IB Mathematics: End-of-Year Revision Checklist | IB 数学：期末复习提纲

📚 IB Mathematics: End-of-Year Revision Checklist | IB 数学：期末复习提纲

As the IB final examinations approach, a structured revision plan is your best tool for success. This checklist covers the core topics across both Analysis & Approaches and Applications & Interpretation, highlighting the skills you must master. Use it to identify weak areas, practise targeted exercises, and build confidence before the exams. Remember that conceptual understanding is just as important as procedural fluency in the IB curriculum.

随着 IB 期末考试临近，一份结构化的复习计划是你取得成功的最佳工具。这份提纲涵盖了分析与方法和应用与解释课程的核心主题，突出了你必须掌握的技能。用它来找出薄弱环节，进行有针对性的练习，并在考前树立信心。请记住，在 IB 课程中概念理解与解题熟练度同等重要。

1. Algebra and Equation Solving | 代数与方程求解

Master algebraic manipulation, including expanding, factorising, and simplifying rational expressions. You must be able to solve linear equations, quadratic equations by factorising, completing the square, and using the quadratic formula. For HL, ensure you can handle polynomial equations of higher degree and use the factor and remainder theorems confidently.

掌握代数变换，包括展开、因式分解以及化简有理式。你必须能够解线性方程，并通过因式分解、配方法和二次公式解二次方程。对于 HL，确保能处理更高次的多项式方程，并自信地运用因式定理和余式定理。

Systems of linear equations should be solved both algebraically and with technology, especially in Applications & Interpretation. Be comfortable with interpreting solutions geometrically and recognising when a system has no unique solution. The use of augmented matrices and row reduction (Gaussian elimination) is expected in appropriate courses.

线性方程组应既能用代数方法求解，也能用技术工具求解，尤其在应用与解释课程中。要能自如地对解进行几何解释，并识别方程组何时无唯一解。在相关课程中，还需要会用增广矩阵和行变换（高斯消元法）。

Factorising quadratics and simplifying a/(b) + c/(d) | 二次三项式因式分解与分式化简
Discriminant Δ = b² – 4ac for nature of roots | 根的判别式 Δ = b² – 4ac
Polynomial division and the Remainder Theorem | 多项式除法与余式定理
Solving 2×2 and 3×3 systems by elimination and matrices | 用消元法与矩阵解 2×2 和 3×3 方程组

2. Functions and Transformations | 函数与图像变换

Understand the concepts of domain, range, composite and inverse functions. A function f is invertible only if it is one-to-one on its domain. Be prepared to restrict a domain to make a function invertible and to find the inverse algebraically by swapping x and y and solving for y.

理解定义域、值域、复合函数与反函数的概念。函数 f 仅在定义域上为单射时才可逆。要准备好限制定义域使函数可逆，并通过交换 x 和 y 再解出 y 来求反函数。

Graphical transformations are frequently assessed: translations (vertical and horizontal), stretches (vertical and horizontal), and reflections in the axes. Know the effect of y = a f(b(x – c)) + d and be able to sketch transformations of basic functions such as polynomials, exponentials, logarithms, and trigonometric graphs without technology.

图像变换是常考内容：平移（垂直与水平）、伸缩（垂直与水平）以及关于坐标轴的反射。要掌握 y = a f(b(x – c)) + d 的效果，并能不借助技术工具画出多项式、指数、对数和三角函数等基本函数的变换草图。

Domain: avoid division by zero and negative inside even roots | 定义域：避免分母为零与偶次根号下为负
Finding f⁻¹(x) and verifying f(f⁻¹(x)) = x | 求 f⁻¹(x) 并验证 f(f⁻¹(x)) = x
Sketching y = 2f(x – 3) + 1 from y = f(x) | 由 y = f(x) 画 y = 2f(x – 3) + 1

3. Sequences and Series | 数列与级数

Arithmetic sequences have a constant common difference d, with the nth term uₙ = u₁ + (n – 1)d. The sum of the first n terms is Sₙ = n/2 (2u₁ + (n – 1)d). Geometric sequences have a constant common ratio r; the nth term is uₙ = u₁ rⁿ⁻¹. For |r| < 1, the infinite geometric series converges to a sum S = u₁/(1 - r).

