Question

Find the average rate of change of f(x) = x^2 + 3x - 8 from x to x + h

Answer 1

f(x) = x^2+3x-8

f(x+h) = (x+h)^2+3(x+h)-8

= x^2+2hx+h^2+3x+3h-8

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

lim(h-->0) [f(x+h)-f(x)]/h

= lim(h-->0) [(x^2+2hx+h^2+3x+3h-8)-(x^2+3x-8)]/h

= lim(h-->0) [x^2+2hx+h^2+3x+3h-8-x^2-3x+8]/h

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

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

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

= 2x+3