📚 IB & CIE Maths: Last-Minute Revision Notes | IB CIE 数学：考前冲刺笔记
This revision guide condenses essential topics from IB and CIE mathematics syllabuses into portable, high‑impact notes for the final days before the exam. Whether you are preparing for IB Analysis & Approaches, Applications & Interpretation, or CIE Pure Mathematics and Statistics, the following pages focus on recurring concepts, common error traps, and efficient problem‑solving shortcuts. Use these notes to consolidate your understanding and to sharpen your technique under timed conditions.
本复习指南将 IB 和 CIE 数学大纲中的核心专题浓缩为考前可随身翻阅的高效笔记。无论你备战的是 IB 分析数学、应用数学，还是 CIE 纯数与统计，接下来的内容都着眼于高频考点、常见失分陷阱和快速解题技巧。请利用这些笔记巩固理解，并在限时训练中打磨你的应试技法。
1. Algebra Essentials | 代数学要点
Factorisation, expansion, and manipulation of algebraic fractions remain the bedrock of every advanced topic. Master the ability to spot hidden common factors and to rewrite expressions in forms that simplify calculus, vectors, and complex numbers later.
因式分解、展开和分式变形是所有进阶专题的基石。务必练就一眼识别公因式和将表达式改写为有利于微积分、向量与复数运算形式的能力。
Key techniques: completing the square for any quadratic ax² + bx + c yields (x + b/(2a))² − (b² − 4ac)/(4a²). For simultaneous equations, never assume a unique solution exists – always check whether the system is consistent, inconsistent, or dependent by inspecting the determinant or by back‑substitution.
关键技巧：对任意二次式 ax² + bx + c 配方得到 (x + b/(2a))² − (b² − 4ac)/(4a²)。解联立方程时，不要默认存在唯一解——一定要通过判别式或回代检验方程组是相容、矛盾还是相依。
The binomial expansion (1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + … is valid for |x| < 1 when n is not a positive integer. In CIE P1 and IB SL papers, you are frequently asked to find the coefficient of a specific term; in IB HL, you may need to expand when n is a rational number and state the range of validity.
二项展开式 (1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + … 当 n 不是正整数时仅在 |x| < 1 时有效。在 CIE P1 和 IB SL 试卷中，经常要求找出特定项的系数；在 IB HL 中，可能需要展开 n 为有理数的情形并说明收敛范围。
| Operation | Common Mistake | Correct Approach |
|---|---|---|
| (a + b)² | a² + b² | a² + 2ab + b² |
| √(a² + b²) | a + b | Cannot be simplified unless a or b = 0 |
| Cancelling in fractions | (x² + 3x)/x = x + 3 for all x | x + 3, x ≠ 0; always state domain restriction |
2. Functions & Graphs | 函数与图像
A function f maps each element of the domain to exactly one element of the range. The vertical line test confirms whether a graph represents a function; the horizontal line test tells you if it is injective (one‑to‑one) and thus invertible over a given interval.
函数 f 将定义域中的每个元素唯一地映射到值域中的一个元素。垂直线检验可确认图像是否表示一个函数；水平线检验则能判断函数在给定区间内是否为单射（一一映射），从而是否可逆。
Transformations follow a strict order: stretch/compression first, then reflection, then translation. For y = a f(b(x − h)) + k, remember that the horizontal shift is h (not −h) and the stretch factor for x is 1/b. Both IB and CIE examiners penalise candidates who forget that f(2x) compresses the graph horizontally by a factor of ½.
变换遵循严格的顺序：先伸缩，次反射，后平移。对于 y = a f(b(x − h)) + k，记住水平移动量为 h（而非 −h），水平伸缩因子为 1/b。IB 和 CIE 阅卷人都常扣掉忘记 f(2x) 将图像在水平方向上压缩为原来的 ½ 的考生的分数。
Inverse functions: write y = f(x), swap x and y, then solve for y. The graph of f⁻¹ is the reflection of f in the line y = x. Restricting the domain is often required to make the inverse a function – for example, y = x² is not invertible over ℝ, but it is invertible over x ≥ 0.
反函数：写出 y = f(x)，交换 x 和 y，然后解出 y。f⁻¹ 的图像是 f 关于直线 y = x 的反射。通常需要限制定义域才能使反函数成为一个函数——例如，y = x² 在整个实数集上不可逆，但在 x ≥ 0 上可逆。
For composite functions f(g(x)), ensure that the range of g lies within the domain of f. Failure to check this leads to domain errors when sketching or evaluating.
