📚 A-Level CIE Mathematics Pure Mathematics 1 Core Topics In-Depth | A-Level CIE 数学纯数1核心知识点精讲
The Cambridge International AS & A Level Mathematics Pure Mathematics 1 coursebook covers the essential building blocks of higher mathematics. Mastery of these topics equips students with analytical skills, proof techniques, and a deep understanding of functions, geometry, calculus, and series. This article walks through the key sections, explaining concepts clearly with English and Chinese pairing, perfect for revision and bilingual learning.
剑桥国际AS与A Level数学纯数1教材涵盖了高等数学的核心基础。掌握这些专题能培养学生具备分析能力、证明技巧以及对函数、几何、微积分和级数的深刻理解。本文逐一讲解关键章节，采用中英对照方式清晰解释概念，非常适合复习和双语学习。
1. Quadratics | 二次函数与方程
The study of quadratics involves expressions of the form ax² + bx + c, where a ≠ 0. Completing the square rewrites the quadratic in the form a(x − h)² + k, immediately revealing the vertex (h, k) and helping to solve equations or sketch curves.
二次函数的研究涉及形式为 ax² + bx + c（a ≠ 0）的表达式。配方法可将二次式改写为 a(x − h)² + k 的形式，直接揭示顶点 (h, k) 的坐标，并帮助解方程或绘图。
< 0. This also connects to the number of intersections between a parabola and the x-axis.
判别式 Δ = b² − 4ac 决定根的性质：若 Δ > 0 则有两个相异实根，Δ = 0 有一个重根，Δ < 0 则无实根。这也对应了抛物线与 x 轴交点的数量。
Quadratic inequalities such as ax² + bx + c > 0 are solved by sketching the graph and identifying the intervals where the curve lies above or below the axis. Factorisation and critical values are key techniques.
解二次不等式，例如 ax² + bx + c > 0，需要通过画图标出曲线在轴上方的区间，利用因式分解和临界值的方法。
Vertex: x = −b/(2a), y = c − b²/(4a) (from completed square)
顶点：x = −b/(2a)，y = c − b²/(4a)（由配方式得到）
2. Functions | 函数
A function f maps each input from its domain to exactly one output. Understanding domain and range is fundamental: domain is the set of all admissible x-values, range is the set of all resulting y-values.
函数 f 将其定义域中的每个输入值映射到唯一的输出值。理解定义域和值域是基础：定义域是所有允许的 x 值的集合，值域是所有得到的结果 y 值的集合。
One-to-one functions pass the horizontal line test and possess inverse functions f⁻¹. The inverse is found by swapping x and y and solving for y. The domain of f⁻¹ is the range of f, and vice versa.
单射函数通过水平线检验后存在反函数 f⁻¹。求反函数的方法是交换 x 与 y 后解出 y。f⁻¹ 的定义域是 f 的值域，反之亦然。
Composite functions such as (g ∘ ) exist only when the range of f is a subset of the domain of g. Transformations of graphs – translations, stretches, and reflections – tably: f(x) + a translates vertically, f(x + a) translates horizontally, a f(x) stretches vertically, f(ax) stretches horizontally.
复合函数，如 (g ∘ f)(x) = g(f(x))，仅在 f 的值域是 g 的定义域的子集时存在。图像的变换——平移、拉伸和反射——可预测地改变函数方程：f(x) + a 垂直平移，f(x + a) 水平平移，a f(x) 垂直拉伸，f(ax) 水平拉伸。
3. Coordinate Geometry | 坐标几何
₁)²], and the midpoint is ((x₁+x₂)/2, (y₁+y₂)/2). These formulae are essential for straight-line and circle geometry.
两点 (x₁, y₁) 与 (x₂, y₂) 之间的距离为 √[(x₂ − x₁)² + (y₂ − y₁)²]，中点为 ((x₁+x₂)/2, (y₁+y₂)/2)。这些公式是直线和圆解析几何的基础。
he gradient. Parallel lines share equal gradients; perpendicular lines satisfy m₁ m₂ = −1.
直线的方程为 y = mx + c，其中 m = (y₂ − y₁)/(x₂ − x₁) 是斜率。平行线斜率相等；垂直线满足 m₁ m₂ = −1。
². Completing the square helps to find centre and radius from the general form. To determine intersections or tangency, solve simultaneous equations of the line and circle; the discriminant of the resulting quadratic indicates the number of intersections.
圆心在 (a, b)、半径为 r 的圆的方程为 (x − a)² + (y − b)² = r²。利用配方法可以从一般式求出圆心和半径。要判断相交或相切，需联立直线与圆的方程，由所得二次方程的判别式确定交点个数。
4. Circular Measure | 弧度制与弧长面积
ending angle θ (in radians) is s = rθ. The area of the corresponding sector is A = ½ r² θ.
弧度制为角度提供了自然的度量方法：π 弧度 = 180°。半径为 r 的圆对应弧长（圆心角 θ 以弧度计）为 s = rθ。相应扇形面积为 A = ½ r² θ。
These relationships are widely applied in geometry and physics problems involving rotation.
弓形面积（弦与弧之间的区域）通过扇形面积减去三角形面积求得：A_弓形 = ½ r² (θ − sin θ)。这些关系广泛应用于涉及旋转的几何与物理问题。
5. Trigonometry | 三角学
values for angles 0°, 30°, 45°, 60°, 90° (and their radian equivalents) must be memorised.
