SQA Advanced Mathematics Formula & Theorem Quick Reference | SQA 进阶数学公式定理速查手册

📚 SQA Advanced Mathematics Formula & Theorem Quick Reference | SQA 进阶数学公式定理速查手册

This quick-reference handbook gathers the essential formulas, theorems, and identities you need for Year 11 SQA Advanced Mathematics. Use it to refresh your memory before exams and during revision sessions. Every topic is presented with clear statements, followed by a Chinese translation to support bilingual learners.

本速查手册汇集了 11 年级 SQA 进阶数学所需的核心公式、定理与恒等式。它适合考前快速回顾与复习使用。每个知识要点均以英文陈述，并配有中文译文，方便双语学习者对照查阅。

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1. Algebraic Essentials | 代数基础

s of the quadratic equation ax² + bx + c = 0 are given by the quadratic formula: x = (-b ± √b² – 4ac) / (2a).

二次方程 ax² + bx + c = 0 的解由求根公式给出：x = (-b ± √b² – 4ac) / (2a)。

Completing the square rewrites ax² + bx + c as a(x + p)² + q, where p = b/(2a) and q = c – (b²)/(4a).

配方法将 ax² + bx + c 写成 a(x + p)² + q，其中 p = b/(2a)，q = c – (b²)/(4a)。

The discriminant Δ = b² – two distinct real roots, Δ = 0 gives a repeated real root, and Δ < 0 gives no real roots.

判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个相异实根，Δ = 0 有一个重根，Δ < 0 无实根。

Factor theorem: (x – a) is a factor of polynomial P(x) if and only if P(a) = 0.

因式定理：当且仅当 P(a) = 0 时，(x – a) 是多项式 P(x) 的因式。

Remainder theorem: When P(x) is divided by (x – a), the remainder is P(a).

余式定理：P(x) 除以 (x – a) 的余式为 P(a)。

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2. Functions & Graphs | 函数与图像

The domain of a function is the set of all possible input values (x-values). The range is the set of all possible output values (y-values).

函数的定义域是所有可能输入值（x 值）的集合。值域是所有可能输出值（y 值）的集合。

f is one-to-one. The graph of y = f⁻¹(x) is the reflection of y = f(x) in the line y = x.

反函数 f⁻¹(x) 仅在 f 是一一映射时存在。y = f⁻¹(x) 的图像是 y = f(x) 关于直线 y = x 的对称图形。

Common graph transformations: f(x + a) shifts horizontally by –a; f(x) + a shifts vertically by a; f(–x) reflects in the y-axis; –f(x) reflects in the x-axis; k f(x) stretches vertically by factor k; f(kx) compresses horizontally by factor 1/k.

常见图像变换：f(x + a) 水平平移 –a；f(x) + a 垂直平移 a；f(–x) 关于 y 轴对称；–f(x) 关于 x 轴对称；k f(x) 垂直拉伸因子 k；f(kx) 水平压缩因子 1/k。

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3. Trigonometry | 三角学

rt degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.

弧度制：π 弧度 = 180°。度转弧度乘以 π/180，弧度转度乘以 180/π。

identities: sin²θ + cos²θ = 1; tanθ = sinθ / cosθ; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

基本恒等式：sin²θ + cos²θ = 1；tanθ = sinθ / cosθ；1 + tan²θ = sec²θ；1 + cot²θ = cosec²θ。

Compound angle formulas: sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB; tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB).

两角和差公式：sin(A ± B) = sinA cosB ± cosA sinB；cos(A ± B) = cosA cosB ∓ sinA sinB；tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)。

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

二倍角公式：sin 2θ = 2 sinθ cosθ；cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ；tan 2θ = (2 tanθ) / (1 – tan²θ)。

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4. Differentiation | 微分

The derivative of a function f(x) is defined as f'(x) = lim_h→0 (f(x+h) – f(x)) / h, giving the gradient of the tangent.

函数 f(x) 的导数定义为 f'(x) = lim_h→0 (f(x+h) – f(x)) / h，它表示切线的斜率。

Power rule: d/dx (xⁿ) = n xⁿ⁻¹.

幂函数法则：d/dx (xⁿ) = n xⁿ⁻¹。

Standard derivatives: d/dx (sin x) = cos x; d/dx (cos x) = –sin x; d/dx (tan x) = sec² x; d/dx (eˣ) = eˣ; d/dx (ln x) = 1/x.

基本导数公式：d/dx (sin x) = cos x；d/dx (cos x) = –sin x；d/dx (tan x) = sec² x；d/dx (eˣ) = eˣ；d/dx (ln x) = 1/x。

Chain rule: If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx.

链式法则：若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。

Product rule: d/dx (u v) = u' v + u v'. Quotient rule: d/dx (u/v) = (u' v – u v') / v².