等差数列有常数公差 d，第 n 项 uₙ = u₁ + (n – 1)d。前 n 项和 Sₙ = n/2 (2u₁ + (n – 1)d)。等比数列有常数公比 r；第 n 项 uₙ = u₁ rⁿ⁻¹。当 |r| < 1 时，无穷等比级数收敛，和为 S = u₁/(1 - r)。

Applications include compound interest, population growth, and depreciation. For HL, proof by induction of series summation formulas may appear, as well as the use of sigma notation and binomial expansions with fractional or negative exponents. Ensure you can move fluidly between sequence and series contexts and use the GDC effectively for sequence problems.

应用题包括复利、人口增长和折旧。HL 可能需要用数学归纳法证明级数求和公式，以及使用 sigma 符号和含有分数指数或负指数的二项展开式。要确保能在数列与级数情境间自如转换，并有效使用图形计算器处理数列问题。

Identify a sequence as arithmetic or geometric | 识别数列是等差还是等比
Sum formulas: Sₙ and S∞ (geometric) | 求和公式：Sₙ 与 S∞（等比）
Compound interest: A = P(1 + r/n)^(nt) | 复利：A = P(1 + r/n)^(nt)
Proof by induction (HL) | 数学归纳法证明（HL）

4. Trigonometry and Circular Functions | 三角学与圆函数

Radian measure is fundamental; be confident converting between degrees and radians and using arc length s = rθ and sector area A = ½ r²θ. The unit circle definitions of sine, cosine, and tangent extend trigonometric functions to any real angle. Know exact values for key angles such as π/6, π/4, π/3 and their multiples.

弧度制是基础；要自信地在角度与弧度之间进行换算，并使用弧长公式 s = rθ 和扇形面积公式 A = ½ r²θ。通过单位圆定义正弦、余弦和正切，可将三角函数推广到任意实数角。要熟记 π/6、π/4、π/3 及其倍数等关键角度的精确值。

Trigonometric identities are essential: Pythagorean identities, double-angle formulas, and compound angle formulas. Solving trigonometric equations within given intervals often requires factorising, using identities, and considering the periodic nature of the functions. For periodic function modelling, be able to determine the amplitude, period, phase shift, and vertical shift of a function f(x) = a sin(b(x – c)) + d.

三角恒等式至关重要：毕达哥拉斯恒等式、二倍角公式与和差角公式。在给定区间内解三角方程通常需要因式分解、利用恒等式，并考虑函数的周期性。对于周期函数建模，要能确定 f(x) = a sin(b(x – c)) + d 的振幅、周期、相位平移和垂直平移。

Radian: π rad = 180°, arc length s = rθ | 弧度：π rad = 180°，弧长 s = rθ
Exact values: sin(π/3) = √3/2, cos(π/4) = √2/2 | 精确值：sin(π/3) = √3/2, cos(π/4) = √2/2
Identity: sin²θ + cos²θ = 1, tanθ = sinθ/cosθ | 恒等式：sin²θ + cos²θ = 1, tanθ = sinθ/cosθ
Double-angle: sin(2θ) = 2sinθcosθ | 二倍角：sin(2θ) = 2sinθcosθ

5. Vectors | 向量

Vectors are used to represent quantities with both magnitude and direction. Know how to add and subtract vectors geometrically and algebraically, multiply by a scalar, and calculate the magnitude. The dot (scalar) product a ⋅ b = |a||b|cosθ is essential for finding the angle between two vectors and for determining perpendicularity.

向量用来表示既有大小又有方向的量。要懂得如何用几何方法和代数方法进行向量的加减、数乘以及计算模长。点积（标量积）a ⋅ b = |a||b|cosθ 对于求两向量间的夹角和判定垂直至关重要。

Vector equations of lines in 2D and 3D appear as r = a + λb. For HL, also work with plane equations using the normal vector and the cross product of two vectors. You should be able to find intersections of lines, distances from a point to a line or plane, and determine whether two lines are parallel, skew, or intersecting.

