A-Level Maths Unit 4 (Jan 2021) Key Topic Revision | A-Level 数学第四单元知识点精讲

📚 A-Level Maths Unit 4 (Jan 2021) Key Topic Revision | A-Level 数学第四单元知识点精讲

Unit 4 of the A-Level Mathematics specification typically covers pure mathematics topics that extend from the first three units. The January 2021 paper tested a wide range of skills, from binomial expansion with rational exponents to advanced integration and 3D vectors. This revision guide distills the essential knowledge points, offering clear explanations in both English and Chinese to support bilingual learners preparing for resits or consolidating their understanding.

A-Level 数学第四单元通常涵盖纯数学中较深入的主题，是对前三个单元的拓展。2021年1月的试卷考查了从有理指数二项展开到高级积分与三维向量等多个核心技能。本篇复习指南提炼了必会知识点，用中英双语对照讲解，帮助学习者梳理思路、备战重考或巩固理解。

1. Binomial Expansion for Rational Powers | 有理指数的二项展开式

The binomial expansion can be extended to any rational power providing |x| < 1. The general form is (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … . When the power is a fraction or a negative integer, the series is infinite.

二项展开式可推广到任意有理指数，但要求 |x| < 1。一般形式为 (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … 。当指数为分数或负整数时，级数无穷延伸。

To expand (a + bx)ⁿ, first rewrite it as aⁿ(1 + bx/a)ⁿ. Always state the validity range, e.g. |bx/a| < 1 ⇒ |x| < |a/b|. For approximations, truncate the series to the required degree of accuracy.

要展开 (a + bx)ⁿ，先将其改写为 aⁿ(1 + bx/a)ⁿ。务必注明有效范围，例如 |bx/a| < 1 ⇒ |x| < |a/b|。做近似计算时，按题目要求的精度截断级数即可。

2. Trigonometric Identities and Equations | 三角恒等式与方程

Key identities include sin²θ + cos²θ = 1, tanθ = sinθ/cosθ, and the double-angle formulas: sin2θ = 2sinθcosθ, cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. Compound-angle formulas such as sin(A ± B) and cos(A ± B) are also essential.

核心恒等式包括 sin²θ + cos²θ = 1，tanθ = sinθ/cosθ，以及倍角公式：sin2θ = 2sinθcosθ，cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ。和差角公式如 sin(A ± B) 和 cos(A ± B) 也必不可少。

When solving trig equations, find all values within the given interval. Use quadrant diagrams or graphs to determine additional solutions. Equations of the form a cosθ + b sinθ can be rewritten as Rcos(θ ± α) or Rsin(θ ± α).

解三角方程时，找出给定区间内的全部解。利用象限图或函数图像求出其余解。形如 a cosθ + b sinθ 的方程可改写为 Rcos(θ ± α) 或 Rsin(θ ± α) 的形式。

3. Parametric Equations | 参数方程

A curve can be defined by x = f(t), y = g(t). To find the gradient, use dy/dx = (dy/dt) / (dx/dt). The second derivative d²y/dx² is obtained by differentiating dy/dx with respect to t and dividing by dx/dt.

曲线可由参数方程 x = f(t), y = g(t) 定义。求曲线斜率时用公式 dy/dx = (dy/dt) / (dx/dt)。二阶导数 d²y/dx² 则通过对 dy/dx 关于 t 求导再除以 dx/dt 得到。

To find the Cartesian equation, eliminate the parameter t, often using trigonometric identities (e.g. cos²t + sin²t = 1) or algebraic substitution.

求直角坐标方程需消去参数 t，通常借助三角恒等式（如 cos²t + sin²t = 1）或代数代换的方法。

4. Differentiation Techniques | 微分技巧

Beyond power rule, the chain rule, product rule, and quotient rule must be applied accurately. For composite functions, dy/dx = dy/du × du/dx. Logarithmic and implicit differentiation are tested when variables cannot be easily separated.

