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\textbf{Reveiw for Test 2}

\newcommand{\R}{\mathbb{R}}

\begin{enumerate}

\item  Test: Wednesday, July 3.
\item  Test will cover: 15.1-16.3.
\item  The test will be closed book, but you can have one $4 \times 6$ note card of notes
	(you can have your calculator, but your work must be as if you had no calculator).
\item  Graphing functions of two or more variables:
	\begin{enumerate}
	\item  Graph level curves and arbitrary slices
	\item  Graph function in $\R^3$.
	\end{enumerate}
\item  Partial derivatives
	\begin{enumerate}
	\item  How to omputing
	\item  What do the partial derivatives tell you?
	\item  What are mixed partials (notation?)
	\end{enumerate}
\item  Limits of functions of more then one variable (and continuous functions)
	\begin{enumerate}
	\item  When is there and when isn't there a limit?
	\item  You do not need to know the $\varepsilon$ - $\delta$ definition of continutity and limits.
	\end{enumerate}
\item  Differentiablity.
	\begin{enumerate}
	\item  Locally linear (what does this mean?)
	\item  What is a tangent plane and how do we find the equation of a tangent plane?
	\end{enumerate}
\item  Directional derivative
	\begin{enumerate}
	\item  How to compute the directional derivative.
	\item  What do directional derivatives tell you?
	\end{enumerate}
\item  Gradiants
	\begin{enumerate}
	\item  How do you compute gradiants
	\item  What does the gradiant tell you.
	\item  What are gradiant fields?
	\end{enumerate}
\item  Chain rule
	\begin{enumerate}
	\item  What is the chain rule for multivariable functions?
	\item  How is the chain rule used?
	\end{enumerate}
\item  Tangent planes
	\begin{enumerate}
	\item  Given an implicit function (something like $x^2+y^2+z^2=4$), or a function (like $z=x^2-y^2$), 
		find the equation of the tangent plane at a given point.
	\item  How is this related to gradiant (or how is the gradiant used?)
	\end{enumerate}
\item  Finding maxima and minima.  
	\begin{enumerate}
	\item  How to find critical points
	\item  How to test critical points
	\item  How to find the maxima and minima on the boundary of a set 
		(or, how to test the critical points from the boundary of a set).
		(we parametrized the boundary curve and turned it into a one variable problem)
	\end{enumerate}
\item  Integrals for functions of more then one variable
	\begin{enumerate}
	\item  Definitions (how is the integral defined)
	\item  How to compute integrals (using iterated integrals)
	\item  What if the domain is a rectangle?
	\item  What if the domain is not a rectangle?
	\item  Switching the order of integration.
	\end{enumerate}


\end{enumerate}


\textbf{Sample Problems:}

\begin{enumerate}
\item  There are old tests on the web to look at.
	The following are some problems from old exams to for 
	you think about some things (these are some of the problems I will try to do in class for the review).
\item  For the following double integral:
	\[
	\int_0^2 \int_{y^2}^{2y} 6x^2y \;dx\;dy
	\]
	\begin{enumerate}
	\item  Sketch the region in the $xy$-plane we are integrating over
	\item  Write the correct integral with the order of integration switched.
	\item  Compute the integral
	\end{enumerate}

\item 	Captain Ralph is in trouble near the sunny side of Mecury.
	The temperature of the ship's hull when he is at location $(x,y,z)$ is given by
	\[
	T(x,y,z) = e^{-x^2-2y^2-3z^2}
	\]
	Ralph is currently at location $(1,1,1)$.  $x,y,z$ are measured in meters.
	\begin{enumerate}
	\item  In what direction should Ralph travel to decrease the temperature most rapidly?
	\item  If Ralph's ship can travel at a speed of $e^8$ meters per second, how fast will the temperature decrease 
		if Ralph travels in that direction (the direction of biggest decrease)?
		(Hint: first compute the directional derivative of $T$ in the direction.)
	\end{enumerate}


\item For the function
	\[
	z = f(x,y) = \frac{3x^2}{y} - 2y
	\]
	\begin{enumerate}
	\item  Find $\nabla f$
	\item  Find $\nabla f(6,3)$
	\item  Find the equation of the tangent plane (in $\R^3$) of the graph at this function when $x=6$ and $y=3$.
	\end{enumerate}





\item Consider the surface defined by
	\[
	2x - x^2 - xy + z^2 = 5
	\]
	Find the point(s) on this surface that is closest to the origin.



\end{enumerate}







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