📚 Part 1: Elementary Methods, Differentiation & Complex Numbers | 第一部分：初等方法、微分与复数
This article covers the essential building blocks of IB Mathematics: foundational algebraic techniques, the principles of differentiation, and the algebra of complex numbers. These topics appear in both Analysis & Approaches (AA) and Applications & Interpretation (AI), forming the backbone of further study in calculus and complex analysis.
本文涵盖IB数学的核心模块：初等代数方法、微分的基本原理以及复数代数。这些主题同时出现在分析与方法（AA）和应用与解释（AI）课程中，构成了深入学习微积分与复分析的基础。
1. Elementary Algebraic Techniques | 初等代数技巧
− b)(a + b), and use the sum/product method for trinomials like x² + 5x + 6 = (x+2)(x+3).
二次式的因式分解是解大量方程的第一步。需要识别平方差 a² − b² = (a − b)(a + b)，并对三项式如 x² + 5x + 6 使用和积法得到 (x+2)(x+3)。
The quadratic formula, x = [−b ± √b² − 4ac eal roots if Δ > 0, one repeated root if Δ = 0, and two complex conjugates if Δ < 0.
二次公式 x = [−b ± √b² − 4ac] / (2a) 能求出任意二次方程 ax² + bx + c = 0 的根。判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 有两个相异实根，Δ = 0 有一个重根，Δ < 0 有一对共轭复根。
Polynomial long division is used to factorise higher‑degree polynomials; synthetic division provides a faster shortcut when dividing by a linear factor (x − r).
多项式长除法可用于高次多项式的因式分解；当除数是一次因式 (x − r) 时，综合除法可提供更快的计算路径。
2. Exponents, Logarithms and Equations | 指数、对数与方程
e natural base e ≈ 2.718 appears extensively in IB calculus and exponential growth models.
指数运算法则：aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ。自然底数 e ≈ 2.718 在IB微积分和指数增长模型中大量出现。
Logarithms are the inverse of exponentials: if aˣ hm ln x = logₑ x. Key rules: ln(xy) = ln x + ln y, ln(x/y) = ln x − ln y, ln(xⁿ) = n ln x.
对数是指数的逆运算：若 aˣ = b，则 x = logₐ b。自然对数 ln x = logₑ x。核心法则：ln(xy) = ln x + ln y，ln(x/y) = ln x − ln y，ln(xⁿ) = n ln x。
Change of base: logₐ b = (ln b) / (ln a). This allows solving equations such as 3ˣ = 7 by taking the natural logarithm of both sides.
换底公式：logₐ b = (ln b) / (ln a)。借助它可以通过对方程两边取自然对数来求解如 3ˣ = 7 的方程。
3. Trigonometric Functions and Identities | 三角函数与恒等式
90° = π/2 rad. The unit circle definitions give sin θ = y/r, cos θ = x/r, and tan θ = y/x.
IB 要求学生熟练使用弧度制：π 弧度 = 180°，因此 90° = π/2 弧度。单位圆定义给出 sin θ = y/r，cos θ = x/r，tan θ = y/x。
al identities: tan θ = sin θ / cos θ, sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, and double‑angle formulas sin 2θ = 2 sin θ cos θ, cos 2θ = cos² θ − sin² θ.
必备恒等式：tan θ = sin θ / cos θ，sin² θ + cos² θ = 1，1 + tan² θ = sec² θ，以及倍角公式 sin 2θ = 2 sin θ cos θ，cos 2θ = cos² θ − sin² θ。
uation using identities and finding all solutions in a given interval, e.g. 0 ≤ θ < 2π.
求解三角方程常需利用恒等式变形，并在给定区间内找出所有解，如 0 ≤ θ < 2π。
4. Limits and the Derivative Concept | 极限与导数概念
it lim (x → a) f(x) = L means that f(x) approaches L as x gets arbitrarily close to a. One‑ are used to check continuity: f is continuous at a if limₓ→ₐ⁻ f(x) = limₓ→ₐ⁺ f(x) = f(a).
极限 lim (x → a) f(x) = L 表示当 x 无限趋近 a 时 f(x) 趋近 L。利用单侧极限可以检验连续性：若 limₓ→ₐ⁻ f(x) = limₓ→ₐ⁺ f(x) = f(a)，则 f 在 a 处连续。
f'(x) = lim (h → 0) [f(x + h) − f(x)] / h
The derivative f' iation from first principles applies this limit definition, e.g. for f(x) = x² we obtain f'(x) = 2x.
导数 f'(x) 给出曲线 y = f(x) 切线的斜率。从第一原理求导就是应用上述极限定义，例如对 f(x) = x² 可得出 f'(x) = 2x。
5. Basic Differentiation Rules | 基本求导法则
n xⁿ⁻¹. This rule applies to any real exponent n, including fractions and negative values.
常数的导数为零。幂函数求导：d/dx (xⁿ) = n xⁿ⁻¹。该法则对任意实数指数 n 都适用，包括分数与负数。
| f(x) | f'(x) |
|---|---|
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
The sum/difference rule: (f ± g)' = f' ± g' and the constant multiple rule: (c·f)' = c·f' allow differentiation of polynomial combinations.
和差法则 (f ± g)' = f' ± g' 以及常数倍法则 (c·f)' = c·f' 使我们能对多项式组合进行求导。
6. Advanced Differentiation: Product, Quotient and Chain Rules | 高级求导：乘积、商和链式法则
Product rule: if y = u·v, then dy/dx = u'·v + u·v'. Use it for expressions like x² sin x.
乘积法则：若 y = u·v，则 dy/dx = u'·v + u·v'。适用于 x² sin x 这类表达式。
Quotient rule: if y = u/v, then dy/dx = (u'·v − u·v') / v². It handles rational functions such as (x² + 1)/(x − 1).
