📚 GCSE OCR Maths: Introduction to Calculus | GCSE OCR 数学：微积分基础 考点精讲
Calculus is the branch of mathematics that deals with change and accumulation. At GCSE level, you will be introduced to the two fundamental ideas: differentiation, which finds the gradient of a curve at a point, and integration, which finds the area under a curve. These concepts are built on the notion of limits and infinitesimally small changes, but at this stage you will learn practical rules for polynomial functions and apply them to tangents, increasing/decreasing behaviour and simple motion.
微积分是研究变化与累积的数学分支。在 GCSE 阶段，你将接触两个核心思想：微分——求曲线上某一点的梯度，以及积分——求曲线下的面积。这些概念建立在极限和无穷小变化的基础上，但本阶段主要学习多项式的实用法则，并将其应用于切线、函数的增减性以及简单运动问题。
1. What is Calculus? | 什么是微积分？
Calculus allows us to study how quantities change. The word ‘calculus’ comes from Latin meaning ‘small pebble’, reflecting the ancient method of counting with stones. Today, calculus provides tools to model everything from the motion of planets to the growth of populations. It is split into two main branches: differential calculus, concerned with rates of change and tangents, and integral calculus, concerned with accumulation and areas.
微积分让我们能够研究量如何变化。“微积分”一词源自拉丁语，意为“小石子”，反映了古人用石子计数的历史。如今，微积分提供了从行星运动到人口增长等各种现象的建模工具。它主要分为两个分支：微分学，研究变化率和切线；积分学，研究累积和面积。
In GCSE OCR, you will focus on the basics: finding derivatives of simple polynomials, using derivatives to find gradients of tangents, determining whether a function is increasing or decreasing, and using integration as the reverse process to find areas under straight–line and simple curves.
在 GCSE OCR 考试中，你将重点学习基础内容：求简单多项式的导数，用导数求切线梯度，判断函数的递增与递减，以及用积分作为逆过程求直线和简单曲线下的面积。
2. Gradient of a Curve | 曲线的梯度
For a straight line, the gradient is constant and can be found as rise over run. For a curve, the gradient changes from point to point. The instantaneous gradient at a point is the gradient of the tangent line that just touches the curve at that point. Finding this gradient is the central problem of differential calculus.
对于直线，梯度是恒定的，可以用垂直变化除以水平变化求得。对于曲线，梯度逐点变化。某一点的瞬时梯度就是恰好与曲线在该点相切的切线的梯度。求出这个梯度是微分学的核心问题。
Consider a parabola y = x². At x = 1, the curve is rising with a certain steepness; at x = 2, it is steeper; at x = 0, the tangent is horizontal, giving a gradient of 0. None of these gradients can be found using just the ‘rise/run’ of a straight line between two points unless the two points are infinitely close.
以抛物线 y = x² 为例。在 x = 1 处，曲线以一定的陡度上升；在 x = 2 处更陡；在 x = 0 处，切线是水平的，梯度为 0。这些梯度都无法仅用两点间的“垂直/水平”比来求得，除非两点无限接近。
3. From Chords to Tangents: The Method | 从割线到切线：方法
To approximate the gradient at a point, we draw a chord connecting two nearby points on the curve and compute its gradient. As the second point moves closer to the first, the chord approaches the tangent and its gradient approaches the true instantaneous gradient.
为了近似某一点的梯度，我们画一条连接曲线上相邻两点的割线，并计算其梯度。当第二个点无限接近第一个点时，割线趋近于切线，其梯度趋近于真实的瞬时梯度。
For f(x) = x², take points (x, f(x)) and (x+h, f(x+h)). The gradient of the chord is:
(f(x+h) – f(x)) / h = ((x+h)² – x²) / h = (x² + 2xh + h² – x²) / h = 2x + h
As h → 0, the term 2x + h tends to 2x. Hence the derivative f'(x) = 2x. This process is called differentiation from first principles.
对于 f(x) = x²，取点 (x, f(x)) 和 (x+h, f(x+h))，割线梯度为上述表达式。当 h → 0 时，2x + h 趋近于 2x。因此导数 f'(x) = 2x。这个过程称为基于第一原理的求导。
4. Derivative Notation and Definition | 导数的符号和定义
The derivative of a function y = f(x) can be written in several ways: f'(x) (pronounced ‘f prime of x’), dy/dx (‘dee y by dee x’), or d/dx [f(x)]. Each notation emphasises a different aspect: f'(x) highlights the function, while dy/dx recalls the limiting ratio of small changes Δy/Δx.
