1. Ch. 5 – Applications of Derivatives 5.2 – Mean Value Theorem

2. Mean Value Theorem for Derivatives: If f is continuous on a closed interval [a, b], and f is differentiable at every point on the interval, then there exists at least one point such that – This theorem is saying that some x-value from a to b will have a tangent line with a slope equal to the slope of the average slope on the interval! Ex: Find the x-coordinate of the point that satisfies the mean value theorem for f(x) = 2x 2 – 5 on the interval [2, 6]. – Use the formula above! This is the slope of the secant line through the endpoints!!!

3. A function f is increasing (or decreasing) if its slope is positive (or negative). – If f’(c) > 0, then f is increasing at c – If f’(c) < 0, then f is decreasing at c Ex: Find the local extrema of f(x) = x 2 – x – 12. Also find where the function is increasing and decreasing. – Our only critical point is at x = ½, and there are no endpoints – Make a sign chart to determine where the function is increasing/decreasing... – f is decreasing on (-∞, ½ ) and increasing on (½, ∞) over the domain of f. There is a local minimum at (½, -49/4). + -

4. Ex: Find the local extrema of. Also find where the function is increasing and decreasing. – Find the critical points... – f‘ = 0 where the numerator is zero, so x = 0 and x = 2 are critical points – Also, f’ doesn’t exist when x = 1, so that’s a critical point – Make a sign chart with all 3 critical points to determine where the function is increasing/decreasing... – f is increasing on (-∞, 0 ) and (2, ∞) and decreasing on (0, 1) and (1, 2) over the domain of f. There is a local minimum at (2, 4) and a local maximum at (0, 0). ++ --

5. Antiderivative: Opposite of a derivative – A function F is an antiderivative of f if F’(x) = f(x) for all x in the domain of f. Ex: Find the antiderivative of the following functions: – Think: which functions gives the following derivatives? – Antiderivative: F(x) = x 2 + C – Antiderivative: F(x) = 4x 4 + C – In general, a function f(x) = x n will have an antiderivative of – Antiderivative: F(x) = -cosx + C The C represents some constant value and must always be included in your answer!

6. Ex: Find the antiderivative of the following functions: – Antiderivative: F(x) = ln(x + 3) + C – Antiderivative: F(x) = x 6 – 4x 5 + x 2 /2 – 9x + C – Antiderivative: F(x) = 2e x + C Ex: Find f if f’(x) = 3x 2 + 5 and f passes through (3, 4). – Find the antiderivative, then solve for C!

7. Ex: Find the position function of a rock dropped from rest from a 540m cliff with an acceleration of -9.8 m/s 2. – We’ll need to do 2 antiderivatives, one at a time! – Since the object is dropped from rest, v(0) = 0 – Now find s(t)... – The rock is dropped from 540m, so s(t) has the point (0, 540)