Question

What is the relationship between the Average rate of change of a fuction and derivatives?

Answer 1

The average rate of change gives the slope of a secant line, but the instantaneous rate of change (the derivative) gives the slope of a tangent line.

Average rate of change:

`(f(x+h)-f(x))/h=(f(b)-f(a))/(b-a)`, where the interval is `[a,b]`

Instantaneous rate of change:

`lim_(h ->0) (f(x+h)-f(x))/h`

Also note that the average rate of change approximates the instantaneous rate of change over very short intervals.