Question

Let f be defined as follows.
y = f(x) = x^2 - 4 x

(a) Find the average rate of change of y with respect to x in the following intervals.
from x = 5 to x = 6
from x = 5 to x = 5.5
from x = 5 to x = 5.1

(b) Find the (instantaneous) rate of change of y at x = 5.
4

Answer 1

(a)
Average rate of change of f(x) on an interval I = [x1,x2] equals (f(x2)-f(x1))/(x2-x1) where x2≠x1.
So if f(x)=x^2 - 4 x,
average rate of change between 5 and 6
= (f(6)-f(5))/(6-2)
= (12-5)/(6-5)
= 7

(b)
Instantaneous rate of change at x=5
=f'(5)
where
f'(x)=2x-4
so
f'(5)=10-4=6