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Lecture

# Mean Value Theorem.pdf

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School
University of Alberta
Department
Statistics
Course
STAT312
Professor
Douglas Wiens
Semester
Fall

Description
81 15. Mean Value Theorem  Recall that ( + ) ( ) ( 0) = lim ( ) for ( ) = 0 0 0 provided this limit exists.  Relation to monotonicity: if % (weakly in- creasing: ( ) ( )) on ( ) and di erentiable there then 0( ) 0 on ( ). Proof: As 0 the numerator of ( ) is 0 and continuous, hence ( ) = lim ( ) 0. 0 (Similarly lim 0 ( ) 0.) ¤  Lab problem: If is di erentiable on ( ) and attains a maximum (or minimum) at ( ) 0 then ( ) = 0. 82  Mean Value Theorem: If is continuous on [ ] and di erentiable on ( ) then ( ) with 0 ( ) = ( ) + ( )( ) (15.1) This is a result of crucial importance in the ap- proximation of functions.  An interpretation is that di erentiable func- tions are locally almost linear: If and are very close, and 0is continuous, we can ap- proximate 0( ) by 0( ): ( ) ( ) + ( )( ); here the rhs is a straight line (as a function of 0 ), with slope ( ).
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