📚 Review Set 18C: Mastering IB Math Key Topics | 复习集 18C：掌握IB数学核心主题
Welcome to Review Set 18C — a curated collection of problems and concept summaries designed to strengthen your understanding of the most frequently tested areas in the IB Mathematics curriculum. This set brings together calculus, vectors, probability, trigonometry, and algebraic reasoning, helping you transition from isolated knowledge to integrated problem‑solving. Each question type is paired with a targeted revision note so you can diagnose weak spots and build the fluency needed for Papers 1, 2, and 3.
欢迎来到复习集 18C——这套精心编排的习题集和概念总结旨在巩固你对 IB 数学课程中高频考察领域的理解。本复习集融合了微积分、向量、概率、三角和代数推理，帮助你从孤立的知识点过渡到综合解题。每一类问题都配有针对性复习提示，让你能诊断薄弱环节，逐步建立在试卷一、二和三中所需的流畅度。
1. Differentiation Techniques and Applications | 求导方法及其应用
The chain rule, product rule, and quotient rule must become second nature. When differentiating composite functions, always identify the inner function u(x) and work from the outside in. For optimisation problems, equate the first derivative to zero, then use the second derivative test to confirm maxima or minima. Pay attention to units of measurement when interpreting rates of change in contextual problems.
链式法则、乘积法则和商法则必须运用自如。对复合函数求导时，始终先识别内层函数 u(x)，然后由外向内计算。在优化问题中，令一阶导数等于零，再用二阶导数检验确认极大值或极小值。在处理情境化问题时，解读变化率时务必注意单位。
2. Integration — Indefinite and Definite Integrals | 不定积分与定积分
Integration is the reverse of differentiation, but the constant of integration +C is essential in indefinite integrals. For definite integrals, apply the fundamental theorem of calculus carefully, substituting upper and lower bounds into the antiderivative. Be systematic with u‑substitution: choose u that simplifies the integrand, express dx in terms of du, and remember to change limits when evaluating definite integrals.
积分是微分的逆运算，但在不定积分中积分常数 +C 必不可少。对于定积分，要仔细应用微积分基本定理，将上下限代入原函数。处理换元积分法时要有条不紊：选择能简化被积函数的 u，用 du 表示 dx，并在计算定积分时转换积分限。
3. Kinematics and the Language of Motion | 运动学与运动的数学语言
ance travelled, integrate the absolute value of velocity or separately integrate the velocity over intervals where it keeps a constant sign. Interpreting initial conditions correctly is vital when finding the constant of integration.
在 IB 运动学中，位移 (s)、速度 (v) 和加速度 (a) 通过微分与积分相互联系。记住 v = ds/dt 且 a = dv/dt。求总路程时，需对速度的绝对值积分，或将速度按符号不变的区间分段积分。在确定积分常数时，正确解读初始条件至关重要。
4. Vector Geometry — Equations of Lines and Planes | 向量几何——直线与平面的方程
ections, equate the vector equations and solve for parameters. To test if a point lies on a plane, substitute its coordinates into the Cartesian equation.
三维空间中的直线可表示为 r = a + λb，其中 a 是直线上某点的位置向量，b 是方向向量。平面则需要法向量 n 和平面上的一个点，得到数量积形式 r·n = a·n。求交点时，令向量方程相等并解参数。要检验点是否在平面上，将其坐标代入笛卡尔方程。
5. Scalar and Vector Product Applications | 数量积与向量积的应用
The scalar (dot) product a·b = |a||b| or perpendicular to both a and b; its magnitude gives the area of the parallelogram spanned by a and b. Always distinguish between scalar and vector results: dot gives a scalar, cross gives a vector.
数量积（点积）a·b = |a||b|cosθ 非常适合求两向量间的夹角以及判断垂直关系（a·b = 0）。向量积（叉积）a×b 得到垂直于 a 和 b 的向量；其大小等于 a 和 b 所张成平行四边形的面积。始终区分标量结果与向量结果：点积得出标量，叉积得出向量。
6. Probability Distributions and the Binomial Setting | 概率分布与二项分布情境
The binomial distribution X ~ . For cumulative probabilities, work with the binomial table or complementary events to save time.
