📚 Approximate Methods and Numerical Approximation in IB Maths | IB数学：近似方法与数值逼近
In IB Mathematics, not every equation can be solved exactly. Many real-world problems involve functions that are too complicated for algebraic manipulation, so we rely on approximate methods. This article explores key numerical techniques, their accuracy, and how to apply them confidently in exams.
在 IB 数学中，并非所有方程都能精确求解。许多现实问题涉及的函数过于复杂，无法通过代数操作处理，因此我们需要借助近似方法。本文将深入探讨核心数值技巧、它们的精确度，以及如何在考试中自信地运用它们。
1. Why Approximate? | 为什么要近似？
Exact solutions are beautiful but often impossible. For example, the equation x = cos(x) has no closed-form algebraic solution. Likewise, integrals such as ∫₀¹ e^(x²) dx cannot be expressed using elementary functions. In these cases, numerical methods give us practical answers.
精确解固然优美，但常常无法得到。例如，方程 x = cos(x) 没有闭式代数解。同样，∫₀¹ e^(x²) dx 这类积分无法用初等函数表示。在这些情况下，数值方法为我们提供了实用的答案。
- Many equations in IB exams are designed to be solved with a GDC.
- Numerical methods are iterative, meaning they produce increasingly accurate estimates.
- IB 考试中的许多方程设计为使用图形计算器（GDC）求解。
- 数值方法是迭代的，即它们能够产生越来越精确的估计值。
2. Absolute and Relative Error | 绝对误差与相对误差
Error measures how far an approximation is from the true value. Absolute error is |true − approximate|. Relative error divides this difference by the true value, often expressed as a percentage. These concepts are foundational for every numerical method.
误差衡量近似值与真实值之间的差距。绝对误差为 |真实值 − 近似值|。相对误差将这一差值除以真实值，通常以百分比表示。这些概念是所有数值方法的基础。
Absolute Error = |x_true − x_approx|, Relative Error = |x_true − x_approx| / |x_true|
In IB papers, you may be asked to determine whether an approximation is reasonable. Remember that a small absolute error may still be significant if the quantity itself is small.
在 IB 试卷中，你可能会被要求判断一个近似值是否合理。请记住，如果量本身很小，即使绝对误差很小也可能意义重大。
3. Linear Approximation (Tangent Line) | 线性近似（切线法）
For a differentiable function f at a point a, the tangent line gives the best linear approximation near a. The formula is L(x) = f(a) + f′(a)(x − a). This is also the first-order Taylor polynomial.
对于在点 a 处可微的函数 f，切线给出了 a 附近的最佳线性近似。公式为 L(x) = f(a) + f′(a)(x − a)。它也是一阶泰勒多项式。
L(x) ≈ f(a) + f'(a)(x − a)
Example: Approximate √4.1 using linear approximation with a = 4. Since f(4) = 2 and f′(4) = 1/4, we get 2 + 0.25 × 0.1 = 2.025. The true value is about 2.02485.
示例：使用 a = 4 的线性近似估算 √4.1。因为 f(4) = 2，f′(4) = 1/4，得到 2 + 0.25 × 0.1 = 2.025。真实值约为 2.02485。
4. Taylor Polynomials | 泰勒多项式
Taylor polynomials extend linear approximation by including higher-order derivatives. The n-th order Taylor polynomial of f centered at a is:
泰勒多项式通过包含高阶导数扩展了线性近似。以 a 为中心的 f 的 n 阶泰勒多项式为：
Pₙ(x) = f(a) + f′(a)(x−a) + f″(a)(x−a)²/2! + … + f⁽ⁿ⁾(a)(x−a)ⁿ/n!
In IB HL, you often need to find series expansions for e^x, sin(x), cos(x), and ln(1+x). A key question is how many terms are needed to achieve a given accuracy.
在 IB 高级水平（HL）中，你经常需要求 e^x、sin(x)、cos(x) 和 ln(1+x) 的级数展开。一个关键问题是需要多少项才能达到给定的精度。
Use the Lagrange error bound: if |f⁽ⁿ⁺¹⁾(t)| ≤ M on the interval, then the error is at most M|x−a|ⁿ⁺¹/(n+1)!.
使用拉格朗日误差界：若在区间上 |f⁽ⁿ⁺¹⁾(t)| ≤ M，则误差至多为 M|x−a|ⁿ⁺¹/(n+1)!。
5. Newton–Raphson Method | 牛顿-拉弗森方法
Newton–Raphson is an iterative method for solving f(x) = 0. Starting from an initial guess x₀, each iteration uses the tangent line to find a better approximation:
牛顿-拉弗森方法是一种求解 f(x) = 0 的迭代方法。从初始猜测 x₀ 出发，每次迭代使用切线寻找更好的近似值：
xₙ₊₁ = xₙ − f(xₙ) / f′(xₙ)
This method converges rapidly when the initial guess is close to the root. However, it may fail if f′(xₙ) is zero or if the function oscillates.
当初始猜测接近根时，该方法收敛速度很快。然而，如果 f′(xₙ) 为零或函数振荡，它可能会失败。
- Use a table in your GDC to iterate efficiently.
- In exams, show the first iteration fully, then give values to 3 decimal places or significant figures.
