partial x\partial y},~\frac{{{\partial }^{2}}f}{\partial {{z}^{2}}}.\]
\vspace{3.5in}
\item With the function $f$ given above, find
\[\frac{\partial f}{\partial x}(1,2,-1)\]
in two ways: (1) plugging the numbers into the function found in the previous problem, (2) using limit definition.
\newpage
\item Let \[A=\left[ \begin{matrix}
   1 & 2 & 1  \\
   0 & 1 & 1  \\
   3 & -1 & 2  \\
\end{matrix} \right],~~~B=\left[ \begin{matrix}
   1 & 1  \\
   2 & 3  \\
   1 & 0  \\
\end{matrix} \right],~~~v=\left[ \begin{matrix}
   -1  \\
   2  \\
\end{matrix} \right]\]
Compute $AB$, $BA$, and $Bv$.\vspace{3.8in}
\item Given four points A(1,0,1), B(2,1,1), C(2,3,0) and D(0,-1,2). Find the volume of the parallelepiped of which A, B, C, D are vertices such that A is adjacent to B, C, D.
\newpage
\item Write the parametric equation of the line which is the intersection of the planes $x+2y+z=0$ and $2x-y=1$.
\vspace{3.8in}
\item Find the area of the quadrilateral whose vertices are A(1,1), B(3,0), C(4,3) and D(1,2).
\newpage
\item Indicate whether each of the following maps is a linear map from $\mathbb{R}^2$ to $\mathbb{R}^2$. If a map is linear, find the matrix associate with it.\begin{enumerate}
\item $f(x,y) = (x,x+y)$
\item $g(x,y) = (xy,0)$
\item $h(x,y) = (y-x,x,y)$
\item $k(x,y) = (0,0)$
\end{enumerate}
\vspace{4in}
\item (Just for fun!) Is there a linear map from $\mathbb{R}^2$ to $\mathbb{R}^2$ which maps a rectangle to a circle?
\end{enumerate}

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