📚 IB Mathematics Analysis and Approaches: Key Concepts Revision | IB 数学分析与方法知识点精讲
The IB Mathematics: Analysis and Approaches (AA) course is designed for students who enjoy developing rigorous mathematical arguments, exploring abstract concepts, and solving problems with precision. This revision guide walks you through the essential topics that form the backbone of the AA syllabus, from algebraic foundations to advanced calculus and proof techniques.
IB 数学分析与方法课程专为喜爱构建严谨数学论证、探索抽象概念和精确解题的学生设计。这份复习指南将带你梳理 AA 课程的核心知识点，从代数基础到高等微积分和证明方法，帮助你系统掌握考试重点。
1. Functions and Transformations | 函数与变换
A function f maps each element x of the domain to a unique output f(x). Understanding domain, range, and basic transformations is essential for sketching graphs and solving equations.
函数 f 将定义域中的每个元素 x 映射到一个唯一的输出 f(x)。理解定义域、值域和基本变换对画图和求解方程至关重要。
Horizontal shifts: y = f(x – h) translates the graph h units to the right. Vertical shifts: y = f(x) + k moves it up by k units. Reflections: y = -f(x) reflects in the x-axis, y = f(-x) reflects in the y-axis.
水平平移：y = f(x – h) 将图像向右平移 h 个单位。垂直平移：y = f(x) + k 向上平移 k 个单位。对称：y = -f(x) 关于 x 轴对称，y = f(-x) 关于 y 轴对称。
Stretches: y = a f(x) with a > 1 stretches vertically by factor a, while y = f(bx) with b > 1 compresses horizontally by factor 1/b. Composite functions (f ∘ g)(x) = f(g(x)) combine two mappings, but always check that the range of g lies within the domain of f.
伸缩：y = a f(x) (a > 1) 垂直方向拉伸为原来的 a 倍，y = f(bx) (b > 1) 水平方向压缩为原来的 1/b。复合函数 (f ∘ g)(x) = f(g(x)) 组合两次映射，但需确保 g 的值域包含在 f 的定义域内。
2. Sequences and Series | 数列与级数
Arithmetic sequences have a constant difference d between consecutive terms: uₙ = u₁ + (n – 1)d. The sum of the first n terms is Sₙ = n/2 (2u₁ + (n – 1)d) or Sₙ = n/2 (u₁ + uₙ).
等差数列相邻两项的差 d 为常数：uₙ = u₁ + (n – 1)d。前 n 项和为 Sₙ = n/2 (2u₁ + (n – 1)d) 或 Sₙ = n/2 (u₁ + uₙ)。
Geometric sequences multiply by a constant ratio r: uₙ = u₁ rⁿ⁻¹. The sum of the first n terms is Sₙ = u₁(1 – rⁿ)/(1 – r) for r ≠ 1. An infinite geometric series converges to S∞ = u₁/(1 – r) when |r| < 1.
等比数列每一项乘以常数 r：uₙ = u₁ rⁿ⁻¹。前 n 项和为 Sₙ = u₁(1 – rⁿ)/(1 – r)（r ≠ 1）。当 |r| < 1 时，无穷等比级数收敛于 S∞ = u₁/(1 - r)。
Sigma notation (Σ) compactly represents sums. You should be comfortable manipulating summation indices and applying formulas for ∑k, ∑k², and ∑k³ when evaluating polynomial series.
求和符号 Σ 可简洁表示级数。你应熟练掌握求和指标的变换，并会运用 ∑k、∑k² 和 ∑k³ 的公式来计算多项式级数。
3. Exponentials and Logarithms | 指数与对数
Exponential functions of the form f(x) = aˣ (a > 0, a ≠ 1) have domain all real numbers and range y > 0. The natural exponential function eˣ appears extensively in calculus and growth models.
指数函数 f(x) = aˣ（a > 0, a ≠ 1）的定义域为全体实数，值域为 y > 0。自然指数函数 eˣ 在微积分和增长模型中频繁出现。
Logarithms are the inverses of exponentials: logₐ x = y ⇔ aʸ = x. Key laws: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x – logₐ y, logₐ (xⁿ) = n logₐ x. Change of base: logₐ b = logₓ b / logₓ a.
对数是指数的逆运算：logₐ x = y ⇔ aʸ = x。核心运算法则：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x – logₐ y，logₐ (xⁿ) = n logₐ x。换底公式：logₐ b = logₓ b / logₓ a。
The natural logarithm ln x = logₑ x has derivative d/dx (ln x) = 1/x. Solving exponential equations often involves taking logarithms on both sides and applying the power rule.