- 使用计算器中的表格功能高效迭代。
- 在考试中，完整展示第一次迭代，然后给出保留三位小数或有效数字的值。
6. Bisection Method | 二分法
The bisection method uses the intermediate value theorem. If f(a) and f(b) have opposite signs, there is at least one root in [a, b]. The method repeatedly halves the interval and selects the subinterval where the sign change occurs.
二分法基于中值定理。如果 f(a) 与 f(b) 异号，则 [a, b] 内至少有一个根。该方法反复将区间减半，并选择发生符号变化的子区间。
After n steps, the error is at most (b − a)/2ⁿ. This guarantees convergence but is slower than Newton–Raphson. It is reliable because it always brackets the root.
经过 n 步后，误差至多为 (b − a)/2ⁿ。这保证了收敛，但比牛顿-拉弗森方法慢。它很可靠，因为始终夹住根。
7. Secant Method | 割线法
The secant method approximates the derivative by a difference quotient, avoiding the need to compute f′. Its recurrence is:
割线法用差商来近似导数，从而避免计算 f′。其递推式为：
xₙ₊₁ = xₙ − f(xₙ)(xₙ − xₙ₋₁) / (f(xₙ) − f(xₙ₋₁))
It requires two initial guesses but does not require the derivative, making it useful when f is complicated or not differentiable. Convergence is faster than bisection but slower than Newton–Raphson.
它需要两个初始猜测，但不需要导数，因此在 f 复杂或不可导时非常有用。收敛速度比二分法快，但比牛顿-拉弗森方法慢。
8. Numerical Integration: Trapezoidal Rule | 数值积分：梯形法则
To approximate ∫ₐᵇ f(x) dx, divide [a, b] into n equal subintervals of width h = (b − a)/n. The trapezoidal rule approximates each strip by a trapezoid:
要近似 ∫ₐᵇ f(x) dx，将 [a, b] 分成 n 个宽度为 h = (b − a)/n 的等长子区间。梯形法则用梯形近似每个条带：
Tₙ = (h/2) [f(x₀) + 2f(x₁) + 2f(x₂) + … + 2f(xₙ₋₁) + f(xₙ)]
The error for the trapezoidal rule is proportional to h², so doubling n reduces the error by a factor of four.
梯形法则的误差与 h² 成正比，因此将 n 加倍可使误差减少为原来的四分之一。
9. Simpson’s Rule | 辛普森法则
Simpson’s rule fits quadratics through pairs of intervals. It requires n to be even. The formula is:
辛普森法则通过每对区间拟合二次多项式。它要求 n 为偶数。公式为：
Sₙ = (h/3) [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + 2f(xₙ₋₂) + 4f(xₙ₋₁) + f(xₙ)]
Simpson’s rule is much more accurate than the trapezoidal rule for smooth functions, with error proportional to h⁴. In IB, you may be asked to compare these methods for a given integral.
对于光滑函数，辛普森法则比梯形法则精确得多，其误差与 h⁴ 成正比。在 IB 考试中，你可能会被要求对给定积分比较这些方法。
10. Truncation and Rounding Errors | 截断误差与舍入误差
Numerical methods introduce two types of errors. Truncation errors come from stopping an infinite process after finitely many steps, such as using a finite Taylor series. Rounding errors arise from recording values with limited decimal places.
数值方法会引入两类误差。截断误差来自有限步后停止无限过程，例如使用有限泰勒级数。舍入误差源于用有限小数位数记录数值。
In iterative methods, rounding errors can accumulate. You should store full calculator precision in your GDC and only round at the final answer.
在迭代方法中，舍入误差可能累积。你应在计算器中保留完整精度，只在最终答案处进行四舍五入。
Total Error ≈ Truncation Error + Rounding Error
11. Choosing the Right Method | 选择合适的方法
In IB exams, you may need to decide which method to use. Newton–Raphson is fast but needs a derivative; bisection is slow but guarantees convergence; Simpson’s rule is accurate for smooth integrals; Taylor polynomials are useful for deriving approximations.
在 IB 考试中，你可能需要决定使用哪种方法。牛顿-拉弗森方法快速但需要导数；二分法慢但保证收敛；辛普森法则对光滑积分精确；泰勒多项式适合推导近似表达式。
| Method | Pros | Cons |
| Newton–Raphson | Fast convergence | Needs derivative; may diverge |
| Bisection | Always converges | Slow |
| Trapezoidal | Simple | Less accurate |
| Simpson’s rule | High accuracy | Requires even n |
Method | 优点 | 缺点
牛顿-拉弗森 | 收敛快 | 需要导数；可能发散
二分法 | 总是收敛 | 慢
梯形法则 | 简单 | 精度较低
辛普森法则 | 精度高 | 需要偶数 n
12. Exam Tips and Common Pitfalls | 考试技巧与常见误区
Always check that your result is sensible. If your Newton–Raphson iteration oscillates, choose a different initial guess. For numerical integration, increase n if accuracy is insufficient. State all answers to the required degree of accuracy.
始终检查结果是否合理。如果牛顿-拉弗森迭代振荡，请换一个初始猜测。对于数值积分，如果精度不足，请增大 n。所有答案都要按要求的精度给出。
- Do not forget to set your calculator to radians when using trigonometric functions.
- In Taylor series questions, write the first few terms explicitly before generalizing.
- Use the error bound to justify the number of terms or subintervals.
- 使用三角函数时，别忘了将计算器设置为弧度制。
- 在泰勒级数问题中，先明确写出前几项再推广。
- 使用误差界来证明项数或子区间数量的合理性。
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