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\begin{center}
{\large Math 331, Fall 2004, Due Monday, Nov 15}
\end{center}
\begin{enumerate}

\item{
 Suppose that a turtle is moving with constant  
velocity 1 unit/sec. The turtle is told, every second, to steer right
by an amount equal to $t^2$ degrees, where $t$ is the time (in secs).
(For example, after the first step, it turns right 1 degree, then   
after the second, turn right by 4 degrees, and so on.) Draw the curve
the turtle describes after 10 and after 100 seconds.
}

\item{
Consider the recursively defined sequence
$S_n = S_{n-1}^2 -4 S_{n-1} + 6$  for $n \ge 1$, with $S_0=5$.
Implement this in Maple using both a   
recursive and a non-recursive procedure. [{\em Hint for the
computation of the non-recursive formula: complete the square}.]


Extra credit: rewrite the recursive procedure adding \cmd{option remember} and
see the difference in terms of computational speed.
}


\item{
 By using only \cmd{TurtleCmd}, draw a random 
walk of $n$ steps. (In a random walk the turtle takes a step forward,
backwards, to the right, to the left, with equal probabilities, and then
repeats the process.) [\emph{Check \cmd{rand}}.]
}
\item{
Construct a Cantor set whose box counting dimension is $1/2$.
Explain a general algorithm for constructing a Cantor set with {\em 
any} given box dimension $0 < d < 1$.
[\textit{You can do this on Maple or by hand.}]
}

\item{
Write a TurtleCmd procedure that draws the $n$-th
approximation of a fractal of your choice (not one saw in class) and
calculate its box-counting dimension. 
}


\end{enumerate}

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