Question

Q1 Given, f(x) = {x + 4,0<x< 2 Lx + 1, 2 <x< 4 (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) (c) Write out f(x) in terms of Fourier coefficients you have found in Q1(b). (4 marks)

Answer 1

Solution f(x) = 4. 0 4 x 42 24 x 4 Y+1 , VIVACE 16 4 -|6 12 (6 ) นน x_e [0, 4] there 0ะ 2 ce 5 : * (1) 41 = ! 41 : (H) 41] = {{ + () + (11) ] G. = it + (2-) + (12 - 43 ) 2 : (8 +8) ะ 8 . Clot 8 + ty - 2 () 2 4 11 it () it . t : J f(x) t25 ทITY ส่วน - :[ 1 + fatis contex de * [ 4 หน (Innit) { (2.1) ที่ Cry it + * {( : 0 1 ) - (0 41" }}] [' - th"] = 3 ( 0 -0) - ИП an = [1 - 80 ] an - 4 . ! 1 'r2 หr2 หrt1 -*)) 4 da bn= 0 4 . xาเuuID + สา ท 2 : fal) C7 11" { / + S1n ng* da + 1) Shºw” dº ] - 3 ปี 9 (ที่2) () ทา" * + { (9L+) (2) (8) + 1) 5 (e) - 1 ) + {(te) -(1) -6)}] - :[ 1 - 600") -2 (5 - 31/72 ] b7 - 9 - €)) - 1, (5-361)) หrt 4 (1-e1") - (S-3e) ] IM 3 ni = bn= รูปนี้ MIC P bn 2 +า ท ย 2 L ใน tex) = ∞o + E ancosnil + E businnitu (1.) = + + 2 8 ft - el)” ] ce) ICA : (+60"), cinni 0 . IC | หน โ ห | ออก