{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 260 "Times " 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 261 270 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 271 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 272 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 273 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 274 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " 259 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 277 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 279 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 259 280 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 282 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 259 283 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "System " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Headi ng 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 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 "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot " -1 13 1 {CSTYLE "" -1 -1 "Times" 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 "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 50 "Using Taylor polynomials to appro ximate functions " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nana imo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Converting expressions to functio ns in Maple . . " }{TEXT 0 7 "unapply" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 243 "It takes a while t o get used to the difference between expressions and functions when us ing Maple.\nSome Maple procedures require functions as input and some \+ require expressions. Some procedures can be used with either expressio ns or functions." }}{PARA 0 "" 0 "" {TEXT -1 101 "For example, the plo t command can be used with either a function or expression as the firs t argument." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 203 "The only difference in the graphs produced by the following co mmands is that, in the first case, both axes remain unlabled because t he independent variable is not used when the1st argument is a function ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(sin,0..Pi);\nplot(sin(x),x=0..Pi);" }}}{PARA 0 " " 0 "" {TEXT -1 19 "\nThe Maple command " }{TEXT 0 7 "unapply" }{TEXT -1 54 " can be used to convert an expression into a function." }} {PARA 0 "" 0 "" {TEXT -1 51 "The following command defines the functio n f where " }{XPPEDIT 18 0 "f(x) = sqrt(x)+2;" "6#/-%\"fG6#%\"xG,&-%%s qrtG6#F'\"\"\"\"\"#F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 59 " It is essentially equivalent to using the arrow notation: " } {XPPEDIT 18 0 "f := proc (x) options operator, arrow; sqrt(x)+2 end;" "6#>%\"fGf*6#%\"xG7\"6$%)operatorG%&arrowG6\",&-%%sqrtG6#F'\"\"\"\"\"# F1F,F,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f := unapply(sqrt(x)+2,x);\nf(x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Here i s unother use of " }{TEXT 0 7 "unapply" }{TEXT -1 67 " to convert a de rivative obtained as an expression into a function." }}{PARA 0 "" 0 " " {TEXT -1 77 "The end result is exactly the same as that obtained by \+ applying the operator " }{TEXT 0 1 "D" }{TEXT -1 17 " to the function \+ " }{XPPEDIT 18 0 "f := proc (x) options operator, arrow; sqrt(x)+2 end ;" "6#>%\"fGf*6#%\"xG7\"6$%)operatorG%&arrowG6\",&-%%sqrtG6#F'\"\"\"\" \"#F1F,F,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "f := x -> sqrt(x)+2;\nDiff(f(x),x); \nvalue(%);\ndf := unapply(%,x);\nD(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%%sqrtG6#9$\"\"\"\" \"#F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,&*$-%%sqrtG6 #%\"xG\"\"\"F,\"\"#F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F %*$-%%sqrtG6#%\"xGF%!\"\"#F%\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#dfGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&\"\"\"F.*$-%%sqrtG6#9$F.! \"\"#F.\"\"#F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%) operatorG%&arrowGF&,$*&\"\"\"F,-%%sqrtG6#9$!\"\"#F,\"\"#F&F&F&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" } {TEXT -1 49 ": A procedure similar to, but much simpler than, " } {TEXT 0 7 "unapply" }{TEXT -1 75 " for functions of one variable, can \+ easily be constructed.by making use of " }{TEXT 0 4 "subs" }{TEXT -1 82 " to substitute for the expression _FX and the variable _X in the t emplate _X->_FX." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 62 "simple_unapply := (fx,x) -> subs(\{'_FX'=fx, ' _X'=x\},_X->_FX);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/simple_unapp lyGf*6$%#fxG%\"xG6\"6$%)operatorG%&arrowGF)-%%subsG6$<$/.%$_FXG9$/.