# quiz3 - MIT OpenCourseWare http/ocw.mit.edu 18.085...

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MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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± ² 18.085 Quiz 3 October 7, 2008 Professor Strang Your PRINTED name is Student Number Grading 1 2 3 4 1. (25 points) (a) The 2 π -periodic function F ( x ) equals 1 for 0 x < π and equals 0 for π x < 2 π . Find its Fourier coeﬃcients c k using complex exponentials: ikx F ( x ) = c k e −∞ Write out the terms for k = 1 , 0 , 1. What is the decay rate of the c k as k → ∞ ? How do you see this from the function F ( x )? (b) The energy equality connects | F ( x ) | 2 dx with | c k | 2 . What is this equation for our particular F ( x )? Find a formula for π . (c) What is the derivative of this F ( x )? Draw the graph of dF/dx !! What is the complex Fourier series for dF/dx ? What is the decay rate of the coeﬃcients?
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quiz3 - MIT OpenCourseWare http/ocw.mit.edu 18.085...

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