IB Maths: Last-Minute Revision Notes | IB 数学：考前冲刺笔记

📚 IB Maths: Last-Minute Revision Notes | IB 数学：考前冲刺笔记

As your IB Mathematics exam approaches, a final burst of structured revision can make all the difference. These last-minute notes distill the essential concepts, common pitfalls, and exam strategies for both Analysis & Approaches (AA) and Applications & Interpretation (AI) students, covering SL and HL where relevant. Use them to sharpen your understanding, refine your calculator skills, and walk into the exam hall with confidence.

随着 IB 数学考试的临近，系统化的最后冲刺复习至关重要。这份考前冲刺笔记浓缩了分析与方法（AA）和应用与解释（AI）的核心概念、常见错误及考试策略，适用于 SL 和 HL 考生。用它来强化理解、提升计算器使用技巧，从而自信迎考。

1. Exam Structure & Assessment Objectives | 考试结构与评估目标

Knowing the exact format of your papers is the first step towards effective revision. For AA SL/HL, Paper 1 is a no-calculator paper that tests algebraic fluency and reasoning. For AI, a calculator is always allowed, but the questions are often context-heavy and require interpretation of results. Paper 2 for all routes requires a graphic display calculator (GDC), and HL students must also sit Paper 3, a problem-solving paper with extended reasoning. The assessment objectives are: Knowledge & Understanding (20-30%), Problem-solving (25-35%), Communication & Interpretation (15-25%), and Use of Technology (15-25%). In last-minute revision, aim to hit all four by explaining your steps aloud and checking your GDC logs.

清楚试卷格式是高效复习的第一步。AA SL/HL 试卷1为无计算器试卷，考验代数熟练度和推理能力。AI 试卷1允许使用计算器，但题目通常侧重情景化，需解读结果。所有路径的试卷2都要求使用图形计算器（GDC），HL学生还需完成试卷3，即包含长篇推理的问题解决题。评估目标为：知识与理解（20-30%）、问题解决（25-35%）、交流与解释（15-25%）以及技术使用（15-25%）。冲刺复习时，可通过出声解释步骤和检查 GDC 记录来覆盖全部目标。

2. Algebra & Functions | 代数与函数

The core of any IB maths course lies in algebraic manipulation and function theory. A function f maps a set of inputs (domain) to outputs (range). The inverse function f⁻¹ reverses this mapping, but only if f is one-to-one; use the horizontal line test to verify. Composite functions like f(g(x)) are read from the inside out. Make sure you can state the natural domain of √(x-a), log(x), and rational functions by identifying restrictions (radicand ≥ 0, argument > 0, denominator ≠ 0). For quadratics, the discriminant Δ = b² – 4ac determines the nature of roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.

所有 IB 数学课程的核心都在于代数运算与函数理论。函数 f 将一组输入（定义域）映射为输出（值域）。反函数 f⁻¹ 能逆转这种映射，但前提是 f 为单射；可用水平线检验判定。复合函数如 f(g(x)) 应从内向外读。务必能够通过识别限制条件（被开方数≥0，真数>0，分母≠0）来确定 √(x-a)、log(x) 和有理函数的自然定义域。对于二次式，判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个相异实根，Δ = 0 有一个重根，Δ < 0 无实数根。

Transformations of graphs are tested heavily. The mapping y = af(b(x-c)) + d causes a vertical stretch by factor a, a horizontal stretch by factor 1/b, a horizontal shift by c, and a vertical shift by d. Remember: f(2x) compresses the graph horizontally, contrary to many students’ intuition. Exponential functions aˣ and logarithmic functions logₐ(x) are inverses, with key properties: aˣ × aʸ = aˣ⁺ʸ and logₐ(xy) = logₐx + logₐy. The natural exponential eˣ and its inverse ln x appear in calculus and modelling questions. When solving exponential equations, take logs of both sides or use substitution to create a quadratic in aˣ.

