{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 "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis " -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 264 "" 0 1 0 0 0 0 0 0 1 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 "" -1 267 "" 0 1 0 0 0 0 1 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 "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 258 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 275 "" 0 1 0 0 0 0 0 0 1 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 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 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 "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 "Map le 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 "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 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" -1 258 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 35 "Fourier Series and Chebyshev Seri es" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Cana da" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{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 52 "load Fourier series and Fourier transform procedur es" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 9 "fo urier.m" }{TEXT -1 37 " contains the code for the procedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a comma nd similar to the one that follows, where the file path gives its loca tion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \"K:\\\\Maple/ procdrs/fourier.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "load function approximation procedures" }}{PARA 0 " " 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 10 "fcnapprx.m" } {TEXT -1 37 " contains the code for the procedure " }{TEXT 0 9 "interp oly" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple session by a command similar to th e one that follows, where the file path gives its location. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procdrs/f cnapprx.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 21 "Chebyshev polynomials" }}{PARA 0 "" 0 "" {TEXT -1 5 "\n The " }{TEXT 259 21 "Chebyshev polynomials" }{TEXT -1 79 " of the 1st \+ kind are based on the following family of trigonometric expansions." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "u := 'u':\ncos(2*u)=expand(cos(2*u));\ncos(3*u)=expand(cos(3*u) );\ncos(4*u)=expand(cos(4*u));\ncos(5*u)=expand(cos(5*u));\ncos(6*u)=e xpand(cos(6*u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$*&\"\" #\"\"\"%\"uGF*F*,&*&F)F*)-F%6#F+F)F*F*F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$*&\"\"$\"\"\"%\"uGF*F*,&*&\"\"%F*)-F%6#F+F) F*F**&F)F*F0F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$*& \"\"%\"\"\"%\"uGF*F*,(*&\"\")F*)-F%6#F+F)F*F**&F.F*)F0\"\"#F*!\"\"F*F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$*&\"\"&\"\"\"%\"uGF*F* ,(*&\"#;F*)-F%6#F+F)F*F**&\"#?F*)F0\"\"$F*!\"\"*&F)F*F0F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,$*&\"\"'\"\"\"%\"uGF*F*,**&\"#KF* )-F%6#F+F)F*F**&\"#[F*)F0\"\"%F*!\"\"*&\"#=F*)F0\"\"#F*F*F*F6" }}} {PARA 0 "" 0 "" {TEXT -1 34 "\nIf we replace each occurrence of " } {XPPEDIT 18 0 "cos(u)" "6#-%$cosG6#%\"uG" }{TEXT -1 4 " by " }{TEXT 265 1 "x" }{TEXT -1 105 ", we get the family of Chebyshev polynomials \+ of the 1st kind.\nThis can be done automatically as follows.\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "expand(cos(2*arccos(x)));\n expand(cos(3*arccos(x)));\nexpand(cos(4*arccos(x)));\nexpand(cos(5*arc cos(x)));\nexpand(cos(6*arccos(x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,&*$)%\"xG\"\"#\"\"\"F'F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& *$)%\"xG\"\"$\"\"\"\"\"%*&F'F(F&F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"%\"\"\"\"\")*&F)F()F&\"\"#F(!\"\"F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"&\"\"\"\"#;*&\"#?F()F&\"\"$F(!\" \"*&F'F(F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"xG\"\"'\"\" \"\"#K*&\"#[F()F&\"\"%F(!\"\"*&\"#=F()F&\"\"#F(F(F(F." }}}{PARA 0 "" 0 "" {TEXT -1 10 "\nThus the " }{TEXT 266 1 "n" }{TEXT -1 57 " th Cheb yshev polynomial of the first kind is defined by " }{XPPEDIT 18 0 "T(n ,x) = cos(n*arccos(x))" "6#/-%\"TG6$%\"nG%\"xG-%$cosG6#*&F'\"\"\"-%'ar ccosG6#F(F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 56 "These poly nomials are available via the Maple procedure " }{TEXT 0 10 "Chebyshev T" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 64 "To see the polynomi als in the same form as above we need to use " }{TEXT 0 6 "expand" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "ChebyshevT(6,x);\nexpand(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%+ChebyshevTG6$\"\"'%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"xG\"\"'\"\"\"\"#K*&\"#[F()F&\"\"%F(!\"\"*&\"#=F ()F&\"\"#F(F(F(F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "They are generally used with the domain restricted to the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 26 " in line with the formula " }{XPPEDIT 18 0 "T(n,x) = cos(n*arccos( x));" "6#/-%\"TG6$%\"nG%\"xG-%$cosG6#*&F'\"\"\"-%'arccosG6#F(F-" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "The values will then als o lie in the interval. 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\"eRFap7$Fdr$\"*c;4z#Fap7$Ffil$\"*Wm))e\"Fap7$Fgr$\")u8OFap7$F]s$!*\"33j[Fap7$F`] n$!*_b!ogFap7$F`s$!*M%pG)F.$!*CgL&)*Fa p7$Fis$!*!*=hv*Fap7$Fj[m$!*W0k=*Fap7$F\\t$!*&*)GO#)Fap7$$\"+()z>W))F.$ !*/<%evFap7$Fb\\m$!*Dh>v'Fap7$$\"+-HWb!*F.$!*0\"Q4eFap7$F_t$!*!>#Hs%Fa p7$Fg`n$!*&p#4a$Fap7$Fj\\m$!*?[E@#Fap7$F_an$!):B2tFap7$Fbt$\")?)R7*Fap 7$$\"+&yh(>'*F.$\"*#*\\X'=Fap7$Fgan$\"*b&pmGFap7$$\"+v7SG(*F.$\"*)[2?R Fap7$Fb]m$\"*qVf-&Fap7$$\"+l2/P)*F.$\"*#Rd&='Fap7$F_bn$\"*:a-S(Fap7$$ \"+b-oX**F.$\"*)*)Gr')FapFdt-Fht6&FjtF[uF]uF[u-%*THICKNESSG6#\"\"#-%+A XESLABELSG6$Q\"x6\"Q!Fh^p-%%VIEWG6$;F(Fet%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 39.000000 0 0 "Curve 1" "Curve 2" "Curve 3" " Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 " Introductory example of a Chebyshev series" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 94 "We are going to u se a suitable cosine series find a polynomial approximation for the fu nction " }{XPPEDIT 18 0 "f(x)=exp(x)" "6#/-%\"fG6#%\"xG-%$expG6#F'" } {TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "-1<=x" "6#1,$\"\"\"!\" \"%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 35 ". \nConsi der the auxiliary function " }{XPPEDIT 18 0 "g(u) = f(cos(u));" "6#/-% \"gG6#%\"uG-%\"fG6#-%$cosG6#F'" }{XPPEDIT 18 0 "``=exp(cos(u))" "6#/%! G-%$expG6#-%$cosG6#%\"uG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "Because " }{XPPEDIT 18 0 "cos(u);" "6#-%$cosG6#%\"uG" }{TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#Pi GF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "g(u);" "6#-%\"gG6#%\"uG" }{TEXT -1 30 " is also periodic with period " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\" #\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 59 "Thus w e can calculate some terms of the Fourier series for " }{XPPEDIT 18 0 "g(u);" "6#-%\"gG6#%\"uG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Using the variable " }{TEXT 267 1 "u" }{TEXT -1 13 " in place of \+ " }{TEXT 268 1 "x" }{TEXT -1 130 " will enable the resulting truncated Fourier series to be connected with the Chebyshev polynomials discuss ed in the first section." }}{PARA 0 "" 0 "" {TEXT -1 70 "Since the Fou rier series is a cosine series, we can use the procedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 14 " with option \"" }{TEXT 262 11 "type= cosine" }{TEXT -1 6 "\" or \"" }{TEXT 262 8 "type=cos" }{TEXT -1 51 " \". The required interval is then the interval from " }{XPPEDIT 18 0 " u=0" "6#/%\"uG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "u=Pi" "6#/%\"u G%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "g := u -> exp(cos(u)):\n'g(u)'=g(u );\nFourierSeries(g(u),u=0..Pi,type=cos,numterms=6):\nF := unapply(%,u ):\n'F(u)'=F(u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"uG-%$e xpG6#-%$cosGF&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"uG,0,$-% $IntG6$-%$expG6#-%$cosGF&/F';\"\"!%#PiG*$F5!\"\"\"\"\",$-F+6$*&F-F8F0F 8F2*(\"\"#F8F5F7F0F8F8,$-F+6$*&F-F8-F16#,$*&F>F8F'F8F8F8F2*(F>F8F5F7FC F8F8,$-F+6$*&F-F8-F16#,$*&\"\"$F8F'F8F8F8F2*(F>F8F5F7FLF8F8,$-F+6$*&F- F8-F16#,$*&\"\"%F8F'F8F8F8F2*(F>F8F5F7FVF8F8,$-F+6$*&F-F8-F16#,$*&\"\" &F8F'F8F8F8F2*(F>F8F5F7FjnF8F8,$-F+6$*&F-F8-F16#,$*&\"\"'F8F'F8F8F8F2* (F>F8F5F7FdoF8F8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "Integrals for which Maple's symbolic integration procedu res cannot obtain analytical expressions are left unevaluated. " }} {PARA 0 "" 0 "" {TEXT -1 24 "The following procedure " }{TEXT 0 11 "ev alf_coeff" }{TEXT -1 123 " can be used to evaluate the coefficients nu merically. This involves numerical evaluation of all the unevaluated i ntegrals." }}{PARA 0 "" 0 "" {TEXT -1 84 "The number of digits to be u sed can be given as an index or as the second argument. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 275 "eval f_coeff := proc(z)\n if nargs>1 then Digits := args[2] end if;\n i f type(op(1,procname),posint) then Digits := op(1,procname) end if;\n \+ map(_z->map(_u->`if`(type(_u,specfunc(algebraic,\{cos,sin\})) and \n nops(indets(op(1,_u),name))=1,_u,evalf(_u)),_z),z);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "evalf_coeff[10](F(u));\nG := unapply(%,u):\n'G(u)'=G(u);\nplo t([g(u),G(u)],u=-Pi..3*Pi,0..2.8,color=[red,green],thickness=[1,3],yti ckmarks=3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,0$\"+ye1m7!\"*\"\"\"*& $\"+3#=.8\"F&F'-%$cosG6#%\"uGF'F'*&$\"+&R`\\r#!#5F'-F,6#,$*&\"\"#F'F.F 'F'F'F'*&$\"+&)\\oLW!#6F'-F,6#,$*&\"\"$F'F.F'F'F'F'*&$\"+U/Cua!#7F'-F, 6#,$*&\"\"%F'F.F'F'F'F'*&$\"+=JEHa!#8F'-F,6#,$*&\"\"&F'F.F'F'F'F'*&$\" +)HKx\\%!#9F'-F,6#,$*&\"\"'F'F.F'F'F'F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"GG6#%\"uG,0$\"+ye1m7!\"*\"\"\"*&$\"+3#=.8\"F+F,-%$cosGF&F,F ,*&$\"+&R`\\r#!#5F,-F16#,$*&\"\"#F,F'F,F,F,F,*&$\"+&)\\oLW!#6F,-F16#,$ *&\"\"$F,F'F,F,F,F,*&$\"+U/Cua!#7F,-F16#,$*&\"\"%F,F'F,F,F,F,*&$\"+=JE Ha!#8F,-F16#,$*&\"\"&F,F'F,F,F,F,*&$\"+)HKx\\%!#9F,-F16#,$*&\"\"'F,F'F ,F,F,F," }}{PARA 13 "" 1 "" {GLPLOT2D 641 271 271 {PLOTDATA 2 "6'-%'CU RVESG6%7aq7$$!3*****4tk#fTJ!#<$\"3MBWr6WzyO!#=7$$!3yMay)3PY+$F*$\"3f<4 IbNS8PF-7$$!3-q3EI:onGF*$\"35FvtZr_=QF-7$$!3'z.'3Js^[FF*$\"3g(zE(4$HNHEF*$\"3M0M,KHh#=%F-7$$!3]Dq0(\\F8O#F*$\"3?t+s`)oG\"\\F -7$$!3))o*QcTD:4#F*$\"3kK$)*=Z6/3'F-7$$!3)H=\"F*$\"3'Q/_3@0'f9F*7$$!3Q77?Ggo\\5F*$\"3Q\"[&**>%p^k \"F*7$$!3;F/\"*pd I?F*7$$!37kc-@fxskF-$\"3M(Hi6:\"\\?AF*7$$!3o(fUUegg5&F-$\"3)[=NfKiFR#F *7$$!3T4aorpD-RF-$\"3\"yZ5i]A9_#F*7$$!39@#G\"fLX)p#F-$\"3A3;kB;n@EF*7$ $!375OJ&[a3-#F-$\"3=;bu(>CNm#F*7$$!33***)\\6cDV8F-$\"3/@!p\\<0Rp#F*7$$ !3c$p\"fuhX/5F-$\"3P&odd,:Yq#F*7$$!3Y!)Q%oPnll'!#>$\"3156#\\PoAr#F*7$$ !3GD3x2IdoKFir$\"3ax`x(HIor#F*7$$\"37*HAIh8U>\"!#?$\"30!QE,*)z#=FF*7$$ \"3W'\\m2;F8_$Fir$\"3_T/uCsf;FF*7$$\"3'HwI-'HBBpFir$\"3IAY`/xx6FF*7$$ \"3&H]pf(Q^K5F-$\"3&[$\\L\"eVQq#F*7$$\"3gHf\"fX/FP\"F-$\"3)>_\\%z8$Gp# F*7$$\"3!Hy3eh&3`?F-$\"3II!oj7)yhEF*7$$\"3@O;qvnYLFF-$\"3*f?*fQ-@>EF*7 $$\"3`\"[TRT8[/%F-$\"3&QwG#Q6b2DF*7$$\"3'oK\"=_+;c`F-$\"3gc*G%))pa'F-$\"3zBTlvy_5AF*7$$\"36o5t0o\"yt(F-$\"3V3!>KyWZ/#F*7$ $\"3[_H5Dp#Q:*F-$\"3BP\"\\)4z[R=F*7$$\"3o$[ZWq$)p0\"F*$\"3P4gc)QoZj\"F *7$$\"3E'f$RN$Qp<\"F*$\"3ScLrGTyn9F*7$$\"3%)3(Rj'H*oH\"F*$\"3$4/ysC31J \"F*7$$\"3X+D$)=(GkV\"F*$\"3*zqDND^L9\"F*7$$\"3/#HD8Zkfd\"F*$\"31yhR-@ X[**F-7$$\"3o\"y/dDx%*p\"F*$\"3K'y*))[^m&z)F-7$$\"3NrU3S+*H#=F*$\"3!Rk &f%zx;z(F-7$$\"3_e$f/C;S4#F*$\"3+t3-$>(HngF-7$$\"3A](H`F(4_BF*$\"3VW

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e\"Fgs7$F[w$\"3e,L%f0;rk\"Fgs7$F`w$\"3!)*f4nP**yr\"Fgs7$Few$\"3yWaS(=V +z\"Fgs7$Fjw$\"3q(z6v$)>)o=Fgs7$F_x$\"37ksNNs'z%>Fgs7$Fdx$\"3:ti\"G?vB .#Fgs7$Fix$\"3))R0yCkp>@Fgs7$F^y$\"3K`?+f'RK?#Fgs7$Fcy$\"31BFbkn/.BFgs 7$Fhy$\"3wFvdC$RhR#Fgs7$F]z$\"3Mb/8cGa*\\#Fgs7$Fbz$\"30EadZ#)o-EFgs7$F gz$\"3%)**\\w>%y#=FFgs-F\\[l6&F^[lFa[lF_[lFa[l-Fc[l6#\"\"#-%+AXESLABEL SG6$Q\"x6\"Q!Fdel-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 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 14 "The following " } {TEXT 259 11 "error curve" }{TEXT -1 48 " shows that the maximum absol ute error in using " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" } {TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"! \"\"F%" }{TEXT -1 18 " is approximately " }{XPPEDIT 18 0 "3*`.`*10^(-6 );" "6#*(\"\"$\"\"\"%\".GF%)\"#5,$\"\"'!\"\"F%" }{TEXT -1 2 ". 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#>$\"3Oir#4-F-7$$!3gKL$e9d;J'Fecl$\"3[r:gF..O:F-7$$!3KMLL3s$QM%Fec l$\"3zSpXh'e,9\"F-7$$!3T,+]ivF@AFecl$\"3a+!*)Rp=N)oFH7$$!3]^omm;zr)*!# @$\"3mF-7$$\"3G KLLe\"*[H7F1$!398u.^7`5BF-7$$\"3ylm;HCjV9F1$!39h'\\o&\\`DEF-7$$\"3I*** ****pvxl\"F1$!3[m9)GB5?)GF-7$$\"3g)***\\7JFn=F1$!3sD)>]$3\"*pIF-7$$\"3 #z****\\_qn2#F1$!3Ih-UbSS!>$F-7$$\"3*z\\(o/aWF@F1$!3v!eAD)R(*3KF-7$$\" 3/)*\\P%G?\"y@F1$!3o')y^!)=LBKF-7$$\"37)\\iS;&zGAF1$!3#\\*p3Q#F1$!3QYLwC\"))zB$F-7$$\"3P)\\7Go%\\JCF1$!3Nzv^*zQ3B$F- 7$$\"3U)***\\i&p@[#F1$!3iY9/4WL>KF-7$$\"3L)**\\(=GB2FF1$!3,ILh0;f:JF-7 $$\"3B)****\\2'HKHF1$!3]]dh0T\"o#HF-7$$\"3uJL$3UDX8$F1$!3;!4n%Qu2(o#F- 7$$\"3ElmmmZvOLF1$!3iw9n\"=xZQ#F-7$$\"3WKLLexn_NF1$!3Ag*=c)R!))*>F-7$$ \"3i******\\2goPF1$!3GA!>-*4_b:F-7$$\"3-mm\"H2fU'RF1$!3uEQ,zth76F-7$$ \"3UKL$eR<*fTF1$!3IWdl!Q9SR'FH7$$\"3/mm\"HiBQP%F1$!3!4r^F- 7$$\"3y)***\\7k.6aF1$\"3stX?%*p)oQ#F-7$$\"3IKLe9as;cF1$\"3)))ooYhp$\\F F-7$$\"3#emmmT9C#eF1$\"3(R\"RN\"QQU.$F-7$$\"38**\\7yI3IfF1$\"3#>M\\BLi y9$F-7$$\"3WKLeR` wE$F-7$$\"3wl;/,/UXhF1$\"3#y$3]:nE$H$F-7$$\"3SK3xJZD*>'F1$\"3y<,))e>T6 LF-7$$\"33****\\i!*3`iF1$\"3eA!**\\F>>K$F-7$$\"3klTN@z$\\I'F1$\"3/_O1: 6nCLF-7$$\"3AK$3-y'ycjF1$\"3HE6e->1?LF-7$$\"3y)\\i!Rcj3kF1$\"3)3 %GF-7$$\"3wKLL3N1#4(F1$\"3Er0O-yx?CF-7$$\"3c***\\(o!*R-tF1$\"3[\"=))*e 7=!)=F-7$$\"3Nmm;HYt7vF1$\"3mZBp\"\\a,B\"F-7$$\"3j**\\(o*GP4wF1$\"3Y!e w`fG:+*FH7$$\"3\"HL$ek6,1xF1$\"3$yD6=fZ'QbFH7$$\"3>m;HK%\\E!yF1$\"3$*[ qK'*)4I%>FH7$$\"3Y*******p(G**yF1$!35=8*>\"*R3v\"FH7$$\"3ymmTgg/5!)F1$ !39K=n9;LagFH7$$\"3)HLL3U/37)F1$!37I!)Rs#*yO5F-7$$\"3=***\\7yi:B)F1$!3 #yir&p>^h9F-7$$\"3]mmmT6KU$)F1$!35h[C/0)4(=F-7$$\"3a****\\P$[/a)F1$!32 fixP)oU`#F-7$$\"3fKLLLbdQ()F1$!3q#f]XZ-G0$F-7$$\"36+]i!*z>W))F1$!3D%H* ['*F1$!3 #fFAxd>JF\"F-7$$\"3K+]7G:3u'*F1$!3yzJRQ/&*=!)FH7$$\"3G+vVt7SG(*F1$!30j =]6[:3FFH7$$\"3A++v=5s#y*F1$\"3!GcX3&=-OKFH7$$\"3v]iS\"*3))4)*F1$\"3#p LR>K\\oX'FH7$$\"3;+D1k2/P)*F1$\"3+pBRdUk\\)*FH7$$\"3e\\(=nj+U')*F1$\"3 D&QR:A6>M\"F-7$$\"35+]P40O\"*)*F1$\"3b)pC9%R+< " 0 "" {MPLTEXT 1 0 387 "p:=x->.9999998017+1. 000022290*x+.5000063475*x^2+.1664888732*x^3+.4163501204e-1*x^4+.868682 0989e-2*x^5+.1439274336e-2*x^6;\ne := x -> exp(x)-p(x);\nx1 := -1.;\nx 2 := fsolve(D(e)(x),x=-0.9);\nx3 := fsolve(D(e)(x),x=-0.62);\nx4 := fs olve(D(e)(x),x=-0.21);\nx5 := fsolve(D(e)(x),x=0.23);\nx6 := fsolve(D( e)(x),x=0.63);\nx7 := fsolve(D(e)(x),x=0.9);\nx8 := 1.;\nevalf(map(e,[ x1,x2,x3,x4,x5,x6,x7,x8]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf *6#%\"xG6\"6$%)operatorG%&arrowGF(,0$\"+%\"eGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%$expG6#9$ \"\"\"-%\"pGF/!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$! \"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$!+37>\"**)!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$!+W&>U<'!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#x4G$!+2DRH@!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#x5G$\"+g?nAB!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x6G$\"+Og`)H' !#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x7G$\"+6d]H!*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x8G$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*$!&,,$!#5$\"&m-$F&$!&r2$F&$\"&::$F&$!%SK!\"*$\"%CLF/$ !%)Q$F/$\"%3MF/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The coefficients of the polynomial " }{XPPEDIT 18 0 "p(x) " "6#-%\"pG6#%\"xG" }{TEXT -1 129 " are not vastly different from the \+ coefficients of the degree 4 Taylor polynomial obtained by truncating \+ the Maclaurin series of " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" } {TEXT -1 9 " namely " }{XPPEDIT 18 0 "t(x) = 1+x+x^2/2!+x^3/3!+x^4/4! +x^5/5!+x^6/6!;" "6#/-%\"tG6#%\"xG,0\"\"\"F)F'F)*&F'\"\"#-%*factorialG 6#F+!\"\"F)*&F'\"\"$-F-6#F1F/F)*&F'\"\"%-F-6#F5F/F)*&F'\"\"&-F-6#F9F/F )*&F'\"\"'-F-6#F=F/F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "taylor(exp(x),x,7);\ncon vert(%,polynom);\nt := unapply(evalf(%),x):\n't(x)'=t(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\"\"\"\"\"!F%F%#F%\"\"#F(#F%\"\"'\"\"$#F %\"#C\"\"%#F%\"$?\"\"\"&#F%\"$?(F*-%\"OG6#F%\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0\"\"\"F$%\"xGF$*&#F$\"\"#F$*$)F%F(F$F$F$*&#F$\"\"'F$* $)F%\"\"$F$F$F$*&#F$\"#CF$*$)F%\"\"%F$F$F$*&#F$\"$?\"F$*$)F%\"\"&F$F$F $*&#F$\"$?(F$*$)F%F-F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"tG6 #%\"xG,0$\"\"\"\"\"!F*F'F**&$\"+++++]!#5F*)F'\"\"#F*F**&$\"+nmmm;F/F*) F'\"\"$F*F**&$\"+nmmmT!#6F*)F'\"\"%F*F**&$\"+LLLL$)!#7F*)F'\"\"&F*F**& $\"+*))))))Q\"F@F*)F'\"\"'F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 81 "The error curve associated with the use o f this Taylor polynomial to approximate " }{XPPEDIT 18 0 "exp(x)" "6#- %$expG6#%\"xG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[-1,1] " "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 110 " has a completely different sha pe, and the maximum absolute error is nearly 3000 times that involved \+ in using " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to a pproximate " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "t := x -> 1.