Average Rate of Change

Core Sentence / 核心句

Average rate of change measures how much output changes per unit input over an interval.

中文理解：平均变化率看的是一段区间里的整体变化，不是某一点瞬间的变化。

Formula / 公式或规则

$$\frac{f(b)-f(a)}{b-a}$$

This is the slope of the secant line through $(a,f(a))$ and $(b,f(b))$.

When To Use / 什么时候用

看到 over an interval、between two points、secant slope，就先想到 average rate of change。

Watch Out / 易错点

不要把 average rate of change 和 derivative 混成一个东西；derivative 是 interval shrink 到一点后的 limit。

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• 微积分BC第2次正课 · Average Rate of Change
  • interval-level change equals the slope through the two endpoint values
• 02 - The Paradox of the Derivative · 2. Average rate of change is computed over an interval.
  • 2. Average rate of change is computed over an interval.
• 02 - The Paradox of the Derivative · Knowledge Point 6: Secant Line Becomes Tangent Line
  • average rate of change -> secant slope
• 微积分BC第13次正课-无穷级数判敛 · 9. Exam Language / 关键句中英对照
  • average rate of change -> 平均变化率/割线斜率 -> secant slope
• 03 - Derivative Formulas through Geometry · 1. What This Episode Is About / 这一集在讲什么
  • Chapter 1 先讲 Integrals 和 accumulated area；Chapter 2 解释 Derivatives 到底是什么意思：不是神秘的 instantaneous change，而是 Limits 下的 Average Rate of Change。
• 02 - The Paradox of the Derivative · 9. After Watching Summary / 看完压缩版
  • average rate of change over an interval -> best constant approximation around a point

Connected Notes / 连接

• 微积分BC第2次正课-极限应用与导数入门
• 02 - The Paradox of the Derivative
• Secant Line
• Tangent Line
• Instantaneous Rate of Change
• Derivatives