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Unformatted text preview: Math 251 Sections 1/2 December 14, 2005 Final Exam Name Please check one of the boxes below Section 1 1st per 8:00am Section 2 3rd per 10:10am There are 10 questions on this exam. Question 2, which has 12 parts, is worth 24 points. The other questions are worth 14 points each. The total number of points is 150. If a question has multiple parts, then the points assigned to the question are divided equally among the parts, unless otherwise indicated. Where appropriate, show your work to receive credit; partial credit may be given. Please turn your cell phone OFF . The use of calculators, books, or notes is not permitted on this exam. Time limit 1 hour and 50 minutes. 1. Consider the function f ( x ) = if x < 3 if x < 1 if 1 x Write down the first seven terms in the Fourier series of f ( x ) on [ 2 , 2]. 2. In Parts a. through e. s n ( x ) denotes the n th partial sum of the Fourier series in Problem 1. a. Find lim n s n (4) b. Find lim n s n (5) c. Find lim n s n (6) d. Is it true that for all sufficiently large n : s n ( x )  . 15 for every x in [1 . 4 , 1 . 7] ? e. Is it true that for all sufficiently large n : s n ( x ) 3 . 15 for every x in [0 . 9 , 1 . 1] ? f. What are the values of the following integrals? Z 6 6 cos 3 x cos 5 6 x dx Z 6 6 sin 6 x sin 6 x dx g. What are the values of the following integrals? Z 6 cos 6 x dx Z 6 6 sin 17 x dx h. If u ( x, t ) is the temperature of a thin rod of length L insulated on its sides and ends, then what are u x (0 , t ) and u x ( L, t ) for any t ?...
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 Spring '08
 CHEZHONGYUAN
 Math, Differential Equations, Equations, Partial Differential Equations

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