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\begin{document}
May 10, 1995 \hfill Math 311, Final Exam \hfill
%\hfill Name:\underline{\makebox[1.75in]{}}
\Sk
\q{25}You have the following data points $\left\{\,(-2,1),\ (-1,0),\
(0,2),\ (1,4),\ (2,3),\ (3,4)\,\right\}$, and want to find a linear
combination of the following functions $\left\{\,1,\ t,\ \sin t,\ \cos
t\,\right\}$ which will pass through these data points.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item You soon realize that this is impossible. Why? Note: comparing
numbers of equations and variables is not sufficient.
\item Find the linear combination of these functions which best fits
the data. Be sure to explain what you do. Just pushing the least
square botton on the HP will be worth zero points.
\end{list}
\sk
\q{30}Let $F(x,y,z)=\left(2x-z,\,y^3\sin xz,\,x^2z\cos y\right)$. Set
up each of the following integrals, being very explicit with the
limits of integration and the integrand. Then evaluate only those
which you are explicitly asked to compute.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item $\ds\int\int\int_\Omega div(F)\,dV$, where $\Omega =
\left\{\,(x,y,z)\,:\, x^2+y^2\leq 9,\ \hbox{and}\ 0\leq z\leq
9-x^2-y^2\,\right\}$. {\bf Do not evaluate}.
\item $\ds \int\int_{\partial\Omega}F\cdot dS$, Note: use the outward
pointing normal to the surface, $\partial\Omega$. Remember, $\partial\Omega$
denotes the boundary of the three dimensional region $\Omega$. {\bf
Evaluate this one}.
\item $\ds\int_\Gamma F\cdot d\vec{x}$, where $\Gamma$ denotes the
broken straight line in $R^3$ which consists of the two paths:
\break\null\hspace{1in}$(0,0,0)\rightarrow(0,1,0)\rightarrow(1,2,3)$.
{\bf Evaluate this one}.
\end{list}
\sk
\q{30}Let $\ds f(x,y)=\left(x^2+y^2\right)e^{-(x^2+y^4)}$. Clearly
the function values of $f$ tend to zero as the point $(x,y)$ moves
away from the origin. Moreover $f$ is nonnegative and continuous.
Thus, $f$ must have a maximum value somewhere in $R^2$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item What is the maximum value attained by $f$, and for which values
of $x\ \hbox{and}\ y$ is the maximum value attained.
\item Find the maximum value of $f$ subject to the constraint
$x^2+y^2=4$.
\end{list}
\sk
\q{30}Let $T(u,v)= (u^2-v^2,\,uv)$ Let $\ds R=\left\{\,(u,v)\,:\,1\leq
u\leq2,\ \hbox{and}\ 1\leq v\leq 2\,\right\}$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Let $T(R)$ denote the image of $R$ in the x-y plane under the
transformation $T$. Graph $T(R)$.
\item What is the area of $T(R)$? 
\item $\ds \int\int_{T(R)}\left(x^2+\sin xy\right)\,dx\,dy =$?
\end{list}
\sk
\q{25}Let $\ds A=\frac14\left[\begin{array}{cccc}
-5&-1&7&1 \\ -25&7&27&1 \\ -1&-1&3&1 \\
-31&5&37&3\end{array}\right]$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item $rank(A)= \underline{\makebox[.5in]{}}$.
\item $dim(NS(A))= \underline{\makebox[.5in]{}}$.
\item basis of $NS(A)= \underline{\makebox[2.5in]{}}$.
\item basis of $CS(A)= \underline{\makebox[2.5in]{}}$
\item Does the equation $A\vec{x}=\vec{b}$ have a solution for
$\vec{b}=(1,1,1,1)$. 
\item What diagonal matrix is $A$ similar to?
\item $A^{100100}= 	\underline{\makebox[1.5in]{}}$.
\end{list}
\end{document}