goldch3 - Contents III The Mean Value Theorem for Rational...

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Unformatted text preview: Contents III The Mean Value Theorem for Rational Functions 1 III.A Derivatives of Rational Functions and Tangent Lines . . . . . . . . . . . . . . . . . . . . . . 1 III.B Mean Value Theorem, or MVT, and Rolle’s Theorem for Rational Functions . . . . . . . . 2 III.C Application of the MVT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 III.D Extended Mean Value Theorem (EMVT) for Rational Functions . . . . . . . . . . . . . . . 6 III The Mean Value Theorem for Rational Functions III.1 III The Mean Value Theorem for Rational Functions III.A Derivatives of Rational Functions and Tangent Lines Without any geometry, one may define derivatives of rational functions in a purely algebraic way: • The derivative of x n is ( x n ) = nx n- 1 . • The derivative of a polynomial P ( x ) = n X i =0 a i x i is given by P ( x ) = n X i =1 ia i x i- 1 . • The derivative of a rational function f ( x ) = P ( x ) /Q ( x ) is given by f ( x ) = P ( x ) Q ( x )- P ( x ) Q ( x ) ( Q ( x )) 2 . f ( x ) is again a rational function and dom( f ) = dom( f ). • If f ( x ), g ( x ) are rational functions and a,b ∈ R then [ af ( x ) + bg ( x )] = af ( x ) + bg ( x ) [ f ( x ) g ( x )] = f ( x ) g ( x ) + f ( x ) g ( x ) (product rule) [ f ( x ) /g ( x )] = f ( x ) g ( x )- f ( x ) g ( x ) ( g ( x )) 2 (quotient rule, g ( x ) not the zero polynomial) The geometric interpretation of the derivative is through the slope of the tangent line: Definition: If f ( x ) is a rational function defined at a ∈ R , then the tangent line to the graph of f at ( a,f ( a )) has equation y = f ( a ) + f ( a )( x- a ) . It is the unique line that passes through the point ( a,f ( a )) and that has slope f ( a ). y (a, f(a)) ( x, f(a) + f (a)(x - a)) (x, f(x)) x x a III.B Mean Value Theorem, or MVT, and Rolle’s Theorem for Rational Functions III.2 Once more: • the slope of the tangent line through ( a,f ( a )) is f ( a ). • the equation of the tangent line is y = f ( a ) + f ( a )( x- a ). III.B Mean Value Theorem, or MVT, and Rolle’s Theorem for Rational Functions The Mean Value Theorem (MVT for rational functions): Let f ( x ) = P ( x ) /Q ( x ) be a rational function defined at each point of a closed interval [ a,b ]. Then for]....
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This note was uploaded on 10/04/2011 for the course MATH 16121 taught by Professor Rachelbelinsky during the Spring '11 term at Georgia State.

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goldch3 - Contents III The Mean Value Theorem for Rational...

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