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\begin{center}{ \Large{Lab worksheet}  \\\large{9/25/2017} }\end{center}
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\begin{enumerate}
\item  Write the equation of the plane that contains point $A(1,2,3)$ and perpendicular to vector $\vec{n}=\langle 2,-1,1\rangle$. Plot the plane.\vspace{1in}
\item Write the equation of the plane that contains point $A(1,2,-1)$ and parallel to two vectors $\vec{a}=\langle 0,2,1\rangle$ and $\vec{b}=\langle -2,1,2\rangle$.\vspace{3in}
\item The \emph{parametric equation} of a plane which contains point $A(x_0,y_0,z_0)$ and parallel to two vectors $\vec{a}=\langle a_1,a_2,a_3\rangle$ and $\vec{b}=\langle b_1,b_2,b_3\rangle$ is
\[
\left\{ 
\begin{array}{l}
x=x_0+ta_1+sb_1 \\ 
y=y_0+ta_2+sb_2 \\ 
z=z_0+ta_3+sb_3
\end{array}
\right. 
\]
In short notations, $P(t,s) = A+t\vec{a}+s\vec{b}.$ 

Now write the parametric equation of the plane in Problem 2. Plot the plane. (You will need the command \emph{ParametricPlot3D})\vspace{3in}
\item Given a single-variable function $y=f(x)$. \begin{enumerate}\item Write coordinates of a point $A$ on the graph whose x-coordinate is $x_0$. \item Write the equation of the line passing through $A$ and tangent to the graph. \item Write a direction vector of that line.\end{enumerate}\vspace{3in}
\item Given a two-variable function $z=f(x,y)$. \begin{enumerate}\item Write coordinates of a point $A$ on the graph (surface) whose (x,y)-coordinates are $(x_0,y_0)$. \item The cross section $x=x_0$ of the surface is a curve whose equation is $z=f(x_0,y)$. Write a direction vector of the tangent line to this curve at point $A$. \item The cross section $y=y_0$ of the surface is a curve whose equation is $z=f(x,y_0)$. Write a direction vector of the tangent line to this curve at point $A$.\item Write a parametric equation of the plane tangent to the surface at $A$.\end{enumerate}
\vspace{3.2in}
\newpage\item (Exercise 2 of Lab 3) Given the function $g(x,y)=x^2-y^2$. Write a parametric equation for the tangent plane of fucntion $g$ at (1,2). Plot the surface and together with the tangent plane. (You will need the command \emph{Show}.)
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\vspace{3.5in}
\item Given a two-variable function $z=f(x,y)$. 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 $f$?\item Using the principle ``a level set is perpendicular to the gradient vector', determine a normal vector of this level set at point $(x_0,y_0,z_0)$?\item What is the equation of the equation of the tangent plane of the graph of $f$ at point $A(x_0,y_0,z_0)$?
\end{enumerate}
\newpage
\item (Exercise 4 of Lab 3) Given the function $f(x,y)=xe^{-x^2-y^2}$. Using the method in Problem 7, find the equation of the plane tangent to the graph of $f$ at point $(0,0,f(0,0))$.
\end{enumerate}
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