%#_ XG9%f*6#F7F)F*F)F3F)F)F)F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f := simple_unapply(sqrt(x)+ 2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operato rG%&arrowGF(,&*$-%%sqrtG6#9$\"\"\"F2\"\"#F2F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Diff(f(x),x) ;\nvalue(%);\nsimple_unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%%DiffG6$,&*$-%%sqrtG6#%\"xG\"\"\"F,\"\"#F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*$-%%sqrtG6#%\"xGF%!\"\"#F%\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)operatorG%&arrowGF&,$*&\"\" \"F,*$-%%sqrtG6#9$F,!\"\"#F,\"\"#F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "Finding Taylor series and Taylor polynomials wi th Maple . . " }{TEXT 0 26 "taylor,convert(..,polynom)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "T he Taylor series expansion of a function " }{XPPEDIT 18 0 "g(x);" "6#- %\"gG6#%\"xG" }{TEXT -1 17 " about the point " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "g(a)+g*`'`(a)*(x-a)+g*`\"`(a)*(x-a)^2/2!+g*`'''`(a)* (x-a)^3/3!+` . . . `+`@@`(g,n)*``(a)/n!*(x-a)^n+` . . . `;" "6#,0-%\"g G6#%\"aG\"\"\"*(F%F(-%\"'G6#F'F(,&%\"xGF(F'!\"\"F(F(**F%F(-%\"\"G6#F'F (,&F.F(F'F/\"\"#-%*factorialG6#F5F/F(**F%F(-%$'''G6#F'F(,&F.F(F'F/\"\" $-F76#F>F/F(%(~.~.~.~GF(**-%#@@G6$F%%\"nGF(-%!G6#F'F(-F76#FFF/),&F.F(F 'F/FFF(F(FAF(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "= " } {XPPEDIT 18 0 "Sum(`@@`(g,n)*``(a)/n!*(x-a)^n,n = 0 .. infinity);" "6# -%$SumG6$**-%#@@G6$%\"gG%\"nG\"\"\"-%!G6#%\"aGF,-%*factorialG6#F+!\"\" ),&%\"xGF,F0F4F+F,/F+;\"\"!%)infinityG" }{TEXT -1 2 ", " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 267 22 "______________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "`@@`(g, n)*``(a);" "6#*&-%#@@G6$%\"gG%\"nG\"\"\"-%!G6#%\"aGF)" }{TEXT -1 13 " \+ denotes the " }{TEXT 268 1 "n" }{TEXT -1 23 " th derivative of g at " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The command " }{TEXT 0 18 "taylor(g(x),x=a,n)" }{TEXT -1 13 " computes a " }{TEXT 260 33 " truncated Taylor series expansion" }{TEXT -1 4 " of " }{XPPEDIT 18 0 " g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 31 ", with respect to the variable \+ " }{TEXT 264 1 "x" }{TEXT -1 18 ", about the point " }{TEXT 265 1 "a" }{TEXT -1 14 ", up to order " }{TEXT 266 1 "n" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 52 "The result is given using the differentia l operator " }{TEXT 0 1 "D" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "g := 'g':\ntaylor( g(x),x=a,6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+1,&%\"xG\"\"\"%\"aG! \"\"-%\"gG6#F'\"\"!--%\"DG6#F*F+F&,$*&#F&\"\"#F&---%#@@G6$F/F4F0F+F&F& F4,$*&#F&\"\"'F&---F86$F/\"\"$F0F+F&F&FB,$*&#F&\"#CF&---F86$F/\"\"%F0F +F&F&FK,$*&#F&\"$?\"F&---F86$F/\"\"&F0F+F&F&FT-%\"OG6#F&F=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "For a specific \+ example consider the Taylor series expansion of order 5 of " } {XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 17 " about the poi nt " }{XPPEDIT 18 0 "x = Pi/4;" "6#/%\"xG*&%#PiG\"\"\"\"\"%!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "Note that the highest po wer of " }{TEXT 269 1 "x" }{TEXT -1 37 " which occurs is less than the order." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "srs := taylor(sin(x),x=Pi/4,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$srsG+/,&%\"xG\"\"\"*&\"\"%!\"\"%#PiGF(F+,$*&\"\"#F+F /#F(F/F(\"\"!F-F(,$*&F*F+F/F0F+F/,$*&\"#7F+F/F0F+\"\"$,$*&\"#[F+F/F0F( F*-%\"OG6#F(\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "We can remove the last \"order\" term " }{XPPEDIT 18 0 "O ((x-Pi/4)^5);" "6#-%\"OG6#*$,&%\"xG\"\"\"*&%#PiGF)\"\"%!\"\"F-\"\"&" } {TEXT -1 109 " and construct a function from the resulting polynomial. The order term represents the \"tail\" of the series. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "convert (srs,polynom);\np := unapply(%,x):\n'p(x)'=p(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&\"\"#!\"\"F%#\"\"\"F%F(*(F%F&F%F',&%\"xGF(*&\"\"%F& %#PiGF(F&F(F(*(F-F&F%F'F*F%F&*(\"#7F&F%F'F*\"\"$F&*(\"#[F&F%F'F*F-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,,*&\"\"#!\"\"F*#\"\" \"F*F-*(F*F+F*F,,&F'F-*&\"\"%F+%#PiGF-F+F-F-*(F1F+F*F,F/F*F+*(\"#7F+F* F,F/\"\"$F+*(\"#[F+F*F,F/F1F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 90 "We can now plot the graph of the truncate d Taylor series and compare it with the graph of " }{XPPEDIT 18 0 "sin (x)" "6#-%$sinG6#%\"xG" }{TEXT -1 35 " in the neighbourhood of the poi nt " }{XPPEDIT 18 0 "x = Pi/4;" "6#/%\"xG*&%#PiG\"\"\"\"\"%!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot([sin(x),p(x)],x=-Pi/2..Pi,color=[brown,gree n],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 332 199 199 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$!3+++lBjzq:!#<$!\"\"\"\"!7$$!3?U#)oa'z!o9F*$!3)p 0tBz#HZ**!