MTH 253 Calculus (Other Topics)Differentiating Power SeriesIntegrating Power SeriesDetermining Taylor/Maclaurin SeriesSlide 5MTH 253Calculus (Other Topics)Chapter 10 – Infinite SeriesSection 10.10 – Differentiating and Integrating Power Series; Modeling with Taylor SeriesCopyright © 2006 by Ron Wallace, all rights reserved.Differentiating Power Series0 0 01If ( ) ( ) converges over ( , )kkkf x c x x x R x R�== - - +�( )0 0 01then '( ) ( ) over ( , )kkkdf x c x x x R x Rdx�== - - +�Examples:Consider Maclaurin series for sinx, cosx, and ex.Integrating Power Series0 0 01If ( ) ( ) converges over ( , )kkkf x c x x x R x R�== - - +�( )0 0 01then ( ) ( ) over ( , )kkkf x c x x dx x R x R�== - - +�� �Examples:Consider Maclaurin series for sinx, cosx, and ex.()010 0and ( ) ( ) where & are in ( , )b bkka akf x c x x dxa b x R x R�== -- +�� �Determining Taylor/Maclaurin SeriesUse Differentiation, Integration, Substitution, Multiplication, Addition, etc.Example 1 2 3111x x xx= + + + +-ggglet x = -x22 4 62111x x xx= - + - ++gggintegrate3 5 71tan3 5 7x x xx x-= - + - +gggMuch easier than finding a general formula for the nth derivative of the tan-1x function.Determining Taylor/Maclaurin SeriesUse Differentiation, Integration, Substitution, Multiplication, Addition, etc.Example 2 2 3 412 6 24xx x xe x= + + + + ggg2 4cos 12 24x xx = - + - gggMultiply3 4cos 13 6xx xe x x= + - - +gggMuch easier than finding derivatives excosx … try
View Full Document