📚 Core Knowledge Review for Year 13 CIE Mathematics | Year 13 CIE 数学核心知识点梳理
The second year of CIE A‑Level Mathematics (9709) deepens your understanding with Pure Mathematics 3 (P3) and applied modules. This article consolidates the essential topics—Algebra, Trigonometry, Calculus, Vectors, Complex Numbers, and more—that underpin exam success. Mastering these core ideas through consistent practice is the key to high marks.
CIE A‑Level 数学第二年（9709大纲）通过纯数3（P3）与应用模块进一步深化你的理解。本文梳理了代数、三角、微积分、向量、复数等核心主题，这些是考试成功的基础。通过持续练习掌握这些核心思想是取得高分的关键。
1. Algebra and Functions | 代数与函数
The modulus function |x| outputs the absolute value, representing distance from zero. Graphs involving modulus often require splitting the domain to reflect negative parts above the x‑axis, producing characteristic V‑shapes for linear expressions.
模函数 |x| 输出绝对值，表示到零的距离。涉及模的图形通常需要分割定义域，将负值部分翻折到 x 轴上方，对于线性表达式产生典型的 V 形。
When solving equations such as |2x − 3| = 5, consider two separate linear equations: 2x − 3 = 5 and 2x − 3 = −5. This yields solutions x = 4 and x = −1; always verify against the original modulus condition.
解方程如 |2x − 3| = 5 时，需考虑两个独立的线性方程：2x − 3 = 5 和 2x − 3 = −5，解得 x = 4 和 x = −1；始终要代回原模条件检验。
Polynomial division, together with the factor theorem, enables factorisation of cubics and higher‑degree polynomials. If f(a) = 0, then (x − a) is a factor. The remainder theorem gives the remainder when dividing by (x − a) directly as f(a).
多项式除法结合因式定理可以分解三次及更高次多项式。若 f(a) = 0，则 (x − a) 是一个因式。余数定理直接给出除以 (x − a) 的余数为 f(a)。
Partial fractions express a rational function as a sum of simpler fractions, crucial for integration. For distinct linear factors, e.g. (2x+1)/((x+1)(x−2)) ≡ A/(x+1) + B/(x−2). Solve for A and B by substituting convenient x‑values or comparing coefficients.
部分分式将有理函数表示为简单分式之和，这对积分至关重要。对于不同线性因子，例如 (2x+1)/((x+1)(x−2)) ≡ A/(x+1) + B/(x−2)，可通过代入方便的 x 值或比较系数求解 A 和 B。
Be comfortable with function transformations and inverses. A composition f(g(x)) requires the range of g to be a subset of the domain of f. The inverse function f⁻¹ exists only if f is one‑to‑one; its graph is a reflection of y = f(x) in the line y = x.
要熟练掌握函数变换与反函数。复合 f(g(x)) 要求 g 的值域是 f 定义域的子集。反函数 f⁻¹ 仅在 f 是单射时存在；其图形是 y = f(x) 关于直线 y = x 的反射。
2. Exponential and Logarithmic Functions | 指数与对数函数
The natural exponential function eˣ and the natural logarithm ln x are inverses, with ln(eˣ) = x and e^(ln x) = x. The derivative of eˣ is itself, while d/dx (ln x) = 1/x.
自然指数函数 eˣ 与自然对数 ln x 互为反函数，满足 ln(eˣ) = x 和 e^(ln x) = x。eˣ 的导数仍为其自身，而 d/dx (ln x) = 1/x。
For a general exponential aˣ = e^(x ln a) and its derivative is aˣ ln a. Similarly, logₐ x = (ln x)/(ln a), and the change‑of‑base formula helps convert between different bases.
一般指数函数 aˣ = e^(x ln a)，其导数为 aˣ ln a。类似地，logₐ x = (ln x)/(ln a)，换底公式有助于在不同底之间转换。
Exponential growth and decay models appear as y = A eᵏˣ or y = A e⁻ᵏˣ. The constant k determines the rate; solving problems often requires using given boundary conditions to find A and k.
指数增长和衰减模型表示为 y = A eᵏˣ 或 y = A e⁻ᵏˣ。常数 k 决定速率；解题时通常需要利用给定的边界条件求出 A 和 k。
Logarithms allow linearisation of exponential relationships. Taking ln of both sides of y = a bˣ gives ln y = ln a + x ln b, a straight line when plotting ln y against x.
对数可将指数关系线性化。对 y = a bˣ 两边取自然对数得 ln y = ln a + x ln b，当绘制 ln y 对 x 的图形时得到一条直线。
Solving equations like 2 e³ˣ = 5 requires isolating the exponential term and then applying the natural logarithm: 3x = ln(2.5), so x = (1/3) ln 2.5.
