2.4: Average Rate of Change of a Function
Average Rate of Change
average rate of change
The average rate of change (ARoC) of $y = f(x)$ on the interval $(a, b)$ is \[\text{ARoC} = \dfrac{\text{change in } y}{\text{change in } x} = \dfrac{f(b) - f(a)}{b - a}\]
The ARoC is the slope of the line through $(a, f(a))$ and $(b, f(b))$.
Calculate the average rate of change for $f(x) = (x-3)^2$ on the interval $(1, 3)$.
If you have a distance function $f(t)$, which tells you how far something has traveled over some time, the average rate of change of $f(t)$ on an interval is the average speed.
Suppose you begin walking. The distance you cover is modeled by $f(t) = 4t^2$, where $t$ is minutes elapsed and $f(t)$ is the total distance traveled in feet.
Calculate the average speed in the first five minutes.
What does $\frac{f(x+h) - f(x)}{h}$ mean?
This fraction will appear again in Calculus. It is the ARoC of $f(x)$ on the interval $(x, x+h)$.
Witness:
Consider $f(t) = 16t^2$. Calculate the average rate of change on the interval $(x, x+h)$.