Rollle’s and Mean Value Theorem French mathematician Michel Rolle (21 April 1652 – 8 November 1719)

Mean Value Theorem for Derivatives Geometric Interpretation: Under the given conditions, there is a point in the open interval where the tangent to the curve is the same as the slope of the line joining the endpoints. French mathematician, Joseph-Louis Lagrange Application: Under the given conditions, there is a point in the open interval where the instantaneous rate of change is the same as the average rate of change on the interval (very important). If the function is a position function, then there is a point in the open interval where the instantaneous velocity is the same as the average velocity on the interval is the average rate of change of the function f(x) on the interval [ a, b] (25 January 1736 – 10 April 1813)

f(a) = f(b) does not need to be mentioned f(a) = f(b) does not need to be checked

Let us check for fun (Mrs. Radja’s kind of fun) These values are not in (2, 3)

f(a) = f(b) does not need to be mentioned

EX: Determine whether the MVT can be applied to f(x) = x 3 – x on [0, 2] ● Since f is a polynomial, it is continuous and differentiable for all x. ● Therefore, by the MVT, there is a number c in (0,2) such that: ● f(2) = 6, f(0) = 0, and f ’(x) = 3x 2 – 1. The tangent line at this value of c is parallel to the secant line OB.

If an object moves in a straight line with position function s = f(t), then the average velocity between t = a and t = b is and the velocity at t = c is f ’(c). Thus, the MVT tells us that, at some time t = c between a and b, the instantaneous velocity f ’(c) is equal to that average velocity. In general, the MVT can be interpreted as saying that there is a number at which the instantaneous rate of change is equal to the average rate of change over an interval.

The mean value theorem is one of the "big" theorems in calculus. It is very simple and intuitive, yet it can be mind blowing. Suppose you're riding your new Ferrari and I'm a traffic officer. I suspect you may be abusing your car's power just a little bit. I know you're going to cross a bridge, where the speed limit is 80km/h (about 50 mph). So, I just install two radars, one at the start and the other at the end. The first one will start a chronometer, and the second one will stop it. I also know that the bridge is 200m long. So, suppose I get: Total crossing time: 8 seconds. Your average speed is just total distance over time: So, your average speed surpasses the limit. Does this mean I can fine you? Unfortunatelly for you, I can use the Mean Value Theorem, which says: "At some instant you where actually travelling at the average speed of 90km/h".

EX: Prove that the equation x 3 + x – 1 = 0 has exactly one real root. First, we use the Intermediate Value Theorem to show that a root exists. ● Let f(x) = x 3 + x – 1 ● f(0) = – 1 0 ● there is a number c between 0 and 1 such that f(c) = 0 ● Since f is a polynomial, it is continuous To show that the equation has no other real root, use Rolle’s Theorem and argue by contradiction Suppose that it had two roots a and b. ● → f(a) = 0 = f(b). ● As f is a polynomial, it is differentiable on (a, b) and continuous on [a, b]. ● Thus, by Rolle’s Theorem, there is a number c betweean a and b such that f ’(c) = 0 ● However, f ’(x) = 3x ≥ 1for all x (since x 2 ≥ 0), so f ’(x) can never be 0. This gives a contradiction.  So, the equation can’t have two real roots.