8 Let f RR be defined by fx x x for x 0 x for x 0 a Suppo

8. Let f: RR be defined by f(x) = |x |= x for x < 0. x for x 0,

(a) Suppose x < 0 and h = 0 is small enough so that x+h < 0 also. Find the average

rate of change of f from x to x+h.

(b) Suppose x > 0 and h = 0 is small enough so that x+h > 0 also. Find the average

rate of change of f from x to x+h.

(c) Suppose h > 0. Find the average rate of change of f from 0 to 0 + h = h.

(d) Suppose h < 0. Find the average rate of change of f from h = 0 + h to 0.

8. Let f : R R be defined by f(x) = |x|- r for 20, -T or r for a for x 0. Find the average rate of change of f from 0 to 0+h h. (d) Suppose h

Solution

average rate of change = [f(x + h) - f(x)]/[(x + h) - x]

a) average rate of change of f from x to x + h both less than zero

==> [f(x + h) - f(x)]/[(x + h) - x] = [-x - (-x)]/h ; since x + h < 0 ==> f(x + h) = -x

==> 0/h = 0

Therefore average rate of change of f from x to x + h both less than zero is ZERO.

b) average rate of change of f from x to x + h both greater than zero

==> [f(x + h) - f(x)]/[(x + h) - x] = [x - (x)]/h ; since x + h > 0 ==> f(x + h) = x

==> 0/h = 0

Therefore average rate of change of f from x to x + h both greater than zero is ZERO.

c) average rate of change of f from 0 to 0 + h = h

==> [f(0 + h) - f(0)]/[(0 + h) - 0] = [h - (0)]/h ; since h > 0 ==> f(h) = h

==> h/h = 1

Therefore average rate of change of f from 0 to 0 + h , h > 0 is ONE.

d) average rate of change of f from 0 + h to 0 , h< 0

==> [f(0) - f(0 + h)]/[(0) - (0 + h)] = [0 - (-h)]/(-h) ; since h < 0 ==> f(h) = -h

==> -h/-h = 1

Therefore average rate of change of f from 0 + h to 0 , h < 0 is ONE.