Mathematics Knowledge Explained | 数学知识点精讲

📚 Mathematics Knowledge Explained | 数学知识点精讲

Welcome to this comprehensive revision note covering essential topics in mathematics for international curricula such as A-Level and IB. This article breaks down core concepts, providing clear explanations and key formulas to help you build a deep understanding and approach problems with confidence. Whether you are reviewing for an exam or consolidating your knowledge, these focused summaries will serve as a valuable quick-reference guide.

欢迎阅读这篇全面的数学知识点精讲，涵盖A-Level、IB等国际课程的核心主题。本文将分解基本概念，提供清晰的解释和关键公式，帮助你建立深刻的理解并自信地应对问题。不论你是在备考还是巩固知识，这些重点总结都将成为宝贵的快速参考指南。

1. Algebraic Expressions and Polynomials | 代数表达式与多项式

An algebraic expression combines variables, constants, and operations. A polynomial is a special expression of the form a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀, where n is a non‑negative integer and the coefficients a_i are constants.

代数表达式由变量、常数和运算符号组合而成。多项式是一种特殊表达式，形式为 a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀，其中 n 为非负整数，系数 a_i 为常数。

Factorization reverses expansion. Common techniques include taking out the greatest common factor, recognising the difference of two squares a² – b² = (a + b)(a – b), and factoring quadratic trinomials such as x² + 5x + 6 = (x + 2)(x + 3).

因式分解是展开的逆运算。常用技巧包括提取最大公因式、利用平方差公式 a² – b² = (a + b)(a – b) 以及对二次三项式进行分解，例如 x² + 5x + 6 = (x + 2)(x + 3)。

The Remainder Theorem states that when a polynomial f(x) is divided by (x – a), the remainder is f(a). The Factor Theorem is a direct consequence: (x – a) is a factor if and only if f(a) = 0.

余数定理指出，当多项式 f(x) 除以 (x – a) 时，余数为 f(a)。因式定理是其直接推论：(x – a) 是因式当且仅当 f(a) = 0。

2. Functions and Graphs | 函数与图像

A function f maps each input x from the domain to exactly one output f(x) in the range. The vertical line test checks whether a graph represents a function: any vertical line should intersect the graph at most once.

函数 f 将定义域中的每个输入 x 映射到值域中唯一的一个输出 f(x)。垂直线检验可以用来判断一个图像是否表示函数：任意一条垂直线与图像最多只能有一个交点。

Composite functions combine two functions: (f ∘ g)(x) = f(g(x)). The order matters, as f(g(x)) is generally different from g(f(x)).

复合函数将两个函数组合在一起：(f ∘ g)(x) = f(g(x))。运算顺序很重要，f(g(x)) 通常与 g(f(x)) 不同。

An inverse function f^-1 reverses the effect of f; its graph is the reflection of y = f(x) in the line y = x. To find the inverse, swap x and y and then solve for y.

反函数 f^-1 可以反转 f 的作用；它的图像是 y = f(x) 关于直线 y = x 的反射。求反函数时，只需交换 x 和 y，然后解出 y 即可。

Transformations of graphs include translations (shifting up/down/left/right), stretches (multiplying x- or y-coordinates), and reflections (across axes). For example, y = f(x) + c shifts the graph vertically by c units.

图像变换包括平移（上下左右移动）、伸缩（x 坐标或 y 坐标乘以常数）和反射（关于坐标轴对称）。例如，y = f(x) + c 将图像垂直平移 c 个单位。

3. Trigonometry and Identities | 三角学与恒等式

Angles can be measured in degrees or radians, where π radians = 180°. Radian measure simplifies calculus because the derivative of sin x is cos x only when x is in radians.

角度可以用度或弧度来度量，其中 π 弧度 = 180°。弧度制可简化微积分运算，因为只有当 x 以弧度为单位时，sin x 的导数才是 cos x。

The fundamental trigonometric ratios are sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. On the unit circle, cos θ and sin θ are the x- and y-coordinates of a point.

基本三角比是 sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。在单位圆上，cos θ 和 sin θ 分别是对应点的 x 坐标和 y 坐标。

Key identities include sin²θ + cos²θ = 1, tan θ = sin θ / cos θ, and double-angle formulas such as sin 2θ = 2 sin θ cos θ. These are essential for simplifying expressions and solving equations.

关键恒等式包括 sin²θ + cos²θ = 1，tan θ = sin θ / cos θ，以及倍角公式如 sin 2θ = 2 sin θ cos θ。这些公式在化简表达式和求解方程时必不可少。

The graphs of y = sin x, y = cos x, and y = tan x show periodic behaviour with specific amplitudes, periods, and asymptotes. Sin and cos have period 2π and amplitude 1, while tan has period π and vertical asymptotes at odd multiples of π/2.

y = sin x、y = cos x 和 y = tan x 的图像展现出周期性行为，具有特定的振幅、周期和渐近线。正弦和余弦的周期为 2π，振幅为 1；而正切的周期为 π，并在 π/2 的奇数倍处存在垂直渐近线。

4. Exponentials and Logarithms | 指数与对数

Exponential expressions follow laws: a^m × aⁿ = a^m+n, (a^m)ⁿ = a^mn, and a⁰ = 1. The natural exponential function e^x has the unique property that its derivative is itself.

