📚 A-Level CCEA Maths: Calculus Basics | A-Level CCEA 数学：微积分基础考点精讲
Welcome to this focused revision guide on Calculus Fundamentals for CCEA A-Level Mathematics. Calculus is a cornerstone of higher mathematics, enabling us to model rates of change and accumulation. Mastering differentiation and integration will empower you to solve a wide range of problems involving gradients, tangents, maxima and minima, as well as areas under curves. This article systematically covers the key concepts, techniques, and exam-style applications you need to succeed.
欢迎阅读这篇 CCEA A-Level 数学微积分基础考点精讲。微积分是高等数学的基石，使我们能够刻画变化率与累积量。掌握微分与积分，你将有能力解决涉及梯度、切线、极大极小值以及曲线下方面积等一系列问题。本文系统梳理了关键概念、计算方法和考试常见应用，助你备考无忧。
1. Introduction to Calculus | 微积分简介
Calculus is divided into two main branches: differential calculus, which studies instantaneous rates of change and slopes of curves, and integral calculus, which deals with accumulation of quantities and areas under curves.
微积分主要分为两大分支：微分学研究瞬时变化率与曲线的斜率；积分学则处理量的累积和曲线下方的面积。
Differential calculus answers ‘how fast?’ questions, such as velocity from displacement, while integral calculus reconstructs the total from the rate, like distance from velocity.
微分学回答“有多快？”的问题，例如从位移求速度；而积分学则从变化率重建总量，如从速度求距离。
2. Limits and the Derivative Concept | 极限与导数概念
The derivative of a function f(x) at a point x is defined as the limit of the difference quotient as h approaches zero: f'(x) = limh→0 [f(x+h) – f(x)] / h, provided this limit exists.
函数 f(x) 在点 x 处的导数定义为差分商的极限当 h 趋近于零：f'(x) = limh→0 [f(x+h) – f(x)] / h，只要该极限存在。
This limit represents the instantaneous rate of change and geometrically gives the slope of the tangent line to the curve y = f(x) at that point. The notation f'(x) is called ‘f prime of x’, and dy/dx is Leibniz notation.
该极限表示瞬时变化率，在几何上给出了曲线 y = f(x) 在该点处切线的斜率。记号 f'(x) 读作“f 撇 x”，而 dy/dx 是莱布尼茨记号。
3. Differentiation of Polynomials and Power Rule | 多项式与幂函数求导
For any real constant n, the derivative of xⁿ is n xⁿ⁻¹. This is known as the power rule. For example, d/dx (x⁵) = 5x⁴.
对于任意实数常数 n，xⁿ 的导数是 n xⁿ⁻¹。这就是幂法则。例如，d/dx (x⁵) = 5x⁴。
Power Rule: d/dx (xⁿ) = n xⁿ⁻¹
Constants differentiate to zero: d/dx (c) = 0. The derivative of a constant multiple is c f'(x), and the derivative of a sum is the sum of the derivatives: d/dx [f(x) ± g(x)] = f'(x) ± g'(x).
常数求导为零：d/dx (c) = 0。常数倍的导数是 c f'(x)，和的导数是各导数之和：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。
A typical CCEA question asks you to differentiate expressions like 4x³ – 2x² + 5x – 7. Applying the rules: f'(x) = 12x² – 4x + 5.
典型的 CCEA 考题要求你对 4x³ – 2x² + 5x – 7 这样的表达式求导。运用法则可得：f'(x) = 12x² – 4x + 5。
4. Chain Rule | 链式法则
The chain rule is used to differentiate composite functions. If y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx).
链式法则用于求复合函数的导数。若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du) × (du/dx)。
Chain Rule: dy/dx = dy/du · du/dx
For example, to differentiate y = (2x² + 3x)⁴, let u = 2x² + 3x. Then y = u⁴, so dy/du = 4u³, du/dx = 4x + 3. Hence dy/dx = 4u³ (4x + 3) = 4(2x² + 3x)³ (4x + 3).
例如，要微分 y = (2x² + 3x)⁴，令 u = 2x² + 3x。则 y = u⁴，因此 dy/du = 4u³，du/dx = 4x + 3。于是 dy/dx = 4u³ (4x + 3) = 4(2x² + 3x)³ (4x + 3)。
In function notation, if h(x) = f(g(x)), then h'(x) = f'(g(x)) g'(x). The chain rule can be extended for nested compositions.
用函数记号表示，若 h(x) = f(g(x))，则 h'(x) = f'(g(x)) g'(x)。链式法则可推广至多层复合函数。
5. Product and Quotient Rules | 乘积法则与商法则
When differentiating a product of two functions, use the product rule: d/dx [u(x) v(x)] = u'(x) v(x) + u(x) v'(x).
