KS3 Maths: Calculus Basics – Key Concepts Explained | KS3 数学：微积分基础考点精讲

📚 KS3 Maths: Calculus Basics – Key Concepts Explained | KS3 数学：微积分基础考点精讲

Calculus is often introduced at higher levels, but the key ideas behind it – discovering how things change and how to add up tiny pieces to find a whole – can be explored even at KS3. This article walks you through the foundational concepts of differentiation and integration using simple graphs, slopes, and areas, giving you a head start on one of mathematics’ most powerful tools.

微积分通常在高年级正式学习，但其核心思想——理解事物如何变化以及如何将细小部分累加为整体——在 KS3 阶段就可以进行探索。本文通过简单的图形、斜率和面积，带你了解微分和积分的基础概念，让你提前掌握数学中最强大的工具之一。

1. What is Calculus? | 什么是微积分？

Calculus is the branch of mathematics that studies two main ideas: the rate at which quantities change (differentiation) and the accumulation of quantities (integration). These two processes are deeply connected, like two sides of the same coin.

微积分是研究两大核心思想的数学分支：一个量变化的速率（微分）和量的累积（积分）。这两个过程紧密相连，就像一枚硬币的两面。

For example, if you know the position of a moving object at every moment, differentiation can tell you its speed at any instant. Conversely, if you know the object’s speed at each moment, integration can tell you how far it has travelled.

比如，如果你知道一个运动物体每一时刻的位置，微分可以帮助你计算它在任意时刻的速度；反过来，如果你知道物体每一时刻的速度，积分可以告诉你它总共走了多远。

2. Understanding Rates of Change | 理解变化率

A rate of change compares how much one quantity changes relative to another. If you earn £5 pocket money each week, the rate of change of your savings is £5 per week. In mathematics, we often look at how y changes when x changes, which is written as the change in y over the change in x (Δy / Δx).

变化率比较一个量相对于另一个量的变化幅度。如果你每周获得 5 英镑零花钱，那么储蓄的变化率就是每周 5 英镑。在数学中，我们常观察当 x 变化时 y 如何变化，写成 y 的变化量除以 x 的变化量 (Δy / Δx)。

For a straight line, this ratio is constant and is simply the slope of the line. On a graph, a steeper line means a greater rate of change.

对于一条直线，这个比值是恒定的，就是直线的斜率。在图像上，线越陡，变化率越大。

3. Slopes of Straight Lines | 直线的斜率

To calculate the slope of a straight line between two points (x₁, y₁) and (x₂, y₂), use the formula: slope = (y₂ − y₁) / (x₂ − x₁). This number tells you how many units y increases or decreases for every 1‑unit increase in x.

要计算直线上两点 (x₁, y₁) 和 (x₂, y₂) 之间的斜率，使用公式：斜率 = (y₂ − y₁) / (x₂ − x₁)。这个数值表示 x 每增加 1 个单位时，y 增加或减少多少个单位。

If the line goes uphill from left to right, the slope is positive; if it goes downhill, the slope is negative. A horizontal line has slope 0.

如果直线从左到右向上走，斜率为正；如果向下走，斜率为负。水平线的斜率为 0。

4. Estimating the Slope of a Curve | 估算曲线的斜率

Unlike a straight line, a curve does not have a constant slope. Its steepness changes from point to point. We can estimate the slope at a particular point by drawing a secant line through two close points on the curve and calculating its slope.

与直线不同，曲线没有固定的斜率，它在不同点处的陡峭程度是变化的。我们可以通过绘制一条经过曲线上两个靠近点的割线，计算其斜率，来估算某点的斜率。

For example, to estimate the slope of the curve y = x² at x = 2, you might pick a second point (2.1, 2.1²). The slope of the secant is (4.41 − 4) / (2.1 − 2) = 0.41 / 0.1 = 4.1. As you move the second point closer to the first, the secant’s slope gets closer to the curve’s true steepness at that point.

