Case Study in Action: A Comprehensive Edexcel Year 13 Mathematics Workout | 案例分析实战演练：Edexcel十三年级数学综合训练

📚 Case Study in Action: A Comprehensive Edexcel Year 13 Mathematics Workout | 案例分析实战演练：Edexcel十三年级数学综合训练

This revision workout brings together core topics from Year 13 Edexcel Mathematics, blending pure content into a single, cohesive set of case studies. Each section tackles a typical exam-style challenge, from rational functions and binomial expansions to integration techniques, parametric equations and differential equations. Work through the paired English–Chinese explanations, test your understanding, and build the fluency that top marks demand.

本次复习演练将 Edexcel 十三年级数学的核心纯数内容整合为一套连贯的案例分析。每个小节针对一道典型的考试风格题目，从有理函数和二项展开到积分技巧、参数方程与微分方程。请通过中英对照的解析逐步推进，检验理解，打造高分所需的能力。

1. Rational Functions and Their Domains | 有理函数及其定义域

Consider the function f(x) = (2x² + 5x − 3) / (x − 1). Before manipulating the expression, determine the domain by setting the denominator not equal to zero: x − 1 ≠ 0 ⇒ x ≠ 1. Although the numerator factorises as (2x − 1)(x + 3), no cancellation is possible. Long division transforms f(x) into 2x + 7 + 4/(x − 1), revealing an oblique asymptote y = 2x + 7 and a vertical asymptote at x = 1.

考虑函数 f(x) = (2x² + 5x − 3)/(x − 1)。在操作表达式之前，先令分母不为零以确定定义域：x − 1 ≠ 0 ⇒ x ≠ 1。尽管分子可分解为 (2x − 1)(x + 3)，但无法约分。长除法将 f(x) 化为 2x + 7 + 4/(x − 1)，由此可知有斜渐近线 y = 2x + 7，并且 x = 1 处有垂直渐近线。

The simplified form is especially useful when studying the end behaviour: as x → ∞, 4/(x − 1) → 0, so f(x) behaves like 2x + 7. Always state domain restrictions clearly before differentiating or integrating.

化简后的形式对于研究

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