Rates of Change | 变化率

📚 Rates of Change | 变化率

Rates of change lie at the heart of differentiation. In A-Level Mathematics, you are expected to move from finding gradients of curves to modelling real-world situations where one quantity changes with respect to another, often time. This article covers average and instantaneous rates, connected rates, parametric and implicit differentiation, exponential models, and kinematics, with Edexcel-style examples.

变化率是微分的核心。在 A-Level 数学中，你需要从求曲线的梯度逐步过渡到对现实情境建模，即一个量相对于另一个量（通常是时间）发生变化。本文涵盖平均变化率与瞬时变化率、关联变化率、参数方程与隐函数求导、指数模型以及运动学，并配有 Edexcel 风格的例题。

1. Average and Instantaneous Rate of Change | 平均变化率与瞬时变化率

For a function y = f(x), the average rate of change over an interval [a, b] is the slope of the secant line joining (a, f(a)) and (b, f(b)). It is calculated as Δy / Δx = [f(b) – f(a)] / (b – a). This tells you the overall change in y per unit change in x over that interval.

对于函数 y = f(x)，在区间 [a, b] 上的平均变化率是连接 (a, f(a)) 和 (b, f(b)) 的割线斜率，计算公式为 Δy / Δx = [f(b) – f(a)] / (b – a)。它表示在该区间内每单位 x 变化对应的 y 的总变化。

The instantaneous rate of change at x = a is found by shrinking the interval until b approaches a. This limit is the derivative dy/dx at x = a, which gives the slope of the tangent line. In practical terms, a speedometer reading is an instantaneous rate of change of distance with respect to time.

在 x = a 处的瞬时变化率通过让 b 趋近于 a 来求取。这个极限就是 x = a 处的导数 dy/dx，它给出切线的斜率。实际中，速度表的读数就是距离相对于时间的瞬时变化率。

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