对于复合函数 f(g(x))，必须确保 g 的值域包含在 f 的定义域内。若忽略这一检验，在画图或求值时就会出现定义域错误。
3. Trigonometry & Circular Functions | 三角学与圆函数
Radians are the natural language of calculus. Always set your GDC or calculator to radian mode when differentiating or integrating trig functions, unless the question explicitly uses degrees. IB and CIE both expect radian answers for arc length (s = rθ) and sector area (A = ½ r²θ) unless stated otherwise.
弧度是微积分的自然语言。在对三角函数求导或积分时，务必将图形计算器或计算器设置为弧度模式，除非题目明确使用角度制。IB 和 CIE 都默认要求用弧度给出弧长 (s = rθ) 和扇形面积 (A = ½ r²θ) 的答案。
Know the exact values of sin, cos, and tan for 0, π/6, π/4, π/3, π/2 and their multiples. The cast diagram (or the unit circle) helps you determine the sign of any trig ratio in the four quadrants.
熟记 sin、cos、tan 在 0、π/6、π/4、π/3、π/2 及其倍数的精确值。CAST 图表（或单位圆）有助于确定任意象限中三角比的符号。
Pythagorean identity: sin²θ + cos²θ = 1. Learn the other two forms: 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ. These identities are indispensable when solving trig equations or proving statements in both IB Analysis and CIE Pure 2/3.
勾股恒等式：sin²θ + cos²θ = 1。同时记住另外两种形式：1 + tan²θ = sec²θ 和 1 + cot²θ = csc²θ。这些恒等式在解三角方程或证明命题时必不可少，在 IB 分析数学和 CIE 纯数 2/3 中都会出现。
Double‑angle formulas: sin(2θ) = 2 sinθ cosθ, cos(2θ) = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ. These allow you to express products as sums and are key to integrating sin²x or cos²x.
二倍角公式：sin(2θ) = 2 sinθ cosθ，cos(2θ) = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ。这些公式可将乘积化为和差，是积分 sin²x 或 cos²x 的关键工具。
4. Vectors & Matrices | 向量与矩阵
Vectors describe quantities with both magnitude and direction. In component form, a vector from A(x₁, y₁, z₁) to B(x₂, y₂, z₂) is AB = (x₂−x₁)i + (y₂−y₁)j + (z₂−z₁)k. Magnitude |v| = √(x² + y² + z²). The dot product v·w = |v||w| cos θ gives a scalar; it is zero when vectors are perpendicular.
向量描述既有大小又有方向的量。以分量形式表示时，从 A(x₁, y₁, z₁) 到 B(x₂, y₂, z₂) 的向量为 AB = (x₂−x₁)i + (y₂−y₁)j + (z₂−z₁)k。向量长度 |v| = √(x² + y² + z²)。点积 v·w = |v||w| cos θ 给出标量；当两向量垂直时点积为零。
In CIE Further Maths and IB HL, the cross product v × w produces a vector perpendicular to both v and w. Its magnitude |v × w| = |v||w| sin θ gives the area of the parallelogram spanned by v and w. Use the right‑hand rule for direction.
在 CIE 进阶数学和 IB 高阶课程中，叉积 v × w 产生一个同时垂直于 v 和 w 的向量。其大小 |v × w| = |v||w| sin θ 等于以 v 和 w 为邻边的平行四边形面积。方向由右手定则确定。
Straight‑line equations: vector form r = a + λb, where a is a point on the line and b is the direction vector. Cartesian form can be derived by eliminating λ. The angle between two lines is the angle between their direction vectors.
直线方程：向量形式为 r = a + λb，其中 a 为直线上一点，b 为方向向量。消去 λ 即可化为笛卡尔形式。两条直线之间的夹角就是它们方向向量之间的夹角。
Matrices transform points in the plane. The transformation matrix [[cosθ, −sinθ],[sinθ, cosθ]] rotates a point anticlockwise by θ. The determinant gives the area scale factor; a determinant of 0 indicates that the transformation collapses the plane onto a line or point.
