📚 Review Set 10C: Mastering Core IB Mathematics | 复习集10C：攻克IB数学核心概念
Review Set 10C is designed to consolidate your understanding of critical topics that frequently appear in the IB Mathematics analysis and approaches syllabus. It serves as a targeted checkpoint before summative assessments, bringing together algebraic manipulation, function analysis, trigonometry, calculus techniques, and probabilistic reasoning. Working through this set systematically will help you identify gaps in your knowledge and build the fluency required for both Paper 1 and Paper 2 style questions.
复习集10C旨在巩固你对IB数学分析与方法课程大纲中高频出现的关键主题的理解。它是总结性评估前的一个针对性检查点，将代数运算、函数分析、三角学、微积分技巧和概率推理整合在一起。系统地完成本复习集将帮助你发现知识漏洞，并建立起应对试卷一和试卷二题型所需的熟练度。
1. Algebraic Simplification and Expansion | 代数化简与展开
Begin by ensuring that you can confidently expand brackets using the distributive law and simplify rational expressions by factoring. In IB problems, you are often required to rewrite expressions such as (ax + b)(cx² + dx + e) without errors in sign or degree. Factorisation of quadratics and cubics, including those requiring the factor theorem, is a foundational skill tested either directly or within broader coordinate geometry and calculus questions.
首先要确保你能熟练运用分配律展开括号，并通过因式分解化简有理式。在IB题目中，你经常需要正确无误地重写如(ax + b)(cx² + dx + e)的表达式，符号和次数都不能出错。二次和三次多项式的因式分解，包括需要使用因式定理的情况，是一项基础技能，可能直接考查，也可能嵌入到更广泛的坐标几何和微积分问题中。
- Always check for a common factor first; it simplifies the remaining factorisation.
- 首先检查是否有公因式；这能简化后续的因式分解过程。
- When dividing polynomials, remember that the remainder theorem provides a quick verification: substituting the root into the original polynomial must yield zero.
- 进行多项式除法时，记住余数定理可提供快速验证：将根代入原多项式，结果必须为零。
2. Functions, Domain, and Range | 函数、定义域与值域
Understanding the formal definition of a function is non‑negotiable at IB level. You must distinguish between one‑to‑one, many‑to‑one, and one‑to‑ inators (cannot equal zero), and logarithms (the argument must be > 0). Composite functions and inverse functions appear in almost every examination session; practice verifying whether an inverse is itself a function by applying the horizontal line test on the original graph.
理解函数的形式化定义在IB阶段是不可或缺的。你必须区分一对一、多对一和一对多映射，并能够用集合符号或区间符号给出给定函数的定义域和值域。要特别注意平方根（被开方式必须≥0）、分母（不能等于零）和对数（真数必须>0）所施加的限制。复合函数和反函数几乎出现在每一轮考试中；通过在原始图像上应用水平线检验来练习判断一个反函数本身是否是一个函数。
If f(x) = √x − 4, then the domain is x ≥ 4, and the range is f(x) ≥ 0.
若 f(x) = √x − 4，则定义域为 x ≥ 4，值域为 f(x) ≥ 0。
3. Quadratic Functions and the Parabola | 二次函数与抛物线
s when Δ > 0, one repeated real root when Δ = 0, and no real roots when Δ < 0. In application problems, the vertex of the parabola represents the maximum or minimum value of a quadratic model, such as the peak height of a projectile or the minimum cost in an optimisation scenario.
二次函数是代数和坐标几何之间的桥梁。你应该能够流畅地在标准式f(x) = ax² + bx + c、顶点式f(x) = a(x − h)² + k和因式分解式f(x) = a(x − p)(x − q)之间转换。判别式Δ = b² − 4ac决定了根的性质：当Δ > 0时有两个不同的实根，当Δ = 0时有一个重复的实根，当Δ < 0时没有实根。在应用题中，抛物线的顶点代表二次模型的最大值或最小值，如抛射体的最高点或优化问题中的最小成本。
| Vertex form | f(x) = a(x − h)² + k | Vertex at (h, k) |
| 顶点式 | f(x) = a(x − h)² + k | 顶点为 (h, k) |
| Discriminant | Δ = b² − 4ac | Determines number of x-intercepts |
| 判别式 | Δ = b² − 4ac | 决定与x轴交点的数量 |
4. Exponential and Logarithmic Functions | 指数函数与对数函数
manipulating expressions involving e and ln. The key relationship to internalise is: y = aˣ ⇔ x = logₐ y, and for the natural logarithm, ln(eˣ nitial quantity, and k is the continuous growth (if positive) or decay (if negative) rate.
