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\begin{center}{ \Large{Lab worksheet}  \\\large{10/3/2017} }\end{center}
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\item  A vector-valued function is also called a \emph{vector field}. To plot a vector field, you can use the command \emph{VectorPlot} or \emph{VectorPlot3D}. Press F1 and type the keyword VectorPlot to learn the syntax. 

Now plot the vector fields $f_1(x,y)=\langle-y,x\rangle$,  $f_2(x,y)=\langle x^2,xy\rangle$ and $f_3(x,y,z)=\langle \sin x,\sin y,\sin z\rangle$.\vspace{1.5in}
\item Let $f(x,y)=y^3-x^3$. Plot the gradient vector field $\nabla f$ together with the level sets of $f$. Based on the plot, what can you tell? \vspace{1in}
\item (Exercises 1 and 3 in Lab 4) Let \[T(x,y,z)=\frac{6000}{1+\frac{1}{5}(x^2+y^2+z^2)}.\]
\begin{enumerate}
\item Calculate the directional derivative at point $A(9,2,1)$ in the direction of vector $\langle -4,-3,5\rangle$.\vspace{2in}
\item At point $A$, \begin{enumerate}[(i)]\item in what direction does the function have the largest increase?\vspace{1in}
\item in what direction does the function have the largest decrease?\vspace{1in}
\item in what direction does the function remain at the same value?\vspace{2.5in}
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\item A vector-valued function that has single-valued variable is called a \emph{curve}. 

Consider the curve $c_1(t)=\langle8\sin t-\cos(2t), t-\sin t\rangle$.\begin{enumerate}\item Plot the curve.\item Compute the tangent vector at point $t=0$. Plot this tangent vector on the same graph with the curve. (You will need the command \emph{Arrow} to draw a vector.)\vspace{1.5in}\item Find all $t$ where the tangent vector is parallel to the x-axis.\vspace{2in}\end{enumerate}
\item Consider the curve $c_2(t)=\langle t^3, t,t^2\rangle$.\begin{enumerate}\item Plot the curve.\item Compute the tangent vector at point $t=1$. Plot this tangent vector on the same graph with the curve.\vspace{1.5in}\item Find all $t$ where the tangent vector is \begin{enumerate}[(i)]\item parallel to the plane $x+6z=0$\vspace{2in}\item perpendicular to the plane $x+y=1$\vspace{1.5in}\end{enumerate}\end{enumerate}
\item Let $f(x,y)=(2y+1,e^{2x-4+y^2})$.\begin{enumerate}
\item Find the derivative (Jacobian) matrix of $f$. Double check your result with Mathematica by using the command $D[f[x,y],\{\{x,y\}\}]$.\vspace{2in}
\item Find the Jacobian matrix at point (2,0).\vspace{1in}
\item Give a linear approximation near point (2,0).\vspace{2in}
\item Estimate $f(1.9,0.01)$. Use Mathematica the compute the exact value of $f(1.9,0.01)$. Are they close?\vspace{1.5in}
\end{enumerate}
\item Let $f(x,y,z)=\langle xy,y+z^2\rangle$ and $g(u,v)=e^{u+v}\cos(u+v)$. Put $h=g\circ f$.\begin{enumerate}
\item Using the chain rule, compute $\nabla h(0,-1,1)$. Double check your result with Mathematica.\vspace{2in}
\item Using the chain rule, compute $\dfrac{\partial h}{\partial z}$. Double check your result with Mathematica.
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