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Unformatted text preview: Math 251 December 14, 2005 Answer Key to 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. ANS. 0 = 2 x xy = x (2 y ) and 0 = 3 y + xy = y ( 3 + x ) gives two critical points x = and x = 3 2 . Give 1pt for each correctly found critical pt. b. 6pt In a neighborhood of each critical point approximate the nonlinear system by a linear system. ANS. At the linearization is x = 2 x y = 3 y which has eigenvalues r 1 = 2 and r 2 = 3 which have opposite sign. Give 2pt for linearizing correctly. At x = 3 2 we set u = x 3 v = y 2 . Substituting this into the original equation gives u = ( u + 3)( v )  3 v v = ( v + 2) u 2 u The eigenvalues of this system are purely imaginary. Give 2pt for substituting new variables correctly. Give 2pt for linearizing correctly. c. 2pt Determine the name and the stability of the the critical points of each of the linear approximations. At the origin the critical pt is a saddle which is always unstable. At 3 2 the critical pt is a center which is always stable. Give 1pt for naming the each critical pt and its stability correctly. d. 2pt Sketch a phase portrait for the original nonlinear system. ANS. To see the phase portrait enter the system into the phase portrait applet. The orientation of the center can be determined by plugging (4 , 2) into the original system. Give 1pt for the phase portrait drawn correctly near each critical point, including orientation. 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. ANS. center, proper and improper node. Give 0pt for 1 correct. Give 1pt for 2 correct. Give 2pt for 3 correct. 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 write the answer or write the first seven terms.) ANS. a = 1 2 Z 2 2 f ( x ) dx = 1 2 Z 1 3 dx = 3 Give 2pt for correct a a n = 1 2 Z 2 2 f ( x ) cos n 2 x dx = 1 2 Z 1 3 cos n 2 x dx = 3 2 2 n h sin n 2 x i 1 = 3 n h sin n 2 i Give 2pt for correct expression for a n after substituting f ( x ) into the integral. Give 1pt for correct evaluation of integral....
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This note was uploaded on 07/23/2008 for the course MATH 251 taught by Professor Chezhongyuan during the Spring '08 term at Pennsylvania State University, University Park.
 Spring '08
 CHEZHONGYUAN
 Math, Differential Equations, Equations, Partial Differential Equations

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