二项分布 X ~ B(n, p) 模拟 n 次独立试验中成功的次数，每次成功的概率为 p。先检查四个条件是否成立：试验次数固定、每次只有两种结果、概率不变、试验相互独立。使用公式 P(X = k) = C(n, k) pᵏ (1−p)ⁿ⁻ᵏ。处理累积概率时，可借助二项分布表或利用互补事件以节省时间。
7. Normal Distribution and Inverse Normal Calculations | 正态分布与逆正态计算
The normal distribution N(μ, σ²) requires standardising to Z ~ For inverse normal problems where a probability is given and you need the corresponding X‑value, first find the Z‑ μ + Zσ. Sketch the bell curve, shade the relevant area, and note whether the region is one‑tailed or two‑tailed.
正态分布 N(μ, σ²) 需要通过 Z = (X − μ)/σ 标准化为 Z ~ N(0, 1)。在已知概率需反求对应 X 值的逆正态问题中，先从标准正态表中找出 Z 分数，再用 X = μ + Zσ 转换回来。画出钟形曲线，涂出相关区域，并注意该区域是单尾还是双尾。
8. Trigonometric Equations and General Solutions | 三角方程及其通解
in and cos have period 2π, while tan has period π. Write answers as θ = base + 2πn or θ = base + πn.
解 sinθ = k、cosθ = k 或 tanθ = k 时，先在计算器上用反三角函数求出主值。然后画出单位圆或相关函数图像，找出给定区间内的所有解。对于通解，记住周期性：sin 和 cos 的周期为 2π，而 tan 的周期为 π。将答案写作 θ = 基础解 + 2πn 或 θ = 基础解 + πn。
9. Logarithmic and Exponential Equations | 对数和指数方程
Link exponentials and logarithms through the equivalence aˣ ffer, use common logarithms or natural logs to bring down exponents. For equations like e²ˣ + eˣ − 6 = 0, substitute y = eˣ ys check that arguments of logarithms remain positive, as extraneous solutions can appear.
通过等价关系 aˣ = b ↔ logₐ b = x 将指数与对数联系起来。底数不同时，可借助常用对数或自然对数把指数降下来。对于 e²ˣ + eˣ − 6 = 0 这种方程，设 y = eˣ 化为一元二次方程。始终检查对数中的真数是否为正，因为可能出现增根。
10. Proof by Induction — Structure and Pitfalls | 数学归纳法——结构及常见陷阱
ypothesis). Write a concluding statement that clearly links the proven step to the principle of induction. Avoid the common mistake of manipulating the conclusion as if it were already true; always work from the hypothesis towards the target statement.
一个严密的归纳法证明包含三个明确阶段：基础情形（n = 1 或最小值）、归纳假设（假设 n = k 时成立）以及归纳步骤（利用假设证明 n = k+1 时成立）。写出结论陈述，清晰地将已证步骤与归纳原理联系起来。避免常见的错误——把结论当作已经成立的式子来变形；始终从假设出发推导目标命题。
11. Complex Numbers — Different Forms and Their Uses | 复数——不同形式及其应用
form z = a + bi, polar form z = r(cosθ + i sinθ), or Euler form z = reⁱ⁰. The modulus r = √a² + b² n and division are simplest in polar/Euler form: multiply moduli, add arguments. De Moivre's theorem (cosθ + i sinθ)ⁿ = cos nθ + i sin nθ is a powerful tool for powers and roots.
复数可以写成笛卡尔形式 z = a + bi、极坐标形式 z = r(cosθ + i sinθ) 或欧拉形式 z = reⁱ⁰。模 r = √a² + b²，辐角 θ = arctan(b/a) 并需做象限修正。乘法和除法在极坐标/欧拉形式下最为简单：模相乘，辐角相加。棣莫弗定理 (cosθ + i sinθ)ⁿ = cos nθ + i sin nθ 是处理幂和方根的强大工具。
12. Building an Effective Revision Routine with Set 18C | 用复习集 18C 打造高效复习计划
Use Review Set 18C diagnostically: attempt all questions under timed conditions, then categorise mistakes into conceptual, algebraic, or careless errors. For each weak area, revisit the corresponding section above, redo the worked examples, and attempt at least three similar problems from your textbook. Spaced repetition — reviewing the same set after two days and again after a week — will significantly improve long‑term retention and confidence going into the exam.
把复习集 18C 作为诊断工具：在限定时间内尝试所有问题，然后将错误分为概念性、代数性或粗心错误。对于每个薄弱环节，重读上述对应小节，重做示例，并从教材中至少找三道类似题目进行练习。间隔重复——两天后再次复习同一套题，一周后再复习——将显著提高长期记忆效果和应考信心。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图