Rates of Change | 变化率

📚 Rates of Change | 变化率

Rates of change lie at the heart of calculus and describe how one quantity varies with respect to another. In A-Level Edexcel Mathematics, mastering connected rates, parametric differentiation, implicit differentiation, and exponential growth allows you to model and solve real-world problems with confidence.

变化率是微积分的核心，描述了一个量相对于另一个量的变化快慢。在 A-Level Edexcel 数学中，掌握相关变化率、参数微分、隐函数微分以及指数增长能帮助你自信地建模并解决现实世界中的问题。

1. What is Rate of Change? | 什么是变化率？

The average rate of change of a function y = f(x) over an interval [x₁, x₂] is given by the gradient of the chord: Δy/Δx = (f(x₂) − f(x₁))/(x₂ − x₁). As the interval shrinks to zero, we obtain the instantaneous rate of change, which is the derivative dy/dx = f'(x).

函数 y = f(x) 在区间 [x₁, x₂] 上的平均变化率由弦的斜率给出：Δy/Δx = (f(x₂) − f(x₁))/(x₂ − x₁)。当区间长度趋近于零时，我们得到瞬时变化率，即导数 dy/dx = f'(x)。

Geometrically, dy/dx represents the slope of the tangent line at a point. If dy/dx > 0, y increases as x increases; if dy/dx < 0, y decreases; if dy/dx = 0, the graph has a stationary point.

在几何上，dy/dx 表示某一点处切线的斜率。如果 dy/dx > 0，y 随 x 增加而增加；如果 dy/dx < 0，y 随 x 增加而减小；如果 dy/dx = 0，图像存在一个驻点。

2. The Derivative as a Rate of Change | 导数作为变化率

In applied contexts, the derivative describes the instantaneous rate of change of one variable with respect to another. For example, if s(t) is displacement in metres at time t seconds, then velocity v = ds/dt (m/s) is the rate of change of displacement, and acceleration a = dv/dt = d²s/dt² (m/s²) is the rate of change of velocity.

在应用情境中，导数描述了一个变量相对于另一个变量的瞬时变化率。例如，若 s(t) 是 t 秒时刻的位移（米），那么速度 v = ds/dt（米/秒）是位移的变化率，加速度 a = dv/dt = d²s/dt²（米/秒²）是速度的变化率。

Other familiar rates include: dV/dr for volume of a sphere V = 4/3 π r³, dA/dr for area of a circle A = π r², and dT/dt for temperature change over time.

其他常见的变化率有：球体体积 V = 4/3 π r³ 对半径的导数 dV/dr，圆面积 A = π r² 对半径的导数 dA/dr，以及温度 T 对时间 t 的导数 dT/dt。

v = ds/dt, a = dv/dt = d²s/dt²

3. Connected Rates of Change and the Chain Rule | 相关变化率与链式法则

Connected rates of change occur when two or more variables are related by an equation, and each depends on a common parameter (often time t). The chain rule allows us to link the rates: if y = f(x) and x = g(t), then dy/dt = (dy/dx) × (dx/dt).

相关变化率出现在两个或多个变量由一个方程关联，且每个变量都依赖于某个共同参数（通常是时间 t）时。链式法则让我们能把变化率联系起来：如果 y = f(x) 且 x = g(t)，那么 dy/dt = (dy/dx) × (dx/dt)。

A classic example involves a spherical balloon being inflated. The volume V and radius r are related by V = 4/3 π r³. Differentiating with respect to t gives dV/dt = (dV/dr)(dr/dt) = 4π r² dr/dt. Given the rate of volume increase, you can find the rate of radius increase at a specific radius.

一个经典例子是充气中的球形气球。体积 V 和半径 r 满足 V = 4/3 π r³。对 t 求导得 dV/dt = (dV/dr)(dr/dt) = 4π r² dr/dt。给定体积的增加率，你就能求出在某一特定半径时半径的增加率。

Always identify the known rate and the required rate, write a geometric equation linking the variables, differentiate implicitly with respect to time, and substitute known values.

始终先明确已知的变化率和待求的变化率，写出联系变量的几何方程，对方程关于时间 t 隐式求导，再代入已知数值。

4. Rates of Change with Parametric Equations | 参数方程中的变化率

When a curve is defined parametrically as x = f(t), y = g(t), the gradient dy/dx is obtained via dy/dx = (dy/dt)/(dx/dt) (provided dx/dt ≠ 0). Here dy/dt and dx/dt are rates of change of y and x with respect to the parameter t.

当曲线由参数方程 x = f(t)、y = g(t) 定义时，斜率 dy/dx 通过 dy/dx = (dy/dt)/(dx/dt) 得到（前提 dx/dt ≠ 0）。这里 dy/dt 和 dx/dt 分别是 y 和 x 关于参数 t 的变化率。

If the parameter represents time, then dx/dt and dy/dt are the horizontal and vertical components of velocity. The speed is √((dx/dt)² + (dy/dt)²). Acceleration components are given by second derivatives d²x/dt² and d²y/dt².

