中五 · 复习
微分，复习笔记
微分的精要复习笔记，每一项内容标准、考试公式，以及如何复习才能得分。
需要复习什么
本笔记涵盖微分，中五SPM高级数学第2章。本章由4项内容标准构成，下表列出每一项要求你掌握的能力。
逐项复习：理解概念、掌握方法、反复练习直到解题成为本能。由于SPM评分是分析式的，清晰有序的步骤能在最终答案出错时仍保住分数，因此把整洁的排版当作复习的一部分。
内容标准
| 代码 | 内容标准 | 你必须掌握的 |
|---|---|---|
| 2.1 | 极限及其与微分的关系 | Investigate and determine the value of limit of a function when its variable approaches zero.; Determine the first derivative of a function f(x) by using the first principle. |
| 2.2 | 一阶导数 | Derive the formula of first derivative inductively for the function n ax y , a is a constant and n is an integer.; Determine the first derivative of an algebraic function. Further exploration using dynamic geometry software to compare the graphs of f(x) and f'(x) (gradient function graph) can be carried out.; Determine the first derivative of composite function. Chain rule needs to be involved. The use of the idea of limit to prove the chain rule can be discussed.; Determine the first derivative of a function involving product and quotient of algebraic expressions. The use of the idea of limit to prove product rule and quotient rule can be discussed. |
| 2.3 | 二阶导数 | Determine the second derivative of an algebraic function. |
| 2.4 | 微分的应用 | Interpret gradient of tangent to a curve at different points.; Determine equation of tangent and normal to a curve at a point.; Solve problems involving tangent and normal.; Determine the turning points and their nature. Notes: The following matters need to be involved: (a) Sketching tangent method (b) Second derivative method (c) Point of Inflection; Solve problems involving maximum and minimum values and interpret the solutions.; Interpret and determine rates of change for related quantities. The use of chain rule needs to be emphasised.; Solve problems involving rates of change for related quantities and interpret the solutions.; Interpret and determine small changes and approximations of certain quantities.; Solve problems involving small changes and approximations of certain quantities. Problems involved are limited to two variables. PERFORMANCE STANDARDS PERFORMANCE LEVEL DESCRIPTOR 1 Demonstrate the basic knowledge of differentiation. 2 Demonstrate the understanding of differentiation. 3 Apply the understanding of differentiation to perform simple tasks. 4 Apply appropriate knowledge and skills of differentiation in the context of simple routine problem solving. Apply appropriate knowledge and skills of differentiation in the context of complex routine problem solving. Apply appropriate knowledge and skills of differentiation in the context of non-routine problem solving in a creative manner. LEARNING AREA CALCULUS TOPIC |
如何为分数而复习
本章的位置，以及顺序为何重要
微分是中五课程的第2章，而高级数学更青睐按课程顺序复习、而非跳到看起来最难部分的学生。几乎每一章都倚赖中四的代数，移项、处理函数、干净地运算指数与根式，因此若其中任何一项不稳，花一小时把它夯实，会在微分以及之后的一切中都得到回报。
复习本章时，随手记下任何你不得不回查的旧技能：这份笔记就是值得修补的基础地图。稳健的复习计划不是一口气冲完全部4项标准，而是短而规律的多次学习：复习一项标准，几天后自测，只有当你能不看笔记复现方法时才继续下一项。
老师如何帮助微分
笔记与练习能让学生走很远，但微分正是那种"第二双眼睛"能把"大致会"变成"稳定会"的章节。一对一时，老师边做边看你的解题，捕捉出错的确切一步，某处漏了符号、某处忘了条件、公式用对了地方却用错了方法。
这是书本上的标准答案永远做不到的，因为错误发生在动手时，而非阅读时。由于微分建立在前面的章节之上，老师还能发现真正的缺口其实不在微分，而在它默认掌握的某项中四技能，从而先重建它，让新内容终于扎根。
每一堂课的重点都相同：理解概念、展示方法，并把步骤写得足够清晰，即便某天最终答案出错也能得分。课程为一对一在线、以英文进行，老师会围绕孩子在微分上的确切水平来安排每一节，从重建薄弱基础到冲刺A+。
由于老师面对的是一名学生而非三十名，节奏由理解而非教学进度表决定：五分钟就领会的概念不会被拖沓，一时未领会的则会给足所需时间，而不是为了让全班往前赶而被撇下。
这一切都不取代本站的笔记、例题与练习，而是让它们发挥更大的作用。读过微分笔记、做过练习的学生，会带着真实的疑问而非一张白纸走进课堂，而针对这些疑问的一小时教学，远胜于用一小时讲解课本已经写明的内容。
这正是我们希望学生把两者结合使用的方式：在这里学习材料，留意它在何处开始讲不通，然后把那一点带给能够彻底弥合缺口的老师。
获取一对一辅导。
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