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\begin{center}{ \Large{Worksheets}  \\\large{9/28/2017} }\end{center}
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\begin{enumerate}
\item  Given a function $f(x,y)=e^{x+y}\cos(xy)$,\begin{enumerate}[(i)]
\item Compute the gradient vector $\nabla f$.\vspace{3.5in}
\item Calculate the directional derivative $D_{\vec{a}}f$ at point (0,0) in the direction of vector $\vec{a}=\langle 2,-1\rangle$.\vspace{2in}
\item Calculate the directional derivative $D_{\vec{b}}f$ at point (0,0) in the direction of vector $\vec{b}=\nabla f(0,0)$.\vspace{2in}
\end{enumerate}
\item Given a function $f:\mathbb{R}^2\to\mathbb{R}^3$, $f(x,y)=(xy,x+y^2,\sin y)$,\begin{enumerate}[(i)]
\item Compute the derivative matrix (or Jacobian matrix) $Df$.
\vspace{3.5in}
\item Find the derivative matrix of $f$ at point (1,0).\vspace{2in}
\item Calculate the linear approximation $L(x,y)$ of $f$ at point (1,0).\vspace{2in}
\item Use the linear approximation above to estimate $f(1,-0.1)$.
\end{enumerate}
\newpage
\item Given a function $z=f(x,y)=x^2-xy+y^2$ and a point $A(1,-1,3)$ which lies on its graph. \begin{enumerate}\item The cross section $x=1$ of the graph is a curve. Write a direction vector of the tangent line to this curve at point $A$. \vspace{1in}\item The cross section $y=-1$ of the graph is a curve. Write a direction vector of the tangent line to this curve at point $A$.\vspace{1in}\item Write a parametric equation of the plane tangent to the graph at $A$.\vspace{1.5in}
\end{enumerate}
\item Let $f$ be the same function as in Problem 3. Define $g(x,y,z)=z-f(x,y)$. \begin{enumerate}\item What is the level set $g(x,y,z)=0$ in relation to the graph of function $f$?\vspace{.5in}\item Using the principle ``a level set is perpendicular to the gradient vector'', determine a normal vector of this level set at point $A(1,-1,3)$.\vspace{1.5in}\item What is the (cartesian) equation of the tangent plane of the graph of $f$ at point $A$?
\end{enumerate}
\newpage
\item Let $f:\mathbb{R}^2\to\mathbb{R}$. Put $x=ts$, $y=t+s$ and $g(t,s)=f(x,y)=f(ts,t+s)$. Express the \[\frac{\partial g}{\partial t},~\frac{\partial g}{\partial s},~\frac{\partial^2 g}{\partial t\partial s}\] in terms $t$, $s$ and partial derivatives of $f$.
\end{enumerate}
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