📚 IB Maths Topic 5: Calculus – SL and HL | IB 数学专题五：微积分 – SL 与 HL
Calculus is the mathematical study of continuous change and motion, forming the core of IB Mathematics Analysis and Approaches. In Topic 5, both Standard Level (SL) and Higher Level (HL) students explore limits, differentiation, integration, and their applications, but HL extends significantly into advanced techniques such as implicit differentiation, integration by parts, Maclaurin series, and volumes of revolution. This bilingual guide breaks down the syllabus section by section, highlighting key concepts, formulas, and differences between SL and HL, all with clear English–Chinese explanations.
微积分是研究连续变化与运动的数学分支，构成 IB 数学分析与方法的基石。在专题五中，标准级别（SL）和高级级别（HL）学生都将探索极限、微分、积分及其应用，但 HL 会显著深入到隐函数微分、分部积分、麦克劳林级数和旋转体体积等进阶技法。本双语指南按章节拆解考纲，突显关键概念、公式以及 SL 与 HL 的区别，全部配以清晰的英文–中文解释。
1. Limits and Continuity (HL) | 极限与连续性（HL）
by factorising, rationalising, or using standard limits such as lim(x → 0) (sin x)/x = 1.
极限的概念写作 lim(x → a) f(x) = L，描述当 x 无限趋近 a 时 f(x) 所趋近的值。这一形式化思想支撑着整个微积分，主要在 HL 中考查。学生须能通过因式分解、有理化或使用标准极限（如 lim(x → 0) (sin x)/x = 1）来代数地计算极限。
can be removable (point), jump, or infinite. HL problems often ask you to determine constants that make a piecewise function continuous, or to use the squeeze theorem to prove a limit.
若 lim(x → a) f(x) = f(a)，则函数 f 在 x = a 处连续。间断点可以是可去（点）间断、跳跃间断或无穷间断。HL 题目常要求确定分段函数连续所需的常数，或使用夹逼定理证明极限。
2. Introduction to Derivatives: From First Principles (SL & HL) | 导数入门：从第一原理（SL 和 HL）
The derivative of a function f at a point x is defined by the limit f'(x) = lim(h → 0) [f(x+h) – dents must be able to apply this definition to simple functions like f(x) = x² to derive f'(x) = 2x, linking the formal concept to the slope of a tangent.
函数 f 在某点 x 的导数由极限 f'(x) = lim(h → 0) [f(x+h) – f(x)] / h 定义，只要该极限存在。这一过程称为从第一原理求导。所有 SL 与 HL 学生都必须能将此定义应用于像 f(x) = x² 这样的简单函数，从而推出 f'(x) = 2x，将形式概念与切线斜率联系起来。
The notation f'(x) or dy/dx is used for the derivative. Geometrically, the derivative at a point gives the gradient of the tangent line to the curve at that point. A derivative that does not exist often indicates a corner, cusp, or vertical tangent.
导数可用 f'(x) 或 dy/dx 表示。从几何上看，某点的导数给出曲线在该点处的切线斜率。导数不存在通常意味着存在角点、尖点或垂直切线。
3. Differentiation Rules (SL & HL) | 微分法则（SL 和 HL）
Once first principles are understood, students apply a set of standard rules to differentiate a wide variety of functions efficiently. The following table summarises the fundamental rules required for both SL and HL. For the chain rule, HL students will encounter more layered compositions, but the core technique is the same.
理解第一原理后，学生运用一组标准法则来高效地为各类函数求导。下表总结了 SL 和 HL 都必需的基本法则。对于链式法则，HL 学生会遇到更多层的复合函数，但核心技法是相同的。
| Rule (English) | 中文名称 | Formula | Example |
|---|---|---|---|
| Power Rule | 幂法则 | d/dx (xⁿ) = n xⁿ⁻¹ | d/dx (x⁵) = 5x⁴ |
| Constant Multiple | 常数倍法则 | d/dx [c f(x)] = c f'(x) | d/dx (7x³) = 21x² |
| Sum/Difference | 和差法则 | d/dx [f(x) ± g(x)] = f'(x) ± g'(x) | d/dx (x² + sin x) = 2x + cos x |
| Product Rule | 乘积法则 | d/dx [u v] = u' v + u v' | d/dx (x² eˣ) = 2x eˣ + x² eˣ |
| Quotient Rule | 商法则 | d/dx (u/v) = (u' v – u v') / v² | d/dx (x² / ln x) = (2x ln x – x) / (ln x)² |
| Chain Rule | 链式法则 | dy/dx = dy/du · du/dx | d/dx (sin(3x)) = 3 cos(3x) |
unctions, students memorise the standard derivatives: d/dx (sin x) = cos x, d/dx (cos x) = –sin x, d/dx (eˣ) = eˣ lso need to differentiate inverse trigonometric functions like d/dx (arctan x) = 1/(1+x²).
对于三角函数、指数函数和对数函数，学生需熟记标准导数：d/dx (sin x) = cos x，d/dx (cos x) = –sin x，d/dx (eˣ) = eˣ，以及 d/dx (ln x) = 1/x。HL 学生还需掌握反三角函数的导数，例如 d/dx (arctan x) = 1/(1+x²)。
4. Higher Derivatives and Implicit Differentiation (HL) | 高阶导数与隐函数微分（HL）
The second derivative, denoted f''(x) or d²y/dx², represents the rate of change of the first derivative and is crucial for analysing concavity and acceleration. HL students routinely compute higher-order derivatives and interpret them in contexts such as kinematics.
二阶导数记作 f''(x) 或 d²y/dx²，表示一阶导数的变化率，对于分析凹性和加速度至关重要。HL 学生需要常规地计算高阶导数，并在运动学等情境中加以解释。
r y, such as x² + y² = 1. Differentiating both sides with respect to x and using the chain
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