Average value

Reference. I will refer to the textbook for this class using indicators such as relevant section numbers or page numbers. However, the contents here are NOT directly from the book, so I am not violating any copyright, to my best judgement.

Recall from Section 4.6 of your book that if you are given the position function $s(t)$ in time $t \geq 0$ of a moving object, the velocity function $v(t)$ is given by the derivative of the position function $$v(t) = s'(t).$$ Note that the velocity is different from the average velocity. If the moving object starts to move at time $t = a$ and stop at time $t = b,$ its average velocity on the time interval is $$\frac{s(b) - s(a)}{b - a}.$$ But look! By 1st FTC (p.293), we have
$$\frac{s(b) - s(a)}{b - a} = \frac{1}{b-a}\int_{a}^{b}s'(t)dt = \frac{1}{b-a}\int_{a}^{b}v(t)dt.$$ Why do we bother to observe this? Note that the first fraction is given by the values of the position function $s(t)$, NOT the velocity function $v(t).$ With the last expression, we now can immediately write a formula of the average velocity, using the velocity function!

More generally, the average value of the (piece-wise continuous) function $f(x)$ defined on the interval $[a, b]$ is defined as $$\frac{1}{b-a}\int_{a}^{b}f(x)dx.$$ Is this definition reasonable? When you are introduced a new word for a new notion, it is important for you to sit down and think about why people call the notion with such a word!

Exercise. Read p.308-309.