三个基本三角函数在单位圆上定义：sin θ = y，cos θ = x，tan θ = sin θ / cos θ（当 cos θ = 0 时无定义）。必须熟记 0°、30°、45°、60°、90°（及其弧度等价角度）的精确值。
Trigonometric graphs exhibit periodic behaviour: sine and cosine have period 2π, amplitude 1; tangent has period π and vertical asymptotes. Transformations stretch, shift, or reflect these graphs following the same rules as other functions.
三角函数图像具有周期性：正弦和余弦的周期为 2π，振幅为 1；正切函数的周期为 π 且有竖直渐近线。图像的拉伸、平移或反射遵循与其他函数相同的变换规则。
Key identities include sin² θ + cos² θ ≡ 1 and tan θ ≡ sin θ / cos θ. Solving trigonometric equations within a given interval requires finding principal values and using CAST diagram or graph symmetry to locate all solutions.
重要的恒等式包括 sin² θ + cos² θ ≡ 1 和 tan θ ≡ sin θ / cos θ。在给定区间内解三角方程需要先求主值，再利用 CAST 图或图像对称性找出所有解。
6. Series: Arithmetic and Geometric Progressions | 数列：等差与等比级数
An arithmetic progression (AP) has common difference d: un = a + (n − 1)d. The sum of the first n terms is Sn = n/2 [2a + (n − 1)d] or Sn = n/2 (a + l), where l is the last term.
等差数列（AP）具有公差 d：un = a + (n − 1)d。前 n 项和为 Sn = n/2 [2a + (n − 1)d] 或 Sn = n/2 (a + l)，其中 l 为末项。
A geometric progression (GP) has common ratio r: un = a rn−1. The sum of the first n terms is Sn = a(1 − rn)/(1 − r) for r ≠ 1. An infinite geometric series converges to S_∞ = a/(1 − r) provided |r| < 1.
等比数列（GP）具有公比 r：un = a rn−1。当 r ≠ 1 时前 n 项和为 Sn = a(1 − rn)/(1 − r)。无穷等比级数收敛到 S_∞ = a/(1 − r)，条件是 |r| < 1。
The binomial expansion for (1 + x)ⁿ with n as a positive integer is given by 1 + nx + [n(n−1)/2!] x² + … + [n(n−1)…(n−r+1)/r!] xr. This expansion is particularly useful for approximations and series problems.
对于正整数 n，(1 + x)ⁿ 的二项展开式为 1 + nx + [n(n−1)/2!] x² + … + [n(n−1)…(n−r+1)/r!] xr。该展开式对近似计算和级数问题特别有用。
7. Differentiation | 微分法
ference rule, and the chain rule for composite functions: if y = f(g(x)), dy/dx = f′(g(x)) g′(x).
微分可求出曲线在某点的斜率。xⁿ 的导数为 n xⁿ⁻¹。基本法则包括常数倍法则、和差法则以及适用于复合函数的链式法则：若 y = f(g(x))，则 dy/dx = f′(g(x)) g′(x)。
0 it is a minimum; if d²y/dx² < 0 it is a maximum; if d²y/dx² = 0, further investigation is needed.
二阶导数 d²y/dx² 描述曲线的凹凸性并有助于给驻点分类。在满足 dy/dx = 0 的驻点处，若 d²y/dx² > 0 则为极小值点；若 d²y/dx² < 0 则为极大值点；若 d²y/dx² = 0，则需进一步分析。
x at that point; the normal is perpendicular. This links geometric properties to calculus.
点 (x₁, y₁) 处的切线方程为 y − y₁ = m(x − x₁)，其中 m 为该点的导数 dy/dx；法线与切线垂直。这建立了几何性质与微积分之间的联系。
d/dx (xⁿ) = n xⁿ⁻¹, d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x
导数公式：d/dx (xⁿ) = n xⁿ⁻¹，d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x
8. Integration | 积分法
Integration is the reverse of differentiation. The indefinite integral of xⁿ (n ≠ −1) is (xⁿ⁺¹)/(n+1) + c. This constant of integration c must always be included for indefinite integrals.
积分是微分的逆运算。xⁿ 的不定积分（n ≠ −1）为 (xⁿ⁺¹)/(n+1) + c。不定积分必须始终加上积分常数 c。
The definite integral ∫_ab x = a to x = b. The fundamental theorem of calculus links f and its antiderivative F: ∫_ab f(x) dx = F(b) − F(a).
定积分 ∫_ab f(x) dx 给出曲线与 x 轴在 x = a 和 x = b 之间的有向面积。微积分基本定理将 f 与其原函数 F 联系起来：∫_ab f(x) dx = F(b) − F(a)。
Area between two curves y = f(x) and y = g(x) from a to b is ∫_ab on [a, b]. Areas below the axis must be treated carefully using absolute value or splitting the integral where the curve crosses the axis.
两条曲线 y = f(x) 与 y = g(x) 在 a 到 b 之间的面积是 ∫_ab [f(x) − g(x)] dx，前提是在区间 [a, b] 上 f(x) ≥ g(x)。位于 x 轴下方的面积需谨慎处理，应在曲线与轴相交处分段积分或取绝对值。
Published by TutorHao | A-Level 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