{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 24 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 " " 1 18 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Helvetica" 1 14 128 0 0 1 0 0 2 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 18 "Taylor Polynomials" }} {PARA 257 "" 0 "" {TEXT 259 19 "Mth 351 Summer 2002" }}{PARA 0 "" 0 " " {TEXT 260 21 "June 24, 2002 Maple 6" }}{PARA 0 "" 0 "" {TEXT 257 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" {TEXT 258 35 "Filename: 351u2002_ taylor_polys.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 203 "Maple has a builtin facility for \+ computing Taylor series. In Maple series are a special data type, but \+ we can easily convert them to polynomials. Here is a simple procedure \+ to compute Taylor polynomials:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "taylorp:=(fun,cent,deg)->con vert(taylor(fun,cent,deg+1),polynom):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 292 "Here fun is an express ion defining a function, cent is an expression of the form z=a whic h specifies that we want to expand fun in the variable z about the center a, and deg is the degree of the desired Taylor polynomial. \+ Let's check our procedure in a couple of well known cases:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "tay lorp(sin(x),x=0,11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.%\"xG\"\"\"* &#F%\"\"'F%*$)F$\"\"$F%F%!\"\"*&#F%\"$?\"F%)F$\"\"&F%F%*&#F%\"%S]F%*$) F$\"\"(F%F%F,*&#F%\"'!)GOF%)F$\"\"*F%F%*&#F%\")+o\"*RF%*$)F$\"#6F%F%F, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "taylorp(cos(x),x=0,10); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.\"\"\"F$*&#F$\"\"#F$*$)%\"xGF'F$ F$!\"\"*&#F$\"#CF$)F*\"\"%F$F$*&#F$\"$?(F$*$)F*\"\"'F$F$F+*&#F$\"&?.%F $)F*\"\")F$F$*&#F$\"(+)GOF$*$)F*\"#5F$F$F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 326 "To get an idea of how su ccessive Taylor polynomials approximate a function we can use the seq command to form the sequence and then plot each of the polynomials t ogether with the original function. Here is an example (where I have o mitted the Taylor polynomial of degree 0 since it is just a constant a nd not very exciting):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fun01:=exp(sin(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&fun01G-%$expG6#-%$sinG6#%\"tG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "tlist01:=[fun01, seq(taylorp(fun01,t=0,k) , k=1..6)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(tlist01G7)-%$expG6#- %$sinG6#%\"tG,&\"\"\"F.F,F.,(F.F.F,F.*&#F.\"\"#F.)F,F2F.F.F/,*F.F.F,F. *&F1F.F3F.F.*&#F.\"\")F.*$)F,\"\"%F.F.!\"\",,F.F.F,F.*&F1F.F3F.F.*&#F. F8F.F9F.F<*&#F.\"#:F.*$)F,\"\"&F.F.F<,.F.F.F,F.*&F1F.F3F.F.*&#F.F8F.F9 F.F<*&#F.FCF.FDF.F<*&#F.\"$S#F.*$)F,\"\"'F.F.F<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "We will plot fun01 \+ in blue and the Taylor polynomials in brown:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "clist01:=[b lue, seq(brown,k=1..6)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(clist01 G7)%%blueG%&brownGF'F'F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot(tlist01,t=-1.4..1.6,color=clist01,thickness=3,title=\"Some \+ Taylor polynomials\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6--%'CURVESG6$7S7$$!3!**************R\"!#<$\"3(zRr(yJrKP! #=7$$!3')****\\P&3YL\"F*$\"3#*p*oCWfBy$F-7$$!3#)**\\ivL_UF-7$$!3v)**\\i+#QU'*F-$\"3'R3$fp(4sR%F-7$$!3/***\\i!3%f+*F -$\"3=8hPM%[rc%F-7$$!3G***\\7oS:P)F-$\"3c'4m]\\\"3eZF-7$$!3s)****\\<#) *=xF-$\"3IZHlf%*Hy\\F-7$$!3.****\\(G3U9(F-$\"3'>mUh#e[$>&F-7$$!3e)**** \\-\\r\\'F-$\"3)=/V)pe'4Y&F-7$$!3K*****\\(GVZeF-$\"3%)G)*Q_]0edF-7$$!3 j)****\\(4J@_F-$\"3oZO8`S-tgF-7$$!3A***\\iIKFl%F-$\"3Q\"f01u'y%Q'F-7$$ !3'*)****\\FPm(RF-$\"3oBG7j'3\"*y'F-7$$!3)*)******4'*QS$F-$\"3-w\"4Dm! 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