📚 IB Maths End-of-Term Revision Checklist | IB 数学期末复习提纲
As the end of term approaches, a structured review of the IB Mathematics syllabus can make a huge difference to your confidence and performance. This revision checklist covers the essential topics you need to master, from algebra and functions to calculus and statistics, with practical reminders about key formulas, common mistakes, and exam technique. Use it to guide your self‑study and identify areas that need extra practice.
随着期末临近，系统梳理 IB 数学课程内容能显著提升你的信心与成绩。这份复习提纲覆盖了你需要掌握的核心主题，从代数、函数直到微积分与统计，同时提醒你关键公式、常见错误和考试技巧。用它来指导你的自主学习，找出需要加强练习的薄弱环节。
1. Algebra and Number | 代数与数
Simplify algebraic expressions confidently, including rational expressions and those involving exponents and radicals. Be sure you can apply the laws of indices and logarithms fluently.
熟练化简代数表达式，包括有理式以及含有指数和根式的表达式。确保自己能灵活运用指数律和对数律。
Laws of exponents: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ
指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ⁿ = 1/aⁿ
Log rules: logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, logₐ(xⁿ) = n logₐx
对数法则：logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx − logₐy，logₐ(xⁿ) = n logₐx
Change of base: logₐb = log_b / log_a (any base).
换底公式：logₐb = log_b / log_a（对任意底数）。
Binomial expansion: (a + b)ⁿ = Σ (ⁿCᵣ) aⁿ⁻ʳ bʳ, for n ∈ ℕ. Know how to find a specific term using the general formula.
二项式展开：(a + b)ⁿ = Σ (ⁿCᵣ) aⁿ⁻ʳ bʳ，n 为自然数。会利用通项公式求特定项。
Complex numbers (AA HL): standard form z = a + bi, modulus |z| = √a² + b², argument arg(z), and polar form z = r(cosθ + isinθ). Euler's form: reⁱᶿ.
复数（AA HL）：标准形式 z = a + bi，模 |z| = √a² + b²，辐角 arg(z)，极坐标形式 z = r(cosθ + isinθ)。欧拉形式：reⁱᶿ。
2. Functions and Equations | 函数与方程
Understand the concept of a function, its domain and range, and be able to work with inverses and composite functions. Practice solving a variety of equations and inequalities, both analytically and graphically.
理解函数的概念、定义域和值域，能处理反函数和复合函数。练习解析法和图像法求解各类方程与不等式。
Composite functions: (f ∘ g)(x) = f(g(x)). Domain restrictions: the range of g must be a subset of the domain of f.
复合函数：(f ∘ g)(x) = f(g(x))。定义域限制：g 的值域必须是 f 的定义域的子集。
unction: swap x and y, solve for y. Graphs of f and f⁻¹ are reflections in the line y = x.
反函数：交换 x 和 y，解出 y。f 与 f⁻¹ 的图像关于直线 y = x 对称。
ons: solve by factorising, completing the square, or using the quadratic formula x = [−b ± √b² − 4ac] / 2a. Discriminant Δ = b² − 4ac determines the nature of roots.
二次方程：因式分解、配方法或求根公式 x = [−b ± √b² − 4ac] / 2a。判别式 Δ = b² − 4ac 决定根的性质。
Polynomial equations: factor theorem, remainder theorem, and use of long division or synthetic division.
多项式方程：因式定理、余式定理，以及长除法或综合除法。
Exponential and logarithmic equations: use logarithms to solve aˣ = b, and be aware of extraneous solutions when solving log equations.
指数与对数方程：利用对数求解 aˣ = b，解对数方程时要注意可能出现的增根。
3. Sequences and Series | 数列与级数
Arithmetic and geometric sequences are core to the syllabus. Memorise the formulas for the nth term and the sum of n terms, and know when to apply the sum to infinity for convergent geometric series.
等差数列和等比数列是课程的核心。熟记第 n 项和前 n 项和的公式，并知道何时对收敛的几何级数应用无穷和公式。
Arithmetic: uₙ = u₁ + (n−1)d, Sₙ = n/2 (u₁ + uₙ) = n/2 [2u₁ + (n−1)d].
