A-Level AQA Mathematics: Mind Map Quick Revision | A-Level AQA 数学：思维导图速记

📚 A-Level AQA Mathematics: Mind Map Quick Revision | A-Level AQA 数学：思维导图速记

Mind maps are a powerful revision tool for A-Level AQA Mathematics, helping you visually organise topics, remember key formulas, and see connections between concepts. This article provides a mind-map-inspired quick revision guide covering the entire AQA specification, structured for efficient recall.

思维导图是A-Level AQA数学复习的强大工具，能帮你直观地组织知识点、记住关键公式，并看清概念间的联系。本文以思维导图为灵感，提供覆盖整个AQA考纲的速记指南，结构清晰，便于快速回忆。

1. Algebra and Functions | 代数与函数

The Factor Theorem: if f(a)=0 then (x-a) is a factor of f(x). The Remainder Theorem states that f(a) gives the remainder when f(x) is divided by (x-a).

因式定理：若 f(a)=0，则 (x-a) 是 f(x) 的因式。余数定理指出 f(a) 等于 f(x) 除以 (x-a) 的余数。

Graph transformations: f(x+a) translates the graph left by a; f(x)+a translates up by a; af(x) stretches vertically by factor a; f(ax) compresses horizontally by factor 1/a. Reflections: -f(x) reflects in the x-axis, f(-x) reflects in the y-axis.

图像变换：f(x+a) 向左平移 a；f(x)+a 向上平移 a；af(x) 垂直拉伸 a 倍；f(ax) 水平方向压缩 1/a。反射：-f(x) 关于 x 轴反射，f(-x) 关于 y 轴反射。

An inverse function f⁻¹(x) exists only if f is one-to-one. The graph of f⁻¹ is the reflection of f in the line y=x. Domain of f⁻¹ = range of f.

反函数 f⁻¹(x) 仅在 f 是一一映射时存在。f⁻¹ 的图像是 f 关于直线 y=x 的反射。f⁻¹ 的定义域等于 f 的值域。

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2. Coordinate Geometry | 坐标几何

Line equation: y – y₁ = m(x – x₁), where gradient m = (y₂ – y₁)/(x₂ – x₁). Perpendicular lines satisfy m₁ m₂ = -1. Midpoint is ((x₁+x₂)/2, (y₁+y₂)/2) and distance between points is √[(x₂-x₁)² + (y₂-y₁)²].

直线方程：y – y₁ = m(x – x₁)，斜率 m = (y₂ – y₁)/(x₂ – x₁)。垂直线斜率满足 m₁ m₂ = -1。中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)，两点间距离为 √[(x₂-x₁)² + (y₂-y₁)²]。

Equation of a circle: (x – a)² + (y – b)² = r², centre (a,b), radius r. The general form x² + y² + 2gx + 2fy + c = 0 has centre (-g, -f) and radius √(g²+f²-c). To find a tangent, set the perpendicular distance from centre to line equal to r, or solve simultaneous equations with discriminant zero.

圆的方程：(x – a)² + (y – b)² = r²，圆心 (a,b)，半径 r。一般式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (-g, -f)，半径为 √(g²+f²-c)。求切线时，令圆心到直线的垂直距离等于半径，或联立方程令判别式为零。

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3. Sequences and Series | 数列与级数

Arithmetic sequences: nth term uₙ = a + (n-1)d, sum Sₙ = n/2 [2a + (n-1)d] = n/2 (a + l), where l is the last term.

等差数列：第 n 项 uₙ = a + (n-1)d，前 n 项和 Sₙ = n/2 [2a + (n-1)d] = n/2 (a + l)，其中 l 为末项。

Geometric sequences: nth term uₙ = ar^(n-1), sum Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1. Sum to infinity S∞ = a/(1 – r) exists only when |r| < 1.

