Matematika formulalari

Maktab matematikasidagi 152 ta asosiy formula bo'limlar bo'yicha: har biri qaysi sinfda o'tilishi bilan.

O'lchov va amallar (5-sinf)

To'g'ri to'rtburchak perimetri: P = 2(a + b)
To'g'ri to'rtburchak yuzi: S = a · b
Kvadrat perimetri va yuzi: P = 4a, S = a²
To'g'ri burchakli parallelepiped hajmi: V = a · b · c
Kub hajmi: V = a³
Tezlik, vaqt, masofa: S = v · t, v = S/t, t = S/v
Daraja — n marta ko'paytma: aⁿ = a · a · … · a
Amallar tartibi: ( ) → aⁿ → · : → + −
O'rta arifmetik: (a + b + …) / n

Bo'linish, nisbat, foiz (6-sinf)

2 ga bo'linish — oxirgi raqam bo'yicha: …0, …2, …4, …6, …8
3 va 9 ga bo'linish — raqamlar yig'indisi bo'yicha: 1+2+3 = 6 ⋮ 3
5 va 10 ga bo'linish — oxirgi raqam bo'yicha: 5: …0, …5 10: …0
Kasrni qisqartirish — EKUB ga: a/b = (a : d) / (b : d)
Kasrlarni qo'shish: a/b + c/d = (ad + cb) / bd
Kasrlarni ko'paytirish: a/b · c/d = ac / bd
Kasrlarni bo'lish: a/b : c/d = a/b · d/c
Proporsiya: a/b = c/d ⇒ a · d = b · c
Sonning foizini topish: a · p / 100
Sonni foizi bo'yicha topish: a = (b · 100) / p
Aylana uzunligi: C = 2πr = πd
Doira yuzi: S = πr²
Qarama-qarshi son va modul: −(−a) = a, |−a| = a

Qisqa ko'paytirish (7-sinf)

Yig'indi kvadrati: (a + b)² = a² + 2ab + b²
Ayirma kvadrati: (a − b)² = a² − 2ab + b²
Kvadratlar ayirmasi: a² − b² = (a − b)(a + b)
Yig'indi kubi: (a + b)³ = a³ + 3a²b + 3ab² + b³
Kublar yig'indisi: a³ + b³ = (a + b)(a² − ab + b²)
Kublar ayirmasi: a³ − b³ = (a − b)(a² + ab + b²)

Daraja va ildiz (7-sinf)

Ko'paytma: aᵐ · aⁿ = aᵐ⁺ⁿ
Bo'linma: aᵐ : aⁿ = aᵐ⁻ⁿ
Darajaning darajasi: (aᵐ)ⁿ = aᵐⁿ
Manfiy ko'rsatkich: a⁻ⁿ = 1 / aⁿ
Ratsional ko'rsatkich: a^(m/n) = ⁿ√(aᵐ)
Ildiz ko'paytmasi: √(ab) = √a · √b

Kvadrat tenglama (8-sinf)

Diskriminant: D = b² − 4ac
Ildizlar: x = (−b ± √D) / 2a
Viyet teoremasi: x₁ + x₂ = −b/a, x₁ · x₂ = c/a
Uchhadni ajratish: ax² + bx + c = a(x − x₁)(x − x₂)
Parabola uchi: x₀ = −b / 2a

Planimetriya (8-sinf)

Pifagor teoremasi: a² + b² = c²
Uchburchak yuzi: S = a · h / 2
Uchburchak yuzi (sinus): S = ½ · a · b · sin C
Geron formulasi: S = √(p(p−a)(p−b)(p−c)), p = P/2
Parallelogramm yuzi: S = a · h
Trapetsiya yuzi: S = (a + b) · h / 2
Romb yuzi: S = d₁ · d₂ / 2
Trapetsiya o'rta chizig'i: m = (a + b) / 2
Ko'pburchak burchaklari: S = (n − 2) · 180°
Aylana uzunligi: C = 2πr
Doira yuzi: S = πr²
Yoy uzunligi: l = πrn / 180
Sektor yuzi: S = πr²n / 360

Trigonometriya (9-sinf)