二维和三维空间中直线的向量方程为 r = a + λb。HL 还要会利用法向量和两向量的叉积处理平面方程。你要会求直线的交点、点到直线或平面的距离，并判断两直线是平行、异面还是相交。

Magnitude: |v| = √(v₁² + v₂² + v₃²) | 模长：|v| = √(v₁² + v₂² + v₃²)
Dot product: v·w = v₁w₁ + v₂w₂ + v₃w₃ | 点积：v·w = v₁w₁ + v₂w₂ + v₃w₃
Line equation: r = (x₀,y₀,z₀) + λ(a,b,c) | 直线方程：r = (x₀,y₀,z₀) + λ(a,b,c)
Scalar and vector projections | 向量投影与标量投影

6. Complex Numbers (HL) | 复数（HL）

Complex numbers in the form z = a + bi extend the real number system, where i² = -1. Operations include addition, subtraction, multiplication, division (using conjugates), and finding the modulus |z| = √(a² + b²) and argument arg(z). Cartesian and polar forms are linked by z = r(cosθ + i sinθ).

形如 z = a + bi 的复数扩展了实数系，其中 i² = -1。运算包括加、减、乘、除（使用共轭复数），以及求模 |z| = √(a² + b²) 和辐角 arg(z)。直角坐标形式与极形式通过 z = r(cosθ + i sinθ) 联系起来。

De Moivre’s theorem is key for raising complex numbers to powers and extracting nth roots. The complex plane and Argand diagrams help visualise geometrical interpretations such as addition as vector addition and multiplication as rotation and scaling. Euler’s form z = re^(iθ) simplifies many operations and linking to exponential and trigonometric functions.

棣莫弗定理对于求复数的幂和开 n 次方根很关键。复平面和阿氏图有助于几何解释可视化，如加法可看作向量加法，乘法可看作旋转和伸缩。欧拉形式 z = re^(iθ) 可简化许多运算，并与指数函数和三角函数关联。

Conjugate: (a + bi)* = a – bi | 共轭复数：(a + bi)* = a – bi
Multiplication in polar form: multiply moduli, add arguments | 极形式乘法：模长相乘，辐角相加
De Moivre: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) | 棣莫弗定理：(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)
Roots of unity and solving zⁿ = c | 单位根与解 zⁿ = c

7. Differentiation | 微分

Differentiation deals with instantaneous rates of change and slopes of tangents. Know the limit definition of the derivative and the rules: power rule, sum/difference rule, product rule, quotient rule, and chain rule. You must be able to differentiate polynomial, rational, radical, exponential, logarithmic, and trigonometric functions.

微分研究瞬时变化率和切线斜率。要理解导数的极限定义以及各种法则：幂法则、和/差法则、乘法法则、除法法则和链式法则。你必须能对多项式、有理式、根式、指数、对数和三角函数求导。

Applications of differentiation include finding equations of tangents and normals to curves, optimisation problems (maximising or minimising quantities), and kinematic problems involving displacement, velocity, and acceleration. Use the first derivative test to find intervals of increase/decrease and the second derivative test for concavity and points of inflexion.

微分的应用包括求曲线的切线和法线方程、最优化问题（最大化或最小化某些量），以及涉及位移、速度和加速度的运动学问题。利用一阶导数测试求递增/递减区间，利用二阶导数测试判断凹性和拐点。

Power rule: d/dx[xⁿ] = n xⁿ⁻¹ | 幂法则：d/dx[xⁿ] = n xⁿ⁻¹
Chain rule: d/dx[f(g(x))] = f'(g(x)) g'(x) | 链式法则：d/dx[f(g(x))] = f'(g(x)) g'(x)
Product: (uv)’ = u’v + uv’ ; Quotient: (u/v)’ = (u’v – uv’)/v² | 乘法：(uv)’ = u’v + uv’；除法：(u/v)’ = (u’v – uv’)/v²
d/dx[sin x] = cos x ; d/dx[eˣ] = eˣ ; d/dx[ln x] = 1/x

8. Integration | 积分

Integration is essential for finding areas under curves, anti-derivatives, and solving differential equations. You must know the integral of standard functions: xⁿ, eˣ, 1/x, sin x, cos x, sec² x, etc. Integration by inspection (reverse chain rule) and substitution are core techniques. For HL, integration by parts is also required.

