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# Calc04_2 - Mean Value Theorem for Derivatives 4.2 Teddy...

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Unformatted text preview: Mean Value Theorem for Derivatives 4.2 Teddy Roosevelt National Park, North Dakota Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002 If f ( x ) is continuous over [ a , b ] and differentiable over ( a , b ), then at some point c between a and b : ( 29 ( 29 ( 29 f b f a f c b a-′ =-Mean Value Theorem for Derivatives The Mean Value Theorem only applies over a closed interval. → The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope . y x A B a b Slope of chord: ( 29 ( 29 f b f a b a--Slope of tangent: ( 29 f c ′ ( 29 y f x = Tangent parallel to chord. c → A function is increasing over an interval if the derivative is always positive. A function is decreasing over an interval if the derivative is always negative. A couple of somewhat obvious definitions: → y x ( 29 y f x = ( 29 y g x = These two functions have the same slope at any value of x ....
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Calc04_2 - Mean Value Theorem for Derivatives 4.2 Teddy...

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