Question

Suppose  g(s)=s^3−3  What is the average rate of change of g(s) with respect to s as s changes from 1 to 1+h ?

Answer 1

g(s) = s^3-3

g(1) = (1)^3-3 = 1-3 = -2

g(1+h) = (1+h)^3-3 = 1+h^3+3h^2+3h-3

Compute the limit as (h-->0) of the average rate of change.

Average rate of change = lim(h-->0) [g(1+h)-g(1)]/h

= lim(h-->0) [(1+h^3+3h^2+3h-3)-(-2)]/h

= lim(h-->0) [1+h^3+3h^2+3h-3+2}/h

= lim(h-->0) [h^3+3h^2+3h]/h

= lim(h-->0) h(h^2+3h+3)/h

= lim(h-->0) h^2+3h+3

= 3