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\centerline{Mathematics 241, section 1 -- Multivariable Calculus}
\centerline{Makeup Midterm Exam 2}
\centerline{November 1, 2013}
\vskip 10pt
\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
\[
{\bf F}(x,y) = (x^2 - 2x, xy - y).
\]
\begin{enumerate}
\item[A.] (10)  Find all critical points of ${\bf F}(x,y)$.
\item[B.] (5)  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.] (20) Show that $\alpha(t) = (0,4e^{-t})$ and $\beta(t) = \displaystyle{\left(\frac{2}{1+e^{2t}},0\right)}$
are both flow lines of $F$.   What are $\lim_{t\to\infty} \alpha(t)$ and  $\lim_{t\to\infty} \beta(t)$?
\item[D.] (5) Is there a scalar-valued function $f(x,y)$ such that ${\bf F}(x,y) = \nabla f(x,y)$?  Why or why not? 
\end{enumerate}
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\noindent
II.  In the neighborhood of Eagle Mountain, the landscape has elevation in feet above sea level given by 
$f(x,y) = \frac{x^2}{4} + y^2 + 1000$.
\begin{enumerate}
\item[A.] (10) Sketch the contours of $f(x,y)$ for $c = 999, 1000, 1001$ on the 
same set of axes.
\item[B.] (10) Compute the directional derivative $D_u f(2,1)$ for a general 
unit vector.  
\item[C.] (5) In the direction of which unit vector $u$ should you 
walk from the point with $(x,y) = (2,1)$ in order to decrease your elevation at
the fastest rate?  
\end{enumerate}
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\noindent
III.  All parts of this problem refer to the function 
\[
f(x,y) = 
\frac{x^3 + x^2y}{x^2+y^2} {\rm\ if\ } (x,y) \ne (0,0) {\rm\ and\ } f(0,0) = 0.
\]
\begin{enumerate}
\item[A.]  (15)  Find the tangent plane to $z = f(x,y)$ at $(1,1,f(1,1))$.  
\item[B.]  (10)  Does $\frac{\partial f}{\partial x}(0,0)$ exist? If so, find it; if not say why not.
\end{enumerate}
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%\noindent
%IV.  (10)  Can the curve $\alpha(t) = (2\cos(t),\sin(t))$ for $t \in (0,\infty)$ be a
%flow line of the vector field $\nabla f(x,y)$ for a differentiable function $f$?  Why or
%why not?
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\noindent
{\it Extra Credit}  (10)  Refer to the function in question III.
 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 $(1,m)$).  Is $\lim_{t\to 0} (f\circ\alpha)(t) = 0$
for \emph{every} differentiable curve $\alpha(t)$ with $\alpha(0) = (0,0)$?  Explain. 
\vfill\eject
\noindent
Vector Field 1:
\begin{center}
\includegraphics[width=3.0truein,height=3.0truein]{VF1.png}
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\noindent
Vector Field 2:
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\includegraphics[width=3.0truein,height=3.0truein]{VF2.png}
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