+x+.5000000000*x^2+.1666666667*x^3+.4166666667e-1 *x^4: plot(exp(x)-t(x),x=-1..1,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 438 291 291 {PLOTDATA 2 "6&-%'CURVESG6#7U7$$!\"\"\"\"!$!3Vqw b)ze07(!#?7$$!3ommm;p0k&*!#=$!3W'p(3m^5NdF-7$$!3wKL$3jAF-7$$!3\"ommT!R=0vF1$!3W*)>OCOZgs%HaF1$!3NHq$GGl>g$FW7$$!3Q+++]$*4)*\\F1$!3;Ae&)yjC(R#FW7$$!39+++ ]_&\\c%F1$!3M&*y$Q\\KS`\"FW7$$!31+++]1aZTF1$!3!*31;PfQg&*!#A7$$!3umm;/ #)[oPF1$!3]A]Rp(*HcfF`p7$$!3hLLL$=exJ$F1$!3%\\;5BZpJ<$F`p7$$!3*RLLLtIf $HF1$!3?]I9$!3y&*yzIW$z>$!#D7$$!3KMLL3s$QM %Fir$!3W'*p*\\7F&z7!#E7$$!3]^omm;zr)*FW$\"3M.x&>EI!R6!#M7$$\"3%pJL$ezw 5VFir$\"3y6jqe8X\\7Fbs7$$\"3s*)***\\PQ#\\\")Fir$\"3svF6!4/i.$F\\s7$$\" 3GKLLe\"*[H7F1$\"3K7Iyr/0!R#Fer7$$\"3I*******pvxl\"F1$\"3_hbpIx\"H2\"F eq7$$\"3#z****\\_qn2#F1$\"38uxc*[@TL$Feq7$$\"3U)***\\i&p@[#F1$\"3CC.UV uj)=)Feq7$$\"3B)****\\2'HKHF1$\"3uXF>8=r)*=F`p7$$\"3ElmmmZvOLF1$\"3u/$ *)>(3A[OF`p7$$\"3i******\\2goPF1$\"3O9\")ohG&\\v'F`p7$$\"3UKL$eR<*fTF1 $\"3&R*[zf9VUFW7$$\"3#emmmT9C #eF1$\"3!yb/Z*ywlhFW7$$\"33****\\i!*3`iF1$\"3=s/BhGkx))FW7$$\"3%QLLL$* zym'F1$\"3pUw)*4>8L7F-7$$\"3wKLL3N1#4(F1$\"3MV`-AS\\\"p\"F-7$$\"3Nmm;H Yt7vF1$\"39([)e4;ltAF-7$$\"3Y*******p(G**yF1$\"3v_6n+PpUHF-7$$\"3]mmmT 6KU$)F1$\"3!f48&\\;V(*QF-7$$\"3fKLLLbdQ()F1$\"3IshmEmS^\\F-7$$\"3[++]i `1h\"*F1$\"3Kl!*Q<]B>jF-7$$\"3Y++++PDj$*F1$\"3gS\"\\'HNnuqF-7$$\"3W++] P?Wl&*F1$\"35=94'QZ?!zF-7$$\"3A++v=5s#y*F1$\"3)G5X@c>t())F-7$$\"\"\"F* $\"3M*4X!*3&\\[**F--%'COLOURG6&%$RGBG$F*F*F_\\l$\"*++++\"!\")-%+AXESLA BELSG6$Q\"x6\"Q!Fg\\l-%%VIEWG6$;F(Fg[l%(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 15 "The polynomial " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 30 " is an example of a truncated " }{TEXT 259 16 "Chebyshev series" }{TEXT -1 128 ". Such \+ series can be computed more efficiently by a direct method, which does not involve finding the associated Fourier series." }}{PARA 0 "" 0 " " {TEXT -1 20 "The Maple procedure " }{TEXT 0 9 "chebyshev" }{TEXT -1 8 " in the " }{TEXT 0 9 "numapprox" }{TEXT -1 30 " package can be used for this." }}{PARA 0 "" 0 "" {TEXT -1 12 "The package " }{TEXT 0 9 "o rthopoly" }{TEXT -1 183 " is required in order to convert truncated Ch ebyshev series to the corresponding polynomial. A 3rd optional argumen t can be used to obtain a polynomial with the specified error bound." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "with(numapprox):with(orthopoly):\nchebyshev(exp(x),x=-1..1,1e-5) ;\nexpand(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,0*&$\"+ye1m7!\"*\"\" \"-%\"TG6$\"\"!%\"xGF(F(*&$\"+3#=.8\"F'F(-F*6$F(F-F(F(*&$\"+'R`\\r#!#5 F(-F*6$\"\"#F-F(F(*&$\"+&)\\oLW!#6F(-F*6$\"\"$F-F(F(*&$\"+U/Cua!#7F(-F *6$\"\"%F-F(F(*&$\"+>JEHa!#8F(-F*6$\"\"&F-F(F(*&$\"+'HKx\\%!#9F(-F*6$ \"\"'F-F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0$\"+ " 0 "" {MPLTEXT 1 0 69 "f := x -> exp(x):\n'f(x)'=f(x);\nchebseries(f(x),x=-1 ..1,6);\nexpand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG- %$expGF&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,0$\"+ye1m7!\"*\"\"\"*&$\" +3#=.8\"F&F'-%+ChebyshevTG6$F'%\"xGF'F'*&$\"+&R`\\r#!#5F'-F,6$\"\"#F.F 'F'*&$\"+&)\\oLW!#6F'-F,6$\"\"$F.F'F'*&$\"+U/Cua!#7F'-F,6$\"\"%F.F'F'* &$\"+>JEHa!#8F'-F,6$\"\"&F.F'F'*&$\"+&HKx\\%!#9F'-F,6$\"\"'F.F'F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,0$\"+ " 0 "" {MPLTEXT 1 0 39 "chebseries(f(x),x=-1..1,6,output=poly);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,0*&$\"+NVFR9!#7\"\"\")%\"xG\"\"'F(F(*&$\"+\"*4# oo)F'F()F*\"\"&F(F(*&$\"+.7]jT!#6F()F*\"\"%F(F(*&$\"+K())[m\"!#5F()F* \"\"$F(F(*&$\"+tM1+]F:F()F*\"\"#F(F(*&$\"+!HA++\"!\"*F(F*F(F($\"+8!)** ****F:F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "We can obtain a better approximation for" }{XPPEDIT 18 0 " f(x)=ex p(x)" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 104 " by increasing the \+ number of terms in the Chebyshev series and so increasing its degree a s a polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 58 "f := x -> exp(x):\n'f(x)'=f(x);\nchebseries(f( x),x=-1..1,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$exp GF&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,2$\"+ye1m7!\"*\"\"\"*&$\"+3#=. 8\"F&F'-%+ChebyshevTG6$F'%\"xGF'F'*&$\"+&R`\\r#!#5F'-F,6$\"\"#F.F'F'*& $\"+&)\\oLW!#6F'-F,6$\"\"$F.F'F'*&$\"+U/Cua!#7F'-F,6$\"\"%F.F'F'*&$\"+ >JEHa!#8F'-F,6$\"\"&F.F'F'*&$\"+&HKx\\%!#9F'-F,6$\"\"'F.F'F'*&$\"+ikV) >$!#:F'-F,6$\"\"(F.F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 16 "The extra term 0" }{XPPEDIT 18 0 ".3198436462*`.`* 10^(-5)*`.`*ChebyshevT(7,x)" "6#*,-%&FloatG6$\"+ikV)>$!#5\"\"\"%\".GF) )\"#5,$\"\"&!\"\"F)F*F)-%+ChebyshevTG6$\"\"(%\"xGF)" }{TEXT -1 70 " \" contains\" most of the error in the previous degree 6 approximation. \+ " }}{PARA 0 "" 0 "" {TEXT -1 145 "This can be seen by plotting the gra ph of this term along with the previous error graph. There is very lit tle difference between the two graphs. " }}{PARA 0 "" 0 "" {TEXT -1 83 "This explains why the error function is virtually a \"scaled\" Che byshev polynomial. 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" }}{PARA 0 "" 0 "" {TEXT -1 13 "(a) Find the " }{TEXT 262 21 "Fourier cosine series" }{TEXT -1 27 " for the periodic function " }{XPPEDIT 18 0 "g(u)=f(cos(u))" "6#/-%\" gG6#%\"uG-%\"fG6#-%$cosG6#F'" }{TEXT -1 26 " up as far as the term in \+ " }{XPPEDIT 18 0 "cos(9*u);" "6#-%$cosG6#*&\"\"*\"\"\"%\"uGF(" }{TEXT -1 54 ", and plot the graph of this truncated Fourier series." }} {PARA 0 "" 0 "" {TEXT -1 20 "(b) By substituting " }{XPPEDIT 18 0 "x=c os(u)" "6#/%\"xG-%$cosG6#%\"uG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0< =u" "6#1\"\"!%\"uG" }{XPPEDIT 18 0 "``<=Pi" "6#1%!G%#PiG" }{TEXT -1 19 ", or equivalently, " }{XPPEDIT 18 0 "u=arccos(x)" "6#/%\"uG-%'arcc osG6#%\"xG" }{TEXT -1 85 ", in the truncated Fourier series found in ( a), construct a polynomial approximation " }{XPPEDIT 18 0 "p(x)" "6#-% \"pG6#%\"xG" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) = \+ ln(1+x/2);" "6#/-%\"fG6#%\"xG-%#lnG6#,&\"\"\"F,*&F'F,\"\"#!\"\"F," } {TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"! \"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 142 "(c) Plot an ab solute error curve for the polynomial approximation found in (b), and \+ give an estimate for the absolute error involved in using " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " in the interval \+ " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 8 "Solution " }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 223 "f := x -> ln(1+x/2):\n'f(x)'=f(x); \ng := u -> f(cos(u)):\n'g(u)'=g(u);\nFourierSeries(g(u),u=0..Pi,type= cos,numterms=9,mode=numeric):\nF := unapply(%,u):\n'F(u)'=F(u);\nplot( [g(u),F(u)],u=-3..10,color=[red,green],thickness=[1,3]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%#lnG6#,&\"\"\"F,*&\"\"#!\"\"F'F, F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"uG-%#lnG6#,&\"\"\"F, *&#F,\"\"#F,-%$cosGF&F,F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"FG6# %\"uG,6$\"+#e\"FdoF1-F36#,$*&\"\"*F1F'F1F1F1F1" }}{PARA 13 "" 1 "" {GLPLOT2D 476 179 179 {PLOTDATA 2 "6&-%'CURVESG6%7jq7$$!\"$\"\"!$!3VI! fi?%*=$o!#=7$$!3qmm\"z\\=$eG!#<$!3t#3Y*)Rj1a'F-7$$!3SLL$e*pj;FF1$!3?22 :OpRzgF-7$$!31+vVy1O$f#F1$!39C[nPS(Qc&F-7$$!3um;/hV3qCF1$!3Q&p@![#F1$!3QZenF1$!3h@'z0(ybS=F-7$$!3/+Dcc^ ![x\"F1$!3I:Cfr8/o5F-7$$!3(omTNN7fj\"F1$!3([lXV3/wI$!#>7$$!3B+]iX492:F 1$\"38>5sIf3JJF[o7$$!3eL$3x`p$y8F1$\"3qcv,[w5K\"*F[o7$$!30n;a.U.X7F1$ \"3%Gd.\\KLV[\"F-7$$!3H+]Pp))p66F1$\"3t6f8&*eO,?F-7$$!3VpmTgF.Q(*F-$\" 3)[ksniFqZ#F-7$$!3+OL3Fo2f$)F-$\"39A*)z=C?\"*GF-7$$!3P++v='4+h&F-$\"3G 0rCRK@INF-7$$!3MM$3_&G8'>%F-$\"3#GgEsM47w$F-7$$!3Komm\"4cAy#F-$\"3$RHsJ@p(RF-7$$!3@,+voE\"p`\"F-$\"3p.>KzF G:SF-7$$!3cwm\"Hd4C9*F[o$\"3yIa3u/sSSF-7$$!3,TLLeCp:HF[o$\"3;x#)*R?MK0 %F-7$$\"35***\\(=i6%4%F[o$\"3/4u#)\\u&=0%F-7$$\"3#RL$e*[#R56F-$\"3\\26 >i:5MSF-7$$\"3$zm\"HdLP6=F-$\"35pb4Lm'***RF-7$$\"3%>++]AaB^#F-$\"3)*H4 TKJX\\RF-7$$\"3********\\D2?RF-$\"3#*>HF*GY&)z$F-7$$\"3/)****\\(3zF`F- $\"3ky%yFAM;e$F-7$$\"3A+++vd)4/)F-$\"3GWp@i2\"z(HF-7$$\"3UI$eRA1HF*F-$ \"3t>g'\\ycOi#F-7$$\"31m;HnE[]5F1$\"3.o[Ft#Q+A#F-7$$\"3c*\\iS#*pp>\"F1 $\"37T\"4;[qqn\"F-7$$\"34LL$3=dMM\"F1$\"39*R(=T<$y1\"F-7$$\"3=LL3-6bn9 F1$\"3E&)yqi[rC]F[o7$$\"3CLLLB]k\"f\"F1$!3e)f&=B*Hy/\"F[o7$$\"3@m\"zWv &*ft\"F1$!3A!)[4<(\\-e)F[o7$$\"3=**\\i&[Y.)=F1$!3!