\magnification=\magstep1
\centerline{Mathematics 241, section 1 -- Multivariable Calculus}
\centerline{Exam III -- December 3, 2010}
\bigskip
\noindent
{\it Directions} 
\bigskip
Do all work in the blue exam booklet.  There are 100 regular points
and 10 Extra Credit points.
\bigskip
\noindent
I.  
\item{A)} (10)  Give a precise statement of the Second Derivative
Test for local extrema of functions of two variables.
\item{B)} (20)  Let $f(x,y) = 3x^4 + y^3 - 4x^3 - 3y$.  Find
all critical points of $f$ and apply the Second
Derivative Test to each.  
\bigskip
\noindent
II.  
\item{A)}  (20) Find the maximum and minimum values of 
$$f(x,y) = xy - y$$
on the closed and bounded set 
$$S = \{(x,y) \in {\bf R}^2 \mid x^2 + y^2 \le 4\}.$$
\item{B)}  (5)  Give a sketch of the contours of $f$ for the 
maximum and minimum values you found in part A, together with $S$.
\bigskip
\noindent
III.  All parts of this problem refer to the region
$$R = \{(x,y) \in {\bf R}^2 \mid x^2 + y^2 \le 9, y \ge 0, y \le 2x\}.$$
and the solid $S \subset {\bf R}^3$ with ``shadow'' $R$ in the $xy$-plane
and $0 \le z \le x^2$.  
\bigskip
\item{A)} (5) Sketch the region $R$ in the plane.
\item{B)} (10)  Set up iterated integral(s) of the form $\int\int f(x,y)\ dy\ dx$
to compute the volume of $S$.  Do not compute the value.
\item{C)}  (10)  Now set up iterated integral(s) of the form $\int\int f(x,y)\ dx\ dy$
to compute the volume of $S$.  Do not compute the value.
\item{D)}  (10)  Finally, set up a polar coordinate integral that 
will compute the volume of $S$.  Do not compute the value.
\item{E)}  (10)  Finally, compute the volume of $S$ using any one
of your answers from parts B,C,D (your choice).   {\it For full credit, you 
must show all work leading to your final answer.}
\bigskip     
\noindent
{\it Extra Credit} (10)  
Evaluate the improper double integral
$$\int\int_{{\bf R}^2} e^{-x^2-y^2}\ dA = \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-x^2 -y^2}\ dy\ dx.$$
\end 
