Formula & Theorem Quick Reference Handbook | 公式定理速查手册

📚 Formula & Theorem Quick Reference Handbook | 公式定理速查手册

This quick reference handbook covers all the essential formulae, identities, and theorems required for the Year 13 Edexcel A-Level Mathematics course. Each section presents key results in both English and Chinese, with clear standardised notation using Unicode characters. Use this guide for rapid revision and as a companions to your exam preparation.

本速查手册涵盖了 Year 13 Edexcel A-Level 数学课程所有必备的公式、恒等式和定理。每个部分均以中英双语呈现关键结果，并使用 Unicode 字符进行标准化记号。请将此指南用于快速复习并作为考试准备的伙伴。

1. Algebraic Formulas & Binomial Theorem | 代数公式与二项式定理

For rational functions, partial fractions decompose expressions like (px+q)/((x-a)(x-b)) into A/(x-a) + B/(x-b). The binomial expansion for (a+b)ⁿ with positive integer n uses Pascal’s triangle or C(n, r). For expressions of the form (1 + x)ⁿ where n is negative or a fraction, the expansion is an infinite series valid for |x| < 1:

(1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + …

对于有理函数，部分分式将形如 (px+q)/((x-a)(x-b)) 的表达式分解为 A/(x-a)+B/(x-b)。二项式展开 (a+b)ⁿ（n为正整数）使用杨辉三角或 C(n, r)。对于 n 为负数或分数的 (1 + x)ⁿ，展开式是无穷级数，要求 |x| < 1:

(1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + …

The general binomial coefficient is C(n, r) = n!/(r!(n-r)!) for integer n. For the infinite series, the expansion can be used to approximate values, e.g. √1.01 or 1/(1-x). The condition |x| < 1 ensures convergence.

通用的二项式系数对于整数 n 为 C(n, r) = n!/(r!(n-r)!)。对于无穷级数，该展开可用于近似值，例如 √1.01 或 1/(1-x)。条件 |x| < 1 确保收敛。

Polynomial division is used to simplify improper algebraic fractions before partial fractions. The remainder theorem states that when a polynomial f(x) is divided by (x-a), the remainder is f(a). The factor theorem says (x-a) is a factor if and only if f(a) = 0.

多项式除法用于在部分分式分解前化简假分式。余数定理指出，当多项式 f(x) 除以 (x-a) 时，余数为 f(a)。因式定理表明 (x-a) 是因式当且仅当 f(a) = 0。

2. Exponentials and Logarithms | 指数与对数

The natural logarithm ln x and exponential function eˣ are inverses: ln(eˣ) = x for all real x, and e^(ln x) = x for x > 0. Logarithms to any base a > 0, a ≠ 1 satisfy logₐ(x) = ln x / ln a.

自然对数 ln x 与指数函数 eˣ 互为反函数：对一切实数 x，ln(eˣ) = x；对 x > 0，e^(ln x) = x。任何底数 a > 0, a ≠ 1 的对数满足 logₐ(x) = ln x / ln a。

The fundamental laws of logarithms are:

ln(ab) = ln a + ln b, ln(a/b) = ln a – ln b, ln(aᵏ) = k ln a

基本对数运算法则为：

ln(ab) = ln a + ln b, ln(a/b) = ln a – ln b, ln(aᵏ) = k ln a

Exponential growth and decay models take the form P = P₀ eᵏᵗ, where k > 0 gives growth and k < 0 gives decay. The doubling time is (ln 2)/k, and half-life is (ln 2)/|k|.

指数增长和衰减模型的形式为 P = P₀ eᵏᵗ，其中 k > 0 为增长，k < 0 为衰减。倍增时间为 (ln 2)/k，半衰期为 (ln 2)/|k|。

3. Trigonometric Identities & R-Formula | 三角恒等式与 R 公式

Radians are the natural measure of angle: 180° = π rad. For a sector of a circle with radius r and angle θ (radians), arc length = rθ and area = ½ r²θ. The basic identities are:

sin²θ + cos²θ ≡ 1, tan θ ≡ sin θ / cos θ

弧度是角度的自然度量：180° = π rad。对于半径为 r、圆心角为 θ（弧度）的扇形，弧长 = rθ，面积 = ½ r²θ。基本恒等式为：

sin²θ + cos²θ ≡ 1, tan θ ≡ sin θ / cos θ

Further identities derived from these are:

1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ cosec²θ

由此导出的其他恒等式为：

1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ cosec²θ

Double-angle formulas express trigonometric functions of 2θ in terms of θ:

sin 2θ ≡ 2 sin θ cos θ

cos 2θ ≡ cos²θ – sin²θ ≡ 2cos²θ – 1 ≡ 1 – 2sin²θ

tan 2θ ≡ 2 tan θ / (1 – tan²θ)

倍角公式用 θ 表示 2θ 的三角函数：

sin 2θ ≡ 2 sin θ cos θ

cos 2θ ≡ cos²θ – sin²θ ≡ 2cos²θ – 1 ≡ 1 – 2sin²θ

tan 2θ ≡ 2 tan θ / (1 – tan²θ)

The R‑formula combines a sine and cosine term into a single sine or cosine function:

a sin θ ± b cos θ = R sin(θ ± α), a cos θ ± b sin θ = R cos(θ ∓ α)

where R = √(a² + b²) and tan α = b/a (adjust quadrant according to signs).

R 公式将正弦项与余弦项合并为单一正弦或余弦函数：

a sin θ ± b cos θ = R sin(θ ± α), a cos θ ± b sin θ = R cos(θ ∓ α)

其中 R = √(a² + b²)，tan α = b/a（根据符号调整象限）。

4. Differentiation Rules & Standard Derivatives | 微分法则与标准导数

The derivative of a power function: if y = xⁿ, then dy/dx = n xⁿ⁻¹ (valid for any real n). The derivative of exponential and logarithmic functions:

d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (aˣ) = aˣ ln a

幂函数的导数：如果 y = xⁿ，则 dy/dx = n xⁿ⁻¹（对任意实数 n 成立）。指数函数和对数函数的导数：

d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (aˣ) = aˣ ln a

Derivatives of the six trigonometric functions must be known:

d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (tan x) = sec² x

d/dx (sec x) = sec x tan x, d/dx (cosec x) = -cosec x cot x, d/dx (cot x) = -cosec² x

必须熟记六个三角函数的导数：

d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (tan x) = sec² x

d/dx (sec x) = sec x tan x, d/dx (cosec x) = -cosec x cot x, d/dx (cot x) = -cosec² x

The three fundamental rules of differentiation are:

Chain rule: d/dx [f(g(x))] = f'(g(x)) · g'(x)
Product rule: d/dx [u v] = u’ v + v’ u
Quotient rule: d/dx [u/v] = (v u’ – u v’) / v²

微分的三条基本法则是：

链式法则： d/dx [f(g(x))] = f'(g(x)) · g'(x)
乘积法则： d/dx [u v] = u’ v + v’ u
商法则： d/dx [u/v] = (v u’ – u v’) / v²

5. Integration Techniques & Standard Integrals | 积分方法与标准积分

The reverse of differentiation gives the fundamental standard integrals:

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1), ∫ 1/x dx = ln|x| + C

∫ eˣ dx = eˣ + C, ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C

微分的逆运算给出基本的积分公式：

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1), ∫ 1/x dx = ln|x| + C

∫ eˣ dx = eˣ + C, ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C

Integration by substitution (the reverse of the chain rule) is used when an integrand contains a function and its derivative. For definite integrals, limits must be changed to the new variable. For rational expressions, partial fractions are often employed before integrating.

代换积分法（链式法则的逆运算）用于被积函数包含一个函数及其导数的情况。对于定积分，必须将积分限更换为新变量。对于有理表达式，通常在积分前先进行部分分式分解。

Integration by parts is the counterpart of the product rule:

∫ u (dv/dx) dx = u v – ∫ v (du/dx) dx

分部积分法是乘积法则的对应：

∫ u (dv/dx) dx = u v – ∫ v (du/dx) dx

The trapezium rule approximates a definite integral using strips of equal width h:

∫ₐᵇ f(x) dx ≈ h/2 [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)]

梯形法则用等宽 h 的条带近似定积分：

∫ₐᵇ f(x) dx ≈ h/2 [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)]

6. Parametric Equations | 参数方程

When a curve is defined by x = f(t) and y = g(t), the first derivative is given by

dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0.

当曲线由 x = f(t) 和 y = g(t) 定义时，一阶导数为

dy/dx = (dy/dt) / (dx/dt)，前提是 dx/dt ≠ 0。

The second derivative with respect to x is obtained by differentiating dy/dx with respect to t and dividing by dx/dt:

d²y/dx² = d(dy/dx)/dx = [d/dt (dy/dx)] / (dx/dt).

关于 x 的二阶导数可以通过对 t 求导 dy/dx 然后除以 dx/dt 得到：

d²y/dx² = d(dy/dx)/dx = [d/dt (dy/dx)] / (dx/dt)。

The area under a parametric curve between t = t₁ and t = t₂ is calculated using

Area = ∫ y dx = ∫ y(t) · (dx/dt) dt,

参数曲线在 t = t₁ 到 t = t₂ 之间的面积用下式计算：

面积 = ∫ y dx = ∫ y(t) · (dx/dt) dt，

limits must be with respect to t.

积分限必须关于 t。

7. Differential Equations | 微分方程

A first-order separable differential equation has the form dy/dx = f(x)g(y). The solution is found by separating variables:

∫ 1/g(y) dy = ∫ f(x) dx.

一阶可分离变量微分方程的形式为 dy/dx = f(x)g(y)。通过分离变量求解：

∫ 1/g(y) dy = ∫ f(x) dx。

For a first-order linear differential equation dy/dx + P(x)y = Q(x), the integrating factor is μ(x) = e^{∫ P(x) dx}. The solution is:

y = (1/μ(x)) ∫ μ(x) Q(x) dx.

对于一阶线性微分方程 dy/dx + P(x)y = Q(x)，积分因子为 μ(x) = e^{∫ P(x) dx}。解为：

y = (1

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Formula & Theorem Quick Reference Handbook | 公式定理速查手册