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Calc04_2 - 4.2 Mean Value Theorem for Derivatives 2 2 1 1 3...

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Mean Value Theorem for Derivatives 4.2
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- 2 , 2 ( ) -∞ ,- 1 æ è ç ö ø ÷ È 1,¥ æ è ç ö ø ÷ - 3 , 3 é ë ê ù û ú - 3 , 3 é ë ê ù û ú - 3 , 3 æ è ç ö ø ÷ C = -5
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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 .
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y x 0 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
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Example Explain why each of the following functions fails to satisfy the conditions of the MVT on the interval [-1,1]: 3 2 2 x +3, for x < 1 a) f(x) = x + 1 b) f(x) = x + 1, for x 1 f(x) = x + 1 which is not differentiable at (0,1) lim x →1 - φ(ξ29 = 4, λιμ ξ→1 + φ(ξ29 = 2 Σινχε τηεσε λιμ ιτσαρε νοτεθυαλ , φ(ξ29 ισδισχοντινυουσατξ = 1
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A function is increasing
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