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

) 327 ocx<2 2 snc4 re fin) = x + 1 3+ fex) = 4 2 1 -3 - 2 - 2 3 2 5 for f(x) ocze <2 cesa Centre de to ER) et Half range cosine series. do + an cosna where 2 f(x) dx 5 24 emergo due 2 4 da 4

an 4 cos n na da sinna 2. 4 sin (22) = O vu vu 0 SO 4 + an cosna 4 n=1 Now, 2. fin) Itx 22x ca 4 do 4. (x + 1) da 16 +4 - xp xxuson Citi 4 xp lexuen x cosnth up 9 4 sinnat 4 sinna + ana 2 2 to a sin nan 2. xprenu 2,18 Sin real + na na vu 0 9 sina na 4 sin(na) MA + I cos ham na + 20 sin 2. na ички, KUULS - h*

Ga (doc contosan C-65] is odd an ethau is oren . Here seriel. auises obubs trou do xcuson be 1 4 + coshT 3 n=1 + I cos 3*+---- # lost t 4.09 - 12 6 12

() cosine serier, Half range do 4 + s cosna n2 nas so, co-efficient lie. fourier cosine 8 co-efficient © COST+ fraid = 4 + Cost+tosan t. +5 X 4 + 2 / 2 ( cosn + - cos 3 n + 2 Cossat And