等差数列：uₙ = u₁ + (n−1)d，Sₙ = n/2 (u₁ + uₙ) = n/2 [2u₁ + (n−1)d]。
uₙ = u₁ rⁿ⁻¹, Sₙ = u₁(1 − rⁿ)/(1 − r) for r ≠ 1. Sum to infinity S∞ = u₁/(1 − r) valid for |r| < 1.
等比数列：uₙ = u₁ rⁿ⁻¹，Sₙ = u₁(1 − rⁿ)/(1 − r)（r ≠ 1）。无穷和 S∞ = u₁/(1 − r)，要求 |r| < 1。
Sigma notation: understand how to shift the index and evaluate sums of simple series.
西格玛符号：理解如何移动指标，会计算简单级数的和。
Applications: compound interest, population growth, depreciation, and modelling with sequences.
应用：复利、人口增长、折旧以及用数列建模。
4. Trigonometry and Circular Functions | 三角学与圆函数
Trigonometry combines geometric reasoning with algebraic manipulation. You must be fluent in radian measure, exact values of special angles, and the graphs of sin, cos, and tan functions, including transformations.
三角学结合了几何推理与代数运算。你必须熟练掌握弧度制、特殊角的精确值，以及正弦、余弦、正切函数的图像及其变换。
Exact values for 0, π/6, π/4, π/3, π/2 and their multiples. Know how to derive from the unit circle.
0、π/6、π/4、π/3、π/2 及其倍数的精确值。知道如何利用单位圆推导。
Identities: tanθ = sinθ/cosθ, sin²θ + cos²θ = 1, double‑angle formulas sin2θ = 2sinθcosθ, cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.
恒等式：tanθ = sinθ/cosθ，sin²θ + cos²θ = 1，倍角公式 sin2θ = 2sinθcosθ，cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ。
Solving trigonometric equations: look for all solutions within a given interval, use cast diagrams or graphs, and be careful with quadrant signs.
解三角方程：在给定区间内寻找所有解，使用象限图或图像，注意各象限的符号。
Graph transformations: y = a sin(b(x − c)) + d. Amplitude = |a|, period = 2π/|b|, phase shift = c, vertical shift = d.
图像变换：y = a sin(b(x − c)) + d。振幅 = |a|，周期 = 2π/|b|，相位移 = c，垂直位移 = d。
Inverse trig functions: arcsin, arccos, arctan and their restricted domains.
反三角函数：arcsin、arccos、arctan 及其限制定义域。
5. Vectors | 向量
Vector concepts are fundamental for geometry in both two and three dimensions. Be comfortable with vector operations, dot product, cross product (HL), and their applications to lines and planes.
向量概念是二维和三维几何的基础。熟练进行向量运算、点积、叉积（HL），并会应用于直线和平面。
Vector operations: addition, scalar multiplication, magnitude |v| = √x² + y² + z².
向量运算：加法、数乘、模 |v| = √x² + y² + z²。
Dot product: v·w = |v||w| = v₁w₁ + v₂w₂ + v₃w₃. Used for finding angles, projections, and testing perpendicularity.
点积：v·w = |v||w|cosθ = v₁w₁ + v₂w₂ + v₃w₃。用于求夹角、投影和判断垂直。
(HL): v × w gives a vector perpendicular to both. Magnitude equals area of parallelogram.
叉积（HL）：v × w 得到垂直于两者的向量。其模等于平行四边形的面积。
ation of a line: r = a + tb, where a is a point on the line and b is the direction vector.
直线的向量方程：r = a + tb，其中 a 是线上一点，b 是方向向量。
duct form r·n = a·n, or Cartesian form ax + by + cz = d. Intersection of lines and planes.
平面方程（HL）：点法式 r·n = a·n，或笛卡尔形式 ax + by + cz = d。直线与平面的交点。
6. Calculus – Differentiation | 微分
Differentiation is a powerful tool for analysing rates of change and the behaviour of functions. Master the rules and be able to apply them to tangents, normals, optimisation, and kinematics.
