Calculus: Early Transcendentals 8th Edition

$$f'\bigg(\frac{p+q}{2}\bigg) = \frac{f'(p)+f'(q)}{2}$$
$f(x) = ax^{2} + bx + x$ Use power rule to determine the derivative of the parabola equation. $f'(x) = 2ax +b$ Find the slopes at the endpoints of the inteval [p,q] and average them. $f'(p) = 2ap+b$ $f'(q) = 2aq+b$ $m_{a\nu e} = \frac{2ap+b+2aq+b}{2} = ap+aq+b$ Find the slope at the midpoint of the interval [p,q] $f'\bigg(\frac{p+q}{2}\bigg) = 2a\bigg(\frac{f'(p)+f'(q)}{2}\bigg)+b$ $f'\bigg(\frac{p+q}{2}\bigg) = aq+ap+b$ $f'\bigg(\frac{p+q}{2}\bigg) = \frac{f'(p)+f'(q)}{2}$