除幂函数求导外，必须熟练运用链式法则、乘法法则和除法法则。复合函数使用 dy/dx = dy/du × du/dx。当变量难以分离时，会涉及对数微分和隐函数微分。

Derivatives of standard functions include d/dx (sinx) = cosx, d/dx (lnx) = 1/x, d/dx (eˣ) = eˣ. The derivative of aˣ is aˣ lna. Trigonometric derivatives with the chain rule appear frequently, e.g. d/dx sin(ax+b) = a cos(ax+b).

基本函数的导数：d/dx (sinx) = cosx，d/dx (lnx) = 1/x，d/dx (eˣ) = eˣ。aˣ 的导数为 aˣ lna。结合链式法则的三角函数求导极为常见，如 d/dx sin(ax+b) = a cos(ax+b)。

5. Integration Techniques | 积分技巧

Antidifferentiation reverses differentiation. Standard integrals include ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠−1), ∫ 1/x dx = ln|x| + C, ∫ eˣ dx = eˣ + C, ∫ cosx dx = sinx + C, ∫ sinx dx = −cosx + C.

积分是微分的逆运算。基本积分公式有 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠−1)，∫ 1/x dx = ln|x| + C，∫ eˣ dx = eˣ + C，∫ cosx dx = sinx + C，∫ sinx dx = −cosx + C。

Advanced techniques include integration by substitution and integration by parts: ∫ u dv = uv − ∫ v du. When integrating rational functions, use partial fractions first. Trig integrals often require identities like sin²x = (1−cos2x)/2.

更高级的技巧包括换元积分法和分部积分法：∫ u dv = uv − ∫ v du。对有理函数积分可先拆分成部分分式。三角函数的积分常需用恒等式，如 sin²x = (1−cos2x)/2。

6. Differential Equations | 微分方程

First-order separable equations can be solved by separating variables: dy/dx = f(x)g(y) → ∫ 1/g(y) dy = ∫ f(x) dx. After integration, use given initial conditions to find the constant.

一阶可分离变量的微分方程通过移项求解：dy/dx = f(x)g(y) → ∫ 1/g(y) dy = ∫ f(x) dx。积分后利用给定的初始条件确定常数。

Modelling contexts include exponential growth/decay, where dy/dt = ky yields y = A eᵏᵗ. Word problems may involve rates of change like temperature or population; interpret the question carefully to set up the equation.

建模场景包括指数增长或衰减，如 dy/dt = ky 的解为 y = A eᵏᵗ。应用题可能涉及温度变化、人口增长等速率问题，需仔细审题建立方程。

7. Vectors in 3D | 三维向量

Vectors in three dimensions are expressed as ai + bj + ck or as column vectors. Magnitude is |v| = √(a² + b² + c²). The scalar product is v·w = |v||w|cosθ = a₁a₂ + b₁b₂ + c₁c₂, used to find angles between vectors.

三维向量可表示为 ai + bj + ck 或列向量。模长为 |v| = √(a² + b² + c²)。数量积为 v·w = |v||w|cosθ = a₁a₂ + b₁b₂ + c₁c₂，用于求向量间的夹角。

Vector equation of a line: r = a + λb, where a is a point on the line and b is a direction vector. To find intersection of lines or distance from point to line, set up parametric forms and solve.

直线的向量方程为 r = a + λb，其中 a 是直线上一点，b 是方向向量。求两直线交点或点到直线的距离时，需写出参数式并列方程求解。

8. Numerical Methods | 数值方法

The Newton-Raphson procedure finds roots of f(x)=0 using the iteration xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). Choose a starting value near the root and continue until convergence. Always check for derivative zero, which causes failure.

牛顿-拉夫森法通过迭代公式 xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ) 求方程 f(x)=0 的根。选择靠近根的初始值，反复迭代直到收敛。注意当导数为零时方法失效。

Another approach is fixed point iteration (rearranging f(x)=0 into x = g(x)). The iteration xₙ₊₁ = g(xₙ) converges if |g'(x)| < 1 near the root. The trapezium rule approximates ∫ f(x) dx ≈ h/2 [y₀ + 2(y₁ + y₂ + …) + yₙ] where h = (b−a)/n.