微分是分析变化率和函数性态的强大工具。掌握求导法则，并能应用于切线、法线、最优化和运动学问题。
Basic derivatives: d/dx (xⁿ) = nxⁿ⁻¹, d/dx (sinx) = cosx, d/dx (cosx) = −sinx, d/dx (eˣ) = eˣ, d/dx (lnx) = 1/x.
基本导数：d/dx (xⁿ) = nxⁿ⁻¹，d/dx (sinx) = cosx，d/dx (cosx) = −sinx，d/dx (eˣ) = eˣ，d/dx (lnx) = 1/x。
Rules: product rule, quotient rule, chain rule. Practise combinations of these.
求导法则：乘法法则、除法法则、链式法则。练习它们的组合使用。
Equation of tangent at x = a: y − f(a) = f '(a)(x − a). Normal has gradient −1/f '(a) if f '(a) ≠ 0.
在 x = a 处的切线方程：y − f(a) = f '(a)(x − a)。法线斜率为 −1/f '(a)（若 f '(a) ≠ 0）。
Second derivative: f ''(x) tells you about concavity and points of inflection. Use it with first derivative to classify stationary points.
二阶导数：f ''(x) 揭示凹凸性和拐点。结合一阶导数用来判定驻点的类型。
Optimisation: set f '(x) = 0 to find critical values, then test using second derivative or sign chart.
最优化：令 f '(x) = 0 求临界值，然后用二阶导数或符号表检验。
7. Calculus – Integration | 积分
Integration is the reverse process of differentiation. Focus on standard integrals, the definite integral as area under a curve, and integration techniques such as substitution and parts (HL).
积分是微分的逆运算。重点掌握标准积分、定积分作为曲线下面积的解释，以及换元法和分部积分法（HL）等技巧。
Standard integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫ 1/x dx = ln|x| + C, ∫ eˣ dx = eˣ + C, ∫ cosx dx = sinx + C, ∫ sinx dx = −cosx + C.
标准积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1），∫ 1/x dx = ln|x| + C，∫ eˣ dx = eˣ + C，∫ cosx dx = sinx + C，∫ sinx dx = −cosx + C。
Definite integral: ∫ₐᵇ f(x) dx = F(b) − F(a). Area between curves: ∫ (upper − lower) dx.
定积分：∫ₐᵇ f(x) dx = F(b) − F(a)。曲线间面积：∫ (上函数 − 下函数) dx。
Integration by substitution: choose u = g(x), replace dx with du/g'(x), and adjust limits for definite integrals.
换元积分法：设 u = g(x)，将 dx 替换为 du/g'(x)，对于定积分要相应地改变积分限。
u dv = uv − ∫ v du. Useful for products of polynomials and exponentials or trig functions.
分部积分法（HL）：∫ u dv = uv − ∫ v du。适用于多项式与指数函数或三角函数的乘积。
Kinematics: given acceleration a(t), integrate to find velocity v(t) and displacement s(t). Use initial conditions to find constants.
运动学：给定加速度 a(t)，积分求速度 v(t) 和位移 s(t)。利用初始条件确定积分常数。
8. Statistics and Probability | 统计与概率
Data handling and probability models are essential, especially for the Applications and Interpretations route. Know how to summarise data, work with distributions, and perform hypothesis tests.
数据处理和概率模型至关重要，对“应用与解释”方向尤为如此。知道如何汇总数据、处理分布并进行假设检验。
Descriptive statistics: mean, median, mode, range, interquartile range, variance, standard deviation. Recognise skewness from box‑plots.
描述统计：平均数、中位数、众数、极差、四分位距、方差、标准差。能通过箱线图识别偏态。
Probability: P(A|B) = P(A ∩ B)/P(B). Independence: P(A ∩ B) = P(A)P(B). Tree diagrams and Venn diagrams.
