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\flushleft\underline{Lab 5 --- MATH 1586 Spring 2018} \\
We will use this webpage today: \href{http://web.monroecc.edu/manila/webfiles/pseeburger/CalcPlot3D/}{http://web.monroecc.edu/manila/webfiles/pseeburger/CalcPlot3D/}

We will look at surfaces and the gradient vector field in this lab. Recall the directional derivative in the direction of the \textbf{unit} vector $\vec{u}=\langle a,b \rangle$:\\
$$D_{\vec{u}} f(x_0,y_0) = \vec{u} \cdot \nabla f(x_0,y_0) = a \dfrac{\partial f}{\partial x}(x_0,y_0) + b \dfrac{\partial f}{\partial y}(x_0,y_0).$$
Recall the following theorem from class: \\
\textbf{Theorem}: The maximum value of $D_u f\left(a,b\right)$ is $\left\lVert \nabla f\left(a,b \right) \right\rVert$ and it occurs in the same direction as the gradient vector $\nabla f\left(a,b \right)$.

\begin{enumerate}[1.]
\item Let $f(x,y)=2-x^2-y^2$. Plot the surface $z=f(x,y)$. We saw in class that the gradient of this function is $\nabla f = \left\langle -2x, -2y \right\rangle$. Plot the vector field as well. Notice that the vector arrows always point in the direction of ``steepest ascent"! Take a screenshot of this final image and attach it to your lab (in Windows: "Print Screen" key copies the screen to clipboard, then open mspaint and paste it there to get the picture).

\item An electric potential $V$ of a charged plate is given by $V(x,y)=x^2y-xy^2$. \\
\begin{enumerate}[a.)]
\item Find the rate of change of the potential at $(1,0)$ in the direction of $\vec{v}=\langle -1,-1 \rangle$. 

\item In which direction does $V$ change most rapidly?

\item What is the maximum rate of change at $(1,1)$?

\item Draw $V$ and $\nabla V$ in CalcPlot3D. Screenshot and observe the results you found above.
\end{enumerate}

\item The temperature on a metal plate at point $(x_0,y_0)$ is given by $T(x,y)=2e^{-x^2-3y^2}$. 
\begin{enumerate}[a.)]
\item Find the rate of change of the temperature of the plate at the point $(1,1)$ in the direction of the vector $\langle -1,-1\rangle$. 
\item In which direction does the temperature change the fastest at $(1,1)$?
\item What is the maximum rate of temperature change at $(1,1)$?
\item Draw $V$ and $\nabla V$ in CalcPlot3D. Screenshot and observe the results you found above.
\end{enumerate}

\end{enumerate}

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