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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.
 Spring '08
 ALL
 Calculus, Mean Value Theorem

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