Small Changes and Errors | 微小变化与误差

📚 2. Small Changes and Errors | 微小变化与误差

In many real-world situations, small changes in the input of a function lead to small changes in the output. Measurements are never exact, and the resulting errors propagate through calculations. Calculus provides a powerful tool—the differential—to estimate these small changes and errors efficiently, a topic that appears frequently in the IB Mathematics syllabus.

在许多现实情况中，函数输入的微小变化会导致输出的微小变化。测量从来都不是完全精确的，由此产生的误差会通过计算传播。微积分提供了一个强大的工具——微分，用来高效地估计这些微小变化和误差，这是 IB 数学大纲中经常出现的主题。

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1. Understanding Differentials: dy and dx | 理解微分：dy 与 dx

For a differentiable function y = f(x), the differential dy is defined as dy = f'(x) dx. Here dx represents an arbitrarily small change in the independent variable x, and dy gives the corresponding change in y along the tangent line at x. While the actual change is Δy = f(x + Δx) – f(x) along the curve, the differential dy approximates Δy when Δx is sufficiently small.

对于可微函数 y = f(x)，微分 dy 定义为 dy = f'(x) dx。这里 dx 表示自变量 x 的任意微小变化，dy 给出在 x 处沿切线方向 y 的相应变化。虽然实际变化 Δy = f(x + Δx) – f(x) 沿曲线发生，但当 Δx 足够小时，微分 dy 可以近似 Δy。

dy = f'(x) dx

Δy ≈ dy

This idea is the essence of linear approximation: near any point, a differentiable curve behaves almost exactly like its tangent line, making it possible to replace complicated functions with simple linear expressions for small changes.

这一思想是线性近似的本质：在任一点附近，可微曲线几乎完全像它的切线，因此对于微小变化，可以用简单的线性表达式替代复杂的函数。

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2. The Linear Approximation Formula | 线性近似公式

If we know the value of a function f at a point a and its derivative f'(a), we can estimate f(x) for x near a using the formula f(x) ≈ f(a) + f'(a)(x – a). Setting Δx = x – a, this becomes f(a + Δx) ≈ f(a) + f'(a) Δx. This linearisation is the foundation for handling both small changes and error propagation.

如果我们知道函数 f 在某点 a 的值及其导数 f'(a)，就可以使用公式 f(x) ≈ f(a) + f'(a)(x – a) 估计 a 附近的 f(x)。令 Δx = x – a，则变为 f(a + Δx) ≈ f(a) + f'(a) Δx。这种线性化是处理微小变化和误差传播的基础。

For example, to estimate √17, we can use f(x) = √x at a = 16. Since f'(x) = 1/(2√x), f'(16) = 1/8. Then √17 ≈ √16 + (1/8)×(17−16) = 4 + 0.125 = 4.125, which is very close to the true value 4.1231.

例如，估计 √17 时，我们可以使用 f(x)=√x，取 a=16。因为 f'(x)=1/(2√x)，f'(16)=1/8。那么 √17 ≈ 4 + 0.125 = 4.125，这与真实值 4.1231 非常接近。

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3. Measurement Errors as Small Changes | 作为微小变化的测量误差

When a quantity x is measured with an uncertainty ±Δx (the absolute error in measurement), any calculated quantity y = f(x) will inherit an error. Using the differential, we approximate the resulting error Δy by Δy ≈ f'(x) Δx. This relationship forms the core of error propagation analysis.

当某个量 x 的测量带有不确定度 ±Δx（测量的绝对误差）时，任何计算量 y = f(x) 都会产生误差。利用微分，我们可以用 Δy ≈ f'(x) Δx 近似计算误差 Δy。这一关系构成了误差传播分析的核心。

Note that Δx is positive when the error is expressed as a maximum possible deviation. To account for the worst-case scenario, we often consider the absolute value: |Δy| ≈ |f'(x)| |Δx|. This gives the maximum likely error in the computed quantity.

注意，当误差表示为最大可能偏差时，Δx 取正值。为了考虑最坏情况，我们通常使用绝对值：|Δy| ≈ |f'(x)| |Δx|。这给出了计算量的最大可能误差。

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4. Absolute, Relative and Percentage Errors | 绝对误差、相对误差与百分比误差

The absolute error in y is the magnitude |Δy|, which estimates how far the computed value could be from the true value. More meaningful in many contexts is the relative error, given by |Δy / y|, and the percentage error, which is the relative error multiplied by 100%.

y 的绝对误差是 |Δy|，它估计计算值与真实值可能相差多远。在许多情况下更有意义的是相对误差，由 |Δy / y| 给出，以及百分比误差，即相对误差乘以 100%。

Absolute error ≈ |f'(x) Δx|

Relative error ≈ |f'(x) Δx / f(x)|

These quantities tell us the significance of the error relative to the size of the measurement. For instance, a 1 mm error in measuring the length of a pencil is far more significant than a 1 mm error in measuring the length of a football pitch.

这些量告诉我们误差相对于测量值大小的显著程度。例如，测量铅笔长度时的 1 mm 误差远比测量足球场长度时的 1 mm 误差更显著。

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5. Error Propagation in Simple Operations | 简单运算中的误差传播

Although the differential method works universally, some basic arithmetic operations have simple error-propagation rules that are worth memorising for efficiency:

尽管微分方法普遍适用，一些基本算术运算有简单的误差传播规则，值得记忆以提高效率：

Sums and differences: The absolute error in a sum or difference is approximately the sum of the absolute errors of the terms. |Δ(x ± y)| ≈ |Δx| + |Δy|.

和与差：和或差的绝对误差近似等于各项绝对误差之和。|Δ(x ± y)| ≈ |Δx| + |Δy|。

Products and quotients: The relative error in a product or quotient is approximately the sum of the relative errors of the factors. |Δ(xy)/(xy)| ≈ |Δx/x| + |Δy/y|, and similarly for division.

积与商：积或商的相对误差近似等于各因子的相对误差之和。|Δ(xy)/(xy)| ≈ |Δx/x| + |Δy/y|，对于除法类似。

These rules can be derived by applying differentials to the functions u = x + y or v = xy. The chain rule and product rule confirm these practical shortcuts.

这些规则可以通过对 u = x + y 或 v = xy 应用微分推导出来。链式法则和乘积法则证实了这些实用的捷径。

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6. Worked Example: Area of a Circle | 实例：圆的面积

The area of a circle is A = πr². Suppose the radius is measured as r = 5.0 cm with a maximum error of 0.1 cm. We can estimate the absolute error in area using dA = 2πr dr. Thus ΔA ≈ 2πr Δr = 2π × 5.0 × 0.1 = π ≈ 3.14 cm².

圆的面积公式为 A = πr²。假设半径测量值为 r = 5.0 cm

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