📚 IB Math SL Cambridge Core Revision: Key Concepts Explained | IB数学SL剑桥知识点精讲
Welcome to this comprehensive revision guide covering the essential topics from the IB Mathematics SL syllabus, as structured by the Cambridge coursebook. This article breaks down the most critical concepts, providing clear explanations, examples, and tips to help you master each area before the examination. Whether you are reviewing algebraic techniques, honing your differentiation skills, or solidifying your understanding of probability, this guide serves as your one-stop resource.
欢迎阅读这份全面的复习指南，涵盖IB数学SL课程的大纲核心内容，以剑桥教材为框架。本文梳理了最关键的知识点，提供清晰的解释、示例和技巧，帮助你在考试前掌握每一个领域。无论你是在复习代数技巧、磨练微分能力，还是巩固对概率的理解，本文都将是你的一站式复习资源。
1. Algebraic Fundamentals | 代数基础
Mastering algebraic manipulation is the bedrock of success in IB Math SL. You must be comfortable expanding products of binomials, factorising quadratic expressions, and simplifying complex rational expressions. Pay special attention to completing the square, as it reappears in functions, quadratics, and integration. Always check for common factors before employing more advanced methods, and practice manipulating expressions with negative and fractional indices to build fluency.
掌握代数运算是IB数学SL成功的基石。你必须熟练展开二项式乘积、因式分解二次表达式以及化简复杂的有理式。要特别注意配方法，因为它会在函数、二次式和积分中反复出现。在使用更高级的方法之前，务必检查公因式，并练习处理带负指数和分数指数的表达式，以培养流畅的运算能力。
When working with sequences and series, distinguish clearly between arithmetic and geometric progressions. For an arithmetic sequence, the n-th term is given by uₙ = u₁ + (n-1)d and the sum of the first n terms by Sₙ = n/2 (2u₁ + (n-1)d) or Sₙ = n/2 (u₁ + uₙ). For geometric sequences, the n-th term is uₙ = u₁ rⁿ⁻¹ and the sum of a finite series is Sₙ = u₁(1 – rⁿ)/(1 – r), provided r ≠ 1. Understand the concept of convergent infinite geometric series: if |r| < 1, S∞ = u₁/(1 - r).
在处理数列与级数时，要清楚区分等差数列和等比数列。对于等差数列，第n项公式为 uₙ = u₁ + (n-1)d，前n项和为 Sₙ = n/2 (2u₁ + (n-1)d) 或 Sₙ = n/2 (u₁ + uₙ)。对于等比数列，第n项为 uₙ = u₁ rⁿ⁻¹，有限项级数和为 Sₙ = u₁(1 – rⁿ)/(1 – r)，其中 r ≠ 1。理解收敛的无穷等比级数的概念：如果 |r| < 1，则 S∞ = u₁/(1 - r)。
2. Functions and Equations | 函数与方程
The concept of a function underpins almost every topic in the course. Know the standard notation f(x) and be able to evaluate functions, find their domains and ranges, and construct composite functions (f∘g)(x) = f(g(x)). An inverse function f⁻¹(x) exists only if f is one-to-one; you must be able to find the inverse by swapping x and y and solving for y. The relationship f(f⁻¹(x)) = f⁻¹(f(x)) = x is frequently tested.
函数的概念几乎贯穿课程所有主题。要熟悉标准记号 f(x)，能够求函数值、找出定义域和值域，并构造复合函数 (f∘g)(x) = f(g(x))。反函数 f⁻¹(x) 仅当 f 是一一映射时才存在；你必须能够通过交换 x 和 y 并解出 y 来求得反函数。关系式 f(f⁻¹(x)) = f⁻¹(f(x)) = x 经常出现在考题中。
Solving equations often involves moving between different representations of the same function. For quadratics, you can factorise, use the quadratic formula x = (-b ± √(b² – 4ac))/(2a), or apply completing the square. The discriminant Δ = b² – 4ac determines the nature of roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots. Graphing calculators are allowed in IB exams, but you must still know how to solve systems of linear equations algebraically (substitution, elimination).