Asosiy ayniyat: sin²α + cos²α = 1
Tangens: tg α = sin α / cos α
Qo'shish (sinus): sin(α ± β) = sin α cos β ± cos α sin β
Qo'shish (kosinus): cos(α ± β) = cos α cos β ∓ sin α sin β
Ikkilangan burchak: sin 2α = 2 sin α cos α
Ikkilangan burchak: cos 2α = cos²α − sin²α
Jadval qiymatlari: sin: 0 · ½ · √2/2 · √3/2 · 1
Jadval qiymatlari: cos: 1 · √3/2 · √2/2 · ½ · 0
Radian: 180° = π, 1 rad = 180°/π
Sinuslar teoremasi: a/sin A = b/sin B = c/sin C = 2R
Kosinuslar teoremasi: c² = a² + b² − 2ab · cos C

Progressiyalar (9-sinf)

Arifmetik: n-had: aₙ = a₁ + (n − 1)d
Arifmetik: yig'indi: Sₙ = (a₁ + aₙ) · n / 2
Geometrik: n-had: bₙ = b₁ · qⁿ⁻¹
Geometrik: yig'indi: Sₙ = b₁(qⁿ − 1) / (q − 1)
Cheksiz kamayuvchi: S = b₁ / (1 − q), |q| < 1

Logarifm va ko'rsatkich (10-sinf)

Ta'rif: logₐ b = c ⇔ aᶜ = b
Ko'paytma: logₐ(xy) = logₐx + logₐy
Bo'linma: logₐ(x/y) = logₐx − logₐy
Daraja: logₐ(xⁿ) = n · logₐx
Asosni almashtirish: logₐb = log꜀b / log꜀a
Murakkab foiz: S = S₀(1 + p/100)ⁿ

Stereometriya (10-sinf)

Eyler formulasi: U − Q + Y = 2
Parallelepiped hajmi: V = a · b · c
Parallelepiped sirti: S = 2(ab + bc + ac)
Fazoviy diagonal: d = √(a² + b² + c²)
Prizma hajmi: V = S · h
Piramida hajmi: V = S · h / 3
Silindr hajmi: V = πr²h
Silindr yon sirti: S = 2πrh
Konus hajmi: V = πr²h / 3
Konus yon sirti: S = πrl
Shar hajmi: V = 4/3 · πr³
Sfera yuzi: S = 4πr²

Hosila va integral (11-sinf)

Daraja: (xⁿ)′ = n · xⁿ⁻¹
Ko'paytma: (uv)′ = u′v + uv′
Bo'linma: (u/v)′ = (u′v − uv′) / v²
Murakkab funksiya: (f(g))′ = f′(g) · g′
Jadval: (sin x)′ = cos x, (cos x)′ = −sin x
Jadval: (eˣ)′ = eˣ, (ln x)′ = 1/x
Urinma: y = f(x₀) + f′(x₀)(x − x₀)
Boshlang'ich: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
Nyuton–Leybnits: ∫ₐᵇ f(x) dx = F(b) − F(a)

Ehtimollik va statistika (11-sinf)

Klassik ta'rif: P = m / n
O'rin almashtirish: Pₙ = n!
O'rinlashtirish: Aₙᵏ = n! / (n − k)!
Kombinatsiya: Cₙᵏ = n! / (k!(n − k)!)
O'rta arifmetik: x̄ = (x₁ + … + xₙ) / n
Dispersiya: D = Σ(xᵢ − x̄)² / n
Chetlanish: σ = √D

Chiziqli algebra (201-sinf)

2×2 determinant: |a b; c d| = ad − bc
3×3 determinant — birinchi satr bo'yicha: a₁₁M₁₁ − a₁₂M₁₂ + a₁₃M₁₃
Matritsalar ko'paytmasi: cᵢⱼ = Σₖ aᵢₖ bₖⱼ
Teskari matritsa (2×2): A⁻¹ = 1/det · [d −b; −c a]
Kramer formulalari: x = Δₓ/Δ, y = Δᵧ/Δ

Vektorlar va analitik geometriya (201-sinf)

