lect116_17rev_f11

lect116_17rev_f11 - Friday, October 14 Lecture 17 : Mean...

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Friday, October 14 Lecture 17 : Mean value theorem (Refers to Section 4.5 in your text) Students who have mastered the content of this lecture know about : The Mean Value theorem, the Increase-decrease theorem Students who have practiced the techniques presented in this lecture will be able to : Apply the MVT to various problems, apply the Increase-decrease theorem to determine intervals on which a function is increasing or decreasing. 17.1 Introduction Suppose Tim runs from home to school every morning. On a particular morning he leaves at time t = t 0 and arrives at school at time t = t 1 . His home is located at a distance p ( t 0 ) directly to the east of the point A where the streets starts. The school is at a distance of p ( t 1 ) directly to the east of the point p ( t 0 ). When Tim tires, he slows down, may walk for a bit, and then run again. From these facts we can compute his “average velocity” Suppose that, during the trip, the maximum speed he attains is v max while his speed is never less than v min . Then we can assume that Most people would conjecture that there must be some time t * between t 0 and t 1 such that v ( t *) = v avg . But there are strict conditions that must be satisfied for this to hold true: 1) The function p ( t ) must be continuous on the interval [ t 0 , t 1 ], 2) The function p ( t ) must be differentiable on the interval ( t 0 , t 1 ). In this section we will prove a statement which guarantees that if both of these conditions are satisfied then there must exist some time t * such that p ( t *) = v ( t *) = v avg . This statement is universally called the Mean value theorem . We must however first prove another theorem called Rolle’s theorem . 17.2
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This note was uploaded on 12/28/2011 for the course MATH 116 taught by Professor Robertandre during the Spring '11 term at Waterloo.

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lect116_17rev_f11 - Friday, October 14 Lecture 17 : Mean...

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