📚 FC Paper 158: Advanced Pure Mathematics Revision Guide | FC 试卷 158：A-Level 纯数学高级复习指南
This comprehensive revision guide is designed for Edexcel A-Level Mathematics students preparing for assessments styled like FC Paper 158. It brings together essential pure mathematics topics, including functions, calculus, trigonometry, exponentials, sequences, vectors, parametric equations, numerical methods, and differential equations. Each section provides clear explanations with worked examples, ensuring you build both fluency and exam confidence through bilingual support.
本综合复习指南专为备战类似于 FC 试卷 158 风格考试的爱德思 A-Level 数学学生设计。它汇集了纯数学核心主题，包括函数、微积分、三角学、指数、数列、向量、参数方程、数值方法和微分方程。每个部分都提供清晰的解释和实例，通过中英双语支持帮助你提升熟练度与考试信心。
1. Functions and Transformations | 函数与图像变换
ontrols vertical stretch, b horizontal stretch (1/b), c horizontal translation, and d vertical translation. For instance, replacing x with (x – 2) shifts the graph 2 units right, while y = 3 f(x) stretches it vertically by factor 3.
函数 f 将定义域中的每个输入 x 映射到值域中唯一的输出 f(x)。理解定义域限制、复合函数 fg(x) 和反函数 f⁻¹(x) 对高等微积分至关重要。图像变换遵循明确的层次结构：y = a f(b(x + c)) + d，其中 a 控制垂直拉伸，b 控制水平拉伸（1/b），c 水平平移，d 垂直平移。例如，用 (x – 2) 替换 x 会使图像右移 2 个单位，而 y = 3 f(x) 则使其垂直拉伸 3 倍。
Key transformations: y = f(x + a) (shift left), y = f(x) + a (shift up), y = –f(x) (reflect in x-axis), y = f(–x) (reflect in y-axis).
ing with composite functions, always consider the range of the inner function as it becomes the domain of the outer function.
反函数只存在于一一对应的函数；要求 f⁻¹(x)，交换 y = f(x) 中的 x 和 y 并解出 y。f 与 f⁻¹ 的图像关于直线 y = x 对称。处理复合函数时，始终要考虑内层函数的值域，因为它会成为外层函数的定义域。
2. Differentiation Rules | 微分法则
ising functions. The standard derivative of xⁿ is n xⁿ⁻¹. Key derivatives include: d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (sin x) = cos x, d/dx (cos x) = – or differentiating expressions like (3x² + 2)⁵ or sin(2x). The product rule (d/dx (uv) = u'v + uv') and quotient rule (d/dx (u/v) = (u'v – uv') / v²) extend this to products and quotients of functions.
微分是分析变化率和优化函数的核心工具。xⁿ 的标准导数为 n xⁿ⁻¹。关键导数包括：d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x，d/dx (sin x) = cos x，d/dx (cos x) = –sin x，d/dx (tan x) = sec² x。链式法则（复合函数求导）指出 dy/dx = dy/du × du/dx，对求 (3x² + 2)⁵ 或 sin(2x) 等表达式至关重要。乘积法则 (d/dx (uv) = u'v + uv') 和商法则 (d/dx (u/v) = (u'v – uv') / v²) 将这些规则拓展到函数的乘积和商。
| f(x) | f'(x) |
|---|---|
| xⁿ | n xⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | –sin x |
| tan x | sec² x |
ntiation treats y as a function of x, applying the chain rule to terms like y² → 2y dy/dx.
二阶导数 (d²y/dx²) 有助于对令 dy/dx = 0 求得的驻点进行分类。若 d²y/dx² > 0，该点为局部极小值；若 d²y/dx² < 0，则为局部极大值。参数方程微分利用 dx/dt 和 dy/dt：dy/dx = (dy/dt) ÷ (dx/dt)。隐函数微分将 y 视为 x 的函数，对 y² 等项应用链式法则得 2y dy/dx。
3. Integration Techniques | 积分技巧
to differential equations. The fundamental integrals include ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1), ∫ 1/x dx = ln|x| + C, ∫ eˣ dx = eˣ + C, ∫ cos x dx = sin x + C, and ∫ sin x dx = – is from x = a to x = b. Substitution (reverse chain rule) simplifies integrals like ∫ 2x ex² then du = 2x dx. Integration by parts, derived from the product rule, follows ∫ u dv = uv – ∫ v du.
