STAT312 Lecture : Mean Value Theorem.pdf
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0 ( ) for ( 0 + ) ( 0) provided this limit exists. Relation to monotonicity: creasing : di erentiable there then 0( ) Proof: as and continuous, hence (similarly lim 0 ( ) If is di erentiable on ( attains a maximum (or minimum) at then 0 ( ) = 0. Mean value theorem: if and di erentiable on ( ) then ( ) = ( ) + 0( )( is continuous on [ This is a result of crucial importance in the ap- proximation of functions. (15. 1) An interpretation is that di erentiable func- tions are locally almost linear : if are very close, and 0 is continuous, we can ap- proximate. 0( ) by 0( ): and ( ) + 0( )( here the rhs is a straight line (as a function of. A consequence of the mvt is that if.