AQA Maths: Calculus Basics – Key Points | AQA 数学：微积分基础考点精讲

📚 AQA Maths: Calculus Basics – Key Points | AQA 数学：微积分基础考点精讲

Calculus is the mathematical study of change and accumulation, split into differentiation (finding rates of change) and integration (finding totals from rates). For AQA A-level Mathematics, mastering the basics of both is essential, as they form the foundation for much of the pure content and applied problems in mechanics. This guide walks you through the core concepts, from derivatives of simple powers to calculating areas under curves, with clear explanations in both English and Chinese.

微积分是研究变化与累积的数学分支，分为微分（求变化率）和积分（由变化率求总量）。对 AQA A-level 数学来说，掌握微积分基础至关重要，因为它们是纯数内容和力学应用问题的基石。本文为你梳理核心概念，从简单幂函数的导数到计算曲线下方的面积，每个要点都配有中英文清晰讲解。

1. What is Differentiation? | 什么是微分？

Differentiation allows us to find the gradient of a curve at any point. The result is a new function called the derivative, often written as dy/dx or f'(x). It measures the instantaneous rate of change of y with respect to x.

微分帮我们求出曲线上任意一点的斜率，结果是一个新函数，称为导数，通常记作 dy/dx 或 f'(x)。它度量了 y 随 x 变化的瞬时变化率。

2. Power Rule for xⁿ | xⁿ 的求导公式

For any real constant n, the derivative of xⁿ is nxⁿ⁻¹. This is the most fundamental rule: multiply by the power, then subtract one from the power.

对任意实数常数 n，xⁿ 的导数为 nxⁿ⁻¹。这是最基本的求导法则：乘以指数，再将指数减一。

If y = xⁿ, then dy/dx = nxⁿ⁻¹.

若 y = xⁿ，则 dy/dx = nxⁿ⁻¹。

Example: y = x⁵ gives dy/dx = 5x⁴. y = x gives dy/dx = 1x⁰ = 1. y = 1/x² (which is x⁻²) gives dy/dx = -2x⁻³ = -2/x³.

例：y = x⁵ 的导数为 dy/dx = 5x⁴；y = x 导数为 1x⁰ = 1；y = 1/x²（即 x⁻²）的导数为 -2x⁻³ = -2/x³。

3. Sum/Difference and Constant Multiple Rules | 和差与常数倍的求导

Differentiation is linear: the derivative of a sum is the sum of the derivatives, and a constant factor can be taken outside. That is, d/dx[f(x) ± g(x)] = f'(x) ± g'(x) and d/dx[k·f(x)] = k·f'(x).

微分是线性运算：和的导数等于导数的和，常数因子可以提到外面。即 d/dx[f(x) ± g(x)] = f'(x) ± g'(x)，且 d/dx[k·f(x)] = k·f'(x)。

Example: If y = 4x³ – 2x + 7, then dy/dx = 4·3x² – 2·1 + 0 = 12x² – 2.

例：若 y = 4x³ – 2x + 7，则 dy/dx = 4·3x² – 2·1 + 0 = 12x² – 2。

4. Equation of a Tangent | 切线方程

The derivative at a point gives the gradient of the tangent. If we know a point (x₁, y₁) on the curve and the gradient m = dy/dx at that point, the tangent line is y – y₁ = m(x – x₁).

函数在某点的导数值就是该点切线的斜率。若知道曲线上一点 (x₁, y₁) 以及该点斜率 m = dy/dx，切线方程就是 y – y₁ = m(x – x₁)。

Example: For y = x² + 1 at x = 1, y = 2, dy/dx = 2x = 2. Tangent: y – 2 = 2(x – 1) ⇒ y = 2x.

例：对 y = x² + 1 在 x = 1 处，y = 2，dy/dx = 2x = 2。切线：y – 2 = 2(x – 1) ⇒ y = 2x。

5. Equation of a Normal | 法线方程

The normal is perpendicular to the tangent. Its gradient is the negative reciprocal: m_normal = -1 / m_tangent, provided m_tangent ≠ 0. Use the same point-slope form.

法线与切线垂直，其斜率为切线斜率的负倒数：m_normal = -1 / m_tangent，只要切线斜率不为零。同样使用点斜式。

Example: Using the curve above at x = 1, tangent gradient = 2, so normal gradient = -½. Normal: y – 2 = -½(x – 1) ⇒ y = -½x + 2.5.

例：以上例 x=1 处，切线斜率 2，则法线斜率 -½。法线：y – 2 = -½(x – 1) ⇒ y = -½x + 2.5。

6. Stationary Points | 驻点

Stationary points occur where dy/dx = 0. They can be local maxima, minima, or points of inflection. To find them, set the derivative to zero and solve for x, then substitute back to get y.

驻点出现在 dy/dx = 0 的地方，可能是局部极大值、极小值或拐点。求驻点时，令导数为零解出 x，再代回原函数求 y。

Example: y = x³ – 3x. dy/dx = 3x² – 3 = 0 ⇒ x² = 1 ⇒ x = ±1. Points: (1, -2) and (-1, 2).

