Pre-U WJEC Further Maths: Quick Reference Formula & Theorem Handbook | Pre-U WJEC 进阶数学：公式定理速查手册

📚 Pre-U WJEC Further Maths: Quick Reference Formula & Theorem Handbook | Pre-U WJEC 进阶数学：公式定理速查手册

This handbook collects the essential formulae and theorems required for the WJEC Pre-U Further Mathematics syllabus. It is designed as a rapid revision aid, pairing each result with its immediate translation and explanation in Chinese to reinforce understanding. The topics cover pure mathematics content from FP1, FP2 and FP3, including calculus, complex numbers, matrices, vectors, series, differential equations, numerical methods, hyperbolic functions, polar coordinates and proof by induction.

本手册汇集了 WJEC Pre-U 进阶数学课程所需的重要公式与定理，旨在作为快速复习工具，每一条结果均配有中文翻译与解释以加深理解。主题覆盖 FP1、FP2 与 FP3 中的纯数内容，包括微积分、复数、矩阵、向量、级数、微分方程、数值方法、双曲函数、极坐标以及归纳法证明。

1. Differentiation Rules & Standard Derivatives | 微分法则与标准导数

1. Power rule: d/dx (xⁿ) = n xⁿ⁻¹.

1. 幂函数法则：d/dx (xⁿ) = n xⁿ⁻¹。

2. Constant multiple rule: d/dx (a f(x)) = a f'(x).

2. 常数倍法则：d/dx (a f(x)) = a f'(x)。

3. Sum rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x).

3. 和差法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。

4. Product rule: d/dx (u v) = u’ v + u v’.

4. 乘积法则：d/dx (u v) = u’ v + u v’。

5. Quotient rule: d/dx (u/v) = (u’ v − u v’) / v².

5. 商法则：d/dx (u/v) = (u’ v − u v’) / v²。

6. Chain rule: dy/dx = dy/du × du/dx.

6. 链式法则：dy/dx = dy/du × du/dx。

7. Derivative of eˣ: d/dx (eˣ) = eˣ. More generally, d/dx (eᵏˣ) = k eᵏˣ.

7. eˣ 的导数：d/dx (eˣ) = eˣ；一般地，d/dx (eᵏˣ) = k eᵏˣ。

8. Derivative of ln x: d/dx (ln x) = 1/x; d/dx (ln|f(x)|) = f'(x)/f(x).

8. ln x 的导数：d/dx (ln x) = 1/x；d/dx (ln|f(x)|) = f'(x)/f(x)。

9. Derivatives of trigonometric functions: d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (tan x) = sec² x; d/dx (cot x) = −csc² x; d/dx (sec x) = sec x tan x; d/dx (csc x) = −csc x cot x.

9. 三角函数的导数：d/dx (sin x) = cos x；d/dx (cos x) = −sin x；d/dx (tan x) = sec² x；d/dx (cot x) = −csc² x；d/dx (sec x) = sec x tan x；d/dx (csc x) = −csc x cot x。

10. Derivatives of inverse trigonometric functions: d/dx (arcsin x) = 1/√(1 − x²); d/dx (arccos x) = −1/√(1 − x²); d/dx (arctan x) = 1/(1 + x²).

10. 反三角函数的导数：d/dx (arcsin x) = 1/√(1 − x²)；d/dx (arccos x) = −1/√(1 − x²)；d/dx (arctan x) = 1/(1 + x²)。

11. Parametric differentiation: if x = f(t), y = g(t), then dy/dx = (dy/dt) / (dx/dt).

11. 参数方程求导：若 x = f(t)，y = g(t)，则 dy/dx = (dy/dt) / (dx/dt)。

12. Implicit differentiation: differentiate both sides with respect to x, treating y as a function of x, and then solve for dy/dx.

12. 隐函数求导：对等式两边关于 x 求导，将 y 视为 x 的函数，然后解得 dy/dx。

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2. Integration Techniques & Standard Integrals | 积分技巧与标准积分

1. Reverse power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1.

1. 幂函数积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ −1。

2. ∫ 1/x dx = ln|x| + C.

