Applications of Differentials in Approximate Calculations | 微分在近似计算中的应用

📚 Applications of Differentials in Approximate Calculations | 微分在近似计算中的应用

Approximate calculation is one of the most practical uses of the derivative. In many scientific and engineering problems we know the value of a function at a certain point, but we need to estimate its value at a nearby point. Since a differentiable function looks almost exactly like its tangent line in a sufficiently small neighbourhood, the slope given by the derivative lets us build a simple linear formula that replaces a complicated curve by a straight line. This idea, called linearization, is the foundation of differential approximation and appears explicitly in IB Mathematics courses.

近似计算是导数最实用的应用之一。在许多科学与工程问题中，我们已知函数在某一点的值，却需要估计它在邻近一点的值。由于可微函数在足够小的邻域内看起来几乎和它的切线完全一样，导数给出的斜率使我们能用一条简单直线替代复杂曲线。这一思想称为线性化，是微分近似的基础，也是 IB 数学课程中明确涉及的考点。

1. The Tangent Line and Local Linearity | 切线与局部线性性

inal function at x = a, which makes it the best linear approximation of f near that point.

如果函数 f 在 x = a 处可微，那么在 a 附近可以用它的切线来近似。a 处的切线方程为 y = f(a) + f′(a)(x − a)。这条直线在 x = a 处与原函数有相同的值和相同的斜率，因此它是 f 在该点附近的最佳线性逼近。

ile the true value is 1.21. The error is only 0.01, which is very small compared with the change from 1 to 1.1.

例如，考虑 f(x) = x² 在 a = 1 处。切线为 y = 2x − 1。当 x = 1.1 时，切线给出 y = 1.2，而真实值为 1.21。误差仅为 0.01，与从 1 到 1.1 的变化相比非常小。

Tangent line: y = f(a) + f′(a)(x − a)

切线方程：y = f(a) + f′(a)(x − a)

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2. The Linearization Formula | 线性化公式

e f(x) ≈ L(x). This formula is exact at x = a, and the error grows as x moves away from a.

f 在 a 处的线性近似通常写成 L(x) = f(a) + f′(a)(x − a)。我们称 L 为 f 在 a 处的线性化。当 x 接近 a 时，可以写成 f(x) ≈ L(x)。这个公式在 x = a 时精确成立，随着 x 远离 a，误差逐渐增大。

Example: estimate √9.04 using the linearization of f(x) = √x at a = 9. Since f(9) = 3 and f′(x) = 1/(2√x), we have f′(9) = 1/6. Therefore

例：利用 f(x) = √x 在 a = 9 处的线性化估计 √9.04。因为 f(9) = 3，f′(x) = 1/(2√x)，所以 f′(9) = 1/6。因此

√9.04 ≈ 3 + (1/6) × 0.04 = 3.006666...

The real value is approximately 3.006659, so the approximation is accurate to about five decimal places.

真实值约为 3.006659，所以该近似精确到约五位小数。

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3. Differentials and the Meaning of dy | 微分记号与 dy 的含义

iable, the actual change in y, namely Δy = f(a + Δx) − f(a), is approximately equal to dy.

记号 dx 表示 x 的小变化，并且我们把它等同于 Δx。微分 dy 定义为 dy = f′(x) dx。当自变量发生小变化时，y 的实际变化量 Δy = f(a + Δx) − f(a) 近似等于 dy。

Geometrically, Δy is the vertical rise of the curve, while dy is the vertical rise of the tangent line. The two values become closer as Δx becomes smaller. In applications, we therefore use the simple formula

从几何上看，Δy 是曲线的竖直增量，而 dy 是切线的竖直增量。当 Δx 越小时，二者越接近。因此在应用中我们采用简单公式

Δy ≈ dy = f′(x) dx

te that dx = Δx, but dy is generally not equal to Δy; we only say that dy approximates Δy.

注意 dx = Δx，但 dy 一般不等于 Δy；我们只说 dy 近似等于 Δy。

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

When we use a differential approximation, we should be able to describe its quality. The absolute error is |Δy − dy|. The relative error is |Δy − dy| / |Δy|, and the percentage error is the relative error multiplied by 100%.

使用微分近似时，我们应当能够描述其精度。绝对误差是 |Δy − dy|。相对误差是 |Δy − dy| / |Δy|，百分比误差是相对误差乘以 100%。

12 × 0.01 = 0.12. The original value is V = 8, so the relative error is about 0.12/8 = 1.5%.

设 V

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