指数增长和衰减模型支撑着从种群生物学到复利的各类主题，IB考生必须熟练处理包含e和ln的表达式。需要内化的关键关系是：y = aˣ ⇔ x = logₐ y，对于自然对数，ln(eˣ) = x。求解指数方程时，若无法将底数统一，通常的方法是对方程两边取自然对数。留意那些要求你解释模型N = N₀ eᵏᵗ中各常数的题目：N₀是初始量，k是连续增长（若为正）或衰减（若为负）的速率。
- Remember that ln 1 = 0 and ln e = 1; these values simplify many expressions immediately.
- 记住 ln 1 = 0 且 ln e = 1；这些值能立刻简化许多表达式。
- When a logarithm appears in a denominator, state the domain restriction explicitly: the argument of the log must be strictly positive, and the denominator cannot be zero.
- 当分母中出现对数时，要明确说明定义域的限制：对数的真数必须严格为正，且分母不能为零。
5. Trigonometric Functions and Equations | 三角函数与三角方程
Trigonometry in the IB curriculum extends well beyond right‑ ing the periodic properties of the functions. The identities sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ are essential tools; more complex equations may require the double‑angle identities for sine and cosine.
IB课程中的三角学远超直角三角形中的比。你必须掌握正弦、余弦和正切的单位圆定义，并用它来找到以弧度和角度表示的30°（π/6）和45°（π/4）倍数的精确值。求解三角方程需要你利用函数的周期性质，写出在指定区间内的所有解。恒等式 sin²θ + cos²θ = 1 和 tan θ = sin θ / cos θ 是必备工具；更复杂的方程可能需要正弦和余弦的倍角恒等式。
Example: Solve sin x = √3/2 for 0 ≤ x < 2π. Solutions: x = π/3, 2π/3.
示例：在 0 ≤ x < 2π 内解 sin x = √3/2。解为 x = π/3, 2π/3。
6. Differentiation Techniques and Applications | 微分技巧及其应用
tions. The power rule, d/dx (xⁿ) = n xⁿ⁻¹, must be automatic, as must the derivatives of eˣ whether a function is increasing or decreasing, and classify stationary points as local maxima, minima, or points of inflection using the first or second derivative tests. Optimisation problems that involve forming a function from a geometric or physical context and then finding its maximum or minimum are a hallmark of IB differentiation questions.
复习集10C通常包含多项式、有理数次幂、指数函数和基本三角函数的微分。幂法则 d/dx (xⁿ) = n xⁿ⁻¹ 必须像条件反射一样熟练，eˣ、ln x、sin x 和 cos x 的导数也同样如此。除了计算导数，你还应能够求出给定点处切线和法线的方程，判断函数是递增还是递减，并利用一阶导数判别法或二阶导数判别法将驻点区分为局部极大值、局部极小值或拐点。从几何或物理情境中构造出一个函数，再求其最大值或最小值的优化问题，是IB微分题的标志性题型。
- The product rule (u v)' = u'v + u v' and quotient rule (u/v)' = (u'v − u v') / v² must be committed to memory.
- 乘积法则 (u v)' = u'v + u v' 和商法则 (u/v)' = (u'v − u v') / v² 必须牢牢记住。
- Always simplify your derivative before setting it to zero; this reduces algebraic mistakes when finding stationary points.
- 在令导数为零之前，始终先化简导数表达式；这样在求驻点时能减少代数运算错误。
7. Integration as the Reverse of Differentiation | 作为微分逆运算的积分
returns a family of functions differing by a constant, and you must never forget to write '+ C' your final answer. Definite integration computes the signed area between a curve and the x‑ a), where F is any antiderivative of f. In IB exams, you are also expected to find the area between two curves by integrating the difference of the functions. Be cautious: if the curves cross within the interval, you must split the integral into sub‑intervals where the ordering of the functions remains unchanged.
