IB Mathematics: Small Changes and Error Estimation | IB数学：微小变化与误差估计

📚 IB Mathematics: Small Changes and Error Estimation | IB数学：微小变化与误差估计

One of the most powerful uses of differentiation is not finding a slope, but predicting how one quantity changes when another quantity changes by a small amount. In IB Mathematics, this idea is called the method of small changes, and it leads naturally to the estimation of errors in measurement.

微分最强大的用途之一并不是求斜率，而是预测当某个量发生微小变化时，另一个量会如何变化。在 IB 数学中，这一思想称为“微小变化法”，并自然地引向对测量误差的估计。

1. The Core Idea of Small Changes | 微小变化的核心思想

If a quantity \( x \) changes by a small amount \( \Delta x \), then a dependent quantity \( y = f(x) \) changes by \( \Delta y \). When \( \Delta x \) is small, the curve \( y = f(x) \) is almost straight, so the actual change in \( y \) can be approximated by the slope times \( \Delta x \).

如果某个量 \( x \) 改变了一个微小量 \( \Delta x \)，那么因变量 \( y = f(x) \) 会改变 \( \Delta y \)。当 \( \Delta x \) 很小时，曲线 \( y = f(x) \) 几乎是一段直线，因此 \( y \) 的实际变化可以用斜率乘以 \( \Delta x \) 来近似。

The derivative \( f'(x) \) measures how fast \( y \) changes relative to \( x \). Multiplying this rate by \( \Delta x \) gives an estimate of \( \Delta y \). This is the essence of linear approximation.

导数 \( f'(x) \) 衡量的是 \( y \) 相对于 \( x \) 的变化快慢。用这个变化率乘以 \( \Delta x \)，就得到 \( \Delta y \) 的估计值。这就是线性近似的本质。

2. The Key Formula: Δy ≈ f'(x) Δx | 核心公式：Δy ≈ f'(x)Δx

For a differentiable function \( y = f(x) \), the small change in \( y \) is approximated by the product of the derivative and the small change in \( x \):

对于可微函数 \( y = f(x) \)，\( y \) 的微小变化可以用导数与 \( x \) 的微小变化之积来近似：

Δy ≈ f'(x) Δx

Equivalently, for a small change \( \delta x \), we often write \( \delta y \approx \frac{dy}{dx} \cdot \delta x \). The symbol \( \delta \) is used to denote a small increment, while \( \Delta \) is used for a finite change.

等价地，对于微小变化 \( \delta x \)，我们常写作 \( \delta y \approx \frac{dy}{dx} \cdot \delta x \)。符号 \( \delta \) 表示微小增量，而 \( \Delta \) 表示有限变化量。

This formula works because for small intervals, the tangent line at \( x \) is almost identical to the curve itself. The smaller \( \Delta x \) is, the better the approximation.

这个公式之所以成立，是因为在小区间内，\( x \) 处的切线几乎与曲线本身重合。\( \Delta x \) 越小，近似效果就越好。

3. Worked Example 1: Volume of a Cube | 例1：正方体的体积

A cube has side length \( x = 10 \) cm. If the side length is increased by \( 0.2 \) cm, estimate the increase in volume.

一个正方体的边长 \( x = 10 \) cm。如果边长增加 \( 0.2 \) cm，估计体积的增加量。

The volume is \( V = x^3 \), so \( \frac{dV}{dx} = 3x^2 \). At \( x = 10 \), \( \frac{dV}{dx} = 300 \). With \( \Delta x = 0.2 \):

体积为 \( V = x^3 \)，所以 \( \frac{dV}{dx} = 3x^2 \)。在 \( x = 10 \) 时，\( \frac{dV}{dx} = 300 \)。取 \( \Delta x = 0.2 \)：

ΔV ≈ 300 × 0.2 = 60 cm³

The actual change is \( 10.2^3 – 10^3 = 1061.208 – 1000 = 61.208 \) cm³. Our estimate of 60 cm³ is very close, with an error of only about 1.2 cm³.

实际变化为 \( 10.2^3 – 10^3 = 1061.208 – 1000 = 61.208 \) cm³。我们估计的 60 cm³ 与实际非常接近，误差仅为约 1.2 cm³。

This example shows that the formula is especially useful when the exact computation is cumbersome or when only a rough prediction is needed.

