📚 IB Mathematics: Approximate Linearisation Methods | IB数学：近似线性化方法
Linearisation is one of the most powerful tools in calculus. It allows us to approximate a complicated function near a known point using a simple straight line, making calculations tractable when exact values are difficult or impossible to obtain. In IB Mathematics, this technique appears in the Analysis and Approaches (AA) and Applications and Interpretation (AI) syllabi, particularly in the context of local behaviour of functions, differentials, and numerical estimation.
线性化是微积分中最强大的工具之一。它让我们能够在已知点附近用一条简单的直线来近似一个复杂的函数，从而在精确值难以或无法获得时，使计算变得可行。在IB数学中，这一技巧出现在分析与方法（AA）和应用与解释（AI）课程大纲中，尤其涉及函数的局部行为、微分和数值估计等主题。
This article explores the theory, formulas, and exam-style applications of approximate linearisation. We will cover tangent line approximation, the differential method, error analysis, and second-order improvements, all with worked examples tailored to IB standards.
本文将探讨近似线性化的理论、公式以及考试风格的应用。我们将涵盖切线近似、微分法、误差分析和二阶改进，并附有符合IB标准的完整例题。
1. What Is Linearisation? | 什么是线性化？
At its core, linearisation is based on a simple observation: if you zoom in closely enough on any smooth curve at a point, it begins to look like a straight line. The tangent line at that point is the best straight-line approximation to the curve locally. The closer the point of interest is to the tangent point, the more accurate the approximation becomes.
线性化的核心基于一个简单的观察：如果你足够近距离地放大某点附近的光滑曲线，它看起来就像一条直线。该点的切线是曲线在局部最佳的直线近似。所关注的点离切点越近，近似就越精确。
at x = a, then the tangent line at (a, f(a)) can be used to approximate f(x) for x near a.
在数学上，如果函数 f 在 x = a 处可微，那么点 (a, f(a)) 处的切线可以用来近似 x 接近 a 时的 f(x) 值。
This idea bridges a complex nonlinear world to a simple linear one — a core theme in science and engineering, and an essential skill for IB Mathematics students.
这个思想将复杂的非线性世界与简单的线性世界连接起来——这是科学和工程中的核心主题，也是IB数学学生必备的重要技能。
2. Tangent Line Approximation | 切线近似
Suppose f is differentiable at x = a. The equation of the tangent line at that point is:
假设 f 在 x = a 处可微。该点的切线方程为：
L(x) = f(a) + f'(a)(x − a)
We call L(x) the linear approximation of f at a, sometimes denoted La(x). For values of x sufficiently close to a, we can write f(x) ≈ L(x).
我们称 L(x) 为 f 在 a 处的线性近似，有时记为 Lₐ(x)。对于足够接近 a 的 x 值，我们可以写出 f(x) ≈ L(x)。
Example 1: Approximate √4.1 using linearisation at x = 4.
例1：使用 x = 4 处的线性化近似计算 √4.1。
Let f(x) = √x. Then f(4) = 2 and f'(x) = 1/(2√x), so f'(4) = 1/4. The linear approximation is:
令 f(x) = √x。则 f(4) = 2，f'(x) = 1/(2√x)，所以 f'(4) = 1/4。线性近似为：
L(x) = 2 + (1/4)(x − 4)
Therefore, at x = 4.1:
因此，在 x = 4.1 处：
L(4.1) = 2 + 0.25(0.1) = 2.025
The true value of √4 ccurate to three decimal places. In IB exams, you are often asked to state whether the approximation is an overestimate or underestimate — this depends on the concavity of f, a point we will return to in Section 5.
√4.1 的真实值约为 2.0248，因此该近似精确到小数点后三位。在IB考试中，常会要求你判断近似值是高估还是低估——这取决于 f 的凹凸性，我们将在第5节中讨论这一点。
3. The Differential Method | 微分法
If y = f(x), then a small change dx in x produces a small change dy in y, approximated by:
微分法是线性化的另一种形式。我们不用切线方程，而是用小变化量来表达近似。如果 y = f(x)，那么 x 的小变化 dx 会引起 y 的小变化 dy，可近似为：
dy = f'(x) dx
Here, dx represents an infinitesimal or very small change in x, and dy is the corresponding estimated change in y. This is useful when estimating the propagation of error in measurements.
这里，dx 表示 x 的无穷小或非常小的变化，dy 是对应 y 的估计变化量。这在估计测量误差传播时非常有用。
Example 2: cm with a possible measurement error of ±0.1 cm. Estimate the maximum error in the volume.
例2：一个立方体的边长 s = 10 厘米，测量误差可能为 ±0.1 厘米。估计体积的最大误差。
V = s³, so dV = 3s² ds. With s = 10 and ds = 0.1:
V = s³，所以 dV = 3s² ds。以 s = 10 和 ds = 0.1 代入：
dV = 3(10)²(0.1) = 30 cm³
.301 cm³, so the differential approximation is very close. This method is favoured in Applications and Interpretation (AI) papers due to its practical relevance.