另一种方法是固定点迭代，将方程改写为 x = g(x)。若在根附近 |g'(x)| < 1，则迭代收敛。梯形法则近似计算 ∫ f(x) dx ≈ h/2 [y₀ + 2(y₁ + y₂ + …) + yₙ]，其中 h = (b−a)/n。

9. Partial Fractions | 部分分式

Rational functions of the form P(x)/Q(x) can be split into simpler fractions if the degree of numerator is less than denominator. Linear factors (ax+b) give numerators A/(ax+b); repeated factors yield A/(ax+b) + B/(ax+b)²; irreducible quadratics give (Ax + B)/(ax²+bx+c).

形如 P(x)/Q(x) 的有理函数，若分子次数低于分母，可拆分为更简单的分式。一次因式 (ax+b) 对应分式 A/(ax+b)；重复因式对应 A/(ax+b) + B/(ax+b)²；不可约二次因式对应 (Ax + B)/(ax²+bx+c)。

Solve for constants by multiplying through by the denominator and equating coefficients or substituting suitable x values. Partial fractions are especially useful for integration and binomial expansion.

求常数时，可通分后比较系数，或代入合适的 x 值求解。部分分式对积分和二项展开式特别有用。

10. Proof by Induction | 归纳法证明

Mathematical induction verifies statements for all positive integers. The three steps are: base case (show true for n=1), inductive hypothesis (assume true for n=k), and inductive step (prove true for n=k+1 using the assumption).

数学归纳法用于证明对所有正整数成立的命题。三步为：基础情形（证明 n=1 时成立），归纳假设（假设 n=k 时成立），归纳递推（利用该假设证明 n=k+1 成立）。

Common induction proofs include summation formulas, divisibility, and matrix powers. Always write a concluding statement: ‘Since true for n=1 and true for n=k implies true for n=k+1, by induction true for all positive integers n.’

常见归纳证明包括求和公式、整除性和矩阵乘方。务必写出总结句：“由于 n=1 成立且 n=k 成立推出 n=k+1 成立，根据归纳法，命题对所有正整数 n 均成立。”

11. Exponentials and Logarithms | 指数与对数函数

The natural exponential function y = eˣ has derivative eˣ and inverse y = ln x. Key logarithm laws: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln aᵇ = b ln a. Using these, solve exponential equations by taking logs on both sides.

自然指数函数 y = eˣ 的导数仍为 eˣ，其反函数为 y = ln x。对数运算法则有：ln(ab) = ln a + ln b，ln(a/b) = ln a − ln b，ln aᵇ = b ln a。解指数方程时常对方程两边取对数。

Exponential growth/decay models appear in forms like P = P₀ eᵏᵗ. The half-life or doubling time can be derived by setting P/P₀ = 1/2 or 2 and solving for t. Logarithmic differentiation simplifies products, quotients, or powers of functions.

指数增长/衰减模型常以 P = P₀ eᵏᵗ 的形式出现。半衰期或倍增时间可通过令 P/P₀ = 1/2 或 2 求解 t 获得。对数微分法则可简化函数的乘积、商或幂的求导。

12. Applications of Calculus | 微积分应用

Tangents and normals: at point (x₀, y₀) on curve y = f(x), the tangent has gradient f'(x₀) and equation y − y₀ = f'(x₀)(x − x₀); the normal has gradient −1/f'(x₀). For parametric curves, use dy/dx as derived above.

切线与法线：在曲线 y = f(x) 上点 (x₀, y₀) 处，切线斜率为 f'(x₀)，方程为 y − y₀ = f'(x₀)(x − x₀)；法线斜率为 −1/f'(x₀)。参数曲线则使用前面推导的 dy/dx。

Stationary points occur where dy/dx = 0; classify using second derivative test or sign change. In kinematics, given displacement s(t), velocity v = ds/dt, acceleration a = dv/dt = d²s/dt². Reverse relationships involve integration with initial conditions.

驻点出现在 dy/dx = 0 处，利用二阶导数检验或斜率符号变化进行判定。在运动学中，给定位移 s(t)，速度 v = ds/dt，加速度 a = dv/dt = d²s/dt²。逆向关系则需要通过积分并结合初始条件求得。

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