📚 Formula Cheat Sheet for IB & OCR Mathematics | IB OCR 数学公式汇总手册
This comprehensive formula handbook brings together the essential equations, identities, and rules required for IB Mathematics (Analysis and Approaches, Applications and Interpretation) and OCR A-Level Mathematics. Whether you are revising for IB Standard or Higher Level exams or preparing for OCR’s pure, statistics, and mechanics papers, this reference covers the common core – algebra, functions, trigonometry, calculus, vectors, probability, complex numbers, and kinematics. Bookmark this page and use it alongside your past paper practice.
这本综合公式手册汇集了 IB 数学（分析与方法、应用与解释）和 OCR A-Level 数学所必需的关键方程、恒等式与法则。无论你是在为 IB 标准或高级水平备考，还是在准备 OCR 的纯数、统计与力学试卷，这份速查表都覆盖了共同的核心——代数、函数、三角、微积分、向量、概率、复数以及运动学。请收藏本页，配合历年真题练习使用。
1. Algebra & Quadratic Formula | 代数与二次公式
For any quadratic equation ax² + bx + c = 0 (a ≠ 0), the roots are given by the quadratic formula. The discriminant Δ = b² – 4ac determines the nature of the solutions.
对于任何一元二次方程 ax² + bx + c = 0 (a ≠ 0)，其根由求根公式给出。判别式 Δ = b² – 4ac 决定解的性质。
x = [-b ± √(b² – 4ac)] / (2a)
当 Δ > 0 时方程有两个不等实根；Δ = 0 时有两个相等实根（一个重根）；Δ < 0 时有一对共轭复根。
The sum of roots α + β = -b/a, and the product of roots αβ = c/a. For polynomial equations of higher degree, the same ideas extend via Vieta’s formulas.
两根之和 α + β = -b/a，两根之积 αβ = c/a。对于高次多项式方程，同样的思想通过韦达定理推广。
Completing the square: x² + bx = (x + b/2)² – (b/2)². This technique is fundamental to deriving the quadratic formula and to handling conic sections.
配方法：x² + bx = (x + b/2)² – (b/2)²。这一技巧是推导求根公式以及处理圆锥曲线的基础。
The binomial theorem for (1 + x)ⁿ, with n rational, states: (1 + x)ⁿ = 1 + nx + [n(n-1)/2!]x² + [n(n-1)(n-2)/3!]x³ + …, valid for |x| < 1. For integer n, the expansion is finite and given by the binomial coefficients nCr.
对于有理数 n，(1 + x)ⁿ 的二项式定理为：(1 + x)ⁿ = 1 + nx + [n(n-1)/2!]x² + [n(n-1)(n-2)/3!]x³ + …，在 |x| < 1 时成立。当 n 为整数时，展开式为有限项，由二项式系数 nCr 给出。
2. Exponentials & Logarithms | 指数与对数
The fundamental relationship: aˣ = b ⇔ x = logₐb, where a > 0, a ≠ 1. The natural exponential function eˣ and natural logarithm ln x = logₑx appear pervasively in calculus.
基本关系：aˣ = b ⇔ x = logₐb，其中 a > 0 且 a ≠ 1。自然指数函数 eˣ 与自然对数 ln x = logₑx 贯穿微积分始终。
ln (xy) = ln x + ln y; ln (x/y) = ln x – ln y; ln (xᵏ) = k ln x
对数的运算法则：ln(xy) = ln x + ln y；ln(x/y) = ln x – ln y；ln(xᵏ) = k ln x。
Change of base: logₐb = log b / log a = ln b / ln a. This is essential for solving equations involving logs of different bases.
换底公式：logₐb = log b / log a = ln b / ln a。对于求解不同底数的对数方程，这一公式至关重要。
Derivative of eᵏˣ is keᵏˣ; derivative of aˣ is aˣ ln a. The integral of 1/x is ln |x| + C.
eᵏˣ 的导数为 keᵏˣ；aˣ 的导数为 aˣ ln a。1/x 的积分为 ln |x| + C。
Exponential growth and decay models: N(t) = N₀ eᵏᵗ, where k > 0 gives growth and k < 0 gives decay. The half-life relates to k via T₁/₂ = ln 2 / |k|.
指数增长与衰减模型：N(t) = N₀ eᵏᵗ，其中 k > 0 表示增长，k < 0 表示衰减。半衰期与 k 的关系为 T₁/₂ = ln 2 / |k|。
3. Functions & Graphs | 函数与图像
A function f: A → B assigns exactly one output to each input. The domain is set A, and the range is the set of all output values f(x).