求解方程往往需要在同一函数的不同表示形式之间转换。对于二次方程，你可以因式分解、使用求根公式 x = (-b ± √(b² – 4ac))/(2a)，或运用配方法。判别式 Δ = b² – 4ac 决定了根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重实根，Δ < 0 没有实根。IB考试允许使用图形计算器，但你仍须掌握用代数方法解线性方程组（代入法、消元法）。
3. Quadratic Functions and Graphs | 二次函数与图像
Parabolas are central to the functions topic. The graph of a quadratic function y = ax² + bx + c can be expressed in vertex form y = a(x – h)² + k, where (h, k) is the vertex. The sign of a determines whether the parabola opens upwards (a > 0) or downwards (a < 0). The axis of symmetry is the vertical line x = h. You should be able to find intercepts, sketch graphs showing key features, and interpret quadratic models in real-world contexts such as projectile motion.
抛物线是函数主题的核心。二次函数 y = ax² + bx + c 的图像可以表示为顶点式 y = a(x – h)² + k，其中 (h, k) 为顶点。a 的符号决定抛物线开口向上 (a > 0) 还是向下 (a < 0)。对称轴是垂直线 x = h。你应能找出截距、画出体现关键特征的草图，并能解释现实情境中的二次模型，如抛体运动。
Transformations of graphs are tested extensively. Know the effects of f(x) + d (vertical translation), f(x + c) (horizontal translation), a f(x) (vertical stretch by factor a), and f(bx) (horizontal stretch by factor 1/|b|). Reflections: -f(x) reflects in the x-axis, f(-x) reflects in the y-axis. Be meticulous in applying transformations in the correct order, particularly when both horizontal and vertical changes are present.
图像的变换是考试重点。要了解 f(x) + d（垂直平移）、f(x + c)（水平平移）、a f(x)（垂直拉伸 a 倍）以及 f(bx)（水平拉伸 1/|b| 倍）的影响。对称变换：-f(x) 关于 x 轴对称，f(-x) 关于 y 轴对称。在同时出现水平和垂直变化时，需格外注意按正确顺序施加变换。
4. Exponentials and Logarithms | 指数与对数
Exponential functions of the form f(x) = aˣ (with a > 0, a ≠ 1) model growth and decay. The base e ≈ 2.718 appears frequently, and you must be comfortable with the natural exponential function eˣ and its inverse, the natural logarithm ln x. Key properties: ln(eˣ) = x for all x, and e^(ln x) = x for x > 0. Change of base formula allows log_b (x) = ln x / ln b.
指数函数 f(x) = aˣ（a > 0，a ≠ 1）用于模拟增长与衰减。底数 e ≈ 2.718 频繁出现，你必须熟练掌握自然指数函数 eˣ 及其反函数自然对数 ln x。关键性质：对所有 x，ln(eˣ) = x；对 x > 0，e^(ln x) = x。换底公式为 log_b (x) = ln x / ln b。
Solving exponential equations often requires taking logarithms on both sides. For example, to solve 3ˣ = 5, take ln: x ln 3 = ln 5 ⇒ x = ln 5 / ln 3. Logarithmic equations benefit from using the laws: log (AB) = log A + log B, log (A/B) = log A – log B, and log (Aⁿ) = n log A. Watch out for extraneous solutions when the variable appears inside a logarithm.
解指数方程常常需要两边取对数。例如，解 3ˣ = 5，取自然对数：x ln 3 = ln 5 ⇒ x = ln 5 / ln 3。对数方程则利用运算律：log (AB) = log A + log B，log (A/B) = log A – log B，log (Aⁿ) = n log A。当变量出现在对数内部时，注意排除增根。
5. Trigonometry and Circular Functions | 三角学与圆函数
You must know exact values of sin, cos, and tan for key angles (0°, 30°, 45°, 60°, 90° and their radian equivalents). The unit circle approach is essential for extending definitions beyond acute angles. The sine rule a/sin A = b/sin B = c/sin C and cosine rule c² = a² + b² – 2ab cos C are used to solve non-right triangles. Area of a triangle can be found using (1/2)ab sin C.
你必须知道特殊角（0°、30°、45°、60°、90° 及其弧度等价）的正弦、余弦和正切的精确值。单位圆方法对于将定义推广到锐角之外至关重要。正弦定理 a/sin A = b/sin B = c/sin C 和余弦定理 c² = a² + b² – 2ab cos C 用于解非直角三角形。三角形面积可用 (1/2)ab sin C 求得。
Circular functions f(x) = A sin(B(x – C)) + D introduce amplitude |A|, period 2π/|B|, phase shift C, and vertical shift D. Be prepared to sketch sine and cosine graphs, identifying maxima, minima, and intercepts. Solving trigonometric equations within a given interval requires finding the principal value and then using symmetry properties (e.g., sin(π – θ) = sin θ, cos(2π – θ) = cos θ) to obtain all solutions.