Vektor uzunligi: |a| = √(x² + y² + z²)
Skalyar ko'paytma: a·b = x₁x₂ + y₁y₂ + z₁z₂ = |a||b|cos φ
Vektor ko'paytma: a × b = (y₁z₂ − z₁y₂; z₁x₂ − x₁z₂; x₁y₂ − y₁x₂)
Perpendikulyarlik sharti: a·b = 0
To'g'ri chiziq tenglamasi: y = kx + b, Ax + By + C = 0
Nuqtadan to'g'ri chiziqqacha masofa: d = |Ax₀ + By₀ + C| / √(A² + B²)

Limitlar (201-sinf)

Birinchi ajoyib limit: lim sin x / x = 1, x → 0
Ikkinchi ajoyib limit: lim (1 + 1/x)ˣ = e, x → ∞
Ekvivalent cheksiz kichiklar (x → 0): sin x ~ tg x ~ ln(1 + x) ~ eˣ − 1 ~ x
Ratsional kasr, x → ∞: lim = bosh hadlar nisbati
Lopital qoidasi: lim f/g = lim f′/g′ (0/0, ∞/∞)

Hosila va integral (201-sinf)

Ko'paytma hosilasi: (uv)′ = u′v + uv′
Bo'linma hosilasi: (u/v)′ = (u′v − uv′) / v²
Murakkab funksiya: (f(g(x)))′ = f′(g) · g′
Ko'rsatkichli va logarifmik: (eˣ)′ = eˣ, (aˣ)′ = aˣ ln a, (ln x)′ = 1/x
Trigonometrik: (sin x)′ = cos x, (cos x)′ = −sin x, (tg x)′ = 1/cos²x
Nyuton–Leybnits: ∫ₐᵇ f(x)dx = F(b) − F(a)
Bo'laklab integrallash: ∫u dv = uv − ∫v du
Asosiy integrallar: ∫dx/x = ln|x| + C, ∫eˣdx = eˣ + C, ∫cos x dx = sin x + C

Ko'p o'zgaruvchili funksiyalar va karrali integral (202-sinf)

Xususiy hosila: ∂f/∂x — y o'zgarmas
To'la differensial: df = f′ₓdx + f′ᵧdy
Gradient: grad f = (∂f/∂x; ∂f/∂y)
Ekstremum sharti: f′ₓ = 0, f′ᵧ = 0; AC − B² > 0
Ikki karrali integral: ∬ f dxdy = ∫ₐᵇ dx ∫ f dy
Qutb koordinatalari: dxdy = r dr dφ

Qatorlar va differensial tenglamalar (202-sinf)

Umumlashgan garmonik qator: Σ 1/nᵖ yaqinlashadi ⇔ p > 1
Dalamber alomati: lim aₙ₊₁/aₙ = l < 1 ⇒ yaqinlashadi
Yaqinlashish radiusi: R = lim |aₙ / aₙ₊₁|
Makloren qatori: f(x) = Σ f⁽ⁿ⁾(0) xⁿ / n!
eˣ yoyilmasi: eˣ = 1 + x + x²/2! + x³/3! + …
Ajraladigan tenglama: y′ = f(x)g(y) ⇒ ∫dy/g = ∫f dx
y″ + py′ + qy = 0 tenglamasi: k² + pk + q = 0 → y = C₁e^(k₁x) + C₂e^(k₂x)
Karrali ildiz: y = (C₁ + C₂x)e^(kx)

Ehtimollar nazariyasi (203-sinf)

Klassik ehtimollik: P(A) = m / n
Yig'indi ehtimoli: P(A + B) = P(A) + P(B) − P(AB)
Ko'paytma ehtimoli: P(AB) = P(A) · P(B|A)
To'la ehtimollik: P(A) = Σ P(Hᵢ) P(A|Hᵢ)
Bayes formulasi: P(Hᵢ|A) = P(Hᵢ)P(A|Hᵢ) / P(A)
Bernulli formulasi: Pₙ(k) = Cₙᵏ pᵏ qⁿ⁻ᵏ

Tasodifiy miqdorlar va statistika (203-sinf)

Matematik kutilma: M(X) = Σ xᵢpᵢ
Dispersiya: D(X) = M(X²) − M(X)²
Binomial taqsimot: M = np, D = npq
Uch sigma qoidasi: P(|X − a| < 3σ) ≈ 0,997
Tanlanma o'rtacha: x̄ = Σ xᵢ / n
Tuzatilgan dispersiya: s² = Σ(xᵢ − x̄)² / (n − 1)