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...Chapter themeanvaluetheorem his s topic is called the mean value theorem or mvt tisnotsomething like say thechainrule thatyouwillusedaily butit doeshavesomeimportantconsequencesthatwewilladdressinsection easy to visual y ize it concerns a function f x that denedonaclosed interval as shown on right we re doing calcu lus so dier b entiable atleaston and continuous drawalinesegmentbetween points this line segment has slope m rise run fromthispicture you would expect at some point tangent c same seg ment in other words there number between for which fact exactly what meanvaluetheoremsays if dierentiable then example consider ontheinterval graphed below here theoremstatesthatthereisanumber cbetween andforwhich nd because equation yields shownabove indeed its checkyourunderstandingbyworkingexercise again asserts possible more than one such happen be three numbers themeanvaluetheoremguaranteesthatthereisalwaysatleastone themeanvaluetheoremcanbeprovedeasilywithfamiliartechniques of nding globa...