图像变换是高频考点。映射 y = af(b(x-c)) + d 实现对图像进行垂直拉伸 a 倍、水平拉伸 1/b 倍、水平平移 c 单位和垂直平移 d 单位。务必注意：f(2x) 会让图像水平压缩，这与多数人的直觉相反。指数函数 aˣ 与对数函数 logₐ(x) 互为反函数，关键性质有：aˣ × aʸ = aˣ⁺ʸ 以及 logₐ(xy) = logₐx + logₐy。自然指数 eˣ 及其反函数 ln x 频繁出现在微积分和建模题中。解指数方程时，可两边取对数，或用换元法将其转化为关于 aˣ 的二次方程。

3. Trigonometry & Circular Functions | 三角学与圆函数

Switch seamlessly between degrees and radians, especially when using the GDC or finding arc lengths and sector areas. The arc length is rθ, the sector area is ½r²θ, with θ in radians. On the unit circle, cos θ is the x-coordinate and sin θ is the y-coordinate. The Pythagorean identity sin²θ + cos²θ = 1 is essential; HL students must also know derived forms such as 1 + tan²θ = sec²θ. When solving trigonometric equations, first find the reference angle, then use CAST or a graph to locate all solutions within the required interval. Always check that the calculator’s angle mode matches the question.

要能在度与弧度间自如切换，尤其是在使用 GDC 或计算弧长与扇形面积时。弧长公式为 rθ，扇形面积公式为 ½r²θ，其中 θ 单位为弧度。在单位圆中，cos θ 是 x 坐标，sin θ 是 y 坐标。毕达哥拉斯恒等式 sin²θ + cos²θ = 1 是根本；HL 考生还须掌握 1 + tan²θ = sec²θ 等派生形式。解三角方程时，先求出一个参考角，再借助 CAST 法或函数图像找出给定区间内的全部解。务必将计算器角度模式与题目要求一致。

Double angle formulas are a must: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ. The sine and cosine rules are needed for non-right-angled triangles: a/sin A = b/sin B = c/sin C and a² = b² + c² – 2bc cos A. The ambiguous case of the sine rule (SSA) can yield two possible triangles; verify by checking if the sum of angles exceeds 180°. The area formula ½ab sin C calculates the area of any triangle when two sides and the included angle are known.

倍角公式必须掌握：sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ。正弦与余弦定理用于非直角三角形：a/sin A = b/sin B = c/sin C 以及 a² = b² + c² – 2bc cos A。正弦定理的歧义情况（SSA）可能产生两个三角形，可通过验证角度和是否超过180°来判断。当已知两边及其夹角时，面积公式 ½ab sin C 可计算任意三角形的面积。

4. Calculus | 微积分

Differentiation gives the instantaneous rate of change. The power rule: d/dx (xⁿ) = nxⁿ⁻¹. The chain rule: if y = f(g(x)) then dy/dx = f'(g(x)) × g'(x). Product rule: (uv)’ = u’v + uv’. Quotient rule: (u/v)’ = (u’v – uv’)/v². HL students must be fluent in differentiating implicitly and with related rates. The derivative of eˣ is eˣ; the derivative of ln x is 1/x. For trigonometric functions, d/dx sin x = cos x, d/dx cos x = -sin x, and d/dx tan x = sec²x. Always write the derivative in simplest factored form when possible.

微分给出瞬时变化率。幂函数求导法则：d/dx (xⁿ) = nxⁿ⁻¹。链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。乘积法则：(uv)’ = u’v + uv’。商法则：(u/v)’ = (u’v – uv’)/v²。HL 考生必须熟练掌握隐函数微分和相关变化率。eˣ 的导数是 eˣ；ln x 的导数是 1/x。三角函数的导数：d/dx sin x = cos x，d/dx cos x = -sin x，d/dx tan x = sec²x。求导后尽可能写成最简因式分解形式。

Integration reverses differentiation. The indefinite integral ∫ f(x) dx gives a family of functions; never forget the constant of integration +C. For definite integrals, ∫ₐᵇ f(x) dx = F(b) – F(a) where F is an antiderivative. The area between a curve and the x-axis is found by integrating the absolute value, so split the interval where the curve crosses the axis. Volume of revolution about the x-axis is V = π ∫ₐᵇ y² dx. Use u-substitution (and integration by parts for HL) to handle composite integrands. When a question asks for the area enclosed by two curves, integrate the difference of the top and bottom functions (right – left for y-axis).