不定积分返回的是一族仅相差一个常数的函数，你绝不能忘记在最终答案中写上'+ C'。定积分计算的是在闭区间[a, b]上曲线与x轴之间带有符号的面积，其结果为数值。微积分基本定理将这两个概念联系起来：∫ₐᵇ f(x) dx = F(b) − F(a)，其中F是f的任意一个原函数。在IB考试中，你还应能通过对两个函数之差进行积分来求出两曲线之间的面积。要小心：如果曲线在区间内相交，你必须将积分拆分成若干子区间，使得在每个子区间内函数的先后顺序保持不变。
∫ (3x² − 2x + 1) dx = x³ − x² + x + C
∫ (3x² − 2x + 1) dx = x³ − x² + x + C
8. Probability and Statistical Distributions | 概率与统计分布
oncept of independence. For events A and B, the defining condition for independence is P(A ∩ B) = P(A) × P(B); alternatively, P(A | onvert any normal variable to the standard normal distribution, enabling you to use the provided probability tables or GDC functions. Knowing how to find an unknown mean or standard deviation given a particular probability is a skill that distinguishes higher‑achieving candidates.
复习集10C中的概率问题常结合树形图、条件概率和独立性的概念。对于事件A和B，独立性的定义条件是 P(A ∩ B) = P(A) × P(B)；或者等价地，P(A | B) = P(A)。在处理连续数据时，正态分布扮演着核心角色。你将使用标准化公式 z = (x − μ) / σ 将任意正态变量转换为标准正态分布，从而能够使用所提供的概率表或图形计算器功能。已知某个特定概率，反求未知的均值或标准差的能力是区分高分考生的基本功。
- Draw a clearly labelled tree diagram even if the question does not explicitly ask for one; it reduces confusion in multi‑stage probability problems.
- 即使题目没有明确要求，也要画一个标注清晰的树形图；这能减少多阶段概率问题中的混淆。
- Remember that the total area under a probability density curve equals 1; this is the continuous analogue of probabilities summing to 1 for all possible discrete outcomes.
- 记住概率密度曲线下的总面积等于1；这是所有可能的离散结果概率之和为1在连续情形下的类比。
9. Vectors: Lines and Scalar Product | 向量：直线与标量积
r to a point on the line and b is a direction vector. You must be able to determine whether two lines intersect, are parallel, or are skew (in 3D). The scalar product (dot product) a ∙ b = |a||b| cos θ is used to find the angle between two vectors and to test for perpendicularity: a ⟂ b ⇔ a ∙ calculating the foot of a perpendicular and the shortest distance from a point to a line.
向量几何是IB分析与方法课程中一个独具特色的部分。二维或三维空间中直线的向量方程表示为 r = a + t b，其中a是直线上一点的位置向量，b是方向向量。你必须能够判断两条直线是相交、平行还是（在三维空间中）为异面直线。标量积（点积）a ∙ b = |a||b| cos θ 用于求两个向量之间的夹角以及检验垂直关系：a ⟂ b ⇔ a ∙ b = 0。应用包括计算垂足和点到直线的最短距离。
Angle between vectors: cos θ = (a ∙ b) / (|a| |b|)
向量间的夹角：cos θ = (a ∙ b) / (|a| |b|)
10. Strategic Review and Exam Preparation | 策略性复习与备考
Completing Review Set 10C is not merely about finding the right answers; it is about evaluating your problem‑solving process. For each question you attempt, ask yourself whether the method you chose was the most efficient. IB Mathematics rewards fluency: the ability to select and execute a technique without hesitation. Time yourself on each section and compare your pace against the mark allocation, aiming for roughly one minute per mark on Paper 1 and Paper 2. After marking your work, categorise your errors—conceptual misunderstanding, algebraic slip, misinterpretation of the question, or calculator mishandling—and focus your subsequent revision on the most frequent category.
完成复习集10C不仅仅是找到正确答案；它更在于评估你的解题过程。对于你所尝试的每一道题，问一问自己所选择的方法是否是最有效率的。IB数学奖励的是流畅性：即能够不假思索地选择和执行某种技巧的能力。为每个部分计时，并将你的速度与分数分配进行对比，目标是在试卷一和试卷二上达到每分钟一分的节奏。在批改完你的作答后，将错误归类——概念性误解、代数运算疏忽、误读题目或计算器操作不当——并将后续的复习重点集中在出现频率最高的类别上。
- Revisit definitions: many marks are lost because candidates cannot precisely state the definition of a fundamental concept like the derivative or a vector.
- 重新回顾定义：许多考生因无法精确陈述基本概念（如导数或向量）的定义而丢分。
- Practise writing clear, logical solutions, as IB examiners award method marks for structured reasoning, even if the final answer is incorrect due to a minor slip.
- 练习书写清晰、逻辑严谨的解题过程，因为IB考官即使在你因微小失误而导致最终答案错误时，仍会为结构化的推理过程授予方法分。
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