积分是微分的逆运算，用于计算面积、体积以及求解微分方程。基本积分公式包括 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1)，∫ 1/x dx = ln|x| + C，∫ eˣ dx = eˣ + C，∫ cos x dx = sin x + C，∫ sin x dx = –cos x + C。定积分 ∫ₐᵇ f(x) dx 给出曲线与 x 轴在 x = a 到 x = b 之间的净面积。代换法（逆链式法则）可简化如 ∫ 2x ex² dx 之类的积分，令 u = x²，则 du = 2x dx。分部积分法源自乘积法则，公式为 ∫ u dv = uv – ∫ v du。
Integration by parts: ∫ u (dv/dx) dx = uv – ∫ v (du/dx) dx
(dx/dt) dt. Always check whether the curve crosses the axis to avoid cancellation errors.
计算面积时，注意 x 轴下方的区域对定积分贡献负值；应取绝对值或相应拆分积分。绕 x 轴旋转的体积公式为 V = π ∫ₐᵇ y² dx。对于参数曲线，使用 V = π ∫ₜ₁ᵗ² y² (dx/dt) dt。务必检查曲线是否穿过坐标轴，以避免正负抵消的误差。
4. Trigonometry | 三角学
1 + cot²θ = cosec²θ. Double-angle formulae include sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ, and tan 2θ = (2 tan θ)/(1 – . The R-formula simplifies expressions like a sin θ ± b cos θ into R sin(θ ± α) or R cos(θ ∓ α), where R = √a² + b² and tan α = b/a.
三角函数、恒等式和方程是纯数学的重要组成部分。微积分中几乎全部使用弧度制：π 弧度 = 180°。勾股恒等式 sin²θ + cos²θ = 1 可推导出 1 + tan²θ = sec²θ 和 1 + cot²θ = cosec²θ。倍角公式包括 sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ，以及 tan 2θ = (2 tan θ)/(1 – tan²θ)。R 公式可将形如 a sin θ ± b cos θ 的表达式化为 R sin(θ ± α) 或 R cos(θ ∓ α)，其中 R = √a² + b²，tan α = b/a。
graphs. For sin θ = k, the principal solution is θ = sin⁻¹ k; secondary solutions are 180° – principal (in degrees) or π – tan θ = k, solutions repeat every π. Always adjust angles for transformations like sin(2θ – 30°) before applying standard methods.
求解三角方程通常需要利用图像的对称性和周期性找出给定区间内的所有解。对于 sin θ = k，主解为 θ = sin⁻¹ k；第二解为 180° – 主解（度制）或 π – 主解（弧度制）。对于 cos θ = k，解为 ± 主解 + 2π 的整数倍。对于 tan θ = k，解每隔 π 重复一次。处理诸如 sin(2θ – 30°) 的变换时，务必先调整角度再套用标准解法。
5. Exponentials and Logarithms | 指数与对数
The natural exponential function eˣ logarithm ln x are inverses; their graphs reflect in the line y = x. Key properties: ln(eˣ) = x for all x, and eln x ithmic laws simplify products, quotients, and powers: ln(ab) = ln a + ln b, ln(a/b) = ln a – n(aᵏ) = k ln a. These are essential for solving exponential equations. The derivative of aˣ is aˣ ln a, and the integral of aˣ is aˣ / ln a + C. For growth and decay models, the equation dy/dt = ky leads to y = A ekt.
