Math 331, Fall 2004, Due Monday, Nov 15

1. 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.

2. Consider the recursively defined sequence S_{n} = S_{n−1}^{2} − 4S_{n−1} + 6 for n ≥ 1, with S_{0} = 5. Implement this in Maple using both a recursive and a non-recursive procedure. [Hint for the computation of the non-recursive formula: complete the square.]

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

3. By using only 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.) [Check rand.]

4. Construct a Cantor set whose box counting dimension is 1/2. Explain a general algorithm for constructing a Cantor set with any given box dimension 0 < d < 1. [You can do this on Maple or by hand.]

5. 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.