例如，要估算曲线 y = x² 在 x = 2 处的斜率，你可以选择第二个点 (2.1, 2.1²)。割线斜率为 (4.41 − 4) / (2.1 − 2) = 0.41/0.1 = 4.1。当你把第二个点移得更近时，割线的斜率就趋近于曲线在该点真正的陡峭程度。

5. Instantaneous Rate of Change | 瞬时变化率

If we imagine bringing the two points infinitely close together, the secant line becomes a tangent line that just touches the curve at one point. The slope of this tangent is called the instantaneous rate of change or the derivative at that point.

如果我们想象两个点无限靠近，割线就会变成恰好接触曲线于一点的切线。这条切线的斜率称为瞬时变化率，即该点的导数。

This is the fundamental idea of differentiation: finding the steepness of a curve exactly, rather than roughly. It tells us how fast y is changing relative to x at a precise instant.

这就是微分的基本思想：准确求出曲线的陡峭程度，而非粗略估算。它告诉我们 y 相对于 x 在精确瞬间的变化速度。

6. The Derivative of x² | x²的导数

Let’s see how we can find the derivative of f(x) = x² without constantly drawing graphs. Consider a small change h in x. The change in y is f(x+h) − f(x) = (x+h)² − x² = x² + 2xh + h² − x² = 2xh + h². The average rate of change is (2xh + h²)/h = 2x + h.

我们来看看如何不用一次次画图而求出 f(x) = x² 的导数。考虑 x 的一个微小变化量 h。y 的变化量为 f(x+h) − f(x) = (x+h)² − x² = x² + 2xh + h² − x² = 2xh + h²。平均变化率为 (2xh + h²)/h = 2x + h。

As h gets smaller and smaller, approaching 0, 2x + h approaches 2x. So the derivative of x² is 2x. We denote the derivative of f(x) as f'(x) or dy/dx.

随着 h 越来越小，趋近于 0，2x + h 趋近于 2x。因此 x² 的导数是 2x。我们将 f(x) 的导数记为 f'(x) 或 dy/dx。

If f(x) = x², then f'(x) = 2x

7. Derivative Rules for Powers | 幂函数的导数规则

The method we used for x² works for other power functions as well. The general rule for differentiating xⁿ is to multiply by the power n and then reduce the power by 1:

我们用于 x² 的方法也适用于其他幂函数。求导 xⁿ 的一般规则是乘以指数 n，然后将指数减 1：

d/dx (xⁿ) = n xⁿ⁻¹

Examples:

例子：

The derivative of x³ is 3x².

x³ 的导数是 3x²。
The derivative of x⁵ is 5x⁴.

x⁵ 的导数是 5x⁴。
If f(x) = x (which is x¹), its derivative is 1 · x⁰ = 1. This makes sense because the line y = x has a constant slope of 1.

若 f(x) = x（即 x¹），其导数为 1·x⁰ = 1。这很合理，因为直线 y = x 的斜率恒为 1。
A constant like f(x) = 7 has derivative 0 because it does not change when x changes.

常数如 f(x) = 7 的导数为 0，因为它不随 x 变化而变化。

8. Introduction to Integration: Area Under a Curve | 积分入门：曲线下的面积

Integration is the reverse process of differentiation. Just as differentiation finds the slope at a point, integration finds the total accumulation of a quantity – often represented as the area under a curve between two x‑values.

积分是微分的逆过程。正如微分求某点的斜率，积分求的是量的总累积——通常表示为曲线下方介于两个 x 值之间的面积。

If a graph shows speed against time, the area under the speed‑time graph between two times gives the total distance travelled. This is a key connection between integration and real‑world problems.

如果图像表示速度随时间的变化，那么速度‑时间图像下方在两个时间之间的面积就给出了总共行驶的距离。这就是积分与实际问题的关键联系。

9. Approximating Area with Rectangles | 用矩形近似面积

To estimate the area under a curve, we can split the interval into several thin vertical strips, treat each strip as a rectangle, and sum their areas. The narrower the rectangles, the better the approximation.

