{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 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 "Tim es" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "Grey Emphasis" -1 265 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 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 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 "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 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 "Ma ple 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 "Time s" 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 "Times" 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 "T imes" 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 65 "Evaluating the inverse tangent fu nction from its Maclaurin series" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Pe ter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Versi on: 25.3.2007" }}{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 50 "Graphing partial sums \+ of the Maclaurin series for " }{XPPEDIT 18 0 "arctan(x);" "6#-%'arctan G6#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 25 "The Maclaurin series for " } {XPPEDIT 18 0 "arctan(x);" "6#-%'arctanG6#%\"xG" }{TEXT -1 5 " is " } {XPPEDIT 18 0 "Sum((-1)^n*x^(2*n+1)/(2*n+1),n = 0 .. infinity) = x-x^3 /3+x^5/5-x^7/7+` . . . `;" "6#/-%$SumG6$*(),$\"\"\"!\"\"%\"nGF*)%\"xG, &*&\"\"#F*F,F*F*F*F*F*,&*&F1F*F,F*F*F*F*F+/F,;\"\"!%)infinityG,,F.F**& F.\"\"$F:F+F+*&F.\"\"&FF+F+%(~.~.~.~GF*" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 54 "We can make a comparison of finite \+ sums approximating " }{XPPEDIT 18 0 "arctan(x);" "6#-%'arctanG6#%\"xG " }{TEXT -1 24 " graphically as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "First we define a function " } {TEXT 0 2 "AT" }{TEXT -1 84 " with two arguments or input parameters. \+ The degree of the polynomial approximation " }{TEXT 265 7 "AT(n,x)" } {TEXT -1 4 " to " }{XPPEDIT 18 0 "arctan(x);" "6#-%'arctanG6#%\"xG" } {TEXT -1 4 " is " }{XPPEDIT 18 0 "2*n+1;" "6#,&*&\"\"#\"\"\"%\"nGF&F&F &F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "x := 'x': i := 'i': n := 'n':\nAT := (n,x) -> sum((-1)^i*x^(2*i+1)/(2*i+1),i=0..n);\nAT(1,x);\nAT(2,x);\nAT(3,x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ATGf*6$%\"nG%\"xG6\"6$%)operat orG%&arrowGF)-%$sumG6$*()!\"\"%\"iG\"\"\")9%,&F3\"\"#F4F4F4F7F2/F3;\" \"!9$F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"*&#F%\"\"$ F%*$)F$F(F%F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"xG\"\"\"*&# F%\"\"$F%*$)F$F(F%F%!\"\"*&#F%\"\"&F%)F$F.F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"xG\"\"\"*&#F%\"\"$F%*$)F$F(F%F%!\"\"*&#F%\"\"&F%)F $F.F%F%*&#F%\"\"(F%*$)F$F2F%F%F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "evalf(AT(12,0.5));\nevalf(ar ctan(0.5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#4wkj%!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!4wkj%!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The degree 1, 3 and 5 approximatio ns " }{XPPEDIT 18 0 "y = x;" "6#/%\"yG%\"xG" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "y = x-x^3/3;" "6#/%\"yG,&%\"xG\"\"\"*&F&\"\"$F)!\"\"F* " }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = x-x^3/3+x^5/5;" "6#/%\"yG,( %\"xG\"\"\"*&F&\"\"$F)!\"\"F**&F&\"\"&F,F*F'" }{TEXT -1 28 " can be p lotted as follows." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = arctan(x);" "6#/%\"yG-%'arctanG6#%\"xG" }{TEXT -1 33 " is also plot ted for comparison. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "AT := (n, x) -> sum((-1)^i*x^(2*i+1)/(2*i+1),i=0..n):\nplot([arctan(x),AT(0,x),A T(1,x),AT(2,x)],x=-1..1,-1..1,\n color=[black,red,green,blue],l inestyle=[2,1$3],\nnumpoints=100);" }}{PARA 13 "" 1 "" {GLPLOT2D 526 299 299 {PLOTDATA 2 "6(-%'CURVESG6%7`q7$$!\"\"\"\"!$!3!G[uRj\")R&y!#=7 $$!3?ccccOj)y*F-$!3+/2$4qtru(F-7$$!35JJJJ\\s/'*F-$!3u>kY*f'Q_wF-7$$!33 NNNN.!zR*F-$!3ef_=8qoVvF-7$$!3/srrrZq*=*F-$!3L(=!GVr(>V(F-7$$!3oDDDD() \\#)*)F-$!3WN$flQt%=tF-7$$!3S&fff>#R!z)F-$!3$[pQ-DJ6@(F-7$$!3Q6777qZ\" f)F-$!3)yHX[h3y4(F-7$$!3IWWWW'edQ)F-$!3)zv=eiU#ypF-7$$!3#z\"===+q!=)F- $!3GjuE\"GDm&oF-7$$!3!fhhh,s(pzF-$!39ggSF0&*GnF-7$$!33<<<>>n*\\d'F-$!30'H*QPMH;eF-7$$ !3S3444$['fjF-$!3=+V+1JXkcF-7$$!3Ixwww%G!phF-$!3zKN_MWbFbF-7$$!3G3444$ )))ffF-$!3UyqS7*[YP&F-7$$!3s5777utgdF-$!3Wi:7/&3kA&F-7$$!3%GMMMMWHb&F- $!3a_r_i%*)*o]F-7$$!3sRSSSa7i`F-$!3+aVmeC'>#\\F-7$$!3Yuuuu9Ic^F-$!3'Hr .n'RtgZF-7$$!3&)>???#3D%\\F-$!3#>Pz\"zpP!f%F-7$$!3wjjjj0ScZF-$!333v3aV qRWF-7$$!3?&fff>*RbXF-$!3y_&\\Bk8XF%F-7$$!3_9:::^uZVF-$!3wI#4lI075%F-7 $$!3#*3444hfWTF-$!3p1&)*)*y!4HRF-7$$!3')*********Q![RF-$!3)yMxFq*=gPF- 7$$!3nTUUUezHPF-$!3M'zeMr\")*pNF-7$$!3jeeeeYpLNF-$!3!z5?N6MnR$F-7$$!34 (yyyy9VK$F-$!31NXAemQ4KF-7$$!3[/000beMJF-$!3'GAk6+zv.$F-7$$!3/%[[[Ghr# HF-$!39@[d,-hZGF-7$$!3ceeee7*>t#F-$!35e:+H=\"pm#F-7$$!3qWXXXB)z_#F-$!3 C9<*pc0hZ#F-7$$!3%GMMMME&GBF-$!3Ug=&*e+w(G#F-7$$!3l$RRRR9(>@F-$!39<2OY hz)3#F-7$$!3%z'oooOg=>F-$!3Sl&o$)zob*=F-7$$!3&R]]]!*QHr\"F-$!31^z&3Nuk p\"F-7$$!3xZ\\\\\\r(*3:F-$!3I![xg#zn(\\\"F-7$$!35yyyyub@8F-$!3Y9\"y,QV RJ\"F-7$$!3H%eeeQ`n5\"F-$!3(y$3&QUnA5\"F-7$$!3#HVWWW+j9*!#>$!3,Sm*pHB4 7*Fgy7$$!3u+444\\'y4(Fgy$!3gWNR.7)f3(Fgy7$$!3/8FFF2DP^Fgy$!3F#3$[r&QF8 &Fgy7$$!3;HYYY1CgHFgy$!33d*)37kPfHFgy7$$!37syyy)4^2\"Fgy$!3oi`1f%o]2\" Fgy7$$\"3A?////`k5Fgy$\"35-h-&>!\\k5Fgy7$$\"3QZUUU-e9IFgy$\"3ts+$RbnO, $Fgy7$$\"3;F...$Gw9&Fgy$\"3r'QTkx)3V^Fgy7$$\"3=]SSS!)3))pFgy$\"3O;Mb*= Yn(pFgy7$$\"3/q[[[o:%3*Fgy$\"35V\"QJy\"Hf!*Fgy7$$\"3w&QQQe&)36\"F-$\"3 ;@uCM%\\j5\"F-7$$\"3j9999KA88F-$\"3F-$\"3g0I.>o$[*=F-7$$\"3*4111E!Q<@F-$\"3kJFiuAc'3#F-7$$\"3z&QQQ)>QF BF-$\"3_\\kGGXn'G#F-7$$\"3:9888T[qS&[NiY#F-7$$\"3%[NNNb#eFFF -$\"3-)H$*f')3Gm#F-7$$\"37\\[[[-#)GHF-$\"3%**o^/x=qgbR$F-7$$\"3z00004;IPF-$\"3O&*39*=-.d$F-7$$\"3w#444HS&[RF- $\"3aQIt'QB1w$F-7$$\"39QPPP\\EYTF-$\"31t1,#*[^IRF-7$$\"3;ECCCaZ[VF-$\" 3Uw7Y'\\>=5%F-7$$\"3y$333G=Tb%F-$\"3[AER1GXtUF-7$$\"3f&GGG3WJu%F-$\"3' o?$)\\1)))GWF-7$$\"3Kpmmm%RY%\\F-$\"3u!=VRm*3#f%F-7$$\"3daaaa5gW^F-$\" 3;xbQ2m[^ZF-7$$\"3')*zzzz(\\e`F-$\"3k#**Q*)\\W\">\\F-7$$\"3A2333CDZbF- $\"3G*p)\\\\zjk]F-7$$\"3%[WWW/O]w&F-$\"38*p%QUajH_F-7$$\"3S(eee)*o:'fF -$\"3yfC4()*))eP&F-7$$\"3mgeee!Gg:'F-$\"3%p.EWDK\"=bF-7$$\"3%>BBB.M^O' F-$\"3!GXO14e$ocF-7$$\"3K%GGG)e3vlF-$\"3*H;\\*)obj\"eF-7$$\"3/zzzz46mn F-$\"3'o:d(Q`U=iF-7$$\"3)GDDDDY:Q(F-$\"3QQi#3mo(ejF-7$$\"3apmmmyBnvF-$ \"3E,yLBY!zZ'F-7$$\"3SPOOO9^!y(F-$\"3dJ[XUZ87mF-7$$\"330///)**4)zF-$\" 38kID%>8et'F-7$$\"3Ousss;bz\")F-$\"3E2u&oHPf&oF-7$$\"3M\"444\\h#y$)F-$ \"3_/Hk?#RQ(pF-7$$\"3gGFFFH$4e)F-$\"3)*eOYI\"R<4(F-7$$\"3_baaa7w&z)F-$ \"3YWBJt\"fT@(F-7$$\"3)fRRRzPF**)F-$\"3oh&o,IPTK(F-7$$\"3MJJJJ*yr=*F-$ \"3?2\\k&e21V(F-7$$\"3;3222L.&R*F-$\"3z73&fPk@a(F-7$$\"31.---_V-'*F-$ \"3?ueE^a>^wF-7$$\"3/B>>>2f)y*F-$\"39Okit<:ZxF-7$$\"\"\"F*$\"3!G[uRj\" )R&yF--%'COLOURG6&%$RGBGF*F*F*-%*LINESTYLEG6#\"\"#-F$6%7`q7$F(F(7$F/F/ 7$F4F47$F9F97$F>F>7$FCFC7$FHFH7$FMFM7$FRFR7$FWFW7$FfnFfn7$F[oF[o7$F`oF `o7$FeoFeo7$FjoFjo7$F_pF_p7$FdpFdp7$FipFip7$F^qF^q7$FcqFcq7$FhqFhq7$F] rF]r7$FbrFbr7$FgrFgr7$F\\sF\\s7$FasFas7$FfsFfs7$F[tF[t7$F`tF`t7$FetFet 7$FjtFjt7$F_uF_u7$FduFdu7$FiuFiu7$F^vF^v7$FcvFcv7$FhvFhv7$F]wF]w7$FbwF bw7$FgwFgw7$F\\xF\\x7$FaxFax7$FfxFfx7$F[yF[y7$F`yF`y7$FeyFey7$F[zF[z7$ F`zF`z7$FezFez7$FjzFjz7$F_[lF_[l7$Fd[lFd[l7$Fi[lFi[l7$F^\\lF^\\l7$Fc\\ lFc\\l7$Fh\\lFh\\l7$F]]lF]]l7$Fb]lFb]l7$Fg]lFg]l7$F\\^lF\\^l7$Fa^lFa^l 7$Ff^lFf^l7$F[_lF[_l7$F`_lF`_l7$Fe_lFe_l7$Fj_lFj_l7$F_`lF_`l7$Fd`lFd`l 7$Fi`lFi`l7$F^alF^al7$FcalFcal7$FhalFhal7$F]blF]bl7$FbblFbbl7$FgblFgbl 7$F\\clF\\cl7$FaclFacl7$FfclFfcl7$F[dlF[dl7$F`dlF`dl7$FedlFedl7$FjdlFj dl7$F_elF_el7$FdelFdel7$FielFiel7$F^flF^fl7$FcflFcfl7$FhflFhfl7$F]glF] gl7$FbglFbgl7$FgglFggl7$F\\hlF\\hl7$FahlFahl7$FfhlFfhl7$F[ilF[il7$F`il F`il7$FeilFeil7$FjilFjil7$F_jlF_jl7$FdjlFdjl-Fijl6&F[[m$\"*++++\"!