当对两个函数的乘积求导时，使用乘积法则：d/dx [u(x) v(x)] = u'(x) v(x) + u(x) v'(x)。
Product Rule: (uv)’ = u’v + uv’
For a quotient, the quotient rule is: d/dx [u(x) / v(x)] = [u'(x) v(x) – u(x) v'(x)] / [v(x)]², provided v(x) ≠ 0.
对于分式求导，商法则为：d/dx [u(x) / v(x)] = [u'(x) v(x) – u(x) v'(x)] / [v(x)]²，其中 v(x) ≠ 0。
Quotient Rule: (u/v)’ = (u’v – uv’) / v²
Example: differentiate (3x+1)/(x²–2). Let u = 3x+1, v = x²–2. Then u’ = 3, v’ = 2x. Plugging in gives f'(x) = [3(x²–2) – (3x+1)(2x)] / (x²–2)², which simplifies to (–3x² – 2x – 6) / (x²–2)².
示例：对 (3x+1)/(x²–2) 求导。令 u = 3x+1，v = x²–2。则 u’ = 3，v’ = 2x。代入得 f'(x) = [3(x²–2) – (3x+1)(2x)] / (x²–2)²，化简后为 (–3x² – 2x – 6) / (x²–2)²。
6. Second Derivative and Higher Derivatives | 二阶导数与高阶导数
The second derivative, denoted f”(x) or d²y/dx², is the derivative of the first derivative. It measures the rate of change of the gradient and indicates concavity.
二阶导数，记作 f”(x) 或 d²y/dx²，是一阶导数的导数。它度量梯度的变化率，并指示曲线的凹性。
If f”(x) > 0 on an interval, the graph is concave up (∪ shape); if f”(x) < 0, it is concave down (∩ shape). Points where concavity changes are inflection points.
如果在某区间上 f”(x) > 0，图像向上凹（∪ 形）；若 f”(x) < 0，则向下凹（∩ 形）。凹性发生变化的点称为拐点。
Higher derivatives like f”'(x) are sometimes used, but the first two are essential for curve sketching and optimisation problems in CCEA.
有时会用到三阶导数 f”'(x)，但前两阶对 CCEA 曲线绘图和最优化问题至关重要。
7. Equations of Tangents and Normals | 切线与法线方程
The gradient of the tangent to the curve y = f(x) at a point (x₁, y₁) is m = f'(x₁). The equation of the tangent line is y – y₁ = m (x – x₁).
曲线 y = f(x) 在点 (x₁, y₁) 处切线的梯度为 m = f'(x₁)。切线方程为 y – y₁ = m (x – x₁)。
The normal line is perpendicular to the tangent, so its gradient is –1/m, provided m ≠ 0. The normal equation is y – y₁ = (–1/m) (x – x₁).
法线垂直于切线，因此其梯度为 –1/m（若 m ≠ 0）。法线方程为 y – y₁ = (–1/m) (x – x₁)。
Typical CCEA question: find the equation of the tangent and normal to y = x³ – 2x at x = 1. First, y(1) = –1, y’ = 3x² – 2 so m = 1. Tangent: y + 1 = 1(x – 1) → y = x – 2. Normal: gradient –1, normal: y + 1 = –1(x – 1) → y = –x.
典型 CCEA 考题：求 y = x³ – 2x 在 x = 1 处的切线和法线方程。首先，y(1) = –1，y’ = 3x² – 2，故 m = 1。切线：y + 1 = 1(x – 1) → y = x – 2。法线梯度为 –1，法线：y + 1 = –1(x – 1) → y = –x。
8. Stationary Points and Optimization | 驻点与最优化
Stationary points occur where the first derivative is zero, f'(x) = 0. They can be classified as local maxima, local minima, or points of inflection using the second derivative test.
驻点出现在一阶导数为零处，即 f'(x) = 0。可利用二阶导数判别法将其分为局部极大值、局部极小值或拐点。
If f”(a) > 0 at a stationary point x = a, the point is a local minimum. If f”(a) < 0, it is a local maximum. If f''(a) = 0, the test is inconclusive; use a gradient sign chart.
若在驻点 x = a 处 f”(a) > 0，则该点为局部极小值点。若 f”(a) < 0，则为局部极大值点。若 f''(a) = 0，该判别法失效，需使用梯度符号表。
Optimisation problems apply these concepts to real-world contexts, like maximising area or minimising cost. Set up a function, differentiate, find stationary points, and verify with the second derivative.
最优化问题将这些概念应用于实际情境，如最大化面积或最小化成本。建立函数、求导、找出驻点，并用二阶导数验证。
9. Indefinite Integration Basics | 不定积分基础
Integration is the reverse process of differentiation. The indefinite integral of a function f(x) is written ∫ f(x) dx = F(x) + C, where F'(x) = f(x) and C is the constant of integration.
积分是微分的逆运算。函数 f(x) 的不定积分写成 ∫ f(x) dx = F(x) + C，其中 F'(x) = f(x) 且 C 为积分常数。
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