IB Math AA Core Topics Summary: Probability, Calculus, Algebra, Geometry | IB数学AA核心考点梳理：概率、微积分、代数、几何

📚 IB Math AA Core Topics Summary: Probability, Calculus, Algebra, Geometry | IB数学AA核心考点梳理：概率、微积分、代数、几何

IB Mathematics: Analysis and Approaches (AA) demands a deep conceptual understanding and strong algebraic fluency. This article synthesises the core topics—Probability, Calculus, Algebra, and Geometry—that frequently appear in examinations, providing a clear, bilingual revision guide to help you target high marks.

IB数学分析与方法（AA）要求深刻的概念理解与纯熟的代数技能。本文梳理了概率、微积分、代数、几何等高频考点，提供一份清晰的双语复习指南，帮助你冲击高分。

1. Algebra and Functions | 代数与函数

Mastering algebra is foundational. You must be confident in manipulating exponents and logarithms, applying the laws of indices and log properties to solve equations like aˣ = b. For logarithmic expressions, remember logₐ(xy) = logₐx + logₐy and the change of base formula: logₐb = log_c b / log_c a.

代数是基础。必须熟练掌握指数与对数运算，运用指数律与对数性质解方程如 aˣ = b。对数表达式要牢记 logₐ(xy) = logₐx + logₐy，以及换底公式：logₐb = log_c b / log_c a。

Binomial expansion is another key skill. For (a + b)ⁿ, use the binomial theorem with nCᵢ coefficients. In HL, you extend this to rational exponents using the infinite series (1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + … for |x| < 1.

二项式展开是另一关键技能。对于 (a + b)ⁿ，使用含 nCᵢ 系数的二项式定理。在 HL 中，需拓展到有理指数，利用无穷级数 (1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + … 其中 |x| < 1。

Partial fractions decomposition aids integration. For rational functions, express as sum of simpler fractions, e.g., (2x+1)/((x-1)(x+2)) = A/(x-1) + B/(x+2). Ensure linear factors are covered, and in HL, repeated and irreducible quadratic factors.

部分分式分解有助于积分。对有理函数，可写为简单分式之和，例如 (2x+1)/((x-1)(x+2)) = A/(x-1) + B/(x+2)。SL 掌握一次因子，HL 需处理重复因子和不可约二次因子。

2. Sequences and Series | 数列与级数

Arithmetic sequences have a common difference d; the nth term uₙ = u₁ + (n-1)d and sum Sₙ = n/2 (2u₁ + (n-1)d). Geometric sequences have common ratio r; uₙ = u₁ rⁿ⁻¹ and sum of finite terms Sₙ = u₁(1 – rⁿ)/(1 – r), with infinite sum S∞ = u₁/(1 – r) for |r| < 1.

等差数列公差为 d，通项 uₙ = u₁ + (n-1)d，求和 Sₙ = n/2 (2u₁ + (n-1)d)。等比数列公比为 r，通项 uₙ = u₁ rⁿ⁻¹，有限项和 Sₙ = u₁(1 – rⁿ)/(1 – r)，当 |r| < 1 时无穷和 S∞ = u₁/(1 - r)。

Sigma notation Σ is used extensively for sums. Understand how to manipulate sums, such as Σ(k²) from k=1 to n = n(n+1)(2n+1)/6. In HL, you also use Maclaurin series to approximate functions: eˣ = Σ xⁿ/n!, sin x = Σ (-1)ⁿ x²ⁿ⁺¹/(2n+1)!, cos x = Σ (-1)ⁿ x²ⁿ/(2n)!.

求和符号 Σ 广泛使用。需灵活处理求和，如 Σ(k²) (k=1到n) = n(n+1)(2n+1)/6。HL 中还使用麦克劳林级数逼近函数：eˣ = Σ xⁿ/n!，sin x = Σ (-1)ⁿ x²ⁿ⁺¹/(2n+1)!，cos x = Σ (-1)ⁿ x²ⁿ/(2n)!。

3. Polynomials and Rational Functions | 多项式与有理函数

The factor and remainder theorems are essential: for polynomial P(x), if P(a)=0 then (x-a) is a factor; the remainder when dividing by (x-a) is P(a). Use these to factorise cubic and quartic equations. The sum and product of roots formulas (Vieta’s formulas) link coefficients to roots: for quadratic ax²+bx+c, sum = -b/a, product = c/a.

因式定理与余数定理至关重要：对于多项式 P(x)，若 P(a)=0 则 (x-a) 为因式；除以 (x-a) 的余数为 P(a)。利用它们分解三次、四次方程。根的和与积公式（韦达定理）连接系数与根：二次 ax²+bx+c，根的和 = -b/a，积 = c/a。

Sketching rational functions requires asymptotes and intercepts. Vertical asymptotes occur where denominator equals zero (and no cancellation). Horizontal asymptotes are determined by comparing degrees of numerator and denominator. Oblique asymptotes arise if degree of numerator exactly one greater.

草绘有理函数需要渐近线与截距。分母为零（且不可约）处出现垂直渐近线。水平渐近线由分子与分母的次数比较决定。若分子次数比分母恰高一次，则出现斜渐近线。

4. Geometry and Trigonometry | 几何与三角学

Radian measure is fundamental: π rad = 180°. Arc length s = rθ, sector area A = ½ r² θ. The unit circle defines sine, cosine, and tangent. Essential identities include sin²θ + cos²θ = 1, double angle formulas sin 2θ = 2 sinθ cosθ, cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ.

弧度制是基础：π 弧度 = 180°。弧长 s = rθ，扇形面积 A = ½ r² θ。单位圆定义正弦、余弦与正切。核心恒等式包括 sin²θ + cos²θ = 1，倍角公式 sin 2θ = 2 sinθ cosθ，cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ。

Solving trigonometric equations requires using identities and periodic properties. Find all solutions within a given interval, typically using the general solution: sin x = sin α ⇒ x = nπ + (-1)ⁿ α. For non-right triangles, apply the sine rule a/sin A = b/sin B = c/sin C and cosine rule a² = b² + c

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