自然指数函数 eˣ 与自然对数 ln x 互为反函数；它们的图像关于直线 y = x 对称。关键性质：对所有 x 均有 ln(eˣ) = x，且对 x > 0 有 eln x = x。对数运算法则可简化乘积、商和幂运算：ln(ab) = ln a + ln b，ln(a/b) = ln a – ln b，ln(aᵏ) = k ln a。这些对于解指数方程至关重要。aˣ 的导数为 aˣ ln a，积分 ∫ aˣ dx = aˣ / ln a + C。对于增长和衰减模型，方程 dy/dt = ky 的解为 y = A ekt。
tions or half-lives. Logarithmic scales are used to linearise exponential data: if y = a bˣ logs gives ln y = ln a + x ln b, which forms a straight line when plotting ln y against x.
指数模型出现在人口增长、放射性衰变和牛顿冷却定律中。你可能需要根据初始条件或半衰期确定未知常数。对数尺度可用于线性化指数数据：若 y = a bˣ，则取对数得 ln y = ln a + x ln b，绘制 ln y 关于 x 的图形时将形成一条直线。
6. Sequences and Series | 数列与级数
Arithmetic sequences have a common difference d: uₙ = a + (n – 1)d. The sum of the first n terms is Sₙ = n/2 [2a + (n – is the last term. Geometric sequences have a common ratio r: uₙ = a rⁿ⁻¹. The sum Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1. An infinite geometric series converges to a/(1 – r) provided |r| < 1. The binomial expansion for (1 + x)ⁿ, where n is rational, is valid for |x| < 1: (1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + n(n – 1)(n – 2)/3! x³ + ... .
等差数列有公差 d：uₙ = a + (n – 1)d。前 n 项和 Sₙ = n/2 [2a + (n – 1)d] 或 n/2 (a + l)，其中 l 为末项。等比数列有公比 r：uₙ = a rⁿ⁻¹。当 r ≠ 1 时，和 Sₙ = a(1 – rⁿ)/(1 – r)。当 |r| < 1 时，无穷等比级数收敛于 a/(1 – r)。二项展开式 (1 + x)ⁿ（n 为有理数）在 |x| < 1 时成立：(1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + n(n – 1)(n – 2)/3! x³ + ... 。
ding the long-term behaviour (convergence or divergence) is important. For arithmetic and geometric sequences, being able to distinguish between them from given terms is frequently examined. The sum to infinity of a convergent geometric series is a classic problem; link it to the condition |r| < 1.
数列也可用迭代方式定义，例如 xₙ₊₁ = f(xₙ)。理解其长期行为（收敛或发散）很重要。对等差和等比数列，从给定项辨别其类型是常见考点。收敛等比级数的无穷和是一个经典问题；须与条件 |r| < 1 关联。
7. Vectors | 向量
en as i, j, k components or as column vectors. The magnitude of vector a = xi + yj + zk is |a| = √x² + y² + z² nt-wise, and scalar multiplication scales each component. The scalar (dot) product a · b = |a||b| cos θ is used to find the angle between two vectors: cos θ = (a · b) / (|a||b|). Two non-zero vectors are perpendicular if and only if a · b = 0.
向量表示既有大小又有方向的量。在二维和三维中，向量用 i、j、k 分量或列向量表示。向量 a = xi + yj + zk 的模为 |a| = √x² + y² + z²。向量加法按分量进行，标量乘法则缩放每个分量。标量积（点积）a · b = |a||b| cos θ 用于求两向量夹角：cos θ = (a · b) / (|a||b|)。两个非零向量垂直当且仅当 a · b = 0。
If a = x₁i + y₁j + z₁k and b = x₂i + y₂j + z₂k, then a · b = x₁x₂ + y₁y₂ + z₁z₂.
e and b is the direction vector. To find the intersection of two lines, set their vector equations equal and solve for the parameters; check consistency for 3D skew lines. The point-to-line distance can be found using the cross product in Further Maths, but at A-Level pure, perpendicular projections are used.
直线的向量方程形式为 r = a + λ b，其中 a 是直线上一点，b 是方向向量。求两直线交点时，令其向量方程相等并解参数；对于三维异面直线需判断是否一致。点到直线的距离可使用叉积（进阶数学内容）求解，但在 A-Level 纯数中通常采用垂直投影的方法。
8. Parametric Equations |
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