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GCSE Maths: Calculus Basics | GCSE 数学：微积分基础 考点精讲

2026-07-14 02:30 · 本文部分内容由AI辅助创作

📚 GCSE Maths: Calculus Basics | GCSE 数学：微积分基础 考点精讲

Calculus is the branch of mathematics that studies continuous change. For GCSE Higher tier, you need to know the basics of differentiation to find gradients of curves and equations of tangents, and the reverse process – integration – to find areas and reconstruct original functions. This guide covers all the essentials you'll need with clear steps, examples, and common pitfalls.

微积分是研究连续变化的数学分支。在 GCSE 高阶课程中，你需要掌握微分的基础，用以求曲线的斜率和切线方程，以及其逆运算——积分——来求面积和还原原函数。本考点精讲覆盖所有必备知识，包含清晰的步骤、例子和常见错误的提醒。

1. What is Calculus? | 什么是微积分？

Calculus is split into two main areas: differentiation and integration. Differentiation finds the gradient of a curve at any point, telling us the instantaneous rate of change. Integration is the reverse process: it takes a rate of change and recovers the original quantity, often used to work out areas under curves.

微积分主要分为两大领域：微分和积分。微分求曲线上任意一点的斜率，给出瞬时变化率。积分是微分的逆运算：它利用变化率恢复原始的量，常用来计算曲线下的面积。

At GCSE, we only work with polynomial functions – expressions made of powers of x, such as x², 3x⁴ − 2x + 5. There are no trigonometric or exponential functions.

在 GCSE 阶段，我们只处理多项式函数——由 x 的幂组成的表达式，例如 x²、3x⁴ − 2x + 5。不涉及三角函数或指数函数。

2. Differentiation: Finding the Gradient Function | 微分：求梯度函数

Differentiation gives a new function called the derivative or gradient function. For a curve y = f(x), the derivative is written as dy/dx or f'(x). It tells you the steepness (gradient) of the curve for any value of x.

微分会得到一个新的函数，称为导数或梯度函数。对于曲线 y = f(x)，导数写作 dy/dx 或 f'(x)。它能给出曲线上任意 x 值对应的陡峭程度（斜率）。

The notation dy/dx is read as "the derivative of y with respect to x". The process of finding the derivative is called differentiating.

符号 dy/dx 读作“y 对 x 的导数”。求导的过程称为微分。

3. The Power Rule for Differentiation | 微分的幂法则

For any single term axⁿ, the derivative is found by multiplying by the power n and then reducing the power by 1:

对于任何单项 axⁿ，求导的方法是先乘以幂指数 n，再将指数减 1：

If y = axⁿ, then dy/dx = naxⁿ⁻¹

Examples:

例子：

iating a constant gives zero, because a constant has no rate of change. y = 7 → dy/dx = 0.

常数的导数为零，因为常数没有变化率。y = 7 → dy/dx = 0。

l terms, differentiate each term separately. For y = 2x³ + 3x − 5, we get dy/dx = 6x² + 3.

若函数含有多个项，则逐项求导。对 y = 2x³ + 3x − 5 求导，得 dy/dx = 6x² + 3。

4. Finding the Gradient at a Point | 计算某一点的梯度

Once you have the derivative dy/dx, you can find the gradient of the curve at a specific point by substituting the x‑coordinate.

求出导数 dy/dx 后，只要代入某点的 x 坐标，就能得到曲线在该点的斜率。

Example: or y = x² − 4x + 1, dy/dx = 2x − 4. At the point where x = 3, the gradient is 2×3 − 4 = 2.

例子：对于 y = x² − 4x + 1，dy/dx = 2x − 4。在 x = 3 处，斜率为 2×3 − 4 = 2。

The gradient of the curve at a point is exactly the same as the gradient of the tangent line drawn at that point.

曲线在某点的斜率，恰好等于该点处切线的斜率。

5. Equation of a Tangent Line | 切线方程

To find the equation of the tangent to a curve at a point, you need two things: the point (x₁, y₁) and the gradient m (found from dy/dx). Then use the straight‑line formula:

要求曲线在某点的切线方程，你需要两个信息：切点 (x₁, y₁) 以及由 dy/dx 求出的斜率 m。然后代入直线方程：

y − y₁ = m(x − x₁)

Worked example: Find the tangent to y = x² at (2, 4).

解题示例：求曲线 y = x² 在 (2, 4) 处的切线。

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6. Introduction to Integration | 积分入门

Integration is the reverse of differentiation. If you know the derivative function (the rate of change), integration lets you recover the original function. Because differentiation loses constant terms, integration introduces an unknown constant, usually called c – the constant of integration.

积分是微分的逆运算。如果你已知导数函数（变化率），积分可以帮助你还原出原函数。由于微分会丢失常数项，积分时会引入一个未知常数，通常记作 c，被称为积分常数。

The integral sign is ∫. An indefinite integral (one without limits) is written as:

积分符号是 ∫。不定积分（不带上下限的积分）写作：

∫ f'(x) dx = f(x) + c

7. The Power Rule for Integration | 积分的幂法则

For a term axⁿ, increase the power by 1 and divide by the new power (n+1). Don't forget to add the constant of integration.

对于项 axⁿ，将指数加 1，然后除以新的指数 (n+1)。千万别忘记加上积分常数。

∫ axⁿ dx = (a/(n+1))xⁿ⁺¹ + c,   where n ≠ −1

Examples:

例子：

8. Definite Integrals and Area Under a Curve | 定积分与曲线下方面积

A definite integral has lower and upper limits, a and b, and gives a numerical value – the exact area between the curve y = f(x), the x‑axis, and the lines x = a and x = b.

定积分带有下限 a 和上限 b，得出的数值表示曲线 y = f(x) 与 x 轴以及直线 x = a、x = b 所围成的准确面积。

Area = ∫ab f(x) dx

To evaluate: integrate f(x) to get the antiderivative F(x) (without +c), then compute F(b) − F(a).

计算方法：先积分 f(x) 得到原函数 F(x)（不带 +c），再计算 F(b) − F(a)。

Example: Find the area under y = 2x between x = 1 and x = 3.

示例：求 y = 2x 在 x = 1 到 x = 3 之间与 x 轴围成的面积。

9. Calculus in Kinematics | 运动学中的微积分

In motion problems, calculus links displacement (s), velocity (v), and acceleration (a).

在运动问题中，微积分连接了位移 (s)、速度 (v) 和加速度 (a)。

If you are given acceleration, integrate to find velocity; integrate velocity to find displacement. Initial conditions are used to find the constant c.

如果给出加速度，积分可得速度；再积分速度可得位移。利用初始条件可求出常数 c。

Typical GCSE question: cceleration at t = 2 s, and the displacement from t = 0 to t = 3 s given s = 0 when t = 0.

典型 GCSE 题型：质点运动的速度为 v = 3t² − 4 m/s。求 t = 2 s 时的加速度，以及从 t = 0 到 t = 3 s 的位移，已知 t = 0 时 s = 0。