Calculus Test review problem

Question

Use limits to compute the following derivatives:

f'(3), where f(x) = x² - 2x + 1

Answer 1

The limit definition of the derivative is:

For f(x) = x² - 2x + 1:

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

So f(x + h) - f(x) is:

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

So [f(x + h) - f(x)]/h = [h(2x + h - 2)]/h = 2x + h - 2

And the limit as h approaches zero makes this "2x - 2"

So f '(3) = 2(3) - 2 = 4

Answer 2