📚 Pdfjoiner (4)-148: Edexcel A-Level Mathematics Revision Notes | Pdfjoiner (4)-148：Edexcel A-Level 数学复习笔记
This document, taken from a compiled PDF labelled 'pdfjoiner_(4)-148' blem sets and key theoretical concepts that appear repeatedly in the Pure and Applied papers. Throughout the notes, you will find step-by-step examples, essential formulas, and tips to avoid common pitfalls. The aim is to strengthen your understanding of functions, calculus, trigonometry, vectors, and their interconnections – all of which are central to success in the Edexcel examinations.
本文选自一份名为“pdfjoiner_(4)-148”的合并PDF文件，为备考Edexcel A-Level 数学的同学提供了一份精简而全面的复习指南。内容涵盖纯数和应用数学试卷中反复出现的典型习题和核心理论概念。在每个要点中，你都会找到分步例题、关键公式以及避免常见错误的技巧。目标是帮助你加深对函数、微积分、三角学、向量及其相互关系的理解——这些都是Edexcel 考试中取得好成绩的关键。
1. Algebraic Fractions and Polynomial Division | 代数分式与多项式除法
actor theorem: if f(a) = 0, then (x − a) is a factor. To divide a cubic or quartic by a linear factor, either apply long division or equate coefficients after splitting into partial fractions. Watch out for improper fractions where the degree of the numerator equals or exceeds that of the denominator – these must first be converted into a mixed polynomial form.
在化简代数分式之前，务必对分子和分母进行彻底因式分解。对于高次多项式，利用因式定理：若 f(a) = 0，则 (x − a) 为一个因式。要用线性因式除以三次或四次多项式，可采用长除法，或在拆分为部分分式后比较系数。务必注意分子次数大于或等于分母次数的假分式——这类分式必须首先转化为多项式带真分式的混合形式。
− x − 6) = (x − 1)(x − 3)(x + 2). The original fraction becomes (x − 3)(x + 2) for x ≠ 1.
例题：化简 (x³ − 2x² − 5x + 6) / (x − 1)。首先代入 x = 1：1 − 2 − 5 + 6 = 0，故 x − 1 为因式。做除法得 (x − 1)(x² − x − 6) = (x − 1)(x − 3)(x + 2)。原分式在 x ≠ 1 时可化简为 (x − 3)(x + 2)。
2. Exponential Functions and Logarithms | 指数函数与对数
The natural exponential function y = eˣ has the unique property that its derivative is itself. For any base a > 0, aˣ = eˣˡ y models, the differential equation dx/dt = kx leads to x = A eᵏᵗ, where the sign of k determines growth (k > 0) or decay (k < 0).
自然指数函数 y = eˣ 具有导数等于自身的独特性质。对于任意底数 a > 0，有 aˣ = eˣˡⁿᵃ。解指数方程时，两边取自然对数，牢记 ln(aᵇ) = b·ln(a)。处理增长与衰减模型时，微分方程 dx/dt = kx 的解为 x = A eᵏᵗ，其中 k 的正负决定增长 (k > 0) 或衰减 (k < 0)。
en solving equations like ln(3x − 2) = ln(x + 4); you must check that the arguments are positive for the solution to be valid.
对数运算中，换底公式至关重要：logₐ b = ln b / ln a。同时牢记 ln 1 = 0，ln e = 1，且 ln x 的定义域为 x > 0。解方程如 ln(3x − 2) = ln(x + 4) 时需格外小心；必须验证解使得各对数参数为正，方为有效。
3. Trigonometric Identities and Equations | 三角恒等式与方程
Edexcel regularly tests the three core identities: sin²θ + cos²θ ≡ 1; 1 + tan²θ ≡ sec²θ; and 1 + cot²θ ≡ for solving equations but also for later integration problems, such as finding ∫ sin²x dx.
Edexcel 试卷经常考查三个核心恒等式：sin²θ + cos²θ ≡ 1；1 + tan²θ ≡ sec²θ；以及 1 + cot²θ ≡ cosec²θ。你还必须掌握倍角公式：sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ。这些公式不仅对解方程至关重要，对后续积分问题（例如求 ∫ sin²x dx）也同样不可或缺。
When solving trigonometric equations within a given interval, first find the principal value using the inverse function, then sketch the CAST diagram or the graph to locate all solutions. Express solutions in radians unless degrees are specified. Always check for extraneous solutions introduced by squaring both sides.
在给定区间内求解三角方程时，先用反三角函数求出主值，再借助 CAST 图或函数图像定位所有解。除非题目明确要求度数，否则一律以弧度表示解。务必检查两边平方是否引入了增根。
4. Differentiation: First Principles and Basic Rules | 微分：定义与基本规则
ons like f(x) = x². Key differentiation rules: power rule d/dx (xⁿ) = n xⁿ⁻¹; constant multiple rule; and the sum rule. For exponential and trigonometric functions: d/dx (eˣ)= eˣ, d/dx (sin x)= cos x, d/dx (cos x)= −sin x.
由第一性原理定义的导数为 f′(x) = lim(h→0) [f(x + h) − f(x)] / h。Edexcel 可能会要求你将此定义应用于诸如 f(x) = x² 这样的简单函数。关键的微分法则：幂法则 d/dx (xⁿ) = n xⁿ⁻¹；常数倍法则；以及和差法则。对于指数函数与三角函数：d/dx (eˣ)= eˣ，d/dx (sin x)= cos x，d/dx (cos x)= −sin x。
simplify your final derivative where possible, as marks are often reserved for the simplest form.