2. ∫ 1/x dx = ln|x| + C。

3. ∫ eˣ dx = eˣ + C; ∫ aˣ dx = aˣ/ln a + C.

3. ∫ eˣ dx = eˣ + C；∫ aˣ dx = aˣ/ln a + C。

4. ∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C; ∫ sec² x dx = tan x + C; ∫ csc² x dx = −cot x + C; ∫ sec x tan x dx = sec x + C; ∫ csc x cot x dx = −csc x + C.

4. ∫ sin x dx = −cos x + C；∫ cos x dx = sin x + C；∫ sec² x dx = tan x + C；∫ csc² x dx = −cot x + C；∫ sec x tan x dx = sec x + C；∫ csc x cot x dx = −csc x + C。

5. ∫ tan x dx = ln|sec x| + C; ∫ cot x dx = ln|sin x| + C; ∫ sec x dx = ln|sec x + tan x| + C; ∫ csc x dx = −ln|csc x + cot x| + C.

5. ∫ tan x dx = ln|sec x| + C；∫ cot x dx = ln|sin x| + C；∫ sec x dx = ln|sec x + tan x| + C；∫ csc x dx = −ln|csc x + cot x| + C。

6. Standard inverse trigonometric integrals: ∫ 1/√(a² − x²) dx = arcsin(x/a) + C; ∫ 1/(a² + x²) dx = (1/a) arctan(x/a) + C; ∫ 1/(x√(x² − a²)) dx = (1/a) arcsec(|x|/a) + C.

6. 标准反三角函数积分：∫ 1/√(a² − x²) dx = arcsin(x/a) + C；∫ 1/(a² + x²) dx = (1/a) arctan(x/a) + C；∫ 1/(x√(x² − a²)) dx = (1/a) arcsec(|x|/a) + C。

7. Integration by substitution: ∫ f(g(x)) g'(x) dx = ∫ f(u) du, where u = g(x).

7. 换元积分法：∫ f(g(x)) g'(x) dx = ∫ f(u) du，其中 u = g(x)。

8. Integration by parts: ∫ u dv = u v − ∫ v du. The LIATE rule helps choose u: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential.

8. 分部积分法：∫ u dv = u v − ∫ v du。选择 u 的 LIATE 顺序：对数函数、反三角函数、代数函数、三角函数、指数函数。

9. Definite integrals: ∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x).

9. 定积分：∫ₐᵇ f(x) dx = F(b) − F(a)，其中 F'(x) = f(x)。

10. Improper integrals: evaluated as limits, e.g., ∫ₐ∞ f(x) dx = lim_{R→∞} ∫ₐᴿ f(x) dx, or if the integrand is unbounded at an endpoint.

10. 反常积分：通过极限求值，例如 ∫ₐ∞ f(x) dx = lim_{R→∞} ∫ₐᴿ f(x) dx，或者当被积函数在端点无界时类似处理。

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3. Complex Numbers | 复数

1. Standard form: z = a + b i, where a, b ∈ ℝ, i² = −1. Real part Re(z) = a; imaginary part Im(z) = b.

1. 标准形式：z = a + b i，其中 a, b ∈ ℝ，i² = −1。实部 Re(z) = a；虚部 Im(z) = b。

2. Complex conjugate: z̄ = a − b i. Properties: z z̄ = a² + b², (z₁ ± z₂)̄ = z₁̄ ± z₂̄, (z₁ z₂)̄ = z₁̄ z₂̄, (z₁/z₂)̄ = z₁̄ / z₂̄.

