IB & AQA Mathematics: Formula Compendium | IB 与 AQA 数学：公式汇总手册

📚 IB & AQA Mathematics: Formula Compendium | IB 与 AQA 数学：公式汇总手册

This reference guide consolidates the most essential formulas required for both IB (Analysis & Approaches / Applications & Interpretation) and AQA A‑level Mathematics. It is designed to support revision, problem‑solving, and exam preparation by presenting each formula with clear notation and concise explanations. Whether you are tackling pure mathematics, statistics, or mechanics, this compendium will serve as your go‑to quick reference.

本参考指南汇总了 IB（分析与方法 / 应用与解释）和 AQA A‑level 数学中最重要的公式。它以清晰的符号和简洁的解释呈现每条公式，旨在辅助复习、解题和备考。无论你正在攻克纯数学、统计学还是力学，这本手册都将是你的首选快速参考。

1. Algebra & Number | 代数与数

The fundamental building blocks include exponent laws, logarithms, and the binomial expansion. These rules underpin manipulation of expressions across all branches of the course.

基本构建块包括指数定律、对数以及二项式展开。这些规则是整个课程中表达式运算的基础。

Exponent Laws: a^m × aⁿ = a^m+n, (a^m)ⁿ = a^mn, a⁰ = 1, a⁻ⁿ = 1/aⁿ.

指数定律： a^m × aⁿ = a^m+n，(a^m)ⁿ = a^mn，a⁰ = 1，a⁻ⁿ = 1/aⁿ。

Logarithms: log_a(xy) = log_ax + log_ay, log_a(x/y) = log_ax − log_ay, log_a(x^k) = k log_ax. Change of base: log_ba = log_ca / log_cb.

对数： log_a(xy) = log_ax + log_ay，log_a(x/y) = log_ax − log_ay，log_a(x^k) = k log_ax。换底公式：log_ba = log_ca / log_cb。

Binomial Expansion: For |x| < 1, (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … (valid for any rational n). The general term for (a + b)ⁿ is _nC_r a^n−r b^r.

二项式展开： 当 |x| < 1 时，(1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + …（对任意有理数 n 成立）。(a + b)ⁿ 的通项是 _nC_r a^n−r b^r。

Arithmetic Series: S_n = n/2 [2a + (n−1)d] = n/2 (a + l), where l = a + (n−1)d.

等差数列： S_n = n/2 [2a + (n−1)d] = n/2 (a + l)，其中 l = a + (n−1)d。

Geometric Series: S_n = a(1 − rⁿ)/(1 − r) for r ≠ 1. Infinite sum: S_∞ = a/(1 − r) provided |r| < 1.

等比数列： 当 r ≠ 1 时，S_n = a(1 − rⁿ)/(1 − r)。无穷和：当 |r| < 1 时，S_∞ = a/(1 − r)。

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2. Functions & Graphs | 函数与图像

Mastery of function transformations, inverses, and key features is essential for modelling and equation solving. This section covers both algebraic and graphical perspectives.

掌握函数变换、反函数和关键特征是建模和方程求解的关键。本节涵盖代数和图像两个视角。

Transformations: y = af(x) + b. Value a gives vertical stretch by factor a; b is vertical translation. For y = f(cx + d), factor c causes horizontal stretch by 1/|c|, and −d/c shifts horizontally.

变换： y = af(x) + b。a 表示垂直方向拉伸 a 倍；b 为垂直平移。对于 y = f(cx + d)，因子 c 导致水平方向拉伸 1/|c|，−d/c 进行水平平移。

Inverse Function: f⁻¹(x) reflects y = f(x) in the line y = x. Domain and range swap. Composition: f(f⁻¹(x)) = x.

反函数： f⁻¹(x) 是 y = f(x) 关于直线 y = x 的反射。定义域和值域互换。复合：f(f⁻¹(x)) = x。

Quadratic Formula: For ax² + bx + c = 0, x = [−b ± √(b² − 4ac)] / (2a). Discriminant Δ = b² − 4ac.

二次公式： 对于 ax² + bx + c = 0，x = [−b ± √(b² − 4ac)] / (2a)。判别式 Δ = b² − 4ac。

Remainder Theorem: When a polynomial f(x) is divided by (x − k), the remainder is f(k). Factor Theorem: (x − k) is a factor if f(k) = 0.

余数定理： 当多项式 f(x) 除以 (x − k) 时，余数为 f(k)。因式定理：若 f(k) = 0，则 (x − k) 为因式。

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3. Trigonometry | 三角学

Trigonometric identities and rules for solving triangles are indispensable for IB and AQA exams. Radians are the default measure in calculus contexts.

三角恒等式和解三角形的规则在 IB 和 AQA 考试中不可或缺。弧度是微积分中的默认度量单位。

Radian Measure: π rad = 180°. Arc length s = rθ, area of sector A = ½ r²θ.