*\\f5kHY_;F-7$$\"3#f ;z>M@\"3?F1$!3o=1&)yB^zBF-7$$\"3mKLL)>'*e8#F1$!3-gfb]SL;JF-7$$\"39m\"z pY&3wAF1$!3%z\\+Z,,v\"RF-7$$\"3h**\\iNZF;CF1$!3_Hrc[TN'o%F-7$$\"3%**** \\72o(\\DF1$!3[7Q0>iyg`F-7$$\"3C+](oShKo#F1$!39Iq,D'\\#\\fF-7$$\"3jm\" H#)pZD#GF1$!3]GO7yk9RkF-7$$\"3+LLe*)R$='HF1$!3A.F(e*[irnF-7$$\"3)H3_v8 )yDIF1$!3)pzb!)e;Z'oF-7$$\"3'H$3_&GU(*3$F1$!3fwEAO:/=pF-7$$\"3'z?0&f$> <7$F1$!3!G?WkK(\\HpF-7$$\"3'He*[Lkp`JF1$!3bv#HRLR2$pF-7$$\"3%z&RZ2Nn&= $F1$!3!G/!R&[i<#pF-7$$\"3%HLe9e]w@$F1$!3-#o=OT.E!pF-7$$\"3_*\\P%ephbLF 1$!3ig.v?m(eq'F-7$$\"35mmTNLe$\\$F1$!3&\\7<+R:kL'F-7$$\"3>mTNE;*oj$F1$ !3;@OP&HPmz&F-7$$\"3Em;H<**>!y$F1$!3d,(RNKvE8&F-7$$\"3(HLLe?]\\!RF1$!3 76iK\\)oD[%F-7$$\"3m**\\P%\\+(HSF1$!3g?)4M!*)z*y$F-7$$\"3[KekBUVkTF1$! 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\")-%+AXESLABELSG6$Q\"x6\"Q!Fcfn-%%VIEWG6$;F(Fcen%(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 44 "The maximum absol ute error is approximately " }{XPPEDIT 18 0 "5*`.`*10^(-7);" "6#*(\"\" &\"\"\"%\".GF%)\"#5,$\"\"(!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "infnorm" }{TEXT -1 16 " in th e package " }{TEXT 0 9 "numapprox" }{TEXT -1 28 " can be used to verif y this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "numapprox[infnorm](f(x)-p(x),x=-1..1);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+`3w[]!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 79 ": Essentially the sa me polynomial can be obtained directly using the procedure " }{TEXT 0 10 "chebseries" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "f := x -> ln(1+x/2);\nchebse ries(f(x),x=-1..1,9,output=poly);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%#lnG6#,&\"\"\"F0*&#F0\"\"#F0 9$F0F0F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,6*&$\"+5]T]S!#8\"\"\" )%\"xG\"\"*F(F(*&$\"+n%\\H])F'F()F*\"\")F(!\"\"*&$\"+&=g*>!*F'F()F*\" \"(F(F(*&$\"+9P5ZA!#7F()F*\"\"'F(F1*&$\"+g%f*\\jF:F()F*\"\"&F(F(*&$\"+ %GVgd\"!#6F()F*\"\"%F(F1*&$\"+^%4[;%FEF()F*\"\"$F(F(*&$\"+)QC)\\7!#5F( )F*\"\"#F(F1*&$\"+R&4++&FPF(F*F(F($\"+E0Y+O!#;F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT 274 8 "Question" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 41 " be the function \+ defined on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F% " }{TEXT -1 12 " defined by " }{XPPEDIT 18 0 "f(x)=``" "6#/-%\"fG6#%\" xG%!G" }{XPPEDIT 18 0 "erf(x) = 2/sqrt(Pi)" "6#/-%$erfG6#%\"xG*&\"\"# \"\"\"-%%sqrtG6#%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-t ^2), t=0..x)" "6#-%$IntG6$-%$expG6#,$*$%\"tG\"\"#!\"\"/F+;\"\"!%\"xG" }{TEXT -1 17 ". erf is the " }{TEXT 259 14 "error function" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "(a) Find the " }{TEXT 262 21 "Fourier cosine series" }{TEXT -1 27 " for the periodic functio n " }{XPPEDIT 18 0 "g(u)=f(cos(u))" "6#/-%\"gG6#%\"uG-%\"fG6#-%$cosG6# F'" }{TEXT -1 26 " up as far as the term in " }{XPPEDIT 18 0 "cos(7*u) ;" "6#-%$cosG6#*&\"\"(\"\"\"%\"uGF(" }{TEXT -1 54 ", and plot the grap h of this truncated Fourier series." }}{PARA 0 "" 0 "" {TEXT -1 20 "(b ) By substituting " }{XPPEDIT 18 0 "x=cos(u)" "6#/%\"xG-%$cosG6#%\"uG " }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0<=u" "6#1\"\"!%\"uG" }{XPPEDIT 18 0 "``<=Pi" "6#1%!G%#PiG" }{TEXT -1 19 ", or equivalently, " } {XPPEDIT 18 0 "u=arccos(x)" "6#/%\"uG-%'arccosG6#%\"xG" }{TEXT -1 85 " , in the truncated Fourier series found in (a), construct a polynomial approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) = erf(x);" "6#/-%\"fG6#%\"xG -%$erfG6#F'" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 142 " (c) Plot an absolute error curve for the polynomial approximation foun d in (b), and give an estimate for the absolute error involved in usin g " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approxim ate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " in the in terval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 8 "Solut ion" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "f := x -> erf(x):\n'f(x)'=f(x);\ng := u -> f(cos(u)):\n'g(u)'=g(u);\nFourierSeries(g(u),u=0..Pi,type=cos ,numterms=7,mode=numeric):\nF := unapply(%,u):\n'F(u)'=F(u);\nplot([g( u),F(u)],u=-3..10,color=[red,green],thickness=[1,3],ytickmarks=3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$erfGF&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"uG-%$erfG6#-%$cosGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"uG,**&$\"+mLYV!*!#5\"\"\"-%$cosGF&F-F- *&$\"+Y6G6m!#6F--F/6#,$*&\"\"$F-F'F-F-F-!\"\"*&$\"+eGNHZ!#7F--F/6#,$*& \"\"&F-F'F-F-F-F-*&$\"+#p#GZF!#8F--F/6#,$*&\"\"(F-F'F-F-F-F9" }}{PARA 13 "" 1 "" {GLPLOT2D 476 179 179 {PLOTDATA 2 "6'-%'CURVESG6%7iq7$$!\"$ \"\"!$!3?'Q1@#*[]Q)!#=7$$!3qmm\"z\\=$eG!#<$!3O\"f!R_%*)[D)F-7$$!3SLL$e *pj;FF1$!3[y))zeR.C!)F-7$$!31+vVy1O$f#F1$!3%[Sy=07bs(F-7$$!3um;/hV3qCF 1$!3]$[![icx#F1$!3^A T^Y4u+fF-7$$!3CLLL[SD`?F1$!3SE'phnHE)[F-7$$!3ALLefzp8>F1$!3qS#e'>ChbOF -7$$!3/+Dcc^![x\"F1$!3I2,zMB;bAF-7$$!3(omTNN7fj\"F1$!34^^dse,Kt!#>7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f := x -> erf(x);\nchebserie s(f(x),x=-1..1,7,output=poly);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$erfG6#9$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&$\"+C4Ee@Rk 5!#5F()F*\"\"&F(F(*&$\"+h3BWPF0F()F*\"\"$F(F,*&$\"+LYDG6!\"*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 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT 263 8 "Question" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 41 " be the function defined on the interval " }{XPPEDIT 18 0 "[-1,1] " "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "f(x)=sin(Pi /2*x)" "6#/-%\"fG6#%\"xG-%$sinG6#*(%#PiG\"\"\"\"\"#!\"\"F'F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "(a) Find the " }{TEXT 262 21 " Fourier cosine series" }{TEXT -1 27 " for the periodic function " } {XPPEDIT 18 0 "g(u)=f(cos(u))" "6#/-%\"gG6#%\"uG-%\"fG6#-%$cosG6#F'" } {TEXT -1 26 " up as far as the term in " }{XPPEDIT 18 0 "cos(5*u)" "6# -%$cosG6#*&\"\"&\"\"\"%\"uGF(" }{TEXT -1 54 ", and plot the graph of t his truncated Fourier series." }}{PARA 0 "" 0 "" {TEXT -1 20 "(b) By s ubstituting " }{XPPEDIT 18 0 "x=cos(u)" "6#/%\"xG-%$cosG6#%\"uG" } {TEXT -1 5 " for " }{XPPEDIT 18 0 "0<=u" "6#1\"\"!%\"uG" }{XPPEDIT 18 0 "``<=Pi" "6#1%!G%#PiG" }{TEXT -1 19 ", or equivalently, " }{XPPEDIT 18 0 "u=arccos(x)" "6#/%\"uG-%'arccosG6#%\"xG" }{TEXT -1 85 ", in the \+ truncated Fourier series found in (a), construct a polynomial approxim ation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x)=sin(Pi/2*x)" "6#/-%\"fG6#%\"xG-%$sinG 6#*(%#PiG\"\"\"\"\"#!\"\"F'F-" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 142 "(c) Plot an absolute error curve for the polyn omial approximation found in (b), and give an estimate for the absolut e error involved in using " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\" \"!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 8 "Solution" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 33 "(a) In this example, if we omit \"" }{TEXT 262 12 "mode=n umeric" }{TEXT -1 115 "\", Maple succeeds in obtaining analytical expr essions for the coefficients which involve certain Bessel functions. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "f := x -> sin(Pi/2*x);\ng := u -> f(cos(u));\nFourierSeries(g (u),u=0..Pi,type=cos,numterms=5);\nG := unapply(%,u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$sinG6# ,$*&#\"\"\"\"\"#F2*&%#PiGF29$F2F2F2F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"uG6\"6$%)operatorG%&arrowGF(-%\"fG6#-%$cos G6#9$F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*(\"\"#\"\"\"-%(Besse lJG6$F&,$*&F%!\"\"%#PiGF&F&F&-%$cosG6#%\"uGF&F&**F%F&,(*&\"#KF&F'F&F,* &F'F&)F-F%F&F&*(\"\")F&-F(6$\"\"!F*F&F-F&F&F&F-!\"#-F/6#,$*&\"\"$F&F1F &F&F&F&**F%F&,,*&\"%WhF&F'F&F&*(\"$)GF&F'F&F7F&F,*&F'F&)F-\"\"%F&F&*( \"%O:F&F:F&F-F&F,*(\"#CF&F:F&)F-FBF&F&F&F-!\"%-F/6#,$*&\"\"&F&F1F&F&F& F&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"GGf*6#%\"uG6\"6$%)operatorG% &arrowGF(,(*(\"\"#\"\"\"-%(BesselJG6$F/,$*&F.!\"\"%#PiGF/F/F/-%$cosG6# 9$F/F/**F.F/,(*&\"#KF/F0F/F5*&F0F/)F6F.F/F/*(\"\")F/-F16$\"\"!F3F/F6F/ F/F/F6!\"#-F86#,$*&\"\"$F/F:F/F/F/F/**F.F/,,*&\"%WhF/F0F/F/*(\"$)GF/F0 F/F@F/F5*&F0F/)F6\"\"%F/F/*(\"%O:F/FCF/F6F/F5*(\"#CF/FCF/)F6FKF/F/F/F6 !\"%-F86#,$*&\"\"&F/F:F/F/F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 90 "The coefficients can be evaluated num erically by using either of the following procedures " }{TEXT 0 11 "ev alf_coeff" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "F := unapply(evalf_coeffs[15](G(u) ),u);\nplot([g(u),F(u)],u=-3..10,color=[red,green],thickness=[1,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"uG6\"6$%)operatorG%&arr owGF(,(*&$\"0v6y<[O8\"!#9\"\"\"-%$cosG6#9$F1F1*&$\"0:sew<2Q\"!#:F1-F36 #,$*&\"\"$F1F5F1F1F1!\"\"*&$\"0*='pC92\\%!#O(*F-7$$!3um;/hV3qCF1$!3A(yC[GdSU*F-7$$!3A+D1\\sWJBF1$!32![Ud^\">L) )F-7$$!3ELL3P,\"G>#F1$!37VsK`X@FzF-7$$!3CLLL[SD`?F1$!3Ogq^&yB'fmF-7$$! 3ALLefzp8>F1$!3(f7+#oOBR]F-7$$!3j;H2e:DW=F1$!3aX;0f4+;TF-7$$!3/+Dcc^![ x\"F1$!3v1c28e$*GJF-7$$!3X$3_]ve`q\"F1$!3a&\\z92m<4#F-7$$!3(omTNN7fj\" F1$!3T!