meanvaluetheorem(更新版)
【正文】 (af)(bF)(bf)()(ddtFtfxy???because: xyoThe slope of the chord The slope of the tangent ? ?? ? ? ?( ) 0 , 1 ,0 1 , in t 0 , 1( ) 2 ( ( 1 ) ( 0 ) )S u p p o s e f x is c o n ti n u o u s o n a n d d e r iv a b lein th e n th e r e e x is ts a t le a s t o n e p os u c hEth a t f fx a m p lef????? ??，? ??mean value the average rate of change of f(x) on a,bsummarize 1. The relations of mean value theorems Rolle’s theorem Lagrange’s theorem Cauchy’s theorem )()( afbf ?xxF ?)( )()( afbf ?xxF ?)(2. The applications (1) Prove the equation (2) Prove inequality (3) Prove the existence of mean value key: set a auxiliary function Fermat’s theorem 1. Suppose ],0[)( ?Cxf ?and derivable in ),0( ?there exists a ,),0( ?? ? such that .c o t)()( ??? ff ???hints: Only need to prove . ? ? 0s in)( ?? ? ?xxxfxxfxF s in)()( ?2 If )(xf is derivable, prove there exists at least one zero point of )()( xfxf ??between the two zero point of f . hint: Suppose ,0)()( 2121 xxxfxf ???We want: ,),( 21 xx??? such that 0)()( ??? ?? ffOnly need to prove 0)()( ??? ?? ff?e ?e. 0])([ ?? ? ?xx xfe()(),.()xaxx a xf x F xfxli m m ay e xis ts m ay n ot e xis tFx???? ? ?when （ or ） ， the functions ( ) a n d ( ) p r o a c h z r e o o r i n f i n i t y ，then the limitThis kind of limit is called an indeterminate form of type 00??or L’Hospital’s rules 0()3 ) li m()xxfx AFx?? ?? (A is a constant or infinity） 00( ) ( )lim lim( ) ( )x x x xf x f xg x g x?????02 ) ( ) ( ) ( )f x g x a r e d if f e r e n ti a b le in U x
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