乘法法则：d/dx (u v) = u' v + u v'。除法法则：d/dx (u/v) = (u' v – u v') / v²。

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5. Integration | 积分

reverse process of differentiation. The indefinite integral ∫ f(x) dx = F(x) + C, where F'(x) = f(x).

积分是微分的逆运算。不定积分 ∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)。

Power rule for integration: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ –1.

幂函数的积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ –1。

Standard integrals: ∫ sin x dx = –cos x + C; ∫ cos x dx = sin x + C; ∫ sec² x dx = tan x + C; ∫ eˣ dx = eˣ + C; ∫ 1/x dx = ln|x| + C.

基本积分公式：∫ sin x dx = –cos x + C；∫ cos x dx = sin x + C；∫ sec² x dx = tan x + C；∫ eˣ dx = eˣ + C；∫ 1/x dx = ln|x| + C。

Definite integral: ∫_a^b f(x) dx = F(b) – F(a), where F is an antiderivative of f. It represents the signed area under the curve.

定积分：∫_a^b f(x) dx = F(b) – F(a)，其中 F 是 f 的一个原函数。它表示曲线下的有向面积。

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6. Exponentials & Logarithms | 指数与对数

aws of exponents: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ / aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1/aⁿ.

指数律：aᵐ × aⁿ = aᵐ⁺ⁿ；aᵐ / aⁿ = aᵐ⁻ⁿ；(aᵐ)ⁿ = aᵐⁿ；a⁰ = 1（a ≠ 0）；a⁻ⁿ = 1/aⁿ。

Logarithm definition: If aˣ = y then x = log_a y. Natural logarithm ln x = log_e x.

对数定义：若 aˣ = y，则 x = log_a y。自然对数 ln x = log_e x。

Logarithm laws: log_a (xy) = log_a x + log_a y; log_a (x/y) = log_a x – log_a y; log_a (xⁿ) = n log_a x; log_a 1 = 0; log_a a = 1.

对数律：log_a (xy) = log_a x + log_a y；log_a (x/y) = log_a x – log_a y；log_a (xⁿ) = n log_a x；log_a 1 = 0；log_a a = 1。

Change of base: log_b x = log_a x / log_a b.

换底公式：log_b x = log_a x / log_a b。

Derivatives and integrals: d/dx (eˣ) = eˣ, ∫ eˣ dx = eˣ + C; d/dx (ln x) = 1/x, ∫ 1/x dx = ln|x| + C.

导数与积分：d/dx (eˣ) = eˣ, ∫ eˣ dx = eˣ + C；d/dx (ln x) = 1/x, ∫ 1/x dx = ln|x| + C。

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7. Vectors | 向量

A vector in component form is written as u = (u₁, u₂, u₃) or as a column. Its magnitude is |u| = √u₁² + u₂² + u₃².

向量用分量形式表示为 u = (u₁, u₂, u₃) 或列向量。其模长为 |u| = √u₁² + u₂² + u₃²。

Vector addition and scalar multiplication: u + v = (u₁+v₁, u₂+v₂, u₃+v₃); k u = (k u₁, k u₂, k u₃).

向量加法与数乘：u + v = (u₁+v₁, u₂+v₂, u₃+v₃)；k u = (k u₁, k u₂, k u₃)。

The scalar (dot) product: u · v = u₁v₁ + u₂v₂ + u₃v₃ = |u| |v| cos θ, where θ is the angle between the vectors.

向量的点积：u · v = u₁v₁ + u₂v₂ + u₃v₃ = |u| |v| cos θ，其中 θ 是两向量间的夹角。

If u · v = 0, the vectors are perpendicular. For parallel vectors, u = k v for some scalar k.

若 u · v = 0，则两向量垂直。若两向量平行，则存在标量 k 使得 u = k v。

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8. Sequences & Series | 数列与级数

Arithmetic sequence: uₙ = a + (n – the first term and d is the common difference. Sum of the first n terms: Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (a + l), where l is the last term.

等差数列：uₙ = a + (n – 1)d，其中 a 为首项，d 为公差。前 n 项和：Sₙ = n/2 [2a + (n – 1)d] 或 Sₙ = n/2 (a + l)，l 为末项。

a is the first term and r is the common ratio. Sum of the first n terms (r ≠ 1): Sₙ = a(1 – rⁿ)/(1 – r).

等比数列：uₙ = a rⁿ⁻¹，a 为首项，r 为公比。前 n 项和（r ≠ 1）：Sₙ = a(1 – rⁿ)/(1 – r)。

Sum to infinity of a geometric series exists when |r| < 1: S∞ = a / (1 – r).

当 |r| < 1 时，无穷等比级数的和为 S∞ = a / (1 – r)。

Sigma notation: Σ_k=1ⁿ u_k denotes the sum of terms u₁ through uₙ.

