Question

Given, ?(?) = { 4 , 0 ≤ ? < 2 ? + 1, 2 ≤ ? < 4 (a) Sketch the graph of ?(?) and its even half-range expansion. Then sketch THREE (3) full periods of the periodic function in the interval −12 < ? < 12. (6 marks) (b) Determine the Fourier cosine coefficients of Q1(a). (10 marks) (c) Write out ?(?) in terms of Fourier coefficients you have found in Q1(b)

Answer 1

Given solution OS X 32 f(x) = {x41 d<x<4 (a) flxe) 5 fo)=4 4 If y=xth y(2) = 2+123 (23) 404-441= (4.5) 3Ť f(x)2 3+1 2 L I 2 3 4 5 6 If f(x) is periodic function have then in interval (~1212), we 4. 4sxe-2 X+1 -2<x<o 4 afl f(x) = 0<x<2 2<x<4 43x36 6 <x< 8 4 x+1

2 of(x) 9 f(x)=x+1 7 5 - (-8,4) (6,4) 4,4) 8.41 (104) o (2,9) 3 2 t + + + 0 2 1 + 6 7 2 3 4 5 -8 7 - 3 -4 -3 ما le ht -24 -34 f(x)=x+2 -4+ -67 7 -9 f(x)=x+1 -10 <- (6) The Courier zesies sesies is is given by doo + E (ancosni + bn sinas) f(x) = nal (1) Then, 4 ao Ź Š flapse 2 1/2 [ f(x) due + Sf(x) dx] 2

3 ao = Ž [ { 4 dx + ( x + 1) an Ž { (4x) + ( x 2 į ( 8 + (432 ++)-( +2)=L(+ = 2 )] 8+8+4-4 il 3 2 5 an = bee 1/2 s f(x) cognitive de = NTR dat 2 s free) cognita dae 2 [ Se fles cos = 4( ! + conta obe *[ *() + (pan) S pe +h) cognitx de 17 dx 2 = 2 hi sin ntr 2 2 2 h 1. S l. sinnita dac 2 2 + 2 ni + (3 CoonTK [460-9 (5 sin ant) - 2 (3 sin sinnt) kon? -1 (C)( C332016-com) + [ 6m) Cinem] 2 asinno Cosant-l Cosnategn. 4 an when nis old 772 when 3 n 18 even

$ by = £ 5 fex) sim More dx = 1/2 f(x) sinnt dat S f(x). sinnte da & Sinne der the+L) sin hrix de 4 2 =*[ $tsimmpex de 15&+1) 4 + ={[ *43. costupa) * (enverdien mogel (sinnte] 2 + ( ) ( Bin ne = { [ -8 {cosnt-1980) 5.2 cosanit ht MT 3.2 nin COSNIT fo 9 sinn 20 Cosntizar Coornt=1 a (-1-1) - 10 NT nh 6 figh NT nimeong Ź la mer f 6 han find ил is bn = when nis even when n is oold (0 using equn (2), (3) &A) in equn. (2), we get- f(x) = cosna & Co sinna (n is odd) (n is even) 을 구 + Ź 4 nzo n272 nzo nIT