L22 - Lecture 22 — Mean Value Theorem Rolle’s Theorem:...

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Unformatted text preview: Lecture 22 — Mean Value Theorem Rolle’s Theorem: Let f be a function that satisfies the following properties: (1) (2) (3) Then there is a c in ( a, b ) such that The conditions of Rolle’s Theorem are necessary: Find the value of c guaranteed by Rolle’s Theorem 2 for f (x) = (2x − x2) 3 on [ 0, 2 ] . Show that f (x) = x3 + 3x − 2 has exactly one real root. Rolle’s Theorem is a special case of. . . Mean Value Theorem: Let f be a function for which (1) (2) Then there is a c in ( a, b ) such that Interpretations: (1) (2) Find the value of c guaranteed by the Mean Value Theorem for f (x) = x3 − x2 − 2x on [ −1, 1 ] . 2 What happens if we try the function f (x) = x + x on [ −1, 1 ] ? The following consequence of Mean Value Theorem will be a fundamental result as we begin our study of integration. Theorem: If f (x) = 0 for all x in an interval ( a, b ), then Corollary: If f (x) = g (x) for all x in an interval ( a, b ), then f − g is i.e. f (x) = Try it! Find the value of c guaranteed by the Mean x on [ −1, 4 ] . Value Theorem for f (x) = x+2 Prove the Pythagorean identity for trigonometry by examining the derivatives of the functions sin2(x) and cos2(x) . ...
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This note was uploaded on 05/27/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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L22 - Lecture 22 — Mean Value Theorem Rolle’s Theorem:...

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