Year 13 CAIE Mathematics: Formula & Theorem Quick Reference Handbook | Year 13 CAIE 数学：公式定理速查手册

📚 Year 13 CAIE Mathematics: Formula & Theorem Quick Reference Handbook | Year 13 CAIE 数学：公式定理速查手册

This quick reference handbook compiles key formulas, theorems, and essential results for the Year 13 CAIE Mathematics syllabus (9709), focusing on Pure Mathematics 3 together with selected Mechanics and Statistics topics. It is designed for rapid revision and exam preparation.

本速查手册汇编了 Year 13 CAIE 数学（9709）课程的核心公式、定理和重要结论，重点涵盖 Pure Mathematics 3 以及选考的力学与统计专题，旨在帮助快速复习与备考。

1. Algebraic & Logarithmic Functions | 代数与对数函数

Laws of indices: a^m × aⁿ = a^m+n; (a^m)ⁿ = a^mn; a^1/n = ⁿ√a (the nth root); a⁰ = 1 (a ≠ 0).

指数法则：a^m × aⁿ = a^m+n; (a^m)ⁿ = a^mn; a^1/n = ⁿ√a (n次方根); a⁰ = 1 (当 a ≠ 0).

Logarithm definition: a^x = b ⇔ g_e x. Change of base: logₐ b = log_c b / log_c a. Special values: logₐ 1 = 0, logₐ a = 1.

对数定义：a^x = b ⇔ x = logₐ b. 自然对数：ln x = log_e x. 换底公式：logₐ b = log_c b / log_c a. 特值：logₐ 1 = 0, logₐ a = 1.

Modulus function: |x| = x for x ≥ 0, |x| = -x for x < 0. Equations |f(x)| = k lead to f(x) = ±k for k > 0. Inequalities: |f(x)| > k ⇔ f(x) < -k or f(x) > k; |f(x)| < k ⇔ -k < f(x) < k.

模函数：|x| = x 当 x ≥ 0, |x| = -x 当 x < 0. 方程 |f(x)| = k (k>0) → f(x)=±k. 不等式: |f(x)| > k ⇔ f(x)<-k 或 f(x)>k; |f(x)|

Binomial expansion for rational n: (1 + x)ⁿ = 1 + nx + n(n-1)/2! x² + ... + n(n-1)...(n-r+1)/r! x^r + ..., valid for |x| < 1. General term: n(n-1)...(n-r+1)/r! x^r.

有理指数二项展开：(1 + x)ⁿ = 1 + nx + n(n-1)/2! x² + ... + n(n-1)...(n-r+1)/r! x^r + ...，收敛域 |x| < 1. 通项: n(n-1)...(n-r+1)/r! x^r.

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2. Trigonometry | 三角学

Pythagorean identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ.

毕达哥拉斯恒等式: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ.

Addition formulas: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B).

和角公式: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B).

inθ cosθ; cos 2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ; tan 2θ = (2 tanθ)/(1 - tan²θ).

二倍角公式: inθ cosθ; cos 2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ; tan 2θ = (2 tanθ)/(1 - tan²θ).

Expressing a sinθ + b cosθ: a sinθ + b cosθ = R sin(θ ± α) or R cos(θ ∓ α), where R = √a² + b² and tan α = b/a (or a/b depending on form). Maximum value = R, minimum = -R.

辅助角公式: a sinθ + b cosθ = R sin(θ ± α) 或 R cos(θ ∓ α)，其中 R = √a² + b²，tan α = b/a (依形式而定), 最大值 R, 最小值 -R.

3. Differentiation | 微分

Standard derivatives: d/dx (xⁿ) = n x^n-1; d/dx (e^x) = e^x = sec² x; d/dx (cot x) = -csc² x; d/dx (sec x) = sec x tan x; d/dx (csc x) = -csc x cot x.

基本导数: d/dx (xⁿ) = n x^n-1; d/dx (e^x) = e^x = sec² x; d/dx (cot x) = -csc² x; d/dx (sec x) = sec x tan x; d/dx (csc x) = -csc x cot x.

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

乘积法则: d/dx (uv) = u' v + u v'; 商法则: d/dx (u/v) = (u' v - u v')/v²; 链式法则: dy/dx = dy/du × du/dx.

x = (dy/dt)/(dx/dt). Second derivative: d²y/dx² = d/dx (dy/dx) = [d/dt (dy/dx)] / (dx/dt).

参数求导: 若 x = f(t), y = g(t), 则 dy/dx = (dy/dt)/(dx/dt). 二阶导数: d²y/dx² = [d/dt (dy/dx)] / (dx/dt).

Implicit differentiation: Differentiate both sides of an equation with respect to x, treating y as a function of x and using the chain rule.

隐函数求导: 对方程两边关于 x 求导，将 y 视为 x 的函数并运用链式法则。

4. Integration | 积分

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 +

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