等比数列：第 n 项 uₙ = ar^(n-1)，和 Sₙ = a(1 – rⁿ)/(1 – r)（r ≠ 1）。无穷等比级数和 S∞ = a/(1 – r) 仅当 |r| < 1 时存在。

Binomial expansion: (a+b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ for positive integer n. For (1+x)ⁿ with |x|<1, the expansion 1 + nx + n(n-1)/2! x² + … is valid for rational n. Use the term independent of x when substituting.

二项式展开：(a+b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ 对正整数 n 适用。对于 (1+x)ⁿ，|x|<1 时，1 + nx + n(n-1)/2! x² + … 对有理数 n 有效。求与 x 无关的项时可代入化简。

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4. Trigonometry | 三角学

Radian measure: π rad = 180°. Exact values: sin 30° = 1/2, cos 45° = √2/2, tan 60° = √3, etc. Fundamental identity: sin²θ + cos²θ = 1; tanθ = sinθ/cosθ.

弧度制：π 弧度 = 180°。精确值：sin 30° = 1/2，cos 45° = √2/2，tan 60° = √3 等。基本恒等式：sin²θ + cos²θ = 1；tanθ = sinθ/cosθ。

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).

和角公式：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: sin2A = 2sinA cosA; cos2A = cos²A – sin²A = 2cos²A – 1 = 1 – 2sin²A; tan2A = 2tanA/(1 – tan²A). Solve equations by writing a sinθ + b cosθ as R sin(θ±α) or R cos(θ±α).

二倍角公式：sin2A = 2sinA cosA；cos2A = cos²A – sin²A = 2cos²A – 1 = 1 – 2sin²A；tan2A = 2tanA/(1 – tan²A)。解三角方程时可将 a sinθ + b cosθ 写成 R sin(θ±α) 或 R cos(θ±α)。

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5. Exponentials and Logarithms | 指数与对数

Exponential laws: aˣ × aʸ = aˣ⁺ʸ, (aˣ)ʸ = aˣʸ, aˣ / aʸ = aˣ⁻ʸ. Natural exponential eˣ: d/dx (eˣ) = eˣ; eˣ is its own derivative.

指数法则：aˣ × aʸ = aˣ⁺ʸ，(aˣ)ʸ = aˣʸ，aˣ / aʸ = aˣ⁻ʸ。自然指数 eˣ：d/dx (eˣ) = eˣ，eˣ 的导数仍为自身。

Logarithm laws: ln x = logₑ x, so ln(eˣ) = x and e^(ln x) = x. Laws: log a + log b = log(ab); log a – log b = log(a/b); n log a = log(aⁿ). Change of base: log₆ a = logₐ a / logₐ b.

对数法则：ln x = logₑ x，故 ln(eˣ) = x，e^(ln x) = x。运算法则：log a + log b = log(ab)；log a – log b = log(a/b)；n log a = log(aⁿ)。换底公式：log₆ a = logₐ a / logₐ b。

Exponential equations: take logs of both sides. Logarithmic equations: combine using laws and solve. Watch domain: argument of log must be >0.

指数方程两边取对数。对数方程利用法则合并后求解，注意对数真数必须大于零。

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6. Differentiation | 微分

First principles: f'(x) = lim(h→0) [f(x+h)-f(x)]/h. Basic derivatives: d/dx (xⁿ) = nxⁿ⁻¹; d/dx (sin x) = cos x; d/dx (cos x) = -sin x; d/dx (tan x) = sec²x; d/dx (ln x) = 1/x; d/dx (aˣ) = aˣ ln a; d/dx (eˣ) = eˣ.

导数定义：f'(x) = lim(h→0) [f(x+h)-f(x)]/h。基本求导公式：d/dx (xⁿ) = nxⁿ⁻¹；d/dx (sin x) = cos x；d/dx (cos x) = -sin x；d/dx (tan x) = sec²x；d/dx (ln x) = 1/x；d/dx (aˣ) = aˣ ln a；d/dx (eˣ) = eˣ。

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

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

Parametric differentiation: dy/dx = (dy/dt) / (dx/dt). Implicit differentiation: differentiate both sides w.r.t. x, treating y as a function of x. Stationary points: f'(x)=0; use second derivative or sign change

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