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\centerline{Mathematics 241, section 1 -- Multivariable Calculus}
\centerline{Midterm Exam 2}
\centerline{November 5, 2010}
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\noindent
{\it Directions}
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Do all work in the blue exam booklet.  There are 100 regular points and 10 Extra Credit points.
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\noindent
I.  All parts of this problem refer to the vector field
\[
{\bf F}(x,y) = (x^2 - 2x, y^2 + 2y).
\]
\begin{enumerate}
\item[A.] (5)  Find all critical points of ${\bf F}(x,y)$.
\item[B.] (10)  There are two vector fields plotted on the back of this sheet.
Say which one shows ${\bf F}(x,y)$ and use that plot to classify each of the 
critical points as a source, sink, saddle, or center.
\item[C.] (10)  Show that $\alpha(t) = \displaystyle{\left(\frac{2}{1-2e^{2t}},\frac{-2}{1 - 3e^{-2t}}\right)}$
is a flow line of ${\bf F}$.  
\item[D.] (10) Show that any $f(x,y) = \frac{x^3}{3} - x^2 + \frac{y^3}{3} + y^2 + c$
satisfies $\nabla f = {\bf F}$.  What are the critical points of such an $f$?
Classify each of them as a local maximum, local minimum, or saddle point.  
(Don't ``start over'' here; use previous parts of this question as appropriate.) 
\end{enumerate}
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\noindent
II.  All parts of this problem refer to the function 
\[
f(x,y) = \frac{x^3 - 3 xy^2}{x^2+y^2} {\rm\ if\ } (x,y) \ne (0,0) {\rm\ and\ }
                          f(0,0) = 0.
\]
\begin{enumerate}
\item[A.]  (10)  Compute $\frac{\partial f}{\partial x}(x,y)$ and $\frac{\partial f}{\partial y}(x,y)$
for $(x,y) \ne (0,0)$.
\item[B.]  (10)  Do $\frac{\partial f}{\partial x}(0,0)$ and $\frac{\partial f}{\partial y}(0,0)$ exist?
If so, find them; if not say why not.
\item[C.]  (10)  Let $m$ be arbitrary and compute
$\lim_{t\to 0} f(t,mt)$ (the limit of the value of $f$ along
the line through the origin in the direction of the vector $(1,m)$).  
Does this {\it prove} anything about the existence
of the limit $\lim_{(x,y) \to (0,0)} f(x,y)$?  Explain your 
answer using the definition of the statement
$\lim_{(x,y) \to (0,0)} f(x,y) = L$.  
\item[D.]  (10 Extra Credit)  Is $f(x,y)$ {\it differentiable} at $(0,0)$?  Explain 
your answer.
\end{enumerate}
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\noindent
III.  All parts of this problem refer to $f(x,y) = 4x^2 - y^2$.
\begin{enumerate}
\item[A.] (15) Sketch the contours of $f(x,y)$ for $c = -1, 0, 1$ on the 
same set of axes.
\item[B.] (10) Consider 
\[
S = \{(x,y) \in \mathbb{R}^2\mid -1 < f(x,y) < 0\}.
\]
Is $S$ open, closed, or neither?  Why?  (An intuitive explanation referring
to the definition of open or closed is OK here.)
\item[C.]  (10) For which unit vectors $u$ is the directional derivative 
$D_u f(1,2) = 0$?
\end{enumerate}

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