解方程如 2 e³ˣ = 5 需要先分离指数项，然后应用自然对数：3x = ln(2.5)，因此 x = (1/3) ln 2.5。
3. Trigonometry | 三角学
In P3, the reciprocal functions sec θ = 1/cos θ, cosec θ = 1/sin θ, and cot θ = cos θ/sin θ are introduced. Their graphs, ranges, and key asymptotes must be memorised.
在 P3 中引入了倒数函数 sec θ = 1/cos θ，cosec θ = 1/sin θ 和 cot θ = cos θ/sin θ。必须记住它们的图形、值域和关键渐近线。
Trigonometric identities extend to these new functions: 1 + tan² θ = sec² θ and 1 + cot² θ = cosec² θ, alongside sin² θ + cos² θ = 1. These are essential for simplifying expressions and proving identities.
三角恒等式扩展到这些新函数：1 + tan² θ = sec² θ 和 1 + cot² θ = cosec² θ，以及 sin² θ + cos² θ = 1。它们在化简表达式和证明恒等式中必不可少。
Inverse trigonometric functions arcsin x, arccos x and arctan x (or sin⁻¹x, etc.) return principal angles. Their restricted domains and ranges ensure they are functions; e.g. arcsin x has range [−π/2, π/2].
反三角函数 arcsin x、arccos x 和 arctan x（或 sin⁻¹x 等）返回主值角度。它们限制定义域和值域以确保成为函数；例如 arcsin x 的值域为 [−π/2, π/2]。
Expressions of the form a sin θ ± b cos θ can be written as R sin(θ ± α) or R cos(θ ± α), where R = √(a² + b²) and tan α = b/a. This technique helps solve equations and find maxima/minima.
形如 a sin θ ± b cos θ 的表达式可写成 R sin(θ ± α) 或 R cos(θ ± α)，其中 R = √(a² + b²)，tan α = b/a。该技巧有助于解方程和求极值。
Solving trigonometric equations in a given interval involves using identities to reach a single trig function, then applying the general solution and selecting appropriate values. Be careful with quadrants and the domain of inverse functions.
在给定区间内解三角方程需要利用恒等式化为一个三角函数，然后应用通解并选择合适的值。注意象限及反函数的定义域。
4. Differentiation Techniques | 微分技巧
The chain rule, product rule and quotient rule form the backbone of differentiation. For y = f(g(x)), dy/dx = f ‘(g(x)) g'(x); for u(x)v(x), use u’v + uv’; for u/v, apply (u’v − uv’)/v².
链式法则、乘法律和除法律是微分的基石。对于 y = f(g(x))，dy/dx = f ‘(g(x)) g'(x)；对于 u(x)v(x)，使用 u’v + uv’；对于 u/v，应用 (u’v − uv’)/v²。
Derivatives of exponential and logarithmic functions: d/dx (eᵏˣ) = k eᵏˣ, d/dx (aˣ) = aˣ ln a, d/dx (ln x) = 1/x, d/dx (logₐ x) = 1/(x ln a). Differentiating trigonometric functions yields d/dx (sin x) = cos x, d/dx (sec x) = sec x tan x, etc.
指数和对数函数的导数：d/dx (eᵏˣ) = k eᵏˣ，d/dx (aˣ) = aˣ ln a，d/dx (ln x) = 1/x，d/dx (logₐ x) = 1/(x ln a)。三角函数的微分给出 d/dx (sin x) = cos x，d/dx (sec x) = sec x tan x 等。
Implicit differentiation treats y as a function of x. When differentiating y², apply chain rule: d(y²)/dx = 2y dy/dx. Collect dy/dx terms and factor to find the derivative.
隐函数微分将 y 视为 x 的函数。对 y² 微分时应用链式法则：d(y²)/dx = 2y dy/dx。收集 dy/dx 项并提取公因式求出导数。
Parametric differentiation: if x = f(t) and y = g(t), then dy/dx = (dy/dt) / (dx/dt). The second derivative is d²y/dx² = d(dy/dx)/dt ÷ dx/dt.
参数微分：若 x = f(t) 且 y = g(t)，则 dy/dx = (dy/dt) / (dx/dt)。二阶导数为 d²y/dx² = d(dy/dx)/dt ÷ dx/dt。
Be familiar with the derivatives of inverse trig functions: d/dx (sin⁻¹ x) = 1/√(1−x²), d/dx (tan⁻¹ x) = 1/(1+x²). These often appear in integration and related rates problems.
