Darslar 6–7 · 2 ta dars · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra va analiz asoslari), B. Q. Xaydarov (geometriya). Matematika 11-sinf, 1- va 2-qismlar, 1-nashr. «ZAMIN NASHR» MChJ, Toshkent, 2018
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Funksiya grafigiga o‘tkazilgan urinma va normal tenglamalari
Darslik, 1-qism: 34–38-betlar
MaqsadFunksiya grafigiga berilgan nuqtada o‘tkazilgan urinma va normal tenglamalarini tuzish hamda berilgan to‘g‘ri chiziqqa parallel urinmani topish.
Yangi so‘zlar
urinma tenglamasi · tangent equationnormal · normal lineparallellik sharti · condition of parallelismperpendikularlik sharti · condition of perpendicularity
Tushuntirish
y = f(x) grafigiga abssissasi x₀ bo‘lgan nuqtada o‘tkazilgan urinmaning burchak koeffitsiyenti f′(x₀) ga teng va u (x₀; f(x₀)) nuqtadan o‘tadi, shuning uchun urinma tenglamasi y − f(x₀) = f′(x₀)(x − x₀). Normal — shu nuqtada urinmaga perpendikular to‘g‘ri chiziq; burchak koeffitsiyentlari ko‘paytmasi −1 bo‘lgani uchun, f′(x₀) ≠ 0 da normal tenglamasi y − f(x₀) = −(1/f′(x₀))(x − x₀). Agar f′(x₀) = 0 bo‘lsa, urinma gorizontal (y = f(x₀)), normal esa vertikal (x = x₀). Grafikka y = kx + b ga parallel urinma kerak bo‘lsa, f′(x₀) = k tenglamadan x₀ topiladi va keyin urinma tenglamasi yoziladi. Urinmaning Ox o‘qining musbat yo‘nalishi bilan burchagi α uchun tg α = f′(x₀).
Namunalar
f(x) = x³ − 3x, x₀ = 2. f(2) = 2, f′(x) = 3x² − 3, f′(2) = 9. Urinma: y − 2 = 9(x − 2), ya’ni y = 9x − 16. Normal: y − 2 = −(1/9)(x − 2), ya’ni y = −x/9 + 20/9.
f(x) = x³ − 2x grafigiga y = x + 4 ga parallel urinmalar: f′(x) = 3x² − 2 = 1 ⇒ x₀ = ±1. f(1) = −1: y + 1 = x − 1, ya’ni y = x − 2; f(−1) = 1: y − 1 = x + 1, ya’ni y = x + 2. Gorizontal urinma, masalan, y = x² − 4x + 5 uchun: 2x − 4 = 0 ⇒ x₀ = 2, f(2) = 1, tenglama y = 1.
Sinfda faoliyat
«Ikki to‘g‘ri chiziq»: juftliklar tanlagan funksiya grafigida (masalan, y = x² − 2x) bir nuqta oladi, urinma va normal tenglamalarini yozadi va chizg‘ich bilan koordinata tekisligida chizadi. Ikkinchi juftlik chizmaning to‘g‘ri burchak hosil qilishini transportir yoki to‘g‘ri burchakli chizg‘ich bilan tekshiradi.