圆函数 f(x) = A sin(B(x – C)) + D 引出了振幅 |A|、周期 2π/|B|、相移 C 和垂直位移 D。准备好绘制正弦和余弦图像，标出最大值、最小值和截距。在一定区间内求解三角方程时，需要先求出主值，再利用对称性质（如 sin(π – θ) = sin θ，cos(2π – θ) = cos θ）得到所有解。
6. Vectors in Two and Three Dimensions | 二维与三维向量
Vectors represent quantities with both magnitude and direction. You must be able to add and subtract vectors, multiply by a scalar, and find the magnitude of a vector v = (x, y) using |v| = √(x² + y²). The scalar (dot) product v · w = x₁x₂ + y₁y₂ = |v||w| cos θ, where θ is the angle between them, is crucial for determining perpendicularity (v · w = 0) and angles.
向量代表既有大小又有方向的量。你必须能够进行向量的加减、标量乘法，并利用 |v| = √(x² + y²) 求向量 v = (x, y) 的模。标量积（点积）v · w = x₁x₂ + y₁y₂ = |v||w| cos θ（θ 为两向量夹角）对于判定垂直（v · w = 0）和求角度至关重要。
In three dimensions, vectors are written as (x, y, z) or column form. The distance between two points A and B is the magnitude of AB vector. Vector equation of a line in 3D is r = a + λ b, where a is a position vector on the line and b is the direction vector. You may be asked to find intersection points, or determine whether two lines are parallel, skew, or intersecting.
在三维空间中，向量写作 (x, y, z) 或列向量形式。两点 A 与 B 之间的距离即向量 AB 的模。三维直线的向量方程为 r = a + λ b，其中 a 为直线上一点的位置向量，b 为方向向量。考题可能会要求找交点，或判断两直线是否平行、异面或相交。
7. Introduction to Differential Calculus | 微分学导论
The derivative f'(x) is the rate of change of a function and gives the gradient of the tangent. From first principles, f'(x) = lim(h→0) [(f(x+h) – f(x))/h]. You must know the power rule: d/dx (xⁿ) = n xⁿ⁻¹, and derivatives of sin x (cos x), cos x (-sin x), eˣ (eˣ), and ln x (1/x). Standard differentiation rules include the sum/difference rule, the product rule, the quotient rule, and the chain rule for composite functions.
导数 f'(x) 是函数的变化率，给出切线的斜率。从第一性原理出发，f'(x) = lim(h→0) [(f(x+h) – f(x))/h]。你必须掌握幂函数求导法则：d/dx (xⁿ) = n xⁿ⁻¹，以及 sin x (cos x)、cos x (-sin x)、eˣ (eˣ)、ln x (1/x) 的导数。标准微分法则包括和/差法则、积法则、商法则以及用于复合函数的链式法则。
Application of derivatives includes finding equations of tangents and normals, determining rates of change, and analysing turning points. The second derivative f”(x) determines concavity: if f”(x) > 0 the graph is concave up (local minimum), and if f”(x) < 0 it is concave down (local maximum). Points of inflection occur where f''(x) = 0 and concavity changes. Optimisation problems involve modelling a situation with a function and finding its maximum or minimum using differentiation.
导数的应用包括求切线和法线方程、确定变化率以及分析驻点。二阶导数 f”(x) 决定凹凸性：若 f”(x) > 0，图像为凹向上（局部极小点）；若 f”(x) < 0，图像为凹向下（局部极大点）。拐点出现在 f''(x) = 0 且凹凸性改变处。优化问题则通过函数建模，并利用微分求出最大值或最小值。
8. Integration and Antiderivatives | 积分与反导数
Integration is the reverse process of differentiation. The indefinite integral ∫ f(x) dx = F(x) + C, where F'(x) = f(x). For basic functions, reverse the power rule: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, n ≠ -1. Also know ∫ sin x dx = -cos x + C and ∫ eˣ dx = eˣ + C. The constant of integration C must always be included unless a definite integral is evaluated.