#=7$$!3/hwCIkqy8F*$!3#e6)=`R2;)*F37$$!3gx/6#*o>y7F*$!3QGfjZK (\\d*F37$$!3a(R6:6@q<\"F*$!3+)R89zrYB*F37$$!3ySkt,iKw5F*$!3D/ffOH?-))F 37$$!37'*QH08pH)*F3$!3-,)4**zv9K)F37$$!3a84Z2H.j))F3$!3[\"**3(**oRZxF3 7$$!3)[DB'4MJjyF3$!3=F[$RHjw2(F37$$!3_NS)[-+o'oF3$!3U[(***R9tRjF37$$!3 F7[h@NwTeF3$!3$pIb%*4D^^&F37$$!34#G'f63\"*Q\\F3$!3(*=D;babSZF37$$!3a(= )>+D^ARF3$!3ZKd#[^'pAQF37$$!38b*z'f2%>!HF3$!3#))>h#=>QhGF37$$!3_udM!yI %=>F3$!3qhl;O[o1>F37$$!3$oXqr\\4`-\"F3$!3n-?<(*R^B5F37$$\"3#z^(\\1$4)p O!#?$\"3=Z+Jp5!)pOFgp7$$\"3S_Lfd#yNO*!#>$\"3qk'flZ,*\\$*F]q7$$\"3I&[^, (f(G)>F3$\"3Uom+9x!*p>F37$$\"3#e#Rh`=A4HF3$\"3E[lj=yNoGF37$$\"3WZmp0,d DRF3$\"3/?:&*=<_DQF37$$\"3qlW5()RP$*[F3$\"3!y0@,]6/q%F37$$\"3u<$f!3e<. fF3$\"3%H)Gic)[ic&F37$$\"3(GB3d5*[IoF3$\"3dZ*[WZ3;J'F37$$\"3y!p(*Gx@2$ yF3$\"3%p'>A$\\,Y0(F37$$\"3exw$QS#op))F3$\"3!49aB`*f^xF37$$\"3IHJG(o+T x*F3$\"3S)p:hO>0H)F37$$\"37u_2@(*3v5F*$\"3?T%=5IGjz)F37$$\"39X0\"*fB+w 6F*$\"3ha=-!Ge2B*F37$$\"3%HH(e*oDZF\"F*$\"3+%=`5<.\\c*F37$$\"3#3e3teX- P\"F*$\"31)Rw;9q&*z*F37$$\"3#[LJ![TIw9F*$\"3OU/y[$*Qb**F37$$\"3M!y&)>[ -;d\"F*$\"2y@p0v'******F*7$$\"3O\\U]%z`Ln\"F*$\"33jf%G%4\")*F37$$\"3GsWYTlNm=F*$\"3Wz%)ea6Rm&*F37$$\"3? OCKlB?h>F*$\"3C&oOZ7YvC*F37$$\"3'3$f$)zNMg?F*$\"3\"R8BhUea#))F37$$\"3I VqQQAFd@F*$\"3?!\\(p1='*G$)F37$$\"3!\\^45cZ(eAF*$\"3F=\\b5^[DxF37$$\"3 KKJKd.[cBF*$\"3[Q@5Ph/pqF37$$\"31[VTdiUcCF*$\"3tMzCJ'=!GjF37$$\"3O77* \\bWbb#F*$\"3rSlS8*G2`&F37$$\"3'y=Ok(RiYEF*$\"3=L>8z3/]ZF37$$\"31m&>6\"*>wgs!QF37$$\"3D,5.8lPWGF*$\"3AFOqCafGHF37$$\"3I@FmLL#R %HF*$\"3c'ztirXQ'>F37$$\"3$=Q&*GM-#RIF*$\"3:,,w#)\\6A5F37$$\"3!)***\\/ l#fTJF*$\"3pawpOMzRJ!#E-%'COLOURG6&%$RGBG$\")#)eqk!\")$\"))eqk\"Fb[lFc [l-F$6$7W7$F($!3LP+759G@ZF37$F/$!3?Ry,XLLLdF37$F5$!3m@%yh%=z#Q'F37$F:$ !3YX\\(H1it(oF37$$!3dP4\"=+4wA\"F*$!36DvS[tgQqF37$F?$!3nK)GVn`[9(F37$$ !30>RicOnE6F*$!3K\"))QS())p)>(F37$FD$!3c[P?&\\EM?(F37$$!3D:H8mwkH5F*$! 3#zTzL8-i;(F37$FI$!35qL!)*4R54(F37$$!3G/CQ1@OY$*F3$!3C-C21JPvpF37$FN$! 3)*[laG[PBoF37$FS$!33FDOqW!HS'F37$FX$!3!yYeNFBm&eF37$Fgn$!39!\\sZS]3=& F37$F\\o$!3-fBb^y_/XF37$Fao$!3[@$)G>>*zm$F37$Ffo$!3mf&\\Q;cRw#F37$F[p$ !3mM]K;_$p%=F37$F`p$!3E%=5`P'[o)*F]q7$Fep$\"3-C4MHS*oe&Fgp7$F[q$\"33Bv -p*fGX*F]q7$Faq$\"3)[.(\\\"[%Qu>F37$Ffq$\"3]Z.4qDBqGF37$F[r$\"3pk^s%*p 5EQF37$F`r$\"3Q^c&e&=b+ZF37$Fer$\"3!\\#G$4.mic&F37$Fjr$\"3D0:PZ\"4;J'F 37$F_s$\"3]s>A$\\,Y0(F37$Fds$\"3A8db#)e*Rn(*F37$Ffv$\"3IH6nj4:(\\*F37$F[w$\"34#3 !QkarW\"*F37$F`w$\"3_vURFkWv')F37$Few$\"3gbp8`>W=\")F37$Fjw$\"3kG^-'Qe NV(F37$F_x$\"3gQlHM5TymF37$Fdx$\"3$3.2N[iG\"eF37$Fix$\"3!4!4YGCfl[F37$ F^y$\"368YAPMO@RF37$Fcy$\"3[gDszy/eFF37$Fhy$\"3'G'4*\\E.*\\;F37$F]z$\" 3#)R,2bw<[SF]q7$Fbz$!3'4mWI4]MT)F]q7$Fgz$!3U=w7toLJAF3-F][l6&F_[l$F-F- $\"*++++\"Fb[lFafl-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!F\\gl-% %VIEWG6$;$!+Fjzq:!\"*$\"+aEfTJFdgl%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "It looks as though the truncated Taylor series provides a reasonably good approximation for \+ " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 29 " over the i nterval from 0 to " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 70 "We can check how good the approximation is by plotting an error graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(sin(x)- p(x),x=0..Pi/2,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 474 268 268 {PLOTDATA 2 "6&-%'CURVESG6#7U7$$\"\"!F)$!3eM!>GA@L'>!#?7$$\"3=9;b\"[W> r\"!#>$!39\"3.7&**4bFJ$!3Msw^eBBoXFM7$$\"3kHX[TMk\"G#F J$!3*[$z\"\\LY\\V$FM7$$\"3Saq%H FJ$!3#GP'[;gA.=FM7$$\"3)pq'4OKt)G$FJ$!3?*[pJ4)G^7FM7$$\"3#f/N#RTo*e$FJ $!3aI$G'yIeh))!#A7$$\"3Ut79wN[GRFJ$!3q$=U&o>&3$eF[p7$$\"3/HF#F[p7$$\"3%[IQ)4z@%*[FJ $!3uj\"GO\"4a,9F[p7$$\"31\\,oR/A[_FJ$!3mN](yg8ZP(!#B7$$\"3uvW/X)FJ$ \"33+Z,/O\\1EF\\t7$$\"3gv)\\AI@S\\)FJ$\"3)z_Oh->4E'Ffs7$$\"3vn()=V,i>) )FJ$\"3IN5nhtom[F`s7$$\"3'G:[dg&*f:*FJ$\"3q^+GQN9c@Fer7$$\"3sSt6rL2&[* FJ$\"3D^paa(oPh'Fer7$$\"3mka(*HIZ.)*FJ$\"3/TTapFeq7$$\"3sJ_ *>P$Q\"3\"Fdv$\"3kjUXP))))p7F[p7$$\"3.M\"G%4t676Fdv$\"339b%[n.*o?F[p7$ $\"3co*Q4iSP`LF[p7$$\"3[cr`&*GLx6Fdv$\"3#Gv(zXlzu]F[p 7$$\"3Hk&>q'*z.@\"Fdv$\"3cR1h1[(pb(F[p7$$\"34Q$[)>&*oU7Fdv$\"3)*[w)e^7 I3\"FM7$$\"3I^lOFY^w7Fdv$\"3]\"[(*)*e.n`\"FM7$$\"3`!)f6EA448Fdv$\"3G(G /pA!R/@FM7$$\"3=;T7EvSU8Fdv$\"3Z9v=OVbWGFM7$$\"3a_wiepWv8Fdv$\"3#*GcgP g\"yw$FM7$$\"3U$obdw1eS\"Fdv$\"3H4ps(>(f5[FM7$$\"3s$)o#3`-1W\"Fdv$\"39 FTudn&=F'FM7$$\"3-#*)4zFC0nabyFM7$$\"3y<;V^l!\\]\"Fdv$ \"3)Q&f8>:ht)*FM7$$\"3\\3Y:@imO:Fdv$\"3'H8][>'e;7F,7$$\"3+++lBjzq:Fdv$ \"3D#*f$fkUt]\"F,-%+AXESLABELSG6$Q\"x6\"Q!F`\\l-%'COLOURG6&%$RGBGF(F($ \"*++++\"!\")-%%VIEWG6$;F($\"+Fjzq:!