真实变化为 (10.1)³ − 10³ = 30.301 cm³，因此微分近似非常接近。这种方法在应用与解释（AI）试卷中因其实用性而备受欢迎。
4. Linearisation for Multivariable Functions | 多元函数的线性化
For a function z = f(x, y) that is differentiable at (a, b), the linear approximation is:
线性化可以扩展到二元函数，这在IB高级水平（HL）学习偏导数时会出现。对于在 (a, b) 处可微的函数 z = f(x, y)，线性近似为：
L(x, y) = f(a, b) + fₓ(a, b)(x − a) + fᵧ(a, b)(y − b)
x, y) describes the tangent plane to the surface z = f(x, y) at the point (a, b, f(a, b)).
其中 fₓ 和 fᵧ 分别表示关于 x 和 y 的偏导数。从几何上讲，L(x, y) 描述了在点 (a, b, f(a, b)) 处曲面 z = f(x, y) 的切平面。
Example 3: Find the linear approximation of f(x, y) = x²y + sin(x) at (π, 2).
例3：求 f(x, y) = x²y + sin(x) 在 (π, 2) 处的线性近似。
ves: fₓ = 2xy + cos(x), so fₓ(π, 2) = 4π + cos(π) = 4π − 1. And fᵧ = x², so fᵧ(π, 2) = π².
计算：f(π, 2) = π²·2 + sin(π) = 2π²。偏导数：fₓ = 2xy + cos(x)，所以 fₓ(π, 2) = 4π + cos(π) = 4π − 1。fᵧ = x²，所以 fᵧ(π, 2) = π²。
L(x, y) = 2π² + (4π − 1)(x − π) + π²(y − 2)
This tangent plane gives an accurate approximation near (π, 2) and can be used to estimate values without a calculator in exam contexts.
这个切平面在 (π, 2) 附近能够给出精确的近似值，在考试场景中可用于在没有计算器的情况下估算数值。
5. Overestimate or Underestimate? | 高估还是低估？
Whether the tangent line approximation overestimates or underestimates the true function value depends on the second derivative. If f''(x) > 0 on the interval of interest, the function is concave up and lies above its tangent line, so L(x) is an underestimate. If f''(x) < 0, the function is concave down and lies below its tangent line, so L(x) is an overestimate.
切线近似是高估还是低估真实函数值，取决于二阶导数。如果 f''(x) > 0（在所考虑的区间上），函数是上凹的，位于其切线上方，因此 L(x) 是低估。如果 f''(x) < 0，函数是下凹的，位于其切线下方，因此 L(x) 是高估。
Example 4: Determine whether the approximation f(x) = √x at x = 4.1 (from Example 1) is an overestimate or underestimate.
例4：判断例1中 f(x) = √x 在 x = 4.1 处的近似是高估还是低估。
We compute f''(x) = −1/(4x3/2). At x = 4.1, f'' and L(4.1) = 2.025 is an overestimate. Indeed, the true value 2.0248 is slightly smaller.
我们计算 f''(x) = −1/(4x3/2)。在 x = 4.1 处，f''(4.1) < 0，因此曲线是下凹的。所以切线位于曲线上方，L(4.1) = 2.025 是高估。确实，真实值 2.0248 略小。
In IB exams, this concavity argument is a common follow-up question worth 2–3 marks. Students should always state the sign of the second derivative and link it to the shape of the curve.
在IB考试中，这个凹凸性论证是一个常见的后续问题，通常占2–3分。学生应始终指出二阶导数的符号，并将其与曲线形状联系起来。
6. Error Analysis and Percentage Error | 误差分析与百分比误差
Linearisation introduces an error known as the truncation or approximation error. The absolute error is |f(x) − L(x)|, and the relative error is |f(x) − L(x)| / |f(x)|. Multiplying the relative error by 100 gives the percentage error. IB exam questions frequently ask students to compute percentage errors when comparing approximations to exact values.
线性化会引入误差，称为截断误差或近似误差。绝对误差为 |f(x) − L(x)|，相对误差为 |f(x) − L(x)| / |f(x)|。将相对误差乘以100即得百分比误差。IB考试题经常要求学生在比较近似值与精确值时计算百分比误差。
Example 5: For Example 1, the exact value of √4.1 is approximately 2.024846. The linear approximation gave 2.025. Calculate the percentage error.
例5：对于例1，√4.1 的精确值约为 2.024846。线性近似给出 2.025。计算百分比误差。
percentage error = |2.024846 − 2.025| / 2.024846 × 100% ≈ 0.0076%
This extremely small error demonstrates the effectiveness of linearisation when x is close to a. Students should note that error grows as x moves away from a — a key limitation of the method.
