Year 11 CAIE Additional Mathematics: Formula & Theorem Quick Reference Handbook | Year 11 CAIE 进阶数学：公式定理速查手册

📚 Year 11 CAIE Additional Mathematics: Formula & Theorem Quick Reference Handbook | Year 11 CAIE 进阶数学：公式定理速查手册

This handbook provides a compact summary of all essential formulas and theorems required for the CAIE IGCSE Additional Mathematics (0606) syllabus. Use it as a quick reference to reinforce your understanding of functions, quadratics, trigonometry, calculus and more.

本手册为 CAIE IGCSE 进阶数学（0606）提供了所有必备公式与定理的简洁汇总。把它当作速查手册，让你快速回顾函数、二次式、三角学、微积分等核心内容。

1. Algebra & Polynomials | 代数与多项式

Remainder Theorem: When a polynomial f(x) is divided by (x – c), the remainder is f(c).

余式定理：多项式 f(x) 除以 (x – c) 的余式为 f(c)。

Factor Theorem: (x – c) is a factor of f(x) if and only if f(c) = 0.

因式定理：(x – c) 是 f(x) 的因式当且仅当 f(c) = 0。

Completing the square: ax² + bx + c = a(x + b/(2a))² – (b² – 4ac)/(4a).

配方法：ax² + bx + c = a(x + b/(2a))² – (b² – 4ac)/(4a)。

Polynomial identities: (a ± b)² = a² ± 2ab + b², (a + b)(a – b) = a² – b², (a ± b)³ = a³ ± 3a²b + 3ab² ± b³.

多项式恒等式：(a ± b)² = a² ± 2ab + b²，(a + b)(a – b) = a² – b²，(a ± b)³ = a³ ± 3a²b + 3ab² ± b³。

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2. Surds & Indices | 根式与指数

Index laws for a > 0: a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m–n, (a^m)ⁿ = a^mn, a⁰ = 1, a^–n = 1/aⁿ, a^1/n = ⁿ√a.

指数律（a > 0）：a^m × aⁿ = a^m+n，a^m ÷ aⁿ = a^m–n，(a^m)ⁿ = a^mn，a⁰ = 1，a^–n = 1/aⁿ，a^1/n = ⁿ√a。

Surd rules: √ab = √a × √b, √a/b = √a / √b, rationalising denominator: 1/(√a + √b) = (√a – √b)/(a – b).

根式运算法则：√ab = √a × √b，√a/b = √a / √b，分母有理化：1/(√a + √b) = (√a – √b)/(a – b)。

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3. Quadratic Functions & Equations | 二次函数与方程

Standard form: f(x) = ax² + bx + c (a ≠ 0). Vertex form: f(x) = a(x – h)² + k, vertex (h, k).

标准式：f(x) = ax² + bx + c（a ≠ 0）。顶点式：f(x) = a(x – h)² + k，顶点为 (h, k)。

Quadratic formula: x = [–b ± √b² – 4ac] / (2a).

二次公式：x = [–b ± √b² – 4ac] / (2a)。

Discriminant Δ = b² – 4ac: gt; 0 → two distinct real roots; Δ = 0 → one repeated real root; Δ < 0 → no real roots.

判别式 Δ = b² – 4ac：Δ > 0 → 有两个不等实根；Δ = 0 → 有一个重根；Δ < 0 → 无实根。

Sum & product of roots (Vieta): if roots are α, β, then α + β = –b/a, αβ = c/a.

根与系数的关系（韦达定理）：若根为 α, β，则 α + β = –b/a，αβ = c/a。

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4. Equations & Inequalities | 方程与不等式

Linear inequalities: Solve as equations but reverse inequality sign when multiplying/dividing by a negative number.

线性不等式：解法同方程，但当乘以或除以负数时，不等号方向改变。

Quadratic inequality: e.g. (x – p)(x – q) > 0 → x < p or x > q (for p < q); use sign diagram or graph.

二次不等式：例如 (x – p)(x – q) > 0 → x < p 或 x > q（设 p < q）；可使用符号表或图像。

Modulus equations & inequalities: |ax + b| = c → ax + b = ±c; |ax + b| < c → –c < ax + b < c; |ax + b| > c → ax + b < –c or ax + b > c.

绝对值方程与不等式：|ax + b| = c → ax + b = ±c；|ax + b| < c → –c < ax + b < c；|ax + b| > c → ax + b < –c 或 ax + b > c。

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5. Functions & Graphs | 函数与图像

Domain & range: Domain = set of all possible inputs; range = set of all possible outputs.