\\JtpQ.-\"F-7$$!3aLLe\\m_r:F1$!3AQt%)**z=Z6!#?7$$!3B+]iX492:F1$ \"35w$4H:$fv**!#>7$$!3\"pmm;CbFW\"F1$\"3i/(3bV^B*>F-7$$!3eL$3x`p$y8F1$ \"3#==S3HO!fHF-7$$!30n;a.U.X7F1$\"3F@5#\\Miz\"[F-7$$!3H+]Pp))p66F1$\"3 nwUp-m:7kF-7$$!3VpmTgF.Q(*F-$\"3,#f%o&>(pExF-7$$!3+OL3Fo2f$)F-$\"3kMVY 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "f := x -> sin(Pi/2*x):\n'f(x )'=f(x);\nchebseries(f(x),x=-1..1,5,output=poly);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinG6#,$*(\"\"#!\"\"%#PiG\"\"\"F'F0F0 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&$\"+%zU^=(!#6\"\"\")%\"xG\"\"& F(F(*&$\"+8R,@k!#5F()F*\"\"$F(!\"\"*&$\"+zqJq:!\"*F(F*F(F(" }}}{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 4 "Let " }{XPPEDIT 18 0 "f(x)" "6 #-%\"fG6#%\"xG" }{TEXT -1 41 " be the function defined on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 4 " by " } {XPPEDIT 18 0 "f(x) = tan(Pi/4*x);" "6#/-%\"fG6#%\"xG-%$tanG6#*(%#PiG \"\"\"\"\"%!\"\"F'F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "( a) Find the " }{TEXT 262 21 "Fourier cosine series" }{TEXT -1 27 " for the periodic function " }{XPPEDIT 18 0 "g(u)=f(cos(u))" "6#/-%\"gG6#% \"uG-%\"fG6#-%$cosG6#F'" }{TEXT -1 26 " up as far as the term in " } {XPPEDIT 18 0 "cos(9*u);" "6#-%$cosG6#*&\"\"*\"\"\"%\"uGF(" }{TEXT -1 56 ", and plot the graph of this truncated Fourier series. " }}{PARA 0 "" 0 "" {TEXT -1 20 "(b) By substituting " }{XPPEDIT 18 0 "x=cos(u) " "6#/%\"xG-%$cosG6#%\"uG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0<=u" " 6#1\"\"!%\"uG" }{XPPEDIT 18 0 "``<=Pi" "6#1%!G%#PiG" }{TEXT -1 19 ", o r equivalently, " }{XPPEDIT 18 0 "u=arccos(x)" "6#/%\"uG-%'arccosG6#% \"xG" }{TEXT -1 85 ", in the truncated Fourier series found in (a), co nstruct a polynomial approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6# %\"xG" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) = tan(Pi /4*x);" "6#/-%\"fG6#%\"xG-%$tanG6#*(%#PiG\"\"\"\"\"%!\"\"F'F-" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F% " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 142 "(c) Plot an absolute error curve for the polynomial approximation found in (b), and give a n estimate for the absolute error involved in using " }{XPPEDIT 18 0 " p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 31 "_______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 31 "_______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\" xG" }{TEXT -1 41 " be the function defined on the interval " } {XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 4 " by " } {XPPEDIT 18 0 "f(x) = 1/(1+x^2);" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)*$F '\"\"#F)!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "(a) Find the " }{TEXT 262 21 "Fourier cosine series" }{TEXT -1 27 " for the pe riodic function " }{XPPEDIT 18 0 "g(u)=f(cos(u))" "6#/-%\"gG6#%\"uG-% \"fG6#-%$cosG6#F'" }{TEXT -1 26 " up as far as the term in " } {XPPEDIT 18 0 "cos(10*u);" "6#-%$cosG6#*&\"#5\"\"\"%\"uGF(" }{TEXT -1 55 ", and plot the graph of this truncated Fourier series. " }}{PARA 0 "" 0 "" {TEXT -1 20 "(b) By substituting " }{XPPEDIT 18 0 "x=cos(u) " "6#/%\"xG-%$cosG6#%\"uG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0<=u" " 6#1\"\"!%\"uG" }{XPPEDIT 18 0 "``<=Pi" "6#1%!G%#PiG" }{TEXT -1 19 ", o r equivalently, " }{XPPEDIT 18 0 "u=arccos(x)" "6#/%\"uG-%'arccosG6#% \"xG" }{TEXT -1 85 ", in the truncated Fourier series found in (a), co nstruct a polynomial approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6# %\"xG" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) = 1/(1+x ^2);" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)*$F'\"\"#F)!\"\"" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 142 "(c) Plot an absolute er ror curve for the polynomial approximation found in (b), and give an e stimate for the absolute error involved in using " }{XPPEDIT 18 0 "p(x )" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f (x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "_______________________________" }}{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 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 31 "_______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\" xG" }{TEXT -1 41 " be the function defined on the interval " } {XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 4 " by " } {XPPEDIT 18 0 "f(x) = arctan(x);" "6#/-%\"fG6#%\"xG-%'arctanG6#F'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "(a) Find the " }{TEXT 262 21 "Fourier cosine series" }{TEXT -1 27 " for the periodic functio n " }{XPPEDIT 18 0 "g(u)=f(cos(u))" "6#/-%\"gG6#%\"uG-%\"fG6#-%$cosG6# F'" }{TEXT -1 26 " up as far as the term in " }{XPPEDIT 18 0 "cos(9*u) ;" "6#-%$cosG6#*&\"\"*\"\"\"%\"uGF(" }{TEXT -1 55 ", and plot the grap h of this truncated Fourier series. " }}{PARA 0 "" 0 "" {TEXT 259 4 "N ote" }{TEXT -1 51 ": Use the option \"mode=numeric\" with the procedur e " }{TEXT 0 13 "FourierSeries" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "(b) By substituting " }{XPPEDIT 18 0 "x=cos(u)" "6#/%\"xG -%$cosG6#%\"uG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0<=u" "6#1\"\"!%\" uG" }{XPPEDIT 18 0 "``<=Pi" "6#1%!G%#PiG" }{TEXT -1 19 ", or equivalen tly, " }{XPPEDIT 18 0 "u=arccos(x)" "6#/%\"uG-%'arccosG6#%\"xG" } {TEXT -1 85 ", in the truncated Fourier series found in (a), construct a polynomial approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) = arctan(x);" " 6#/-%\"fG6#%\"xG-%'arctanG6#F'" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 142 "(c) Plot an absolute error curve for the polyn omial approximation found in (b), and give an estimate for the absolut e error involved in using " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\" \"!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "___________ ____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 31 "____________ ___________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 262 14 "gamma function" }{TEXT -1 15 " is defined by " } }{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "GAMMA(x) = Int(exp(- t)*t^(x-1),t = 0 .. infinity);" "6#/-%&GAMMAG6#%\"xG-%$IntG6$*&-%$expG 6#,$%\"tG!\"\"\"\"\")F0,&F'F2F2F1F2/F0;\"\"!%)infinityG" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 44 "and is available in Maple via the p rocedure " }{TEXT 0 5 "GAMMA" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "assume(x_,po sitive);\nInt(exp(-t)*t^(x-1),t=0..infinity);\n``=subs(x_=x,value(subs (x=x_,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#,$% \"tG!\"\"\"\"\")F+,&%\"xGF-F-F,F-/F+;\"\"!%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&GAMMAG6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(GAMMA(x),x=0..5, y=0..30);" }}{PARA 13 "" 1 "" {GLPLOT2D 246 191 191 {PLOTDATA 2 "6%-%' CURVESG6$7ho7$$\"3(*******R`!eS$!#?$\"3*)[h/MXUIH!#:7$$\"3&*******z1h6 oF*$\"33K*[6NwBY\"F-7$$\"3*******>gT<-\"!#>$\"3%yAU1***[I(*!#;7$$\"3++ ++O@Ki8F6$\"3eI0zCq,%G(F97$$\"3;+++qE!Hq\"F6$\"3vNI+\"3ji\"eF97$$\"3** *****R?$[V?F6$\"3LoB!Gmny$[F97$$\"3;+++QP1%Q#F6$\"3gU&[`&[5RTF97$$\"3) ******>FWYs#F6$\"3\">_Q%paFWF6$\"3&*Q(z\")zu]?#F97$$ \"3K+++wu7oZF6$\"3O^)z`edS/#F97$$\"38+++5!3(3^F6$\"3,d/z;*[X!>F97$$\"3 /+++X&)G\\aF6$\"3s<[=?c^#y\"F97$$\"3I+++7'\\/8'F6$\"3],ndzPAz:F97$$\"3 k+++!o5;\"oF6$\"3\"obx)\\]q;9F97$$\"3G+++[+]%F^r7$$\"35+ ++(RwX5$Fbr$\"37gOH\\s'e)GF^r7$$\"3G+++sZ3yTFbr$\"3;82J3lj@@F^r7$$\"35 +++]4\\Y_Fbr$\"3/z\"p=1x6p\"F^r7$$\"3S+++U-/PiFbr$\"3-Y%>]BEsV\"F^r7$$ \"3s******empisFbr$\"3-UIqO)=$e7F^r7$$\"3&******>*>VB$)Fbr$\"36&*y9!>% yH6F^r7$$\"3c*****R`l2Q*Fbr$\"3&RYnyfm(R5F^r7$$\"3-+++0j$o/\"F^r$\"3' \\F!p*fi/v*Fbr7$$\"3!******>&>jU6F^r$\"3)4\\()RrU^N*Fbr7$$\"3%******H; v/D\"F^r$\"3!\\eF6VYI1*Fbr7$$\"3!******z=h(e8F^r$\"3QZ4bb3*H!*)Fbr7$$ \"35+++Q[6j9F^r$\"3[-*='f8/c))Fbr7$$\"35+++\\z(yb\"F^r$\"3q-u4O%*o%*)) Fbr7$$\"3%******\\Xg0n\"F^r$\"3Rvr[7,!R.*Fbr7$$\"3)******pJpW`(> F^r$\"3AlBP!y_#)*)*Fbr7$$\"3#******4f#=$3#F^r$\"3-\\'e;Zo!Q5F^r7$$\"3% )*****Hxpe=#F^r$\"33QFOYPU$4\"F^r7$$\"35+++uI,$H#F^r$\"33P`3'=Y=;\"F^r 7$$\"3=+++rSS\"R#F^r$\"3X@S%p*4CN7F^r7$$\"3-+++`?`(\\#F^r$\"3W?%pe\\Pq K\"F^r7$$\"3!********>pxg#F^r$\"3y!f\"pN6,Q9F^r7$$\"38+++g4t.FF^r$\"3) o(*Gv[*G\\:F^r7$$\"3*********Gst!GF^r$\"3;r![HYJpo\"F^r7$$\"38+++ERW9H F^r$\"3qZ)f2Aw3&=F^r7$$\"3@+++KE>>IF^r$\"3iO)\\5c%)e.#F^r7$$\"3%****** >RU07$F^r$\"3JrJ_G'Q;C#F^r7$$\"36+++?S2LKF^r$\"3S\"RR-4!)e]#F^r7$$\"3? +++$p)=MLF^r$\"3Q^0biok!y#F^r7$$\"3\"*******)=]@W$F^r$\"3Bc\"3G4C'>JF^ r7$$\"3')******\\$z*RNF^r$\"3'p,Yek:TZ$F^r7$$\"3#)*****RYKpk$F^r$\"3oP 'p)z&))=#RF^r7$$\"3))*****z+nvu$F^r$\"3X4!Q!)G&G5WF^r7$$\"3)******R5fF &QF^r$\"3*z+Q+Z\\C+&F^r7$$\"3')*****\\g.c&RF^r$\"3/Pu0D/:wcF^r7$$\"3K+ ++nAFjSF^r$\"3Kh8FEm***\\'F^r7$$\"3q*****\\)*pp;%F^r$\"3'>&zK5Q)*GuF^r 7$$\"3#)*****z(e,tUF^r$\"3*)oeh?e#[Z%F^r$\"3.h&\\ZW*GB6F97$$\"3)******pG!e&e%F^r $\"3$)4\">\"=[;68F97$$\"3N+++'37^j%F^r$\"3*36n)4PU19F97$$\"3%)*****\\) Qk%o%F^r$\"3GOW\\4:F97$$\"3A+++8^XPZF^r$\"39!Q5\"e#p(G;F97$$\"3y*** **>Mm-z%F^r$\"3#))4.li?'eVS?F97$$\"3;+++c-oX\\F^r$\"3;QWt\"p$>7AF97$$ \"\"&\"\"!