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p2spring10

p2spring10 - Prelim 2 Math 2930 show work 6 pages no...

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Prelim 2 Math 2930 Spring 2010 show work, 6 pages, no calculators 1a) (4 points) Write your name and section number at the top of this page. 1b) (8 points) A frictionless spring and mass system is forced at two angular frequencies 0 < a < 1 < b and the mass position y ( t ) is described by y = - y + cos( at ) + 2 sin( bt ). Find the solutions. 1c) (4 points) In class we discussed the geometric series 1 + x + x 2 + x 3 + · · · which converges to 1 / (1 - x ) when | x | < 1. To what does the series 1 + x 2 3 + x 4 9 + x 6 27 + x 8 81 + · · · converge, and for what numbers x ? 1d) (4 points) Estimate the error in the common approximation sin( x ) . = x , when 0 < x < 1 10 . (Remember sin( x ) = x - x 3 3! + x 5 5! - · · · for all x .) 1

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2a) (4 points) Prove that π - π cos(2 t ) cos(3 t ) dt = 0 using whatever you know about trig functions. b) (6 points) Show that the functions e it , e - it and sin( t ) are linearly de- pendent. You are allowed to use complex coefficients. (You must say what “linearly dependent” means. No credit for any statement about Wronskian.) c) (6 points) Find (only) the coefficient a 3 in the Fourier series f ( t ) = a 0 2 + n =0 ( a n cos( nt ) + b n sin( nt )) for the period 2 π
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p2spring10 - Prelim 2 Math 2930 show work 6 pages no...

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