变化率与相关变化率
爱德思国际GCSE进阶纯数学· 9G· 25 分钟阅读
1. 1. 基础变化率★★☆☆☆⏱ 5 min
物理量(y)相对于另一物理量(x)的变化率由一阶导数(\frac{dy}{dx})给出。当自变量为时间(t)时，该导数表示(y)随时间增减的快慢，单位为y的单位每单位时间。
变化率
因变量相对于自变量的导数，用于衡量因变量的变化快慢。正值表示该物理量在增大，负值表示该物理量在减小。
例:
If (V) is the volume of a balloon in cm³ and (t) is time in seconds, (\frac{dV}{dt}) has units cm³/s, measuring how fast the balloon inflates or deflates.
立方体的表面积公式为(A = 6x^2)，其中(x)为边长，单位为厘米。求当边长(x=3)厘米时，表面积相对于边长的变化率。
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步骤1：将面积函数对(x)求导
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步骤2：将(x = 3)厘米代入导数表达式
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最终答案：表面积的变化率为36 cm²每厘米边长增量。
Exam tip:
作答变化率类题目时最终答案务必写明单位；正确的数值和正确的单位都会单独给分。
2. 2. 相关变化率★★★☆☆⏱ 8 min
✓ 计算器
当两个物理量由已知公式关联时，它们的变化率可以通过链式法则建立联系。这是相关变化率问题的核心，这类题目会给出一个已知变化率，要求求解另一个相关的变化率。
相关变化率
两个或多个相关物理量的变化率，通过链式法则建立关联，可由已知变化率计算未知变化率。
例:
For an expanding sphere, the rate of change of volume (\frac{dV}{dt}) can be linked to the rate of change of radius (\frac{dr}{dt}) via the chain rule: (\frac{dV}{dt} = \frac{dV}{dr} \times \frac{dr}{dt}).
一个球形气球正在充气，其半径以2 cm/s的恒定速率增加。求当半径为5 cm时气球体积的增长速率。（球体体积公式：(V = \frac{4}{3}\pi r^3)，可在公式表中查阅。）
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步骤1：列出已知量和未知量
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步骤2：写出关联两个变化率的链式法则关系式
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步骤3：将体积公式对(r)求导
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步骤4：将所有已知数值代入链式法则关系式
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最终答案：体积的增长速率为(200\pi) cm³/s，若题目要求数值结果，保留三位有效数字为628 cm³/s。
3. 3. 小量近似★★★☆☆⏱ 7 min
当自变量(x)发生极微小的变化时，我们可以利用该点处函数的梯度近似计算因变量(y)对应的变化量。该近似方法在4PM1考试的估算类题目中应用十分广泛。
小量近似
用于计算自变量x发生小变化(\delta x)时，因变量y对应变化量(\delta y)的近似公式。当(\delta x)越小时，该近似的准确度越高。
例:
If (y = x^2), a small increase of (\delta x = 0.01) at (x = 10) gives (\delta y \approx 2(10)(0.01) = 0.2), very close to the exact change of (10.01^2 - 10^2 = 0.2001).
立方体的体积由(V = x^3)给出，其中(x)为边长。使用小增量近似法，估算边长从8 cm增加到8.02 cm时体积的变化量。
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步骤1：确定(x)和(\delta x)的取值
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步骤2：将体积函数对(x)求导
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步骤3：代入小增量近似公式
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最终答案：体积的近似增量为3.84 cm³。
When is the small change approximation most accurate?
When (\delta x) is very large
When (\delta x) is very small
When (\frac{dy}{dx}) is zero
When (y) is a linear function
显示答案
1 —Correct! The approximation relies on the curve being almost straight over very small intervals of (x).
4. 4. 求解考试风格的混合变化率问题★★★★☆⏱ 5 min
大多数考试题会结合实际情境、单位换算，以及相关变化率或小增量近似的知识点。遵循统一的分步解题流程以避免出错
Read the question carefully to identify known values, unknown values, and the relationship between quantities.
Write down any relevant formulae (use the formula sheet for mensuration values where allowed).
Set up the chain rule (for connected rates) or small change approximation formula explicitly.
Substitute values, paying close attention to units and the sign of rates (negative for decreasing quantities).
Check your final answer has the correct units and makes sense in the context of the problem.
5. 常见陷阱
错误做法:
Forgetting to include the chain rule link in connected rates problems, e.g. writing (\frac{dV}{dt} = 4\pi r^2) without multiplying by (\frac{dr}{dt}).
原因:
You are directly differentiating with respect to time instead of linking via the intermediate variable.
正确做法:
Always write out the full chain rule relationship explicitly before substituting values.
错误做法:
Using a positive rate for a decreasing quantity (e.g. water leaking from a tank).
原因:
The sign of the rate indicates direction of change, so incorrect signs lead to wrong final answers.
正确做法:
Assign negative values to rates of decreasing quantities, and check your final answer sign matches the context.
错误做法:
Calculating the exact change instead of using the small change approximation when instructed.
原因:
The question specifically tests your knowledge of the approximation formula, so exact calculations get zero marks.
正确做法:
Use only the (\delta y \approx \frac{dy}{dx}\delta x) formula when the question explicitly asks for the small change approximation.
错误做法:
Forgetting to include units in the final answer.
原因:
Exam marks are explicitly awarded for correct units for rates questions.
正确做法:
Always check units match the quantity being measured (e.g. cm³/s for volume rate of change).
错误做法:
Mixing up the order of variables in the chain rule, e.g. writing (\frac{dV}{dt} = \frac{dr}{dV} \times \frac{dr}{dt}).
原因:
The chain rule requires the intermediate variable to cancel out (e.g. (dr) in numerator and denominator).
正确做法:
Write the chain rule so the intermediate variable appears once in the numerator and once in the denominator to cancel correctly.
6. 速查表
Concept | Formula | Key Notes |
|---|---|---|
Basic Rate of Change | (\frac{dy}{dx}) | Units = (units of y) / (units of x) |
Connected Rates (Chain Rule) | (\frac{dy}{dt} = \frac{dy}{du} \times \frac{du}{dt}) | Intermediate variable (u) cancels out |
Small Change Approximation | (\delta y \approx \frac{dy}{dx} \delta x) | Only valid for very small (\delta x) |
Sign Convention | Negative = decreasing, Positive = increasing | Check sign matches problem context |
7. 常见问题
在相关变化率题目中我是否需要明确写出链式法则步骤？
是的，你必须在代入数值前写出链式法则的关系式（例如(\frac{dV}{dt} = \frac{dV}{dr} \times \frac{dr}{dt})）才能获得全部分数。即使最终计算结果错误，只要链式法则的关系式书写正确也可获得部分分数。
深入阅读
下一步
Now that you have mastered rates of change and connected rates for Edexcel IGCSE Further Pure Maths 4PM1, you are ready to move to more advanced calculus topics. These skills are foundational for all subsequent calculus work, including integration applications and kinematics problems. Make sure to practice past paper questions to get comfortable with the context and problem structures commonly tested, and always double-check your chain rule setup and units to avoid easy mark losses.