函数 y = f(x) 的导数有几种写法：f'(x)（读作 f 撇 x）、dy/dx 或 d/dx [f(x)]。每种符号强调不同的侧面：f'(x) 突出函数本身，而 dy/dx 让人联想到微小变化 Δy/Δx 的极限比值。
The formal definition is: f'(x) = limh→0 [f(x+h) – f(x)] / h, provided this limit exists. If the limit exists at a point, we say the function is differentiable there.
正式的定义是：f'(x) = limh→0 [f(x+h) – f(x)] / h，只要该极限存在。如果极限在某点存在，我们就说函数在该点可导。
5. Differentiating Power Functions | 幂函数的求导
For any function of the form f(x) = xⁿ, where n is a real number, the derivative is given by the power rule:
d/dx (xⁿ) = n xⁿ⁻¹
This rule works for positive integers, negative integers and fractional powers (though GCSE usually covers only positive integer powers initially). Examples:
- y = x³ → dy/dx = 3x²
- y = x → dy/dx = 1x⁰ = 1
- y = x⁷ → dy/dx = 7x⁶
- y = 1/x = x⁻¹ → dy/dx = –1 x⁻² = –1/x²
- y = √x = x½ → dy/dx = ½ x–½ = 1/(2√x)
对于形如 f(x) = xⁿ 的函数，导数为 n xⁿ⁻¹。此法则适用于正整数、负整数和分数指数（不过 GCSE 通常先学习正整数指数）。示例包括 y = x³ 的导数为 3x²，y = x 的导数为 1，y = 1/x 的导数为 –1/x²，等等。
6. Linearity Rules for Differentiation | 求导的线性法则
Differentiation is a linear operator, which means two simple rules make it easy to handle sums and constant multiples:
- Constant multiple rule: d/dx [c f(x)] = c f'(x), where c is a constant.
- Sum rule: d/dx [f(x) + g(x)] = f'(x) + g'(x).
微积分中，微分是线性算子，有两个简单法则可以方便地处理常数倍与和差：
- 常数倍法则： d/dx [c f(x)] = c f'(x)，其中 c 为常数。
- 加法法则： d/dx [f(x) + g(x)] = f'(x) + g'(x)。
Example: f(x) = 4x³ + 2x² – 5x + 7. Differentiating term by term: f'(x) = 4·3x² + 2·2x – 5·1 + 0 = 12x² + 4x – 5. The derivative of a constant is always zero.
示例：f(x) = 4x³ + 2x² – 5x + 7。逐项求导得 f'(x) = 12x² + 4x – 5。常数的导数始终为零。
7. Finding the Equation of a Tangent | 求切线方程
Once the derivative dy/dx is known, finding the equation of a tangent at a specific point is straightforward. At x = a, compute the y-coordinate y = f(a) and the gradient m = f'(a). Then use the line equation y – y₁ = m (x – x₁).
一旦知道导数 dy/dx，求某点处的切线方程就很简单。在 x = a 处，计算 y = f(a) 和梯度 m = f'(a)，然后使用直线方程 y – y₁ = m (x – x₁)。
Example: Find the tangent to y = x² at x = 3. f(3) = 9, f'(x) = 2x, so m = 6. Equation: y – 9 = 6 (x – 3) → y = 6x – 9.
示例：求 y = x² 在 x = 3 处的切线。f(3) = 9, f'(x) = 2x, m = 6。切线方程：y – 9 = 6 (x – 3) → y = 6x – 9。
8. Increasing and Decreasing Functions | 递增与递减函数
A function is increasing on an interval if its derivative is positive (f'(x) > 0), and decreasing if f'(x) < 0. Points where f'(x) = 0 are called stationary points, which can be a local maximum, minimum or point of inflection.
如果在某区间上导数 f'(x) > 0，则函数递增；如果 f'(x) < 0，则函数递减。使得 f'(x) = 0 的点称为驻点，可能是局部极大值、极小值或拐点。
For example, f(x) = x³ – 3x has f'(x) = 3x² – 3 = 3(x² – 1). Setting f'(x) = 0 gives x = ±1. For x < –1, f'(x) > 0, so increasing; between –1 and 1, f'(x) < 0, so decreasing; for x > 1, f'(x) > 0, so increasing again. Hence x = –1 is a local maximum and x = 1 a local minimum.