熟悉反三角函数的导数：d/dx (sin⁻¹ x) = 1/√(1−x²)，d/dx (tan⁻¹ x) = 1/(1+x²)。它们常出现在积分和相关变化率问题中。
5. Applications of Differentiation | 微分的应用
Equations of tangents and normals to a curve at a point (x₁, y₁) use dy/dx as the gradient m: y − y₁ = m (x − x₁) for the tangent, and gradient −1/m for the normal.
曲线在某点 (x₁, y₁) 的切线与法线方程使用 dy/dx 作为梯度 m：切线为 y − y₁ = m (x − x₁)，法线梯度为 −1/m。
Stationary points occur where dy/dx = 0. Use the second derivative to classify: d²y/dx² > 0 indicates a minimum, d²y/dx² < 0 a maximum, and if d²y/dx² = 0, check sign change of first derivative.
驻点发生在 dy/dx = 0 处。用二阶导数判断：d²y/dx² > 0 为极小值，d²y/dx² < 0 为极大值，若 d²y/dx² = 0，则检查一阶导数的符号变化。
Connected rates of change: if two quantities x and y are related, and one rate dx/dt is known, then dy/dt = (dy/dx)(dx/dt). Setup the relationship equation first, then differentiate with respect to t.
相关变化率：若两个量 x 与 y 相关联，且已知一个变化率 dx/dt，则 dy/dt = (dy/dx)(dx/dt)。首先建立关系式，然后对 t 微分。
Optimisation problems involve modelling a situation with a function of one variable and finding its maximum or minimum using derivatives, always checking endpoints and physical constraints.
优化问题涉及用一个一元函数对情境建模，并利用导数求其最大值或最小值，始终检查端点与物理约束。
Small increments: δy ≈ (dy/dx) δx for small δx. This linear approximation is useful for estimating changes and errors in measurement.
小增量：对于小 δx，δy ≈ (dy/dx) δx。这种线性近似在估计变化和测量误差时非常有用。
6. Integration Techniques | 积分技巧
Integration reverses differentiation: ∫ f ‘(x) dx = f(x) + C. Standard integrals include ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫ 1/x dx = ln|x| + C, ∫ eᵏˣ dx = (1/k) eᵏˣ + C.
积分是微分的逆：∫ f ‘(x) dx = f(x) + C。基本积分包括 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1），∫ 1/x dx = ln|x| + C，∫ eᵏˣ dx = (1/k) eᵏˣ + C。
Integration by substitution: given ∫ f(g(x)) g'(x) dx, let u = g(x) to obtain ∫ f(u) du. For definite integrals, remember to change the limits. Choose u strategically to simplify the integrand.
代换积分法：对于 ∫ f(g(x)) g'(x) dx，令 u = g(x) 得到 ∫ f(u) du。对于定积分，记得变换积分限。策略性地选取 u 以简化被积函数。
Integration by parts: ∫ u dv = uv − ∫ v du, derived from the product rule. Usually applied to products like x eˣ, x sin x, or ln x (set dv = dx). Use LIATE to decide u.
分部积分法：∫ u dv = uv − ∫ v du，来源于乘法律。常用于形如 x eˣ、x sin x 或 ln x 的乘积（令 dv = dx）。用 LIATE 规则选择 u。
Using partial fractions and trigonometric identities transforms rational or trig expressions into standard integrable forms. E.g., ∫ (2x+1)/((x+1)(x−2)) dx splits into simpler logs.
利用部分分式和三角恒等式可将有理式或三角表达式化为标准可积形式。例如，∫ (2x+1)/((x+1)(x−2)) dx 拆分为更简单的对数积分。
Integrals leading to inverse trig functions: ∫ 1/√(a²−x²) dx = sin⁻¹(x/a) + C and ∫ 1/(a²+x²) dx = (1/a) tan⁻¹(x/a) + C. Recognising these patterns is a common P3 skill.
积分得到的反三角函数：∫ 1/√(a²−x²) dx = sin⁻¹(x/a) + C，∫ 1/(a²+x²) dx = (1/a) tan⁻¹(x/a) + C。识别这些模式是 P3 常见技能。
7. Numerical Solutions of Equations | 方程的数值解
When an exact root cannot be found, numerical methods approximate solutions. Locate roots by sign change: if f(a) and f(b) have opposite signs and f is continuous, a root lies in [a, b].
当无法找到精确根时，数值方法可近似求解。通过符号变化确定根的位置：若 f(a) 与 f(b) 异号且 f 连续，则 [a, b] 内存在根。
The iterative formula xₙ₊₁ =
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