指数表达式遵循以下法则：a^m × aⁿ = a^m+n，(a^m)ⁿ = a^mn，且 a⁰ = 1。自然指数函数 e^x 有一个独特的性质：它的导数等于它本身。

A logarithm answers the question: to what power must the base be raised to obtain a given number? That is, log_b a = c means b^c = a. The natural logarithm, ln x, has base e.

对数回答这样一个问题：底数需要乘方到多少次才能得到给定的数？即 log_b a = c 意味着 b^c = a。自然对数 ln x 的底数为 e。

Log laws mirror exponent laws: log_b(xy) = log_b x + log_b y, log_b(x/y) = log_b x – log_b y, and log_b(x^k) = k log_b x. The change‑of‑base formula is log_a x = (log_b x) / (log_b a).

对数法则与指数法则相对应：log_b(xy) = log_b x + log_b y，log_b(x/y) = log_b x – log_b y，log_b(x^k) = k log_b x。换底公式为 log_a x = (log_b x) / (log_b a)。

Exponential and logarithmic functions are inverses: e^{ln x} = x for x > 0, and ln(e^x) = x. Their graphs are reflections across the line y = x.

指数函数与对数函数互为反函数：对于 x > 0 有 e^{ln x} = x，且 ln(e^x) = x。它们的图像关于直线 y = x 对称。

5. Differentiation | 微分

The derivative f ‘(x) measures the instantaneous rate of change of a function at a point and gives the gradient of the tangent. It is defined as the limit of the difference quotient: f ‘(x) = lim_h→0 [f(x+h) – f(x)] / h.

导数 f ‘(x) 度量了函数在某点处的瞬时变化率，并给出切线的斜率。它定义为差商的极限：f ‘(x) = lim_h→0 [f(x+h) – f(x)] / h。

d/dx (xⁿ) = n x^n-1

The power rule is the most fundamental: if f(x) = xⁿ then f ‘(x) = n x^n-1. For sums, differentiate term by term.

幂函数求导法则是最基本的：如果 f(x) = xⁿ，那么 f ‘(x) = n x^n-1。对于多项式，可以对每一项分别求导。

The chain rule handles composite functions: if y = f(g(x)), then dy/dx = f ‘(g(x)) × g'(x). The product rule and quotient rule are also essential: (uv)’ = u’v + uv’ and (u/v)’ = (u’v – uv’) / v².

链式法则用于处理复合函数：如果 y = f(g(x))，那么 dy/dx = f ‘(g(x)) × g'(x)。乘法法则和除法法则同样重要：(uv)’ = u’v + uv’，(u/v)’ = (u’v – uv’) / v²。

Derivatives help find stationary points (where f ‘(x) = 0), which can be maxima, minima, or points of inflection, and are used in optimisation problems.

导数有助于寻找驻点（满足 f ‘(x) = 0 的点），这些点可能是极大值、极小值或拐点，并被用于优化问题中。

6. Integration | 积分

Integration is the reverse process of differentiation. The indefinite integral ∫ f(x) dx yields a family of functions F(x) + C, where F ‘(x) = f(x) and C is an arbitrary constant.

积分是微分的逆运算。不定积分 ∫ f(x) dx 得到一族函数 F(x) + C，其中 F ‘(x) = f(x)，C 为任意常数。

∫ xⁿ dx = (xⁿ⁺¹) / (n+1) + C, for n ≠ -1

The definite integral ∫_a^b f(x) dx represents the signed area between the curve and the x‑axis from x = a to x = b. The Fundamental Theorem of Calculus links differentiation and integration: it can be evaluated as F(b) – F(a).

定积分 ∫_a^b f(x) dx 表示曲线与 x 轴之间从 x = a 到 x = b 的有向面积。微积分基本定理将微分与积分联系起来：它可以计算为 F(b) – F(a)。

Basic integration rules mirror those of derivatives, but with the addition of a constant. Common integrals include ∫ e^x dx = e^x + C, ∫ 1/x dx = ln|x| + C, and ∫ cos x dx = sin x + C.

基本的积分法则与导数法则相呼应，但需要加上常数。常见的积分包括 ∫ e^x dx = e^x + C，∫ 1/x dx = ln|x| + C，以及 ∫ cos x dx = sin x + C。

Integration by substitution and integration by parts extend the range of functions that can be integrated, often by recognising a function and its derivative.

换元积分法和分部积分法拓展了可积函数的范围，通常需要识别一个函数及其导数。

7. Sequences and Series | 数列与级数

An arithmetic sequence has a constant difference d between successive terms: the nth term is given by u_n = a + (n – 1)d. The sum of the first n terms is S_n = n/2 [2a + (n – 1)d].