2. 共轭复数：z̄ = a − b i。性质：z z̄ = a² + b²，(z₁ ± z₂)̄ = z₁̄ ± z₂̄，(z₁ z₂)̄ = z₁̄ z₂̄，(z₁/z₂)̄ = z₁̄ / z₂̄。

3. Modulus: |z| = √(a² + b²). Argument: arg(z) = θ such that z = |z|(cos θ + i sin θ), usually −π < θ ⩽ π.

3. 模：|z| = √(a² + b²)。辐角：arg(z) = θ，满足 z = |z|(cos θ + i sin θ)，通常取主值 −π < θ ⩽ π。

4. Polar form / modulus-argument form: z = r(cos θ + i sin θ), where r = |z|, θ = arg(z). Exponential form: z = r eⁱᶿ.

4. 极坐标形式/模辐形式：z = r(cos θ + i sin θ)，其中 r = |z|，θ = arg(z)。指数形式：z = r eⁱᶿ。

5. Multiplication and division in polar form: if z₁ = r₁ eⁱᶿ¹, z₂ = r₂ eⁱᶿ², then z₁ z₂ = r₁ r₂ eⁱ⁽ᶿ¹⁺ᶿ²⁾, z₁/z₂ = (r₁/r₂) eⁱ⁽ᶿ¹⁻ᶿ²⁾.

5. 极坐标形式的乘除：若 z₁ = r₁ eⁱᶿ¹，z₂ = r₂ eⁱᶿ²，则 z₁ z₂ = r₁ r₂ eⁱ⁽ᶿ¹⁺ᶿ²⁾，z₁/z₂ = (r₁/r₂) eⁱ⁽ᶿ¹⁻ᶿ²⁾。

6. de Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for integer n. Also holds for rational n, giving multiple values.

6. 棣莫弗定理：(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ), n 为整数。对有理数 n 也成立，但会得到多值。

7. nth roots of unity: zⁿ = 1 has solutions zₖ = e⁽²πⁱᵏ/ⁿ⁾ for k = 0,1,…, n−1. The roots lie on the unit circle. For a general complex number w, the nth roots are given by z = r^(1/n) e^(i(θ + 2πk)/n).

7. n 次单位根：zⁿ = 1 的解为 zₖ = e⁽²πⁱᵏ/ⁿ⁾，k = 0,1,…, n−1。这些根位于单位圆上。对于一般复数 w，n 次根为 z = r^(1/n) e^(i(θ + 2πk)/n)。

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4. Matrices & Linear Transformations | 矩阵与线性变换

1. Determinant of 2×2 matrix: det [[a, b], [c, d]] = ad − bc. For 3×3, use Sarrus’ rule or cofactor expansion.

1. 2×2 矩阵行列式：det [[a, b], [c, d]] = ad − bc。3×3 矩阵使用萨鲁斯法则或代数余子式展开。

2. Inverse of 2×2 matrix: A⁻¹ = (1/det A) [[d, −b], [−c, a]], provided det A ≠ 0. For larger matrices, use adjugate/determinant or row reduction.

2. 2×2 矩阵的逆：A⁻¹ = (1/det A) [[d, −b], [−c, a]]，要求 det A ≠ 0。更大矩阵用伴随行列式或行化简。

3. Eigenvalues λ are found from det(A − λ I) = 0. Eigenvectors v satisfy (A − λ I) v = 0.

3. 特征值 λ 由 det(A − λ I) = 0 求得。特征向量 v 满足 (A − λ I) v = 0。

4. Diagonalisation: if matrix A has n linearly independent eigenvectors, then A = P D P⁻¹, where D is a diagonal matrix of eigenvalues, and P is the matrix of corresponding eigenvectors as columns.

4. 对角化：若矩阵 A 有 n 个线性无关的特征向量，则 A = P D P⁻¹，其中 D 是特征值组成的对角矩阵，P 的列为对应的特征向量。

5. Cayley-Hamilton theorem: every square matrix satisfies its own characteristic equation, p(A) = 0. This can be used to find powers of a matrix.

5. 凯莱-哈密顿定理：每个方阵满足自身的特征方程 p(A) = 0，可用于求矩阵的幂。

6. Matrix transformations in 2D: rotation by angle θ: [[cos θ, −sin θ], [sin θ, cos θ]]; reflection in line y = x: [[0, 1], [1, 0]]; shear parallel to x-axis: [[1, k], [0, 1]]. Compositions correspond to matrix multiplication (right to left).

