O‘ZBEKISTON RESPUBLIKASI AXBOROT TEXNOLOGIYALARI VA KOMMUNIKATSIYALARINI RIVOJLANTIRISH VAZIRLIGI
MUHAMMAD AL-XORAZMIY NOMIDAGI
TOSHKENT AXBOROT TEXNOLOGIYALARI UNIVERSITETI SAMARQAND FILIALI
"Dasturiy injiniring" kafedrasi
№ _____ мustaqil ta’lim ish hisoboti
Fan___________“ Algoritmlarni loyihalash”___________________
Guruh 21-08
Talaba
Abdullayev G’ulom
Quyidagi funksiyalarni Furye qatorlariga yoyishda a0,a1 va b1 koeffitsentlar qiymatlarini toping. Algortim va dastur tuzing
2.y=x2+2x+3
F(x) = x^2 + 2x + 3
Furye qatori formulasi:
a0 = (1/T) * integral(F(x), dx, -T/2, T/2)
a1 = (2/T) * integral(F(x)*cos(2*pi*x/T), dx, -T/2, T/2)
b1 = (2/T) * integral(F(x)*sin(2*pi*x/T), dx, -T/2, T/2)
Biz T=2 olib, integrallarni hisoblaymiz:
a0 = (1/2) * integral(x^2 + 2x + 3, dx, -1, 1)
= (1/2) * [(1/3)*x^3 + x^2 + 3x] from -1 to 1
= (1/2) * [(1/3)*1^3 + 1^2 + 3*1 - ((1/3)*(-1)^3 + (-1)^2 + 3*(-1))]
= (1/2) * [4/3 + 4]
= 5/3
a1 = (2/2) * integral((x^2 + 2x + 3)*cos(2*pi*x/2), dx, -1, 1)
= integral((x^2 + 2x + 3)*cos(pi*x), dx, -1, 1)
= [(1/pi)*(x^2+4)*sin(pi*x) - (2/pi)*x*cos(pi*x) - (3/pi)*sin(pi*x)] from -1 to 1
= [(1/pi)*(1^2+4)*sin(pi*1) - (2/pi)*1*cos(pi*1) - (3/pi)*sin(pi*1)]
- [(1/pi)*((-1)^2+4)*sin(pi*(-1)) - (2/pi)*(-1)*cos(pi*(-1)) - (3/pi)*sin(pi*(-1))]
= [(5/pi)*0 - (2/pi)*(-1) - (3/pi)*0] - [(5/pi)*0 - (2/pi)*1 - (3/pi)*0]
= 4/pi
b1 = (2/2) * integral((x^2 + 2x + 3)*sin(2*pi*x/2), dx, -1, 1)
= integral((x^2 + 2x + 3)*sin(pi*x), dx, -1, 1)
= [(1/pi)*(-x^2-2)*cos(pi*x) + (2/pi)*x*sin(pi*x) + (3/pi)*(1-cos(pi*x))] from -1 to 1
= [(1/pi)*(-1^2-2)*cos(pi*1) + (2/pi)*1*sin(pi*1) + (3/pi)*(1-cos(pi*1))]
- [(1/pi)*((-1)^2+2)*cos(pi*(-1)) + (2/pi)*(-1)*sin(pi*(-1)) + (3/pi)*(1-cos(pi*(-1)))]
= [(3/pi)*0 + (2/pi)*0 + (3/pi)*2] - [(3/pi)*0 - (2/pi)*0 + (3/pi)*2]
= 12/(pi)
Jami qator:
F(x) = 5/3 + 4/pi*cos(pi*x) + 12/(pi)*sin(pi*x)
2.Quyidagi funksiyalarni Furye qatorlariga yoying va yetakchi garmonikalarini aniqlang
2.y=x2+2x+3
F(x) = a0/2 + ∑(n=1, ∞)[an*cos(2*pi*n*x/T) + bn*sin(2*pi*n*x/T)]
Bu formulaga ko'ra, y=x^2+2x+3 funksiyasining Furye qatorining yetakchi garmonikalari quyidagicha bo'ladi:
a0 = 5/3
an = 0 (har bir toq sonli n uchun)
bn = 4/(pi*n)^2 * (-1)^n * sin(pi*n)
Shu bilan y=x^2+2x+3 funksiyasining Furye qatorining yetakchi garmonikalari quyidagicha bo'ladi:
a0 = 5/3
an = 0 (har bir toq sonli n uchun)
bn = 4/(pi*n)^2 * (-1)^n * sin(pi*n)
Bizning funksiya y=x^2+2x+3 kvadratik funksiya bo'lgani uchun, bu funksiyani y(x) = ax^2 + bx + c ko'rinishida yozib olishimiz mumkin. Shuningdek, y(x) = a(x+h)^2 + k ko'rinishida ham yozib olishimiz mumkin, bu yerda h va k - ko'chirish va o'tkazish ko'ordinatalari bo'ladi.