这个例子表明，当精确计算比较麻烦或只需要粗略预测时，这个公式尤其有用。

4. Worked Example 2: Surface Area of a Sphere | 例2：球的表面积

The radius of a sphere is measured as \( r = 5 \) cm with a possible error of \( \pm 0.1 \) cm. Estimate the resulting error in the surface area.

一个球的半径测得为 \( r = 5 \) cm，可能误差为 \( \pm 0.1 \) cm。估计由此引起的表面积误差。

Recall the surface area formula \( A = 4\pi r^2 \). Differentiate with respect to \( r \):

表面积公式为 \( A = 4\pi r^2 \)。对 \( r \) 求导：

\(\frac{dA}{dr} = 8\pi r\)

At \( r = 5 \), \( \frac{dA}{dr} = 40\pi \approx 125.664 \). The small change in area is approximately:

在 \( r = 5 \) 时，\( \frac{dA}{dr} = 40\pi \approx 125.664 \)。面积的小变化约为：

ΔA ≈ 40π × 0.1 = 4π ≈ 12.57 cm²

Since the error can be positive or negative, we write \( A = 100\pi \pm 4\pi \) cm², or \( A \approx 314.16 \pm 12.57 \) cm².

由于误差可能为正也可能为负，我们写作 \( A = 100\pi \pm 4\pi \) cm²，即 \( A \approx 314.16 \pm 12.57 \) cm²。

5. Relative Change and Percentage Change | 相对变化与百分比变化

Sometimes we are more interested in how the fractional change in one quantity affects another. The relative change in \( y \) is \( \frac{\Delta y}{y} \), and the percentage change is this value multiplied by 100.

有时我们更关心一个量的“相对变化”如何影响另一个量。\( y \) 的相对变化为 \( \frac{\Delta y}{y} \)，百分比变化则是该值乘以 100。

If \( y = f(x) \), then the relative change in \( y \) can be approximated using the derivative:

若 \( y = f(x) \)，则 \( y \) 的相对变化可借助导数近似：

\(\frac{\Delta y}{y} \approx \frac{f'(x)}{f(x)} \Delta x\)

The quantity \( \frac{f'(x)}{f(x)} \) is often called the logarithmic derivative, because \( \frac{d}{dx} (\ln f(x)) = \frac{f'(x)}{f(x)} \).

量 \( \frac{f'(x)}{f(x)} \) 常被称为对数导数，因为 \( \frac{d}{dx} (\ln f(x)) = \frac{f'(x)}{f(x)} \)。

In the sphere example, the percentage error in radius is \( \frac{0.1}{5} \times 100 = 2\% \). Since area is proportional to \( r^2 \), the percentage error in area is approximately \( 2 \times 2\% = 4\% \). This matches our earlier result: \( 4\pi / 100\pi = 4\% \).

在球的例子中，半径的百分比误差为 \( \frac{0.1}{5} \times 100 = 2\% \)。由于面积与 \( r^2 \) 成正比，面积的百分比误差约为 \( 2 \times 2\% = 4\% \)。这与之前的结果一致：\( 4\pi / 100\pi = 4\% \)。

6. Error Propagation in Measurements | 测量中的误差传播

In real experiments, every measurement carries an uncertainty. When a calculated quantity depends on measured values, the uncertainty propagates through the calculation. The small-change formula becomes a tool for estimating this propagation.

在真实实验中，每次测量都存在不确定性。当计算量依赖于测得值时，不确定性会通过计算过程传播。微小变化公式因此成为估计误差传播的工具。

For a function of one variable \( y = f(x) \), the absolute error in \( y \) is approximately \( |f'(x)|\, \Delta x \), where \( \Delta x \) is the maximum error in \( x \).

对单变量函数 \( y = f(x) \)，\( y \) 的绝对误差约为 \( |f'(x)|\, \Delta x \)，其中 \( \Delta x \) 是 \( x \) 的最大误差。

Absolute error in \( y \approx |f'(x)| \times \text{error in } x \).

绝对误差 \( y \approx |f'(x)| \times x \) 的误差。
Relative error in \( y \approx \left| \frac{f'(x)}{f(x)} \right| \times \text{error in } x \).