积分是微分的逆过程。不定积分 ∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)。对于基本函数，反向使用幂法则：∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，n ≠ -1。还要知道 ∫ sin x dx = -cos x + C 与 ∫ eˣ dx = eˣ + C。积分常数 C 必须始终包含，除非计算定积分。
The definite integral ∫[a to b] f(x) dx computes the signed area between the curve and the x-axis. The Fundamental Theorem of Calculus links differentiation and integration: if F is an antiderivative of f on [a, b], then ∫[a to b] f(x) dx = F(b) – F(a). Area between two curves y = f(x) and y = g(x) is found by integrating |f(x) – g(x)| over the interval, taking care to split the integral where the curves cross.
定积分 ∫[a to b] f(x) dx 计算曲线与 x 轴之间的有符号面积。微积分基本定理将微分与积分联系起来：若 F 是 f 在 [a, b] 上的反导数，则 ∫[a to b] f(x) dx = F(b) – F(a)。两曲线 y = f(x) 与 y = g(x) 之间的面积可通过在区间上积分 |f(x) – g(x)| 求得，并注意在曲线相交处将积分分段。
9. Probability and Discrete Distributions | 概率与离散分布
Probability in IB SL covers basic principles: if A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B); otherwise P(A ∪ B) = P(A) + P(B) – P(A ∩ B). For independent events, P(A ∩ B) = P(A)P(B). Conditional probability uses P(B|A) = P(A ∩ B)/P(A). Tree diagrams and Venn diagrams are powerful tools for organising information and solving complex problems.
IB SL 的概率涵盖基本原理：若 A 与 B 互斥，则 P(A ∪ B) = P(A) + P(B)；否则 P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。对于独立事件，P(A ∩ B) = P(A)P(B)。条件概率使用 P(B|A) = P(A ∩ B)/P(A)。树形图和维恩图是整理信息、解决复杂问题的有力工具。
Discrete random variables have probability distributions where ∑ P(X = xᵢ) = 1. The expected value E(X) = ∑ xᵢ P(X = xᵢ) and the variance Var(X) = E(X²) – [E(X)]². The binomial distribution X ~ B(n, p) models the number of successes in n independent trials: P(X = k) = (n choose k) pᵏ (1-p)ⁿ⁻ᵏ. Be ready to find probabilities using technology, but also know the formula for manual calculation of simple cases.
离散随机变量的概率分布满足 ∑ P(X = xᵢ) = 1。期望值 E(X) = ∑ xᵢ P(X = xᵢ)，方差 Var(X) = E(X²) – [E(X)]²。二项分布 X ~ B(n, p) 模拟 n 次独立试验中的成功次数：P(X = k) = (n 选 k) pᵏ (1-p)ⁿ⁻ᵏ。准备好用技术工具求概率，但也要知道简单情况下手动计算的公式。
10. Basic Statistics and the Normal Distribution | 基础统计与正态分布
Descriptive statistics include measures of central tendency (mean, median, mode) and spread (range, interquartile range, variance, standard deviation). For a data set {x₁, x₂, …, xₙ}, the sample variance s² = Σ(xᵢ – x̄)²/(n-1). Box-and-whisker plots display the five-number summary. Outliers are often defined as values more than 1.5 × IQR below Q₁ or above Q₃.
描述性统计包括集中趋势量数（均值、中位数、众数）和离散量数（全距、四分位距、方差、标准差）。对于数据集 {x₁, x₂, …, xₙ}，样本方差 s² = Σ(xᵢ – x̄)²/(n-1)。箱线图展示五数概括。异常值通常定义为低于 Q₁ – 1.5 × IQR 或高于 Q₃ + 1.5 × IQR 的值。
The normal distribution is a continuous probability model with parameters μ (mean) and σ (standard deviation). The curve is bell-shaped and symmetric about μ. Approximations include: 68% of data within 1σ of μ, 95% within 2σ, and 99.7% within 3σ. Standardising to Z = (X – μ)/σ allows use of the standard normal table or calculator. Inverse normal calculations find the value x for a given cumulative probability.
正态分布是一种连续概率模型，参数为 μ（均值）和 σ（标准差）。曲线呈钟形，关于 μ 对称。近似法则：约 68% 数据落在 μ ± σ 内，95% 落在 μ ± 2σ 内，99.7% 落在 μ ± 3σ 内。标准化为 Z = (X – μ)/σ 后，可使用标准正态表或计算器。逆正态计算可用于求给定累积概率下的 x 值。
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