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
- 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
- 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(ex -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.)
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