\"*%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 68 "The error in creases as we move away from the series expansion point " }{XPPEDIT 18 0 "x = Pi/4;" "6#/%\"xG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 61 ", and the maximum absolute error over the interval from 0 to " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 14 " occurs where \+ " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 53 "An estimate for this maximum absolute error is 0.002 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note " }{TEXT -1 39 ": We can say that this curve shows the " }{TEXT 260 14 "absolute error" }{TEXT -1 247 ", even though, technically, it only shows the difference in the two function values, with the sign depend ing on which way we subtract the functions. We can certainly see the a bsloute error by mentally taking the absolute value of the errors show n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 6 "Rem ark" }{TEXT -1 16 ": The procedure " }{TEXT 0 6 "series" }{TEXT -1 24 " can be used instead of " }{TEXT 0 6 "taylor" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "series(sin(x),x=Pi/4,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/,&% \"xG\"\"\"*&\"\"%!\"\"%#PiGF&F),$*&\"\"#F)F-#F&F-F&\"\"!F+F&,$*&F(F)F- F.F)F-,$*&\"#7F)F-F.F)\"\"$,$*&\"#[F)F-F.F&F(-%\"OG6#F&\"\"&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "However, \+ " }{TEXT 0 6 "series" }{TEXT -1 58 " can provide more general types of series expansions than " }{TEXT 0 6 "taylor" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "se ries(exp(x)/x,x=0,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\"\"\" !\"\"F%\"\"!#F%\"\"#F%#F%\"\"'F)#F%\"#C\"\"$#F%\"$?\"\"\"%#F%\"$?(\"\" &#F%\"%S]F+-%\"OG6#F%\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "taylor(exp(x)/x,x=0,8);" }}{PARA 8 "" 1 "" {TEXT -1 53 "Error, does not have a taylor expansion, try seri es()" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Some stand ard Maclaurin series" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 16 "Maclaurin series" } {TEXT -1 16 " for a function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG " }{TEXT -1 28 " is the Taylor series about " }{XPPEDIT 18 0 "x=0" "6# /%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 "When the e xpansion is about " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 35 ", the resulting series is called a " }{TEXT 260 9 "Maclaurin" } {TEXT -1 8 " series." }}{PARA 0 "" 0 "" {TEXT -1 24 "The Maclaurin ser ies of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 5 " is: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(0)+g*`'`(0)*x+g*`' '`(0)*x^2/2!+g*`'''`(0)*x^3/3!+` . . . `+`@@`(g,n)*``(0)/n!*x^n+` . . \+ . ` = Sum(`@@`(g,n)*``(0)/n!*x^n,n = 0 .. infinity);" "6#/,0-%\"gG6#\" \"!\"\"\"*(F&F)-%\"'G6#F(F)%\"xGF)F)**F&F)-%#''G6#F(F)F.\"\"#-%*factor ialG6#F3!\"\"F)**F&F)-%$'''G6#F(F)F.\"\"$-F56#F " 0 "" {MPLTEXT 1 0 21 "taylor(exp(x),x=0,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\"\"\"\"\"!F%F%#F%\"\"#F(#F%\"\"'\"\"$#F%\"#C\" \"%#F%\"$?\"\"\"&#F%\"$?(F*#F%\"%S]\"\"(-%\"OG6#F%\"\")" }}}{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 271 36 "_____________________________ _______" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x=x-x^3/3!+x^5/5!-x^7/7!+x^9 /9!-x^11/11!+` . . . `" "6#/*&%$sinG\"\"\"%\"xGF&,0F'F&*&F'\"\"$-%*fac torialG6#F*!\"\"F.*&F'\"\"&-F,6#F0F.F&*&F'\"\"(-F,6#F4F.F.*&F'\"\"*-F, 6#F8F.F&*&F'\"#6-F,6#F " 0 "" {MPLTEXT 1 0 29 "sin(x)=tay lor(sin(x),x=0,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%\"xG +/F'\"\"\"F)#!\"\"\"\"'\"\"$#F)\"$?\"\"\"&#F+\"%S]\"\"(#F)\"'!)GO\"\"* -%\"OG6#F)\"#5" }}}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT 263 30 "______________________________" }{TEXT -1 0 "" }{TEXT 262 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x=1-x^2/2!+x^4/4!-x^6/6!+x^8/8!-x^10/10!+` . . . `" "6#/*&%$cosG\"\"\"%\"xGF&,0F&F&*&F'\"\"#-%*factorialG6#F*!\"\" F.*&F'\"\"%-F,6#F0F.F&*&F'\"\"'-F,6#F4F.F.*&F'\"\")-F,6#F8F.F&*&F'\"#5 -F,6#F " 0 "" {MPLTEXT 1 0 29 "cos(x)=taylor(cos(x),x=0,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%\"xG+/F'\"\"\"\"\"!#!\"\"\"\"#F-#F)\"#C\"\"% #F,\"$?(\"\"'#F)\"&?.%\"\")-%\"OG6#F)\"#5" }}}{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 272 31 "_______________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x+1)=x-x^2/2+x^3 /3-x^4/4+x^5/5-x^6/6+x^7/7+` . . . `" "6#/-%#lnG6#,&%\"xG\"\"\"F)F),2F (F)*&F(\"\"#F,!\"\"F-*&F(\"\"$F/F-F)*&F(\"\"%F1F-F-*&F(\"\"&F3F-F)*&F( \"\"'F5F-F-*&F(\"\"(F7F-F)%(~.~.~.~GF)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "`` = Sum((-1)^n*x^(n+1)/(n+1),n = 0 .. infinity);" "6#/%!G-%$SumG6$ *(),$\"\"\"!\"\"%\"nGF+)%\"xG,&F-F+F+F+F+,&F-F+F+F+F,/F-;\"\"!%)infini tyG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-1 < x;" "6#2,$\"\"\"!\"\"%\"xG " }{XPPEDIT 18 0 "`` <= 1;" "6#1%!G\"\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ln(x +1)=taylor(ln(x+1),x=0,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6 #,&\"\"\"F(%\"xGF(+3F)F(F(#!