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\begin{center}
{\Large Final

Math 2210 - 2}
\end{center}


\vspace{.2in}
Name: \hrulefill
\vspace{.1in}

Instructions:
\begin{itemize}
\item  You have 2 hours to complete this exam.
\item  Show all your work.  If I can't follow your work or it is missing you WILL NOT get credit.  
	(This holds for graphing too!)
\item  Use the back of these sheets if you run out of room.  Make sure I can find your work.
\item  No calculators, but you are allowed one ``cheat sheet.''
\item  Each problem is worth 10 points.
\end{itemize}

\hrule

\begin{enumerate}

\item  Let $C$ be the curve in $\R^2$ determined by the parametric equation 
\[ 
\sigma(t)=(t,t^n) \qquad 0 \leq t \leq 1
\]
(where $n$ is some unknown positive integer).
Compute the line integral:
\[
\int_C y \;dx + (3y^3-x) \; dy
\]
\newpage


\item  Consider the vector field:
\[
\mbf{F}(x,y,z) = \left( \frac{2x}{yz}, \frac{-x^2}{y^2z}, \frac{-x^2}{yz^2} \right)
\]
Is $F$ a conservative vector field?  Justify your answer!
\newpage






\item  Consider the conservative vector field (you don't need to check this!):
\[
\mbf{F}(x,y) = \left( \frac{2x}{y^2+1}, -\frac{2y(x^2+1)}{(y^2+1)^2} \right)
\]
\begin{enumerate}
\item  Find a potential function for $\mbf{F}$.
\item  Let $C$ be the curve determined by
	\[
	\sigma(t)=(t^3-1, t^6-t) \qquad 0 \leq t \leq 1
	\]
	Compute the line integral
	\[
	\int_C \mbf{F} \dotprod d\mbf{r}
	\]
\end{enumerate}
\newpage



\item  Let $C$ be the curve made up of the four sides of the square:
\[
\sigma(t) =
\left\{ \begin{array}{cc}
	(t,0) & \mbox{if $0\leq t \leq 2$} \\
	(2,t-2) & \mbox{if $2 < t \leq 4$} \\
	(6-t,2) & \mbox{if $4 < t \leq 6$} \\
	(0,8-t) & \mbox{if $6 < t \leq 8$}
\end{array} \right.
\]
Compute the line integral by applying Green's Theorem
\[
\oint_C (x^2+y^2)\; dx - 2xy \; dy
\]
\newpage


\item  Consider the curve defined by
\[
\mbf{r}(t) = \left( t, t^2, \frac{2}{3}t^3 \right)
\]
At the point $\mbf{r}(1)=(1,1,2/3)$ (so when $t=1$), find
\begin{enumerate}
\item  $\mbf{T}$, the unit tangent vector,
\item  $\mbf{N}$, the unit normal vector,
\item  $\mbf{B}$, the unit bi-normal vector,
\item  $\kappa$, the curvature of the curve.
\end{enumerate}
\newpage



\item  Consider the function
\[
f(x,y) = x^3 + y^3 - 6xy
\]
Find all local maxima, local minima and saddle points for $f$.
\newpage

\item  Let $R$ be the region of the $xy$-plane defined by
\[
x \geq 0, \quad y \geq x^2, \quad y \leq 10-x^2
\]
Find the integral:
\[
\iint_R xy \; dA
\]
\newpage


\item  Find the surface area of the part of the 
sphere $x^2+y^2+z^2=9$ that is trapped between the planes $z=1$ and $z=2$.


\end{enumerate}




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