积分对于求曲线下方面积、反导数以及解微分方程至关重要。你必须熟记标准函数的积分：xⁿ、eˣ、1/x、sin x、cos x、sec² x 等。逆链式法则（观察法）和换元积分是核心技巧。HL 还需要掌握分部积分法。

Definite integrals give the signed area between a curve and the x-axis. Be careful with areas crossing the x-axis; you may need to split the integral. Kinematics problems involve using integration to move from acceleration to velocity and displacement. Also be able to find volumes of revolution for solids generated by rotating a curve about an axis.

定积分给出曲线与 x 轴之间的有号面积。注意处理穿过 x 轴的情况；可能需要拆分积分。运动学问题需要用积分从加速度得到速度再到位移。还要能够求曲线绕某轴旋转生成的旋转体体积。

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1) | ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
∫ 1/x dx = ln|x| + C ; ∫ eˣ dx = eˣ + C | ∫ 1/x dx = ln|x| + C；∫ eˣ dx = eˣ + C
Substitution method: let u = g(x) | 换元法：设 u = g(x)
Integration by parts (HL): ∫ u dv = uv – ∫ v du | 分部积分法（HL）：∫ u dv = uv – ∫ v du
Area between curves and volume of revolution | 曲线间面积与旋转体体积

9. Probability and Counting | 概率与计数

Use sample space diagrams, Venn diagrams, and tree diagrams to represent events. Understand mutually exclusive events P(A ∪ B) = P(A) + P(B), conditional probability P(A|B) = P(A ∩ B)/P(B), and independent events where P(A ∩ B) = P(A)P(B). Bayes’ theorem is used for reversing conditional probabilities, particularly in HL and Applications courses.

使用样本空间图、维恩图和树状图表示事件。理解互斥事件 P(A ∪ B) = P(A) + P(B)，条件概率 P(A|B) = P(A ∩ B)/P(B)，以及独立事件中 P(A ∩ B) = P(A)P(B)。贝叶斯定理用于逆向条件概率，特别在 HL 和应用课程中出现。

Counting techniques include the product principle, permutations (order matters) and combinations (order does not matter). The formulas nPr = n!/(n – r)! and nCr = n!/(r!(n – r)!) must be used fluently. For discrete distributions, focus on the binomial distribution for a fixed number of trials and the Poisson distribution (HL) for events occurring randomly over time or space.

计数技巧包括乘法原理、排列（顺序重要）和组合（顺序不重要）。必须熟练运用公式 nPr = n!/(n – r)! 和 nCr = n!/(r!(n – r)!)。对于离散分布，重点掌握二项分布（固定试验次数）和泊松分布（HL，描述随机事件在时间或空间上的发生）。

Conditional probability: P(A|B) and tree diagrams with replacement | 条件概率：P(A|B) 和有放回/无放回树状图
Permutation vs Combination keywords: arrangement vs selection | 排列与组合关键词：排列与选择
Binomial: X ~ B(n, p), P(X = k) = nCk p^k (1-p)^(n-k) | 二项分布：X ~ B(n, p)，P(X = k) = nCk p^k (1-p)^(n-k)
Poisson distribution and normal approximation (HL) | 泊松分布与正态近似（HL）

10. Descriptive and Inferential Statistics | 描述性与推断统计

Understand measures of central tendency (mean, median, mode) and dispersion (variance, standard deviation, interquartile range). Box-and-whisker plots, histograms, and cumulative frequency graphs are used to display data. For bivariate data, scatter plots, correlation coefficients (Pearson’s r), and least squares regression lines are used to model linear relationships.

理解集中趋势的度量（平均数、中位数、众数）和离散程度的度量（方差、标准差、四分位距）。箱线图、直方图和累积频率图用于展示数据。对于双变量数据，散点图、相关系数（皮尔逊 r）和最小二乘回归线用于建立线性关系模型。

Inferential statistics in IB often involve confidence intervals and hypothesis tests for means and proportions. Be able to formulate null and alternative hypotheses, interpret p-values, and draw conclusions in context. The normal distribution Z ~ N(0,1²) is extensively used, and you must know how to use inverse normal calculations on the GDC. For AI courses, t-distributions and chi-squared tests for independence/goodness of fit are also important.