弧度制： π 弧度 = 180°。弧长 s = rθ，扇形面积 A = ½ r²θ。

Pythagorean Identity: sin²θ + cos²θ = 1. Derived forms: 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ.

勾股恒等式： sin²θ + cos²θ = 1。衍生形式：1 + tan²θ = sec²θ，1 + cot²θ = csc²θ。

Double Angle Formulas: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ, tan 2θ = 2 tan θ / (1 − tan²θ).

倍角公式： sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ，tan 2θ = 2 tan θ / (1 − tan²θ)。

Sine Rule: a/sin A = b/sin B = c/sin C. Cosine Rule: a² = b² + c² − 2bc cos A. Area = ½ ab sin C.

正弦定理： a/sin A = b/sin B = c/sin C。余弦定理： a² = b² + c² − 2bc cos A。面积 = ½ ab sin C。

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4. Coordinate Geometry & Vectors | 坐标几何与向量

Straight lines, circles, and vector operations form the geometric backbone. Both 2D and 3D applications appear across pure and applied modules.

直线、圆和向量运算是几何的支柱。二维和三维应用贯穿纯数学和应用模块。

Distance & Midpoint: Between (x₁, y₁) and (x₂, y₂): distance d = √[(x₂−x₁)² + (y₂−y₁)²]; midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).

距离与中点： 点 (x₁, y₁) 和 (x₂, y₂) 之间：距离 d = √[(x₂−x₁)² + (y₂−y₁)²]；中点 = ((x₁+x₂)/2, (y₁+y₂)/2)。

Equation of a Circle: Centre (a, b), radius r: (x − a)² + (y − b)² = r².

圆方程： 圆心 (a, b)，半径 r：(x − a)² + (y − b)² = r²。

Vector Operations: For vectors a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k:

向量运算： 对于向量 a = a₁i + a₂j + a₃k 和 b = b₁i + b₂j + b₃k：

• Dot product: a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cos θ
• 点积：a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cos θ
• Cross product (AQA Further/IB HL): a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k
• 叉积（AQA 进阶 / IB HL）：a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k

Equation of a Line: r = a + tb, where a is a point on the line and b the direction vector.

直线方程： r = a + tb，其中 a 为直线上一点，b 为方向向量。

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5. Differentiation | 微分

Differentiation rules enable the study of rates of change, gradients, and optimisation. The chain, product, and quotient rules must be applied fluently.

微分法则用于研究变化率、梯度和最优化问题。必须熟练应用链式法则、乘积法则和商法则。

Basic Derivatives: d/dx (xⁿ) = n xⁿ⁻¹; d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (e^x) = e^x; d/dx (ln x) = 1/x.

基本导数： d/dx (xⁿ) = n xⁿ⁻¹；d/dx (sin x) = cos x；d/dx (cos x) = −sin x；d/dx (e^x) = e^x；d/dx (ln x) = 1/x。

Chain Rule: If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x).

链式法则： 若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。

Product Rule: d/dx [u(x) v(x)] = u'(x)v(x) + u(x)v'(x).

乘积法则： d/dx [u(x) v(x)] = u'(x)v(x) + u(x)v'(x)。

Quotient Rule: d/dx [u/v] = (u’v − uv’) / v².

商法则： d/dx [u/v] = (u’v − uv’) / v²。

Second Derivative: f”(x) denotes the rate of change of gradient; used to determine concavity and points of inflection.

二阶导数： f”(x) 表示梯度的变化率，用于判断凹凸性和拐点。

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6. Integration | 积分

Integration recovers quantities from rates, such as area under a curve. Definite integrals and integration by substitution are core techniques.

积分从变化率中恢复总量，例如曲线下面积。定积分和换元积分法是核心技巧。

Basic Integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1); ∫ 1/x dx = ln|x| + C; ∫ e^x dx = e^x + C; ∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C.

基本积分： ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)；∫ 1/x dx = ln|x| + C；∫ e^x dx = e^x + C；∫ sin x dx = −cos x + C；∫ cos x dx = sin x + C。

Definite Integral: ∫_a^b f(x) dx = F(b) − F(a), where F'(x) = f(x).

定积分： ∫_a^b f(x) dx = F(b) − F(a)，其中 F'(x) = f(x)。

Integration by Substitution: For ∫ f(g(x)) g'(x) dx, let u = g(x), then du = g'(x) dx, giving ∫ f(u) du.

换元积分法： 对于 ∫ f(g(x)) g'(x) dx，令 u = g(x)，则 du = g'(x) dx，得到 ∫ f(u) du。

Area Between Curves: Area = ∫_a^b [f(x) − g(x)] dx, where f(x) ≥ g(x) on [a, b].

曲线间面积： 若在 [a, b] 上 f(x) ≥ g(x)，则面积 = ∫_a^b [f(x) − g(x)] dx。

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7. Exponentials & Logarithms in Calculus | 微积分中的指数与对数

These functions appear frequently in growth and decay models. Their unique differentiation and integration properties deserve special attention.