函数 f: A → B 将每个输入恰好对应一个输出。定义域为集合 A，值域为所有输出值 f(x) 构成的集合。
Transformations of graphs: y = a f(b(x – h)) + k. a gives vertical stretch (by factor |a|) and reflection in x-axis if a < 0; b gives horizontal stretch (by factor 1/|b|) and reflection in y-axis if b < 0; h is horizontal shift; k is vertical shift.
图像变换：y = a f(b(x – h)) + k。a 实现垂直拉伸（因子 |a|），若 a < 0 则关于 x 轴反射；b 实现水平拉伸（因子 1/|b|），若 b < 0 则关于 y 轴反射；h 为水平平移；k 为垂直平移。
Inverse function f⁻¹ exists if f is one-to-one. Its graph is the reflection of y = f(x) in the line y = x. Domain and range swap.
反函数 f⁻¹ 在 f 为一一映射时存在。其图像是 y = f(x) 关于直线 y = x 的对称。定义域与值域互换。
Composite function (f ∘ g)(x) = f(g(x)). The domain of f ∘ g is the set of x in the domain of g for which g(x) is in the domain of f.
复合函数 (f ∘ g)(x) = f(g(x))。f ∘ g 的定义域为 g 的定义域中使得 g(x) 落入 f 的定义域的那些 x 的集合。
Even and odd functions: f(-x) = f(x) for even functions (symmetric about y-axis); f(-x) = -f(x) for odd functions (rotational symmetry about origin).
奇偶函数：偶函数满足 f(-x) = f(x)（关于 y 轴对称）；奇函数满足 f(-x) = -f(x)（关于原点旋转对称）。
4. Trigonometry | 三角学
The unit circle defines sine and cosine. Exact values for 0°, 30°, 45°, 60°, 90° must be memorised: sin 30° = 1/2, cos 30° = √3/2, tan 45° = 1, etc. Radian measure: π rad = 180°.
单位圆定义正弦和余弦。必须熟记 0°、30°、45°、60°、90° 的精确值：sin 30° = 1/2，cos 30° = √3/2，tan 45° = 1 等。弧度制：π rad = 180°。
Pythagorean identity: sin²θ + cos²θ = 1. Variants: 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ.
毕达哥拉斯恒等式：sin²θ + cos²θ = 1。变体：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 and cosine rules for triangles. For any triangle ABC: a/sin A = b/sin B = c/sin C (sine rule). Cosine rule: a² = b² + c² – 2bc cos A. Area = ½ ab sin C.
三角形中的正弦定理和余弦定理。对于任意三角形 ABC：a/sin A = b/sin B = c/sin C（正弦定理）。余弦定理：a² = b² + c² – 2bc cos A。面积 = ½ ab sin C。
Trigonometric equations: general solutions. For sin θ = k, θ = nπ + (-1)ⁿ α; for cos θ = k, θ = 2nπ ± α; for tan θ = k, θ = nπ + α, where n ∈ ℤ and α is the principal solution.
三角方程的通解：sin θ = k 时，θ = nπ + (-1)ⁿ α；cos θ = k 时，θ = 2nπ ± α；tan θ = k 时，θ = nπ + α，其中 n ∈ ℤ，α 为主值解。
5. Sequences & Series | 数列与级数
Arithmetic sequence: uₙ = a + (n-1)d. Sum of first n terms: Sₙ = n/2 [2a + (n-1)d] = n/2 (a + l), where l is the last term.
等差数列：uₙ = a + (n-1)d。前 n 项和：Sₙ = n/2 [2a + (n-1)d] = n/2 (a + l)，其中 l 为末项。
Geometric sequence: uₙ = arⁿ⁻¹. Sum of first n terms: Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1. Sum to infinity exists when |r| < 1: S∞ = a/(1 - r).
等比数列：uₙ = arⁿ⁻¹。前 n 项和：Sₙ = a(1 – rⁿ)/(1 – r)，r ≠ 1。当 |r| < 1 时存在无穷和：S∞ = a/(1 - r)。
Convergence tests for infinite series include the ratio test and comparison test. The harmonic series Σ 1/n diverges, while the p-series Σ 1/nᵖ converges for p > 1.
无穷级数的收敛性判别法包括比值判别法和比较判别法。调和级数 Σ 1/n 发散，而 p 级数 Σ 1/nᵖ 当 p > 1 时收敛。
Binomial series: (a + b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ for n ∈ ℕ. For rational n, the infinite expansion holds for |x| < 1 in (1 + x)ⁿ.
二项级数：对于 n ∈ ℕ，(a + b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ。对于有理数 n，(1 + x)ⁿ 的无穷展开在 |x| < 1 时成立。
Arithmetico-geometric series and the method of differences appear frequently in IB HL and OCR further sequences topics.
