NAME NetID MATH 285 E1 F1 Exam 3 C November 14 2014 Instructor Pascaleff Problem Possible Actual 1 20 Do all work on these sheets 2 20 Show all work 3 20 The exam is 50 minutes 4 20 Do not discuss this exam with anyone until after 3 00 pm on Nov 14 2014 5 20 Total 100 INSTRUCTIONS 1 Orthogonality formulas Z L m t n t cos cos dt L L L 0 m 6 n L m n 0 m 6 n m t n t sin sin dt L L L m n L Z L m t n t cos sin dt 0 L L L 1 L Z 2 3 Some integral formulas Z u cos u du u sin u cos u C 4 u sin u du u cos u sin u C 5 Z 2 1 20 points Find the general solution of the differential equation y 0 4y xe4x 3 2 20 points Consider the forced oscillator with mass m 1 spring constant k 7 no damping c 0 and forcing function F t F t 2 sin 2t cos 4t cos 6t Find a particular solution of the differential equation mx00 kx F t 4 3 a 10 points Suppose that a function f t which is periodic of period 2 has the Fourier series X 1 n 1 f t sin nt n2 3 n 1 Use the orthogonality formulas to evaluate the integral Z f t sin 2t dt b 10 points Let g t be the function which is periodic of period 24 and which is defined on the interval 12 t 12 by the formula g t 4 2t 7t2 Set up but do not evaluate an integral expression for the coefficient of cos 5 t 12 in the Fourier series of g t also known as a5 in our standard notation 5 4 a 5 points Consider the function which is periodic of period 2 defined on the interval t t 0 800 f t 609250 t 0 t 0 t If we take the Fourier series of f t and put t 0 in that series what number does it converge to Put another way what is the sum of the Fourier series of f t at t 0 Explain your answer briefly b 15 points Consider the function defined by the Fourier series g t X 2e 3n sin n t n 1 Find a Fourier series expression for the antiderivative address the question of convergence 6 R g t dt You are not expected to 5 20 points Find the Fourier series of the periodic function of period 2 defined on the interval 1 t 1 by f t 3 t 1 t 1 Hint You should use the fact that f t is an even function 7 This page is for work that doesn t fit on other pages 8
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