\"\"\"\"#F-#F(\"\"$F/#F,\"\"%F1#F(\"\"&F3# F,\"\"'F5#F(\"\"(F7-%\"OG6#F(\"\")" }}}{PARA 256 "" 0 "" {TEXT -1 4 " \+ " }{TEXT 274 34 "__________________________________" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan*x = x-x^3/3+x^ 5/5-x^7/7+x^9/9-x^11/11+` . . . `" "6#/*&%'arctanG\"\"\"%\"xGF&,0F'F&* &F'\"\"$F*!\"\"F+*&F'\"\"&F-F+F&*&F'\"\"(F/F+F+*&F'\"\"*F1F+F&*&F'\"#6 F3F+F+%(~.~.~.~GF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "`` = Sum((-1)^(n+1) *x^(2*n-1)/(2*n-1),n = 1 .. infinity);" "6#/%!G-%$SumG6$*(),$\"\"\"!\" \",&%\"nGF+F+F+F+)%\"xG,&*&\"\"#F+F.F+F+F+F,F+,&*&F3F+F.F+F+F+F,F,/F.; F+%)infinityG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-1 < x;" "6#2,$\"\"\"! \"\"%\"xG" }{XPPEDIT 18 0 "`` <= 1;" "6#1%!G\"\"\"" }{TEXT -1 2 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "arctan(x)=taylor(arctan(x),x=0,12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'arctanG6#%\"xG+1F'\"\"\"F)#!\"\"\"\"$F,#F)\"\"&F.#F +\"\"(F0#F)\"\"*F2#F+\"#6F4-%\"OG6#F)\"#7" }}}{PARA 256 "" 0 "" {TEXT -1 4 " " }{TEXT 273 33 "_________________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 62 "Using Tayl or polynomials to approximate functions .. examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT 279 8 "Question" }{TEXT -1 2 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 31 "(a) Find the Taylor polynomial " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 25 " of degree 16 cen tred at " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 18 " for \+ the function " }{XPPEDIT 18 0 "f(x) = sqrt(x);" "6#/-%\"fG6#%\"xG-%%sq rtG6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "(b) Plot the g raph of the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG " }{TEXT -1 39 " found in (a) along with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " over the interval " } {XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 5/2;" "6 #1%!G*&\"\"&\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "(c) Plot a graph to show the absolute error when the Tayl or polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 37 " found in (a) is used to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "1/2 <= x;" " 6#1*&\"\"\"F%\"\"#!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 3/2;" "6#1%!G*&\" \"$\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 " \+ What is the maximum absolute error when " }{XPPEDIT 18 0 "p(x)" " 6#-%\"pG6#%\"xG" }{TEXT -1 24 " is used to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "1/2 <= x;" "6#1*&\"\"\"F%\"\"#!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= \+ 3/2;" "6#1%!G*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 4 " ? " }}{PARA 0 "" 0 "" {TEXT 280 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "f := x -> sqr t(x):\n'f(x)'=f(x);\ntaylor(f(x),x=1,17):\nconvert(%,polynom):\np := u napply(%,x):\n'p(x)'=p(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6 #%\"xG*$F'#\"\"\"\"\"#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\" xG,D#\"\"\"\"\"#F**&F+!\"\"F'F*F**&\"\")F-,&F'F*F*F-F+F-*&\"#;F-F0\"\" $F**(\"\"&F*\"$G\"F-F0\"\"%F-*(\"\"(F*\"$c#F-F0F5F**(\"#@F*\"%C5F-F0\" \"'F-*(\"#LF*\"%[?F-F0F9F**(\"$H%F*\"&oF$F-F0F/F-*(\"$:(F*\"&Ob'F-F0\" \"*F**(\"%JCF*\"'W@EF-F0\"#5F-*(\"%*>%F*\"')GC&F-F0\"#6F**(\"&$RHF*\"( /V>%F-F0\"#7F-*(\"&.?&F*\"(3')Q)F-F0\"#8F**(\"'Dd=F*\")KWbLF-F0\"#9F-* (\"'0VLF*\")k)3r'F-F0\"#:F**(\"(X[p*F*\"+[O[Z@F-F0F2F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot([f(x),p(x)],x=0..5/2,color=[red,mage nta],legend=[`f(x)`,`p(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6&-%'CURVESG6%7W7$$\"\"!F)F(7$$\"3gmmTN@Ki8!#>$\"3')* f*ohb=n6!#=7$$\"3ALL$3FWYs#F-$\"3!3\\Z6l\\1l\"F07$$\"3%)***\\iSmp3%F-$ \"3-0sq%pC;-#F07$$\"3WmmmT&)G\\aF-$\"3s*>zL7rVL#F07$$\"3O**\\7`p)*>yF- $\"3?_W/eRU'z#F07$$\"3PL$ek`o!>5F0$\"3=93s)G&G#>$F07$$\"3omm\"z>)G_:F0 $\"3#\\B^&z)3*RRF07$$\"3-nmT&QU!*3#F0$\"3C?bX?VgqXF07$$\"3HL$eRZXKi#F0 $\"3(GM!oaGw@^F07$$\"3xm;z>,_=JF0$\"35O.5b6P%e&F07$$\"3v**\\7G$[8j$F0$ \"3[(y/G)p1EgF07$$\"35n;z%*frhTF0$\"3/&[u*Hj8^kF07$$\"3A+]ilFQ!p%F0$\" 3S^6e>pj[oF07$$\"3@ML$3_\"=M_F0$\"3'oyt0NkZB(F07$$\"3HnmTg(fJr&F0$\"3o `H,vYaevF07$$\"3k++]7eP_iF0$\"3c]8T.m>2zF07$$\"3Q++]Pf!Qz'F0$\"3F'QONp aCC)F07$$\"3@++](=ubJ(F0$\"3'oMKK*H6`&)F07$$\"37n;zW(*Q*y(F0$\"3Azk,[> vD))F07$$\"3#QLL3F-GN)F0$\"3#=/-qkm$R\"*F07$$\"3=MLL$e'3I))F0$\"37lH;R S&oR*F07$$\"3?+]7.\"3M &eq\"Q**F07$$\"3;+DJ&H\"fT5!#<$\"3a>*>7!Qe?5F\\s7$$\"35+v$f)[$H4\"F\\s $\"3Q^gZh_VX5F\\s7$$\"3cL$ek`1l9\"F\\s$\"3iL=![X]22\"F\\s7$$\"3OLe*[.- d>\"F\\s$\"3mTZf6;[$4\"F\\s7$$\"3km;/Egw[7F\\s$\"3B\"GeD+#[<6F\\s7$$\" 3zm\"z%*f%)QI\"F\\s$\"3RWC(Rmx=9\"F\\s7$$\"3/+voza'=N\"F\\s$\"3;Z!o8]( pi6F\\s7$$\"3(om\"zWho.9F\\s$\"3oBo68Ex%=\"F\\s7$$\"3-++]i>Ad9F\\s$\"3 ,6f!