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lecture23

# lecture23 - Lecture 23 Mean Value Theorem(Chapter 4 Sec 23...

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Lecture 23: Mean Value Theorem (Chapter 4, Sec. 23) Rolle’s Theorem Let f be a function satisfying the following: 1) 2) 3) Then there is a number c in ( a; b ) such that Consider the graphs: 6 - ? 6 - ?

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2 ex. Find the value of c implied by Rolle’s Theorem for f ( x ) = p 2 x ¡ x 2 on [0 ; 2].
3 ex. Find the value of c implied by Rolle’s Theorem for f ( x ) = cos(2 x ) on [0 ; … ].

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4 ex. Show that the equation x 3 + 6 x + 4 = 0 has exactly one real root.
5 We can use Rolle’s Theorem to prove the important Mean Value Theorem : Let f be a function that satisfles the following con- ditions: 1) 2) Then there is a number c in the interval ( a; b ) such that 6 - ?

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6 ex. Find the value of c implied by the Mean Value Theorem for f ( x ) = x 3 ¡ x 2 ¡ 2 x on [ ¡ 1 ; 1].
7 ex. The position of an object dropped from 800 ft is s ( t ) = 800 ¡ 16 t 2 , where t is in seconds. Find the average velocity on the time interval [0 ; 5]. Use the Mean Value Theorem to verify that at some time in the flrst flve seconds, average velocity equals instantaneous velocity.

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8 ex. Suppose that f (0) = ¡ 2 and f 0 ( x ) 2 for all values of x . Find the largest possible value of f (4).
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lecture23 - Lecture 23 Mean Value Theorem(Chapter 4 Sec 23...

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