$\"3O+++++++CF9-%'COLOURG6&%$RGBG$\"#5!\"\"$FaclFaclF[dl-%+A XESLABELSG6$Q\"x6\"Q\"yF`dl-%%VIEWG6$;F[dlF_cl;F[dl$\"#IFacl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 271 1 "n" }{TEXT -1 24 " i s a positive integer, " }{XPPEDIT 18 0 "GAMMA(n+1)=n!" "6#/-%&GAMMAG6# ,&%\"nG\"\"\"F)F)-%*factorialG6#F(" }{TEXT -1 61 ", so the gamma funct ion generalises the factorial operation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(n!);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&GAMMAG6#,&%\"nG\"\"\"F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#- %\"fG6#%\"xG" }{TEXT -1 41 " be the function defined on the interval \+ " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 4 " by " } {XPPEDIT 18 0 "f(x) = GAMMA(x/2+1);" "6#/-%\"fG6#%\"xG-%&GAMMAG6#,&*&F '\"\"\"\"\"#!\"\"F-F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "(a) Find the " }{TEXT 262 21 "Fourier cosine series" }{TEXT -1 27 " for the periodic function " }{XPPEDIT 18 0 "g(u)=f(cos(u))" "6#/-%\" gG6#%\"uG-%\"fG6#-%$cosG6#F'" }{TEXT -1 26 " up as far as the term in \+ " }{XPPEDIT 18 0 "cos(11*u);" "6#-%$cosG6#*&\"#6\"\"\"%\"uGF(" }{TEXT -1 55 ", and plot the graph of this truncated Fourier series. " }} {PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 18 ": Use the option \"" }{TEXT 262 12 "mode=numeric" }{TEXT -1 21 "\" with the procedure " } {TEXT 0 13 "FourierSeries" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "(b) By substituting " }{XPPEDIT 18 0 "x=cos(u)" "6#/%\"xG-%$cos G6#%\"uG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0<=u" "6#1\"\"!%\"uG" } {XPPEDIT 18 0 "``<=Pi" "6#1%!G%#PiG" }{TEXT -1 19 ", or equivalently, \+ " }{XPPEDIT 18 0 "u=arccos(x)" "6#/%\"uG-%'arccosG6#%\"xG" }{TEXT -1 85 ", in the truncated Fourier series found in (a), construct a polyno mial approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) = GAMMA(x/2+1);" "6#/- %\"fG6#%\"xG-%&GAMMAG6#,&*&F'\"\"\"\"\"#!\"\"F-F-F-" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 142 "(c) Plot an absolute error cu rve for the polynomial approximation found in (b), and give an estimat e for the absolute error involved in using " }{XPPEDIT 18 0 "p(x)" "6# -%\"pG6#%\"xG" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" " 6#-%\"fG6#%\"xG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[-1, 1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "_____________________________" }}{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 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 29 "_____________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 262 23 "Fresnel cosine functi on" }{TEXT -1 1 " " }{XPPEDIT 18 0 "C(x)" "6#-%\"CG6#%\"xG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "FresnelC(x)" "6#-%)FresnelCG6#%\"xG" }{TEXT -1 15 " is defined by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "C(x) = Int(cos(Pi/2*t^2),t=0..x)" "6#/-%\"CG6#%\"xG-%$IntG6$-%$c osG6#*(%#PiG\"\"\"\"\"#!\"\"%\"tGF1/F3;\"\"!F'" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 44 "and is available in Maple via the procedu re " }{TEXT 0 8 "FresnelC" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Int(cos(Pi/2*t^2), t=0..x);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$ cosG6#,$*(\"\"#!\"\"%#PiG\"\"\"%\"tGF+F./F/;\"\"!%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%)FresnelCG6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(FresnelC(x),x =-2..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 395 246 246 {PLOTDATA 2 "6%-%'C URVESG6$7jt7$$!\"#\"\"!$!+hS`#)[!#57$$!+g*4P#>!\"*$!+lkqZTF-7$$!+@*>u% =F1$!+fm.oNF-7$$!+W,B9=F1$!+3;*RR$F-7$$!+m./\"y\"F1$!+aY2wKF-7$$!+xaWk Wwj$F-7$$!+0$ f`c\"F1$!+%oTt*QF-7$$!+K\"o]T\"F1$!+v-P!G&F-7$$!+o7\\l7F1$!+h@BymF-7$$ !+=G:'>\"F1$!+-XoyrF-7$$!+nV\"o7\"F1$!+Uh\\RvF-7$$!+#R;44\"F1$!+K-9mwF -7$$!+=%=]0\"F1$!+9)R1v(F-7$$!+I%pq.\"F1$!+#z2rx(F-7$$!+V/7>5F1$!+uk:$ z(F-7$$!+c9<,5F1$!+RC\"*)z(F-7$$!*oCA$)*F1$!+p\"QXz(F-7$$!+)*ffY'*F-$! +\\gczxF-7$$!+:t'4Y*F-$!+sc@axF-7$$!+L'Q`F*F-$!+[Vw=xF-7$$!+]*4(*3*F-$ !+1Z^twF-7$$!+&e_%=()F-$!+Xa$\\b(F-7$$!*A&>Z$)F1$!+d!o7S(F-7$$!+S<12wF -$!+BJ7-qF-7$$!*EGp'oF1$!+8$G'*\\'F-7$$!*v\"HW`F1$!+_'>xB&F-7$$!*n_J+% F1$!+f5'y(RF-7$$!*tZL\\#F1$!+'4r4\\#F-7$$!)QVt(*F1$!+vP@t(*!#67$$\")s2 O[F1$\"+t12O[F]w7$$\"++&\\p9\"F-$\"+F0!p9\"F-7$$\"*G\"H5=F1$\"+D;\")4= F-7$$\"+:)o!*f#F-$\"+uR9'f#F-7$$\"*NYyQ$F1$\"+46&oP$F-7$$\"+!RWg0%F-$ \"+#[T!HSF-7$$\"*VUUs%F1$\"+?(3lm%F-7$$\"*w)yyiF1$\"+vlCUgF-7$$\"*oD[l (F1$\"+!Rg7.(F-7$$\"+q4p4%)F-$\"+OjYHuF-7$$\"*EcX;*F1$\"+I\"=Hp(F-7$$ \"+G)eUM*F-$\"+zp3LxF-7$$\"+&RhR_*F-$\"+!=lRw(F-7$$\"+jRm.(*F-$\"+8fG& y(F-7$$\"+IlO$))*F-$\"+Nd!oz(F-7$$\"+5pI15F1$\"+$*yI)z(F-7$$\"+mrFC5F1 $\"+Kdg*y(F-7$$\"+BuCU5F1$\"+hhaqxF-7$$\"+!o<-1\"F1$\"+!=85u(F-7$$\"+! *z@N6F1$\"+Y<\"Q](F-7$$\"+,$=-@\"F1$\"+P*Qx3(F-7$$\"+(plzM\"F1$\"+H0?U fF-7$$\"+s[a'\\\"F1$\"+H\"RY[%F-7$$\"+v3rt:F1$\"+J***\\$QF-7$$\"+yo(3l \"F1$\"+FpH%Q$F-7$$\"+'=qwm\"F1$\"+Jc$3K$F-7$$\"+%\\jWo\"F1$\"+bpFrKF- 7$$\"+-oD,!G5A$F-7$$\"+M+VoF1$\"+buA2UF-7$$\"+% \\@-3#F1$\"+*H3-l&F-7$$\"+TN)o6#F1$\"+^2sWfF-7$$\"+*eXN:#F1$\"+Ja:yhF- 7$$\"+7m(=<#F1$\"+))o2niF-7$$\"+Ow?!>#F1$\"+4FKNjF-7$$\"+g'Q&3AF1$\"+R +k\"Q'F-7$$\"+$opoA#F1$\"+4<50kF-7$$\"+#*egWAF1$\"+8b]0kF-7$$\"++@MiAF 1$\"+!)fz$Q'F-7$$\"+3$y+G#F1$\"+CH9SjF-7$$\"+:X\"yH#F1$\"+\"fw]F'F-7$$ \"+JpGLBF1$\"+5GF1$\"+@z8i[F-7$$\"+u1 \\()GF1$\"+Ven>bF-7$$\"+(3rf&HF1$\"+XOenfF-7$$\"+$)youHF1$\"+e=CFgF-7$ $\"+yYS$*HF1$\"+X-;bgF-7$$\"+t977IF1$\"+8AG]gF-7$$\"+n#Q3.$F1$\"+_!eD, 'F-7$$\"+d=FoIF1$\"+wKdVeF-7$$\"+Yaq0JF1$\"+6APobF-7$$\"+F'\\h<$F1$\"+ /:I$*[F-7$$\"+4QfYKF1$\"+)\\[aG%F-7$$\"+DC+lKF1$\"+ZMYxTF-7$$\"+U5T$G$ F1$\"+q+K*4%F-7$$\"+f'>=I$F1$\"+3[)R0%F-7$$\"+w#G-K$F1$\"+H;GVSF-7$$\" +$*ojQLF1$\"+wBznSF-7$$\"+5b/dLF1$\"+i-!o7%F-7$$\"+FTXvLF1$\"+')3G=UF- 7$$\"+VF'QR$F1$\"+-N%*QVF-7$$\"+%*Q&eY$F1$\"+pWR$*\\F-7$$\"+X]%y`$F1$ \"+;)yaj&F-7$$\"+gqocNF1$\"+B(>+v&F-7$$\"+w!Hbd$F1$\"+\"fb3$eF-7$$\"+# 4rVf$F1$\"+3R@ueF-7$$\"+3J@8OF1$\"+e9&z(eF-7$$\"+;TjAOF1$\"+yWxkeF-7$$ \"+C^0KOF1$\"+AapTeF-7$$\"+KhZTOF1$\"+!ya*3eF-7$$\"+Sr*3l$F1$\"+'32pw& F-7$$\"+c\"R(pOF1$\"+W9'ol&F-7$$\"+r6e)o$F1$\"+9(Rl^&F-7$$\"+u&p6w$F1$ \"+dzyD[F-7$$\"+vzvLQF1$\"+%)ywqUF-7$$\"+3gJ_QF1$\"+]Y-0UF-7$$\"+RS(3( QF1$\"+X:\\zTF-7$$\"+r?V*)QF1$\"+&4\\c>%F-7$$\"+-,*z!RF1$\"+on'GD%F-7$ $\"+lh5XRF1$\"+9\"4uZ%F-7$$\"+FAA#)RF1$\"+(y_z![F-7$$\"+u'Re0%F1$\"+XD \\(\\&F-7$$\"+>rXHTF1$\"+>**fodF-7$$\"+2R5(>%F1$\"+9AMTaF-7$$\"+%p]ZE% F1$\"+b_I&z%F-7$$\"+?h^.VF1$\"+.AvpWF-7$$\"+Y:GUVF1$\"+Oz:*G%F-7$$\"+f UmhVF1$\"+m'z3F%F-7$$\"+sp/\"Q%F1$\"+W8u.VF-7$$\"+&oH/S%F1$\"+PDk&Q%F- 7$$\"+)R7)>WF1$\"+b-'4^%F-7$$\"+#o&F-7$$\"+EK&Rh% F1$\"+0%3Vi&F-7$$\"+crVKYF1$\"+H_ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 41 " be the function defined on the interval " }{XPPEDIT 18 0 "[-1,1] " "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "f(x)=Fresne lC(x)" "6#/-%\"fG6#%\"xG-%)FresnelCG6#F'" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 13 "(a) Find the " }{TEXT 262 21 "Fourier cosine serie s" }{TEXT -1 27 " for the periodic function " }{XPPEDIT 18 0 "g(u)=f(c os(u))" "6#/-%\"gG6#%\"uG-%\"fG6#-%$cosG6#F'" }{TEXT -1 26 " up as far as the term in " }{XPPEDIT 18 0 "cos(11*u);" "6#-%$cosG6#*&\"#6\"\"\" %\"uGF(" }{TEXT -1 55 ", and plot the graph of this truncated Fourier \+ series. " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 18 ": Use the option \"" }{TEXT 262 12 "mode=numeric" }{TEXT -1 21 "\" with the pro cedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 20 "(b) By substituting " }{XPPEDIT 18 0 "x=cos(u)" "6#/%\" xG-%$cosG6#%\"uG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0<=u" "6#1\"\"!% \"uG" }{XPPEDIT 18 0 "``<=Pi" "6#1%!G%#PiG" }{TEXT -1 19 ", or equival ently, " }{XPPEDIT 18 0 "u=arccos(x)" "6#/%\"uG-%'arccosG6#%\"xG" } {TEXT -1 85 ", in the truncated Fourier series found in (a), construct a polynomial approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 19 " for the function " }{XPPEDIT 18 0 "f(x)=FresnelC(x)" " 6#/-%\"fG6#%\"xG-%)FresnelCG6#F'" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 142 "(c) Plot an absolute error curve for the polyn omial approximation found in (b), and give an estimate for the absolut e error involved in using " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\" \"!