等差数列的相邻两项之间的差 d 为常数：第 n 项公式为 u_n = a + (n – 1)d。前 n 项和为 S_n = n/2 [2a + (n – 1)d]。

A geometric sequence has a constant ratio r between successive terms: u_n = a r^n-1. Its sum for finite n is S_n = a(1 – rⁿ) / (1 – r), provided r ≠ 1.

等比数列的相邻两项之比 r 为常数：u_n = a r^n-1。其有限项求和公式为 S_n = a(1 – rⁿ) / (1 – r)，前提是 r ≠ 1。

An infinite geometric series converges to a/(1 – r) if |r| < 1; otherwise it diverges. This has applications in recurring decimals and finance.

无穷等比级数当 |r| < 1 时收敛于 a/(1 - r)；否则发散。这在循环小数和金融中有应用。

The binomial expansion expresses (1 + x)ⁿ as an infinite series when n is not a positive integer, valid for |x| < 1. For integer n, the expansion is finite and uses binomial coefficients.

二项式展开可将 (1 + x)ⁿ 表示为一个无穷级数，当 n 不是正整数时，该展开在 |x| < 1 时有效。对于整数 n，展开式为有限项，并使用二项式系数。

8. Vectors | 向量

A vector has both magnitude and direction, often written in column form or using unit vectors i, j, k. The magnitude of v = a i + b j is √(a² + b²).

向量具有大小和方向，通常以列向量形式或使用单位向量 i、j、k 表示。向量 v = a i + b j 的大小为 √(a² + b²)。

The scalar (dot) product of two vectors a and b is a · b = |a||b| cos θ, where θ is the angle between them. It equals a₁b₁ + a₂b₂ + a₃b₃ in component form.

两个向量 a 与 b 的标量积（点积）为 a · b = |a||b| cos θ，其中 θ 为它们之间的夹角。在分量形式下，它等于 a₁b₁ + a₂b₂ + a₃b₃。

If a · b = 0 and both vectors are non‑zero, they are perpendicular. The dot product is used to find projections and the angle between vectors.

如果 a · b = 0 且两个向量均非零，则它们垂直。点积可用于求投影和向量间的夹角。

The vector equation of a line is r = a + t b, where a is a fixed position vector on the line, b is a direction vector, and t is a scalar parameter.

直线的向量方程为 r = a + t b，其中 a 是直线上一个固定的位置向量，b 是方向向量，t 为标量参数。

9. Probability | 概率

Probability measures the likelihood of events on a scale from 0 (impossible) to 1 (certain). The sum of probabilities of all mutually exclusive outcomes in a sample space is 1.

概率度量事件发生的可能性，取值范围从 0（不可能）到 1（必然）。样本空间中所有互斥结果的概率之和为 1。

For two events A and B, the addition rule is P(A or B) = P(A) + P(B) – P(A and B). If events are mutually exclusive, P(A and B) = 0.

对于两个事件 A 和 B，加法法则为 P(A 或 B) = P(A) + P(B) – P(A 且 B)。若事件互斥，则 P(A 且 B) = 0。

Conditional probability P(A|B) is the probability of A given B, defined as P(A and B) / P(B). Events A and B are independent if P(A and B) = P(A)P(B), which also implies P(A|B) = P(A).

条件概率 P(A|B) 是在 B 发生的条件下 A 发生的概率，定义为 P(A 且 B) / P(B)。若 P(A 且 B) = P(A)P(B)，则事件 A 与 B 独立，此时亦有 P(A|B) = P(A)。

Tree diagrams are helpful for multistage experiments, allowing probabilities to be multiplied along branches and added for combined outcomes.

树形图对于多阶段试验很有帮助，可沿分支将概率相乘，并将组合结果的概率相加。

10. Statistics | 统计

Measures of central tendency include the mean, median, and mode. For a data set x₁, x₂, …, x_n, the mean is Σx_i / n. The median is the middle value when data are ordered.

集中趋势的度量包括平均数、中位数和众数。对于数据集 x₁, x₂, …, x_n，平均数为 Σx_i / n。中位数是数据排序后位于中间的值。

Variance measures the spread of data: σ² = Σ(x_i – μ)² / n for a population, and sample variance uses n – 1 in the denominator. The standard deviation σ is the square root of variance.

方差衡量数据的离散程度：总体方差 σ² = Σ(x_i – μ)² / n，样本方差的分母则用 n – 1。标准差 σ 是方差的平方根。

The normal distribution is a symmetric, bell‑shaped curve defined by its mean μ and standard deviation σ. About 68% of data lie within 1σ of the mean, and 95% within 2σ.

正态分布是一种对称的钟形曲线，由其均值 μ 和标准差 σ 确定。大约 68% 的数据落在均值 ± 1σ 范围内，95% 落在 ± 2σ 范围内。

Standardising using z = (x – μ) / σ allows probabilities to be found from the standard normal distribution tables, a key technique for solving problems involving normally distributed variables.

通过 z = (x – μ) / σ 进行标准化，可以利用标准正态分布表查找概率，这是解决涉及正态分布变量问题的关键技术。

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Mathematics Knowledge Explained | 数学知识点精讲