\")$ F*F*F\\bm-F][m6#Fejl-F$6%7`q7$F($!3SnmmmmmmmF-7$F/$!3WA%R+dIAm'F-7$F4$ !3'*[c1\"H[7l'F-7$F9$!3%31D8&=9JmF-7$F>$!3==xmQAy-mF-7$FC$!3am3:4skmlF -7$FH$!3R./wD5DElF-7$FM$!3q4(*>-xexkF-7$FR$!3g$*3*zJ5,U'F-7$FW$!3ANSYs MvbjF-7$Ffn$!3_Z%QFDyBG'F-7$F[o$!3![+X@8p=@'F-7$F`o$!3r'y-;Mpg7'F-7$Fe o$!3'>wfxc\\K.'F-7$Fjo$!3t0BtI;lPfF-7$F_p$!35$*oS2>vXeF-7$Fdp$!3\"z)>C @([.t&F-7$Fip$!38$Q)eRp_FcF-7$F^q$!3nb0zQ!fA]&F-7$Fcq$!35ce2!*yW'Q&F-7 $Fhq$!3i3(yg>KUD&F-7$F]r$!3X8W:0G[B^F-7$Fbr$!3-8Y#3o!>#)\\F-7$Fgr$!3A9 7/\"Q7#[[F-7$F\\s$!3edZh._K*p%F-7$Fas$!3]1Qnm)\\+a%F-7$Ffs$!3]DeJ1\\r( R%F-7$F[t$!3(Q#)RQ=#HSUF-7$F`t$!3q)=*fC`ztSF-7$Fet$!3Pv3^**4G2RF-7$Fjt $!3!3R&['*=\"Hu$F-7$F_u$!3#)=cx@0%ob$F-7$Fdu$!3WIm.$F-7$Fcv$!3v6o+$*)eN%GF-7$Fhv$!37r3!QN@Sm #F-7$F]w$!3)\\'Q?n/8uCF-7$Fbw$!3Ct)>&R=W'G#F-7$Fgw$!3/\"HA=!o'z3#F-7$F \\x$!3[S%=c9i]*=F-7$Fax$!3%Q#yu]a='p\"F-7$Ffx$!3ax:Q_R_(\\\"F-7$F[y$!3 S9^C\\P'QJ\"F-7$F`y$!3'H!f]+XB-6F-7$Fey$!3#)zn-Vgz?\"*Fgy7$F[z$!3A:7=- `%f3(Fgy7$F`z$!3exNsG9tK^Fgy7$Fez$!3)R,Ox&fPfHFgy7$Fjz$!3WtG>c%o]2\"Fg y7$F_[l$\"31'=#H#>!\\k5Fgy7$Fd[l$\"3^2xKcqm8IFgy7$Fi[l$\"3nX0Ih:3V^Fgy 7$F^\\l$\"33JFSwHrwpFgy7$Fc\\l$\"31Jo+%yo\"f!*Fgy7$Fh\\l$\"3)fK_Q*eJ16 F-7$F]]l$\"3-8zoFTn08F-7$Fb]l$\"34LiA1)HK]\"F-7$Fg]l$\"3aFA\\/9'=p\"F- 7$F\\^l$\"3?%zarb$F-7$F^al$\"3O_!*>T]LVPF-7$Fcal$\"3q\\6UYIm3RF-7$Fhal$\"3[cN* 4c(QuSF-7$F]bl$\"3!\\_L^+x#RUF-7$Fbbl$\"3i$)*4U8\\uQ%F-7$Fgbl$\"36->Yf -mTXF-7$F\\cl$\"3MB!eDdG2p%F-7$Facl$\"3A$o,q1Fc%[F-7$Ffcl$\"3C_o;m?Dy \\F-7$F[dl$\"3!zm+X#QNE^F-7$F`dl$\"3)=**zNs:`D&F-7$Fedl$\"3E@9k5YQy`F- 7$Fjdl$\"3wAk%*RT_0bF-7$F_el$\"3O!\\].2^mF-7$F_jl$\"3i0%QRxGAm'F-7$Fdjl$\"3Snmmmmmmm F--Fijl6&F[[mF\\bmFiamF\\bmF]bm-F$6%7`q7$F($!3'pmmmmmmm)F-7$F/$!3')otR ,Dhf%)F-7$F4$!3%4V4Fu5gG)F-7$F9$!3qaNJ<8J(4)F-7$F>$!33(eRWM'e8zF-7$FC$ !3iuyy$>*=OxF-7$FH$!3Yr5Xrc'fd(F-7$FM$!3&*=(=^#))y8uF-7$FR$!3w;:+TyY\\ sF-7$FW$!39dyG'>[&)3(F-7$Ffn$!3TC;u$3]a#pF-7$F[o$!3m'RAU*3S$y'F-7$F`o$ !3\">]2`eM[i'F-7$Feo$!3'RG,,W.mY'F-7$Fjo$!3v:J+Trk9jF-7$F_p$!3Q(ozJa*z whF-7$Fdp$!3\\bmWa!4F,'F-7$Fip$!3Cvcxe\\GteF-7$F^q$!3zdgOWBK5dF-7$Fcq$ !3'R#)H[\\V^c&F-7$Fhq$!3'4\"*fVwAYS&F-7$F]r$!397Aue2P]_F-7$Fbr$!3)\\<@ 'pky(3&F-7$Fgr$!3^**o%4vpo$\\F-7$F\\s$!3C*o'=#>CAx%F-7$Fas$!3GOdO=$Q!* f%F-7$Ffs$!3M!4$RDHSYWF-7$F[t$!3;_Iy[h_zUF-7$F`t$!3EM)pr(f'[5%F-7$Fet$ !3/mmwCF-7$Fbw$!3=6=UT4\"yG#F-7$Fgw$!3mjOV*oA ))3#F-7$F\\x$!3%=>_$*3#e&*=F-7$Fax$!3%yw4YR!['p\"F-7$Ffx$!33hxyE/o(\\ \"F-7$F[y$!3!=5\"\\sV%RJ\"F-7$F`y$!3J-[o6xE-6F-7$Fey$!3H]3)p0C47*Fgy7$ F[z$!3/3YaK8)f3(Fgy7$F`z$!3$ohX\\eQF8&Fgy7$Fez$!3YMLP7kPfHFgy7$Fjz$!3! )*fl!f%o]2\"Fgy7$F_[l$\"3YBj-&>!\\k5Fgy7$Fd[l$\"37eIDavm8IFgy7$Fi[l$\" 3uK`4!z)3V^Fgy7$F^\\l$\"3-C$o`IYn(pFgy7$Fc\\l$\"3#>yox]#Hf!*Fgy7$Fh\\l $\"3awzmH(\\j5\"F-7$F]]l$\"3!\\;'eSAv08F-7$Fb]l$\"3lS?sJ$*Q.:F-7$Fg]l$ \"3q!p=L`_@p\"F-7$F\\^l$\"3kX+bt+&[*=F-7$Fa^l$\"3g+_s:')e'3#F-7$Ff^l$ \"3yo#yzBDnG#F-7$F[_l$\"3%prrouAjY#F-7$F`_l$\"3syE,m0'Hm#F-7$Fe_l$\"3% pZ%QMA$F-7$Fd`l$\"3wfx^t YX'R$F-7$Fi`l$\"3mBraS\")frNF-7$F^al$\"39)fz]8JDw$F-7$Fcal$\"3n[aIi9mF-7$F]gl$\"3)>d&p1ev!y'F-7$Fbgl$\"3fm*f U!)yS$pF-7$Fggl$\"3=HR:$3aw3(F-7$F\\hl$\"3')\\[z)HPNC(F-7$Fahl$\"3B`bd htG0uF-7$Ffhl$\"3SE\"*3]JR!e(F-7$F[il$\"3c^$)R&QQ[u(F-7$F`il$\"3:BRy)e #R6zF-7$Feil$\"37UVHD1u%4)F-7$Fjil$\"3!)H!\\d7&)QG)F-7$F_jl$\"3-iC*\\G r&f%)F-7$Fdjl$\"3'pmmmmmmm)F--Fijl6&F[[mF\\bmF\\bmFiamF]bm-%+AXESLABEL SG6$Q\"x6\"Q!Feho-%%VIEWG6$;F(FdjlFjho" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 214 "Successively higher degree Taylor polyn omials produce progressively better approximations over a fixed interv al centred at the origin.\nThe next picture shows the graphs of the de gree 7, 9 and 11 Taylor polynomials." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "AT := (n,x) -> sum((-1)^i *x^(2*i+1)/(2*i+1),i=0..n):\nplot([arctan(x),AT(3,x),AT(4,x),AT(5,x)], x=-1..1,-1..1,\n color=[black,magenta,coral,cyan],linestyle=[2, 1$3],\nnumpoints=100);" }}{PARA 13 "" 1 "" {GLPLOT2D 523 342 342 {PLOTDATA 2 "6(-%'CURVESG6%7`q7$$!\"\"\"\"!$!3!G[uRj\")R&y!#=7$$!3?ccc cOj)y*F-$!3+/2$4qtru(F-7$$!35JJJJ\\s/'*F-$!3u>kY*f'Q_wF-7$$!33NNNN.!zR *F-$!3ef_=8qoVvF-7$$!3/srrrZq*=*F-$!3L(=!GVr(>V(F-7$$!3oDDDD()\\#)*)F- $!3WN$flQt%=tF-7$$!3S&fff>#R!z)F-$!3$[pQ-DJ6@(F-7$$!3Q6777qZ\"f)F-$!3) yHX[h3y4(F-7$$!3IWWWW'edQ)F-$!3)zv=eiU#ypF-7$$!3#z\"===+q!=)F-$!3GjuE \"GDm&oF-7$$!3!fhhh,s(pzF-$!39ggSF0&*GnF-7$$!33<<<>>n*\\d'F-$!30'H*QPMH;eF-7$$!3S3444 $['fjF-$!3=+V+1JXkcF-7$$!3Ixwww%G!phF-$!3zKN_MWbFbF-7$$!3G3444$)))ffF- $!3UyqS7*[YP&F-7$$!3s5777utgdF-$!3Wi:7/&3kA&F-7$$!3%GMMMMWHb&F-$!3a_r_ i%*)*o]F-7$$!3sRSSSa7i`F-$!3+aVmeC'>#\\F-7$$!3Yuuuu9Ic^F-$!3'Hr.n'RtgZ F-7$$!3&)>???#3D%\\F-$!3#>Pz\"zpP!f%F-7$$!3wjjjj0ScZF-$!333v3aVqRWF-7$ $!3?&fff>*RbXF-$!3y_&\\Bk8XF%F-7$$!3_9:::^uZVF-$!3wI#4lI075%F-7$$!3#*3 444hfWTF-$!3p1&)*)*y!4HRF-7$$!3')*********Q![RF-$!3)yMxFq*=gPF-7$$!3nT UUUezHPF-$!3M'zeMr\")*pNF-7$$!3jeeeeYpLNF-$!3!z5?N6MnR$F-7$$!34(yyyy9V K$F-$!31NXAemQ4KF-7$$!3[/000beMJF-$!3'GAk6+zv.$F-7$$!3/%[[[Ghr#HF-$!39 @[d,-hZGF-7$$!3ceeee7*>t#F-$!35e:+H=\"pm#F-7$$!3qWXXXB)z_#F-$!3C9<*pc0 hZ#F-7$$!3%GMMMME&GBF-$!3Ug=&*e+w(G#F-7$$!3l$RRRR9(>@F-$!39<2OYhz)3#F- 7$$!3%z'oooOg=>F-$!3Sl&o$)zob*=F-7$$!3&R]]]!*QHr\"F-$!31^z&3Nukp\"F-7$ $!3xZ\\\\\\r(*3:F-$!3I![xg#zn(\\\"F-7$$!35yyyyub@8F-$!3Y9\"y,QVRJ\"F-7 $$!3H%eeeQ`n5\"F-$!3(y$3&QUnA5\"F-7$$!3#HVWWW+j9*!#>$!3,Sm*pHB47*Fgy7$ $!3u+444\\'y4(Fgy$!3gWNR.7)f3(Fgy7$$!3/8FFF2DP^Fgy$!3F#3$[r&QF8&Fgy7$$ !3;HYYY1CgHFgy$!33d*)37kPfHFgy7$$!37syyy)4^2\"Fgy$!3oi`1f%o]2\"Fgy7$$ \"3A?////`k5Fgy$\"35-h-&>!\\k5Fgy7$$\"3QZUUU-e9IFgy$\"3ts+$RbnO,$Fgy7$ $\"3;F...$Gw9&Fgy$\"3r'QTkx)3V^Fgy7$$\"3=]SSS!)3))pFgy$\"3O;Mb*=Yn(pFg y7$$\"3/q[[[o:%3*Fgy$\"35V\"QJy\"Hf!*Fgy7$$\"3w&QQQe&)36\"F-$\"3;@uCM% \\j5\"F-7$$\"3j9999KA88F-$\"3F-$ \"3g0I.>o$[*=F-7$$\"3*4111E!Q<@F-$\"3kJFiuAc'3#F-7$$\"3z&QQQ)>QFBF-$\" 3_\\kGGXn'G#F-7$$\"3:9888T[qS&[NiY#F-7$$\"3%[NNNb#eFFF-$\"3- )H$*f')3Gm#F-7$$\"37\\[[[-#)GHF-$\"3%**o^/x=qgbR$F-7$$\"3z00004;IPF-$\"3O&*39*=-.d$F-7$$\"3w#444HS&[RF-$\"3aQ It'QB1w$F-7$$\"39QPPP\\EYTF-$\"31t1,#*[^IRF-7$$\"3;ECCCaZ[VF-$\"3Uw7Y' \\>=5%F-7$$\"3y$333G=Tb%F-$\"3[AER1GXtUF-7$$\"3f&GGG3WJu%F-$\"3'o?$)\\ 1)))GWF-7$$\"3Kpmmm%RY%\\F-$\"3u!=VRm*3#f%F-7$$\"3daaaa5gW^F-$\"3;xbQ2 m[^ZF-7$$\"3')*zzzz(\\e`F-$\"3k#**Q*)\\W\">\\F-7$$\"3A2333CDZbF-$\"3G* p)\\\\zjk]F-7$$\"3%[WWW/O]w&F-$\"38*p%QUajH_F-7$$\"3S(eee)*o:'fF-$\"3y fC4()*))eP&F-7$$\"3mgeee!Gg:'F-$\"3%p.EWDK\"=bF-7$$\"3%>BBB.M^O'F-$\"3 !GXO14e$ocF-7$$\"3K%GGG)e3vlF-$\"3*H;\\*)obj\"eF-7$$\"3/zzzz46mnF-$\"3 'o:d(Q`U=iF-7$$\"3)GDDDDY:Q(F-$\"3QQi#3mo(ejF-7$$\"3apmmmyBnvF-$\"3E,y LBY!zZ'F-7$$\"3SPOOO9^!y(F-$\"3dJ[XUZ87mF-7$$\"330///)**4)zF-$\"38kID% >8et'F-7$$\"3Ousss;bz\")F-$\"3E2u&oHPf&oF-7$$\"3M\"444\\h#y$)F-$\"3_/H k?#RQ(pF-7$$\"3gGFFFH$4e)F-$\"3)*eOYI\"R<4(F-7$$\"3_baaa7w&z)F-$\"3YWB Jt\"fT@(F-7$$\"3)fRRRzPF**)F-$\"3oh&o,IPTK(F-7$$\"3MJJJJ*yr=*F-$\"3?2 \\k&e21V(F-7$$\"3;3222L.&R*F-$\"3z73&fPk@a(F-7$$\"31.---_V-'*F-$\"3?ue E^a>^wF-7$$\"3/B>>>2f)y*F-$\"39Okit<:ZxF-7$$\"\"\"F*$\"3!G[uRj\")R&yF- -%'COLOURG6&%$RGBGF*F*F*-%*LINESTYLEG6#\"\"#-F$6%7`q7$F($!3.R_4Q_4QsF- 7$F/$!3`mJG!*pYHsF-7$F4$!3p[j=./\")3sF-7$F9$!3-$\\(3zAOsrF-7$F>$!3p4MV tU)G7(F-7$FC$!3Qi^O>Z:iqF-7$FH$!3!oV%[12f'*pF-7$FM$!3'*f*>f'p=?pF-7$FR $!3**y'o*)y()G$oF-7$FW$!3J.[&=:_#QnF-7$Ffn$!3cHqfl;>\" H[g`'F-7$F`o$!3Xf5))4(=/U'F-7$Feo$!31?zJ$o2()H'F-7$Fjo$!3qT/+!f.l<'F-7 $F_p$!3)yP.#=#Q;1'F-7$Fdp$!3F(\\'G,+a?fF-7$Fip$!3yR'y=P(R(z&F-7$F^q$!3 I;.Q`T@]cF-7$Fcq$!3,O;Nfxc;bF-7$Fhq$!3GwT:SiYm`F-7$F]r$!3WnD5*y#H?_F-7 $Fbr$!3)>XlJ&)GX1&F-7$Fgr$!3+LSu8=m=\\F-7$F\\s$!3EO!))=#*z$eZF-7$Fas$! 3w.K'H^X()e%F-7$Ffs$!3f34%)o^`QWF-7$F[t$!3'fSw*[1rtUF-7$F`t$!3mH9A)zq1 5%F-7$Fet$!3D^F#R0R(GRF-7$Fjt$!3_ih,_*f*fPF-7$F_u$!3O\\$)G7B%)pNF-7$Fd u$!3kZ='oZZmR$F-7$Fiu$!3@l0vBhL4KF-7$F^v$!3&Q!)R'[*[v.$F-7$Fcv$!3giVs< QfZGF-7$Fhv$!3u#=9(\\H!pm#F-7$F]w$!3/X2E965wCF-7$Fbw$!3'zR>x\"zv(G#F-7 $Fgw$!3C*o)R?_z)3#F-7$F\\x$!3E$=(R=%ob*=F-7$Fax$!3U(o#48UZ'p\"F-7$Ffx$ !3yd)H=)yn(\\\"F-7$F[y$!34[zqmL%RJ\"F-7$F`y$!3%R)*46UnA5\"F-7$Fey$!32w M0#HB47*Fgy7$F[z$!3))zv)G?