例如 f(x) = x³ – 3x 的导数 f'(x) = 3x² – 3。解 f'(x)=0 得 x = ±1。在 x < –1 时导数大于零，递增；在 –1 < x < 1 时导数小于零，递减；在 x > 1 时导数又大于零，递增。因此 x = –1 是局部极大值，x = 1 是局部极小值。
9. Simple Kinematics Application | 简单运动学应用
If the displacement s of a particle moving in a straight line is given as a function of time t, s = f(t), then the derivative ds/dt represents the instantaneous velocity v. The second derivative d²s/dt² (or dv/dt) represents the acceleration a.
如果质点直线运动的位移 s 是时间 t 的函数 s = f(t)，则导数 ds/dt 表示瞬时速度 v。二阶导数 d²s/dt²（或 dv/dt）表示加速度 a。
Example: s = t³ – 6t² + 9t. Then v = ds/dt = 3t² – 12t + 9, a = dv/dt = 6t – 12. The particle is instantaneously at rest when v = 0, i.e. 3(t² – 4t + 3) = 0 → t = 1 or 3. The displacement and acceleration at those instants can be evaluated by substitution.
示例：s = t³ – 6t² + 9t。则 v = 3t² – 12t + 9, a = 6t – 12。当 v = 0 时质点瞬间静止，解得 t = 1 或 3。此时可通过代入求出位移和加速度。
10. Introduction to Integration as Area | 积分作为面积的引入
Integration is the reverse process of differentiation. If F'(x) = f(x), we say F is an antiderivative of f. The indefinite integral is written ∫ f(x) dx = F(x) + C, where C is an arbitrary constant. Geometrically, integration can be used to compute the area under a curve between two limits.
积分是微分的逆过程。如果 F'(x) = f(x)，则称 F 是 f 的一个原函数。不定积分写作 ∫ f(x) dx = F(x) + C，其中 C 为任意常数。从几何上看，积分可用于计算两界限之间曲线下的面积。
To approximate the area under y = f(x) from x = a to x = b, the region can be divided into n thin rectangular strips. As n → ∞, the sum of such rectangles approaches the true area. This limiting sum is the definite integral ∫ab f(x) dx.
要近似求 y = f(x) 从 x = a 到 x = b 下方的面积，可将区域分成 n 个狭长矩形条。当 n → ∞ 时，这些矩形面积之和趋近于真实面积。这个极限和就是定积分 ∫ab f(x) dx。
11. Integrating Power Functions | 幂函数的积分
The power rule for integration is the reverse of the differentiation rule. For any n ≠ –1:
∫ xⁿ dx = (xⁿ⁺¹) / (n+1) + C
When n = –1, the integral is the natural logarithm (not required at GCSE). Examples: ∫ x² dx = x³/3 + C, ∫ x⁵ dx = x⁶/6 + C, ∫ √x dx = ∫ x½ dx = (x3/2)/(3/2) + C = ⅔ x3/2 + C. Constant multiples and sums can be integrated termwise.
幂函数的积分法则是微分法则的逆运算。对于 n ≠ –1，∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C。当 n = –1 时积分是对数函数（GCSE 不要求）。示例：∫ x² dx = x³/3 + C，∫ √x dx = ⅔ x3/2 + C。常数倍和和差可以逐项积分。
12. Definite Integrals and Area Under a Curve | 定积分与曲线下面积
A definite integral has limits: ∫ab f(x) dx. To evaluate it, find an antiderivative F(x), then compute F(b) – F(a). This is the Fundamental Theorem of Calculus.
定积分带有上下限：∫ab f(x) dx。计算方法是先求一个原函数 F(x)，再求 F(b) – F(a)。这就是微积分基本定理。
Example: Find the area under y = 2x + 1 from x = 1 to x = 3. F(x) = x² + x, so area = F(3) – F(1) = (9+3) – (1+1) = 12 – 2 = 10 square units. For a curve y = x² from 0 to 2, F(x) = x³/3, area = 8/3 – 0 = 8/3 ≈ 2.67.
示例：求直线 y = 2x+1 从 x=1 到 x=3 下的面积。F(x)=x²+x，面积 = F(3)–F(1) =12–2=10 平方单位。对于曲线 y=x² 从 0 到 2，面积 = 8/3。
If the curve goes below the x-axis, the integral gives a negative value there. To find the total shaded area, split the interval into parts where the function is positive and negative, and take absolute values of each definite integral.
如果曲线在 x 轴下方，该部分的积分值为负。求总面积时，需按函数正负将区间分开，并对各部分定积分取绝对值。
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