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Cauchy MVT

# Cauchy MVT - Cauchy Mean Value Theorem Hypotheses f and g...

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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 L’Hospital’s 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 Rolle’s Theorem, but to what function? Notice f ( x ) g ( x ) is not the derivative of anything nice. Let’s 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 ( a )) · f ( x ) - ( f ( b ) - f ( a )) · g ( x ) , which luckily turns out to have all the properties we will need (see the proof).

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Cauchy MVT - Cauchy Mean Value Theorem Hypotheses f and g...

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