等差-等比级数与差分法在 IB HL 和 OCR 进阶序列主题中频繁出现。
6. Differentiation | 微分
The derivative from first principles: f'(x) = limₕ→₀ [f(x+h) – f(x)] / h. This defines the gradient of the tangent to the curve y = f(x).
从第一原理出发的导数：f'(x) = limₕ→₀ [f(x+h) – f(x)] / h。它定义了曲线 y = f(x) 的切线斜率。
d/dx (xⁿ) = n xⁿ⁻¹, d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (eˣ) = eˣ
基本导数：d/dx (xⁿ) = n xⁿ⁻¹，d/dx (sin x) = cos x，d/dx (cos x) = -sin x，d/dx (eˣ) = eˣ。
Product rule: d/dx (uv) = u’v + uv’. Quotient rule: d/dx (u/v) = (u’v – uv’) / v². Chain rule: dy/dx = dy/du · du/dx.
乘法法则：d/dx (uv) = u’v + uv’。商法则：d/dx (u/v) = (u’v – uv’) / v²。链式法则：dy/dx = dy/du · du/dx。
Parametric differentiation: if x = f(t), y = g(t), then dy/dx = (dy/dt) / (dx/dt). The second derivative d²y/dx² = d/dx (dy/dx) = d/dt (dy/dx) / (dx/dt).
参数方程求导：若 x = f(t)，y = g(t)，则 dy/dx = (dy/dt) / (dx/dt)。二阶导数 d²y/dx² = d/dx (dy/dx) = d/dt (dy/dx) / (dx/dt)。
Implicit differentiation: differentiate each term with respect to x, treating y as a function of x and appending dy/dx where necessary.
隐函数求导：对每一项关于 x 求导，将 y 视为 x 的函数，并在必要时附上 dy/dx。
Stationary points: solve f'(x) = 0. Nature determined by second derivative: f”(x) > 0 → local minimum; f”(x) < 0 → local maximum; f''(x) = 0 requires further investigation.
驻点：解 f'(x) = 0。性质由二阶导数决定：f”(x) > 0 → 局部极小；f”(x) < 0 → 局部极大；f''(x) = 0 需进一步判断。
7. Integration | 积分
Indefinite integration is the reverse of differentiation: ∫ f(x) dx = F(x) + C, where F'(x) = f(x). The constant of integration C is essential.
不定积分是微分的逆运算：∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)。积分常数 C 不可或缺。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1), ∫ 1/x dx = ln |x| + C, ∫ eˣ dx = eˣ + C
基本积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)，∫ 1/x dx = ln |x| + C，∫ eˣ dx = eˣ + C。
Definite integral: ∫ₐᵇ f(x) dx = F(b) – F(a). It represents the signed area between the curve and the x-axis from x = a to x = b.
定积分：∫ₐᵇ f(x) dx = F(b) – F(a)。它表示曲线与 x 轴之间从 x = a 到 x = b 的有号面积。
Area between two curves: ∫ₐᵇ [f(x) – g(x)] dx, where f(x) ≥ g(x) on [a, b]. Volume of revolution about the x-axis: V = π ∫ₐᵇ [f(x)]² dx.
两曲线之间的面积：∫ₐᵇ [f(x) – g(x)] dx，其中在 [a, b] 上 f(x) ≥ g(x)。绕 x 轴旋转体的体积：V = π ∫ₐᵇ [f(x)]² dx。
Integration by substitution: ∫ f(g(x)) g'(x) dx = ∫ f(u) du, where u = g(x). Integration by parts: ∫ u dv = uv – ∫ v du.
换元积分法：∫ f(g(x)) g'(x) dx = ∫ f(u) du，其中 u = g(x)。分部积分法：∫ u dv = uv – ∫ v du。
Trapezium rule for numerical integration: ∫ₐᵇ f(x) dx ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ], where h = (b – a)/n.
梯形法则用于数值积分：∫ₐᵇ f(x) dx ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]，其中 h = (b – a)/n。
8. Vectors | 向量
A vector has magnitude and direction. In component form: v = a i + b j + c k. Magnitude: |v| = √(a² + b² + c²). Unit vector: v̂ = v / |v|.
向量具有大小和方向。分量形式：v = a i + b j + c k。模长：|v| = √(a² + b² + c²)。单位向量：v̂ = v / |v|。
Scalar (dot) product: u · v = |u||v| cos θ = u₁v₁ + u₂v₂ + u₃v₃. Used to find the angle between vectors and to test perpendicularity (u · v = 0).
数量积（点积）：u · v = |u||v| cos θ = u₁v₁ + u₂v₂ + u₃v₃。用于求向量间的夹角以及判断垂直（u · v = 0）。
Vector (cross) product (for 3D): u × v = |u||v| sin θ n̂, where n̂ is the unit vector perpendicular to both. Component formula via determinant. Area of parallelogram = |u × v|.
