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exam1-sol

# exam1-sol - Exam 1 MAP 4484 February 6 2009 Name Student ID...

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Exam 1: MAP 4484 * February 6, 2009 Name : Student ID : This is a closed book exam and the use of calculators is not allowed. 1. A population model is described by the following equation: x n +1 = 2 1 + x n Calculate all fixed points and all period 2 points (if any). Fixed points are solutions of: x = 2 1 + x , or equivalently of: x 2 + x - 2 = 0 . This quadratic equation has two solutions: x 1 , 2 = - 1 ± 3 2 = 1 , - 2 We discard the negative solution, and thus there is a single fixed point: x * = 1 . Period two points must satisfy: x = 2 1 + 2 1+ x , which reduces to x 2 + x - 2 = 0 , the same quadratic equation we solved before. Thus, there are no period two points, as the only solution of the above quadratic equation only yields the fixed point determined earlier. What happens to the solution sequences as n → ∞ ? ( Hint : Use a global result from our cobwebbing notes) Since the function f ( x ) = 2 / (1 + x ) is decreasing and bounded, and since there are no period two points, we conclude that all solutions converge to the unique fixed point x * = 1.

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