📚 IB CIE Mathematics: End-of-Term Revision Guide | IB CIE 数学：期末复习提纲
Success in IB and CIE mathematics examinations depends on structured revision that connects theory, problem-solving techniques and common pitfalls. This guide outlines the essential topics, key formulas and strategic tips you need to master before the end of term. Use it as a checklist to ensure no topic is left unrevised.
在 IB 和 CIE 数学考试中取得成功，关键在于结构化的复习，将理论、解题技巧和常见易错点有机结合。这份提纲列出了期末前必须掌握的核心主题、关键公式与策略要诀。请将它作为一份自查清单，确保没有遗漏任何一个知识点。
1. Algebraic Manipulation and Equation Solving | 代数运算与方程求解
Proficiency in algebra underpins almost every other topic. Ensure you can confidently simplify rational expressions, factorise polynomials, apply the remainder and factor theorems, and solve quadratic, exponential and logarithmic equations. Pay special attention to the laws of indices and logarithms, as they appear frequently in calculus and functions problems.
熟练的代数运算能力是几乎所有其他主题的基础。确保你能够自信地化简有理式、因式分解多项式、运用余式定理与因式定理，并求解二次方程、指数方程和对数方程。要特别关注指数与对数的运算法则，因为它们在微积分和函数问题中频繁出现。
- Review completing the square and the discriminant for quadratics.
- 复习二次方程的配方法和判别式。
- Practice hidden quadratics (e.g., 4x − 2x+1 + 1 = 0) by substitution.
- 通过换元法练习隐藏二次方程（如 4x − 2x+1 + 1 = 0）。
- Know logab = logcb / logca and the change-of-base formula.
- 熟悉对数换底公式。
| Key Laws: am × an = am+n; (am)n = amn; loga(xy) = logax + logay; loga(x/y) = logax − logay |
2. Functions and Graph Transformations | 函数与图像变换
Understanding domain, range, composition and inverses is essential. You must be able to sketch graphs of basic functions and apply transformations: translations, stretches and reflections. For IB and CIE, linking transformations to changes in the function expression – f(x+a), f(x)+b, f(kx), kf(x) – is frequently examined alongside modulus functions.
理解定义域、值域、复合函数与反函数至关重要。你必须能够绘制基本函数的图像，并应用平移、伸缩和反射等变换。在 IB 与 CIE 考试中，将图像变换与函数表达式的变化（f(x+a)、f(x)+b、f(kx)、kf(x)）联系起来是常见考点，经常与绝对值函数一同出现。
- Find the inverse by swapping x and y and rearranging; ensure domain restrictions.
- 求反函数时交换 x 与 y 并整理，注意定义域的限制。
- Composite functions: fg(x) means apply g first, then f.
- 复合函数：fg(x) 表示先作用 g，再作用 f。
- Modulus: |f(x)| reflects negative parts of f(x) above the x‑axis.
- 绝对值：|f(x)| 将 f(x) 负的部分翻折到 x 轴上方。
formation order: horizontal shift → horizontal stretch → vertical stretch → vertical shift
变换顺序：水平平移 → 水平伸缩 → 垂直伸缩 → 垂直平移
3. Trigonometry and Circular Functions | 三角函数与圆函数
Master identities such as sin²θ + cos²θ = 1, tanθ = sinθ/cosθ, and double-angle formulas.
在角度制与弧度制之间自如转换。记住 0、π/6、π/4、π/3、π/2 及其整数倍的正弦、余弦和正切的精确值。理解单位圆定义并用来解三角方程。掌握基本恒等式，如 sin²θ + cos²θ = 1、tanθ = sinθ/cosθ 以及倍角公式。
- Solve equations using CAST diagram or graphical method.
- 使用 CAST 图或图像法求解三角方程。
- Sine and cosine rules: a/sinA = b/sinB = c/sinC; a² = b² + c² − 2bc cosA.
- 正弦定理与余弦定理。
- Area of triangle: ½ ab sinC.
- 三角形面积公式：½ ab sinC。
| Exact values: sin30°=½, cos45°=√2/2, tan60°=√3, etc.; radian equivalents |
4. Sequences, Series and Binomial Expansion | 数列、级数与二项式展开
Arithmetic and geometric sequences appear frequently in both IB and CIE. Know the formulas for the nth term and the sum of the first n terms. For geometric series, be able to find the sum to infinity when |r| < 1. The binomial theorem for positive integer powers is essential, and for rational powers the expansion is valid only when |x| < 1.
等差数列和等比数列在 IB 与 CIE 考试中频繁出现。熟记第 n 项与前 n 项和的公式。对于等比级数，当 |r| < 1 时能够计算无穷和。正整数幂的二项式定理是必备知识，而对于有理数幂的展开，只有在 |x| < 1 时才有效。
- A.P.: un = a + (n−1)d, Sn = n/2 (2a + (n−1)d).
- G.P.: un = arn−1, Sn = a(1−rn)/(1−r), S∞ = a/(1−r).
- Binomial: (1+x)n = 1 + nx + [n(n−1)/2!]x² + ...
- 二项式展开： (1+x)n = 1 + nx + [n(n−1)/2!]x² + ...
5. Differentiation Techniques and Applications | 微分技巧与应用
Derivatives measure rates of change. You should be able to differentiate polynomials, trigonometric, exponential and logarithmic functions from first principles and using rules. Chain, product and quotient rules are frequently tested. Applications include finding equations of tangents and normals, determining stationary points and solving optimisation problems.
