Question

1. Determine whether the function f(x) = (x2 - 1) sin 5x is even, odd, or neither. A. Even B. Odd C. Neither 2. a). Find the Fourier sine series of the function f(x) shown below. b). Sketch the extended function f(x) that includes its two periodic extensions. TT/2 TT Formula to use: The sine series is f(x) = 6 sin NIT P where b. - EL " (x) sin " xd

Answer 1

The function is given as, fou a even function to be f(x) = (x²-1) sin sa f(x) = f(x) fl-x) = -(x²-1) sir sx f(x) + f(x) hence function is not even for a function to be odd, the condition of f(x) = f(-x) should setisfy Now, f(-x) = (x² - 1) sin(5x) fl-x) = -(x²-x) sin (5x) {s: Sin (0) --sine} -f(-x) (x²-1) sinds) f(x) = -fl-x) Hence the given function Option (B) selected us on Onswer. in ODD function (2) first of all sketching the function for 2 extended feriad. time period of fla) = π see. Mere 1 flu) - TT -T2 TW 37 2T → ST 377 Tine furiod T= I. 2

for os</ for Thcx <T f(x) = π f(x) = (TT-0) 22 TT f(x) = 2x time fuiad T3 T 27/2 = 21 .: f= 21 f(x) 2a o cu & T/ Lo = 2 T ; T{ {UXT The fourie series for any periodic function fin) with itime period te geven as, fla) a + (an cos neon + bn sin nw x) n=1 where Oo, ani bn care +34) de fourice Co Coefficients f How come seda ; bon focal limanda an f(x) da ; TT Th TT ao it i fox) dx + | anda & ft dx ㅠ ch Th T 1 x + TT 11 ㅠ The ao 2 11 - -0) + TT ㅠ 4 ao ㅠ + TT 2 +(31) Ao 3/4

TT One The 21 x Sin 2nx Sin 2nx da 21 + 24 / fon) cas n 2x dx [an en som du ta on... falu 24 as ) - (0 - 1) = sino - cima π sin anx 2n an Th 2 5 sin ana ㅠ + Cos anx 4n2 + π I sin anx an 12 24 2 TT sin ni + COST 2x 20 On - { .. sinni Sinanit=0 } 42 Q = -2 for {(1, -1} مد n=2 q=0 π n² 9=3, g - - 2 - e so on T bn 2 filx) sin 20x dx O T bn- 2 T an sin anadat π sin anada bn 32 -x Cos anx + sin anx TT IT Cos anx an 4n2 12

--=[x{17 I COSNT 21) + sin nit 4n2 Il cos 2nT - CONT bon -) -(0+0)} - (a * (213 60] [- 10 pt + LOMMY {:: Losant T=1} for na Cont 2n I las ant + π Cosnt 2n 2n Txt 2n bn 1 n ns1 Ib, 3-| n=2 1 b2 = -₂ n=3 I b₂ = -² I so on. is written as, .. Overall fouin series for flow {E0"-1} cos anx + 8 ( 4 ) sin 2nx 3 T ta Rn2 n=1 f)= 37 2 Cos za 2 Cos 6x + ga ५ I sin 6x sin 2x - 2 sin 4x This is a required required fourie series of flor)