**[ar?\"F\\s7$$\"32+]i:jf4:F\\s$\"3$*p@#\\Ic'G7F\\s7$$\"39+DJ&>r-c \"F\\s$\"3'pat*f\"3\"\\7F\\s7$$\"3++]P4q`;;F\\s$\"3i.9:(3J9F\"F\\s7$$ \"3;LL$eM%4n;F\\s$\"3m!4Sv2g6H\"F\\s7$$\"37++v$4v5s\"F\\s$\"34v8[;v*=J \"F\\s7$$\"3cm\"zWn*)*pF \\s$\"3I/^Qf2%zQ\"F\\s7$$\"3CLL3-=!y(>F\\s$\"3M%y:aTVjS\"F\\s7$$\"3))* \\7G8O;.#F\\s$\"3X4')GRZND9F\\s7$$\"3!pmm;*\\[$3#F\\s$\"3nTd+3#GMW\"F \\s7$$\"3*pmT&Qz]O@F\\s$\"31Pb[M)z;Y\"F\\s7$$\"3iLekG=4*=#F\\s$\"3[?@@ %)zbz9F\\s7$$\"3F++]i4TPAF\\s$\"3Qo^\"zw(z&\\\"F\\s7$$\"3qL$3F9!z#H#F \\s$\"3n%z/N?'>9:F\\s7$$\"3'pmm;%>KUBF\\s$\"3*=)RF;YYI:F\\s7$$\"3/+DJq J8&R#F\\s$\"3Q_6$)yK(pg'>2zF07$Fgp$\"3.B'pPpaCC)F07$F\\q$\"3c5IC$*H6`&)F07 $Faq$\"39^o,[>vD))F07$Ffq$\"3$H/-qkm$R\"*F07$F[r$\"3CmH;RS&oR*F0F_r7$F er$\"3583M&eq\"Q**F07$Fjr$\"35>*>7!Qe?5F\\sF_s7$Fes$\"3SL=![X]22\"F\\s 7$Fjs$\"3aQZf6;[$4\"F\\s7$F_t$\"3/-\"eD+#[<6F\\s7$Fdt$\"3vos'Rmx=9\"F \\s7$Fit$\"35nvI,vpi6F\\s7$F^u$\"3)zM8DhsZ=\"F\\s7$Fcu$\"3iwh1&[ar?\"F \\s7$Fhu$\"3xnfLvilG7F\\s7$F]v$\"3!G?di,3\"\\7F\\s7$Fbv$\"3%G-&)*z.Vr7 F\\s7$Fgv$\"3S= \"QI8F\\s7$Ffw$\"3u%=q=Rr-N\"F\\s7$F[x$\"3/WB5&HJ'o8F\\s7$F`x$\"3'phEV [LtQ\"F\\s7$Fex$\"3i3\"=%e(f[S\"F\\s7$Fjx$\"3oXo7R\"e8F\\s7$Fhz$\"3m(Qk^DJ[D\"F \\s7$$\"3]$e*)fbF(oBF\\s$\"3O;IPJ6Lf6F\\s7$F][l$\"3j#3%o%o0x-\"F\\s7$$ \"3*)\\iST*pxS#F\\s$\"3k1)*e?35([*F07$$\"3<++]7nS?CF\\s$\"3%Hl2'[#)pt& )F07$$\"3W]Pf$[VIV#F\\s$\"3AK`L)42*>vF07$Fb[l$\"3&)\\![&*oUmI'F07$$\"3 !Hc^GAqCX#F\\s$\"3w*3p9Ir9e&F07$$\"35Dc,\">g#fCF\\s$\"3S81P%[i/![F07$$ \"3u(oz\"f,0mCF\\s$\"3!*[bEP+ufRF07$$\"3O]PMF,%GZ#F\\s$\"3YF@hO(y^0$F0 7$$\"3c7y]&4I'zCF\\s$\"3'\\Brni*Q#3#F07$$\"3=v=nj+U'[#F\\s$\"3;j#=q^/n .\"F07$$\"3#y$f$=.5K\\#F\\s$!3a-KQl`.%o)!#?7$Fg[l$!3S=JSC:_$H\"F0-F\\ \\l6&F^\\lF_\\lF(F_\\l-Fc\\l6#%%p(x)G-%+AXESLABELSG6$Q\"x6\"Q!F[[m-%%V IEWG6$;F($\"+++++D!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "f(x)" "p(x)" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(f(x)-p(x),x=1/2..3/2,color=blue);" }}{PARA 13 " " 1 "" {GLPLOT2D 411 306 306 {PLOTDATA 2 "6&-%'CURVESG6#7\\o7$$\"3++++ ++++]!#=$!3t'o)4f\"yi\"e!#D7$$\"3*HL3FWYs-&F*$!3%)GL>Gfnv_F-7$$\"3)fm; a)G\\a]F*$!3;CN)=I&)Hy%F-7$$\"3'*)*\\7G$R<3&F*$!3Ot3zvdW)!#E7$$\"3Anm ;z_\"4i&F*$!3$o.)Gk(oy^&Fen7$$\"32nmm;hEGdF*$!3Kpb'H;\"RfNFen7$$\"3$pm mT&phNeF*$!3Ur0\")Ra8sAFen7$$\"3X**\\(=ddC%fF*$!3nU([^$QnP9Fen7$$\"35L Le*=)H\\gF*$!3ipRmzlR#**)!#F7$$\"3o++v=JN[hF*$!3Gr$>BQrsv&F_p7$$\"3;nm \"z/3uC'F*$!3EQ>$o,Sbk$F_p7$$\"37++DJ$RDX'F*$!3?.ZZnxgi8F_p7$$\"3'fm;z R'okmF*$!3Z+5(Gx[GErR4Xk!#J7$$\"3` +++v'Hi#zF*$!3*4p$yIClO7F`s7$$\"3jmm\"z*ev:\")F*$!3^@,r!3F-K#!#K7$$\"3 kKLL347T$)F*$!3AS?9U.2kA!#L7$$\"3,LLLLY.K&)F*$!3QCt)HR>F\\\"!#M7$$\"3? ***\\7o7Tv)F*$!3ae2Lm%QJ)RFgt7$$\"3IKLL$Q*o]*)F*$!3s6i&oq#eah!#N7$$\"3 A++D\"=lj;*F*$\"3)=S7%**\\0[LFgt7$$\"3]***\\PaRU%>#Fgt7$$\"3GLLeR\"3Gy*F*$\"37X8\"3(37oCF gt7$$\"3cmm;/T1&***F*$!3w'**oOM8,Y*Fbu7$$\"3em;zRQb@5!#<$!3[i7]<\"p+*e Fgt7$$\"3%)**\\(=>Y2/\"F_w$\"3%G!)43waIh&Fgt7$$\"3imm\"zXu91\"F_w$\"3Y &H)>N%*eY#)Fbu7$$\"3'******\\y))G3\"F_w$!3;evNHZpzBFgt7$$\"3!****\\i_Q Q5\"F_w$!3E2<^q]2#*HFgt7$$\"3#***\\7y%3T7\"F_w$!3KzU_yC*y3*Fgt7$$\"3#* ***\\P![hY6F_w$!3_BRGd+E#3)Fgt7$$\"3ELLLQx$o;\"F_w$\"3o@zh5tZ.KFat7$$ \"3')****\\P+V)=\"F_w$\"3Vb]hht=%F_ p7$$\"3rm;/@-/19F_w$\"3e]\"*zwCH^mF_p7$$\"3@LL3dg6<9F_w$\"3[@5h?()GV5F en7$$\"3()**\\(oTAqU\"F_w$\"3CcP$RdeZa\"Fen7$$\"3_mmmw(GpV\"F_w$\"3')R )f(\\l_mAFen7$$\"3GLeRA5\\Z9F_w$\"3;kg$\\AF&yLFen7$$\"3-+]7oK0e9F_w$\" 3q^@AEu%)))\\Fen7$$\"3-+++&oi\"o9F_w$\"3)3<@gt1Q=(Fen7$$\"3-+](=5s#y9F _w$\"3()*or>gAj-\"F-7$$\"3-+v$40O\"*[\"F_w$\"3)G7cH]:K\\\"F-7$$\"3++++ ++++:F_w$\"3_iOT)4hW:#F--%'COLOURG6&%$RGBG$\"\"!Fc`lFb`l$\"*++++\"!\") -%+AXESLABELSG6$Q\"x6\"Q!F[al-%%VIEWG6$;$\"+++++]!#5$\"+++++:!\"*%(DEF AULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve \+ 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Th e previous graph indicates that the maximum absolute error when the de gree 17 Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 24 " is used to approximate " }{XPPEDIT 18 0 "f(x) = sqrt(x); " "6#/-%\"fG6#%\"xG-%%sqrtG6#F'" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "1/2 <= x;" "6#1*&\"\"\"F%\"\"#!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 3/2;" "6#1%!G*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 12 " is about 6 " }{TEXT 281 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-8)" "6#)\"# 5,$\"\")!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "This ca n be confirmed by using the procedure " }{TEXT 0 7 "infnorm" }{TEXT -1 10 " from the " }{TEXT 0 9 "numapprox" }{TEXT -1 10 " package. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "numapprox[infnorm](f(x)-p(x),x=1/2..3/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+f\"yi\"e!