📚 Using Derivatives to Estimate Small Changes | IB数学：用导数估算微小变化
In IB Mathematics, one of the most practical applications of differentiation is estimating how a small change in one variable affects another. Instead of calculating the exact new value, we use the derivative as an instantaneous rate of change to predict the approximate change.
在IB数学中，求导最实用的应用之一就是估计一个变量的微小变化如何影响另一个变量。我们并不计算精确的新值，而是将导数视为瞬时变化率，用它来预测近似的改变量。
1. What Is a Small Change? | 什么是微小变化？
When we say a ‘small change’, we mean a small increment or decrement in the independent variable, often denoted by Δx (delta x) or δx. The corresponding change in the dependent variable is Δy or δy.
当我们说“微小变化”时，指的是自变量的一个小的增加量或减少量，通常用 Δx（德尔塔 x）或 δx 表示。因变量的相应变化则记为 Δy 或 δy。
For a function y = f(x), if x increases by a tiny amount Δx, the exact change in y is given by Δy = f(x + Δx) − f(x). However, this exact calculation may be complicated, and we can use differentiation to approximate it efficiently.
对于函数 y = f(x)，如果 x 增加一个微小量 Δx，y 的精确变化为 Δy = f(x + Δx) − f(x)。然而，这种精确计算可能很复杂，我们可以利用求导来高效地近似它。
2. The Derivative as an Instantaneous Rate of Change | 导数作为瞬时变化率
The derivative f'(x) is defined as the limit of the average rate of change as Δx approaches zero:
导数 f'(x) 定义为当 Δx 趋近于零时平均变化率的极限：
f'(x) = lim (Δx → 0) [f(x + Δx) − f(x)] / Δx
This means that for very small Δx, the average rate of change is almost equal to the instantaneous rate of change. Therefore, we can write the approximation:
这意味着，当 Δx 非常小时，平均变化率几乎等于瞬时变化率。因此，我们可以写出近似关系：
Δy / Δx ≈ f'(x) ⇒ Δy ≈ f'(x) Δx
This fundamental idea is the basis for estimating small changes using derivatives.
这一基本思想是利用导数估计微小变化的基础。
3. The Linear Approximation Formula | 线性近似公式
The formula Δy ≈ f'(x) Δx is also called the linear approximation or the “differentials” method. It works because near a point, the tangent line is very close to the curve.
公式 Δy ≈ f'(x) Δx 也被称为线性近似或“微分”方法。它的原理是在某一点附近，切线非常接近曲线本身。
If we write dx to represent a small change in x (instead of Δx) and dy to represent the corresponding change in y, the classical differential notation gives:
如果我们用 dx 表示 x 的微小变化（而不是 Δx），用 dy 表示 y 的相应变化，经典的微分记号给出：
dy = f'(x) dx
This is an exact relationship between the differentials, and for small actual changes we use dy ≈ Δy and dx ≈ Δx. Thus Δy ≈ f'(x) Δx is a practical version.
这是微分之间的精确关系，而对于小的实际变化，我们使用 dy ≈ Δy 和 dx ≈ Δx。因此 Δy ≈ f'(x) Δx 是实用的版本。
4. Worked Example 1: Polynomial Function | 实例1：多项式函数
Problem: Let y = x³ + 2x. Estimate the change in y when x changes from 2 to 2.01.
题目：设 y = x³ + 2x，估算当 x 从 2 变到 2.01 时 y 的改变量。
Solution: First find the derivative:
解答：首先求导：
f'(x) = 3x² + 2
At x = 2, f'(2) = 3(4) + 2 = 14. The change in x is Δx = 0.01. Thus:
在 x = 2 处，f'(2) = 3(4) + 2 = 14。x 的变化量为 Δx = 0.01。因此：
Δy ≈ f'(2) × Δx = 14 × 0.01 = 0.14
So the estimated change is 0.14. The exact change is f(2.01) − f(2) = 0.140601, showing the approximation is very close.
所以估计的变化量是 0.14。精确变化为 f(2.01) − f(2) = 0.140601，可见近似非常接近。
5. Using Percentage Change | 使用百分比变化
In many problems, we are given the percentage change of x, not the absolute change. If x increases by p%, then Δx = (p/100) × x, and hence:
在许多问题中，我们得到的是 x 的百分比变化，而不是绝对变化。如果 x 增加 p%，则 Δx = (p/100) × x，因此：
Δy ≈ f'(x) × (p/100) x
The percentage change in y is then approximately:
那么 y 的百分比变化近似为：
(Δy / y) × 100% ≈ [f'(x) x / f(x)] × p%
The quantity f'(x) x / f(x) is sometimes called the elasticity of y with respect to x in economics, and it links relative changes beautifully.
数量 f'(x) x / f(x) 在经济学中有时被称为 y 关于 x 的弹性，它很好地联系了相对变化。
6. Example 2: Finding a Percentage Change | 实例2：求百分比变化
Problem: The surface area of a sphere is A = 4πr². If the radius r increases by 2%, find the approximate percentage increase in the surface area.
题目：球体的表面积公式为 A = 4πr²。如果半径 r 增加 2%，求表面积的大致百分比增加。
Solution: Differentiate with respect to r:
解答：对 r 求导：
dA/dr = 8πr
Here Δr = 0.02r. Therefore:
这里 Δr = 0.02r。因此：
ΔA ≈ dA/dr × Δr = 8πr × 0.02r = 0.16πr²
Since A = 4πr², the fractional increase is:
因为 A = 4πr²，所以增加的比例为：
ΔA / A = (0.16πr²) / (4πr²) = 0.04
Hence the surface area increases by approximately 4%. This matches the exact result very closely for small changes.
