This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Tambua kazi na muda. Tunahitaji kupata mfululizo wa Fourier sine wa nusu-masafa kwa kazi f(x) = x kwenye muda 0 < x < . Kwa mfululizo wa Fourier sine wa nusu-masafa, urefu wa muda ni L = . Step 2: Kokotoa vigawo vya Fourier sine b_n. Fomula ya vigawo vya Fourier sine ni: b_n = (2)/(L) _0^L f(x) ((n x)/(L)) dx Badilisha L= na f(x)=x: b_n = (2)/() _0^ x ((n x)/()) dx b_n = (2)/() _0^ x (nx) dx Tutatumia ujumuishaji kwa sehemu ( u\,dv = uv - v\,du). Chagua u = x du = dx. Chagua dv = (nx) dx v = (nx) dx = -(1)/(n)(nx). b_n = (2)/() [ . -(x)/(n)(nx) |_0^ - _0^ -(1)/(n)(nx) dx ] b_n = (2)/() [ ( -()/(n)(n) - (-(0)/(n)(0)) ) + (1)/(n)_0^ (nx) dx ] b_n = (2)/() [ -()/(n)(n) + (1)/(n) . (1)/(n)(nx) |_0^ ] Tunajua kuwa (n) = (-1)^n na (n) = 0 kwa nambari kamili n. b_n = (2)/() [ -()/(n)(-1)^n + (1)/(n^2) ((n) - (0)) ] b_n = (2)/() [ -()/(n)(-1)^n + (1)/(n^2) (0 - 0) ] b_n = (2)/() [ -()/(n)(-1)^n ] b_n = -(2)/(n)(-1)^n Hii inaweza kuandikwa kama: b_n = (2)/(n)(-1)^n+1 Step 3: Andika mfululizo wa Fourier sine. Mfululizo wa Fourier sine umetolewa na: f(x) = _n=1^ b_n ((n x)/(L)) Badilisha L= na b_n: f(x) = _n=1^ (2)/(n)(-1)^n+1 (nx) Hivyo, mfululizo wa Fourier sine wa nusu-masafa kwa f(x)=x ni: f(x) = _n=1^ (2)/(n)(-1)^n+1 (nx) That's 2 down. 3 left today — send the next one.