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "(d) Is the \+ function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 77 " an e ven or odd function? How does this affect the approximating polynomia l " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT -1 29 "_____________________________" }}{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 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 29 "_____________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 262 21 "Fresnel sine f unction" }{TEXT -1 1 " " }{XPPEDIT 18 0 "S(x);" "6#-%\"SG6#%\"xG" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "FresnelS(x);" "6#-%)FresnelSG6#%\"xG " }{TEXT -1 15 " is defined by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "S(x) = Int(sin(Pi/2*t^2),t = 0 .. x);" "6#/-%\"SG6#%\"x G-%$IntG6$-%$sinG6#*(%#PiG\"\"\"\"\"#!\"\"%\"tGF1/F3;\"\"!F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 44 "and is available in Maple via the procedure " }{TEXT 0 8 "FresnelS" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Int(sin (Pi/2*t^2),t=0..x);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$IntG6$-%$sinG6#,$*(\"\"#!\"\"%#PiG\"\"\"%\"tGF+F./F/;\"\"!%\"xG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%)FresnelSG6#%\"xG" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot( FresnelS(x),x=-2..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 386 266 266 {PLOTDATA 2 "6%-%'CURVESG6$7et7$$!\"#\"\"!$!+%ycTV$!#57$$!+@*>u%=!\"*$ !+AOK(4%F-7$$!+53m986o'F-7$$!+= G:'>\"F1$!+M&*=/iF-7$$!+nV\"o7\"F1$!+'RUUh&F-7$$!+=%=]0\"F1$!+Oj#*H\\F -7$$!*oCA$)*F1$!+`G*[@%F-7$$!+]*4(*3*F-$!+gR$Q[$F-7$$!*A&>Z$)F1$!+9_S% z#F-7$$!+S<12wF-$!+vFQs@F-7$$!*EGp'oF1$!+\"fg,j\"F-7$$!*v\"HW`F1$!+[$y !yy!#67$$!+5AstYF-$!+wKq+`Fas7$$!*n_J+%F1$!+)o*yVLFas7$$!++-D[KF-$!+\\ p*4z\"Fas7$$!*tZL\\#F1$!+!Qp06)!#77$$!+bbMN.CJ?Fas7$$\"+!RWg0%F-$\"+\"GRsZ$Fas7$$\"*VUUs%F1$\"+TgVsaFas7$$\"*w)y yiF1$\"+!f#*4E\"F-7$$\"*oD[l(F1$\"+'oH-@#F-7$$\"+q4p4%)F-$\"+hp<]GF-7$ $\"*EcX;*F1$\"+O*>hb$F-7$$\"+IlO$))*F-$\"+tQ)fE%F-7$$\"+!o<-1\"F1$\"+A $>5)\\F-7$$\"+!*z@N6F1$\"+KDK!p&F-7$$\"+,$=-@\"F1$\"+,]Y6jF-7$$\"+**>4 z7F1$\"+QMxdnF-7$$\"+(plzM\"F1$\"+@NTWqF-7$$\"+%4QlO\"F1$\"+j#y**3(F-7 $$\"+!\\5^Q\"F1$\"+091@rF-7$$\"+()Go.9F1$\"+I_EPrF-7$$\"+%GbAU\"F1$\"+ \")HGQrF-7$$\"+\"oF3W\"F1$\"+cu!R7(F-7$$\"+y+Sf9F1$\"+L#RS4(F-7$$\"+vC (zZ\"F1$\"+%Q%p[qF-7$$\"+s[a'\\\"F1$\"+c:,))pF-7$$\"+v3rt:F1$\"+;3')zl F-7$$\"+yo(3l\"F1$\"+F;bffF-7$$\"+ULA&y\"F1$\"+)*Q))[YF-7$$\"+uAxd=F1$ \"+?!fn,%F-7$$\"+07KI>F1$\"+rCf#e$F-7$$\"+U(e!\\>F1$\"+=mJ9NF-7$$\"+yi zn>F1$\"+_QXmMF-7$$\"+9Q`')>F1$\"+Mv$)RMF-7$$\"+]8F0?F1$\"+H/.NMF-7$$ \"+'))3S-#F1$\"+6MI_MF-7$$\"+AkuU?F1$\"+qvh\"\\$F-7$$\"+eR[h?F1$\"+D.h _NF-7$$\"+%\\@-3#F1$\"+)3'eMOF-7$$\"+*eXN:#F1$\"+hw#H8%F-7$$\"+$opoA#F 1$\"+Vi+A[F-7$$\"+:X\"yH#F1$\"+#3:=^&F-7$$\"+Y$f(oBF1$\"+\"[R!\\gF-7$$ \"+\"Q_%)Q#F1$\"+#*)z(\\hF-7$$\"+:a93CF1$\"+D&QXA'F-7$$\"+]%QyU#F1$\"+ BSXriF-7$$\"+&[JvW#F1$\"+z!\\#*G'F-7$$\"+>XAnCF1$\"+*f%GxiF-7$$\"+av\" p[#F1$\"+?5gNiF-7$$\"+*e5m]#F1$\"+:[%\\;'F-7$$\"+DOIEDF1$\"+2%yn1'F-7$ $\"+'*Q3(f#F1$\"+Dh(o_&F-7$$\"+nT'ym#F1$\"+h&4L$[F-7$$\"+9sVVFF1$\"+\" GdH=%F-7$$\"+h-,>GF1$\"+&4BO)QF-7$$\"+k.8OGF1$\"+*oFB)QF-7$$\"+n/D`GF1 $\"+**\\.2RF-7$$\"+r0PqGF1$\"+$=@t&RF-7$$\"+u1\\()GF1$\"+m`9KSF-7$$\"+ \")3t@HF1$\"+Ee@[UF-7$$\"+(3rf&HF1$\"+=BBNXF-7$$\"+n#Q3.$F1$\"+Cg:n_F- 7$$\"+Yaq0JF1$\"+q5l\\eF-7$$\"+\"\\;L7$F1$\"+SVXHfF-7$$\"+Pv#49$F1$\"+ *f;6)fF-7$$\"+#eQ&eJF1$\"+j(HH+'F-7$$\"+F'\\h<$F1$\"+dk1%*fF-7$$\"+=

$e)HMF1$\"+;F) e9%F-7$$\"+2h&yW$F1$\"+iWJ(4%F-7$$\"+%*Q&eY$F1$\"+?/=$3%F-7$$\"+\"o^Q[ $F1$\"+2l>/TF-7$$\"+p%\\=]$F1$\"+25tfTF-7$$\"+cs%)>NF1$\"+6VzZUF-7$$\" +X]%y`$F1$\"+yF3lVF-7$$\"+3J@8OF1$\"+SJ4b]F-7$$\"+r6e)o$F1$\"+Kr%)*o&F -7$$\"+s#Gnq$F1$\"+4&H*ydF-7$$\"+t`([s$F1$\"+Mw%H$eF-7$$\"+tC-VPF1$\"+ _GJ\\eF-7$$\"+u&p6w$F1$\"+R`5FeF-7$$\"+vPY(z$F1$\"+$ob?n&F-7$$\"+vzvLQ F1$\"+**o'\\R&F-7$$\"+-,*z!RF1$\"+1%=yn%F-7$$\"+FAA#)RF1$\"+nS!\\A%F-7 $$\"+!fE1+%F1$\"+7A=0UF-7$$\"+^4.>SF1$\"+@5%yA%F-7$$\"+8`VPSF1$\"+z\\& =H%F-7$$\"+u'Re0%F1$\"+iP%RR%F-7$$\"+(R[E4%F1$\"+yL%))o%F-7$$\"+>rXHTF 1$\"+\\B1\\]F-7$$\"+90GjTF1$\"+)\\]$p`F-7$$\"+2R5(>%F1$\"+yv+;cF-7$$\" +/c,9UF1$\"+nnt&p&F-7$$\"+,t#4B%F1$\"+R/XSdF-7$$\"+(**QyC%F1$\"+LjwZdF -7$$\"+%p]ZE%F1$\"+s%orr&F-7$$\"+Y:GUVF1$\"+H5$p<&F-7$$\"+)R7)>WF1$\"+ :k&>Z%F-7$$\"+F&[rV%F1$\"+$*)RGP%F-7$$\"+dY[aWF1$\"+P0_5VF-7$$\"+'y?=Z %F1$\"+H1y)G%F-7$$\"+&F-7$$\"+7Z!z%[ F1$\"+MfjSXF-7$$\"+Bm\"p'[F1$\"+&o]zU%F-7$$\"+M&Gf)[F1$\"+h^kjVF-7$$\" +X/%\\!\\F1$\"+MqK`VF-7$$\"+cB&R#\\F1$\"+N[2)R%F-7$$\"+yh(>'\\F1$\"+&p fPj%F-7$$\"\"&F*$\"+>Q\">*\\F--%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F\\\\n- %+AXESLABELSG6$Q\"x6\"Q!Fa\\n-%%VIEWG6$;F(Fa[n%(DEFAULTG" 1 2 0 1 10 0 2 9 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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 41 " be the function defined on the interval " } {XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 4 " by " } {XPPEDIT 18 0 "f(x) = FresnelS(x);" "6#/-%\"fG6#%\"xG-%)FresnelSG6#F' " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "(a) Find the " } {TEXT 262 21 "Fourier cosine series" }{TEXT -1 27 " for the periodic f unction " }{XPPEDIT 18 0 "g(u)=f(cos(u))" "6#/-%\"gG6#%\"uG-%\"fG6#-%$ cosG6#F'" }{TEXT -1 26 " up as far as the term in " }{XPPEDIT 18 0 "co s(11*u);" "6#-%$cosG6#*&\"#6\"\"\"%\"uGF(" }{TEXT -1 55 ", and plot th e graph of this truncated Fourier series. " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 18 ": Use the option \"" }{TEXT 262 12 "mode=nu meric" }{TEXT -1 21 "\" with the procedure " }{TEXT 0 13 "FourierSerie s" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "(b) By substituting " }{XPPEDIT 18 0 "x=cos(u)" "6#/%\"xG-%$cosG6#%\"uG" }{TEXT -1 5 " fo r " }{XPPEDIT 18 0 "0<=u" "6#1\"\"!%\"uG" }{XPPEDIT 18 0 "``<=Pi" "6#1 %!G%#PiG" }{TEXT -1 19 ", or equivalently, " }{XPPEDIT 18 0 "u=arccos( x)" "6#/%\"uG-%'arccosG6#%\"xG" }{TEXT -1 85 ", in the truncated Fouri er series found in (a), construct a polynomial approximation " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 19 " for the function " }{XPPEDIT 18 0 "f(x) = FresnelS(x);" "6#/-%\"fG6#%\"xG-%)FresnelSG 6#F'" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$ \"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 142 "(c) Plo t an absolute error curve for the polynomial approximation found in (b ), and give an estimate for the absolute error involved in using " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " in the interv al " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 20 "(d) Is the function " }{XPPEDIT 18 0 "f(x )" "6#-%\"fG6#%\"xG" }{TEXT -1 77 " an even or odd function? How does this affect the approximating polynomial " }{XPPEDIT 18 0 "p(x)" "6#- %\"pG6#%\"xG" }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT -1 29 "________ _____________________" }}{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 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 29 "__ ___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 262 20 "exponential integral" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Ei(x);" "6#-%#EiG6#%\"xG" }{TEXT -1 4 " or " } {XPPEDIT 18 0 "Ei(1,x);" "6#-%#EiG6$\"\"\"%\"xG" }{TEXT -1 15 " is def ined by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Ei(x) = I nt(exp(-x*t)/t,t = 1 .. infinity);" "6#/-%#EiG6#%\"xG-%$IntG6$*&-%$exp G6#,$*&F'\"\"\"%\"tGF1!\"\"F1F2F3/F2;F1%)infinityG" }{TEXT -1 2 ", " } }{PARA 0 "" 0 "" {TEXT -1 44 "and is available in Maple via the proced ure " }{TEXT 0 7 "Ei(1,x)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "assume(x_,positive );\nInt(exp(-x*t)/t,t=1..infinity);\n``=subs(x_=x,value(subs(x=x_,%))) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#,$*&%\"xG\"\" \"%\"tGF-!\"\"F-F.F//F.;F-%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%#EiG6$\"\"\"%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(Ei(1,x),x=0..3,y=0..2);" }} {PARA 13 "" 1 "" {GLPLOT2D 317 246 246 {PLOTDATA 2 "6%-%'CURVESG6$7ao7 $$\"+-K[V?!#7$\"+]i#zh&!\"*7$$\"+0k'p3%F*$\"+C%>o#\\F-7$$\"+2'\\/8'F*$ \"+)e#RBXF-7$$\"+5G$R<)F*$\"+4nuPUF-7$$\"+-;u@5!#6$\"+tsj;SF-7$$\"+A** 3E7F@$\"+fxMOQF-7$$\"+U#Q/V\"F@$\"+_qA%o$F-7$$\"+ilyM;F@$\"+jNs_NF-7$$ \"+.K[V?F@$\"+E'HOL$F-7$$\"+V)z@X#F@$\"+W%\\`:$F-7$$\"+%[w3'GF@$\"+C>B 0IF-7$$\"+DJdpKF@$\"+'\\Dd(GF-7$$\"+1k'p3%F@$\"+#Q21m#F-7$$\"+(ofV!\\F @$\"+_(yi[#F-7$$\"+pHv@dF@$\"+k')3SBF-7$$\"+]i9RlF@$\"+/d[9AF-7$$\"+(H l:'zF@$\"+3vQJ?F-7$$\"+XV)RQ*F@$\"+bNj!)=F-7$$\"+R.k!3\"!#5$\"+V*GIv\" F-7$$\"+WA)GA\"Faq$\"+&z2Gk\"F-7$$\"+TS\"Ga\"Faq$\"+b.HS9F-7$$\"+Qeui= Faq$\"+,ZE\"G\"F-7$$\"+]$)z%=#Faq$\"+;)[4:\"F-7$$\"+i3&o]#Faq$\"++@:U5 F-7$$\"+:FPFGFaq$\"+/ql*\\*Faq7$$\"+pX*y9$Faq$\"+Vb(Gq)Faq7$$\"+WTAUPF aq$\"+v%*>wuFaq7$$\"+%*zhdVFaq$\"+HlzfkFaq7$$\"+%>fS*\\Faq$\"+P*[\\g&F aq7$$\"+>$f%GcFaq$\"+l_G,\\Faq7$$\"+Dy,\"G'Faq$\"+.,1'H%Faq7$$\"+7Jh,MFaq7$$\"+Drc_\")Faq$\"+LQs@IF aq7$$\"+D!*oy()Faq$\"+z\")>/FFaq7$$\"+%pnsM*Faq$\"+y&R0X#Faq7$$\"+siL- 5F-$\"+?ZE&=#Faq7$$\"+!R5'f5F-$\"+pt*p)>Faq7$$\"+/QBE6F-$\"+34Z#y\"Faq 7$$\"+:o?