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**Unformatted text preview: **Math 251 December 14, 2005 Final Exam Name Section There are 10 questions on this exam. Many of them have multiple parts. The point value of each question is indicated either at the beginning of each question or at the beginning of each part where there are multiple part Where appropriate, show your work to receive credit; partial credit may be given. The use of calculators, books, or notes is not permitted on this exam. Please turn off your cell phone. Time limit 1 hour and 50 minutes. Question Score 1 18pt 2 16pt 3 12pt 4 14pt 5 12pt 6 14pt 7 14pt 8 16pt 9 20pt 10 14pt Total 150pt 1. Consider the nonlinear system: x = 2 x- xy y =- 3 y + xy a. 2pt Find all the critical points of the nonlinear system. b. 6pt In a neighborhood of each critical point approximate the nonlinear system by a linear system. c. 2pt Determine the name and the stability of the the critical points of each of the linear approximations. d. 2pt Sketch a phase portrait for the original nonlinear system. e. 2pt The linearization of a nonlinear system may have a critical point that is not guaranteed to reflect the behavior of the original nonlinear system. List the three types of critical points for which this may happen. 2. Consider the function f ( x ) = if x < 3 if ≤ x < 1 if 1 ≤ x a. 10pt Find the Fourier series of f ( x ) on [- 2 , 2]. (Either use summation notation to write2]....

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