By Richard A. Holmgren

Given the convenience with which pcs can do generation it really is now attainable for nearly somebody to generate attractive pictures whose roots lie in discrete dynamical platforms. pictures of Mandelbrot and Julia units abound in courses either mathematical and never. the math at the back of the images are appealing of their personal correct and are the topic of this article. Mathematica courses that illustrate the dynamics are incorporated in an appendix.

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Why aren't these two sets the same? c) Find 8([0,~]) and 8- 1 (8([0, ~])). 2 Let 9 : IR - t Z be the step function defined such that g(x) is the largest integer less than or equal to x. 1]) and g-l( {2,4}). 3 a) Let A be a subset of the domain of the function f . Prove r1(f(A)) :::> A and show by example that it may happen that f-1(f(A)) =I A. b) Let B be a subset of the codomain of the function f . 4 Suppose f : [-1 , IJ -t [-1 , IJ is defined so that f(x) = ~x. a) Find the range of f. b) Find the range of f 0 f.

Recall from calculus that a function that is differentiable at a point is also continuous at that point. An important property of differentiable functions , which we will often exploit, is the Mean Value Theorem. 2. MEAN VALUE THEOREM . Let f : [a, bJ entiable on [a , bJ. Then there exists c in [a, bJ such that --+ lR be differ- f(b) - f(a) = f'(c)(b - a). After examining a few graphs, the reader will realize that the Mean Value Theorem is an intuitively clear result. A proof can be found in any good calculus text.

44 5. 3. SARKOVSKII ' S ORDERING. the natural numbers is Sarkovskii's ordering of 3>-5>-7>- ·· · >- 2·3>-2·5>-2·7>-··· ... >- 22 . 3 >- 22 . 5 >- >- 22 . 7 >- ... >- 2n . 3 >- 2n . 5 >- 2n . 7 >- .. . . >- 23 >- 22 >- 2 >- 1. The relation a >- b indicates a precedes b in the order. When writing the order, all odd numbers except one are listed in ascending order, then two times every odd, then four times every odd, and so on. The order is completed by listing the powers of two in descending order.