Question

Q1 Given, f(x) = {x +1, 25x<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) (c) Write out f(x) in terms of Fourier coefficients you have found in Q1(b). (4 marks) [Total: 20 marks]

Answer 1

Salation :- Ginen function flaw S 4 XAL ocxL2 25x24. (a). Graph of the funtion fex) Y 4 2 4 Graph of even extension + -12 12 oh raye 12 L&L 12.

(b) we need to find Fouries cosine coefficient Co Jens + Con cas (nting n=1 where cosine coefficient en 2 2 8 fix cos(hT2 ) sx, MEN. ewhere L= 4 4 so со ti f(x) cos ( "Txan 4 with f(x1= 0< x L2 24 x 24 Xf1 4 Cn an Es 4 cos (DTX) dx + 1 f ( x + 1) (b. [ sinfning. + 1 S to Txar 4 2 4 a los dnt 49 2 [ Sinf ) (mp3* *]; + [x Sin (nita 4 3 [ Sin (44) Sinf 4 [Sinfont) x 1 9 + Sin (m) x りの ni

8 C. ht sin (30) + a 17 [+ sin(96) – 3 sin[ 20)] [ cos (had) + 4 - 4. * ht HT + 2 no [ Sin (br) - sin ( 47 ) Since ntn then sinlui) VnEN.. enit 4 nm na + h2pL & sin(nI) + Lin (nI) [cos (hal - Cos (37)] ( -fialno Santiago de integral certum tacosferi 2 hTI In a ho an

4 4 & L + (x) dx 2 1 2 4 2.dx + £ $*(X+3) ds. + 1 2 +! [ 8 + 4 - 2 -2] 2 [a x] 4 4 - 2 2 + 4 = 6. (©). Fouries Loefficient. + nal f(x) = 6 Z cn Cos (nik) 3 + {{( 2 ) sin(at) + Gofry to cart - Gol 953)} n=1 Ans.