这些函数经常出现在增长与衰减模型中。它们独特的微积分性质值得特别注意。

Derivative of a^x: d/dx (a^x) = a^x ln a. Consequently, d/dx (e^kx) = k e^kx.

a^x 的导数： d/dx (a^x) = a^x ln a。因此，d/dx (e^kx) = k e^kx。

Integral of a^x: ∫ a^x dx = a^x/ln a + C (a > 0, a ≠ 1).

a^x 的积分： ∫ a^x dx = a^x/ln a + C (a > 0, a ≠ 1)。

Logarithmic Differentiation: Useful when variable appears in both base and exponent. Take ln of both sides and differentiate implicitly.

对数微分法： 当变量同时出现在底数和指数中时有用。对两边取 ln 并隐式求导。

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8. Probability & Statistics | 概率与统计

IB and AQA statistics topics cover probability rules, distributions, and hypothesis testing. The following formulas are essential for data analysis.

IB 和 AQA 的统计部分涵盖概率规则、分布和假设检验。以下公式对于数据分析至关重要。

Probability Rules: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For independent events, P(A ∩ B) = P(A)P(B). Conditional probability: P(A|B) = P(A ∩ B)/P(B).

概率规则： P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。对于独立事件，P(A ∩ B) = P(A)P(B)。条件概率：P(A|B) = P(A ∩ B)/P(B)。

Binomial Distribution: X ~ B(n, p). P(X = k) = _nC_k p^k (1 − p)^n−k. Mean = np, variance = np(1 − p).

二项分布： X ~ B(n, p)。P(X = k) = _nC_k p^k (1 − p)^n−k。均值 = np，方差 = np(1 − p)。

Normal Distribution: Z = (X − μ)/σ ~ N(0, 1). Use standardisation to find probabilities. Approximately 68% within 1σ, 95% within 2σ, 99.7% within 3σ.

正态分布： Z = (X − μ)/σ ~ N(0, 1)。利用标准化求概率。大约 68% 的数据落在 1σ 内，95% 在 2σ 内，99.7% 在 3σ 内。

Expected Value & Variance: E[aX + b] = aE[X] + b. Var[aX + b] = a² Var[X]. For a discrete variable, E[X] = Σ x P(X = x), Var[X] = Σ (x − μ)² P(X = x) = E[X²] − (E[X])².

期望与方差： E[aX + b] = aE[X] + b。Var[aX + b] = a² Var[X]。对于离散变量，E[X] = Σ x P(X = x)，Var[X] = Σ (x − μ)² P(X = x) = E[X²] − (E[X])²。

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9. Mechanics (AQA & IB HL Applications) | 力学（AQA 与 IB HL 应用）

Kinematics and Newton’s laws are central to AQA mechanics and the IB Applications HL option. SUVAT equations assume constant acceleration.

运动学和牛顿定律是 AQA 力学和 IB 应用 HL 选项的核心。SUVAT 方程假设加速度恒定。

SUVAT Equations: v = u + at; s = ut + ½ at²; s = ½ (u + v)t; v² = u² + 2as; s = vt − ½ at². These link displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).

SUVAT 方程： v = u + at；s = ut + ½ at²；s = ½ (u + v)t；v² = u² + 2as；s = vt − ½ at²。它们关联位移 s、初速度 u、末速度 v、加速度 a 和时间 t。

Newton’s Second Law: F = ma, where F is the resultant force. For connected particles, draw force diagrams and solve simultaneously.

牛顿第二定律： F = ma，其中 F 为合外力。对于连接体，画出受力图并联立求解。

Momentum: p = mv. Impulse = change in momentum = F Δ t. In collisions, total momentum is conserved if no external forces act.

动量： p = mv。冲量 = 动量变化 = F Δ t。若无外力作用，碰撞中总动量守恒。

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10. Sequences & Series (Further) | 数列与级数（进阶）

Sigma notation and mathematical induction recur in IB HL and AQA Further Pure. Knowing these summation formulas saves time.

求和符号和数学归纳法在 IB HL 和 AQA 进阶纯数中反复出现。掌握这些求和公式能节省时间。

Summation Formulas: Σ_r=1ⁿ r = n(n+1)/2; Σ r² = n(n+1)(2n+1)/6; Σ r³ = n²(n+1)²/4.

求和公式： Σ_r=1ⁿ r = n(n+1)/2；Σ r² = n(n+1)(2n+1)/6；Σ r³ = n²(n+1)²/4。

Proof by Induction: 1) Base case: prove true for n = 1. 2) Inductive step: assume true for n = k, prove for n = k+1. 3) Conclusion.

数学归纳法证明： 1) 基础步骤：证明 n = 1 时成立。2) 归纳步骤：假设 n = k 时成立，证明 n = k+1 时成立。3) 结论。

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