Question

Given, f(x) = {x #1, 2 5x<4 4,0<x< 2 (a) Sketch the graph of f(x) and its even half-range expansion. Then sketch THREE (3) full periods of the periodic function in the interval -12 < x < 12. (6 marks) (b) Determine the Fourier cosine coefficients of Ql(a). (10 marks)

Answer 1

S4@, ococca froc) T= 4 Costa, acXL4 Wo = QT = 1/2 9) fex) 5 4 3 v +21L10-9-8-7 6-14-4-33 + 1 + a 34 5878 १ 10 11 12 soc T b) AID = olt f's f(0) Cosnoox da o V an fcoe) Cos Inde dae a 4 བ 4 cos nac x dac t 6C+1) COS TO dac [ S [+ a 4 sin Ix II LD 878 t- S X COS TTT OC doc o a t Stopx 6] (se sin un cos mp3 = -10 à [a a sinato TD + cos in a ) + TTDIQ T12 a + Le Bin TL 9 Taj Tot 2

sin The = el- de la asi 211 TD + a sinto Tola cosaitha Tola Si TV to/a COSITO Da Sinan TO 12 sinto =0&o for all o) n= 0,1,2,3,--- Sin 2170 =0 for all n =0,1,2,3,---- a C08 2111 = 1 for all 1 for all Dao, 1,2,3,.. Cos ITD =(-) for all n=0, 1, 2, an = [o to tamo asy] 2-2 G1)? CEO = *42c0"] 160 DI 1- GO TO D-odd an a oj D-even