如果参数代表时间，那么 dx/dt 和 dy/dt 就是速度的水平分量和垂直分量。速率为 √((dx/dt)² + (dy/dt)²)。加速度分量则由二阶导数 d²x/dt² 和 d²y/dt² 给出。

Example: x = t² − 1, y = t³ + 2t. Then dy/dx = (3t² + 2)/(2t). At t = 2, the rate dy/dx = (3×4+2)/(4) = 14/4 = 3.5.

示例：x = t² − 1, y = t³ + 2t。那么 dy/dx = (3t² + 2)/(2t)。在 t = 2 时，变化率 dy/dx = (3×4+2)/(4) = 14/4 = 3.5。

5. Implicit Differentiation and Rates | 隐函数求导与变化率

Many problems involve equations where y is not explicitly expressed in terms of x, such as x² + y² = r². To find dy/dx, we differentiate term-by-term with respect to x, treating y as a function of x and applying the chain rule: d/dx(y²) = 2y dy/dx.

许多问题涉及无法显式解出 y 的方程，如 x² + y² = r²。为求 dy/dx，我们逐项对 x 求导，将 y 视为 x 的函数并运用链式法则：d/dx(y²) = 2y dy/dx。

When the variables also change with time, implicit differentiation can be performed with respect to t. For instance, from x² + y² = 25 we obtain 2x dx/dt + 2y dy/dt = 0, enabling us to find an unknown rate when one rate and coordinates are known.

当变量还随时间变化时，可以关于 t 进行隐式求导。例如，从 x² + y² = 25 得到 2x dx/dt + 2y dy/dt = 0，从而在已知其中一个变化率和坐标的情况下，求出另一个变化率。

A typical question: ‘A ladder 5 m long leans against a wall. Its foot slides away at 0.5 m/s. How fast is the top descending when the foot is 3 m from the wall?’ Using x² + y² = 25 gives the connected rates.

一道典型题目：“一架长 5 m 的梯子斜靠在墙上。其底端以 0.5 m/s 的速度向外滑动。当底端离墙 3 m 时，顶端下降速率是多少？”利用 x² + y² = 25 即可得到相关变化率。

6. Displacement, Velocity and Acceleration | 位移、速度与加速度

For particle motion along a straight line, displacement s(t), velocity v(t) = ds/dt, and acceleration a(t) = dv/dt = d²s/dt² are central rates of change. The sign of v indicates direction; the sign of a indicates whether the particle is accelerating or decelerating relative to the positive direction.

对于沿直线运动的质点，位移 s(t)、速度 v(t) = ds/dt 和加速度 a(t) = dv/dt = d²s/dt² 是核心的变化率。速度的正负表示方向；加速度的正负表示质点是加速还是减速（相对于正方向而言）。

If velocity and acceleration have opposite signs, the particle is decelerating (speed decreasing). If they have the same sign, the particle is accelerating (speed increasing). Distance travelled requires integrating |v(t)| over the interval.

如果速度和加速度符号相反，则质点在做减速运动（速率减小）；如果符号相同，则在做加速运动（速率增大）。运动距离需要在整个区间对 |v(t)| 进行积分。

Quantity	Symbol	Rate relationship
Displacement	s(t)	v = ds/dt
Velocity	v(t)	a = dv/dt = d²s/dt²
Acceleration	a(t)	integral of a gives v

7. Exponential Growth and Decay | 指数增长与衰减

Many natural processes obey the law that the rate of change of a quantity is proportional to the quantity itself: dy/dt = ky, where k is a constant. If k > 0, it is exponential growth; if k < 0, exponential decay. The general solution is y = A eᵏᵗ, where A is the initial value.

许多自然过程遵循一个规律：某个量的变化率与该量本身成正比，即 dy/dt = ky，其中 k 为常数。若 k > 0，则是指数增长；若 k < 0，则是指数衰减。通解为 y = A eᵏᵗ，其中 A 为初始值。

In modelling problems, you may be given a rate and an initial condition to find particular solutions, or asked to calculate half-life (for decay) or doubling time (for growth). For decay, half-life T = ln 2 / |k|.

在建模问题中，你可能得到某个变化率和初始条件，从而求出特解；或者计算半衰期（衰减情形）或倍增时间（增长情形）。对于衰减，半衰期 T = ln 2 / |k|。

Example: A population P grows at a rate proportional to P. Initially P = 1000, and after 2 hours P = 1200. Set up dP/dt = kP, solve to find P = 1000 eᵏᵗ, use P(2)=1200 to find k.