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT 282 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 31 "(a) Find the Taylor pol ynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 25 " of de gree 15 centred at " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) = sin(x);" "6#/-%\"fG6 #%\"xG-%$sinG6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "(b) \+ Plot the graph of the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-% \"pG6#%\"xG" }{TEXT -1 39 " found in (a) along with the graph of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " over the interva l " }{XPPEDIT 18 0 "-15/2 <= x;" "6#1,$*&\"#:\"\"\"\"\"#!\"\"F)%\"xG" }{XPPEDIT 18 0 "`` <= 15/2;" "6#1%!G*&\"#:\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "(c) Plot a graph to show the abs olute error when the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\" pG6#%\"xG" }{TEXT -1 37 " found in (a) is used to approximate " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval \+ " }{XPPEDIT 18 0 "-Pi/2 <= x;" "6#1,$*&%#PiG\"\"\"\"\"#!\"\"F)%\"xG" } {XPPEDIT 18 0 "`` <= Pi/2;" "6#1%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 " What is the maximum absolu te error when " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 24 " is used to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "-Pi/2 <= x;" "6#1,$*&% #PiG\"\"\"\"\"#!\"\"F)%\"xG" }{XPPEDIT 18 0 "`` <= Pi/2;" "6#1%!G*&%#P iG\"\"\"\"\"#!\"\"" }{TEXT -1 4 " ? " }}{PARA 0 "" 0 "" {TEXT 283 8 " Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "f := x -> sin(x):\n'f(x)'=f (x);\ntaylor(f(x),x=0,16):\nconvert(%,polynom):\np := unapply(%,x):\n' p(x)'=p(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinGF& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,2F'\"\"\"*&#F)\"\"' F)*$)F'\"\"$F)F)!\"\"*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"%S]F)*$)F'\" \"(F)F)F0*&#F)\"'!)GOF)*$)F'\"\"*F)F)F)*&#F)\")+o\"*RF)*$)F'\"#6F)F)F0 *&#F)\"++3-FiF)*$)F'\"#8F)F)F)*&#F)\".+!oVn28F)*$)F'\"#:F)F)F0" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "plot([f(x),p(x)],x=-15/2..15 /2,color=[red,magenta],legend=[`f(x)`,`p(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 576 279 279 {PLOTDATA 2 "6&-%'CURVESG6%7]s7$$!3++++++++v!#<$ !3'*)QZxw***z$*!#=7$$!3J++vV8_OtF*$!3y\"yW*3Jy!p)F-7$$!3u****\\(oUI<(F 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" }} {PARA 0 "" 0 "" {TEXT -1 45 "This can be confirmed by using the proced ure " }{TEXT 0 7 "infnorm" }{TEXT -1 10 " from the " }{TEXT 0 9 "numap prox" }{TEXT -1 10 " package. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "numapprox[infnorm](f(x)-p(x) ,x=-Pi/2..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+MXPBg!#@" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 " " 0 "" {TEXT 275 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 31 "(a) Find the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6 #-%\"pG6#%\"xG" }{TEXT -1 25 " of degree 11 centred at " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 18 " for the function " } {XPPEDIT 18 0 "f(x) = x^x;" "6#/-%\"fG6#%\"xG)F'F'" }{XPPEDIT 18 0 "`` =exp(x*ln(x))" "6#/%!G-%$expG6#*&%\"xG\"\"\"-%#lnG6#F)F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "(b) Plot the graph of the Taylor p olynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 39 " fou nd in (a) along with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6# %\"xG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "0 <= x;" "6# 1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 3;" "6#1%!G\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "(c) Plot a graph to show the absolute e rror when the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\" xG" }{TEXT -1 37 " found in (a) is used to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "1/2 <= x;" "6#1*&\"\"\"F%\"\"#!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= \+ 3/2;" "6#1%!G*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 " What is the maximum absolute error when " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 24 " is used to appro ximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "1/2 <= x;" "6#1*&\"\"\"F%\"\"#!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 3/2;" "6#1%!G*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 4 " ? " }}{PARA 0 "" 0 "" {TEXT 276 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "f := x -> x^x:\n'f(x)'=f(x);\ntaylor(f(x),x=1,12):\nconvert(% ,polynom):\np := unapply(%,x):\n'p(x)'=p(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG)F'F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# /-%\"pG6#%\"xG,8F'\"\"\"*$),&F'F)F)!\"\"\"\"#F)F)*&F.F-F,\"\"$F)*&F0F- F,\"\"%F)*&\"#7F-F,\"\"&F)*(F0F)\"#SF-F,\"\"'F)*&\"$?\"F-F,\"\"(F-*(\" #fF)\"%?DF-F,\"\")F)*(\"#rF)\"%S]F-F,\"\"*F-*(\"$J\"F)\"&!35F-F,\"#5F) *(\"#`F)FBF-F,\"#6F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "plot( [f(x),p(x)],x=0..3,color=[red,magenta],legend=[`f(x)`,`p(x)`],numpoint s=75);" }}{PARA 13 "" 1 "" {GLPLOT2D 408 340 340 {PLOTDATA 2 "6&-%'CUR VESG6%7ap7$$\"\"!F)$\"\"\"%*undefinedG7$$\"3;VKC=E]D8!#?$\"3H%pS[BdD\" **!#=7$$\"3K'['[O_+^EF0$\"301yy16&R%)*F37$$\"3]H(HZ&y]wRF0$\"31i0fYBg# 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LL3dg6<9Fjw$\"3cGf(=.7y$=Fio7$$\"3_mmmw(GpV\"Fjw$\"3'Q:1k9A\"oJFio7$$ \"3-+]7oK0e9Fjw$\"3b)*)fD&f86bFio7$$\"3-+++&oi\"o9Fjw$\"3>K%\\4w)y#*=\"F-7$$\"3++++++++:Fjw$\"3b8RWd\"Gz`\"F--%'COLOURG6&%$RGBG$\"\"!F` `lF_`l$\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!Fh`l-%%VIEWG6$;$\"+++++]! #5$\"+++++:!\"*%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 98 "The previous graph indicates that the maximum abso lute error when the degree 11 Taylor polynomial " }{XPPEDIT 18 0 "p(x) " "6#-%\"pG6#%\"xG" }{TEXT -1 24 " is used to approximate " }{XPPEDIT 18 0 "f(x) = x^x;" "6#/-%\"fG6#%\"xG)F'F'" }{TEXT -1 17 " on the inter val " }{XPPEDIT 18 0 "1/2 <= x;" "6#1*&\"\"\"F%\"\"#!\"\"%\"xG" } {XPPEDIT 18 0 "`` <= 3/2;" "6#1%!G*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 12 " is about 4 " }{TEXT 278 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^ (-6);" "6#)\"#5,$\"\"'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "This can be confirmed by using the procedure " }{TEXT 0 7 "infn orm" }{TEXT -1 10 " from the " }{TEXT 0 9 "numapprox" }{TEXT -1 10 " p ackage. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "numapprox[infnorm](f(x)-p(x),x=1/2..3/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+a#)4RQ!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 31 "(a) Find the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 25 " of degree 14 centred at " } {XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 18 " for the functio n " }{XPPEDIT 18 0 "f(x) = ln(x);" "6#/-%\"fG6#%\"xG-%#lnG6#F'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "(b) Plot the graph of t he Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 39 " found in (a) along with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 5/2;" "6#1%!G*&\"\"&\"\" \"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "(c) Plot a graph to show the absolute error when the Taylor polynomial " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 37 " found in (a) is \+ used to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "1/2 <= x;" "6#1*&\"\"\"F%\" \"#!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 3/2;" "6#1%!G*&\"\"$\"\"\"\"\"#! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 " What is the maximum absolute error when " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG " }{TEXT -1 24 " is used to approximate " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "1/2 <= x ;" "6#1*&\"\"\"F%\"\"#!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 3/2;" "6#1%!G* &\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 4 " ? " }}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________" }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 33 "____________ _____________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 31 "(a) Find the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6# %\"xG" }{TEXT -1 25 " of degree 23 centred at " }{XPPEDIT 18 0 "x = 0; " "6#/%\"xG\"\"!" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f( x) = arctan(x);" "6#/-%\"fG6#%\"xG-%'arctanG6#F'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 44 "(b) Plot the graph of the Taylor polynomi al " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 39 " found in ( a) along with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "-6/5 <= x;" "6#1,$* &\"\"'\"\"\"\"\"&!\"\"F)%\"xG" }{XPPEDIT 18 0 "`` <= 6/5;" "6#1%!G*&\" \"'\"\"\"\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "( c) Plot a graph to show the absolute error when the Taylor polynomial \+ " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 37 " found in (a) \+ is used to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "-1/2 <= x;" "6#1,$*&\" \"\"F&\"\"#!\"\"F(%\"xG" }{XPPEDIT 18 0 "`` <= 1/2;" "6#1%!G*&\"\"\"F& \"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 " What is the maximum absolute error when " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6 #%\"xG" }{TEXT -1 24 " is used to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "-1/ 2 <= x;" "6#1,$*&\"\"\"F&\"\"#!\"\"F(%\"xG" }{XPPEDIT 18 0 "`` <= 1/2; " "6#1%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 4 " ? " }}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 33 "__ _______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0 " 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }