What does average rate of change mean in calculus?

The average rate of change of a continuous function, ##f(x)## , on the closed interval ## [a, b]## is given by

##(f(b)-f(a))/(b-a)##

Basically, the average rate of change tells us how, on average, the y-value of a function is changing per unit change in the x-value.

The average rate of change formula is just a variation of the slope formula

##(y_2-y_1)/(x_2-x_1)##

since ## y = f(x)##.

Let's let ##f(x)=-x^2+8x-9##.

The average rate of change on the closed interval ##[2,5]## is given by

##(f(5)-f(2))/(5-2)=(6-3)/(5-2)=3/3=1##

This is also the slope of the secant line to the graph of ##f## through the points ##(2,3)## and ##(5,6)##

See image below.