📚 IB Edexcel Mathematics: Last-Minute Revision Notes | IB Edexcel 数学：考前冲刺笔记
This last-minute revision guide distills essential topics from IB Mathematics (both Analysis & Approaches and Applications & Interpretation) and Edexcel A-Level Mathematics. It highlights key formulas, common pitfalls, and exam tips to boost your confidence before the test. Work through each section systematically and use the paired Chinese explanations to reinforce understanding.
这份考前冲刺笔记浓缩了 IB 数学（分析与方法、应用与解释）以及 Edexcel A-Level 数学的核心要点。它突出关键公式、常见陷阱和考试技巧，助你在考前提升信心。请系统性地过一遍每个部分，并利用配对的中文解释加深理解。
1. Algebra & Functions | 代数与函数
Algebraic fluency is non-negotiable. Ensure you can manipulate indices, logarithms, quadratics, and function notation rapidly and accurately.
代数运算的熟练度至关重要。务必能够迅速且准确地处理指数、对数、二次函数以及函数记号。
Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ. For rational exponents, a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ.
指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1（a ≠ 0），a⁻ⁿ = 1/aⁿ。对于有理指数，a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ。
Logarithm rules: logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx – logₐy, logₐ(xⁿ) = n logₐx. The change-of-base formula is logₐb = log_c b / log_c a; often c = 10 or e.
对数法则：logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx – logₐy，logₐ(xⁿ) = n logₐx。换底公式为 logₐb = log_c b / log_c a；常取 c = 10 或 e。
For quadratics ax² + bx + c = 0, the quadratic formula is:
对于二次方程 ax² + bx + c = 0，其求根公式为：
x = (-b ± √(b² – 4ac)) / (2a)
The discriminant Δ = b² – 4ac tells us: Δ > 0 two distinct real roots; Δ = 0 one repeated real root; Δ < 0 no real roots (complex roots appear in IB HL and Further Maths).
判别式 Δ = b² – 4ac 告诉我们：Δ > 0 有两个不等实根；Δ = 0 有一个重根；Δ < 0 无实根（在 IB HL 和进阶数学中出现复根）。
Functions: f(x) notation, domain (input x values) and range (output y values). Composite function f(g(x)) means apply g first, then f; the domain of f(g(x)) requires x in domain of g and g(x) in domain of f.
函数：f(x) 记号，定义域（输入 x 的值）和值域（输出 y 的值）。复合函数 f(g(x)) 表示先作用 g 再作用 f；f(g(x)) 的定义域要求 x 在 g 的定义域内且 g(x) 在 f 的定义域内。
Inverse function f⁻¹(x) exists only if f is one-to-one. Graphically reflect in the line y = x. Do not confuse f⁻¹(x) with [f(x)]⁻¹ = 1/f(x).
反函数 f⁻¹(x) 仅在 f 为一一映射时存在。图像上关于直线 y = x 对称。切勿混淆 f⁻¹(x) 与 [f(x)]⁻¹ = 1/f(x)。
Transformations: y = f(x) + a shifts vertically by a; y = f(x + a) shifts horizontally by -a; y = a f(x) stretches vertically by factor a; y = f(ax) stretches horizontally by factor 1/a. Reflections: y = -f(x) reflects in x-axis; y = f(-x) reflects in y-axis.
函数变换：y = f(x) + a 竖直平移 a；y = f(x + a) 水平平移 -a；y = a f(x) 竖直拉伸 a 倍；y = f(ax) 水平拉伸 1/a 倍。反射：y = -f(x) 关于 x 轴对称；y = f(-x) 关于 y 轴对称。
2. Trigonometry | 三角学
Trigonometry appears heavily in both curricula. Radian measure, trigonometric identities, and solving equations are fundamental.
三角学在两个课程体系中占比都很重。弧度制、三角恒等式以及解三角方程是基础。
Radians: π rad = 180°. Arc length s = rθ, sector area A = ½ r²θ. Always check whether your angle is in radians or degrees, especially in calculus where radians are essential.
弧度：π rad = 180°。弧长 s = rθ，扇形面积 A = ½ r²θ。始终确认角度使用弧度还是度数，微积分中务必使用弧度。
Fundamental identities:
sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ
Double-angle formulas: sin(2θ) = 2 sinθ cosθ, cos(2θ) = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ, tan(2θ) = 2 tanθ / (1 – tan²θ).
倍角公式：sin(2θ) = 2 sinθ cosθ，cos(2θ) = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ，tan(2θ) = 2 tanθ / (1 – tan²θ)。
Compound angle formulas are in the IB formula booklet; for Edexcel they must be memorised: sin(A ± B) = sinA cosB ± cosA sinB, cos(A ± B) = cosA cosB ∓ sinA sinB.
和差角公式在 IB 公式表中提供；Edexcel 学生需要记忆：sin(A ± B) = sinA cosB ± cosA sinB，cos(A ± B) = cosA cosB ∓ sinA sinB。
Solving trig equations: use identities to reduce to a single trig function, find the principal value, then use CAST diagram or graphs to find all solutions in the given interval. Express general solutions using + 2πn or + 360°n as appropriate.
解三角方程：利用恒等式化为单一三角函数，求出主值，再借助 CAST 图或图像找出给定区间内的所有解。根据情况使用 + 2πn 或 + 360°n 表示通解。
Graphs: know the shapes, amplitudes, periods and asymptotes of y = sin x, cos x, tan x. For y = a sin(bx + c) + d, amplitude = |a|, period = 2π/|b|, phase shift = -c/b.
图像：熟记 y = sin x、cos x、tan x 的形状、振幅、周期及渐近线。对于 y = a sin(bx + c) + d，振幅 = |a|，周期 = 2π/|b|，相位移动 = -c/b。
3. Differentiation | 微分
Differentiation is central to both analysis of functions and applied problems. Memorise the basic derivatives and rules, then practise applications to tangents, normals, optimisation, and kinematics.
微分是函数分析与应用问题的核心。熟记基本导数公式和运算法则，接着通过练习切线、法线、优化问题和运动学来强化应用。
| f(x) | f'(x) |
|---|---|
| xⁿ | nxⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | -sin x |
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