矩阵可对平面上的点进行变换。变换矩阵 [[cosθ, −sinθ],[sinθ, cosθ]] 使点逆时针旋转 θ。行列式给出面积缩放因子；行列式为零则表明变换将平面压缩到一条直线或一个点上。
5. Differential Calculus | 微分学
The derivative f ′(x) measures the instantaneous rate of change. From first principles, f ′(x) = lim(h→0) [f(x+h) − f(x)]/h. In practice, rely on rules: power rule d/dx (xⁿ) = n xⁿ⁻¹, product rule d/dx (uv) = u′v + uv′, quotient rule d/dx (u/v) = (u′v − uv′)/v², and chain rule dy/dx = dy/du · du/dx.
导数 f ′(x) 用来度量瞬时变化率。从第一性原理出发，f ′(x) = lim(h→0) [f(x+h) − f(x)]/h。实际解题中可依赖各种求导法则：幂法则 d/dx (xⁿ) = n xⁿ⁻¹，乘法法则 d/dx (uv) = u′v + uv′，除法法则 d/dx (u/v) = (u′v − uv′)/v²，以及链式法则 dy/dx = dy/du · du/dx。
Know the derivatives of standard functions by heart: d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x. For IB HL and CIE P3, d/dx (aˣ) = aˣ ln a, d/dx (tan x) = sec² x, and d/dx (arctan x) = 1/(1+x²).
牢记基本函数的导数：d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x。在 IB HL 和 CIE P3 中，d/dx (aˣ) = aˣ ln a，d/dx (tan x) = sec² x，d/dx (arctan x) = 1/(1+x²)。
Stationary points occur where f ′(x) = 0. Use the second derivative test: if f ″(a) > 0, it is a local minimum; if f ″(a) < 0, a local maximum; if f ″(a) = 0, check the sign change of f ′(x) or use higher derivatives. Both boards frequently include optimisation problems where you must model a real‑world situation with a function and then find its extreme values.
驻点出现在 f ′(x) = 0 处。用二阶导数检验：若 f ″(a) > 0，为局部极小值点；若 f ″(a) < 0，为局部极大值点；若 f ″(a) = 0，则需检查 f ′(x) 的符号变化或利用高阶导数判断。两套课程都经常出现优化问题，要求你先用函数对现实情境建模，然后求其极值。
6. Integral Calculus | 积分学
Indefinite integration reverses differentiation. The general power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1. The special case ∫ 1/x dx = ln|x| + C. Never omit the constant of integration unless evaluating a definite integral.
不定积分是微分的逆运算。一般幂法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ −1。特殊情况 ∫ 1/x dx = ln|x| + C。除非计算定积分，否则切勿遗漏积分常数。
Integration by substitution (the reverse chain rule) is the most versatile method. Choose u = g(x) so that du = g′(x) dx appears in the integrand. For definite integrals, remember to change the limits to values of u, or to substitute back before evaluating.
换元积分法（链式法则的逆用）是最灵活的积分方法。选取 u = g(x)，使得被积函数中出现 du = g′(x) dx。计算定积分时，要么将积分限换成 u 的值，要么在计算前代回原变量。
Integration by parts: ∫ u dv = uv − ∫ v du. Use the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) to decide which part to differentiate and which to integrate. This appears in IB HL and CIE P3/FM.
分部积分法：∫ u dv = uv − ∫ v du。利用 LIATE 规则（对数、反三角、代数、三角、指数）来决定哪一部分求导、哪一部分积分。这一方法见于 IB HL 和 CIE P3/进阶数学。
The area under a curve y = f(x) from a to b is ∫ₐᵇ f(x) dx. Area between two curves: ∫ₐᵇ (top − bottom) dx. Volume of revolution about the x‑axis: V = π ∫ₐᵇ y² dx. Volumes about the y‑axis require you to express x in terms of y.
曲线 y = f(x) 从 a 到 b 下方的面积为 ∫ₐᵇ f(x) dx。两曲线之间的面积：∫ₐᵇ (上方函数 − 下方函数) dx。绕 x 轴旋转的体积为 V = π ∫ₐᵇ y² dx。绕 y 轴旋转的体积则需将 x 表示为 y 的函数。
7. Sequences, Series & Proof | 数列、级数与证明
Arithmetic sequences have a constant difference d: uₙ = a + (n−1)d. Sum of n terms: Sₙ = n/2 (2a + (n−1)d) = n/2 (a + l). Geometric sequences have a constant ratio r: uₙ = arⁿ⁻¹. Sum: Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. The infinite sum converges to a/(1 − r) when |r| < 1.