∑ 符号：Σ_k=1ⁿ u_k 表示 u₁ 到 uₙ 的连加。

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9. Binomial Theorem | 二项式定理

For any positive integer n, (a + b)ⁿ = Σ_r=0ⁿ ⁿCᵣ aⁿ⁻ʳ bʳ, where ⁿCᵣ = n! / [r! (n – r)!].

对任意正整数 n，有 (a + b)ⁿ = Σ_r=0ⁿ ⁿCᵣ aⁿ⁻ʳ bʳ，其中 ⁿCᵣ = n! / [r! (n – r)!]。

The general term in the expansion is T_r+1 = ⁿCᵣ aⁿ⁻ʳ bʳ.

展开式中的通项为 T_r+1 = ⁿCᵣ aⁿ⁻ʳ bʳ。

Pascal's triangle can be used to find binomial coefficients for small n.

当 n 较小时，可使用杨辉三角查找二项式系数。

For rational or negative n (where |x| < 1), the binomial series is (1 + x)ⁿ = 1 + nx + n(n–1)x²/2! + n(n–1)(n–2)x³/3! + ...

对于有理数或负数 n（要求 |x| < 1），二项式级数为 (1 + x)ⁿ = 1 + nx + n(n–1)x²/2! + n(n–1)(n–2)x³/3! + …

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10. Differential Equations | 微分方程

parating variables is a method used when the equation can be written as dy/dx = g(x) h(y).

微分方程含有未知函数及其导数。若方程可写为 dy/dx = g(x) h(y)，则可采用分离变量法求解。

Procedure: rewrite as (1/h(y)) dy = g(x) dx and integrate both sides. Don't forget the constant of integration.

步骤：将方程改写为 (1/h(y)) dy = g(x) dx，然后两边积分。切勿漏掉积分常数。

General solution contains an arbitrary constant C. A particular solution is found by substituting initial conditions to determine C.

通解包含一个任意常数 C。将初始条件代入确定 C 值，即可得到特解。

Example: dy/dx = ky leads to y = A e^kx, where A is a constant. This models exponential growth or decay.

例如：dy/dx = ky 的解为 y = A e^kx，其中 A 为常数。它模拟指数增长或衰减。

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11. Polynomials & Remainder Theorem | 多项式与余式定理

For a polynomial P(x), if divisor is linear (x – a), the remainder is P(a). If divisor is (ax – b), the remainder is P(b/a).

对于多项式 P(x)，若除式为一次式 (x – a)，则余数为 P(a)。若除式为 (ax – b)，则余数为 P(b/a)。

Synthetic division is a quick method to divide by (x – a). It also verifies the remainder theorem.

综合除法是除以 (x – a) 的快捷方法，也能验证余式定理。

A polynomial equation of degree n has at most n real roots. Complex roots occur in conjugate pairs when coefficients are real.

n 次多项式方程最多有 n 个实根。当系数为实数时，复根成共轭对出现。

The relationship between roots and coefficients: for ax² + bx + c = 0, sum of roots = –b/a, product = c/a. For cubic ax³ + bx² + cx + d = 0, sum = –b/a, sum of pairwise products = c/a, product = –d/a.

根与系数的关系：对 ax² + bx + c = 0，根之和 = –b/a，根之积 = c/a。对三次方程 ax³ + bx² + cx + d = 0，根之和 = –b/a，两两积之和 = c/a，根之积 = –d/a。

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12. Key Derivative & Integral Table | 常用导数与积分表

Memorising these frequently occurring results speeds up problem-solving.

熟记以下高频结果可大幅提升解题速度。

d/dx (xⁿ) = n xⁿ⁻¹ → ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ –1

d/dx (sin x) = cos x → ∫ sin x dx = –cos x + C

d/dx (cos x) = –sin x → ∫ cos x dx = sin x + C

d/dx (tan x) = sec² x → ∫ sec² x dx = tan x + C

d/dx (eˣ) = eˣ → ∫ eˣ dx = eˣ + C

d/dx (ln x) = 1/x → ∫ 1/x dx = ln|x| + C

d/dx (aˣ) = aˣ ln a → ∫ aˣ dx = aˣ / ln a + C, a > 0, a ≠ 1

d/dx (sin⁻¹ x) = 1/√1 – x² → ∫ 1/√a² – x² dx = sin⁻¹(x/a) + C

d/dx (tan⁻¹ x) = 1/(1 + x²) → ∫ 1/(a² + x²) dx = (1/a) tan⁻¹(x/a) + C

These pairs cover the core integration techniques required in the SQA Advanced course.

这些对照关系涵盖了 SQA 进阶课程所要求的核心积分技巧。

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