导数衡量变化率。你应当能够从第一性原理出发，并使用运算法则对多项式、三角函数、指数函数和对数函数求导。链式法则、乘法法则和除法法则是常考内容。应用部分包括求切线与法线方程、确定驻点以及解决最优化问题。
- d/dx (xn) = nxn−1; d/dx (sinx) = cosx; d/dx (lnx) = 1/x; d/dx (ex) = ex.
- Chain rule: dy/dx = dy/du × du/dx.
- 链式法则。
- Second derivative test: f''(x) > 0 → local minimum.
- 二阶导数判定：f''(x) > 0 为局部极小。
Tangent line at x = a: y − f(a) = f'(a)(x − a)
切线方程：y − f(a) = f'(a)(x − a)
6. Integration Methods, Area and Volume | 积分方法、面积与体积
Integration reverses differentiation. Learn standard integrals and apply them to find areas under curves and volumes of revolution. Techniques include substitution, integration by parts and partial fractions (depending on syllabus). Always include the constant of integration for indefinite integrals. For definite integrals, careful evaluation of limits is crucial.
积分是微分的逆运算。学习标准积分，并用其求曲线下方的面积和旋转体的体积。常用技巧包括换元法、分部积分法和部分分式法（视考纲而定）。不定积分务必加上积分常数；对于定积分，准确代入上下限至关重要。
- ∫ xn dx = xn+1/(n+1) + C (n ≠ −1); ∫ 1/x dx = ln|x| + C.
- Area = ∫ab [f(x) − g(x)] dx for region between curves.
- 两曲线间的面积 = ∫ab [f(x) − g(x)] dx。
- Volume of revolution about x‑axis: V = π ∫ab [f(x)]² dx.
- 绕 x 轴旋转体体积：V = π ∫ab [f(x)]² dx。
7. Vectors, Lines and Planes | 向量、直线与平面
Vectors describe both magnitude and direction. In IB and CIE, vector questions often involve position vectors, scalar (dot) product and, for HL/Further Maths, vector (cross) product. Be able to write equations of lines in parametric and Cartesian forms. If planes are in your syllabus, understand normal vectors and intersections.
向量同时描述大小和方向。在 IB 与 CIE 中，向量问题常涉及位置向量、标量积（点乘），以及 HL/进阶数学中的向量积（叉乘）。要能够写出直线的参数方程与直角坐标方程。如果你所在课程包含平面，需理解法向量及平面的交线。
- Dot product: a·b = |a||b|cosθ; used for angle between vectors.
- 点乘：a·b = |a||b|cosθ，用于求向量间夹角。
- Line equation: r = a + t b; distance from point to line using cross product.
- 直线方程：r = a + t b；点线距离可用叉乘公式。
If a·b = 0, vectors are perpendicular.
若 a·b = 0，则两向量垂直。
8. Probability and Statistical Distributions | 概率与统计分布
Probability theory underpins statistical inference. Review set notation, Venn diagrams, tree diagrams, conditional probability and Bayes’ theorem. For distributions, master the binomial and normal models. Know how to use the normal distribution table, apply continuity corrections, and find probabilities and inverse normal values.
概率论是统计推断的基础。复习集合符号、文氏图、树形图、条件概率和贝叶斯定理。分布方面，要掌握二项分布与正态分布。熟悉如何使用正态分布表、进行连续性校正，以及求概率和逆正态值。
- P(A|B) = P(A∩B)/P(B); independent events: P(A∩B) = P(A)P(B).
- 条件概率公式；独立事件检验。
- X ~ B(n, p): E(X)=np, Var(X)=np(1−p).
- 二项分布期望与方差。
- Standardise to Z: Z = (X − μ)/σ.
- 标准化：Z = (X − μ)/σ。
| Normal approximation to binomial: use when np > 5 and n(1−p) > 5; apply continuity correction ±0.5 |
9. Complex Numbers and Polynomial Roots | 复数与多项式根
For advanced streams, complex numbers extend the real number system. Understand the Argand diagram, modulus and argument, polar form and De Moivre’s theorem. Solving polynomial equations with real coefficients yields conjugate pairs. These concepts often link to roots of unity and trigonometric identities.
对于高阶课程，复数扩展了实数系统。理解阿干特图、模与辐角、极坐标形式和棣莫弗定理。求解实系数多项式方程会产生共轭复根对。这些概念常与单位根和三角恒等式关联。
- z = a + bi; |z| = √a²+b²; arg(z) = θ where tanθ = b/a.
- Polar form: z = r(cosθ + i sinθ) = reiθ.
- 极坐标形式。
- De Moivre: (cosθ + i sinθ)n = cos(nθ) + i sin(nθ).
- 棣莫弗定理。
10. Exam Strategy and Error Prevention | 考试策略与易错点防范
Revision is not complete without practising past papers under timed conditions. Identify command words: 'state', 'find', 'show that', 'hence'. Always check units, rounding instructions and domain restrictions. In multi-step problems, show clear working to earn method marks even if the final answer is wrong.
没有限时刷过真题的复习是不完整的。识别指令词：“写出”、“求”、“证明”、“由此”。务必检查单位、修约指令和定义域限制。在多步解题过程中，即使最终答案有误，也要清晰呈现解题步骤以获取过程分。
- Sketch graphs – label intercepts and asymptotes.
- 绘制图像——标注截距和渐近线。
- Check for extraneous solutions when squaring or taking logs.
- 平方或取对数时检验增根。
- Read the question carefully: answer exactly what is asked.
- 仔细读题：准确回答题目所问。
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