Rational Expressions/ Intro to Chaos

1. Rational Expressions/Intro to Chaos Macon State College Gaston Brouwer, Ph.D. July 2009

2. PART 1

3. Rational Expressions • Basics • Simplifying Rational Expressions • Rational Functions • Horizontal & Vertical Asymptotes • Graphing a Rational Function

4. Basics Multiplying fractions: Adding fractions: Simplifying fractions:

5. Rational Expressions Definition A rational expression can be written in the form Where and are both polynomial expressions and

6. Examples Rational expression Rational expression Not a rational expression

7. Simplifying rational expressions

8. Rational Functions Definition A rational function can be written in the form Where and are both polynomial functions and

9. Domain of a Rational Function The domain of a rational function is given by: Examples Domain: Domain:

10. End Behavior Let be a rational function. The line is a horizontal asymptote (HA) if:

11. How to find a horizontal asymptote 1. Divide and by the highest power of that shows up in . Call the resulting functions and . 2. HA:

12. HA Examples HA:

13. HA Examples (Continued) HA: No Horizontal Asymptote

14. Vertical Asymptotes Let be a rational function in lowest terms. The line is a vertical asymptote (VA) if:

15. How to find vertical asymptotes 1. Reduce the function to lowest terms. 2. The vertical asymptote(s) is (are): where is (are) the solution(s) to

16. VA Example Solve: VA: (Note that is not a vertical asymptote!)

17. Graphing a Rational Function Graph: 1. Reduce to lowest terms: 2. Find y-intercepts (set x=0): 3. Find x-intercepts (solve f(x)=0): 4. Find the horizontal asymptote: 5. Find the vertical asymptote(s):

18. Graphing a Rational Function (Cont’d) 6. Create a table for Not in the domain! (open circle)

19. Graphing a Rational Function (Cont’d)

20. PART 2

21. Intro to Chaos • Sequences • Compositions of Functions • Dynamical Systems • Cobweb Diagrams • Attracting Fixed Points

22. Sequences A (real)sequence can be defined as a function from the whole numbers W = {0, 1, 2, 3, …} to the real numbers. Example Let be defined by: . etc. Then: Alternatively, we could write this as:

23. Compositions of Functions Let be any interval on the real number line and let be a function from into . Then this function can be used to construct a sequence as follows: Let be any point in . Then:

24. Dynamical Systems A dynamical system is a combination of a space and a function acting on that space. Important question: What happens to when (for different choices of )? Examples: “An Introduction to Chaotic Dynamical Systems” by Robert Devaney.

25. The Weather Suppose that represents today’s weather and that is a function that, given any weather input, can compute the weather for the next day. Tomorrow’s weather: Day after tomorrow’s weather: Weather n days from now:

26. Example Dynamical System Consider the dynamical system: Find out what happens to for the following choices of : Conclusion:

27. Cobweb Diagram

28. A Chaotic Example is called the logistic equation

29. Attracting Fixed Points A point is called a fixed point if To find the fixed point(s) of we need to solve: A (fixed) point is attracting if Attracting when: Attracting when:

30. Fixed Points Summary If There is one fixed point in [0,1], namely: This point is attracting. If Fixed points: Attracting fixed point: