The Derivative

Substituting 1 for x ₀ in the formula for the derivative, we have

Observing the graph of f (x) = x ² , we see that this is the slope of the tangent to the graph at the point (1, f (1)) = (1, 1) .

At the vertex, the tangent line to the graph will be horizontal, with slope 0 . Therefore, we search for an x such that f'(x) = 0 . We have

Thus f'(x) = 0 if and only if x = - 1 . Now f (x) = (- 1)² + 2(- 1) + 2 = 1 , so the vertex of the parabola is (- 1, 1) . We can check this by noting that f (x) - 1 = (x + 1)² , so the graph of f (x) is the graph of x ² translated 1 unit to the left and 1 unit up.

First we compute f'(2) :

The equation of the line through (2, f (2)) = (2, 8) with slope 12 is given by y = 12(x - 2) + 8 .