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\title{Final Exam}
\target{Math 207: Fall 2014}

\begin{document}


\begin{exam}{\today}

\bp The \emph{Hessian} matrix of a function $f:\R^2 \to \R$ at a point $p\in
\R^2$ is defined to be 
$$H_p=\begin{pmatrix}
\frac{\partial^2 f(p)}{\partial x^2} & \frac{\partial^2 f(p)}{\partial
  y \partial x} \\[8pt]
\frac{\partial^2 f(p)}{\partial x \partial y} & \frac{\partial^2 f(p)}{\partial y^2}
\end{pmatrix}$$
You can think of the Hessian as a kind of second derivative of $f$
since it's the
derivative of the function $\R^2 \to \R^2$ defined by $p\mapsto \left(
\frac{\partial f(p)}{\partial x}, \frac{\partial f(p)}{\partial y} \right)$.
\bigskip

Define $f:\R^2 \to \R$ by $$f(x,y)=6 \sqrt{36-9 (x-y+1)^2-4
  (x+y-3)^2}.$$
There is one critical point $p\in \R^2$ of $f$.  
Find the eigenvectors
of the Hessian at that 
critical point and explain why they are orthogonal.

\begin{center}
\includegraphics[scale=0.5]{surface2.pdf}
\end{center}


\ep


\bp Find a $3\times 3$ matrix $A$ with nonzero integer entries so that
$A^2=I$
\ep


\bp Describe all the different similarity classes of matrices 
that have  the characteristic polynomial $\chi(t)=t^5-2 t^3+t$?  
Which are invertible?  
\ep


\bp Consider the function $f:\R^2 \to \R^2$ defined by $f(x,y)=(u,v)$ where
\begin{align*}
u&=\sin(x+y)\\
v&=\exp(x-2y)
\end{align*}
Notice that $(0,0)\overset{f}{\mapsto}(0,1)$.  In a neighborhood of
the point $(x,y)=(0,0)$, the function $f$ is invertible.  That is,
there exist an open set $U$ containing $(u,v)=(0,1)$ and a function $g:U\to V$ so
that $g(u,v)=(x,y)$.   

\bigskip 
Use the fact that $f$ and $g$ are inverses 
to compute the total derivative $Dg$ implicitly at the point $(u,v)=(0,1)$.

\ep

\bp Compute a determinant.  Your choice:
$$\begin{pmatrix}
 \cos(1) & \cos(6) &  \cos(11) &  \cos(16) &  \cos(21) \\[4pt]
 \cos(2) & \cos(7) &  \cos(12) &  \cos(17) &  \cos(22) \\[4pt]
 \cos(3) & \cos(8) &  \cos(13) &  \cos(18) &  \cos(23) \\[4pt]
 \cos(4) & \cos(9) &  \cos(14) &  \cos(19) &  \cos(24) \\[4pt]
 \cos(5) & \cos(10) &  \cos(15) &  \cos(20) &  \cos(25) 
\end{pmatrix} 
\text{ or }
\begin{pmatrix}
 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\[4pt]
 1 & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\[4pt]
 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\[4pt]
 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\[4pt]
 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} \\[4pt]
 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} \\[4pt]
 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{7} & \frac{1}{7} & \frac{1}{7} \\[4pt]
 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{8} & \frac{1}{8} \\[4pt]
 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} & \frac{1}{9} \\[4pt]
 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} & \frac{1}{10} \\[4pt]
\end{pmatrix}
$$

\ep


\end{exam}
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