Learning Guide Learning Guide

212 • Chapter 6: Examples from CalculusYou can use it to find a polynomial approximation of a function fnear x = a.> f := x -> exp( sin(x) );f := x → e sin(x)> a := Pi;a := π> taylor( f(x), x=a );1 − (x − π) + 1 2 (x − π)2 − 1 8 (x − π)4 + 115 (x − π)5 + O((x − π) 6 )Before you can plot the Taylor approximation, you must convert itfrom a series to a polynomial.> poly := convert( %, polynom );poly := 1 − x + π + 1 2 (x − π)2 − 1 8 (x − π)4 + 1 (x − π)515Now plot the function f along with poly.> plot( [f(x), poly], x=0..10, view=[0..10, 0..3] );32.521.510.502 4 6 8 10xThe expression (1/6!)f (6) (ξ)(x − a) 6 gives the error of the approximation,where ξ is some number between x and a. The sixth derivative of fis> diff( f(x), x$6 );