等差数列具有常数公差 d：uₙ = a + (n−1)d。前 n 项和：Sₙ = n/2 (2a + (n−1)d) = n/2 (a + l)。等比数列具有常数公比 r：uₙ = arⁿ⁻¹。前 n 项和：当 r ≠ 1 时 Sₙ = a(1 − rⁿ)/(1 − r)。无穷级数当 |r| < 1 时收敛于 a/(1 − r)。
Sigma notation is a compact way to represent sums. Become fluent in shifting indices and using standard results such as Σ k = n(n+1)/2, Σ k² = n(n+1)(2n+1)/6, and Σ k³ = (n(n+1)/2)². These are tested in both IB SL/HL and CIE P1/P2.
∑ 符号是表示求和的简洁方式。要熟练掌握指标的平移以及使用标准结果，如 Σ k = n(n+1)/2，Σ k² = n(n+1)(2n+1)/6，Σ k³ = (n(n+1)/2)²。这些在 IB SL/HL 和 CIE P1/P2 中都有考查。
Mathematical induction is a core proof technique. Structure your proof clearly: (1) Base case – verify the statement for n = 1. (2) Induction hypothesis – assume true for n = k. (3) Induction step – prove it holds for n = k+1 using the hypothesis. (4) Conclusion – by PMI, the statement is true for all positive integers n. Both IB and CIE award method marks specifically for this structure.
数学归纳法是一种核心证明技法。条理清晰地展开证明：(1) 基础步骤——验证 n = 1 时命题成立。(2) 归纳假设——假设 n = k 时命题为真。(3) 归纳步骤——利用假设证明 n = k+1 时命题也成立。(4) 结论——根据数学归纳法原理，命题对所有正整数 n 均成立。IB 和 CIE 阅卷都会专门为此结构给出方法分数。
8. Probability & Statistics | 概率与统计
Probability models assign a number between 0 and 1 to each outcome. For mutually exclusive events, P(A ∪ B) = P(A) + P(B). For independent events, P(A ∩ B) = P(A) × P(B). Conditional probability: P(A|B) = P(A ∩ B)/P(B). Bayes’ theorem appears in IB HL and CIE Statistics 2: P(A|B) = P(B|A)·P(A)/P(B).
概率模型给每个结果赋予 0 到 1 之间的一个数。对于互斥事件，P(A ∪ B) = P(A) + P(B)。对于独立事件，P(A ∩ B) = P(A) × P(B)。条件概率：P(A|B) = P(A ∩ B)/P(B)。贝叶斯定理出现在 IB HL 和 CIE 统计 2 中：P(A|B) = P(B|A)·P(A)/P(B)。
Discrete random variables: E(X) = Σ x·P(X=x), Var(X) = E(X²) − [E(X)]². For a linear transformation Y = aX + b, E(Y) = aE(X) + b and Var(Y) = a² Var(X). The binomial distribution X ~ B(n, p) has mean np and variance np(1−p). The Poisson distribution X ~ Po(λ) has mean λ and variance λ; it approximates the binomial when n is large and p is small.
离散型随机变量：E(X) = Σ x·P(X=x)，Var(X) = E(X²) − [E(X)]²。对于线性变换 Y = aX + b，有 E(Y) = aE(X) + b 和 Var(Y) = a² Var(X)。二项分布 X ~ B(n, p) 的均值为 np，方差为 np(1−p)。泊松分布 X ~ Po(λ) 的均值和方差均为 λ；当 n 很大而 p 很小时，可用泊松分布逼近二项分布。
The normal distribution N(μ, σ²) is continuous and symmetric. Standardise to Z = (X − μ)/σ to use tables or calculator functions. The Central Limit Theorem states that for large samples, the sample mean X̅ follows approximately N(μ, σ²/n) regardless of the original distribution. This is a favourite for linking topics in CIE S2 and IB AI HL papers.
正态分布 N(μ, σ²) 是连续的且对称的。标准化为 Z = (X − μ)/σ 后即可查表或使用计算器功能。中心极限定理指出，对于大样本，无论原始分布如何，样本均值 X̅ 都近似服从 N(μ, σ²/n)。这是 CIE 统计 2 和 IB 应用数学 HL 试卷中常见的跨主题考点。
9. Complex Numbers (IB HL) | 复数 (IB 高阶)
A complex number z = a + ib can be represented on the Argand diagram as the point (a, b). The modulus |z| = √(a² + b²) gives its distance from the origin; the argument arg(z) = θ, where tan θ = b/a, gives its direction from the positive real axis.
