Formula Cheat Sheet for IB & OCR Mathematics | IB OCR 数学公式汇总手册

📚 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。

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1. Algebra & Quadratic Formula | 代数与二次公式

e 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。

/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。

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。

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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。

= 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。

ere 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|。

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3. Functions & Graphs | 函数与图像

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。

-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。

mmetric 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）。

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4. Trigonometry | 三角学

osine. 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。

s 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)。

= 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。

in 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。

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5. Sequences & Series | 数列与级数

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。

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。

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6. Differentiation | 微分

The derivative from first principles: f' imₕ→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。

(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。

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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。

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。

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。

l 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。

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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| ₂ + 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 t 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|。

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。

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。

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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。

ue 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 ~ 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 ct 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。

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10. Complex Numbers (IB HL & OCR Further) | 复数（IB HL 及 OCR 进阶）

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 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。

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。

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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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