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

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Answer 1 · 2014-09-19T20:52:17

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.