AP Calculus AB Exam Question

Let f(x) be a function defined by f(x) = 3x^2 - 2x + 4. Find the derivative of f(x) using the definition of the derivative.

Step-by-Step Solution

To find the derivative of f(x) using the definition of the derivative, we need to apply the formula:

f'(x) = lim(h->0) [f(x + h) - f(x)] / h

First, let's find f(x + h):

f(x + h) = 3(x + h)^2 - 2(x + h) + 4 = 3(x^2 + 2xh + h^2) - 2x - 2h + 4 = 3x^2 + 6xh + 3h^2 - 2x - 2h + 4

Now we can substitute back into the definition of the derivative formula:

f'(x) = lim(h->0) [(3x^2 + 6xh + 3h^2 - 2x - 2h + 4) - (3x^2 - 2x + 4)] / h

Simplifying the numerator:

= lim(h->0) [3x^2 + 6xh + 3h^2 - 2x - 2h + 4 - 3x^2 + 2x - 4] / h = lim(h->0) [6xh + 3h^2 - 2h] / h = lim(h->0) (6x + 3h - 2)

Since "h" is approaching zero, we can remove it from the expression:

f'(x) = 6x + 3(0) - 2 = 6x - 2

Therefore, the derivative of f(x) is f'(x) = 6x - 2.

Answer: The derivative of f(x) is given by f'(x) = 6x - 2.