IB 中的推断统计通常涉及均值和比例的置信区间与假设检验。要会建立原假设和备择假设，解读 p 值，并结合情境得出结论。正态分布 Z ~ N(0,1²) 被广泛使用，你必须懂得如何在图形计算器上使用逆正态计算。对于 AI 课程，t 分布和用于独立性/拟合优度的卡方检验也很重要。

Mean μ = Σx/n ; Standard deviation σ = √(Σ(x-μ)²/n) | 均值 μ = Σx/n；标准差 σ = √(Σ(x-μ)²/n)
Scatterplots, r, and line of best fit y = a + bx | 散点图、r 与最佳拟合线 y = a + bx
Confidence interval for mean: x̄ ± z* (σ/√n) | 均值的置信区间：x̄ ± z* (σ/√n)
Hypothesis test: p-value method and conclusion | 假设检验：p 值法与结论

11. Calculus Integration: Differential Equations and Modelling (HL/AI) | 积分应用：微分方程与建模（HL/AI 重点）

Differential equations model dynamic relationships between variables and their rates of change. You should be able to verify that a function satisfies a given differential equation and separate variables to solve simple first-order ODEs such as dy/dx = f(x)g(y). For some courses, logistic differential equations and slope fields are introduced.

微分方程对变量及其变化率之间的动态关系进行建模。你应能验证某函数满足给定的微分方程，并用分离变量法求解简单的一阶常微分方程，如 dy/dx = f(x)g(y)。某些课程会引入逻辑斯蒂微分方程和斜率场。

Coupled differential equations (HL AA) require the use of eigenvalues and eigenvectors for solving systems. In Applications & Interpretation, emphasis is on piecewise models, sinusoidal models, and the trapezoidal rule or Simpson’s rule for numerical integration. Modelling tasks will ask you to reflect on the reasonableness of the model and its limitations.

耦合微分方程（HL AA）需要用特征值与特征向量来解方程组。在应用与解释课程中，重点在于分段模型、正弦模型以及用于数值积分的梯形法则或辛普森法则。建模任务会要求你反思模型的合理性及其局限。

Separation of variables: ∫ 1/g(y) dy = ∫ f(x) dx | 分离变量法：∫ 1/g(y) dy = ∫ f(x) dx
Slope fields and equilibrium solutions | 斜率场与平衡解
Euler’s method for approximation (numerical) | 欧拉方法的数值近似
Trapezoidal rule: ∫ₐᵇ f(x) dx ≈ (b-a)/(2n)[f(x₀)+2f(x₁)+…+f(xₙ)] | 梯形法则

12. Exam Technique and GDC Use | 考试技巧与图形计算器使用

Your Graphical Display Calculator (GDC) is a powerful tool. Know how to solve equations, find numerical derivatives and integrals, compute statistics, generate probability distributions, and graph functions efficiently. However, you must also be able to show analytical work when required, especially on Paper 1 (non-calculator for AA) where step-by-step reasoning is assessed.

图形计算器（GDC）是一个强大工具。要知道如何解方程、求数值导数和积分、计算统计量、生成概率分布以及高效地绘制函数图像。但你也必须能在需要时展示解析过程，特别是在试卷一（AA 课程不允许使用计算器）中，逐步推理是评分依据。

Read each question carefully identifying command terms like “find”, “hence”, “determine”, or “justify”. Manage your time by allocating roughly one minute per mark. For longer structured questions, the parts often build on each other; check your earlier answers for consistency. In the exploration (IA), demonstrate personal engagement, mathematical rigour, and clear communication.

仔细阅读每道题，识别指令词，如 “find”, “hence”, “determine” 或 “justify”。按照大约一分钟一分的标准分配时间。对于结构化长问题，各部分通常是递进的；检查前面的答案以确保一致性。在内部评估（IA）中，要展现出个人投入、数学严谨性和清晰的表达。

GDC skills: graphing window, solver, financial functions | GDC 技能：绘图窗口、解算器、金融函数
Show work: write down formula, substitution, GDC input, and result | 展示过程：列出公式、代入、GDC 输入和结果
Plan coursework: choose a focused topic and collect data early | 规划课程作业：选定一个重点主题并尽早收集数据
Time management: skip and return to difficult parts | 时间管理：先跳过难题，之后返回

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IB Mathematics: End-of-Year Revision Checklist | IB 数学：期末复习提纲