y(x) = ax^2 + bx + c
y(x) = a(x^2+2hx+h^2) + bx + c - h^2a + h^2a + k
y(x) = ax^2 + 2ahx + ah^2 + bx + c - h^2a + k
Bu yerda, a = 1, b = 2, va c = 3 bo'lgani uchun:
y(x) = x^2 + 2x + 3
Shu funksiyani Furye qatoriga yoyish uchun, biz T = 2 ko'rsatkichni tanlaymiz (ya'ni funksiya T = 2 oralig'ida to'g'ri chizilgan o'qda yinayapti) va quyidagi formulalarni ishlatamiz:
a0 = (1/T) * integral(y(x), x, -T/2, T/2)
an = (2/T) * integral(y(x)*cos(2*pi*n*x/T), x, -T/2, T/2)
bn = (2/T) * integral(y(x)*sin(2*pi*n*x/T), x, -T/2, T/2)
a0 ni topish:
a0 = (1/2) * integral(x^2+2x+3, x, -1, 1)
a0 = (1/2) * [((1)^3)/3 + 2*(1)^2 + 3 - ((-1)^3)/3 - 2*(-1)^2 - 3]
a0 = (1/2) * [4 + 6]
a0 = 5/3
an ni topish:
an = (2/2) * integral((x^2+2x+3)*cos(2*pi*n*x/2), x, -1, 1)
an = integral((x^2+2x+3)*cos(pi*n*x), x, -1, 1)
an = [(1/2)*((1)^3)/3 + (1)*2*(1)^2 + (1)*3 - (1/2)*((-1)^3)/3 - (1)*2*(-1)^2 - (1)*3] * integral(cos(pi*n*x), x, -1, 1)
an = [(1/2)*((1)^3)/3 + (1)*2*(1)^2 + (1)*3 - (1/2)*((-1)^3)/3 - (1)*2*(-1)^2 - (1)*3] * [sin(pi*n) - sin(-pi*n)] / pi*n
an = 0
bn ni topish:
bn = (2/2) * integral((x^2+2x+3)*sin(2*pi*n*x/2), x, -1, 1)
bn = integral((x^2+2x+3)*sin(pi*n*x), x, -1, 1)
bn = [(1/2)*((1)^3)/3 + (1)*2*(1)^2 + (1)*3 - (1/2)*((-1)^3)/3 - (1)*2*(-1)^2 - (1)*3] * integral(sin(pi*n*x), x, -1, 1)
bn = [(1/2)*((1)^3)/3 + (1)*2*(1)^2 + (1)*3 - (1/2)*((-1)^3)/3 - (1)*2*(-1)^2 - (1)*3] * [cos(-pi*n) - cos(pi*n)] / pi*n
bn = 4/(pi*n)^2 * (-1)^n * sin(pi*n)
Shu bilan y=x^2+2x+3 funksiyasining Furye qatorining yetakchi garmonikalari quyidagicha bo'ladi:
a0 = 5/3
an = 0 (har bir toq sonli n uchun)
bn = 4/(pi*n)^2 * (-1)^n * sin(pi*n)
3.Amaliy mashg’ulot topshiriqlari. Quyidagi masalalar uchun chiziqli regressiya tenglamasidan (y=ax+b) foydalanib, algoritm va dastur tuzing
Quyidagi jadvalda vagon g’ildiragining yillardagi xizmat muddati (x) va g'ildirak gardishining eskirishi va qalinligi (y, mm) bo‘yicha ma’lumotlar ko‘rsatilgan.
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