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Rolle&acirc;€™s Theorem and The Mean Value Theorem

# Rolle&acirc;€™s Theorem and The Mean Value Theorem...

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Rolle’s Theorem & The Mean Value Theorem Section 3.2

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Rolle’s Thm. Rolle’s Thm. gives conditions that guarantee the existence of an extreme value in the interior of a closed interval.
Exploration Sketch a graph on a piece of paper. Label points (1,3) and (5,3). Using a pencil, draw the graph of a differentiable function f that starts at (1,3) and ends at (5,3). Is there at least one point on the graph for which the derivative is zero? Would it be possible to draw the graph so that there isn’t a point for which the derivative is zero?

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Theorem 3.3 Rolle’s Thm . Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If f(a) = f(b) Then there is at least one number c in (a,b) such that f ‘ (c) = 0. a b c f ‘ (c) = 0
If the differentiability requirement is dropped from Rolle’s Thm. f will still have a

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Unformatted text preview: critical number in (a,b), but it may not yield a horizontal tangent. a c b The Mean Value Thm. If f is continuous o the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a # c in (a,b) such that f ‘ (c) = a b a f b f--) ( ) ( Why is it called mean? At some point, instantaneous rate of change = average rate of change (a,f(a)) (b,f(b)) f ‘ (c) c slope of tangent line = slope of secant line Geometrically, the mvt guarantees the existence of a tangent line that is parallel to the secant line through the points (a, f(a)) and (b,f(b)). Joke Time What would a math student say to a fat parrot? poly-no-mial What do you get if you divide the circumference of a jack-o-lantern by its diameter? pumpkin “pi” What are eyeglasses good for? di-vision...
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Rolle&acirc;€™s Theorem and The Mean Value Theorem...

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