定义域与值域：定义域是所有可能输入值的集合；值域是所有可能输出值的集合。

Composite function: fg(x) = f(g(x)), apply g first then f.

复合函数：fg(x) = f(g(x))，先作用 g 再作用 f。

Inverse function f⁻¹(x): exists only if f is one-one; graph of f⁻¹ is reflection of f in y = x.

反函数 f⁻¹(x)：仅当 f 为一一对应时存在；f⁻¹ 的图像是 f 的图像关于 y = x 的对称图形。

Graph transformations: y = f(x) + a → translate up a units; y = f(x + a) → translate left a units; y = –f(x) → reflect in x-axis; y = f(– lect in y-axis; y = a f(x) → vertical stretch × a; y = f(ax) → horizontal stretch × (1/a).

图像变换：y = f(x) + a → 向上平移 a 单位；y = f(x + a) → 向左平移 a 单位；y = –f(x) → 关于 x 轴反射；y = f(–x) → 关于 y 轴反射；y = a f(x) → 纵向拉伸为原来的 a 倍；y = f(ax) → 横向伸缩为原来的 1/a 倍。

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6. Logarithmic & Exponential Functions | 对数与指数函数

Definition: y = a^x ⇔ x = logₐ y (a > 0, a ≠ 1). ln x = logₑ x, e ≈ 2.718.

定义：y = a^x ⇔ x = logₐ y（a > 0，a ≠ 1）。ln x = logₑ x，e ≈ 2.718。

Log laws: logₐ (MN) = logₐ M + logₐ N; logₐ (M/N) = logₐ M – logₐ N; logₐ M^k = k logₐ M.

对数运算律：logₐ (MN) = logₐ M + logₐ N；logₐ (M/N) = logₐ M – logₐ N；logₐ M^k = k logₐ M。

Change of base: logₐ b = logₓ b / logₓ a = ln b / ln a.

换底公式：logₐ b = logₓ b / logₓ a = ln b / ln a。

Exponential growth/decay: y = Ae^kx; k > 0 growth, k < 0 decay.

指数增长/衰减：y = Ae^kx；k > 0 增长，k < 0 衰减。

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7. Trigonometry | 三角学

Radian measure: π rad = 180°. Arc length s = rθ, sector area A = ½ r²θ (θ in rad).

弧度制：π rad = 180°。弧长 s = rθ，扇形面积 A = ½ r²θ（θ 以弧度计）。

Trigonometric ratios: sinθ = opp/hyp, cosθ = adj/hyp, tanθ = opp/adj.

三角函数比：sinθ = 对边/斜边，cosθ = 邻边/斜边，tanθ = 对边/邻边。

Fundamental identities: sin²θ + cos²θ = 1; tanθ = sinθ/cosθ; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ.

基本恒等式：sin²θ + cos²θ = 1；tanθ = sinθ/cosθ；1 + tan²θ = sec²θ；1 + cot²θ = csc²θ。

Double angle: sin 2θ = 2 sinθ cosθ; cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ; tan 2θ = 2 tanθ / (1 – tan²θ).

倍角公式：sin 2θ = 2 sinθ cosθ；cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ；tan 2θ = 2 tanθ / (1 – tan²θ)。

Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² – 2bc cos A. Area of triangle: 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。三角形面积：½ ab sin C。

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8. Differentiation | 微分

First principles: f'(x) = lim_h→0 [f(x+h) – f(x)]/h.

第一原理：f'(x) = lim_h→0 [f(x+h) – f(x)]/h。

Standard derivatives: d/dx (xⁿ) = n x^n–1; d/dx (e^x) = e^x; d/dx (ln x) = 1/x; d/dx (sin x) = cos x; d/dx (cos x) = –sin x; d/dx (tan x) = sec² x.

常用导数：d/dx (xⁿ) = n x^n–1；d/dx (e^x) = e^x；d/dx (ln x) = 1/x；d/dx (sin x) = cos x；d/dx (cos x) = –sin x；d/dx (tan x) = sec² x。

Chain rule: dy/dx = dy/du × du/dx. Product rule: d/dx (uv) = u'v + uv'. Quotient rule: d/dx (u/v) = (u'v – uv')/v².