复数 z = a + ib 可在阿冈图上表示为点 (a, b)。模 |z| = √(a² + b²) 给出了它到原点的距离；辐角 arg(z) = θ，其中 tan θ = b/a，给出了它从正实轴出发的方向。
Polar form: z = r(cos θ + i sin θ) = r cis θ. Euler’s formula connects exponential and trig functions: e^(iθ) = cos θ + i sin θ. Thus z = r e^(iθ) is a compact exponential form. Multiplication becomes z₁z₂ = r₁r₂ e^(i(θ₁+θ₂)); division becomes z₁/z₂ = (r₁/r₂) e^(i(θ₁−θ₂)).
极坐标形式：z = r(cos θ + i sin θ) = r cis θ。欧拉公式将指数函数与三角函数联系起来：e^(iθ) = cos θ + i sin θ。因此 z = r e^(iθ) 是一种紧凑的指数形式。乘法即 z₁z₂ = r₁r₂ e^(i(θ₁+θ₂))；除法即 z₁/z₂ = (r₁/r₂) e^(i(θ₁−θ₂))。
De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for any integer n. This theorem is the engine for finding powers and roots of complex numbers. The n distinct n‑th roots of a complex number are given by z_k = r^(1/n) cis((θ + 2πk)/n) for k = 0, 1, …, n−1.
棣莫弗定理：对任何整数 n，(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。该定理是求复数的乘方与方根的根本工具。一个复数的 n 个不同的 n 次方根由 z_k = r^(1/n) cis((θ + 2πk)/n) 给出，k = 0, 1, …, n−1。
Complex conjugates: if z = a + ib, then z* = a − ib. Properties: z·z* = |z|², and (z₁ + z₂)* = z₁* + z₂*. In polynomial equations with real coefficients, non‑real roots occur in conjugate pairs. This fact alone often lets you deduce a polynomial’s remaining roots without further calculation.
共轭复数：若 z = a + ib，则 z* = a − ib。性质：z·z* = |z|²，(z₁ + z₂)* = z₁* + z₂*。在实系数多项式方程中，非实根成对以共轭形式出现。仅凭这一事实，往往无需更多计算就能推断出多项式的其余根。
10. Exam Strategy & Common Pitfalls | 考试策略与常见陷阱
Time management: allocate roughly one minute per mark and stick to it. If a question takes longer, mark it, move on, and return at the end. Both IB and CIE papers often place a relatively accessible “blocker” early on to tempt candidates into spending too long; do not fall for it.
时间管理：大约按每分钟 1 分的比例分配时间，并严格遵守。若某道题耗时过多，做好标记，继续前进，最后再回来解决。IB 和 CIE 的试卷往往会在开头设置一道相对可解的“拦路题”诱惑考生花去过多时间；千万不要上当。
Rounding and exact answers: in CIE, unless the question specifies a degree of accuracy, give exact answers – fractions, surds, π, e. IB often requires answers either exact or to three significant figures; check the front of the paper for the instruction. Premature rounding of intermediate results leads to cumulative errors.
取整与精确答案：在 CIE 中，除非题目指定了精度，否则应给出精确答案——分数、根式、π、e。IB 常要求答案或为精确值，或保留三位有效数字；请查看卷首的指引。对中间结果过早取整会导致累积误差。
Calculator fluency: know how to graph functions, solve equations numerically, and find derivatives at a point. But do not let the GDC replace algebraic working – many questions ask you to show exactly what you typed and the reason for each step. On screen‑shot style answers, include annotations.
计算器熟练度：要懂得如何绘制函数图像、数值求解方程以及求某点的导数值。但不要让图形计算器取代代数推导——许多题目要求你明确展示输入了什么以及每一步的理由。在屏显答案旁，加上注释。
Final checks: with any remaining time, re‑read the question to ensure you have answered exactly what was asked. Substituting your solution back into the original equation can catch arithmetic slips. Check for missing units, domain restrictions, and whether all parts of a vector or coordinate have been stated.
最后检查：如有剩余时间，重新审题以确认自己回答的正好是所问。将解代回原方程可发现计算错误。检查是否有遗漏的单位、定义域限制，以及是否陈述了向量或坐标的全部分量。
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