因此表面积大约增加 4%。对于微小变化，这与精确结果非常吻合。
7. Estimating New Values | 估计新值
The linear approximation can also be used to estimate the value of a function near a known point:
线性近似也可用于估计已知点附近的函数值：
f(x + Δx) ≈ f(x) + f'(x) Δx
This is sometimes presented as the tangent line approximation, and it is equivalent to the first-order Taylor polynomial.
这有时被称为切线近似，它等价于一阶泰勒多项式。
For example, to estimate √(9.04), use f(x) = √x at x = 9:
例如，要估算 √(9.04)，在 x = 9 处使用 f(x) = √x：
f'(x) = 1/(2√x), f'(9) = 1/6, Δx = 0.04
f(9.04) ≈ 3 + (1/6)(0.04) = 3 + 0.006666… = 3.006666…
So √(9.04) ≈ 3.0067, which is very accurate.
所以 √(9.04) ≈ 3.0067，非常精确。
8. Small Changes in Exponential and Logarithmic Functions | 指数与对数函数中的微小变化
For y = eˣ, the derivative is itself, so Δy ≈ eˣ Δx = y Δx. This means the absolute change is proportional to the current value.
对于 y = eˣ，导数仍然是它自身，所以 Δy ≈ eˣ Δx = y Δx。这意味着绝对变化与当前值成正比。
For y = ln x, the derivative is 1/x, so Δy ≈ (1/x) Δx. This is often used in financial mathematics and biology to model relative growth.
对于 y = ln x，导数是 1/x，所以 Δy ≈ (1/x) Δx。这在金融数学和生物学中常用于模拟相对增长。
Example: Estimate the change in ln x when x changes from 1 to 1.005.
例：估算当 x 从 1 变到 1.005 时 ln x 的变化。
Here f'(1) = 1, and Δx = 0.005, so Δy ≈ 1 × 0.005 = 0.005. Indeed, ln(1.005) ≈ 0.00499.
这里 f'(1) = 1，Δx = 0.005，所以 Δy ≈ 1 × 0.005 = 0.005。事实上，ln(1.005) ≈ 0.00499。
9. Connection to Marginal Functions in Economics | 经济学中边际函数的联系
In economics, the marginal cost is the derivative of the cost function C(q) with respect to quantity q. The marginal cost approximates the cost of producing one more unit:
在经济学中，边际成本是成本函数 C(q) 关于产量 q 的导数。边际成本近似于多生产一单位所需要的成本：
C(q + 1) − C(q) ≈ C'(q)
Similarly, marginal revenue and marginal profit are used to estimate changes from small changes in output. This is a direct application of the linear approximation.
同样，边际收益和边际利润用于估计产量微小变化带来的改变。这是线性近似的一个直接应用。
10. Accuracy and Limitations | 精确度与局限性
The approximation Δy ≈ f'(x) Δx is most accurate when Δx is very small. The missing part of the exact change is related to the second derivative, roughly (1/2)f”(x)(Δx)².
近似 Δy ≈ f'(x) Δx 在 Δx 非常小时最准确。被忽略的精确变化部分与二阶导数有关，约为 (1/2)f”(x)(Δx)²。
Thus, the error is of order (Δx)², which becomes negligible as Δx → 0. However, if Δx is large, the tangent line may diverge significantly from the curve, and we should use a better approximation.
因此，误差为 (Δx)² 量级，当 Δx → 0 时误差可以忽略。但如果 Δx 较大，切线可能与曲线明显偏离，我们就应该使用更好的近似方法。
Also, when the second derivative is large or changes quickly, the linear approximation may be poor even for moderately small Δx. Always consider the context and the required precision.
此外，当二阶导数很大或变化很快时，即使 Δx 中等大小，线性近似也可能较差。始终要考虑问题的背景和所需精度。
11. Summary Table of Key Formulas | 关键公式总结表
| Description | Formula |
| Linear approximation for a small change | Δy ≈ f'(x) Δx |
| Estimating a new value | f(x + Δx) ≈ f(x) + f'(x) Δx |
| Percentage change of y | %Δy ≈ [x f'(x) / f(x)] × %Δx |
| Differential notation | dy = f'(x) dx |
These formulas are not only exam favourites but also powerful tools for real-world estimation.
这些公式不仅是考试中的常客，更是现实世界中强大的估算工具。
12. Practice Advice for IB Exams | IB考试练习建议
When solving a “small change” question, always follow these steps: identify the function and the point, compute the derivative, determine Δx (or the percentage change), then apply Δy ≈ f'(x) Δx.
在解答“微小变化”问题时，务必遵循以下步骤：确定函数及其自变量的取值，计算导数，确定 Δx（或百分比变化），然后应用 Δy ≈ f'(x) Δx。
Remember to convert percentages to decimals before multiplying, and state whether you are estimating an absolute change, a relative change, or a new value.
记住在相乘之前要将百分比转换为小数，并说明你是估计绝对变化、相对变化还是新值。
Practice with polynomials, trigonometric functions, exponentials, and logarithms. In each case, the derivative rules remain the same; only the interpretation of the change changes.
要多练习多项式、三角函数、指数函数和对数函数。在每种情况下，求导法则相同，只是对变化的解释有所不同。
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