相对误差 \( y \approx \left| \frac{f'(x)}{f(x)} \right| \times x \) 的误差。
Percentage error in \( y \approx \left| \frac{f'(x)}{f(x)} \right| \times \text{error in } x \times 100\% \).

百分比误差 \( y \approx \left| \frac{f'(x)}{f(x)} \right| \times x \) 的误差 × 100%。

These formulas assume that the errors are small. They are derived from the first-order Taylor approximation and ignore higher-order terms.

这些公式假设误差很小。它们由一阶泰勒近似导出，忽略了高阶项。

7. Maximum Error and Bounds | 最大误差与边界

When a measured value is rounded, its true value lies between the lower and upper bounds. For example, \( x = 5.0 \) cm measured to 1 decimal place means \( 4.95 \le x < 5.05 \) cm. The maximum error is \( 0.05 \) cm.

当测量值被四舍五入时，真实值位于上下界之间。例如，\( x = 5.0 \) cm 精确到一位小数意味着 \( 4.95 \le x < 5.05 \) cm，最大误差为 \( 0.05 \) cm。

To estimate the maximum error in a dependent quantity, evaluate \( |f'(x)| \) at the appropriate value of \( x \) and multiply by the maximum error in \( x \).

要估计因变量的最大误差，应在适当的 \( x \) 值处计算 \( |f'(x)| \)，再乘以 \( x \) 的最大误差。

If \( f'(x) \) is not constant, the maximum error may occur at an endpoint of the interval. In such cases, check both the lower bound and the upper bound of \( x \).

如果 \( f'(x) \) 不是常数，最大误差可能出现在区间的端点。此时应同时检查 \( x \) 的下界和上界。

8. Common Mistakes and Exam Tips | 常见错误与考试提示

Many IB students lose marks on this topic through small but avoidable mistakes. The table below summarizes the most common issues and how to avoid them.

许多 IB 学生在这个主题上因为一些细小但可避免的错误而失分。下表总结了最常见的问题及避免方法。

Common Mistake	Correction
Using \(\Delta y \approx f'(x)\) without multiplying by \(\Delta x\).	Always multiply the derivative by the small change in \( x \).
Forgetting to use the absolute value for errors.	Errors are positive quantities, so take \(\|f'(x)\| \cdot \Delta x\).
Applying the formula when \(\Delta x\) is large.	The approximation is only valid for small \(\Delta x\).
Confusing absolute, relative, and percentage error.	Use units for absolute error, a unitless fraction for relative error, and percent for percentage error.

Remember to state whether an approximation is an overestimate or underestimate whenever possible. If \( f(x) \) is concave up, the tangent line lies below the curve, so the small-change estimate is an underestimate; if concave down, it is an overestimate.

只要可以，请说明近似值是高估还是低估。若 \( f(x) \) 是凹向上的，切线在曲线下方，因此微小变化估计是低估；若凹向下，则是高估。

9. Multi-Step Functions and Nested Errors | 多步函数与复合误差

Sometimes a quantity is obtained through a chain of calculations. For example, the area of a triangle might require both a base and a height measurement. In such cases, apply the small-change formula step by step or use a common technique: express the final quantity directly in terms of the measured variables.

有时一个量需要通过一系列计算得到。例如，三角形的面积可能需要同时测量底和高。此时可以逐步应用微小变化公式，或者采用常用技巧：直接把最终量表示为测量变量的函数。

If \( C = A \cdot B \), where both \( A \) and \( B \) have small percentage errors \( p_A\% \) and \( p_B\% \), then the percentage error in \( C \) is approximately \( p_A + p_B \). For a quotient \( C = A / B \), the percentage error is also approximately \( p_A + p_B \).

若 \( C = A \cdot B \)，其中 \( A \) 和 \( B \) 分别有百分比误差 \( p_A\% \) 和 \( p_B\% \)，则 \( C \) 的百分比误差约为 \( p_A + p_B \)。对于商 \( C = A / B \)，百分比误差也约为 \( p_A + p_B \)。

\(\frac{\Delta C}{C} \approx \frac{\Delta A}{A} + \frac{\Delta B}{B}\)

This rule is a direct consequence of logarithmic differentiation and is extremely useful in IB Internal Assessments.