示例：种群数量 P 的增长速率与 P 成正比。初始时 P = 1000，2 小时后 P = 1200。建立 dP/dt = kP，解得 P = 1000 eᵏᵗ，利用 P(2)=1200 求出 k。

8. Solving Connected Rates Problems | 解决相关变化率问题

A systematic approach ensures success. Step 1: Identify the given rate and the unknown rate. Step 2: Find an equation linking the variables (geometry or physical law). Step 3: Differentiate with respect to t using the chain rule. Step 4: Substitute known values and solve for the required rate.

一套系统的方法能确保成功。第 1 步：识别已知变化率和未知变化率。第 2 步：找到关联各变量的方程（几何关系或物理定律）。第 3 步：用链式法则对 t 求导。第 4 步：代入已知数值并解出所需的变化率。

Worked example: Water runs into a conical tank (vertex down) with base radius 5 m and height 10 m at 2 m³/min. Find the rate at which the water level rises when the depth is 6 m. Volume V = (1/3)π r² h. By similar triangles, r/h = 5/10 ⇒ r = h/2. So V = (1/3)π (h/2)² h = π h³/12. Then differentiate: dV/dt = (π/4)h² dh/dt. Given dV/dt = 2, h = 6, solve for dh/dt = 2 / (π/4 × 36) = 2/(9π) ≈ 0.0707 m/min.

例题解析：水以 2 m³/min 的速率注入一个倒置的圆锥形容器（顶点在下），底面半径 5 m、高 10 m。当水深 6 m 时，求水面上升的速率。体积 V = (1/3)π r² h。由相似三角形，r/h = 5/10 ⇒ r = h/2。所以 V = (1/3)π (h/2)² h = π h³/12。然后求导：dV/dt = (π/4)h² dh/dt。已知 dV/dt = 2，h = 6，解得 dh/dt = 2 / (π/4 × 36) = 2/(9π) ≈ 0.0707 m/min。

Always check units are consistent and clearly state the final rate with proper units and sign.

务必检查单位保持一致，并清晰地写明最终变化率及其单位和正负号。

9. Common Pitfalls and How to Avoid Them | 常见陷阱及避免方法

Forgetting the chain rule is the most frequent error. When differentiating V = (4/3)π r³ with respect to t, you must write dV/dt = 4π r² dr/dt, not just 4π r².

忘记使用链式法则是最常见的错误。在将 V = (4/3)π r³ 对 t 求导时，必须写成 dV/dt = 4π r² dr/dt，而不是只写 4π r²。

Confusing signs: a rate of decrease must be negative. If a ladder’s foot is moving away, dx/dt is positive, but dy/dt (top descending) will be negative.

符号混淆：下降的变化率必须是负值。如果梯子底端向外移动，dx/dt 为正，但 dy/dt（顶端下降）应为负。

Using the wrong ‘moment’ for substitution: do not substitute numerical values of variables before differentiating; differentiate first, then plug in instantaneous values.

在错误的“时刻”代入数值：不要在求导之前就代入变量的数值；先求导，再代入瞬时值。

Misidentifying independent variable: in rates problems, time t is often the ultimate independent variable, even if you differentiate with respect to an intermediate variable.

误判自变量：在变化率问题中，即使你针对中间变量求导，时间 t 往往是最终的自变量。

10. Exam Tips and Worked Examples | 考试技巧与实例分析

Start by writing down all given information in mathematical notation: e.g., ‘Given dV/dt = 5, find dr/dt when r = 2’. Draw a diagram if possible. State the relevant formula exactly as you will use it.

首先用数学符号写下所有已知信息，如：“已知 dV/dt = 5，求当 r = 2 时的 dr/dt”。若有可能，画出示意图。精确写出将要使用的相关公式。

For parametric or implicit differentiation, clearly show the step of applying the chain rule. If finding a connected rate, express the final answer in exact form (e.g., 1/(8π) cm/s) unless a decimal is requested.

对于参数或隐函数微分，要清晰地展示运用链式法则的步骤。如果求相关变化率，请给出精确形式的答案（例如 1/(8π) cm/s），除非题目要求以小数表示。

Another classic problem: ‘The radius of a circular oil spill increases at 0.2 m/s. Find the rate at which the area increases when the radius is 10 m.’ A = π r² ⇒ dA/dt = 2π r dr/dt. With r = 10, dr/dt = 0.2, so dA/dt = 2π × 10 × 0.2 = 4π m²/s.

另一经典问题：“圆形浮油半径以 0.2 m/s 的速度增加。当半径为 10 m 时，求面积增加率。” A = π r² ⇒ dA/dt = 2π r dr/dt。代入 r = 10、dr/dt = 0.2 可得 dA/dt = 2π × 10 × 0.2 = 4π m²/s。

When dealing with exponential growth, if asked to find the time to double, set up equations in terms of e and use natural logarithms. Remember to keep k as it appears in the differential equation without changing its sign unless the rate is explicitly negative.

处理指数增长问题时，如果要求出倍增时间，需用 e 建立方程并运用自然对数。记住保持 k 的符号与微分方程中一致，除非变化率明显为负。

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Rates of Change | 变化率