📚 A-Level OCR Maths: Mind Map Quick Revision | A-Level OCR 数学：思维导图速记
Mastering A-Level OCR Mathematics requires more than just solving problems; it demands a clear, interconnected understanding of core concepts. A mind map approach can condense vast syllabi into a visual, logical framework for quick revision. This article breaks down the essential topics into bite-sized English-Chinese explanations, helping you memorize key formulas, theorems, and problem-solving strategies.
掌握 A-Level OCR 数学不仅需要解题技巧，还需要对核心概念有清晰、相互关联的理解。思维导图方法可以将庞大的课程内容浓缩成直观、逻辑清晰的复习框架。本文把重要主题拆解成简明易懂的英中双语讲解，助你牢记关键公式、定理和解题策略。
1. Algebra and Functions | 代数与函数
posite functions: fg(x) = f(g(x)). Inverse function f⁻¹(x) exists only if f is one-to-one.
函数将输入映射到输出。主要类型：一次、二次、三次、倒数、指数函数。定义域是可能输入的集合，值域是输出的集合。复合函数：fg(x) = f(g(x))。反函数 f⁻¹(x) 仅当 f 是一对一时才存在。
Quadratic formula: x = [-b ± √b² - 4ac ated root), Δ<0 (no real roots). Completing the square transforms ax²+bx+c into a(x+p)²+q.
二次公式：x = [-b ± √b² - 4ac] / (2a)。判别式 Δ = b² - 4ac 决定根的性质：Δ>0（两相异实根），Δ=0（一个重根），Δ<0（无实根）。配方法将 ax²+bx+c 化为 a(x+p)²+q。
lex rational expressions into simpler sums, e.g., (px+q)/((x+a)(x+b)) = A/(x+a) + B/(x+b).
多项式除法、因式定理和余式定理：若 f(a)=0 则 (x-a) 是因子。部分分式将复杂有理式分解为简单和式，例如 (px+q)/((x+a)(x+b)) = A/(x+a) + B/(x+b)。
| Graph transformation: f(x+a) | Horizontal shift left by a (a>0) | 图形变换：f(x+a) 水平左移 a |
| Graph transformation: f(x)+a | Vertical shift up by a | 图形变换：f(x)+a 垂直上移 a |
| Graph transformation: af(x) | Vertical stretch by factor a | 图形变换：af(x) 垂直拉伸 a 倍 |
2. Trigonometry | 三角学
sinθ/cosθ. Compound angle formulae: sin(A±B) = sinA cosB ± cosA sinB, cos(A±B) = cosA cosB ∓ sinB. Double angle: sin2θ = 2 sinθ cosθ, cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ.
基本恒等式：sin²θ + cos²θ = 1；tanθ = sinθ/cosθ。和角公式：sin(A±B) = sinA cosB ± cosA sinB，cos(A±B) = cosA cosB ∓ sinA sinB。倍角公式：sin2θ = 2 sinθ cosθ，cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ。
a² = b² + c² - 2bc cosA. Solve trigonometric equations using quadrant diagrams or graphs.
弧度：π rad = 180°。弧长 s = rθ，扇形面积 A = ½r²θ。正弦与余弦定理：a/sinA = b/sinB = c/sinC；a² = b² + c² - 2bc cosA。利用象限图或图像解三角方程。
3. Differentiation | 微分
First principles: f' r rule: d/dx (xⁿ) = n xⁿ⁻¹. 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. Derivatives: d/dx(eˣ)=eˣ, d/dx(ln x)=1/x, d/dx(sin x)=cos x, d/dx(cos x)=-sin x.
第一原理：f'(x) = lim[h→0] (f(x+h)-f(x))/h。幂法则：d/dx (xⁿ) = n xⁿ⁻¹。链式法则：dy/dx = dy/du × du/dx。乘法法则：d/dx (uv) = u'v + uv'。除法法则：d/dx (u/v) = (u'v - uv')/v²。常用导数：d/dx(eˣ)=eˣ, d/dx(ln x)=1/x, d/dx(sin x)=cos x, d/dx(cos x)=-sin x。
Stationary points occur where f'(x)=0. Second derivative f''(x) determines nature: f''(x)>0 local minimum, f''(x)<0 local maximum, f''(x)=0 possible inflection. Modelling: rates of change, optimisation problems.
