lec16 - 16 The Mean Value Theorem Lecture Notes P K...

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16. The Mean Value Theorem P. K. Lamm 10/21/11 (15:06) p. 1 / 7 Lecture Notes: 16. The Mean Value Theorem These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. The Mean Value Theorem and Rolle’s Theorem In this section we will look at two important theorems, the Mean Value Theorem and its special case, Rolle’s Theorem. These theorems answer the following question: Is it possible for a boat to sail from point A to point B without ever sailing in the exact (straight-line) di- rection from A to B ? Another way to ask the same question is as follows: For a sufficiently smooth function f , does there exist a point c [ a, b ] such that f ( b ) - f ( a ) b - a = f 0 ( c ) ? The Mean Value Theorem answers these types of questions, while Rolle’s Theorem addresses the special case when f ( a ) = f ( b ) = 0. Theorem (Rolle’s Theorem) : Suppose f is continuous on [ a, b ] and differentiable on ( a, b ) and that f ( a ) = f ( b ) = 0. Then there is a c ( a, b ) such that f 0 ( c ) = 0 . 1
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16. The Mean Value Theorem P. K. Lamm 10/21/11 (15:06) p. 2 / 7 Proof of Rolle’s Theorem: We will look at three cases: Case #1: Assume f ( x ) = 0 , all x [ a, b ] . In this case clearly f 0 ( x ) = 0 for all x ( a, b ) so any c ( a, b ) gives us the result that f 0 ( c ) = 0. Case #2: Assume f ( x ) > 0 , for some x [ a, b ] . Since f is a continuous function on a closed interval, it must attain a max at some point c [ a, b ], with f ( c ) > 0. First of all, c cannot be an endpoint because f ( a ) = f ( b ) = 0 for endpoints a and b ; so it must be that c is a critical point of f in ( a, b ). From the assumptions of the theorem, f is differentiable on all of ( a, b ), meaning that f 0 ( x ) exists as a real number for all x ( a, b ). So since c ( a, b ) is a critical point of f , the only remaining possibility is that f 0 ( c ) = 0. Case #3: Assume f ( x ) < 0 , for some x [ a, b ] . Then as in Case #2, f must attain its min at some point c ( a, b ) with f ( c ) < 0. Using similar arguments to those used in Case #2, it follows that f 0 ( c ) = 0. Note: Rolle’s Theorem guarantees the existence of at least one c ( a, b ) satisfying f 0 ( c ) = 0, but in fact neither the statement of the theorem nor its proof give any indication of how to find c . The same will be true of the c defined in the Mean Value Theorem, stated next.
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