Question

Q1 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. (b) Determine the Fourier cosine coefficients of Ql(a). (c) Write out f(x) in terms of Fourier coefficients you have found in Q1(b).

Answer 1

som? Givens 03232 HX)= 14 2+1 25*24 (6) Sketching of fla 4 2 6 O 8 lo 12 -12 -lo-8 oven half - range - expansion ie: cogine series expansion in 0.4) 9 n1 { x=0 2 l = 4 a=2 f(x)= aot of ancos An costante (21+x) a = 1 é francosna a de sli 4.cernanda + a 4 2 11 2 2 (en) Cen(97) [of someone) fentent content ) "{$sec) * in fase) - } + e-> haft Isiol NA 2 (-1 an 24 n²2 52 CS

Nous w loc -drat Q EU 2 Jearida á 40x)+(154 +4:) [8+ 8+8 18 lolas 021 En 222 D²2 이리 (+9 fla + Ź in Colonna = 4+ S Plona coolean) 4+2EU cos (272 4 + 2 s 2.) connect flaf 4 + 2 2 cosax + 2 cosshat 9 expansion This is required half-sange 2222 2222 21 12 nz Nowo, sketch of fla) with three ful periods in (-12, 12) 6 -/-/ 2 + + >> 2 4 -2 6 8 10 12 14 8

[b] tourier cosine coefficiend of fla): X-20 An= f l s froe cooner da from we god. ao=4 An = 2-1) 2 272 5272 an (4-6)} where, n212 2272 sc1 f(n) = 4+ 3 (2003mx + 2 cos since consgx + Sketch of cosine series: 4 nnel forn=2 > And soron
I have taken some rough sketch.

Thank you!