向量积（叉积，适用于三维）：u × v = |u||v| sin θ n̂，其中 n̂ 为同时垂直于两者的单位向量。可通过行列式计算分量。平行四边形面积 = |u × v|。
Equation of a line: r = a + λ d, where a is a point on the line and d is the direction vector. Equation of a plane: r · n = a · n, or using two direction vectors.
直线方程：r = a + λ d，其中 a 为直线上一点，d 为方向向量。平面方程：r · n = a · n，或使用两个方向向量表示。
Intersection of lines and planes: solve parametric equations simultaneously. The scalar triple product a · (b × c) gives the volume of a parallelepiped and tests for coplanarity.
直线与平面的交点：联立参数方程求解。标量三重积 a · (b × c) 给出平行六面体的体积，并用于共面性检验。
9. Probability & Statistics | 概率与统计
Probability of event A: P(A) = n(A)/n(S) for equally likely outcomes. Addition rule: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). Mutually exclusive events: P(A ∩ B) = 0.
事件 A 的概率：在等可能结果下 P(A) = n(A)/n(S)。加法法则：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。互斥事件：P(A ∩ B) = 0。
Conditional probability: P(A|B) = P(A ∩ B) / P(B). Independent events: P(A ∩ B) = P(A) P(B) or equivalently P(A|B) = P(A). Bayes’ theorem connects the reverse conditional.
条件概率：P(A|B) = P(A ∩ B) / P(B)。独立事件：P(A ∩ B) = P(A) P(B)，或等价地 P(A|B) = P(A)。贝叶斯定理连接反向条件概率。
Discrete random variable X: expected value E(X) = Σ x P(X = x); variance Var(X) = E(X²) – [E(X)]². Standard deviation σ = √Var(X).
离散随机变量 X：期望 E(X) = Σ x P(X = x)；方差 Var(X) = E(X²) – [E(X)]²。标准差 σ = √Var(X)。
Binomial distribution: X ~ B(n, p). P(X = r) = nCr pʳ qⁿ⁻ʳ, where q = 1 – p. E(X) = np, Var(X) = npq.
二项分布：X ~ B(n, p)。P(X = r) = nCr pʳ qⁿ⁻ʳ，其中 q = 1 – p。E(X) = np，Var(X) = npq。
Normal distribution: X ~ N(μ, σ²). Standardise using Z = (X – μ)/σ. Use tables or calculator to find probabilities. The total area under the curve is 1.
正态分布：X ~ N(μ, σ²)。使用 Z = (X – μ)/σ 标准化。利用表格或计算器求概率。曲线下总面积为 1。
Linear regression: least squares regression line y = a + bx, where b = Sxy / Sxx and a = ȳ – b x̄. The product moment correlation coefficient r measures strength and direction of linear association.
线性回归：最小二乘回归线 y = a + bx，其中 b = Sxy / Sxx，a = ȳ – b x̄。积矩相关系数 r 度量线性关联的强度与方向。
10. Complex Numbers (IB HL & OCR Further) | 复数（IB HL 及 OCR 进阶）
A complex number is z = a + bi, where i² = -1. Real part Re(z) = a, imaginary part Im(z) = b. Complex conjugate: z̄ = a – bi.
复数为 z = a + bi，其中 i² = -1。实部 Re(z) = a，虚部 Im(z) = b。共轭复数：z̄ = a – bi。
Modulus: |z| = √(a² + b²). Argument: arg(z) = θ, where tan θ = b/a, adjusted by quadrant. Polar form: z = r (cos θ + i sin θ) = r cis θ.
模：|z| = √(a² + b²)。辐角：arg(z) = θ，其中 tan θ = b/a，需根据象限调整。极坐标形式：z = r (cos θ + i sin θ) = r cis θ。
Euler’s formula: eⁱᶿ = cos θ + i sin θ. Exponential form: z = r eⁱᶿ. Multiplication and division become simple in polar/exponential form.
欧拉公式：eⁱᶿ = cos θ + i sin θ。指数形式：z = r eⁱᶿ。在极坐标/指数形式下乘法和除法变得简单。
De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for integer n. It is used to find powers and roots of complex numbers.
棣莫弗定理：对于整数 n，(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。它用于求复数的幂与方根。
The n nth roots of unity are solutions to zⁿ = 1, given by z = e²ᵏπⁱ/ⁿ, k = 0, 1, …, n-1. They are equally spaced on the unit circle.
n 次单位根是 zⁿ = 1 的解，由 z = e²ᵏπⁱ/ⁿ 给出，k = 0, 1, …, n-1。它们在单位圆上等间隔分布。
11. Kinematics & Mechanics Basics | 运动学与力学基础
For constant acceleration a, the SUVAT equations link displacement s, initial
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