&=\"F-$\"+*fBNq07Faq7$$\"+UC%[V\"F-$\"+LBh-6Faq7$$ \"+J#>&)\\\"F-$\"+S7S-5Faq7$$\"+>:mk:F-$\"+MN%)*3*F@7$$\"+w&QAi\"F-$\" +hj@b$)F@7$$\"+uLU%o\"F-$\"+qB%\\j(F@7$$\"+bjm[F-$\"+rm5;`F@7$$\" +:K^+?F-$\"+-'zl)[F@7$$\"+7,Hl?F-$\"+M77pWF@7$$\"+4w)R7#F-$\"+T3+CTF@7 $$\"+y%f\")=#F-$\"+m`EzPF@7$$\"+/-a[AF-$\"+5G0$[$F@7$$\"+ial6BF-$\"+w= %)*>$F@7$$\"+i@OtBF-$\"+c4kYHF@7$$\"+fL'zV#F-$\"+XEJ/FF@7$$\"+!*>=+DF- $\"+5pi#F-$\"+*oh&4@F@7$$ \"+bJ*[o#F-$\"+Z;=c>F@7$$\"+r\"[8v#F-$\"+vEw%z\"F@7$$\"+Ijy5GF-$\"+\"> uAm\"F@7$$\"+/)fT(GF-$\"+!Q#HK:F@7$$\"+1j\"[$HF-$\"+TU'yT\"F@7$$\"\"$ \"\"!$\"+4\"Q[I\"F@-%'COLOURG6&%$RGBG$\"#5!\"\"$F\\alF\\alFfal-%+AXESL ABELSG6$Q\"x6\"Q\"yF[bl-%%VIEWG6$;FfalFj`l;Ffal$\"\"#F\\al" 1 2 0 1 10 0 2 9 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 4 "Le t " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 41 " be the func tion defined on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"! \"\"F%" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "f(x) = Ei(1,x/2+1);" "6#/-% \"fG6#%\"xG-%#EiG6$\"\"\",&*&F'F+\"\"#!\"\"F+F+F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 13 "(a) Find the " }{TEXT 262 21 "Fourier cos ine series" }{TEXT -1 27 " for the periodic function " }{XPPEDIT 18 0 "g(u)=f(cos(u))" "6#/-%\"gG6#%\"uG-%\"fG6#-%$cosG6#F'" }{TEXT -1 26 " \+ up as far as the term in " }{XPPEDIT 18 0 "cos(11*u);" "6#-%$cosG6#*& \"#6\"\"\"%\"uGF(" }{TEXT -1 55 ", and plot the graph of this truncate d Fourier series. " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 18 ": Use the option \"" }{TEXT 262 12 "mode=numeric" }{TEXT -1 21 "\" wi th the procedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 20 "(b) By substituting " }{XPPEDIT 18 0 "x=c os(u)" "6#/%\"xG-%$cosG6#%\"uG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0< =u" "6#1\"\"!%\"uG" }{XPPEDIT 18 0 "``<=Pi" "6#1%!G%#PiG" }{TEXT -1 19 ", or equivalently, " }{XPPEDIT 18 0 "u=arccos(x)" "6#/%\"uG-%'arcc osG6#%\"xG" }{TEXT -1 85 ", in the truncated Fourier series found in ( a), construct a polynomial approximation " }{XPPEDIT 18 0 "p(x)" "6#-% \"pG6#%\"xG" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) = \+ Ei(1,x/2+1);" "6#/-%\"fG6#%\"xG-%#EiG6$\"\"\",&*&F'F+\"\"#!\"\"F+F+F+ " }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\" \"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 142 "(c) Plot an absolute error curve for the polynomial approximation found in (b), a nd give an estimate for the absolute error involved in using " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " in the interv al " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 29 "_____________________________" }}{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 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 29 "_____________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q8" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 262 13 "s ine integral" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Si(x);" "6#-%#SiG6#%\"xG " }{TEXT -1 15 " is defined by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Si(x)=Int(sin(t)/t,t=0..x)" "6#/-%#SiG6#%\"xG-%$IntG6$* &-%$sinG6#%\"tG\"\"\"F/!\"\"/F/;\"\"!F'" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 44 "and is available in Maple via the procedure " } {TEXT 0 5 "Si(x)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Int(sin(t)/t,t=0..x);\n``=va lue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$sinG6#%\"tG\" \"\"F*!\"\"/F*;\"\"!%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%#Si G6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(Si(x),x=-25..25);" }}{PARA 13 "" 1 "" {GLPLOT2D 450 194 194 {PLOTDATA 2 "6%-%'CURVESG6$7gt7$$!#D\"\"!$!+^D[J:!\"*7$$!+ HU,\"R#!\")$!+:D?e:F-7$$!+$H'='H#F1$!+Rl%of\"F-7$$!+EV'GC#F1$!+!\\e=h \"F-7$$!+gBa*=#F1$!+Jn(eh\"F-7$$!+Up'e8#F1$!+f#=rg\"F-7$$!+B:>#3#F1$!+ A&)Q(e\"F-7$$!+04Nv>F1$!+OmlP:F-7$$!+SM#e#>F1$!+n4MA:F-7$$!+wfHw=F1$!+ iLB=:F-7$$!+bJ,D=F1$!+,E[F:F-7$$!+M.tt;F-7$$!+C,z9;F1$!+PL,G;F -7$$!+'yc$)e\"F1$!+FI*Hj\"F-7$$!+ZM#>c\"F1$!+lMrL;F-7$$!+4NtM:F1$!+wh! )H;F-7$$!+rNa2:F1$!+N*48i\"F-7$$!+LON![\"F1$!+ZLp3;F-7$$!+&pjJX\"F1$!+ yyu#f\"F-7$$!+seE09F1$!+c=%*f:F-7$$!+[!otN\"F1$!+5eRF:F-7$$!+XsSI8F1$! +;937:F-7$$!+UkW.8F1$!+<`^+:F-7$$!+Sc[w7F1$!+ytq$\\\"F-7$$!+P[_\\7F1$! +pJO#\\\"F-7$$!+ILXA7F1$!+A1&o\\\"F-7$$!+C=Q&>\"F1$!+[Y62:F-7$$!+=.Jo6 F1$!+%4bE_\"F-7$$!+7)Q79\"F1$!+v2cU:F-7$$!+()>1*3\"F1$!+\"e#G)e\"F-7$$ !+i^)o.\"F1$!+4OdL;F-7$$!+%Q%>85F1$!+^T^];F-7$$!+lg.&*)*F-$!+=Chj;F-7$ $!+(GG\"e'*F-$!+cE$>n\"F-7$$!*^?7U*F1$!+F6wu;F-7$$!+xtO!G*F-$!+]fkt;F- 7$$!+XU^R\"*F-$!+%[&Qq;F-7$$!+76m)**)F-$!+s'y\\m\"F-7$$!+!)z!y&))F-$!+ ulYd;F-7$$!+:<5w&)F-$!+FO]O;F-7$$!*X&R%H)F1$!+Mtm3;F-7$$!+S666\\\"F-7$$!+q#HA1(F-$!+W[hg9F-7$$!+5(p9F-7$$!*3`lC&F1$!+ro*e]\"F-7$$!+&ejtq%F-$!+T(R%4;F-7$$!*4u\"oTF1 $!+Jw$\\s\"F-7$$!+NhX6RF-$!+$)=Zu&=F-7$$!+VJP2IF-$!+x**)*[=F-7$$!+`bFibl$!+J4$*eaFibl7$$!+]U****GFibl$!+A!zk )GFibl7$$!(ZzY#F1$!+#\\jyY#!#67$$\"+vR84DFibl$\"+%[u.]#Fibl7$$\"+]E1l_ Fibl$\"+\\#\\Y=&Fibl7$$\"+D8*4-)Fibl$\"+i^xRxFibl7$$\"*+#px5F1$\"+#GJ0 ,\"F-7$$\"++3]d:F-$\"+*>iAO\"F-7$$\"*g4t.#F1$\"+`D2A;F-7$$\"+DHT'H#F-$ \"+6Y\"GF-$\"+2_!Q$=F-7$$\"+Q7^=F -7$$\"+XCc2KF-$\"+,UD^=F-7$$\"+!*>STLF-$\"+m-'e%=F-7$$\"+N:CvMF-$\"+gH `N=F-7$$\"+!3\"34OF-$\"+:I#3#=F-7$$\"+q,wwQF-$\"+P6c!y\"F-7$$\"*ERW9%F 1$\"+UwyH>&F1$\"+;m/::F-7$$\"+?2 IXaF-$\"+)\\mfZ\"F-7$$\"+?^n)p&F-$\"+rcKY9F-7$$\"+?BODeF-$\"+o6QN9F-7$ $\"+?&\\?&fF-$\"+-O6F9F-7$$\"+?ntygF-$\"+P*R:U\"F-7$$\"*#RU0iF1$\"+Ilj =9F-7$$\"+b%)3YjF-$\"+gUY=9F-7$$\"+!*Hv'['F-$\"+D'o8U\"F-7$$\"+DvTFmF- $\"+k9;F9F-7$$\"+g?3onF-$\"+*R4cV\"F-7$$\"+I6T\\qF-$\"+qnMf9F-7$$\"*?S 2L(F1$\"+#\\V+\\\"F-7$$\"+lNJOyF-$\"+-!zNb\"F-7$$\"*$p)=M)F1$\"+[Ax8;F -7$$\"+q1z6')F-$\"+;.dR;F-7$$\"+5Wp\")))F-$\"+%o&))e;F-7$$\"+!GYm,*F-$ \"+xvyl;F-7$$\"+]\")f^\"*F-$\"+S'\\2n\"F-7$$\"+?+b'G*F-$\"+a(RPn\"F-7$ $\"**=]@%*F1$\"+H7wu;F-7$$\"+$zugm*F-$\"++$R?:F- 7$$\"+O(\\s>\"F1$\"+'*)Gi]\"F-7$$\"+s$3CA\"F1$\"+eH'o\\\"F-7$$\"+3qcZ7 F1$\"+4**[#\\\"F-7$$\"+K]'QF\"F1$\"+(fGL\\\"F-7$$\"+cI;+8F1$\"+8XT*\\ \"F-7$$\"+!3hkK\"F1$\"+rt75:F-7$$\"+/\"fFN\"F1$\"+xKcC:F-7$$\"+a8=/9F1 $\"+HH*e\"F1$\"+&[)*Gj\"F-7$$\"+E67:;F1$\"+Td#zi\"F-7$$\"+cb/T;F1$ \"+]JK>;F-7$$\"+&)*ppm\"F1$\"+[/x2;F-7$$\"+KH**>F1$\"+EA[A:F-7$$\"+E>#[(>F1$\"+CmWP:F-7$$\"+(G!e&3#F1$\"+&Gu) )e\"F-7$$\"+'37^8#F1$\"+E#3pg\"F-7$$\"+&)Qk%=#F1$\"+K\"3ch\"F-7$$\"+8^ XPAF1$\"+Y=#Gh\"F-7$$\"+UjE!H#F1$\"+>E$*)f\"F-7$$\"+60O\"R#F1$\"+wk1e: F-7$$\"#DF*$\"+^D[J:F--%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fd\\n-%+AXESLAB ELSG6$Q\"x6\"Q!Fi\\n-%%VIEWG6$;F(Fi[n%(DEFAULTG" 1 2 0 1 10 0 2 9 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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 41 " be the function defined on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "f(x) = Si(4*x);" "6#/-%\"fG6#%\"xG-%#SiG6#*&\" \"%\"\"\"F'F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "(a) Fin d the " }{TEXT 262 21 "Fourier cosine series" }{TEXT -1 27 " for the p eriodic function " }{XPPEDIT 18 0 "g(u)=f(cos(u))" "6#/-%\"gG6#%\"uG-% \"fG6#-%$cosG6#F'" }{TEXT -1 26 " up as far as the term in " } {XPPEDIT 18 0 "cos(11*u);" "6#-%$cosG6#*&\"#6\"\"\"%\"uGF(" }{TEXT -1 55 ", and plot the graph of this truncated Fourier series. " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 18 ": Use the option \"" } {TEXT 262 12 "mode=numeric" }{TEXT -1 21 "\" with the procedure " } {TEXT 0 13 "FourierSeries" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "(b) By substituting " }{XPPEDIT 18 0 "x=cos(u)" "6#/%\"xG-%$cos G6#%\"uG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0<=u" "6#1\"\"!%\"uG" } {XPPEDIT 18 0 "``<=Pi" "6#1%!G%#PiG" }{TEXT -1 19 ", or equivalently, \+ " }{XPPEDIT 18 0 "u=arccos(x)" "6#/%\"uG-%'arccosG6#%\"xG" }{TEXT -1 85 ", in the truncated Fourier series found in (a), construct a polyno mial approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) = Si(4*x);" "6#/-%\"fG 6#%\"xG-%#SiG6#*&\"\"%\"\"\"F'F-" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 142 "(c) Plot an absolute error curve for the polyn omial approximation found in (b), and give an estimate for the absolut e error involved in using " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\" \"!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "(d) Is the \+ function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 77 " an e ven or odd function? How does this affect the approximating polynomia l " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT -1 29 "_____________________________" }}{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 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 29 "_____________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }