📚 IB Mathematics: High-Frequency Exam Topics Summary | IB 数学：高频考点总结
The IB Mathematics curriculum covers a broad range of topics, but certain concepts and question types appear consistently across both Analysis & Approaches (AA) and Applications & Interpretation (AI) at Standard and Higher Levels. By focusing on these high-frequency exam topics, you can streamline your revision and boost your performance. This guide summarises the most critical areas, blending theoretical understanding with practical problem-solving strategies.
IB 数学课程涵盖广泛的主题，但某些概念和题型在分析与方法（AA）以及应用与解释（AI）的标准水平和高水平考试中反复出现。通过聚焦这些高频考点，你可以优化复习并提高成绩。本指南总结了最关键的领域，将理论理解与实际解题策略相结合。
1. Functions and Equations | 函数与方程
Mastering the concept of a function, including domain, range, and the vertical line test, is fundamental. In exams, you are frequently asked to find the domain of composite functions or to sketch transformed graphs by applying translations, reflections, and stretches.
掌握函数的概念，包括定义域、值域和垂直线检验，是基础中的基础。考试中经常要求求解复合函数的定义域，或通过平移、反射和伸缩来绘制变换后的函数图像。
Solving quadratic equations using the quadratic formula or completing the square, and analysing the discriminant to determine the nature of roots, appears in almost every IB Maths paper. Be prepared to tackle disguised quadratics through substitution.
使用求根公式或配方法解二次方程，以及通过判别式分析根的性质，几乎出现在每一份IB数学试卷中。请准备好通过换元解决隐二次方程问题。
Exponential and logarithmic equations require a firm grasp of index and log laws. You may need to combine them with algebraic manipulation or graphing to find intersections; pay attention to domain restrictions, especially inside logarithms.
指数和对数方程需要牢固掌握指数与对数运算律。你可能需要结合代数操作或图像来求解交点；特别注意定义域限制，尤其是对数内部的限制。
Rational functions and their asymptotes (vertical, horizontal, and oblique) are common in HL AA and also feature in AI models. Sketching these graphs by analysing intercepts and asymptotic behaviour is a recurrent skill.
有理函数及其渐近线（垂直、水平、斜渐近线）在HL AA中常见，也出现在AI模型中。通过分析截距和渐近行为来绘制这些函数的图像是一项常见的技能。
2. Trigonometry and Circular Functions | 三角学与圆函数
The unit circle definitions of sine, cosine, and tangent are the building blocks. Exams expect you to know exact trigonometric values for angles like 30°, 45°, and 60°, and to convert between degrees and radians fluently.
单位圆定义的正弦、余弦和正切是基础。考试要求你掌握30°、45°和60°等特殊角的精确三角值，并能熟练地在度与弧度之间转换。
Trigonometric identities are extremely high-yield. You must be able to apply Pythagorean identities, double angle formulas, and compound angle formulas to simplify expressions and solve equations. A summary table of the most tested identities is shown below.
三角恒等式是出题率极高的考点。你必须能够应用毕达哥拉斯恒等式、倍角公式和和角公式来化简表达式和求解方程。下表总结了最常考的恒等式。
| Identity | 恒等式 |
|---|---|
| sin²θ + cos²θ = 1 | sin²θ + cos²θ = 1 |
| tanθ = sinθ / cosθ | tanθ = sinθ / cosθ |
| sin(2θ) = 2 sinθ cosθ | sin(2θ) = 2 sinθ cosθ |
| cos(2θ) = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ | cos(2θ) = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ |
| sin(A ± B) = sinA cosB ± cosA sinB | sin(A ± B) = sinA cosB ± cosA sinB |
| cos(A ± B) = cosA cosB ∓ sinA sinB | cos(A ± B) = cosA cosB ∓ sinA sinB |
Solving trigonometric equations in a specified interval is a staple extended-response question. Always check the required domain and use the symmetry of the unit circle to find all possible solutions.
在指定区间内解三角方程是必考的拓展题。务必检查要求的定义域，并利用单位圆的对称性找出所有可能的解。
Graphs of y = a sin(bx + c) + d and their transformations are frequently assessed. Understand how parameters affect amplitude, period, phase shift, and vertical shift. For HL, be comfortable with the reciprocal functions sec, csc, cot and their graphs, as well as inverse trigonometric functions and their restricted domains.
y = a sin(bx + c) + d 的图像及其变换是常考内容。要理解参数如何影响振幅、周期、相移和垂直位移。对于HL，需熟悉倒数函数 sec、csc、cot 及其图像，以及反三角函数及其限定定义域。
3. Differentiation | 微分
Differentiation rules – power, sum, product, quotient, and especially the chain rule – must become second nature. Exams often combine these rules when differentiating complicated functions like esin x or ln(x² + 1).
微分法则——幂函数、和、乘积、商，尤其是链式法则——必须成为你的第二天性。考试中经常组合这些法则来求复杂函数的导数，如 esin x 或 ln(x² + 1)。
Finding equations of tangents and normals at a point is a high-frequency short question. Remember that the derivative gives the gradient of the tangent; the normal’s gradient is the negative reciprocal.
求解某一点处的切线和法线方程是高频短问题。记住导数给出切线的斜率；法线的斜率是负倒数。
Stationary points, intervals of increase/decrease, concavity, and points of inflection are tested both analytically and graphically. You are expected to construct sign diagrams for the first and second derivatives and to classify stationary points as local maxima, minima, or points of inflection.
驻点、增减区间、凹凸性和拐点既从解析也从图像角度进行考查。你需要构建一阶和二阶导数的符号图，并将驻点分类为局部极大值、极小值或拐点。
Optimisation problems, where you must model a real-life scenario and use calculus to find maximum or minimum values, are classic in SL and HL. Steps include writing a primary equation, reducing it to one variable using a constraint, differentiating, and verifying the nature of the extremum.
优化问题是SL和HL中的经典题型，需要你对实际情境建模并用微积分求最大值或最小值。步骤包括写出主方程、利用约束条件消元至单变量、求导并验证极值性质。
For HL, implicit differentiation, related rates, and parametric differentiation are essential. You should also be able to derive Maclaurin series for standard functions like sin x, cos x, and ex, and use them to approximate function values.
对于HL，隐函数求导、相关变化率和参数微分是必须掌握的。你还应该能推导 sin x、cos x 和 ex 等标准函数的麦克劳林级数，并用它们近似函数值。
4. Integration | 积分
Integration as the reverse of differentiation is the core idea. You must memorise the standard integrals: ∫ xn dx (n ≠ -1), ∫ 1/x dx, ∫ ex dx, ∫ sin x dx, ∫ cos x dx, and ∫ sec² x dx.
积分作为微分的逆运算是核心思想。你必须熟记标准积分：∫ xn dx (n ≠ -1)、∫ 1/x dx、∫ ex dx、∫ sin x dx、∫ cos x dx 和 ∫ sec² x dx。
Definite integrals represent the signed area under a curve. Be ready to calculate the area between a curve and the x-axis, as well as the area enclosed between two curves. Watch out for intervals where the function changes sign – you may need to split the integral.
定积分表示曲线下的带符号面积。要准备好计算曲线与 x 轴之间的面积，以及两条曲线之间围成的面积。注意函数改变符号的区间——你可能需要分割积分区间。
Integration by substitution (reverse chain rule) is a heavily examined technique. Choose u to simplify the integrand, and remember to change the limits when evaluating definite integrals. Trigonometric substitutions like letting u = cos x when ∫ sin x cos³ x dx appears are very common.
换元积分法（反链式法则）是考查极多的技巧。选择 u 来简化被积函数，并记住在计算定积分时更换积分限。三角换元，如当出现 ∫ sin x cos³ x dx 时令 u = cos x，非常常见。
HL candidates must additionally master integration by parts, partial fractions decomposition, and calculating volumes of revolution around both axes. Volumes of revolution questions often combine sketching, integration, and algebraic manipulation.
HL 考生还必须掌握分部积分法、部分分式分解以及计算绕两坐标轴旋转的旋转体体积。旋转体体积问题常常结合了作图、积分和代数处理。
Simple differential equations of the form dy/dx = f(x)g(y) appear at SL, while HL extends to first-order linear equations using an integrating factor, and homogeneous differential equations. Be able to solve and interpret the particular solution given initial conditions.
简单的可分离变量微分方程 dy/dx = f(x)g(y) 出现在SL中，而HL延伸到使用积分因子的一阶线性方程和齐次微分方程。要能够求解并解释给定初始条件下的特解。
5. Sequences and Series | 数列与级数
Arithmetic sequences (un = u1 + (n-1)d) and series sum (Sn = n/2(u1 + un)) are straightforward but frequently used in problem-solving contexts, including financial applications like simple interest.
等差数列（un = u1 + (n-1)d）及其级数和（Sn = n/2(u1 + un)）虽然简单，但常用于解决问题场景，包括单利等金融应用。
Geometric sequences and the sum of a finite geometric series, as well as the sum to infinity for |r| < 1, are exam favourites. You must be able to find the common ratio, the first term, or the number of terms from given information.
等比数列、有限项等比级数和以及无穷项和（|r| < 1）是考试的热门。你需要根据给定信息求出公比、首项或项数。
Sigma notation is used extensively. Practice expanding sums like Σk=15 (2k + 1) and converting series into sigma notation. This skill is essential for mathematical induction proofs at HL.
西格玛记法被广泛使用。练习展开如 Σk=15 (2k + 1) 的求和，并将级数转换为西格玛记法。这项技能对于HL中的数学归纳法证明至关重要。
The binomial theorem for positive integer exponents, nCr = n!/(r!(n-r)!), and using Pascal’s triangle are directly tested. At HL, the extension to rational exponents (1 + x)n with its infinite series and validity condition |x| < 1 is frequently examined.
正整数指数的二项式定理 nCr = n!/(r!(n-r)!) 和帕斯卡三角形是直接考查点。在HL中，扩展到有理数指数 (1 + x)n 的无穷级数及有效性条件 |x| < 1 也常被考查。
6. Vectors | 向量
Vector basics – notation, magnitude, unit vectors, and addition/scalar multiplication – form the foundation. The scalar (dot) product is heavily used to find the angle between two vectors and to determine perpendicularity.
向量基础——记法、模、单位向量以及加法和标量乘法——构成基石。数量积（点积）被大量用于求两向量之间的夹角和判定垂直关系。
The vector equation of a line in 2D and 3D, r = a + λb, is a core topic. You must be able to find intersections of two lines and, in 3D, determine whether lines are parallel, intersecting, or skew.
二维和三维中直线的向量方程 r = a + λb 是核心主题。你必须能够求出两条直线的交点，并在三维中判断直线是平行、相交还是异面。
HL candidates work extensively with the vector (cross) product to find a vector perpendicular to two given vectors, and to form the equation of a plane: r·n = a·n. Finding the intersection of a line and a plane, or the line of intersection between two planes, requires systematic algebraic solving.
HL 考生广泛使用向量积（叉积）来求垂直于两给定向量的向量，并建立平面方程：r·n = a·n。求直线与平面的交点，或两平面的交线，需要系统的代数求解。
Kinematics with vectors is another high-frequency area: position, velocity, and acceleration vectors are differentiated/integrated with respect to time. Questions often ask for the speed (magnitude of velocity) or the time when an object is at a certain location.
向量运动学是另一高频领域：位置、速度和加速度向量对时间进行微分/积分。题目常会求速率（速度的模）或物体在某一位置的时间。
7. Complex Numbers | 复数
Complex numbers are primarily an HL AA topic. The Cartesian form z = a + bi, complex conjugates, modulus |z| = √(a² + b²), and argument Arg(z) are the starting points. Operations like addition, subtraction, multiplication, and division (using conjugates) are tested routinely.
复数主要是HL AA的内容。笛卡尔形式 z = a + bi、共轭复数、模 |z| = √(a² + b²) 和辐角 Arg(z) 是起点。加、减、乘、除（使用共轭）等运算常被考查。
Polar form z = r(cosθ + i sinθ) and Euler’s form z =
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