例：y = x³ – 3x。dy/dx = 3x² – 3 = 0 ⇒ x² = 1 ⇒ x = ±1。驻点：(1, -2) 和 (-1, 2)。

7. Increasing and Decreasing Functions | 函数的递增与递减

The sign of dy/dx tells us where the function is increasing (dy/dx > 0) or decreasing (dy/dx < 0). This helps sketch graphs and confirm the nature of stationary points.

dy/dx 的正负号告诉我们函数在何处递增 (dy/dx > 0) 或递减 (dy/dx < 0)。这有助于概画函数图像并确认驻点的性质。

For y = x³ – 3x, dy/dx = 3(x² – 1). For x < -1, dy/dx > 0 (increasing); for -1 < x < 1, dy/dx < 0 (decreasing); for x > 1, dy/dx > 0 (increasing).

对 y = x³ – 3x，dy/dx = 3(x² – 1)。当 x < -1，dy/dx > 0（递增）；当 -1 < x < 1，dy/dx < 0（递减）；当 x > 1，dy/dx > 0（递增）。

8. Second Derivative and Nature of Stationary Points | 二阶导数与驻点判定

The second derivative, d²y/dx², is the derivative of dy/dx. At a stationary point, if d²y/dx² > 0, it’s a local minimum; if d²y/dx² < 0, it's a local maximum; if d²y/dx² = 0, the test is inconclusive, and we check the sign change of dy/dx.

二阶导数 d²y/dx² 是 dy/dx 的导数。在驻点处，若二阶导大于零则为极小值；若小于零则为极大值；若等于零则无法判定，需检查一阶导的符号变化。

Example: y = x³ – 3x. dy/dx = 3x² – 3, d²y/dx² = 6x. At x = 1, d²y/dx² = 6 > 0 ⇒ minimum. At x = -1, d²y/dx² = -6 < 0 ⇒ maximum.

例：y = x³ – 3x。dy/dx = 3x² – 3，d²y/dx² = 6x。在 x=1，二阶导 6 > 0 ⇒ 极小值；在 x=-1，二阶导 -6 < 0 ⇒ 极大值。

9. Introduction to Integration (Indefinite Integral) | 积分初步（不定积分）

Integration is the reverse of differentiation. The indefinite integral of xⁿ with respect to x is (xⁿ⁺¹)/(n+1) + C, for n ≠ -1. The constant C accounts for the fact that the derivative of any constant is zero.

积分是微分的逆运算。xⁿ 关于 x 的不定积分是 (xⁿ⁺¹)/(n+1) + C，其中 n ≠ -1。常数 C 表示任意常数的导数为零的信息损失。

∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, n ≠ -1.

∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，n ≠ -1。

Example: ∫ x² dx = x³/3 + C. ∫ 4 dx = 4x + C. The rule works for negative and fractional powers too, as long as n ≠ -1.

例：∫ x² dx = x³/3 + C；∫ 4 dx = 4x + C。此法则同样适用于负指数和分数指数，只要指数不为 -1。

10. Finding the Constant of Integration | 确定积分常数

If we know one point on the original curve, we can find the value of C. This comes from ‘initial condition’ problems where the function passes through a given point.

如果知道原曲线上的一点，就能求出 C 的值。这常见于给定函数经过某一点的“初始条件”问题。

Example: A curve has gradient dy/dx = 3x² and passes through (1, 5). Integrate: y = x³ + C. Substitute x=1, y=5 ⇒ 5 = 1 + C ⇒ C = 4. So y = x³ + 4.

例：曲线斜率 dy/dx = 3x² 且过 (1, 5)。积分：y = x³ + C。代入 x=1, y=5 ⇒ 5 = 1 + C ⇒ C = 4。所以 y = x³ + 4。

11. Definite Integrals and the Fundamental Theorem | 定积分与基本定理

A definite integral has limits a and b and represents the net area between the curve and the x-axis from x=a to x=b. Evaluate it as F(b) – F(a), where F is any antiderivative. The constant C cancels out, so we can ignore it.

定积分带有上下限 a 和 b，表示曲线与 x 轴之间从 x=a 到 x=b 的净面积。计算时取原函数 F 的差值 F(b) – F(a)，常数 C 会消去，因此可以忽略。

∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) – F(a).

∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) – F(a)。

Example: ∫₁³ 2x dx = [x²]₁³ = 9 – 1 = 8.

例：∫₁³ 2x dx = [x²]₁³ = 9 – 1 = 8。

12. Area Under a Curve | 曲线下方的面积

To find the area bounded by y = f(x), the x-axis, and vertical lines x = a and x = b, compute ∫ₐᵇ f(x) dx. If the curve dips below the x-axis, the integral yields a negative contribution; to find total area, split into regions above and below and take absolute values.

求曲线 y = f(x)、x 轴以及直线 x=a 和 x=b 围成的面积时，计算 ∫ₐᵇ f(x) dx。若曲线部分在 x 轴下方，积分会贡献负值；要求总面积，需分段取绝对值相加。

Example: Area under y = 4 – x² between x = -2 and x = 2: ∫₋₂² (4 – x²) dx = [4x – x³/3]₋₂² = (8 – 8/3) – (-8 + 8/3) = 32/3 square units.

例：y = 4 – x² 在 x = -2 到 x = 2 之间的面积：∫₋₂² (4 – x²) dx = [4x – x³/3]₋₂² = (8 – 8/3) – (-8 + 8/3) = 32/3 平方单位。

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AQA Maths: Calculus Basics – Key Points | AQA 数学：微积分基础 考点精讲