Average Value of a Function

Core Sentence / 核心句

The average value of a function is its signed area over an interval divided by the interval width.

中文理解：连续函数的平均值是越来越密的有限采样平均之极限，也就是带符号面积除以区间宽度。

Formula / 公式

$$f_{avg}=\frac{1}{b-a}{\int }_a^{b}f(x)dx$$

If $F^{\prime }=f$:

$$f_{avg}=\frac{F(b)-F(a)}{b-a}$$

Memory Sentence / 记忆句

Average height of $f$ equals secant slope of an antiderivative $F$.

Watch Out / 易错点

• 不要只平均 endpoints。
• 不要忘记除以 $b-a$。
• 使用 signed area，below-axis values 为负。
• 它等于 $F$ 的 average rate of change，不是通常意义下 $f$ 自身的 average rate of change。

Source / 出现位置

• Chapter 9 · General Formula
• Chapter 9 · Average Slopes

Connected Notes / 连接

• 09 - What Does Area Have to Do With Slope
• Integrals
• Riemann Sum
• Signed Area
• Antiderivative
• Fundamental Theorem of Calculus
• Average Rate of Change
• 微积分BC第11次正课 · Average Value