概率：P(A|B) = P(A ∩ B)/P(B)。独立事件：P(A ∩ B) = P(A)P(B)。树状图和维恩图。
bility mass function, expected value E(X) = Σ x·P(X=x), variance Var(X) = E(X²) − [E(X)]².
离散随机变量：概率质量函数，期望值 E(X) = Σ x·P(X=x)，方差 Var(X) = E(X²) − [E(X)]²。
Binomial distribution: X ~ B(n, p), P(X = k) = ⁿCₖ pᵏ (1−p)ⁿ⁻ᵏ, mean = np, variance = np(1−p).
二项分布：X ~ B(n, p)，P(X = k) = ⁿCₖ pᵏ (1−p)ⁿ⁻ᵏ，均值 = np，方差 = np(1−p)。
Normal distribution: X ~ e using Z = (X − μ)/σ. Use calculator or tables for probabilities. Know how to perform one‑tailed and two‑tailed hypothesis tests.
正态分布：X ~ N(μ, σ²)。标准化 Z = (X − μ)/σ。用计算器或表格查概率。会进行单尾和双尾假设检验。
9. Proof and Mathematical Induction | 证明与数学归纳法
The IB values logical reasoning. Proof by induction is a key method for sequences and divisibility. Also practise direct proof, contrapositive, and contradiction where relevant.
IB 重视逻辑推理。归纳法证明是处理数列和整除问题的关键方法。同时练习直接证明、逆否命题证明和反证法（视情况而定）。
for n = k), 3) inductive step (prove true for n = k+1). Conclude statement holds for all n ∈ ℕ.
归纳法的结构：1）奠基（通常 n = 1），2）归纳假设（假设 n = k 成立），3）归纳步骤（证明 n = k+1 成立）。得出结论对所有自然数 n 成立。
Common induction proofs: sums of series, divisibility (e.g. 3ⁿ − 1 is divisible by 2), inequalities, and matrix powers.
常见归纳证明：级数求和、整除性（如 3ⁿ − 1 能被 2 整除）、不等式以及矩阵幂。
Direct proof: start from known facts and deduce the result step by step. Used for many algebraic identities.
直接证明：从已知事实出发，逐步推导出结果。常用于代数恒等式。
Proof by contradiction: assume the opposite and show an impossibility. Typical for irrationality of √2 or infinitude of primes.
反证法：假设相反情形，推出不可能的结果。常用于证明 √2 的无理性或素数有无穷多个。
10. Exam Techniques and Common Pitfalls | 考试技巧与常见错误
Even the best-prepared students can lose marks by overlooking practical details. Develop a systematic approach to reading questions, managing time, and checking answers.
即使准备最充分的学生也可能因为忽略实操细节而丢分。养成系统化的读题、时间管理和检查答案的习惯。
Read the command terms: “Write down” means no working needed; “Hence” implies using the previous result; “Find” and “Solve” require clear steps.
阅读指令词：“写下来”意味着无需写出过程；“由此”意味着要利用前一个结果；“求”与“解”则需要清晰的步骤。
Show all working: even if the final answer is wrong, method marks can be awarded. Make sure your reasoning is logically presented.
写出全部演算过程：即使最终答案错了，也可能获得方法分。确保推理逻辑清晰地呈现出来。
Check domain and range when sketching graphs, especially with rational and logarithmic functions.
画草图时检查定义域和值域，尤其是有理函数和对数函数。
Avoid calculator over‑reliance: know when to use exact values (fractions, surds, π) rather than decimal approximations, unless the question asks for a decimal answer.
避免过度依赖计算器：知道何时使用精确值（分数、根号、π）而非小数近似值，除非题目明确要求小数结果。
Time management: allocate roughly one minute per mark. Leave harder problems for the end of the section and return to them if time permits.
时间管理：大约按每分钟一分来分配。把较难的问题留到该部分的最后，如果还有时间再回头处理。
Review common algebraic slip‑ errors, missing solutions when solving quadratics, forgetting ± when taking square roots.
回顾常见的代数疏漏：符号错误、解二次方程时漏解、开平方时忘记写 ±。
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