6. 二维矩阵变换：旋转角度 θ：[[cos θ, −sin θ], [sin θ, cos θ]]；关于直线 y = x 的反射：[[0, 1], [1, 0]]；平行于 x 轴的剪切：[[1, k], [0, 1]]。复合变换对应矩阵乘法（从右到左）。

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5. Vectors | 向量

1. Dot product (scalar product): a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. For perpendicular vectors, a·b = 0.

1. 点积（数量积）：a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。垂直向量满足 a·b = 0。

2. Cross product (vector product): a × b = |a||b| sin θ n̂, where n̂ is a unit vector perpendicular to both a and b. In components: a × b = (a₂b₃ − a₃b₂) i − (a₁b₃ − a₃b₁) j + (a₁b₂ − a₂b₁) k. Area of parallelogram = |a × b|.

2. 叉积（向量积）：a × b = |a||b| sin θ n̂，其中 n̂ 是垂直于 a 和 b 的单位向量。分量形式：a × b = (a₂b₃ − a₃b₂) i − (a₁b₃ − a₃b₁) j + (a₁b₂ − a₂b₁) k。平行四边形面积 = |a × b|。

3. Vector equation of a line: r = a + t b, where a is a point on the line, b is the direction vector. Cartesian form: (x − x₀)/b₁ = (y − y₀)/b₂ = (z − z₀)/b₃.

3. 直线向量方程：r = a + t b，其中 a 为直线上一点，b 为方向向量。笛卡尔形式：(x − x₀)/b₁ = (y − y₀)/b₂ = (z − z₀)/b₃。

4. Vector equation of a plane: r·n = d, or r = a + s u + t v, where n is a normal vector, a is a point, and u, v are direction vectors. Cartesian form: n₁x + n₂y + n₃z = d.

4. 平面向量方程：r·n = d，或 r = a + s u + t v，其中 n 为法向量，a 为平面上一点，u、v 为方向向量。笛卡尔形式：n₁x + n₂y + n₃z = d。

5. Angle between two vectors: cos θ = (a·b)/(|a||b|). Angle between two planes: cos θ = |n₁·n₂|/(|n₁||n₂|). Angle between line and plane: sin θ = |b·n|/(|b||n|).

5. 两向量夹角：cos θ = (a·b)/(|a||b|)。两平面夹角：cos θ = |n₁·n₂|/(|n₁||n₂|)。直线与平面夹角：sin θ = |b·n|/(|b||n|)。

6. Shortest distance: from point P to line r = a + t b is |(P − a) × b|/|b|. From point P to plane r·n = d is |(P·n − d)|/|n|. Between two skew lines: |(b₁ × b₂)·(a₂ − a₁)| / |b₁ × b₂|.

6. 最短距离：点 P 到直线 r = a + t b 的距离为 |(P − a) × b|/|b|。点 P 到平面 r·n = d 的距离为 |(P·n − d)|/|n|。两异面直线间距离：|(b₁ × b₂)·(a₂ − a₁)| / |b₁ × b₂|。

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6. Series Expansions | 级数展开

1. Maclaurin series: f(x) = f(0) + f'(0) x + f”(0) x²/2! + f”'(0) x³/3! + … + f⁽ⁿ⁾(0) xⁿ/n! + … . Valid within the radius of convergence.

1. 麦克劳林级数：f(x) = f(0) + f'(0) x + f”(0) x²/2! + f”'(0) x³/3! + … + f⁽ⁿ⁾(0) xⁿ/n! + … . 在收敛半径内成立。

2. Standard Maclaurin expansions: eˣ = Σₙ₌₀∞ xⁿ/n!; sin x = Σₙ₌₀∞ (−1)ⁿ x²ⁿ⁺¹/(2n+1)!; cos x = Σₙ₌₀∞ (−1)ⁿ x²ⁿ/(2n)!; ln(1+x) = Σₙ₌₁∞ (−1)ⁿ⁺¹ xⁿ/n, for −1 < x ⩽ 1; (1+x)ᵏ = Σₙ₌₀∞ (ᵏₙ) xⁿ, for |x| < 1 (binomial series).

2. 常用麦克劳林展开：eˣ = Σₙ₌₀∞ xⁿ/n!；sin x = Σₙ₌₀∞ (−1

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