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Unformatted text preview: Cauchy Mean Value Theorem Hypotheses f and g are continuous on [ a, b ] and di ff erentiable on ( a, b ) . g ( x ) = 0 for all x in ( a, b ) . Conclusion There is a c in ( a, b ) such that f ( c ) g ( c ) = f ( b )- f ( a ) g ( b )- g ( a ) . Interpretation of Theorem This roughly says that at at least one time c , speed of f speed of g = change in f change in g , which is the key idea to proving LHospitals Rule (when f ( a ) = g ( a ) = 0 ). Idea of Proof We are trying to show there is a solution to f ( x ) g ( x ) = f ( b )- f ( a ) g ( b )- g ( a ) , where x is the variable and a and b are constants. We would like to apply Rolles Theorem, but to what function? Notice f ( x ) g ( x ) is not the derivative of anything nice. Lets get rid of fractions and rearrange to get ( g ( b )- g ( a )) f ( x )- ( f ( b )- f ( a )) g ( x ) = 0 , which is of the form constant f ( x )- constant g ( x ) = 0 . The left-hand-side is just the derivative of ( g ( b )- g...
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This note was uploaded on 06/26/2010 for the course MATH 1A taught by Professor Wilkening during the Fall '08 term at University of California, Berkeley.
- Fall '08