积分是微分的逆运算。不定积分 ∫ f(x) dx 得到一族函数，千万不要遗漏积分常数 +C。对于定积分，∫ₐᵇ f(x) dx = F(b) – F(a)，其中 F 是一个原函数。曲线与 x 轴之间的面积需对被积函数取绝对值并分段积分，需在曲线穿越 x 轴处拆分区间。绕 x 轴旋转体的体积公式为 V = π ∫ₐᵇ y² dx。用换元积分法（HL 还需分部积分）处理复合被积函数。题目若要求两条曲线围成的面积，就用上曲线减下曲线（对 y 轴则是右曲线减左曲线）做定积分。

Kinematics links calculus to motion: displacement s(t), velocity v = ds/dt, acceleration a = dv/dt = d²s/dt². Total distance travelled is the integral of speed |v|, not velocity. HL calculus further covers Maclaurin series, first-order differential equations with separation of variables, and Euler’s method for approximation. For the Maclaurin series, the nth term uses the nth derivative evaluated at 0 divided by n!, so memorise the series for eˣ, sin x, cos x, and ln(1+x).

运动学将微积分与运动联系起来：位移 s(t)，速度 v = ds/dt，加速度 a = dv/dt = d²s/dt²。总路程是速率 |v| 的积分，而非速度的积分。HL 微积分还涵盖麦克劳林级数、一阶微分方程的分离变量法以及欧拉近似法。麦克劳林级数的第 n 项用到 0 处的第 n 阶导数除以 n!，因此要熟记 eˣ、sin x、cos x 和 ln(1+x) 的级数展开式。

5. Statistics & Probability | 统计与概率

Probability calculations begin with the fundamental principle: P(A) = n(A)/n(U) for equally likely outcomes. Venn diagrams and tree diagrams help organise compound events. For conditional probability, P(A|B) = P(A ∩ B)/P(B). Check if events are independent by testing P(A ∩ B) = P(A)P(B) or P(A|B) = P(A). Bayes’ theorem, P(A|B) = [P(B|A)P(A)]/P(B), often appears in AI papers and HL problem-solving. When tackling probability with two or more stages, a tree diagram with probabilities on the branches and multiplication along the paths is the safest approach.

概率计算始于等可能结果的基本原理：P(A) = n(A)/n(U)。维恩图和树状图有助于整理复合事件。条件概率公式为 P(A|B) = P(A ∩ B)/P(B)。可通过检验 P(A ∩ B) = P(A)P(B) 或 P(A|B) = P(A) 来判断事件是否独立。贝叶斯定理 P(A|B) = [P(B|A)P(A)]/P(B) 常出现在 AI 试卷和 HL 的问题解决题中。处理多阶段概率时，使用在分支上标注概率、沿路径相乘的树状图最为稳妥。

For discrete random variables, the expected value E(X) = Σ x·P(X=x). The variance Var(X) = E(X²) – [E(X)]². The binomial distribution B(n, p) models the number of successes in n independent trials; its mean is np, variance is np(1-p). The normal distribution N(μ, σ²) requires standardising to Z ∼ N(0,1) using Z = (X – μ)/σ. Use the GDC’s Normal CDF function to find probabilities directly, rather than consulting tables. Know that for a sample mean, the distribution is X̄ ∼ N(μ, σ²/n) by the Central Limit Theorem, which is essential for hypothesis testing in AI. Always interpret the p-value or critical region in context.

