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Unformatted text preview: MATH 2004 TEST 1 September 2006 I N S T R U C T I O N S This test has one page and is worth 30 marks . It cannot be taken out of the exam room. Test duration: 50 minutes . Write your full name and student number onto the exam booklet provided by your tutor. Show all your work in this booklet and explain carefully your solutions. No calculators are allowed. 1. The 2-periodic even function f ( x ) is given on the interval 0 x 1 by the formula f ( x ) = 2- x. [2 marks] (a) Sketch the graph of f ( x ) and display at least three periods. [2 marks] (b) State a formula for f ( x ) on the interval- 1 < x < 0. [5 marks] (c) Determine all Fourier coefficients of the function f ( x ). [2 marks] (d) State the Fourer series for f ( x ) explicitly, using your results from part (c). 2. The 2 -periodic function f ( x ) is given on the interval- x by the formula f ( x ) = cos( x 2 ) . The Fourier series of this function is: f ( x ) = 4 + X n =1 4(- 1) n +1 (4 n 2- 1) cos( nx ) . [3 marks] (a) Is the Integration Theorem on Fourier Series applicable for this function? Justify your decision! If so, write down the Fourier series for an antiderivative of f ( x ). [6 marks] (b) Is the Differentiation Theorem on Fourier Series applicable for this function? Jus- tify your decision! If so, write down the Fourier series for the derivative of f ( x )....
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- Fall '08