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\begin{center}
{\Large Take home II

Math 2210 - 2}
\end{center}

Instructions:
\begin{itemize}
\item  Due on Wednesday, March 7, at the beginning of class.
\item  There is no reason for sloppy or missing work.  I will take off points for sloppy or missing work.
\item  Show all your work.  If I can't follow your work or it is missing you WILL NOT get credit.
\item  You must work alone, but you are free to use your book and calculator.
\item  I must be able to follow your work as if you had no calculator or book available.
\item  You are free to ask me questions.  It is possible (likely?) that I will not answer, but you can always ask.
\end{itemize}

\hrule

\begin{enumerate}[1.]

\item  (20 points)
Consider the function and the point:
\[
f(x,y) = x^y \qquad \mbf{p} = (x_0,y_0) = (e,1)
\]
\begin{enumerate}[(a)]
\item  Compute $\nabla f$.
\item  Compute $\nabla f(\mbf{p})$.
\item  Find the directional derivative of $f$ at $\mbf{p}$ in the direction $\mbf{v}=(-1,2)$.
\item  Find the direction from $\mbf{p}$ in which $f$ is increasing the most rapidly.
\item  Find a direction, $\mbf{u}$, from $\mbf{p}$ so that the directional derivative $D_{\mbf{u}}f(\mbf{p}) = e$.
	(Hint: you can make this hard or easy, its up to you).
\end{enumerate}





\item  (20 points)
As we have discussed, it is often useful to convert from Cartesian to other coordinates 
(like spherical or cylindrical coordinate).
This problem will discuss the chain rule for this sort of conversion.

Let $f:\R^3 \ra \R$ be a differentiable function.
Unfortunately, our chain rule (Theorem~B of section~15.6) is only written for two variables 
	($f(x,y)$, but we have $f(x,y,z)$).
\begin{enumerate}[(a)]
\item  \label{theorem b}
	How should Theorem~B be generalized so that it works for our case in hand?
	To get you started, $f(x,y,z)$ is our function.
	Suppose we also have $x,y,z$ as functions of $\rho, \theta, \phi$:
	\[
	x = x(\rho, \theta, \phi) \quad y = y(\rho, \theta, \phi) \quad z = z(\rho, \theta, \phi)
	\]
	\begin{enumerate}[i.]
	\item  Find an expression for $\partial f/ \partial \rho$.
	\item  Find an expression for $\partial f/ \partial \theta$.
	\item  Find an expression for $\partial f/ \partial \phi$.
	\end{enumerate}

\item  \label{partials a}
	Now consider the substituting spherical coordinates:
	\[
	x=\rho \cos \theta \sin \phi \qquad 
	y=\rho \sin \theta \sin \phi \qquad 
	z=\rho \cos \phi
	\]
	Using part~\ref{theorem b}, these particular substitutions ($f$ is still arbitrary), compute:
	\[
	\frac{\partial f}{\partial \rho}, \qquad 
	\frac{\partial f}{\partial \theta}, \qquad 
	\frac{\partial f}{\partial \phi}
	\]
	\\(Your answer should be in terms of 
	$\rho$, $\theta$, $\phi$ and the partials of $f$ with respect to $x$, $y$, $z$.)
	\\ (Essentially you are getting a statement of Theorem~B for the special case of 
	spherical coordinates.)
\item  Using your answer for part~\ref{partials a}, and the function
	\[
	f(x,y,z)=x^2+y^2+z^2
	\]
	compute 
	\[
	\frac{\partial f}{\partial \rho}, \qquad 
	\frac{\partial f}{\partial \theta}, \qquad 
	\frac{\partial f}{\partial \phi}
	\]
	(You could do this by first converting $f$ into spherical coordinates and then taking the derivatives,
	but I am asking you to do it using the previous parts.
	But, this comment should provide you with a way to check your work.)
\item  Explain what $\partial f/\partial \rho$ tells you geometrically 
	(explain what it means for an arbitrary function as well as our specific function here).
\end{enumerate}	








\item  (20 points)
Consider the function
\[
f(x,y,z) = \frac{18xyz}{x^2+y^2+z^2}
\]
\begin{enumerate}[(a)]
\item  Find $\nabla f$.
\item  Find $\nabla f(1,-1,2)$.
\item  Consider the surface $f(x,y,z)=-6$.  Find the equation of the tangent plane at $(1,-1,2)$.
\end{enumerate}

\item  (20 points)
Consider the function:
\[
f(x,y) = x^3 + y^2 - 6xy + 6x + 3y
\]
\begin{enumerate}[(a)]
\item  Find all critical points of $f$.
\item  Determine if these critical points are local maximum, local minimum, saddle points, or something else.
\end{enumerate}






\end{enumerate}

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