对于离散随机变量，期望值 E(X) = Σ x·P(X=x)。方差 Var(X) = E(X²) – [E(X)]²。二项分布 B(n, p) 模拟 n 次独立试验中的成功次数；其均值为 np，方差为 np(1-p)。正态分布 N(μ, σ²) 需通过 Z = (X – μ)/σ 标准化为 Z ∼ N(0,1)。直接用 GDC 的 Normal CDF 功能求概率，无需查阅表格。根据中心极限定理，样本均值的分布为 X̄ ∼ N(μ, σ²/n)，这对 AI 中的假设检验至关重要。务必结合语境解读 p 值或拒绝域。

6. Geometry & Vectors | 几何与向量

Coordinate geometry in 2D and 3D relies on the distance formula, midpoint formula, and gradients. The equation of a straight line can be expressed in vector form: r = a + λb, where a is a position vector and b is the direction vector. In 3D, the scalar (dot) product a·b = |a||b|cos θ is used to find the angle between vectors or to determine orthogonality (a·b = 0). HL students need the vector (cross) product a×b, which gives a vector perpendicular to both a and b, with magnitude |a||b|sin θ. Vector product is essential for finding normals to planes and areas of parallelograms. The Cartesian equation of a plane is ax + by + cz = d, where (a,b,c) is a normal vector.

二维和三维的坐标几何依赖于距离公式、中点公式和斜率。直线的方程可用向量形式表示：r = a + λb，其中 a 是位置向量，b 是方向向量。在三维中，数量积（点积）a·b = |a||b|cos θ 用于求向量夹角或判断正交性（a·b = 0）。HL 考生需要掌握向量积（叉积）a×b，它得出一个同时垂直于 a 和 b 的向量，模为 |a||b|sin θ。向量积对求平面的法向量以及平行四边形面积至关重要。平面的笛卡尔方程为 ax + by + cz = d，其中 (a,b,c) 是法向量。

When solving vector intersection problems, set the parametric equations equal and solve the system. For two lines in 3D, they may be intersecting, parallel, or skew (not parallel and not intersecting). To find the shortest distance from a point to a line or plane, use projection formulas or the formula d = |(AP)·n| / |n| for a plane. In complex numbers (HL only), represent numbers as a + bi or in polar form r(cos θ + i sin θ); be comfortable with De Moivre’s theorem and using complex numbers in polynomial roots.

解决向量交点问题时，令参数方程相等并求解方程组。对于三维空间中的两条直线，它们可能相交、平行或为异面直线（不平行且不相交）。求点到直线或平面的最短距离时，使用投影公式或平面距离公式 d = |(AP)·n| / |n|。复数（仅 HL）可用 a + bi 或极坐标形式 r(cos θ + i sin θ) 表示；要能熟练运用棣莫弗定理以及复数在多项式求根中的应用。

7. Key Formulae to Memorise | 必记公式

While the IB formula booklet provides many equations, you still need to know the ones that aren’t listed or are quicker to recall from memory. Here is a table of high-priority items:

虽然 IB 公式手册给出了许多公式，但有些要么不在册内，要么记忆下来对做题速度至关重要。以下表格列出了高优先级项目：

Formula / 公式	Use / 用途
x = [-b ± √(b² – 4ac)] / 2a	Quadratic formula / 二次方程求根公式
aˣ × aʸ = aˣ⁺ʸ	Exponent law / 指数律
logₐ(xy) = logₐx + logₐy	Log property / 对数性质
sin²θ + cos²θ = 1	Pythagorean identity / 毕达哥拉斯恒等式
Δ = b² – 4ac	Discriminant / 判别式
sin 2θ = 2 sin θ cos θ	Double angle / 倍角公式
V = π ∫ₐᵇ y² dx	Volume of revolution / 旋转体体积
Z = (X – μ)/σ	Standardisation / 标准化

