SAMPLE EXAM 1

MA 471-671

Sample Exam #1

This sample exam is not intended to reflect the length of the actual exam. Rather, it is intended to show the types of questions that you should expect.

1. True/False.
- A. The function F(x) = x² + 1 has an attracting fixed point at the origin. _________
- B. Let (001001.....) and (1111....) be two sequences in the sequence space Sigma. Then the distance between these two points is 2/3. __________
- C. The point with ternary expansion .01010101..... is an endpoint of the Cantor Middle-Thirds set. _________
- D. The point 1 lies on a cycle of period 2 for the function F(x) = x² - 1. __________
- E. If a continuous function F has a cycle of period 27, then it must have a cycle of period 28 too. ___________
2. Quickies. Answers only -- no partial credit.
- A. The function F(x) = |x| has eventually fixed points at __________
- B. For which values of A does the function F(x) = A arctan x have an attracting fixed point at the origin? ____________
- C. List all cycles of prime period 4 for the shift map on Sigma.________
- D. The number whose ternary expansion is 121212.... is __________
- E. Give an example of a cycle of period 2 for the doubling map (i.e., find such a cycle explicitly). ____________ Also find a 3-cycle.K _________
3. Use graphical analysis to give a complete orbit analysis of the function defined in three pieces:
F(x) = x+2 (if x <= -1),
F(x) = -x (if -1 <= x <= 1),
F(x) = x-2 (if x >= 1.

List all fixed points and cycles and tell if they are attracting, repelling, or indifferent. List all points whose orbits tend to cycles or fixed points, all points whose orbits tend to infinity, and all points that are eventually fixed or periodic.
4. Definitions. Give the precise definitions of each of the following.
- A. Repelling Fixed Point.
- B. One-to-one.
- C. The shift map.
5. Examples. Give an example of a continuous function F which has the following property (or prove that such a function does not exist). Note: a different function for each property.
- Exactly 3ⁿ fixed points for Fⁿ.
- A repelling fixed point at the origin and an attracting fixed point at 1 (and no other fixed points or cycles).
- All orbits escaping to infinity except for a repelling cycle of period two at 0 and at 1.
6. State the Mean Value Theorem. Use it to prove that if F(p) = p and |F'(p)| < 1, then there is an interval about p in which all orbits tend to p.
7. Consider the family of functions F_A(x) = A(e^x -1) where A > 0. A bifurcation occurs for this family near 0 when A = 1.
- For which values of A is the fixed point at the origin attracting? For which values is it repelling?
- Describe the bifurcation that occurs at A = 1.
- Use graphical analysis on these functions at, before, and after the bifurcation.
- Sketch the phase portraits for these functions at, before, and after the bifurcation.
8. In an essay, describe the sequence space Sigma. You should discuss the distance function on Sigma and give an example or two of how it is used. You should describe which points in Sigma are close together and why. In general, your essay should be a cohesive treatment of what Sigma is. You will be graded on the accuracy of your statements as well as the use of English in your essay (spelling, grammar, etc.)