驻点出现在 f'(x)=0 处。二阶导数 f''(x) 判断性质：f''(x)>0 局部极小，f''(x)<0 局部极大，f''(x)=0 可能为拐点。建模：变化率、最优化问题。
4. Integration | 积分
Integration reverses differentiation. ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1). ∫ 1/x dx = ln|x| + C. ∫ eˣ dx = eˣ ∫ cos x dx = sin x + C. Definite integral ∫ₐᵇ f(x) dx gives area between curve and x-axis.
积分是微分的逆运算。∫ 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。定积分 ∫ₐᵇ f(x) dx 给出曲线与 x 轴间的面积。
Integration by substitution: let u = g(x), then ∫ f(g(x)) g' imates area with n ordinates: width h = (b-a)/n, area ≈ h/2 [y₀ + 2(y₁+y₂+...+yₙ₋₁) + yₙ].
换元积分法：设 u = g(x)，则 ∫ f(g(x)) g'(x) dx = ∫ f(u) du。分部积分法：∫ u dv = uv - ∫ v du。梯形法则用 n 个纵坐标近似面积：宽度 h = (b-a)/n，面积 ≈ h/2 [y₀ + 2(y₁+y₂+...+yₙ₋₁) + yₙ]。
5. Sequences and Series | 序列与级数
term = arⁿ⁻¹, sum Sₙ = a(1-rⁿ)/(1-r) for r≠1. Sum to infinity S∞ = a/(1-r) valid only when |r|<1.
等差数列：第 n 项 = a+(n-1)d，和 Sₙ = n/2 [2a + (n-1)d] 或 n/2 (a+l)。等比数列：第 n 项 = arⁿ⁻¹，和 Sₙ = a(1-rⁿ)/(1-r) (r≠1)。无穷和 S∞ = a/(1-r) 仅当 |r|<1 时成立。
Binomial expansion: (1+x)ⁿ = 1 + nx + n(n-1)/2! x² + ... + n!/(r!(n-r)!) xʳ + ... for |x|<1, n∈ℚ n ∑ to express series compactly. Sequences can be defined iteratively, e.g., xₙ₊₁ = f(xₙ).
二项式展开：(1+x)ⁿ = 1 + nx + n(n-1)/2! x² + ... + n!/(r!(n-r)!) xʳ + ... 对于 |x|<1, n∈ℚ。使用求和符号 ∑ 紧凑表达级数。序列可以迭代定义，例如 xₙ₊₁ = f(xₙ)。
6. Vectors | 向量
Vectors have magnitude and direction. In 2D: a = a₁i + a₂j, magnitude |a| = √a₁² + a₂². Unit vector has magnitude 1. Scalar (dot) product: a·b = |a||b| cosθ = a₁b₁ + a₂b₂. Angle between vectors: cosθ = (a·b) / (|a||b|).
向量具有大小和方向。二维：a = a₁i + a₂j，模 |a| = √a₁² + a₂²。单位向量模为 1。数量积（点乘）：a·b = |a||b| cosθ = a₁b₁ + a₂b₂。向量夹角：cosθ = (a·b) / (|a||b|)。
a + t d = b + u e. Calculate distances and angles in 3D similarly, using i,j,k components.
直线向量方程：r = a + t d，其中 a 为一点的位置向量，d 为方向向量，t 为标量。若存在 t 和 u 使得 a + t d = b + u e，则两直线相交。三维中类似地用 i,j,k 分量计算距离和夹角。
7. Statistics: Probability Distributions | 统计：概率分布
∑ x P(X=x). Variance Var(X) = E(X²) - [E(X)]². Binomial distribution B(n,p): P(X=r)=ⁿCᵣ pʳ (1-p)ⁿ⁻ʳ, mean np, variance np(1-p).
离散随机变量：概率函数 P(X=x) 之和为 1。期望 E(X) = ∑ x P(X=x)。方差 Var(X) = E(X²) - [E(X)]²。二项分布 B(n,p)：P(X=r)=ⁿCᵣ pʳ (1-p)ⁿ⁻ʳ，均值 np，方差 np(1-p)。
The normal distribution N(μ, σ²) is continuous. Standardise: Z = (X - μ)/σ ~ N(0,1). Use tables to
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