要估算曲线下的面积，我们可以将区间分成若干个细长的竖直条带，把每个条带看作一个矩形，然后将它们的面积相加。矩形越窄，近似效果越好。

For the function y = x² between x = 0 and x = 3, we might use six rectangles of width 0.5. The height of each rectangle could be taken from the left endpoint, right endpoint, or midpoint – each giving a slightly different estimate. As the number of rectangles increases, the sum gets closer to the true area.

对于函数 y = x² 在 x = 0 和 x = 3 之间的情况，我们可以使用 6 个宽度为 0.5 的矩形。每个矩形的高度可以取左端点、右端点或中点处的函数值——每种方式给出的估计稍有不同。随着矩形数量的增加，总和会越来越接近真实的面积。

Area ≈ Σ f(xᵢ) · Δx

10. The Integral of x² | x²的积分

When we let the number of rectangles approach infinity (so each width becomes infinitesimally small), we obtain the exact area, called the definite integral. For y = x² from x = 0 to x = a, the exact area is a³/3.

当矩形数量趋近无穷大（即每个宽度变得无限小），我们就得到了精确的面积，称为定积分。对于 y = x² 从 x = 0 到 x = a，精确面积为 a³/3。

More generally, the process of integration reverses differentiation: if F'(x) = f(x), then the integral of f(x) from a to b is F(b) − F(a). Because the derivative of x³/3 is x², we can say that x³/3 is an antiderivative of x².

更一般地说，积分是微分的逆过程：如果 F'(x) = f(x)，那么 f(x) 从 a 到 b 的积分就是 F(b) − F(a)。因为 x³/3 的导数是 x²，所以我们可以说 x³/3 是 x² 的一个原函数。

11. The Fundamental Theorem of Calculus (Intuitive) | 微积分基本定理（直观理解）

The Fundamental Theorem of Calculus reveals that differentiation and integration are inverse operations. If you integrate a function and then differentiate the result, you get back the original function. This is why we can evaluate areas by finding antiderivatives.

微积分基本定理揭示了微分和积分是互为逆运算的。如果你先对一个函数积分，再对结果微分，就会得到原来的函数。这就是为什么我们可以通过寻找原函数来计算面积。

In simple terms, the theorem says: the rate at which the accumulated area changes as you move the right‑hand boundary is exactly the height of the curve at that boundary. So differentiation ‘undoes’ what integration does, and vice versa.

简单来说，这个定理表明：当你移动右边界时，累积面积的变化速率恰好等于曲线在该边界处的高度。所以微分“撤销”了积分的操作，反之亦然。

12. Real-life Example: Speed and Distance | 实际例子：速度与距离

Imagine you are cycling and your speed (in metres per second) is recorded as v(t) = 2t, where t is time in seconds. How far have you travelled after 5 seconds? This is an integration problem: the distance is the area under the speed‑time graph.

假设你正在骑自行车，速度（米/秒）记录为 v(t) = 2t，其中 t 是以秒为单位的时间。5 秒后你总共骑行了多远？这是一个积分问题：距离就是速度‑时间图像下方的面积。

The graph of v = 2t from t = 0 to t = 5 is a straight line. The area under it is a triangle with base 5 and height 10, so the area = ½ × 5 × 10 = 25 metres. You can also find the antiderivative: the antiderivative of 2t is t². Evaluating from 0 to 5 gives 5² − 0² = 25 metres.

v = 2t 从 t = 0 到 t = 5 的图像是一条直线。其下方的面积是一个底为 5、高为 10 的三角形，面积 = ½ × 5 × 10 = 25 米。你也可以求原函数：2t 的原函数是 t²。从 0 到 5 求值得 5² − 0² = 25 米。

Conversely, if you knew the distance function s(t) = t², you could differentiate it to find the speed: s'(t) = 2t, which matches our original speed function.

反过来，如果你已知距离函数 s(t) = t²，就可以对其微分得到速度：s'(t) = 2t，与原始速度函数吻合。

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KS3 Maths: Calculus Basics – Key Concepts Explained | KS3 数学：微积分基础考点精讲