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\begin{center}{ \Large{Worksheets}  \\\large{12/12/2017} }\end{center}
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\item Find and classify all critical points of the function\[g(x,y)=\frac{1}{4}{{x}^{4}}-\frac{5}{3}{{x}^{3}}+{{y}^{3}}+3{{x}^{2}}-\frac{3}{2}{{y}^{2}}+20\]
\newpage
\item Find maximum and minimum values of $f(x,y)=x(x^2+y^2-1)$ on the disk $x^2+y^2\le 1$.\vspace{3.5in}
\item Let $f(x,y)=\dfrac{1}{x^2+y^2}$
\begin{enumerate}
\item Find the equation of a plane tangent to the graph $z=f(x,y)$ at $x=0$, $y=1$, $z=f(0,1)$.\vspace{3in}
\item Find linear approximation of $f$ at $(x,y)=(0,1)$.\vspace{3.5in}
\end{enumerate}
\item Let \[f(x,y)=({{x}^{2}}-3,~3xy-{{y}^{3}}),~~~~g(u,v)=(2uv,~{{u}^{2}}-{{v}^{2}},~u+v).\]
Compute $D(g\circ f)(1,2)$.\vspace{4in}
\item Evaluate the integral\[\int_{0}^{2}{\int_{2y}^{4}{8\sqrt{{{x}^{2}}+1}dxdy}}\]by changing the order of integration.\newpage
\item Evaluate $\iiint_{E}{zdV}$ where $V$ is the region between the cylinder $x^2+y^2=1$ above the $xy$-plane and below the cone $z=\sqrt{x^2+y^2}$.\vspace{4in}
\item Let $S$ be the boundary of the tetrahedron with vertices at (0,0,0), (1,0,0), (0,1,0), (0,0,1). Use the Divergence Theorem to compute\[\iint_{S}{\vec{F}\cdot d\vec{S}}\]
where $\vec{F}(x,y,z)=(xy,~xz,~y)$.
\end{enumerate}
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