📚 IB Mathematics: Numerical Methods – Key Exam Points | IB 数学：数值方法 考点精讲
Numerical methods play a crucial role in the IB Mathematics curriculum, particularly in the Applications and Interpretation (AI) and Analysis and Approaches (AA) courses at both Standard and Higher Level. These techniques provide approximate solutions to problems that cannot be solved analytically, such as finding roots of complex equations, evaluating definite integrals, or solving differential equations. Mastering numerical methods not only helps you answer specific exam questions but also deepens your understanding of calculus and algebraic reasoning. This article summarises the essential concepts, formulas, error analysis, and exam strategies you need to excel.
数值方法在 IB 数学课程中扮演着关键角色，尤其是在应用与解释 (AI) 以及分析与方法 (AA) 的标准水平和高级水平中。这些技术能为无法解析求解的问题提供近似解，例如寻找复杂方程的根、计算定积分或求解微分方程。掌握数值方法不仅有助于回答具体的考试题目，还能加深你对微积分和代数推理的理解。本文总结了必备的核心概念、公式、误差分析以及考试策略，帮助你轻松应对考试。
1. Errors and Accuracy | 误差与精度
When using numerical methods, we must be aware of the inevitable errors that arise. The true value is often unknown, so we work with approximations. The absolute error is the absolute difference between the true value x and its approximation a: |x - a|. The relative error normalises this by the true value: |x - a|/|x|. Multiplying by 100% gives the percentage error. In IB exams, you may be asked to calculate these from given data or to interpret the accuracy of a result.
在使用数值方法时，我们必须认识到不可避免的误差。真实值往往未知，因此我们使用近似值。绝对误差是真实值 x 与其近似值 a 的绝对差：|x - a|。相对误差通过除以真实值对其进行归一化：|x - a|/|x|。再乘以 100% 即得到百分比误差。在 IB 考试中，你可能需要根据给定数据计算这些误差，或解释结果的精度。
When the true value is not available, we often estimate error using the difference between successive approximations, or by using error bounds derived from the method (e.g., interval width in bisection). Understanding these concepts is critical for justifying the number of iterations needed.
当真实值无法获得时，我们通常利用连续近似值之间的差来估计误差，或使用由方法推导出的误差界（例如，二分法中的区间宽度）。理解这些概念对于证明所需迭代次数至关重要。
2. Significant Figures and Decimal Places | 有效数字与小数位
In numerical answers, precision is indicated by significant figures (s.f.) or decimal places (d.p.). For instance, rounding 3.14159 to 3 s.f. gives 3.14; to 2 d.p. gives 3.14 as well but the rules differ when zeros are involved. You must be able to round correctly and state answers to the required accuracy. A common exam instruction is "Give your answer correct to 3 significant figures".
在数值答案中，精度由有效数字 (s.f.) 或小数位 (d.p.) 表示。例如，将 3.14159 保留 3 位有效数字得到 3.14；保留 2 位小数亦为 3.14，但当涉及零时规则有所不同。你必须能够正确舍入并按要求精度给出答案。常见的考试指令是“将
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