8. Common Mistakes & How to Avoid Them | 常见错误及避免方法

Many marks are lost through avoidable errors. (1) Forgetting the constant of integration +C is the classic; every indefinite integral must end with +C. (2) Misusing the chain rule, especially with fractional or negative exponents—rewrite the function before differentiating. (3) Confusing radians and degrees: when plotting trig functions or evaluating derivatives of trig functions, the angle must be in radians. (4) Solving inequalities by multiplying by a negative without flipping the sign. (5) Cancelling terms incorrectly in rational expressions: you can only cancel factors, not terms. (6) For the binomial distribution, forgetting that p + q = 1 and using the wrong number of trials. (7) Missing the ambiguous case in the sine rule. A simple self-check can catch these: after solving, substitute the answer back or do a quick mental estimate.

许多失分源于可避免的错误。（1）遗漏积分常数 +C 是经典问题，每个不定积分都必须以 +C 结尾。（2）误用链式法则，尤其是对分数或负指数的情况——求导前先将函数重写。（3）混淆弧度与度数：画三角函数的图像或求三角函数的导数时，角度必须用弧度。（4）解不等式时两边同乘负数却没有反转不等号。（5）在有理式中非法消去项——只能消去公因式，不能消去随意项。（6）二项分布中忘记 p + q = 1，或错误设定试验次数。（7）遗漏正弦定理的歧义情况。简单的自我检查即可避免这些错误：解完题后将答案代回或快速心算估计。

9. Calculator Skills & Technology | 计算器与科技使用技巧

Your GDC is a powerful ally, but only if you know its capabilities. Practice the following before the exam: graphing functions and using the trace/analyse features to find intersections, zeros, and turning points; solving equations numerically (the solver function); computing definite integrals and derivatives at a point; and performing statistical calculations (1-Var Stats, 2-Var Stats, linear regression). For HL AI, master the GDC’s matrix operations, Chi-squared test, t-test, and confidence intervals. In AA Paper 1, you must rely on algebraic manipulation alone, so avoid calculator dependency. A common trap is trusting the GDC’s default window when a graph has asymptotic behaviour—always adjust the window or use zoom-fit carefully.

你的 GDC 是强大的伙伴，但前提是你要熟知其功能。考前务必练习以下操作：绘制函数图像并使用追踪/分析功能寻找交点、零点和极值点；使用求解器数值解方程；计算定积分和某点导数；执行统计计算（单变量统计、双变量统计、线性回归）。HL AI 考生还需掌握 GDC 的矩阵运算、卡方检验、t 检验和置信区间。在 AA 试卷1中，你只能依靠代数运算，所以要避免对计算器的依赖。一个常见陷阱是当函数具有渐近行为时，直接信任 GDC 的默认窗口——务必调整窗口范围或谨慎使用 zoom-fit。

10. Exam Day Strategy & Time Management | 考试日策略与时间管理

Read the entire paper in the first five minutes. Identify the questions you find easiest and tackle them first to build confidence and secure marks. Allocate time proportionally to the marks: for a 90-minute, 90-mark paper, that’s roughly one minute per mark. If you get stuck on a part, draw a diagram, list what you know, and move on after a couple of minutes—you can return later. Leave 10 minutes at the end for checking. Use a ruler for graphs, label axes with units and scales, and always write the general solution before simplifying. For ‘show that’ questions, work to the given result step by step, and if you can’t reach it, at least present your working for partial credit. Stay calm: deep breathing can reset your focus if anxiety strikes.

最初五分钟通读整份试卷。找出你认为最简单的题目，优先作答以建立信心并确保得分。按照分值分配时间：90分钟90分的试卷大致为每分钟1分。若在某小题上卡住，可画出示意图，列出已知信息，几分钟后先跳过去——后面再回来。最后留出10分钟检查。作图要用尺规，坐标轴标注单位和尺度，并且先写出通解再化简。对于“证明”题，要逐步推导至给定结果；即便无法得出，至少要呈现解题步骤以获取部分分数。保持冷静：若感到焦虑，深呼吸可重置专注力。

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