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{\large Math 331, Fall 2004: Problems 8-9}
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\textbf{NOTE 1:} {\itshape Each exercise is worth 10 points and can be
turned in at any time before its ``expiration date''.
At the end of the semester, I will expect you to have
turned in at least half of the exercises assigned. If you do more, I
will pick your best grades. If you do less, the missing grades will be
counted as zeros. Altogether, these will count the same as one project.
}


\textbf{NOTE 2:} {\it Maple is not necessary to the next two exercises, although you can use is as a word processor.
}

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\begin{enumerate}   
\prob{11}{ 10/5}
Following Section 4 of the notes, prove
that if we describe the circle of center $(a,b)$ and radius $r$ using
the parameters $(a,b,k)$, with $k = a^2 + b^2 - r^2$, rather than the
more natural parameters $(a,b,r)$, then the error function $H(a,b,k) =
E(a,b,\sqrt{a^2 + b^2 -k})$ is quadratic in $a,b$ and $k$. What does
this imply about the number of critical points?

\prob{12}{ 10/5}
With reference to Problem \#11, show that, for $r > 0$, the
transformation $(a,b,r) \mapsto (a,b,k)$ is a
valid change of variables, that is, it is one-to-one. This should help you
prove that $E(a,b,r)$ has only one ``physical'' critical point, which
is a minimum, and is mapped, through the transformation, into the unique
critical point of $H(a,b,k)$.

\end{enumerate}

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