Question

For every function f(x) that is smoothly continuous on (a,b], the average rate of change of f(x) from x = a to x = b, approaches the instantaneous rate of change at x = a as x approaches a from the left.

Please explain this with a graph

Answer 1

Average rate of change:

$(f(b)-f(a))/(b-a)=f'(a)$

since $x=a^+$ , which is in the interval [a,b].

If b=a+h where h is very small,

$f'(a)=(f(a+h)-f(a))/h\ as\ h→0$

by definition of derivative (instantaneous rate of change).