Lesson 3.2 Rolle’s Theorem Mean Value Theorem 12/7/16

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Lesson 3.2 Rolle’s Theorem Mean Value Theorem 12/7/16

Between a and b, will f’(c) ever have to equal zero? Given: Point A and Point B lie on a continuous differentiable function. f(a) = f(b) Either the function is horizontal between a & b, in which case f ’(c) = 0… or somewhere on the interval the function increases & then decreases (or vice versa), & at that point has a f ’(c) = 0. f(a) = f(b) a b Between a and b, will f’(c) ever have to equal zero? That’s a definitive YES. That YES is dependent on the conditions that f is continuous, differentiable & f(a) = f(b).

Dat’s just how I “Rolle”!

Let f(x) = sin 2x. Find all values of c in the interval Example Let f(x) = sin 2x. Find all values of c in the interval such that f’(c) = 0 Does it satisfy Rolle’s Theorem? Find any c values: Sure. . And it’s continuous & differentiable.

“mean” slope over the interval [a, b] Mean Value Theorem For any secant of a function there exists a point on the function where the tangent is parallel to the secant. “mean” slope over the interval [a, b] slope of tangent at c

Example Find a value for c within the interval [-1, 1] where the tangent line at c will be parallel to the secant line through the endpoints of the interval. f(x) = x3 – x2 – 2x

Day 1 Page 172 #1 – 3, 7, 9, 11, 19, 25, 27, 28 There are kinds of people in the world: those that can do calculus and those that can not. Day 2 #29 – 31, 35, 38, 43 – 47 odd