📚 A-Level Edexcel Mathematics: Key Points Summary | 爱德思A-Level数学核心要点总结
This article provides a concise summary of the essential topics covered in the Edexcel A-Level Mathematics syllabus, including pure mathematics, statistics, and mechanics. It is designed as a quick revision guide for students preparing for their exams.
本文简要总结了爱德思A-Level数学大纲涵盖的核心内容，包括纯数学、统计和力学部分，旨在为备考学生提供快速复习指南。
1. Algebraic Techniques and Functions | 代数技巧与函数
is a factor. The remainder theorem: when f(x) is divided by (x-a), the remainder is f(a).
因子定理指出，对于多项式f(x)，若f(a)=0则(x-a)为因子。余式定理：f(x)除以(x-a)的余数为f(a)。
Partial fractions are used to split rational expressions into simpler sums. For distinct linear factors, write as A/(x-a) + B/(x-b). For repeated factors, include terms like A/(x-a) + B/(x-a)². For irreducible quadratic factors, use (Ax+B)/(ax²+bx+c).
部分分式用于将有理式拆分为简单分式之和。不同线性因子时写作A/(x-a)+B/(x-b)。重复因子则包含A/(x-a)+B/(x-a)²等形式。二次因子时用(Ax+B)/(ax²+bx+c)。
. Inverse function f⁻¹(x) exists only if f is one-to-one; to find it, swap x and y and solve for y.
函数：定义域是允许输入的集合，值域是输出的集合。复合函数fg(x)=f(g(x))。反函数f⁻¹(x)仅当f为一一对应时存在；求解时交换x和y并解出y。
2. Coordinate Geometry and Graphs | 坐标几何与图形
llel lines have equal gradients; perpendicular lines have gradients satisfying m₁·m₂ = −1.
直线方程：y - y₁ = m(x - x₁)，其中斜率m = (y₂ - y₁)/(x₂ - x₁)。平行线斜率相等；垂直线斜率满足m₁·m₂ = −1。
Circle with centre (a,b) and radius r: (x - a)² + (y - b)² = r². The tangent to a circle is perpendicular to the radius at the point of contact. Use the discriminant to find intersections of a line and a circle.
圆心(a,b)半径r的圆：(x-a)²+(y-b)² = r²。切线在切点处垂直于半径。利用判别式求直线与圆的交点。
n form by eliminating the parameter t. The derivative is found as dy/dx = (dy/dt)/(dx/dt).
参数方程：x = f(t), y = g(t)。消去参数t可化为直角坐标形式。求导公式为dy/dx = (dy/dt)/(dx/dt)。
tally by factor 1/a. Reflections: y = −f(x) reflects in x-axis, y = f(−x) reflects in y-axis.
图形变换：y=f(x)+a向上平移a；f(x+a)向左平移a；a f(x)垂直拉伸a倍；f(ax)水平拉伸1/a倍。反射：y=−f(x)关于x轴对称，y=f(−x)关于y轴对称。
3. Differentiation and Its Applications | 微分及其应用
The derivative from first principles: f'(x) = lim_(h→0) [f(x+h)−f(x)]/h d/dx (xⁿ) = nxⁿ⁻¹; d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (ln x) = 1/x; d/dx (eˣ) = eˣ.
导数定义：f'(x)=lim_(h→0) [f(x+h)−f(x)]/h。基本公式：d/dx(xⁿ)=nxⁿ⁻¹；d/dx(sin x)=cos x；d/dx(cos x)=−sin x；d/dx(ln x)=1/x；d/dx(eˣ)=eˣ。
Chain rule: dy/dx = dy/du × du/dx. Product rule: d/dx(uv) = u'v + uv'. Quotient rule: d/dx(u/v) = (u'v − uv')/v².
链式法则：dy/dx = dy/du × du/dx。乘积法则：d/dx(uv)=u'v+uv'。商法则：d/dx(u/v)=(u'v−uv')/v²。
Implicit differentiation: differentiate both sides with respect to x, treating y as a function of x and multiplying by dy/dx when differentiating y terms.
隐函数求导：两边对x求导，将y视为x的函数，对y项求导时乘以dy/dx。
mum; d²y/dx² < 0 indicates a local maximum; if d²y/dx² = 0, use the first derivative test.
驻点位于dy/dx=0处。用二阶导数分类：d²y/dx²>0为局部极小值；d²y/dx²<0为局部极大值；若d²y/dx²=0，则用一阶导数判定。
at the point; normal line has gradient −1/m. The equation of tangent: y − y₀ = m(x − x₀).
切线斜率m=dy/dx在该点的
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