该规则是对数求导的直接结果，在 IB 内部评估中非常有用。

10. Worked Example 3: Pendulum Period | 例3：单摆周期

The period of a simple pendulum is \( T = 2\pi \sqrt{\frac{L}{g}} \), where \( L \) is the length and \( g = 9.81 \) m/s². If \( L = 1.00 \) m is measured with an error of \( \pm 0.01 \) m, estimate the percentage error in \( T \).

单摆的周期为 \( T = 2\pi \sqrt{\frac{L}{g}} \)，其中 \( L \) 是摆长，\( g = 9.81 \) m/s²。若 \( L = 1.00 \) m，测量误差为 \( \pm 0.01 \) m，估计 \( T \) 的百分比误差。

Since \( T \propto \sqrt{L} \), we have \( \frac{dT}{dL} = \frac{\pi}{\sqrt{gL}} \). The relative error in \( T \) is:

因为 \( T \propto \sqrt{L} \)，所以 \( \frac{dT}{dL} = \frac{\pi}{\sqrt{gL}} \)。\( T \) 的相对误差为：

\(\frac{\Delta T}{T} \approx \frac{1}{2} \frac{\Delta L}{L} = \frac{1}{2} \times \frac{0.01}{1.00} = 0.005\)

Thus the percentage error in the period is \( 0.5\% \). Notice that the error in the period is only half the error in the length, because of the square root.

因此周期的百分比误差为 \( 0.5\% \)。注意周期的误差仅为长度误差的一半，因为这里存在平方根。

The absolute error in the period is approximately \( T \times 0.005 \). Since \( T \approx 2.006 \) s, the absolute error is about \( 0.010 \) s.

周期的绝对误差约为 \( T \times 0.005 \)。由于 \( T \approx 2.006 \) s，绝对误差约为 \( 0.010 \) s。

11. Practice Questions | 练习题

Try these questions to consolidate your understanding. Cover the solutions until you have attempted them yourself.

尝试以下题目来巩固你的理解。请先自己动笔，再查看答案。

Question	Answer Hint
If \( y = x^2 – 3x \), estimate \( \Delta y \) when \( x = 4 \) and \( \Delta x = 0.1 \).	\( \frac{dy}{dx} = 2x – 3 = 5 \), so \( \Delta y \approx 5 \times 0.1 = 0.5 \).
A circle has radius \( r = 8 \) cm with error \( \pm 0.05 \) cm. Find the approximate error in its area.	\( A = \pi r^2 \), \( \frac{dA}{dr} = 16\pi \), error \( \approx 0.8\pi \approx 2.51 \) cm².
For \( y = e^{0.2x} \), estimate the percentage change in \( y \) when \( x \) increases from 5 to 5.01.	\( \frac{dy}{dx} = 0.2e^{0.2x} \), relative change \( \approx 0.2 \times 0.01 = 0.002 = 0.2\% \).

After attempting each question, check whether your estimate is close to the exact change. This will strengthen your intuition for when linear approximation is reliable.

完成每道题后，检查你的估计值是否接近精确变化值。这会增强你对线性近似何时可靠的理解。

12. Summary | 总结

The method of small changes provides a fast and intuitive way to estimate how an output responds to small perturbations in its input. It is one of the most practical applications of differentiation in IB Mathematics, appearing in both Paper 1 and Paper 2, as well as in modelling tasks.

微小变化法提供了一种快速直观的方法来估计输出对输入微小扰动的响应。它是 IB 数学中微分最具实际应用价值的内容之一，在卷一、卷二以及建模任务中都会出现。

Core formula: \( \Delta y \approx f'(x) \Delta x \).

核心公式：\( \Delta y \approx f'(x) \Delta x \)。
Relative error: \( \frac{\Delta y}{y} \approx \frac{f'(x)}{f(x)} \Delta x \).

相对误差：\( \frac{\Delta y}{y} \approx \frac{f'(x)}{f(x)} \Delta x \)。
Products and quotients: relative errors add.

乘积与商：相对误差相加。
Always check that \( \Delta x \) is small enough for the approximation to be valid.

始终检查 \( \Delta x \) 是否足够小，以保证近似成立。

Mastering small changes and error estimation will not only improve your exam score but also give you a valuable tool for real-world data analysis.

掌握微小变化与误差估计不仅能提高你的考试成绩，也能为你提供一项在现实数据分析中极有价值的工具。

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IB Mathematics: Small Changes and Error Estimation | IB数学：微小变化与误差估计