Hello, jhonwashington!

. . . . . . . . . . . . . . . . . . . . . . . . . change in f(x)Find the average rate of the function: .f(x) .= .x - 2x² from x=1 to x=2.

Average rate of change of f(x) . = . -----------------

. . . . . . . . . . . . . . . . . . . . . . . . . .change in x

We have: . f(1) .= .1 - 2(1²) .= .-1

. . . .and: . f(2) .= .2 - 2(2²) .= .-6

The change in f(x) is: .(-6) - (-1) .= .-5

The change in x is: .2 - 1 .= .1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -5

Therefore, average rate of change of f(x): . -- . = . -5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

Given: .f(x) .= .x² - 2

. . . . f(x + h) - f(x)

Find -----------------

. . . . . . . . h

There are three steps to this "difference quotient":

. . (1) Find f(x + h) ... and simplify

. . (2) Subtract f(x) ... and simplify

. . (3) Divide by h ... and simplify

We have:

(1) f(x+h) .= .(x+h)² - 2 .= .x² + 2xh + h² - 2

(2) f(x+h) - f(x) .= .(x² + 2xh + h² - 2) - (x² - 2) .= .2xh + h²

. . .f(x+h) - f(x) . . .2xh + h² . . h(2x + h)

(3) -------------- .= .---------- .= .----------- .= .2x + h

. . . . . . h . . . . . . . . . h . . . . . . . .h