这个极小的误差证明了当 x 接近 a 时线性化的有效性。学生应注意，随着 x 远离 a，误差会增大——这是该方法的一个关键局限。
A useful bound for the error is given by the Lagrange remainder formula (HL only):
误差的一个有用界限由拉格朗日余项公式给出（仅HL）：
|Error| ≤ (1/2)M(x − a)², where M = max|f''(t)| for t between a and x
This formula provides an upper bound on the error and is a powerful tool for rigorous analysis.
这个公式给出了误差的上界，是进行严谨分析的强大工具。
7. Second-Order (Quadratic) Approximation | 二阶（二次）近似
tion to a second-order Taylor approximation. The quadratic approximation of f at x = a is:
当需要更高精度时，我们可以将线性化扩展到二阶泰勒近似。f 在 x = a 处的二次近似为：
Q(x) = f(a) + f'(a)(x − a) + (1/2)f''(a)(x − a)²
This parabola matches the function's value, slope, and curvature at x = a, providing a better fit over a wider interval.
这条抛物线在 x = a 处匹配了函数的值、斜率和曲率，在更宽的区间内提供了更好的拟合。
Example 6: Find the quadratic approximation of f(x) = eˣ at x = 0, and use it to estimate e0.1.
例6：求 f(x) = eˣ 在 x = 0 处的二次近似，并用它估计 e0.1。
For f(x) = eˣ, we have f(0) = 1, f'(0) = 1, f''(0) = 1. Thus:
对于 f(x) = eˣ，我们有 f(0) = 1，f'(0) = 1，f''(0) = 1。因此：
Q(x) = 1 + x + (1/2)x²
At x = 0.1: Q(0.1) = 1 + 0.1 + 0.005 = 1.105. The true value e0.1 ≈ 1.10517, so the quadratic approximation is correct to four decimal places — far more accurate than the linear approximation 1.1.
在 x = 0.1 处：Q(0.1) = 1 + 0.1 + 0.005 = 1.105。真实值 e0.1 ≈ 1.10517，因此二次近似精确到小数点后四位——远比线性近似 1.1 更精确。
In IB HL Paper 3 style questions, students may be asked to derive or apply such quadratic approximations in unfamiliar contexts.
在IB HL Paper 3风格的题目中，学生可能会被要求在陌生情境中推导或应用这样的二次近似。
8. Exam-Style Problem: Newton–Raphson Connection | 考试风格问题：牛顿–拉夫逊法的联系
Linearisation is also the theoretical foundation of the Newton– root of f(x) = 0, we linearise f at x₁ and find where the tangent line crosses the x-axis:
线性化也是牛顿–拉夫逊法求解方程根的数学基础。给定方程 f(x) = 0 根的初始估计值 x₁，我们在 x₁ 处线性化 f，并找到切线与 x 轴的交点：
xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ)
Example 7: Use linearisation (the Newton– a) to find the second approximation of a root of f(x) = x³ − 2x − 5, starting with x₁ = 2.
例7：使用线性化（牛顿–拉夫逊公式）求 f(x) = x³ − 2x − 5 的根的第二近似值，初始值 x₁ = 2。
We have f(2) = 8 − 4 − 5 = −1, and f'(x) = 3x² − 2, so f'(2) = 12 − 2 = 10. Thus:
我们有 f(2) = 8 − 4 − 5 = −1，f'(x) = 3x² − 2，所以 f'(2) = 12 − 2 = 10。因此：
x₂ = 2 − (−1)/10 = 2 + 0.1 = 2.1
This iterative use of linearisation converges rapidly when the initial guess is reasonably close to the root. IB students should recognise this as a direct application of the tangent line equation.
当初始猜测值接近根时，这种线性化的迭代使用会迅速收敛。IB学生应认识到这是切线方程的直接应用。
9. Summary of Key Formulas | 关键公式总结
The table below summarises the essential linearisation formulas and their uses, providing a quick revision reference for IB examinations.
下表总结了基本的线性化公式及其用途，为IB考试提供快速复习参考。
| Method | 方法 | Formula | 公式 | When to Use | 使用时机 |
| Linear approximation 线性近似 |
L(x) = f(a) + f'(a)(x − a) | Estimating f(x) near x = a 估算 x = a 附近的 f(x) |
| Differential method 微分法 |
dy = f'(x) dx | Propagation of error 误差传播 |
| Quadratic approximation 二次近似 |
Q(x) = f(a) + f'(a)(x − a) + ½f''(a)(x − a)² | Higher accuracy required 需要更高精度 |
| Multivariable linearisation 多元线性化 |
L(x, y) = f(a, b) + fₓ(a, b)(x − a) + fᵧ(a, b)(y − b) | Functions of two variables 二元函数（HL） |
Mastering these formulas, alongside their geometric interpretations, is essential for success in IB Mathematics examinations.
掌握这些公式及其几何解释，是IB数学考试取得成功的关键。
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