自然对数 ln x = logₑ x 的导数为 d/dx (ln x) = 1/x。求解指数方程时常常需要对两边取对数并运用幂法则。
4. Trigonometry | 三角学
The unit circle defines sine, cosine, and tangent for any real angle. Radian measure is essential: π rad = 180°. Arc length s = rθ, sector area A = ½ r²θ when θ is in radians.
单位圆定义了任意实角的三角函数。弧度制很关键：π 弧度 = 180°。弧长 s = rθ，扇形面积 A = ½ r²θ，其中 θ 以弧度为单位。
Key identities: sin²θ + cos²θ = 1, tan θ = sin θ / cos θ. Double-angle formulas: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ.
核心恒等式：sin²θ + cos²θ = 1，tan θ = sin θ / cos θ。倍角公式：sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ。
Trigonometric functions can be transformed like any other: y = a sin(b(x – c)) + d has amplitude |a|, period 2π/|b|, phase shift c, and vertical shift d. The inverse trig functions arcsin, arccos, arctan have restricted domains and ranges.
三角函数也可进行一般变换：y = a sin(b(x – c)) + d 的振幅为 |a|，周期为 2π/|b|，相位平移为 c，垂直平移为 d。反三角函数 arcsin、arccos、arctan 具有受限的定义域和值域。
5. Complex Numbers (HL) | 复数（高水平）
Complex numbers extend the real numbers by introducing i² = -1. A complex number z = a + bi has real part a and imaginary part b. The complex conjugate is z* = a – bi.
复数通过引入 i² = -1 扩展了实数。复数 z = a + bi 的实部为 a，虚部为 b。共轭复数为 z* = a – bi。
Argand diagrams represent complex numbers as points or vectors. The modulus |z| = √(a² + b²) gives distance from origin; the argument arg(z) = θ satisfies tan θ = b/a. Polar form: z = r(cos θ + i sin θ) = r cis θ.
阿尔冈图将复数表示为点或向量。模 |z| = √(a² + b²) 表示到原点的距离；辐角 arg(z) = θ 满足 tan θ = b/a。极坐标形式：z = r(cos θ + i sin θ) = r cis θ。
De Moivre’s theorem: (r cis θ)ⁿ = rⁿ cis (nθ). This is used to find powers and roots of complex numbers. The n distinct nth roots of unity are given by cis (2πk/n) for k = 0, 1, …, n-1.
棣莫弗定理：(r cis θ)ⁿ = rⁿ cis (nθ)，用于求复数的乘方与开方。n 个不同的 n 次单位根为 cis (2πk/n)，k = 0, 1, …, n-1。
6. Differential Calculus | 微分学
The derivative f'(x) = limₕ→₀ [f(x+h) – f(x)]/h represents the instantaneous rate of change and the slope of the tangent. Differentiation rules: power rule d/dx (xⁿ) = nxⁿ⁻¹, constant multiple, sum/difference.
导数 f'(x) = limₕ→₀ [f(x+h) – f(x)]/h 表示瞬时变化率和切线的斜率。求导法则：幂法则 d/dx (xⁿ) = nxⁿ⁻¹，常数倍数法则，和差法则。
Product rule: (uv)’ = u’v + uv’. Quotient rule: (u/v)’ = (u’v – uv’)/v². Chain rule: d/dx f(g(x)) = f'(g(x))·g'(x). These are fundamental for differentiating composite, rational, and trigonometric functions.
乘积法则：(uv)’ = u’v + uv’。商法则：(u/v)’ = (u’v – uv’)/v²。链式法则：d/dx f(g(x)) = f'(g(x))·g'(x)。它们是求复合函数、有理函数和三角函数导数的基础。
Derivatives of special functions: d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x. Tangents and normals: equation of tangent at (a, f(a)) is y – f(a) = f'(a)(x – a).
特殊函数的导数：d/dx (sin x) = cos x，d/dx (cos x) = -sin x，d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x。切线与法线：在点 (a, f(a)) 处的切线方程为 y – f(a) = f'(a)(x – a)。
7. Integral Calculus | 积分学
Integration reverses differentiation. Indefinite integral ∫ f(x) dx = F(x) + C, where F'(x) = f(x). Basic rules: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1), and ∫ 1/x dx = ln|x| + C.
积分是微分的逆运算。不定积分 ∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)。基本法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ -1），∫ 1/x dx = ln|x| + C。
Definite integrals compute the signed area under a curve between limits a and b: ∫ₐᵇ f(x) dx = F(b) – F(a). Integration by substitution is the reverse of the chain rule; for HL, integration by parts ∫ u dv = uv – ∫ v du is also required.
定积分计算曲线在 a 和 b 之间的带符号面积：∫ₐᵇ f(x) dx = F(b) – F(a)。换元积分法是链式法则的逆运算；高水平还需掌握分部积分法 ∫ u dv = uv – ∫ v du。
Area between curves: A = ∫ₐᵇ [f(x) – g(x)] dx where f(x) ≥ g(x). Volumes of revolution about the x-axis use V = π ∫ₐᵇ [f(x)]² dx.
曲线间的面积：A = ∫ₐᵇ [f(x) – g(x)] dx，其中 f(x) ≥ g(x)。绕 x 轴旋转的体积为 V = π ∫ₐᵇ [f(x)]² dx。
8. Kinematics | 运动学
Kinematics links displacement s(t), velocity v(t) = s'(t), and acceleration a(t) = v'(t) = s”(t) as functions of time. The total distance travelled is the integral of the absolute value of velocity.
运动学将位移 s(t)、速度 v(t) = s'(t) 和加速度 a(t) = v'(t) = s”(t) 作为时间函数联系起来。总路程是对速度绝对值积分。
Given a(t), integrating gives v(t) = ∫ a(t) dt, using initial velocity to find the constant. Integrating again yields s(t). Applications often require finding when the particle changes direction (v(t) = 0) or maximum speed.
已知 a(t)，积分得 v(t) = ∫ a(t) dt，利用初速度确定常数。再次积分得到 s(t)。应用时常需求解粒子何时改变运动方向 (v(t) = 0) 或最大速率。
9. Probability and Statistics | 概率与统计
Probability axioms: for any event A, 0 ≤ P(A) ≤ 1, P(S) = 1. The complement rule: P(A’) = 1 – P(A). For mutually exclusive events, P(A ∪ B) = P(A) + P(B); otherwise, P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
概率公理：对任一事件 A，0 ≤ P(A) ≤ 1，P(S) = 1。补集规则：P(A’) = 1 – P(A)。互斥事件满足 P(A ∪ B) = P(A) + P(B)；一般有 P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。
Conditional probability: P(A|B) = P(A ∩ B)/P(B). Independence means P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A). Bayes’ theorem often appears in HL problems.
条件概率：P(A|B) = P(A ∩ B)/P(B)。独立性指 P(A ∩ B) = P(A)P(B)，等价于 P(A|B) = P(A)。高水平题目常涉及贝叶斯定理。
Random variables: discrete distributions use probability mass functions; expectation E(X) = Σ x·P(X = x), Var(X) = E(X²) – [E(X)]². The binomial distribution B(n, p) models number of successes with fixed number of independent trials.
随机变量：离散分布使用概率质量函数；期望 E(X) = Σ x·P(X = x)，方差 Var(X) = E(X²) – [E(X)]²。二项分布 B(n, p) 模拟固定次数的独立试验中成功的次数。
For the normal distribution X ~ N(μ, σ²), use standardization Z = (X – μ)/σ to find probabilities. The inverse normal function is essential for solving for unknown means or bounds.
对于正态分布 X ~ N(μ, σ²)，通过标准化 Z = (X – μ)/σ 求概率。逆正态函数对于求解未知均值或边界至关重要。
10. Proof and Mathematical Induction (HL) | 证明与数学归纳法（高水平）
Direct proof starts from known facts to deduce a conclusion. Proof by contradiction assumes the negation of the statement and derives an impossibility. Proof by contrapositive proves “if P then Q” by showing “if not Q then not P”.
直接证明从已知事实出发推导结论。反证法假设命题的否定成立并推出矛盾。逆否命题证明通过证明“若非 Q 则非 P”来证明“若 P 则 Q”。
Mathematical induction is used to prove statements P(n) for all positive integers n. The base case verifies P(1). The inductive step assumes P(k) and deduces P(k+1). Common applications include sums of series, divisibility, and inequalities.
数学归纳法用于证明对所有正整数 n 成立的命题 P(n)。基础步骤验证 P(1)；归纳步骤假设 P(k) 成立并推出 P(k+1)。常见应用包括级数求和、整除性和不等式证明。
For example, to prove 1 + 2 + 3 + … + n = n(n+1)/2, check n=1 gives 1 = 1. Assume for k, then add (k+1) to both sides and simplify to k(k+1)/2 + (k+1) = (k+1)(k+2)/2, completing the induction.
例如，证明 1 + 2 + 3 + … + n = n(n+1)/2：验证 n=1 得 1 = 1。假设 k 成立，两边加 (k+1) 并化简得 k(k+1)/2 + (k+1) = (k+1)(k+2)/2，完成归纳。
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