链式法则：dy/dx = dy/du × du/dx。积法则：d/dx (uv) = u'v + uv'。商法则：d/dx (u/v) = (u'v – uv')/v²。

Tangents & normals: Gradient of tangent m_t = f'(x₁); normal m_n = –1/m_t.

切线与法线：切线斜率 m_t = f'(x₁)；法线斜率 m_n = –1/m_t。

Second derivative: d²y/dx² determines concavity and points of inflection.

二阶导数：d²y/dx² 判定凹凸性与拐点。

Stationary points: solve f'(x) = 0. Nature: use first derivative test (sign change) or second derivative test (f''(x) > 0 ⇒ minimum; f''(x) < 0 ⇒ maximum; f''(x) = 0 possibly inflection).

驻点：解 f'(x) = 0。判定法：一阶导数符号变化法，或二阶导数测试（f''(x) > 0 ⇒ 极小点；f''(x) < 0 ⇒ 极大点；f''(x) = 0 可能为拐点）。

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9. Integration | 积分

Indefinite integral as reverse of differentiation: ∫ f'(x) dx = f(x) + C.

不定积分作为微分的逆运算：∫ f'(x) dx = f(x) + C。

Standard integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1); ∫ e^x dx = e^x + C; ∫ 1/x dx = ln|x| + C; ∫ sin x dx = –cos x + C; ∫ cos x dx = sin x + C; ∫ sec² x dx = tan x + C.

基本积分公式：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ –1）；∫ e^x dx = e^x + C；∫ 1/x dx = ln|x| + C；∫ sin x dx = –cos x + C；∫ cos x dx = sin x + C；∫ sec² x dx = tan x + C。

Integration of (ax + b)ⁿ: ∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / [a(n+1)] + C, n ≠ –1.

(ax + b)ⁿ 的积分：∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / [a(n+1)] + C，n ≠ –1。

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)。

Area under curve: Area between y = f(x) (above x-axis) and x-axis from a to b is ∫_a^b f(x) dx. If curve lies below x-axis, area = –∫ f(x) dx. Area between two curves: ∫_a^b [f(x) – g(x)] dx.

曲线下方面积：位于 x 轴上方的曲线 y = f(x) 与 x 轴从 a 到 b 围成的面积为 ∫_a^b f(x) dx。若曲线在 x 轴下方，面积为 –∫ f(x) dx。两曲线间的面积：∫_a^b [f(x) – g(x)] dx。

Kinematics: v = ds/dt, a = dv/dt = d²s/dt²; s = ∫ v dt, v = ∫ a dt.

运动学：v = ds/dt，a = dv/dt = d²s/dt²；s = ∫ v dt，v = ∫ a dt。

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10. Vectors & Series | 向量与级数

Vector notation: Position vector of point A is a = OA; may be written as a₁ i + a₂ j, where i and j are unit vectors along x and y axes.

向量表示：点 A 的位置向量为 a = OA；可写作 a₁ i + a₂ j，i, j 为 x, y 轴方向单位向量。

Magnitude: |a| = √a₁² + a₂². Unit vector in direction of a: â = a / |a|.

模：|a| = √a₁² + a₂²。a 方向上的单位向量：â = a / |a|。

Vector addition & subtraction: a + b = (a₁+b₁)i + (a₂+b₂)j. Multiplication by scalar: k a = k a₁ i + k a₂ j. Vector AB = b – a.

向量加减：a + b = (a₁+b₁)i + (a₂+b₂)j。数乘：k a = k a₁ i + k a₂ j。向量 AB = b – a。

Arithmetic progression (AP): nth term u_n = a + (n–1)d; sum S_n = n/2 [2a + (n–1)d] = n/2 (a + l), where l is last term.

等差数列（AP）：第 n 项 u_n = a + (n–1)d；求和 S_n = n/2 [2a + (n–1)d] = n/2 (a + l)，l 为末项。

Geometric progression (GP): nth term u_n = ar^n–1; sum S_n = a(1 – rⁿ) / (1 – r) for r ≠ 1. Sum to infinity S_∞ = a/(1 – r) valid for |r| < 1.

等比数列（GP）：第 n 项 u_n = ar^n–1；求和 S_n = a(1 – rⁿ) / (1 – r)，r ≠ 1。无穷项和 S_∞ = a/(1 – r)，当 |r| < 1 时成立。

Binomial expansion (n natural): (1 + x)ⁿ = 1 + nx + n(n–1)/2! x² + … + n!/(

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