Question

Q1 Given, f(x) = { 1, 95*<2 (x +1,25x<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. (b) Determine the Fourier cosine coefficients of Q1(a). (c) Write out f(x) in terms of Fourier coefficients you have found in Q1(b).

pls solve as quickly as possible

hand written also ok but pls be fast

Answer 1

PI) solution osa 22 f(x)= {aziti 41 2+1, 24 24 a) → Graph of f(x): 5 LP 3 2 1 1 2 3 4 5 6 Graph of even half-range expausion of f(x): 16 5 -4 -6 3 4

Sketch of in the function three interval full periods of the -12<x< 12. periodic S 6 51 3 2 If -12-11 ci -7 -6 - 4 -9 - 3 - - 2 2 3 lo 4 6 7 00 9 12 3 -15 -21 -3

(b) solution: f(1) = {ati, OX<2 2 < a <4 Here C=4 an = f(x) cos ntx da с 4 an= 2 4 를 SFCx) cos natx dx Ź f(x) cos ntx dx 2. = = * = = [f asx + 'cxa 1) dx] : G. = } [*087+[***1'] T: +4+4 0752] = { [8 + 8 +4-2-2 2] = { [18] = 8 1. 90 = 8 4 - an= 2 o 4 2 [ naa 4 COS .da 4 + (x+1) cos nota 4 da 2 sin (as) 1 - 16 h IT DIT/4 4 nT/4 n/4 74 s f(xcos nst x dx il! s 4 cos not 4 cos nix dx = 4 [sin ni sex+1) cos nixe da =[(x+1) sin(na) s sin(nya) 4 (X+1) sin na Al cos (nome) (haga 16 COS(no 12 sin ( ) (h) 16 facos (nome) dx + } (x+1) cos (net) (nita 16 cos(n) 12 sinnu) 16 cos () (6r2 ( 62 (sin(n)=0) + 2 20 sin(n) ht + hi cos y 4 16 sin ni + HIT

16 cos (5) + IT 4 sin (no) 16 cos (nit) 4nt sin (n) + 16 coschot) – 16 cos (nl) (ht)? ( n2 (n.)2 2n sin(n) + 8 cos Cht) an = 8 cos (1) (h)? :.

co Fourier cosine coefficients of Qica) are ao = an = 2n i sin (D) + 8 casinot) - 8 cos (no) (nr2 n=1,2,- (c) To write f (x) in terms of Fourier cosine coefficients in Q1(b). ao f(2)= tz an cos (none) 2 n=1 8 2nst sin (1T) + 8 cos(nit) - 8 cos() + E cos(mix) nel (nr2 :. f(x) = 4 + E nel 2nt sin(n) + 8 cos(nt) – 8 cos(n) n² T2 cos ( npr.