商法则：若 y = u/v，则 dy/dx = (u'·v − u·v') / v²。用于处理如 (x² + 1)/(x − 1) 的有理函数。
dx = (dy/du) × (du/dx). It is essential for composite functions like y = sin(3x²) or y = e2x. Choose u = inner function, differentiate, then multiply.
链式法则：dy/dx = (dy/du) × (du/dx)。对复合函数如 y = sin(3x²) 或 y = e2x 必不可少。选取 u = 内层函数，求导后相乘。
7. Implicit Differentiation and Related Rates | 隐函数求导与相关变化率
h respect to x and apply the chain rule to terms involving y, e.g. d/dx (y²) = 2y (dy/dx).
当 y 未显式解出时，可对等式两边关于 x 求导，并对含 y 的项使用链式法则，例如 d/dx (y²) = 2y (dy/dx)。
o find the gradient of curves defined by equations like x² + y² = 25, giving dy/dx = −x/y.
隐函数求导可用于求出由方程如 x² + y² = 25 所确定曲线的斜率，得到 dy/dx = −x/y。
Related rates problems link the rates of change of two or more variables using the chain rule, for instance relating dr/dt and dV/dt for a growing sphere.
相关变化率问题借助链式法则将两个或多个变量的变化率联系起来，例如连接球体膨胀时 dr/dt 与 dV/dt。
8. Higher-Order Derivatives and Applications | 高阶导数及其应用
The second derivative f''(x) measures the rate of change of the gradient; it indicates concavity and helps classify stationary points. If f''(a) > 0, the point is a local minimum; f''(a) < 0 gives a local maximum.
二阶导数 f''(x) 量度斜率的变化率；它表征曲线的凹凸性并帮助对驻点进行分类。若 f''(a) > 0，驻点为局部极小；f''(a) < 0 则为局部极大。
Stationary points occur where f' al optima in a closed interval; evaluate f(x) at endpoints and stationary points to find the absolute maximum and minimum.
驻点出现在 f'(x) = 0 处。在闭区间内求全局最优解时，需计算端点及所有驻点的函数值，比较得出绝对极大与极小。
Kinematics: if s(t) is displacement, then v(t) = s'(t) is velocity and a(t) = v'(t) = s''(t) is acceleration. This models motion in a straight line.
运动学中：若 s(t) 为位移，则速度 v(t) = s'(t)，加速度 a(t) = v'(t) = s''(t)。这用于直线运动建模。
9. Complex Numbers: Cartesian Form | 复数：代数形式
A complex number is written as z = a + bi, where a, b ∈ ℝ and i² = −1. The real part is Re(z) = a, the imaginary part Im(z) = b.
复数记作 z = a + bi，其中 a, b ∈ ℝ，i² = −1。实部 Re(z) = a，虚部 Im(z) = b。
ltiplication uses distributive law and i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i.
加法：(a + bi) + (c + di) = (a + c) + (b + d)i。乘法利用分配律及 i² = −1：(a + bi)(c + di) = (ac − bd) + (ad + bc)i。
The complex conjugate z̄ = a − bi satisfies z·z̄ = a² + b², a real number. Division: (a + bi)/(c + di) = (a + bi)(c − di)/(c² + d²).
共轭复数 z̄ = a − bi 满足 z·z̄ = a² + b²，这是一个实数。除法：(a + bi)/(c + di) = (a + bi)(c − di)/(c² + d²)。
10. Polar Form and de Moivre’s Theorem | 极坐标形式与棣莫弗定理
The modulus of z = a + bi is r = |z| = √a² + b² the origin. The argument θ = arg(z) satisfies tan θ = b/a, with attention to the quadrant.
复数 z = a + bi 的模为 r = |z| = √a² + b²，即到原点的距离。辐角 θ = arg(z) 满足 tan θ = b/a，并需注意象限。
θ. Multiplication: z₁ z₂ = r₁ r₂ cis(θ₁ + θ₂); division: z₁ / z₂ = (r₁ / r₂) cis(θ₁ − θ₂).
极坐标形式：z = r (cos θ + i sin θ) = r cis θ。乘法：z₁ z₂ = r₁ r₂ cis(θ₁ + θ₂)；除法：z₁ / z₂ = (r₁ / r₂) cis(θ₁ − θ₂)。
nθ, for integer n. It is used to compute powers and roots of complex numbers efficiently.
棣莫弗定理：(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ，对整数 n 成立。它被用于高效计算复数的乘幂与方根。
11. Solving Polynomial Equations with Complex Roots | 用复数求解多项式方程
By the fundamental theorem of algebra, a polynomial of degree n has exactly n complex roots (counting multiplicity). Non‑real roots of real polynomials occur in conjugate pairs.
根据代数基本定理，n 次多项式恰有 n 个复数根（计入重数）。实系数多项式的非实数根以共轭对形式出现。
To find nth roots of a complex number, express z in polar form and apply: wₖ = r1/n cis[(θ + 2πk)/n] for k = 0, 1, …, n−1. The roots are evenly spaced around a circle of radius r1/n.
求复数的 n 次方根时，先将 z 写成极坐标形式，然后用 wₖ = r1/n cis[(θ + 2πk)/n]（k = 0, 1, …, n−1）得到所有根。它们在半径为 r1/n 的圆上等距分布。
For quadratic equations with negative discriminant, the roots are complex conjugates given by the quadratic formula. This connects back to elementary methods and shows the unity of the IB syllabus.
当二次方程判别式为负时，其根为共轭复数，可由二次公式给出。这联系起初等方法，展现了IB课程体系的整体性。
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