Question

2. The function of f(x) is given by TT X+ - 1<xs- 2 7 π -X, <x< 2 2 π X-TT, f(x)= <x<s, 2 f(x+27). a) Sketch the graph of f(x) for the range -1<x<. b) Based on a), determine the type of function f (x) and state your reason. c) Find the Fourier series of f(x).

Answer 1

PAGE NO Given thall 1+11 -31 -I ara - I - I << I <4 < TI (- TT a) so so the graph of given function is given as T TT - TT 7T 2 7.7.) 6 Now, we will see the type of this function as the graph of function is continuous so, flad is continuous on [-T IT lalso the given function is a function of period 217 from the definition also we can check continuity, on l-016] for eg. tim f(x) = tim 74 + 1 = LITO - T I f 2 wil 7 - H - IT f120) = lim H~ It 1 1 1 1 50 LH limit at IT - RH limit at - > funn is continues at - - We can check for n = 1 2 similarly Also the funn is increcying on (-1, -1) then decreasing from ( 1 , I J & again increasing them on (410)

Also We will check whether the function is even or old consider - att -T.C-HS-II I <-3 3 I fl-1) = lie. I <3 < TT -II <n< -ME HTT - M-TT T <-1 <T 2 (at) fl-ul 34 - ICH <-11/2 - 1/2 <n<11/2 sasi - (2-T) go f1-90) = flat) Y HE ERIT) = function is lodd function] on l-MID) cs, in the fourier geries f1H) - I got E ancosna t. Ebnáin na nel 1710 ng) ull the an = 0 (since from part be fli) is odd TT Jao = f f(x) da 2 Sifir) dal y a T Tū TT TI f(H) cosna dn = 2 TT 뮤 f(u) cosnada & bn = o. In

DATE PAGE NO c) As, from part (6) the funn flot is add the fourier series of f(a) will be of the flu) = { basinna where bn = 1 flasin nada = 2 flansinado nel are 0 since all the vatues of ai's in the general Fourier series À flalla Lo + E ancosnat s bosinna, Do nal so, we will calculate only values on bri Tot Now bn =2f flor) sin na dar f14) sinna dai t floosin nu du ㅠ To 0 T (11 Ti - a sin nadal tf (n-itsinna da TT e TT c & {-(H-IT cosna 2 (-a cosna + sinne 12 0 T & linna n2 ร์ก 12 -11/2 cos n112 0 2 f + to T. 12 t sinn 12 All los ni 2 :) 12 n 50 МЛ sinn I cosnit ſ ro SD 2 2 2 12 2 a sin(17/12) 1 1

PAGE NO -sint NTT m = 1; 2;,,.. na E bnsin na where bns - siofano b nol I ²007 11 1 & It ti 72 32 52. View ifr=1 1-sina t I singa - sinsh ta singa TL is to share oost 72 nd woll is the required required Fourier series of flore