要熟练运用乘积法则：若 y = u v，则 dy/dx = u dv/dx + v du/dx。对 y = u/v 使用商法则时，dy/dx = (v du/dx − u dv/dx) / v²。尽可能将最终导数简化至最简形式，因为评分标准常对简化后的结果单独设分。
5. Applications of Differentiation: Tangents, Normals, and Stationary Points | 微分的应用：切线、法线与驻点
l, use the negative reciprocal gradient −1/m. In both cases, exact values are preferred unless a decimal approximation is specifically requested.
要求曲线在点 (a, f(a)) 处的切线方程，只需计算梯度 m = f′(a)。切线方程为 y − f(a) = m (x − a)。法线则使用负倒数梯度 −1/m。两种情况均应优先给出精确值，除非题目明确要求取小数近似。
local maximum; if f″(x) = 0, you must examine the gradient sign on either side or use higher derivatives. For modelling problems, always remember to state the nature of the stationary point in context (e.g., maximum volume).
驻点出现在 f′(x) = 0 处。利用二阶导数判别其性质：若 f″(x) > 0，该点为局部极小值点；若 f″(x) < 0，则为局部极大值点；若 f″(x) = 0，则需检查该点两侧梯度的符号或利用更高阶导数。对于建模问题，务必结合情境说明驻点的性质（例如最大体积）。
6. Integration: Indefinite and Definite Integrals | 积分：不定积分与定积分
F(x) + C, where F′(x) = f(x). Basic results include ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln|x| n x dx = −cos x + C. Always remember the constant of integration for indefinite integrals.
积分是微分的逆运算。不定积分 ∫ f(x) dx = F(x) + C，其中 F′(x) = f(x)。基本积分公式包括 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)，∫ eˣ dx = eˣ + C，∫ 1/x dx = ln|x| + C，∫ cos x dx = sin x + C，以及 ∫ sin x dx = −cos x + C。对不定积分，切勿遗漏积分常数 C。
l; areas between two curves require ∫ (upper − lower) dx. Pay close attention to intervals where the function crosses the axis, as you may need to split the integral to avoid negative areas cancelling out positive ones, unless finding the signed area.
定积分利用微积分基本定理：∫ₐᵇ f(x) dx = F(b) − F(a)。曲线与 x 轴围成的面积直接由积分给出；两曲线间的面积则需计算 ∫ (上曲线 − 下曲线) dx。务必留意函数穿过 x 轴的区间，此时可能需要拆分积分，以免负面积抵消正面积——除非题目所求正是有符号的面积。
7. Parametric Equations and Implicit Differentiation | 参数方程与隐函数微分
= d/dx (dy/dx) = d/dt (dy/dx) ÷ dx/dt. Typical Edexcel questions ask for equations of tangents and normals to parametric curves; many involve simple trigonometric or polynomial forms.
当一条曲线由参数方程 x = f(t), y = g(t) 定义时，其梯度为 dy/dx = (dy/dt) / (dx/dt)。要求二阶导数，则用 d²y/dx² = d/dx (dy/dx) = d/dt (dy/dx) ÷ dx/dt。典型的 Edexcel 考题会要求求解参数曲线上某点的切线或法线方程；许多题目涉及简单的三角函数或多项式形式。
ect to x, treating y as an implicit function and applying the chain rule to terms involving y, e.g., d/dx (y²) = 2y dy/dx. Afterwards, rearrange to find dy/dx. This technique also appears in connected rates of change problems.
当 y 难以显式表达为 x 的函数时，就需要使用隐函数微分法。对等式两边同时关于 x 求导，将 y 视为隐函数，并对含 y 的项应用链式法则，例如 d/dx (y²) = 2y dy/dx。之后整理方程求出 dy/dx。这一技巧也会出现在相关变化率问题中。
8. Vectors: Fundamentals and Applications | 向量：基础与应用
In Edexcel' s), as a position vector a = xi + yj + zk, or as a directed line segment. The magnitude is |a| = √x² + y² + z². The scalar product a · b = |a||b| cos θ is essential for finding the angle between two vectors.
在 Edexcel 的纯数内容中，向量主要学习三维情形。向量可用列向量形式（i, j, k 分量）、位向量 a = xi + yj + zk 或有向线段表示。其模长为 |a| = √x² + y² + z²。数量积 a · b = |a||b| cos θ 是求两向量夹角的核心工具。
d the angle between two vectors, rearrange the scalar product formula: cos θ = (a · b) / (|a||b| is a direction vector. Solving intersection problems requires setting up simultaneous equations in the parameters.
求两向量夹角时，将数量积公式变形：cos θ = (a · b) / (|a||b|)。若两向量平行，则 a = λ b；若垂直，则 a · b = 0。直线的向量方程写作 r = a + λ d，其中 a 为直线上一点，d 为方向向量。解决直线交点问题时，需要针对参数建立并求解联立方程组。
form the backbone of the Edexcel A-Level Mathematics specification. Return to the original 'pdfjoiner_(4)-148' file to attempt targeted exercises after mastering these notes.
诸如统计分布与假设检验等其他主题同样重要，但上述纯数技巧构成了 Edexcel A-Level 数学考纲的骨架。请在掌握这些笔记后，回归原始的“pdfjoiner_(4)-148”文件尝试针对性练习。
Published by TutorHao | 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