\")f3(Fgy7$F`z$!3!4Xb9dQF8&Fgy7$Fez$!39j()37 kPfHFgyFizF^[l7$Fd[l$\"3WW)HRbnO,$Fgy7$Fi[l$\"3]\\KTw()3V^Fgy7$F^\\l$ \"3a>O6*=Yn(pFgy7$Fc\\l$\"3iQ()[yJiY#F-7$F`_l$\"3O`u@8,!Gm#F-7$Fe_l$\"3a:(fSq@\"\\GF-7$Fj_l$\"3jz !z8A!)H.$F-7$F_`l$\"3$*z[4xqxAKF-7$Fd`l$\"3'zV!3]VZ&R$F-7$Fi`l$\"3Y#Q( *ymi,d$F-7$F^al$\"3?A2&)zLRgPF-7$Fcal$\"3]H759>;IRF-7$Fhal$\"3![>nM?%G ,TF-7$F]bl$\"33h\"f!y_%G9Sl&F-7$F_el$\"3i8^/\\uX(z&F-7$Fdel$\"3c3[o8uYCfF-7 $Fiel$\"3%z7]'4P[bgF-7$F^fl$\"3e%Q!p:+FzhF-7$Fcfl$\"32d-0=Gf3jF-7$Fhfl $\"3pM8Ei.6;kF-7$F]gl$\"3wNWMl^ OuO(4-xt'F-7$F\\hl$\"3>al&>Kd&HoF-7$Fahl$\"3rgS.5/\"f\"pF-7$Ffhl$\"3;2 J.=l`)*pF-7$F[il$\"37\\5i.sSlqF-7$F`il$\"3607Ll0@ArF-7$Feil$\"3K]T62]w rrF-7$Fjil$\"35+$))Q5\"[3sF-7$F_jl$\"3)*yE?XNYHsF-7$Fdjl$\"3.R_4Q_4QsF --Fijl6&F[[m$\"*++++\"!\")$F*F*F[^n-F][m6#Fejl-F$6%7`q7$F($!3k]j?\\j? \\$)F-7$F/$!3'*[lJp!Gi9)F-7$F4$!3'zi5^&yq\")zF-7$F9$!3.Jh0BWu2yF-7$F>$ !3T\"eeT*yCUwF-7$FC$!3m\"RW#Rl9&[(F-7$FH$!3DQDY1JzWtF-7$FM$!3c;W&R9\"yPah_mF-7$F`o$!3#ofE%)GW;^'F-7$Feo$!3+0VA'=Q&pjF-7$Fjo$!3mc!H^^B; B'F-7$F_p$!3ANKP*Hg_5'F-7$Fdp$!3_Py&>Z+L&fF-7$Fip$!3A'G8;l8H#eF-7$F^q$ !3UahyzD7pcF-7$Fcq$!3!H\"R'**3Y4`&F-7$Fhq$!3Qn\"Qcp2qP&F-7$F]r$!3a]ZXO j0G_F-7$Fbr$!3yTa;5n5q]F-7$Fgr$!3Q00<\\OtA\\F-7$F\\s$!39wOo&zU7w%F-7$F as$!3O'zl78,2f%F-7$Ffs$!3(>n]Ud>*RWF-7$F[t$!3!z2'>\"G\\YF%F-7$F`t$!38* QYPe(G,TF-7$Fet$!37Knd5+9HRF-7$Fjt$!3Od:!o))=-w$F-7$F_u$!3y!o![Bv**pNF -7$Fdu$!3qz`DOHu'R$F-7$Fiu$!3kGpD97R4KF-7$F^v$!3!f#p)3T\"ePIF-7$Fcv$!3 cp0aZ8hZGF-7$Fhv$!3Y+`:qB\"pm#F-7$F]w$!3S/UT*z0hZ#F-7$Fbw$!3=]A!Q:gxG# F-7$Fgw$!3U!yf.='z)3#F-7$F\\x$!3U#)3!)4)ob*=F-7$Fax$!3%[eiTNukp\"F-7$F fx$!3Y+8!p#zn(\\\"F-7$F[y$!3c00P!QVRJ\"F-7$F`y$!3:(HyQUnA5\"F-7$Fey$!3 Mk/.(HB47*Fgy7$F[z$!3UIcR.7)f3(Fgy7$F`z$!3&>9$[r&QF8&FgyFdzFizF^[lFc[l 7$Fi[l$\"3xZ9Ww()3V^Fgy7$F^\\l$\"3!>xR4Gm# F-7$Fe_l$\"3#G$Q\\B$R\"\\GF-7$Fj_l$\"37GC'*=A,LIF-7$F_`l$\"3'>p>cVMGA$ F-7$Fd`l$\"3#34b!*[pbR$F-7$Fi`l$\"3#=Boe,=.d$F-7$F^al$\"3')[Ro5ElgPF-7 $Fcal$\"3B5-6EVcIRF-7$Fhal$\"3k.b0A>!>5%F-7$F]bl$\"35n`dP!)etUF-7$Fbbl $\"3;&ezO'p4HWF-7$Fgbl$\"3!eTE\"4`T#f%F-7$F\\cl$\"3G'y4\"*G$)>v%F-7$Fa cl$\"37bsvq,\"*>\\F-7$Ffcl$\"3KVlwGJul]F-7$F[dl$\"3*)4r:7jHJ_F-7$F`dl$ \"39n1,#za#y`F-7$Fedl$\"3&Q$*R+[\\9_&F-7$Fjdl$\"3%\\Qv\"4+2tcF-7$F_el$ \"3E;))=Vo(H#eF-7$Fdel$\"3%))[EG))*[dfF-7$Fiel$\"3ABRC*oj&)4'F-7$F^fl$ \"3TvI\\BBqMiF-7$Fcfl$\"375wg7P)3Q'F-7$Fhfl$\"3PuV%on:l]'F-7$F]gl$\"3s X(*f&yu-l'F-7$Fbgl$\"3!)pao@VS&y'F-7$Fggl$\"3%)*=X$ov!)>pF-7$F\\hl$\"3 kQ>JX^dbqF-7$Fahl$\"33(F-7$Ffhl$\"3;Iu1*pd'[tF-7$F[il$\"3O_q9 V$eF\\(F-7$F`il$\"3_EbN[3HSwF-7$Feil$\"3^(*z_ZZS0yF-7$Fjil$\"3!ez#QM=s zzF-7$F_jl$\"3+CJ\"HV)=Y\")F-7$Fdjl$\"3k]j?\\j?\\$)F--Fijl6&F[[mF[^n$ \")AR!)\\F]^nF^^nF_^n-F$6%7`q7$F($!3CTa6Sa6SuF-7$F/$!3anrJPR_FuF-7$F4$ !3o0S^^7M)R(F-7$F9$!3ODfYuHg[tF-7$F>$!3-Ys]w$)Q$G(F-7$FC$!3M(QC(3v!f?( F-7$FH$!3.[L:\\NlCrF-7$FM$!3<#)*>!3aUKqF-7$FR$!3.`:nh7kHpF-7$FW$!3QvRv L$[2#oF-7$Ffn$!3WqMo)*=#Hq'F-7$F[o$!3qjB&*Q%G[f'F-7$F`o$!3e`.pax\")okF -7$Feo$!3S+m[1X5QjF-7$Fjo$!3&[]yr!z[3iF-7$F_p$!37Z*GIFyy3'F-7$Fdp$!3jW T!QA^5%fF-7$Fip$!3/Jq*>U))Q\"eF-7$F^q$!33A#eS]lGm&F-7$Fcq$!3Vma/T!pk_& F-7$Fhq$!3u/0%Q8WRP&F-7$F]r$!3\"e*R*yN[fA&F-7$Fbr$!3mzZ>%[*po]F-7$Fgr$ !3N;1$Gwv<#\\F-7$F\\s$!35DeKC+igZF-7$Fas$!3;CX_k-J!f%F-7$Ffs$!3[0Am?Lm RWF-7$F[t$!37<=G9**[uUF-7$F`t$!3+sOq\">#>,TF-7$Fet$!3(eXs#eO3HRF-7$Fjt $!3]oI#['e=gPF-7$F_u$!3s$G(Gd)z*pNF-7$Fdu$!3cwy^$=LnR$F-7$Fiu$!3([d#4L iQ4KF-7$F^v$!39#e)>,)yv.$F-7$Fcv$!3%*)RH'=,hZGF-7$Fhv$!3EsZ([z6pm#F-7$ F]w$!3eJ#RWb0hZ#F-7$Fbw$!375\\ga+w(G#F-7$Fgw$!3q_#p]9'z)3#F-7$F\\x$!3C UF,)zob*=F-7$Fax$!3g`fx]VZ'p\"F-7$Ffx$!3#\\hhg#zn(\\\"F-7$F[y$!3Hs_'zfY@p\"F-7$F\\^l$\"3K***y'=o$[*=F-7$Fa^l$\"3'y_\\ LFil3#F-7$Ff^l$\"3)))*p'R_umG#F-7$F[_l$\"3%eS5NZNiY#F-7$F`_l$\"3!e,pD$ )3Gm#F-7$Fe_l$\"3e)f**o3Q\"\\GF-7$Fj_l$\"3VDt&[l4I.$F-7$F_`l$\"3hm0o-# HGA$F-7$Fd`l$\"3/^LJw(fbR$F-7$Fi`l$\"3#p>X1L+.d$F-7$F^al$\"33*eaDa>1w$ F-7$Fcal$\"3CD!4Ps20$RF-7$Fhal$\"3yNsg`j!=5%F-7$F]bl$\"3#>7)*G;HMF%F-7 $Fbbl$\"3K!)3sd%[)GWF-7$Fgbl$\"3;\\9R%eA?f%F-7$F\\cl$\"3cphN()eP^ZF-7$ Facl$\"3$py(y)Qf*=\\F-7$Ffcl$\"3I^Wm)o^V1&F-7$F[dl$\"3v!)pIm4]wV?$) yR(F-7$F_jl$\"3^*yF$y*=vU(F-7$Fdjl$\"3CTa6Sa6SuF--Fijl6&F[[mF^^nF[^nF[ ^nF_^n-%+AXESLABELSG6$Q\"x6\"Q!F[cp-%%VIEWG6$;F(FdjlF`cp" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "An \+ adaptive graph plotting procedure: " }{TEXT 0 5 "graph" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The procedure in thi s section is an alternative to the standard Maple function " }{TEXT 0 4 "plot" }{TEXT -1 195 ", for plotting the graph of a single function. It allows control over how the plotting is performed, in a manner whi ch is a bit different from the control one has over the standard Maple routine " }{TEXT 0 4 "plot" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "In particular, the precision o f the floating point calculations is not changed internally, whereas, \+ the Maple procedure " }{TEXT 0 4 "plot" }{TEXT -1 137 " often uses har dware floating point arithmetic whereby the effective precision is inc reased to the equivalent of about 15 decimal digits." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "graph: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 18 "Calling S equence:\n" }}{PARA 0 "" 0 "" {TEXT 272 4 " " }{TEXT -1 17 "graph( \+ f, xrng )\n" }{TEXT 273 1 "\n" }{TEXT -1 26 " graph( f, xrng, yrng \+ )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 269 11 "P arameters:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 23 4 "f - " }{TEXT -1 83 " an expression involvin g a single variable, say x, or a function x -> f(x)\n\n " }{TEXT 23 7 "xrng - " }{TEXT -1 79 " horizontal plotting range in the for m x=a..b, when x is an expression in x" }}{PARA 0 "" 0 "" {TEXT -1 19 " " }{TEXT 256 2 "OR" }{TEXT -1 48 " in the form a. .b when f is afunction x -> f(x)" }}{PARA 0 "" 0 "" {TEXT -1 12 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 23 7 "yrng - " } {TEXT -1 92 " vertical range (optional), which can be given in the form c..d, or in the form y=c..d.\n" }}{PARA 256 "" 0 "" {TEXT 268 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 5 "graph" }{TEXT -1 22 " plots t he graph of a " }{TEXT 258 15 "single function" }{TEXT -1 26 " using a n adaptive method." }}{PARA 15 "" 0 "" {TEXT -1 88 "An even spacing is aimed for along sections of the curve which are approximately linear. " }}{PARA 15 "" 0 "" {TEXT -1 79 "More points are plotted along sectio ns of the curve where the graph is bending." }}{PARA 15 "" 0 "" {TEXT -1 107 "Any maximum or minimum points are located approximately by par abolic interpolation and added to the graph.\n" }}{PARA 256 "" 0 "" {TEXT 270 8 "Options:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "adaptive=true/false" }}{PARA 0 "" 0 "" {TEXT -1 144 "A daptive plotting will, where necessary, sub-divide the plotting interv al in an attempt to get a good graphical representation of the functio n. " }}{PARA 0 "" 0 "" {TEXT -1 111 "By default, this option is set to \"true\", but it can be turned off by setting the \"adaptive\" option to \"false\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 312 "numpoints=n\nFor non-adaptive plotting the interval for \+ the plot is subdivided into a fixed number of sub-intervals of equal w idth by means \"numpoints\" points. \nFor adaptive plotting \"numpoint s\" controls the spacing of points along the curve, that is, in the di rection of the curve rather than just horizontally. " }}{PARA 0 "" 0 " " {TEXT -1 57 "The spacing between points is generally no greater than " }{XPPEDIT 18 0 "plotwidth/numpoints;" "6#*&%*plotwidthG\"\"\"%*num pointsG!\"\"" }{TEXT -1 97 ". Note that, in general, the number of poi nts plotted could be vastly different from \"numpoints\"." }}{PARA 0 " " 0 "" {TEXT -1 36 "The default value is \"numpoints=33\"." }}{PARA 0 "" 0 "" {TEXT -1 12 "\nlinearity=n" }}{PARA 0 "" 0 "" {TEXT -1 82 "\"l inearity\" controls the tolerance for the allowed deviation from a str aight line." }}{PARA 0 "" 0 "" {TEXT -1 93 "Along any arc between two \+ points on the graph, the curve will generally deviate no more than " } {TEXT 274 16 "tol * plotheight" }{TEXT -1 62 " from the straight line \+ segment joining the two points, where " }{XPPEDIT 18 0 "tol = 10^(-4)/ linearity;" "6#/%$tolG*&)\"#5,$\"\"%!\"\"\"\"\"%*linearityGF*" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 98 "Thus increasing \"linearity\" reduces the tolerance, and so gives more points at bends in the graph ." }}{PARA 0 "" 0 "" {TEXT -1 36 "The default value is \"linearity=10 \"." }}{PARA 0 "" 0 "" {TEXT -1 12 "\nmaxpoints=n" }}{PARA 0 "" 0 "" {TEXT -1 108 "This option provides a cut-off for the adaptive subdivis ion by specifying a minimum horizontal distance of " }{XPPEDIT 18 0 " plotwidth/maxpoints" "6#*&%*plotwidthG\"\"\"%*maxpointsG!\"\"" }{TEXT -1 65 " between points for the plot. \nThe default is \"maxpoints=1000 \". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "plotdata=true/false\nSetting the option \"plotdata\" to \"true\" caus es the data points to be returned instead of the graph." }}{PARA 0 "" 0 "" {TEXT -1 61 "This option can also be specified using the word \"p lot_data\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 31 "Available standard plot options" }{TEXT -1 184 ": color, line style, thickness, scaling, xtickmarks, ytickmarks, tickmarks, labels, \+ style, symbol, symbolsize, title, axes, font, labelfont, titlefont, ax esfont, view, labeldirections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to a ctivate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure activ e open the subsection, place the cursor anywhere after the prompt [ > \+ and press [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "graph: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "graph" {MPLTEXT 1 0 9031 "graph := proc(ff,rng)\n local fx,x,y,t1,t2,eps,xL,yL,xR,yR,h,ymin,ymax,yM in,yMax,\n width,height,adaptdiv,xrange,yrange,Options,startoptio ns,rs,\n mxpts,lnrty,nmpts,delta,pdat,adptv,curve,dev,m,addmaxmin ,\n fn,n1,n2,proctype,vars,y1,y2,testvals,aa,bb,i,opt;\n\n if n args<2 then\n error \"at least 2 arguments are required; the basi c syntax is: 'graph(f(x),x=a..b)'.\"\n end if;\n\n # Collect all t he input data.\n if type(ff,procedure) or \n (op(0,ff)=`@@` an d nops(ff)=2 and type(op(1,ff),procedure)) then\n proctype := tru e;\n if type(rng,range) then\n rs := rng;\n else\n \+ error \"the 2nd argument, %1, is invalid .. it should have the \+ form 'a..b' to provide a horizontal range over which to plot the graph of %1\",rng,ff;\n end if;\n elif type(ff,algebraic) then \n \+ vars := indets(ff,name) minus indets(ff,realcons);\n if nops(v ars)<>1 then \n if not type(ff,realcons) and not has(indets(ff ),\{Int,Sum\}) then\n error \"the 1st argument, %1, is inva lid .. it should be an expression which depends only on a single varia ble\",ff;\n end if;\n else\n x := op(1,vars);\n \+ end if;\n if type(rng,name=range) then\n proctype := \+ false;\n x := op(1,rng);\n if not member(x,vars) and n ot type(ff,realcons) then\n error \"the 1st argument, %1, i s invalid .. it should be an expression which depends only on the vari able %2\",ff,x;\n end if;\n rs := op(2,rng);\n el se\n error \"the 2nd argument, %1, is invalid .. it should hav e the form '%2=a..b' to provide a horizontal range over which to plot \+ the graph of %3\",rng,x,ff;\n end if;\n else\n error \"the 1st argument, %1, is invalid .. it should be an algebraic expression \+ in a single variable, or a numerical valued procedure with a single ar gument\",ff; \n end if;\n \n xL := evalf(op(1,rs));\n xR \+ := evalf(op(2,rs));\n if not type(xL,numeric) or not type(xR,numeric ) then\n error \"each end point of the horizontal range %1 must e valuate to a numeric\",rs;\n end if;\n if xL>=xR then\n error \"2nd argument horizontal range is invalid\";\n end if;\n if proc type then\n xrange := xL..xR;\n else\n xrange := x=xL..xR; \n end if;\n\n startoptions := 3; \n yrange := NULL;\n yMax : = infinity;\n yMin := -infinity;\n if nargs>2 then\n if type( args[3],range) or type(args[3],name=range) then\n startoptions := 4;\n if type(args[3],range) then\n rs := args[3 ]; \n else\n rs := op(2,args[3]);\n y := op(1,args[3]);\n end if;\n yMin := evalf(op(1,rs));\n yMax := evalf(op(2,rs));\n if not type(yMin,numeric) \+ or not type(yMax,numeric) then\n error \"each end point of \+ the vertical range %1 must evaluate to a numeric\",rs;\n end i f;\n if yMin>=yMax then\n error \"the 3rd argument \+ vertical range is invalid\";\n end if;\n if type(args[ 3],range) then\n yrange := yMin..yMax; \n else\n \+ yrange := y=yMin..yMax;\n end if;\n end if;\n \+ end if;\n\n # Get the options, but first set default\n nmpts := 33 ;\n lnrty := 10;\n Options := [];\n mxpts := 1000;\n pdat := f alse;\n adptv := true;\n\n if nargs>=startoptions then\n Opti ons := [args[startoptions..nargs]];\n if not type(Options,list(eq uation)) then\n error \"each optional argument after the %-1 a rgument must be an equation\",startoptions-1;\n end if;\n if hasoption(Options,'adaptive','adptv','Options') then\n if not adptv=true then adptv := false end if;\n end if;\n if hasop tion(Options,'numpoints','nmpts','Options') then\n if not type (nmpts,posint) or nmpts<2 then\n error \"\\\"numpoints\\\" \+ must be an integer greater than 1\"\n end if;\n end if;\n if hasoption(Options,'linearity','lnrty','Options') then\n \+ if not type(lnrty,posint) then\n error \"\\\"linearity\\ \" must be a positive integer\"\n end if;\n end if;\n \+ if hasoption(Options,'maxpoints','mxpts','Options') then\n i f not type(mxpts,posint) then\n error \"\\\"maxpoints\\\" m ust be a positive integer\"\n end if;\n end if;\n if hasoption(Options,'plotdata','pdat','Options') or\n hasoption (Options,'plot_data','pdat','Options') then\n if not pdat=true then pdat := false end if;\n end if;\n for i to nops(Option s) do\n opt := op(i,Options); \n if not member(op(1,op t),\n \{'color','colour','linestyle','line_style','thicknes s',\n 'scaling','xtickmarks','ytickmarks','tickmarks','labe ls',\n 'style','symbol','symbolsize','title','axes','font', \n 'labelfont','label_font','titlefont','title_font',\n \+ 'axesfont','axes_font','view','labeldirections'\}) then\n \+ error \"unknown or invalid option: %1\",opt;\n end if; \n end do; \n end if; \n\n # Recursively define d procedure to construct plotting data.\n adaptdiv := proc(pL,pR)\n \+ local x0,x1,x2,y0,y1,y2,p1,dx,dy,divL,divR;\n \n x0 := pL[1] ;\n x2 := pR[1];\n\n x1 := (x0+x2)/2;\n y1 := evalf(fn( x1));\n\n if y1<=yMax and y1>=yMin then\n # Update estima te of the height.\n if y1>ymax then ymax := y1; height := ymax -ymin end if;\n if y1 0 then p := -p end if;\n q := abs(q);\n \+ x3 := x0+p/q;\n y3 := evalf(fn(x3));\n if y3 <=yMax and y3>=yMin then\n # Update estimate of the heig ht.\n if y3>ymax then ymax := y3; height := ymax-ymin en d if;\n if y34 then\n \+ error \"1st argument %1 does not evaluate to a numeric at some point , or points, in the plotting interval\",ff;\n end if;\n\n width := xR-xL;\n\n if adptv then \n ymin := max(yMin,min(yL,yR));\n ymax := min(yMax,max(yL,yR));\n\n height := ymax-ymin;\n \+ eps := evalf(1/lnrty*0.0001);\n delta := evalf(1/mxpts);\n \+ nmpts := iquo(nmpts,2);\n h := evalf(1/nmpts)^2;\n curve : = addmaxmin(adaptdiv([xL,yL],[xR,yR]));\n if pdat=true then\n \+ return curve;\n else\n return plot(curve,xrange,yran ge,op(Options));\n end if;\n else\n nmpts := nmpts-1;\n \+ h := width/nmpts;\n n1 := iquo(nmpts,2);\n n2 := nmpts-n1 -1;\n curve := [[xL,yL],seq([xL+i*h,evalf(fn(xL+i*h))],i=1..n1), \n seq([xR+(i-n2)*h,evalf(fn(xR+(i-n2)*h))],i=0..n2-1),[ xR,yR]];\n if pdat=true then\n return curve;\n else \n return plot(curve,xrange,yrange,op(Options));\n end if ;\n end if;\nend proc: # of graph" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Examples are given in the next section. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 82 "A p rocedure for evaluating the inverse tangent function from its Maclauri n series " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 741 "arctan_series := proc(xx::realcons)\n local x,z,po w,sum,term,eps,k,maxit,even;\n\n maxit := Digits*20;\n eps := Floa t(2,-Digits);\n\n x := evalf(xx);\n if abs(x)>1 then\n error \+ \"Maclaurin series only converges in the interval [-1,1]\"\n end if; \n\n pow := x;\n z := x*x;\n sum := x;\n even := false; \n f or k from 3 to maxit by 2 do\n pow := pow*z;\n term := pow/k ;\n if even then\n sum := sum+term;\n else\n \+ sum := sum-term;\n end if;\n if abs(term)<=eps*abs(sum) the n break end if;\n even := not even;\n end do:\n \n if k > ma xit then\n print(`sum of`,k-1,`terms of series is`,sum);\n e rror \"reached maximum iterations without convergence\"\n end if;\n \n evalf(sum);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "When using the default precision of 10 digits, \+ we can get about 10 digit accuracy in computing " }{XPPEDIT 18 0 "arct an(x);" "6#-%'arctanG6#%\"xG" }{TEXT -1 27 " using this procedure with " }{XPPEDIT 18 0 "-9/10<=x" "6#1,$*&\"\"*\"\"\"\"#5!\"\"F)%\"xG" } {XPPEDIT 18 0 "``<=9/10" "6#1%!G*&\"\"*\"\"\"\"#5!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "xx := 0.9;\nevalf(arctan_series(xx));\nevalf(arctan(x x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+=5:Gt!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+=5:Gt!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We can check the accuracy of the procedure " }{TEXT 0 13 "arctan_series" }{TEXT -1 12 " as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "plot(arctan(x)-'ev alf[10]@arctan_series'(x),x=-0.9..0.9,color=blue,numpoints=80);" }} {PARA 13 "" 1 "" {GLPLOT2D 467 257 257 {PLOTDATA 2 "6&-%'CURVESG6#7\\] r7$$!\"*!\"\"$\"\"!F,7$$!+S.b#**)!#5F+7$$!+!o+^)*)F0$F*F07$$!+?5lx*)F0 $!\"$F07$$!+g8?q*)F0$\"\"&F07$$!++5lx)F0FW7$$!+N01p()F0FO7$$!+r3hh( )F0F47$$!+l*G^v)F0F=7$$!+bqk[()F0$\"\"%F07$$!+X^;U()F0FO7$$!+NKoN()F0F O7$$!+D8?H()F0FB7$$!+:%>Fs)F0FO7$$!+0vB;()F0F+7$$!+&fb(4()F0FG7$$!+!pt Kq)F0F87$$!+!y\"z'p)F0F+7$$!+q)4.p)F0FO7$$!+gz#Qo)F0F+7$$!+]gMx')F0F+7 $$!+ST'3n)F0$!\"%F07$$!+IAQk')F0FG7$$!+?.!zl)F0FO7$$!+5%=9l)F0FW7$$!++ l$\\k)F0F+7$$!+!fa%Q')F0FB7$$!+!os>j)F0F87$$!+q2\\D')F0Fcu7$$!+g)3!>') F0F+7$$!+]p_7')F0F47$$!+S]/1')F0$\"\"'F07$$!+IJc*f)F0FO7$$!+?73$f)F0FG 7$$!+5$*f'e)F0F47$$!++u6!e)F0F+7$$!+!\\NOd)F0F+7$$!+!e`rc)F0Fcu7$$!+q; ng&)F0F+7$$!+k(*=a&)F0F+7$$!+v,!pa)F0FB7$$!+!f5'R&)F0FO7$$!+05KK&)F0F8 7$$!+?9.D&)F0FO7$$!+N=u<&)F0F=7$$!+]AX5&)F0FB7$$!+lE;.&)F0F+7$$!+!3te \\)F0FB7$$!+&\\$e)[)F0F47$$!+5RH\"[)F0F47$$!+DV+u%)F0FB7$$!+SZrm%)F0F4 7$$!+b^Uf%)F0FG7$$!+qb8_%)F0FO7$$!+&)f%[W)F0FW7$$!++kbP%)F0F+7$$!+:oEI %)F0F+7$$!+Is(HU)F0F+7$$!+Xwo:%)F0FB7$$!+g!)R3%)F0F47$$!+v%36S)F0FW7$$ !+!*)=QR)F0FB7$$!+0$HlQ)F0FB7$$!+?(R#z$)F0FO7$$!+N,&>P)F0FW7$$!+]0mk$) F0F87$$!+l4Pd$)F0F47$$!+!Q\"3]$)F0F47$$!+&z\"zU$)F0FG7$$!+5A]N$)F0F]s7 $$!+DE@G$)F0FB7$$!+OI#4K)F0FG7$$!+I^e8$)F0FG7$$!+DsC1$)F0FB7$$!+?$4*)H )F0FG7$$!+:9d\"H)F0F+7$$!+5NB%G)F0F+7$$!+0c*oF)F0F+7$$!++xbp#)F0F87$$! +&z>AE)F0F]s7$$!+!*=)[D)F0FG7$$!+!)RaZ#)F0FO7$$!+vg?S#)F0FB7$$!+q\"oGB )F0F47$$!+l-`D#)F0F47$$!+gB>=#)F0FW7$$!+bW&3@)F0F47$$!+]l^.#)F0F47$$!+ X'yh>)F0FB7$$!+S2%))=)F0FO7$$!+NG]\"=)F0FO7$$!+I\\;u\")F0FG7$$!+Dq#o;) F0FB7$$!+?\"*[f\")F0FB7$$!+:7:_\")F0F47$$!+5L\"[9)F0F47$$!+0aZP\")F0F+ 7$$!++v8I\")F0FO7$$!+&f*zA\")F0F47$$!+!pha6)F0F=7$$!+&yB\"3\")F0F47$$! +!)ey+\")F0F47$$!+qzW$4)F0FO7$$!+l+6'3)F0FG7$$!+Sq!)y!)F0FO7$$!+5S]r!) F0F+7$$!+!)4?k!)F0F+7$$!+]z*o0)F0F47$$!+?\\f\\!)F0F+7$$!+!*=HU!)F0FB7$ $!+g)))\\.)F0FB7$$!+IeoF!)F0F+7$$!++GQ?!)F0F+7$$!+q(zI,)F0F87$$!+Snx0! )F0Fcu7$$!+5PZ)*zF0F+7$$!+!oq6*zF0FB7$$!+]w'Q)zF0FO7$$!+?YcwzF0F47$$!+ !fh#pzF0FO7$$!+g&e>'zF0F47$$!+IblazF0FO7$$!++DNZzF0F47$$!+q%\\+%zF0F47 $$!+SkuKzF0F+7$$!+5MWDzF0F47$$!+!QS\"=zF0FG7$$!+bt$3\"zF0FG7$$!+DV`.zF 0F47$$!+&HJi*yF0FB7$$!+l#G*))yF0F87$$!+N_i\")yF0F+7$$!+0AKuyF0FB7$$!+v \">q'yF0Fcu7$$!+XhrfyF0FB7$$!+>JT_yF0F+7$$!+vAkXyF0FO7$$!+I9()QyF0F+7$ $!+!f+@$yF0F47$$!+X(H`#yF0Fcu7$$!++*e&=yF0Fcu7$$!+g!)y6yF0FG7$$!+:s,0y F0FB7$$!+qjC)z(F0F47$$!+IbZ\"z(F0FB7$$!+&o/Zy(F0FW7$$!+SQ$zx(F0F47$$!+ +I;rxF0FG7$$!+b@RkxF0FO7$$!+58idxF0F+7$$!+q/&3v(F0FO7$$!+D'zSu(F0FW7$$ !+!y3tt(F0FB7$$!+Sz`IxF0FB7$$!+&4nPs(F0FO7$$!+]i*pr(F0FG7$$!+5aA5xF0FG 7$$!+lXX.xF0F87$$!+?Po'p(F0F47$$!+!)G\"**o(F0FO7$$!+S?9$o(F0F+7$$!+&>r jn(F0Fcu7$$!+].gpwF0F87$$!+5&HGm(F0F+7$$!+l'egl(F0FG7$$!+?yG\\wF0F47$$ !+!)p^UwF0F47$$!+NhuNwF0Fcu7$$!+?`tGwF0FO7$$!++Xs@wF0FG7$$!+&o8Zh(F0FG 7$$!+qGq2wF0F+7$$!+b?p+wF0FG7$$!+S7o$f(F0F47$$!+D/n'e(F0F+7$$!+5'f'zvF 0FG7$$!+&z[Ed(F0F+7$$!+!)zjlvF0FO7$$!+grievF0FB7$$!+Xjh^vF0FB7$$!+IbgW vF0F]s7$$!+5ZfPvF0F+7$$!+&*QeIvF0F+7$$!+!3tN_(F0FG7$$!+lAc;vF0FO7$$!+] 9b4vF0FG7$$!+I1a-vF0FO7$$!+:)Hb\\(F0FB7$$!++!>&)[(F0F]s7$$!+!=3:[(F0FO 7$$!+lt\\uuF0FB7$$!+]l[nuF0FG7$$!+NdZguF0F47$$!+?\\Y`uF0FB7$$!++TXYuF0 F47$$!+&GV%RuF0F+7$$!+b;UDuF0F+7$$!+A+S6uF0F+7$$!+D%\\TS(F0FW7$$!+I))* oR(F0FG7$$!+N#['*Q(F0FO7$$!+SwR#Q(F0F47$$!+Xq9vtF0F47$$!+]k*yO(F0F47$$ !+]ekgtF0F]s7$$!+b_R`tF0F+7$$!+gY9YtF0F+7$$!+gS*)QtF0F+7$$!+lMkJtF0FB7 $$!+qGRCtF0FG7$$!+vA9( F0FB7$$!+]:j'=(F0FO7$$!+_4QzrF0F47$$!+5O:srF0F+7$$!+li#\\;(F0FB7$$!+?* )pdrF0FW7$$!+!er/:(F0FB7$$!+NUCVrF0F47$$!+!*o,OrF0F87$$!+]&*yGrF0FW7$$ !+0Ac@rF0F+7$$!+g[L9rF0FW7$$!+?v52rF0FB7$$!+v,))*4(F0F47$$!+IGl#4(F0F+ 7$$!+!\\Da3(F0F47$$!+X\")>yqF0FG7$$!++3(42(F0F+7$$!+gMujqF0FW7$$!+?h^c qF0FB7$$!+v()G\\qF0F47$$!+I91UqF0FB7$$!+!4M[.(F0F+7$$!+XngFqF0FG7$$!++ %z.-(F0F+7$$!+g?:8qF0FG7$$!+:Z#f+(F0F47$$!+qtp)*pF0FO7$$!+I+Z\"*pF0F47 $$!+&oUU)pF0FB7$$!+S`,xpF0F+7$$!++!)yppF0FB7$$!+b1cipF0F47$$!+5LLbpF0F W7$$!+qf5[pF0FO7$$!+g\")oF0FG7$$!+gQwtoF0F47$$!+]'Hj'oF0F47$$!+Sa*)eoF0FB7$$!+I7Y^oF0 FB7$$!+?q-WoF0F+7$$!+5GfOoF0FG7$$!+&fe\"HoF0F+7$$!+!QC<#oF0FG7$$!+q,H9 oF0FB7$$!+gf&o!oF0FB7$$!+]z'F0FG7$$!+ILb%y'F0FB7$$!+ ?\">rx'F0F+7$$!+5\\opnF0FO7$$!++2DinF0F47$$!+!\\;[v'F0FO7$$!+!G#QZnF0F B7$$!+l![*RnF0FB7$$!+bQ^KnF0F47$$!+X'z]s'F0F47$$!+Nakh'F0F+7$$ !+5BW0mF0F+7$$!+by0`lF0F+7$$!+,Mn+lF0F+7$$!+br=rkF0F+7$$!+54qTkF0F+7$$ !+]$HVV'F0F+7$$!+!zdpU'F0FO7$$!+Iie>kF0F47$$!+lY@7kF0FB7$$!++J%[S'F0F4 7$$!+S:Z(R'F0FB7$$!+!)**4!R'F0FG7$$!+?%GFQ'F0F87$$!+goNvjF0FG7$$!++`)z O'F0F+7$$!+SPhgjF0F47$$!+v@C`jF0FB7$$!+51(eM'F0F+7$$!+]!*\\QjF0FB7$$!+ !\\F6L'F0F47$$!+IfvBjF0F+7$$!+qVQ;jF0F+7$$!+5G,4jF0F+7$$!+]7k,jF0F47$$ !+&opUH'F0F47$$!+D\")*oG'F0F47$$!+ll_ziF0F+7$$!+0]:siF0FO7$$!+UMykiF0F 47$$!+Dh'F0FB7$$!+]^w!>'F0FO7$$!+?LO$ ='F0F+7$$!+!\\hf<'F0FB7$$!+g'f&ohF0F+7$$!+Iy:hhF0FB7$$!++gv`hF0F+7$$!+ qTNYhF0FO7$$!+SB&*QhF0FO7$$!+50bJhF0FB7$$!+!o[T7'F0FG7$$!+]ou;hF0F87$$ !+?]M4hF0F47$$!+!>V>5'F0FB7$$!+g8a%4'F0FO7$$!+N&Rr3'F0FB7$$!+0xtzgF0F+ 7$$!+veLsgF0F+7$$!+XS$\\1'F0FB7$$!+:A`dgF0F+7$$!+&QI,0'F0F47$$!+b&GF/' F0FB7$$!+DnKNgF0F+7$$!+'*[#z-'F0FG7$$!+l=z?gF0F47$$!+I)eO,'F0FB7$$!+&z Dl+'F0F+7$$!+gFR**fF0F47$$!+D(fA*fF0F+7$$!+!pE^)fF0F+7$$!+bO*z(fF0F47$ $!+?1'3(fF0F47$$!+&eFP'fF0FB7$$!+]XfcfF0F47$$!+::Y\\fF0F47$$!+![GB%fF0 F87$$!+Xa>NfF0F+7$$!+5C1GfF0FB7$$!+!QH4#fF0F47$$!+Xjz8fF0FO7$$!+5Lm1fF 0FW7$$!+!GI&**eF0FB7$$!+XsR#*eF0F+7$$!+5UE&)eF0FO7$$!+v68yeF0F+7$$!+S \")*4(eF0F+7$$!+0^'Q'eF0FB7$$!+q?tceF0FW7$$!+N!*f\\eF0FB7$$!++gYUeF0F+ 7$$!+qHLNeF0FO7$$!+N**>GeF0FO7$$!++p1@eF0F+7$$!+lQ$R\"eF0F+7$$!+I3!o!e F0F47$$!+)zn'*z&F0F+7$$!+?.>$z&F0FO7$$!+SGr'y&F0FG7$$!+g`B!y&F0F+7$$!+ &)yvtdF0FB7$$!+5/GndF0FB7$$!+IH!3w&F0F+7$$!+]aKadF0F+7$$!+qz%yu&F0FO7$ $!+!\\q8u&F0F+7$$!+5I*[t&F0F47$$!+IbTGdF0F+7$$!+b!Q>s&F0F47$$!+!egar&F 0FO7$$!++J)*3dF0F+7$$!+?c]-dF0FB7$$!+S\"Ggp&F0F47$$!+g1b*o&F0F+7$$!+&= tIo&F0FG7$$!+5dfwcF0FG7$$!+I#=,n&F0FO7$$!+]2kjcF0F47$$!+qK;dcF0FG7$$!+ !z&o]cF0F+7$$!+:$3Uk&F0F+7$$!+S3tPcF0F+7$$!+gLDJcF0F+7$$!+!)exCcF0FG7$ $!++%)H=cF0FB7$$!+?4#=h&F0FB7$$!+XMM0cF0FO7$$!+lf'))f&F0FB7$$!+'[)Q#f& F0F+7$$!+Sho%e&F0F47$$!+!z$)pd&F0F+7$$!+S9GpbF0FO7$$!+&4z:c&F0FO7$$!+] n(Qb&F0F+7$$!++W`F0FB7$$!+S))G8`F0F+7$$!+SRw1`F0FB7$$!+S!R-I&F0 F+7$$!+STr$H&F0F47$$!+S#*=(G&F0FB7$$!+SVm!G&F0FB7$$!+S%RTF&F0FB7$$!+SX hn_F0FB7$$!+S'*3h_F0FB7$$!+SZca_F0F47$$!+S)R![_F0FG7$$!+X\\^T_F0F+7$$! +]+*\\B&F0FB7$$!+]^YG_F0F47$$!+]-%>A&F0F87$$!+]`T:_F0F+7$$!+]/*)3_F0FB 7$$!+]bO-_F0F+7$$!+]1%e>&F0FO7$$!+]dJ*=&F0FO7$$!+]3z#=&F0FB7$$!+]fEw^F 0FO7$$!+]5up^F0F+7$$!+]h@j^F0F+7$$!+]7pc^F0FO7$$!+]j;]^F0FO7$$!+]9kV^F 0FB7$$!+_l6P^F0FB7$$!+Xl_H^F0FB7$$!+Nl$>7&F0F+7$$!+DlM9^F0F47$$!+:lv1^ F0F47$$!+5l;*4&F0F47$$!++ld\"4&F0F+7$$!+!\\')R3&F0F+7$$!+!['Rw]F0FB7$$ !+qk!)o]F0FB7$$!+gk@h]F0FG7$$!+]ki`]F0FB7$$!+Sk.Y]F0F+7$$!+?k&3.&F0F+7 $$!+0kn:]F0F+7$$!+ojJ&)\\F0F+7$$!+Jj&\\&\\F0F+7$$!+AjOZ\\F0FG7$$!+7jxR \\F0F47$$!+.j=K\\F0F47$$!+%H'fC\\F0F+7$$!+&G1q\"\\F0F47$$!+wiT4\\F0FB7 $$!+mi#=!\\F0FB7$$!+diB%*[F0F47$$!+7y^()[F0F+7$$!+o$*z!)[F0F+7$$!+C43u [F0FB7$$!+![it'[F0F+7$$!+OSkg[F0F+7$$!+#fDR&[F0FO7$$!+[r?Z[F0F47$$!+.( )[S[F0F+7$$!+9=0F[F0F+7$$!+E\\h8[F0F+7$$!+#['*o![F0FG7$$!+Q!y,![F0FG7$ $!+%ffMz%F0FB7$$!+\\6u'y%F0FG7$$!+/F-!y%F0F+7$$!+gUItZF0F47$$!+;eemZF0 F47$$!+st')fZF0FG7$$!+G*[Jv%F0F+7$$!+%[Iku%F0FG7$$!+S?rRZF0F47$$!+&f$* Ht%F0F47$$!+]^FEZF0F+7$$!+1nb>ZF0F+7$$!+i#QGr%F0F+7$$!+=)>hq%F0FB7$$!+ u8S*p%F0FB7$$!+IHo#p%F0FB7$$!+'[kfo%F0F+7$$!+TgCzYF0F47$$!+U[(=n%F0FB7 $$!+WO]kYF0FO7$$!+YC8dYF0FB7$$!+[7w\\YF0F+7$$!+]+RUYF0F+7$$!+_)=]j%F0F G7$$!+awkFYF0F+7$$!+ckF?YF0FB7$$!+e_!Hh%F0FB7$$!+gS`0YF0F47$$!+iG;)f%F 0F+7$$!+j;z!f%F0F+7$$!+k/U$e%F0FO7$$!+m#\\gd%F0F47$$!+o!y'oXF0F47$$!+q oIhXF0FB7$$!+sc$Rb%F0F47$$!+uWcYXF0F+7$$!+wK>RXF0F+7$$!+y?#=`%F0FB7$$! +!)3XCXF0FB7$$!+#ozq^%F0F+7$$!+%[3(4XF0F+7$$!+&GPB]%F0F+7$$!+'3m\\\\%F 0FG7$$!+))[f([%F0F+7$$!+!pB-[%F0FB7$$!+#\\_GZ%F0FB7$$!+%H\"[lWF0FB7$$! +'45\"eWF0F87$$!+)*)Q2X%F0F+7$$!++xOVWF0F47$$!+w&[jV%F0F+7$$!+_%H$HWF0 FB7$$!+G.JAWF0FB7$$!+07H:WF0FB7$$!+#3s#3WF0F+7$$!+eHD,WF0F47$$!+MQB%R% F0F+7$$!+5Z@(Q%F0F+7$$!+ik@N%F0F47$$!+o **4XVF0FG7$$!+W33QVF0F47$$!+?<1JVF0F47$$!+'fUSK%F0FG7$$!+sM-G%F0F B7$$!+I(3\\F%F0F+7$$!+1'*)yE%F0F47$$!+#[q3E%F0FB7$$!+e8&QD%F0F+7$$!+MA $oC%F0FB7$$!+5J\")RUF0F+7$$!+')RzKUF0FB7$$!+i[xDUF0F+7$$!+Rdv=UF0F+7$$ !+%)f;gTF0F+7$$!+Iid,TF0F+7$$!+6DD%4%F0F+7$$!+#zGp3%F0FB7$$!+s]gzSF0F+ 7$$!+`8GsSF0FB7$$!+9RjdSF0FB7$$!+wk)H/%F0FB7$$!+cFmNSF0FB7$$!+P!R$GSF0 FG7$$!+=`,@SF0F+7$$!+)f\"p8SF0FO7$$!+zyO1SF0FB7$$!+gT/**RF0FB7$$!+S/s \"*RF0FB7$$!+@nR%)RF0FB7$$!+g7nxRF0F+7$$!++e%4(RF0FG7$$!+R.AkRF0F47$$! +y[\\dRF0FB7$$!+=%p2&RF0FO7$$!+dR/WRF0FO7$$!+'\\=t$RF0F47$$!+OIfIRF0F4 7$$!+wv'Q#RF0F+7$$!+:@9\"p.RF0F+7$$!+Md'p*Q F0FB7$$!+t-C!*QF0F47$$!+7[^$)QF0F+7$$!+_$*ywQF0FB7$$!+J%QL'QF0FB7$$!+5 v))\\QF0FB7$$!+]?;VQF0F47$$!+*eOk$QF0F+7$$!+G6rHQF0F+7$$!+oc)H#QF0F+7$ $!+3-E;QF0F47$$!+ZZ`4QF0F+7$$!+'G4G!QF0F+7$$!+EQ3'z$F0FB7$$!+l$e$*y$F0 F47$$!+/Hj#y$F0F47$$!+Wu!fx$F0FB7$$!+$)>=pPF0F+7$$!+)*G6`OF0F+7$$!+9Q/ PNF0F+7$$!+NAwwMF0F+7$$!+c1[;MF0F+7$$!+ea%*3MF0FB7$$!+h-T,MF0F+7$$!+k] (QR$F0FG7$$!+m)RjQ$F0F47$$!+pY!)yLF0F47$$!+s%p7P$F0F+7$$!+uUtjLF0FB7$$ !+x!*>cLF0F+7$$!+!)Qm[LF0FB7$$!+#oG6M$F0FO7$$!+&[$fLLF0FB7$$!+)GegK$F0 F+7$$!+!4B&=LF0FB7$$!+$*y)4J$F0F+7$$!+'p_MI$F0F+7$$!+)\\$F0F+7$$!+9p'4>$F0FB7$$!+-vS%=$F 0F+7$$!+\"4[y<$F0F+7$$!+!o)GrJF0F+7$$!+o#HZ;$F0FB7$$!+c)p\"eJF0FB7$$!+ X/h^JF0F+7$$!+M50XJF0FO7$$!+A;\\QJF0F+7$$!+5A$>8$F0FO7$$!+*zs`7$F0FB7$ $!+)Q8)=JF0F47$$!+wRD7JF0FG7$$!+lXp0JF0F+7$$!+a^8*4$F0F+7$$!+Udd#4$F0F +7$$!+Jj,'3$F0F+7$$!+Uw%=2$F0F+7$$!+a*yw0$F0F+7$$!+5Yf]IF0F47$$!+m-^VI F0F47$$!+AfUOIF0F47$$!+y:MHIF0F47$$!+MsDAIF0F+7$$!+!*G<:IF0FB7$$!+Y&)3 3IF0F47$$!+-U+,IF0F+7$$!+e)>R*HF0F47$$!+9b$o)HF0F+7$$!+q6vzHF0F+7$$!+E omsHF0FO7$$!+#[#elHF0FW7$$!+Q\")\\eHF0F47$$!+%z89&HF0F+7$$!+]%HV%HF0FB 7$$!+1^CPHF0FB7$$!+i2;IHF0F47$$!+=k2BHF0F47$$!+t?*f\"HF0FB7$$!+Gx!*3HF 0F+7$$!+%QB=!HF0F47$$!+S!RZ*GF0F+7$$!+'paw)GF0F+7$$!+_.d!)GF0F+7$$!+3g [tGF0FG7$$!+k;SmGF0F47$$!+?tJfGF0F+7$$!+z'zY%GF0F+7$$!+Q?/IGF0F+7$$!+< KsAGF0FB7$$!+'R/a\"GF0F+7$$!+wb33GF0F+7$$!+bnw+GF0FB7$$!+MzW$z#F0FB7$$ !+9\"Hhy#F0FB7$$!+$H5)yFF0FG7$$!+s9\\rFF0F+7$$!+_E;Au#F0F+7$$!+pt*[t#F0FB7$$!+[&yvs#F0F+7$$!+G(f -s#F0F+7$$!+24%Hr#F0F47$$!+'3Acq#F0F+7$$!+mKI)p#F0F47$$!+XW)4p#F0F+7$$ !+Ccm$o#F0FB7$$!+/oMwEF0F+7$$!+$)z-pEF0F+7$$!+i\"4RDF0F+7$$!+kX.KDF0FB7$$!+EX([_#F0F+7$$!+)[9x^#F0 FB7$$!+\\Wb5DF0F47$$!+5WR.DF0F+7$$!+sVB'\\#F0F47$$!+MV2*[#F0F+7$$!+'H9 >[#F0F+7$$!+eUvuCF0FB7$$!+?UfnCF0F47$$!+#=M/Y#F0F+7$$!+WTF`CF0FO7$$!+1 T6YCF0FB7$$!+oS&*QCF0F+7$$!+ISzJCF0F+7$$!+#*RjCCF0FB7$$!+aRZs0M#F0F+7$$!+$\\WOL#F0FB7$$!+)y;nK#F0FB7$$!+$3*y>BF0F+7$$!+y8' GJ#F0F+7$$!+tO$fI#F0F+7$$!+of+*H#F0FB7$$!+j#y?H#F0FB7$$!+e0:&G#F0FB7$$ !+Q(RuD#F0FB7$$!+<*G(HAF0FB7$$!+77!GA#F0F+7$$!+1N(e@#F0F+7$$!+,e%*3AF0 F+7$$!+'4=??#F0F+7$$!+\"R!4&>#F0F+7$$!+'oi\")=#F0FB7$$!+\")\\B\"=#F0FB 7$$!+wsIu@F0F47$$!+S_hm@F0F47$$!+0K#*e@F0F+7$$!+q6B^@F0F47$$!+M\"RN9#F 0FB7$$!+*4Ze8#F0F+7$$!+k]:G@F0F+7$$!+GIY?@F0F+7$$!+$*4x7@F0F+7$$!+e*y] 5#F0F+7$$!+ApQ(4#F0F+7$$!+()[p*3#F0F47$$!+_G+#3#F0FB7$$!+;3Ju?F0F+7$$! +\"y=m1#F0F47$$!+Yn#*e?F0F+7$$!+5ZB^?F0F+7$$!+vEaV?F0FB7$$!+S1&e.#F0FB 7$$!+/'e\"G?F0F+7$$!+plY??F0F+7$$!+MXx7?F0F+7$$!+)\\#30?F0F+7$$!+j/R(* >F0FO7$$!+G%)p*)>F0F47$$!+#R1?)>F0F+7$$!+dVJu>F0F+7$$!+ABim>F0FB7$$!+' GI*e>F0FG7$$!+^#Q7&>F0FB7$$!+;iaV>F0F47$$!+!=ae$>F0F47$$!+X@;G>F0F+7$$ !+50D@>F0F47$$!+u)QV\">F0F+7$$!+QsU2>F0F+7$$!+-c^+>F0FG7$$!+mRg$*=F0F4 7$$!+IBp')=F0F+7$$!+%p!yz=F0FB7$$!+e!pG(=F0F+7$$!+Au&f'=F0F47$$!+'yX!f =F0FB7$$!+]T8_=F0F+7$$!+9DAX=F0F+7$$!+y3JQ=F0F+7$$!+U#*RJ=F0F47$$!+1w[ C=F0F+7$$!+rfd<=F0F47$$!+OVm5=F0F47$$!++Fv.=F0F+7$$!+k5%oz\"F0F+7$$!+G %H**y\"F0FB7$$!+#zp\"F0FB7$$!+C)e5>\"F0F+7$$!+/\"[ P=\"F0F+7$$!+&QPk<\"F0F+7$$!+mm7p6F0F+7$$!+Zf\"=;\"F0FB7$$!+G_]a6F0F47 $$!+4X>Z6F0F+7$$!+!z$))R6F0FB7$$!+rIdK6F0F+7$$!+_BED6F0F+7$$!+K;&z6\"F 0F+7$$!+84k56F0FB7$$!+%>IL5\"F0FB7$$!+u%>g4\"F0FO7$$!+b(3()3\"F0F+7$$! +O!)R\"3\"F0F47$$!+=&))* Fc[vF+7$$!+5QR;)*Fc[v$F*Fc[v7$$!+ScgZ(*Fc[vF+7$$!+qu\")y'*Fc[vF+7$$!++ $H+h*Fc[vF+7$$!+I6CT&*Fc[vFd[v7$$!+gHXs%*Fc[vF[\\v7$$!+!zkOS*Fc[vF[\\v 7$$!+?m([L*Fc[vF+7$$!+]%)3m#*Fc[vF+7$$!+!G+t>*Fc[vF+7$$!+:@^G\"*Fc[vF[ \\v7$$!+gn?y&)Fc[vF+7$$!*S,z-)F0Fd[v7$$!+qgWxoFc[vF+7$$!*u!*ps&F0F[\\v 7$$!+]czXaFc[vF+7$$!+g0gk^Fc[vFd[v7$$!+!z,V4&Fc[vF[\\v7$$!+:I+C]Fc[vF[ \\v7$$!+UUq`\\Fc[vF+7$$!+oaS$)[Fc[vF+7$$!+Az!Gu%Fc[vF+7$$!+v.@-YFc[vF+ 7$$!+#>?)RSFc[vF+7$$!*,IuZ$F0F+7$$!+3'f'))GFc[vF+7$$!+0#*))*H#Fc[vF+7$ $!+/S]0?Fc[vF+7$$!+-)=6r\"Fc[vF+7$$!+-D_P;Fc[vFd[v7$$!+-i#Rc\"Fc[vF+7$ $!+-*H.\\\"Fc[vFd[v7$$!+,Ot;9Fc[vF[\\v7$$!++t8V8Fc[vF+7$$!++5ap7Fc[vFd [v7$$!++Z%f>\"Fc[vF+7$$!*S[B7\"F0Fd[v7$$!+;kY^5Fc[vFd[v7$$!+?V%e!)*!#7 F+7$$!+![Cq4*F^cvF+7$$!+SY?)Q)F^cvF+7$$!+g\\cqpF^cvF+7$$!+v_#Hb&F^cvF+ 7$$!+7fk)y&F^cvF+7$$\"*4ve9\"F0F+7$$ \"+B#\\3H\"Fc[vF+7$$\"+cL#eV\"Fc[vF+7$$\"+A/J3:Fc[vF[\\v7$$\"+*[(z!e\" Fc[vF[\\v7$$\"+cXG`;Fc[vF+7$$\"+A;xDd,#Fc[vF[\\v7$$\"+ASpg@Fc[vF+7$$\"+b\"ocI#Fc[vFd[v7$$\"+)GU1X#F c[vF+7$$\"+Akh&f#Fc[vF[\\v7$$\"+)[.\"oEFc[vF+7$$\"+b0fSFFc[vF+7$$\"+Aw 28GFc[vF+7$$\"+)okb)GFc[vF+7$$\"+a<0eHFc[vF+7$$\"+@)Q0.$Fc[vF+7$$\"+)) e-.JFc[vF+7$$\"+aH^vJFc[vF[\\v7$$\"+?++[KFc[vFd[v7$$\"+(3([?LFc[vF+7$$ \"+aT(HR$Fc[vF+7$$\"*AhaY$F0F+7$$\"+:1l:YFc[vF+7$$\"*,Sew&F0F+7$$\"+]l *=$eFc[vF+7$$\"+&3`z*eFc[vF+7$$\"+?'4S'fFc[vF+7$$\"+gh1IgFc[vF[\\v7$$ \"+N#z@;'Fc[vF+7$$\"+5BH%H'Fc[vFd[v7$$\"+])[.O'Fc[vF[\\v7$$\"+&Q0kU'Fc [vF+7$$\"+?>Y#\\'Fc[vF[\\v7$$\"+g%=&elFc[vFd[v7$$\"++]dCmFc[vFd[v7$$\" +N:j!p'Fc[vFd[v7$$\"+q!)ocnFc[vFd[v7$$\"+5YuAoFc[vF+7$$\"+]6!)))oFc[vF +7$$\"+&od[&pFc[vF+7$$\"+?U\"4-(Fc[vFd[v7$$\"+g2(p3(Fc[vF[\\v7$$\"++t- `rFc[vF+7$$\"+NQ3>sFc[vF[\\v7$$\"+q.9&G(Fc[vF+7$$\"+5p>^tFc[vF+7$$\"+& )*4L[(Fc[vF+7$$\"+gIU:wFc[vF+7$$\"++'z9o(Fc[vF[\\v7$$\"+Nh`ZxFc[vF+7$$ \"+qEf8yFc[vFd[v7$$\"*@\\'zyF0F[\\v7$$\"+buNbzFc[vF+7$$\"++d1J!)Fc[vF+ 7$$\"+]Rx1\")Fc[vF+7$$\"+&>#[#=)Fc[vF+7$$\"+!p)*QL)Fc[vF+7$$\"+!=:`[)F c[vF+7$$\"+IM-h&)Fc[v$FHFc[v7$$\"+v;tO')Fc[vF+7$$\"+?*RCr)Fc[vF[\\v7$$ \"+q\"[\")y)Fc[vF[\\v7$$\"+:k&Q'))Fc[vF[\\v7$$\"+gYcR*)Fc[vF+7$$\"+5HF :!*Fc[vFd[v7$$\"+b6)44*Fc[vF[\\v7$$\"++%*om\"*Fc[vF+7$$\"+]wRU#*Fc[vFd [v7$$\"+&*e5=$*Fc[vF+7$$\"+ST\"QR*Fc[vFd[v7$$\"+!RA&p%*Fc[vF[\\v7$$\"+ N1BX&*Fc[vFd[v7$$\"+!))Q4i*Fc[vF+7$$\"+Irk'p*Fc[vF+7$$\"+:,[****Fc[vF+ 7$$\"*JJ-.\"F)F+7$$\"+YF+P5F0F+7$$\"+#=uP/\"F0F+7$$\"+=ca]5F0F+7$$\"+a qJd5F0F47$$\"+!\\)3k5F0F+7$$\"+E*f32\"F0F+7$$\"+i8jx5F0F+7$$\"+)z-W3\" F0FB7$$\"+MU<\"4\"F0F47$$\"+qc%z4\"F0F+7$$\"+1rr/6F0F+7$$\"+U&)[66F0F4 7$$\"+y*f#=6F0F47$$\"+99.D6F0F+7$$\"+]G!=8\"F0F+7$$\"+&Gu&Q6F0F+7$$\"+ cr6_6F0F+7$$\"+G+ml6F0F+7$$\"+k9Vs6F0FB7$$\"++H?z6F0F+7$$\"+OV(f=\"F0F +7$$\"+sdu#>\"F0FB7$$\"+3s^*>\"F0F+7$$\"+W')G17F0F+7$$\"+!3gI@\"F0F+7$ $\"+;:$)>7F0F47$$\"+_HgE7F0F+7$$\"+)QuLB\"F0F47$$\"+Ce9S7F0F47$$\"*E

8F0F+7$$\"+^YLE8F0F+7$$\"+9WbL8F0F47$$\"+xTxS8F 0F+7$$\"+SR*zM\"F0F+7$$\"+-P@b8F0F47$$\"+lMVi8F0F+7$$\"+GKlp8F0F+7$$\" +!*H(oP\"F0F47$$\"+`F4%Q\"F0F+7$$\"+;DJ\"R\"F0F47$$\"+zA`)R\"F0F47$$\" +U?v09F0F+7$$\"+0=(HT\"F0F47$$\"+o:>?9F0F+7$$\"+J8TF9F0F+7$$\"+%4JYV\" F0F+7$$\"+c3&=W\"F0FB7$$\"+>12\\9F0FO7$$\"+#Q!Hc9F0F47$$\"+W,^j9F0FB7$ $\"+2*H2Z\"F0F+7$$\"*n\\zZ\"F)FB7$$\"+/*f[[\"F0FO7$$\"+R,x\"\\\"F0F+7$ $\"+u.o)\\\"F0FB7$$\"+31f0:F0F+7$$\"+U3]7:F0FB7$$\"+w5T>:F0F47$$\"+58K E:F0F+7$$\"+X:BL:F0F47$$\"+!yT,a\"F0F+7$$\"+9?0Z:F0FB7$$\"+[A'Rb\"F0F4 7$$\"+#[s3c\"F0FB7$$\"+;Fyn:F0F47$$\"+^Hpu:F0F+7$$\"+'=.;e\"F0F47$$\"+ ?M^)e\"F0F47$$\"+aOU&f\"F0FO7$$\"+*)QL-;F0FB7$$\"+CTC4;F0FB7$$\"+eV:;; F0F47$$\"+#fkIi\"F0F+7$$\"+E[(*H;F0F+7$$\"+g])oj\"F0F47$$\"+&H&zV;F0F+ 7$$\"+KiVr;F0F+7$$\"*F)F+7$$\"+& *y\"40#F0F+7$$\"*&\\Ad@F)F+7$$\"+mhwk@F0F47$$\"+#Q2B<#F0F47$$\"+)f[)z@ F0F+7$$\"+:)*Q(=#F0F47$$\"+K5$\\>#F0F+7$$\"+[AZ-AF0FB7$$\"+kM,5AF0FB7$ $\"+!oav@#F0F+7$$\"+'*e4DAF0FG7$$\"+7rjKAF0FG7$$\"+G$y,C#F0F47$$\"+X&> xC#F0F+7$$\"+i2EbAF0F+7$$\"+y>!GE#F0FB7$$\"+%>V.F#F0F47$$\"+5W)yF#F0F4 7$$\"+ST@QBF0F47$$\"*(Qa)R#F)F47$$\"+qe.ECF0F47$$\"+qy_`CF0F47$$\"+q3S gCF0FB7$$\"+qQFnCF0FG7$$\"+qo9uCF0FB7$$\"+q)>5[#F0F+7$$\"+qG*y[#F0F+7$ $\"+qew%\\#F0FB7$$\"+q)Q;]#F0F+7$$\"+q=^3DF0FB7$$\"+q[Q:DF0FB7$$\"+qyD ADF0F47$$\"+q38HDF0FB7$$\"+qQ+ODF0F+7$$\"+qo(Ga#F0F+7$$\"+q)\\(\\DF0FB 7$$\"+qGicDF0FB7$$\"+qe\\jDF0F+7$$\"+q)o.d#F0FO7$$\"+q=CxDF0F+7$$\"+q[ 6%e#F0F47$$\"+qy)4f#F0FB7$$\"+q3'yf#F0F+7$$\"+qQt/EF0F47$$\"+qog6EF0F+ 7$$\"*()z%=EF)F+7$$\"+_y*fi#F0F47$$\"+Me^LEF0F+7$$\"+;Q.TEF0F47$$\"+*z ^&[EF0F47$$\"+#ypgl#F0FB7$$\"+kxejEF0FB7$$\"+Yd5rEF0F+7$$\"+GPiyEF0F+7 $$\"+#pfOp#F0F+7$$\"+ccp3FF0F+7$$\"+QO@;FF0F+7$$\"+?;tBFF0F+7$$\"+-'\\ 7t#F0F+7$$\"+&en(QFF0F47$$\"+obGYFF0FB7$$\"+]N!Qv#F0FB7$$\"+K:KhFF0F+7 $$\"+9&R)oFF0F+7$$\"+'\\djx#F0FB7$$\"+ya(Qy#F0FO7$$\"+gMR\"z#F0F+7$$\" +U9\"*)z#F0F+7$$\"+rL)*GGF0F+7$$\"*Ib!fGF)F+7$$\"*?Jm1$F)F+7$$\"+gN$[= $F0F+7$$\"*#f..LF)F+7$$\"+E\"3tJ$F0F+7$$\"+J.eJLF0F+7$$\"+MkrQLF0F+7$$ \"+OD&eM$F0F47$$\"+R'))HN$F0FB7$$\"+UZ7gLF0FB7$$\"+X3EnLF0FB7$$\"+[pRu LF0FB7$$\"+^I`\"Q$F0FB7$$\"+a\"p')Q$F0F47$$\"+d_!eR$F0F+7$$\"+g8%HS$F0 FO7$$\"+iu25MF0F47$$\"+lN@r^$F0FB7$$\"+2^DCNF0F+7$$\"*@\"RJNF)F47$$\"+^E_QNF0FG 7$$\"+#4aca$F0F47$$\"+Lby_NF0FO7$$\"+up\"*fNF0FB7$$\"+:%[qc$F0F47$$\"+ c)zTd$F0F+7$$\"+(H68e$F0FB7$$\"+QFW)e$F0F47$$\"+zTd&f$F0FB7$$\"+?cq-OF 0F+7$$\"+hq$)4OF0FB7$$\"+-&oph$F0F+7$$\"+%QJ7j$F0F+7$$\"+lU\\XOF0F+7$$ \"+1di_OF0F+7$$\"+YrvfOF0FO7$$\"+(e))om$F0FB7$$\"+G+-uOF0F47$$\"+p9:\" o$F0F+7$$\"+5HG)o$F0FB7$$\"+^VT&p$F0F+7$$\"+#zXDq$F0FB7$$\"+u'3or$F0FB 7$$\"+c:2JPF0FB7$$\"+(*H?QPF0F+7$$\"+QWLXPF0F47$$\"+zeY_PF0F47$$\"*K(f fPF)F47$$\"+EDqmPF0FB7$$\"+Kx!Qx$F0F47$$\"+QH\"4y$F0F+7$$\"+W\"=!)y$F0 F+7$$\"+c&GA!QF0F+7$$\"+o*Qk\"QF0F+7$$\"+#zf[%QF0F+7$$\"+:1GtQF0F+7$$ \"+iA7IRF0F+7$$\"*\"R'p)RF)F+7$$\"+)y*y$*RF0FB7$$\"+mch+SF0F47$$\"+W:W 2SF0FO7$$\"+AuE9SF0F+7$$\"++L4@SF0F47$$\"+y\">z-%F0FO7$$\"+c]uMSF0F47$ $\"+N4dTSF0FB7$$\"+9oR[SF0FG7$$\"+#pA_0%F0FG7$$\"+q&[?1%F0F47$$\"+[W() oSF0F47$$\"+E.qvSF0F47$$\"+/i_#3%F0FG7$$\"+#3_$*3%F0FB7$$\"+gz<'4%F0FB 7$$\"+QQ+.TF0F+7$$\"+;(H)4TF0F+7$$\"+%fbm6%F0FB7$$\"+s9[BTF0FB7$$\"+]t IITF0FB7$$\"+GK8PTF0FB7$$\"+1\"fR9%F0FO7$$\"+&)\\y]TF0F47$$\"+k3hdTF0F 47$$\"+UnVkTF0FO7$$\"+?EErTF0F+7$$\"+)\\)3yTF0FB7$$\"+wV\"\\=%F0FB7$$ \"+a-u\">%F0F+7$$\"+Khc)>%F0FB7$$\"*,#R0UF)FG7$$\"+l4x7UF0F+7$$\"+?*\\ ,A%F0FB7$$\"+w)GvA%F0F87$$\"+Jy!\\B%F0FB7$$\"+'y'GUUF0F47$$\"+Udm\\UF0 FB7$$\"+(pWqD%F0F+7$$\"+_OUkUF0F+7$$\"+3E!=F%F0F+7$$\"+j:=zUF0F47$$\"+ =0c'G%F0FB7$$\"+u%RRH%F0FW7$$\"+H%=8I%F0FO7$$\"+%Q(p3VF0F+7$$\"+Sj2;VF 0FB7$$\"+&HbMK%F0FB7$$\"+;6(HN%F0FB7$$\"+Qp[#Q%F0FB7$$\"+$*e')*Q%F0FO7 $$\"+[[C(R%F0FB7$$\"+/Qi/WF0F+7$$\"+fF+7WF0F+7$$\"+9WF0F47$$\"+q1wE WF0F47$$\"+D'RTV%F0FG7$$\"*e=:W%F)F+7$$\"+g7b[WF0F47$$\"+RRebWF0F+7$$ \"+=mhiWF0F+7$$\"+)H\\'pWF0F47$$\"+x>owWF0FB7$$\"+cYr$[%F0FB7$$\"+Otu! \\%F0F+7$$\"+:+y(\\%F0F47$$\"+%p7[]%F0FB7$$\"+u`%=^%F0F+7$$\"+`!y)=XF0 F47$$\"+K2\"f_%F0F+7$$\"+7M%H`%F0F+7$$\"+\"4w*RXF0F47$$\"+q(3qa%F0F47$ $\"+]9/aXF0F47$$\"+IT2hXF0F+7$$\"+4o5oXF0FB7$$\"+)[R^d%F0FB7$$\"+o@<#e %F0FO7$$\"+Z[?*e%F0F47$$\"+EvB'f%F0F+7$$\"+1-F.YF0F+7$$\"+&)GI5YF0F+7$ $\"+kbLU1&F0FO7$$\"+5$>42&F0FB7$$\"+0&>w2&F0F+7$$\"++ (>V3&F0F47$$\"++*>54&F0Fcu7$$\"++,s(4&F0FB7$$\"+&H?W5&F0FB7$$\"+!\\?66 &F0F47$$\"*p?y6&F)FO7$$\"+gcAD^F0F+7$$\"+N1jK^F0FO7$$\"+5c.S^F0F47$$\" +!eSu9&F0FB7$$\"+]b%[:&F0F47$$\"+D0Di^F0F+7$$\"++blp^F0F+7$$\"+q/1x^F0 F47$$\"+SaY%=&F0FB7$$\"+:/(=>&F0F47$$\"+!Rv#*>&F0FB7$$\"+g.o1_F0FB7$$ \"+N`39_F0FO7$$\"+5.\\@_F0FB7$$\"+!G&*)G_F0F47$$\"+b-IO_F0FW7$$\"+I_qV _F0FB7$$\"++-6^_F0FB7$$\"+v^^e_F0FO7$$\"+],#fE&F0FB7$$\"+?^Kt_F0FB7$$ \"+&4I2G&F0F+7$$\"+q]8)G&F0FO7$$\"+S+a&H&F0F+7$$\"+5]%HI&F0FB7$$\"+&)* \\.J&F0FB7$$\"+g\\v<`F0F+7$$\"+I*f^K&F0F47$$\"++\\cK`F0F+7$$\"+v)p*R`F 0FG7$$\"+][PZ`F0F47$$\"*#)zZN&F)FO7$$\"+&\\s=O&F0F+7$$\"+q^'*o`F0F+7$$ \"+Sy0w`F0FB7$$\"+:0:$Q&F0F+7$$\"+!>V-R&F0FB7$$\"+geL(R&F0FB7$$\"+N&GW S&F0FB7$$\"+57_6aF0FB7$$\"+!)Qh=aF0F47$$\"+blqDaF0F47$$\"+I#*zKaF0F+7$ $\"++>*)RaF0FB7$$\"+vX)pW&F0F+7$$\"+]s2aaF0F47$$\"+?*p6Y&F0FB7$$\"+&fi #oaF0FG7$$\"+q_NvaF0FO7$$\"+SzW#[&F0FB7$$\"+:1a*[&F0FG7$$\"+!HLm\\&F0F B7$$\"+gfs.bF0F+7$$\"+N'=3^&F0F+7$$\"+58\"z^&F0F+7$$\"+!)R+DbF0FG7$$\" +bm4KbF0FG7$$\"+I$*=RbF0FG7$$\"++?GYbF0F+7$$\"+vYP`bF0FB7$$\"+]tYgbF0F 47$$\"+?+cnbF0F+7$$\"+&p_Yd&F0FG7$$\"*PX'*ef&F0FB7$$\"++;(Hg&F0FB7$$\"+5q/5cF0FB7$$\"+?C7CcF0 F+7$$\"+SKFJcF0F47$$\"+]'[$QcF0FO7$$\"+gSUXcF0F47$$\"+q%*\\_cF0FB7$$\" +!)[dfcF0F+7$$\"+!H]mm&F0FO7$$\"++dstcF0FB7$$\"+56!3o&F0FO7$$\"+?l(yo& F0F+7$$\"+D>&\\p&F0F47$$\"+It--dF0FB7$$\"+SF54dF0FB7$$\"+]\"yhr&F0F+7$ $\"+gNDBdF0FW7$$\"+q*G.t&F0FB7$$\"+!Q/ut&F0FO7$$\"+!zzWu&F0FB7$$\"++_b ^dF0F+7$$\"+51jedF0FB7$$\"+?gqldF0F47$$\"+I9ysdF0FO7$$\"+So&)zdF0F+7$$ \"+]A$py&F0F47$$\"+gw+%z&F0FG7$$\"+qI3,eF0F47$$\"*[e\"3eF)FG7$$\"+q([b \"eF0F+7$$\"+l!RH#eF0FB7$$\"+g$H.$eF0FW7$$\"+]'>x$eF0FB7$$\"+X*4^%eF0F G7$$\"+S-]_eF0F+7$$\"+I0*)feF0FG7$$\"+D3GneF0F+7$$\"+?6nueF0FG7$$\"+59 1#)eF0F47$$\"+0fF0F47$$\"+qJSEfF0F+7$$\"+gMzLfF0F+7$$\"+bP=TfF0FB7$$\"+ ]Sd[fF0F47$$\"+SV'f&fF0F47$$\"+NYNjfF0F+7$$\"+I\\uqfF0FO7$$\"+?_8yfF0F 47$$\"+:b_&)fF0F47$$\"+5e\"H*fF0FO7$$\"++hI+gF0FW7$$\"+&R'p2gF0F+7$$\" +!p'3:gF0FB7$$\"+!)pZAgF0F+7$$\"+vs')HgF0F47$$\"+qvDPgF0F+7$$\"*'ykWgF )FB7$$\"+!fa90'F0F47$$\"+?8EegF0F+7$$\"+]!o]1'F0F47$$\"+!yu=2'F0F47$$ \"+5:oygF0F47$$\"+S#)[&3'F0F47$$\"+q\\H#4'F0F47$$\"+&p,\"*4'F0F+7$$\"+ ?%3f5'F0FB7$$\"+]^r7hF0F+7$$\"+!)=_>hF0FO7$$\"+5'Gj7'F0FB7$$\"+S`8LhF0 FB7$$\"+q?%*RhF0F47$$\"++)[n9'F0FG7$$\"+Ibb`hF0FO7$$\"+gAOghF0F+7$$\"+ !**or;'F0FG7$$\"+?d(R<'F0F47$$\"+]Cy!='F0FG7$$\"+!=*e(='F0FB7$$\"+5fR% >'F0F47$$\"+SE?,iF0F+7$$\"+l$4!3iF0F47$$\"+!4;[@'F0FB7$$\"+?Gi@iF0FG7$ $\"+]&H%GiF0F47$$\"+!GO_B'F0FO7$$\"+5I/UiF0FO7$$\"+S(\\)[iF0FB7$$\"+qk lbiF0F47$$\"*?jCE'F)F47$$\"+?UVpiF0FO7$$\"+X_SwiF0FO7$$\"+qiP$G'F0FB7$ $\"+!HZ.H'F0FW7$$\"+5$=tH'F0FO7$$\"+N$*G/jF0FB7$$\"+g.E6jF0F47$$\"+!QJ #=jF0F+7$$\"++C?DjF0FB7$$\"+DMpF0F+7$$\"+IE'p#pF0F 47$$\"+S.!4%pF0F47$$\"*0Q[&pF)F47$$\"+5]'>'pF0FO7$$\"+q>4ppF0F+7$$\"+I *=i(pF0F47$$\"+!*eM$)pF0F+7$$\"+]GZ!*pF0F47$$\"+5)*f(*pF0F47$$\"+qns/q F0FB7$$\"+IP&=,(F0FW7$$\"+!p!)*=qF0FO7$$\"+]w5EqF0FO7$$\"+5YBLqF0F47$$ \"+q:OSqF0F47$$\"+I&)[ZqF0FB7$$\"+!\\:Y0(F0FB7$$\"+]CuhqF0FO7$$\"+5%p) oqF0F+7$$\"+qj*f2(F0F+7$$\"+IL7$3(F0FO7$$\"+!H]-4(F0FB7$$\"+]sP(4(F0F4 7$$\"+5U]/rF0F87$$\"+q6j6rF0FB7$$\"+I\"e(=rF0F47$$\"+!4&)e7(F0F47$$\"+ ]?,LrF0FW7$$\"+5!R,9(F0F+7$$\"+qfEZrF0F47$$\"+IHRarF0F47$$\"+!*)>:;(F0 F+7$$\"+]okorF0FB7$$\"+5QxvrF0FB7$$\"*x+H=(F)F87$$\"+0([,>(F0F+7$$\"+S mR(>(F0F+7$$\"+vXk/sF0FO7$$\"+5D*=@(F0FB7$$\"+X/9>sF0FW7$$\"+!Q)QEsF0F 47$$\"+5jjLsF0F+7$$\"+XU)3C(F0FB7$$\"+!=K\"[sF0FW7$$\"+5,QbsF0F+7$$\"+ X!GEE(F0F47$$\"+!)f()psF0FB7$$\"+:R7xsF0FW7$$\"+]=P%G(F0F47$$\"+&y>;H( F0F47$$\"+?x'))H(F0F47$$\"+bc61tF0F47$$\"+!fjLJ(F0FG7$$\"+D:h?tF0FB7$$ \"+g%fyK(F0FB7$$\"+&R2^L(F0F47$$\"+I`NUtF0F+7$$\"+gKg\\tF0F+7$$\"+&>^o N(F0F47$$\"+I\"*4ktF0FO7$$\"+gqMrtF0FW7$$\"+&*\\fytF0FW7$$\"+IH%eQ(F0F 47$$\"+l34$R(F0F+7$$\"++)Q.S(F0FO7$$\"+Nne2uF0F87$$\"*nM[T(F)FB7$$\"+l 3V@vF0FB7$$\"*1F!GwF)FB7$$\"*D$GbyF)FB7$$\"+X'y$pyF0FB7$$\"+SSZ$)yF0FB 7$$\"+N<_!*yF0F+7$$\"+I%pv*yF0FB7$$\"+Irh/zF0F87$$\"+D[m6zF0F+7$$\"+?D r=zF0F47$$\"+?-wDzF0F47$$\"+:z!G$zF0F47$$\"+5c&)RzF0F47$$\"+5L!p%zF0FB 7$$\"+05&R&zF0F+7$$\"++()*4'zF0FO7$$\"++k/ozF0F87$$\"++T4vzF0F+7$$\"+& zT@)zF0FW7$$\"+!\\*=*)zF0FB7$$\"+!>Pi*zF0FW7$$\"+&)[G.!)F0F+7$$\"+!eK. ,)F0F87$$\"+!G!Qc\")F0FG7$$\"+Xotj\") F0F[p7$$\"+!pv7<)F0F47$$\"+NX\")y\")F0FB7$$\"+!Q`j=)F0F47$$\"+?A*Q>)F0 F+7$$\"+l5V,#)F0FB7$$\"+5*p*3#)F0F47$$\"+](3l@)F0F87$$\"+&fZSA)F0FW7$$ \"+SkeJ#)F0F+7$$\"+&GD\"R#)F0FB7$$\"+ITmY#)F0FO7$$\"+vH?a#)F0FO7$$\"+? =uh#)F0F87$$\"+l1Gp#)F0F=7$$\"+5&>oF)F0F47$$\"+b$eVG)F0F87$$\"++s*=H)F 0F47$$\"+XgV*H)F0F]s7$$\"+!*[(pI)F0F+7$$\"+NP^9$)F0F]s7$$\"*e_?K)F)F47 $$\"+&G0(G$)F0FG7$$\"+!*zNN$)F0$!\"'F07$$\"+!p5?M)F0FG7$$\"+&Rj'[$)F0F cu7$$\"++hJb$)F0FB7$$\"++)o>O)F0Fcu7$$\"+0:io$)F0FO7$$\"+5UFv$)F0F[p7$ $\"+:p#>Q)F0FG7$$\"+?'z&)Q)F0FO7$$\"+DBB&R)F0F47$$\"+I])=S)F0F+7$$\"+S />:%)F0F+7$$\"+Xe\\G%)F0F+7$$\"+]&[^V)F0F]s7$$\"+]7!=W)F0F[p7$$\"+bRX[ %)F0FB7$$\"+gm5b%)F0FB7$$\"+l$f')F0FB7$$\"+!=\\qi)F0F47 $$\"+?]sM')F0FO7$$\"+g3SU')F0FO7$$\"++n2]')F0F87$$\"+SDvd')F0F]s7$$\"+ !QGam)F0F+7$$\"+?U5t')F0FB7$$\"+g+y!o)F0FB7$$\"++fX)o)F0F+7$$\"+S<8'p) F0F+7$$\"+!e2Qq)F0F47$$\"+?M[6()F0FG7$$\"+b#f\">()F0F47$$\"+!4Nos)F0F] s7$$\"+I4^M()F0F]s7$$\"+qn=U()F0FG7$$\"+5E')\\()F0F47$$\"+]%Qvv)F0FO7$ $\"+!H9_w)F0FB7$$\"+I,*Gx)F0FB7$$\"*(fc!y)F)F+7$$\"\"*F*F+-%'COLOURG6& %$RGBGF+F+$\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!F]d`l-%%VIEWG6$;F(F`c `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 24 "Alternatively we may use" }{TEXT -1 24 " the graphing procedure " }{TEXT 0 5 "graph" }{TEXT -1 6 " . . " }{HYPERLNK 17 "graph" 1 "" "graph" }{TEXT -1 131 " from the previous section, which uses 10 digi t precision for evaluating functions, instead of using the standard Ma ple procedure " }{TEXT 0 4 "plot" }{TEXT -1 51 ", which increases the \+ precision to about 15 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "graph(arctan(x)-'arctan_seri es'(x),x=-0.9..0.9,color=blue,numpoints=80);" }}{PARA 13 "" 1 "" {GLPLOT2D 463 231 231 {PLOTDATA 2 "6&-%'CURVESG6#7ein7$$!3A+++++++!*!# =$\"\"!F,7$$!3c*****\\(=U#)*)F*F+7$$!3w*****\\*>(o'*)F*F+7$$!3++++]P%[ '*)F*$\"3K+++++++]!#F7$$!3W+++DcEZ*)F*$!3)***************HF87$$!3y**** ***\\(oH*)F*F<7$$!3A+++v$4@\"*)F*$!33+++++++?F87$$!3c******\\7`%*))F*$ \"33+++++++?F87$$!3U+++qcF$*))F*$!3/+++++++5F87$$!3++++DJ&p())F*FN7$$! 3W++++]Pf))F*$\"3)***************HF87$$!3k*****>c,T&))F*F+7$$!3y***** \\(ozT))F*FI7$$!3x******4:)o$))F*$\"3/+++++++5F87$$!3A+++](=U#))F*F[o7 $$!3#)*****z=+x)F*FD7$$!3y****** \\i!Rv)F*FD7$$!3A+++D\"Gjt)F*FI7$$!3c*********\\(=()F*FI7$$!3++++v=<,( )F*F<7$$!3C+++M^S(p)F*FD7$$!3W+++]Pf$o)F*FN7$$!3]+++JY\"om)F*FN7$$!3y* ****\\i:gm)F*$\"39+++++++SF87$$!3A++++vV[')F*FD7$$!3_*****R\\Qwk)F*FN7 $$!37+++&>;9j)F*F+7$$!3b*****\\Pf3j)F*FV7$$!3++++]7G8')F*FD7$$!3u***** >1RDg)F*F+7$$!3W+++DJq&f)F*F<7$$!3)******R+U'y&)F*F[o7$$!3y********\\7 y&)F*F67$$!3A+++voag&)F*$!39+++++++SF87$$!3?+++mk;d&)F*Fjq7$$!3b****** \\(oHa)F*F+7$$!3:+++e)[o`)F*F[o7$$!3++++D1RD&)F*F[o7$$!3w*****4.2A^)F* F+7$$!3W++++D\"y])F*F+7$$!3-+++b**[\"\\)F*FI7$$!3y*****\\PM-\\)F*FV7$$ !3A+++]ils%)F*FN7$$!3_*****>cE(p%)F*FN7$$!3I+++j)fjX)F*FN7$$!3c*****\\ 7y]X)F*F[o7$$!3++++++]P%)F*F<7$$!3u******>WCO%)F*FV7$$!3W+++v=#*>%)F*F +7$$!3P+++b#R^T)F*F+7$$!3i*****HbdDS)F*F+7$$!3y******\\PM-%)F*F[o7$$!3 A+++Dcw%Q)F*FN7$$!3!)*****fw%Qp$)F*F+7$$!3c*******\\(=n$)F*Fjs7$$!3+++ +v$4'\\$)F*F[o7$$!3E+++ME%eM)F*F[o7$$!3W+++]7.K$)F*FN7$$!3y*****\\7`WJ )F*FN7$$!3o*****>\"Gj)H)F*F+7$$!3A++++](oH)F*$!3K+++++++]F87$$!3c***** \\(oHz#)F*F[o7$$!3u*****4\\&[l#)F*FD7$$!3++++](=+++fL%>C)F*FD7$$!3y*******\\ilA)F*F[o7$$!3A+++vV)*3#)F*FI7$$ !3c******\\iS\">)F*F[o7$$!3E+++0P3v\")F*FV7$$!3++++D\"GQ<)F*FD7$$!3W++ +++Dc\")F*FI7$$!3u*****>J?L:)F*FN7$$!3y*****\\(=nQ\")F*F+7$$!3p*****>c ^G7)F*FD7$$!3A+++]P4@\")F*FI7$$!3c*****\\i:N5)F*FN7$$!3[*****>J&p(3)F* FD7$$!3+++++v$f3)F*F<7$$!3W+++v$f$o!)F*F+7$$!3y******\\7y]!)F*F[o7$$!3 A+++DJ?L!)F*F+7$$!3c********\\i:!)F*F<7$$!34+++)=nQ,)F*F+7$$!3)****** \\(o/)*zF*FN7$$!3q******oz;(*zF*FD7$$!3W+++](o/)zF*F+7$$!3y*****\\i!*G 'zF*FI7$$!3E+++Q4'*fzF*F[o7$$!3A++++DJXzF*F[o7$$!3c*****\\PMx#zF*F[o7$ $!3++++]i:5zF*F[o7$$!3W+++D\"yD*yF*F[o7$$!3y***********\\(yF*F+7$$!3A+ ++v=UdyF*F[o7$$!3F+++QMxUyF*FN7$$!3c******\\P%)RyF*FI7$$!3++++DcEAyF*F +7$$!3W++++vo/yF*FN7$$!3y*****\\P4ry(F*F+7$$!3A+++]7`pxF*F+7$$!3E+++il AdxF*F+7$$!3c*****\\7`>v(F*FN7$$!3=+++i:NNxF*FI7$$!3+++++]PMxF*FV7$$!3 W+++voz;xF*FD7$$!3.+++IVZ+xF*FN7$$!3w******\\(=#*p(F*F]y7$$!3A+++D1k\" o(F*FN7$$!3c*******\\iSm(F*F[o7$$!3++++vV[YwF*F[o7$$!38+++)ozTj(F*F[o7 $$!3W+++]i!*GwF*FI7$$!3y*****\\7G8h(F*FD7$$!3A+++++v$f(F*FI7$$!3W++++] 7`uF*FI7$$!3c**********\\7tF*FI7$$!3++++v=#\\H(F*F+7$$!3g******H$*fysF *FD7$$!3W+++]PMxsF*F<7$$!3y*****\\il(fsF*F[o7$$!3=+++)=#\\asF*FN7$$!3A ++++v=UsF*F+7$$!3A+++)=njA(F*F+7$$!3c*****\\P4YA(F*Fjq7$$!3++++]7.2sF* FD7$$!3S+++i:5/sF*FD7$$!3Y+++DJX*=(F*F[o7$$!31+++1)=k=(F*F67$$!3y***** ***\\(=<(F*FN7$$!3A+++voHarF*FN7$$!3T+++\"\\&[SrF*FN7$$!3c******\\(=n8 (F*F<7$$!3++++D19>rF*FI7$$!3v******eL%p6(F*FN7$$!3W++++Dc,rF*FN7$$!3y* ****\\P%)R3(F*F+7$$!3!******zoz;2(F*F+7$$!3A+++]iSmqF*FN7$$!3c*****\\7 G)[qF*FV7$$!3U+++XDdZqF*FD7$$!3++++++DJqF*F+7$$!3W+++v=n8qF*F+7$$!3w** ****\\P4'*pF*F[o7$$!3A+++Dc^ypF*F[o7$$!3c*******\\P4'pF*FN7$$!3'****** >\"y+epF*FV7$$!3++++v$fL%pF*F+7$$!30++++K2QpF*FN7$$!3W+++]7yDpF*F[o7$$ !3y*****\\7.#3pF*F[o7$$!3A++++]i!*oF*FN7$$!3c*****\\(o/toF*FN7$$!3++++ ](oa&oF*FN7$$!3X+++D1*y$oF*F<7$$!3y*******\\7.#oF*F<7$$!3A+++vVt-oF*F+ 7$$!3!)*****R%)R$)z'F*FN7$$!3c******\\i:&y'F*FN7$$!3++++D\"yvw'F*F[o7$ $!3W++++++]nF*F[o7$$!3y*****\\(=UKnF*F[o7$$!3A+++]P%[r'F*FV7$$!3c***** \\ilsp'F*F[o7$$!3u*****4CaMo'F*FI7$$!3+++++vozmF*FV7$$!3W+++v$4@m'F*FD 7$$!3A+++++DcmF*F+7$$!3y******\\7`WmF*FN7$$!3A+++DJ&pi'F*FN7$$!3c***** ***\\P4mF*FD7$$!3++++voz\"f'F*FN7$$!3W+++](=Ud'F*FN7$$!3y*****\\iSmb'F *FD7$$!3#)*****z=#*>a'F*FN7$$!3B++++D1RlF*Fjs7$$!3b*****\\P%[@lF*F+7$$ !3++++]i!R]'F*F[o7$$!3W+++D\"Gj['F*FI7$$!3y*********\\(okF*F[o7$$!3A++ +v=<^kF*FN7$$!3c******\\PfLkF*F[o7$$!3'******>1k1V'F*F+7$$!3++++Dc,;kF *F+7$$!3W++++vV)R'F*FN7$$!3y*****\\Pf3Q'F*F+7$$!3A+++]7GjjF*F+7$$!3c** ***\\7.dM'F*F+7$$!3+++++]7GjF*F<7$$!3W+++voa5jF*F+7$$!3-+++WB:1jF*F+7$ $!3w******\\(oHH'F*FN7$$!3_*****R[r2H'F*F<7$$!3A+++D1RviF*F+7$$!3c**** ***\\7yD'F*F[o7$$!3++++vVBSiF*F[o7$$!3X+++]ilAiF*FI7$$!3y*****\\7y]?'F *F[o7$$!3A+++++](='F*F+7$$!3c*****\\(=#*phF*FN7$$!3++++]PM_hF*FN7$$!3W +++DcwMhF*F[o7$$!3y*******\\(=fF*F[o7$$!3c*********\\i!fF*F[o7$$!3++++v=n))eF*FN7$$!3W+++]P4reF*F [o7$$!3u*****>1k\"oeF*FI7$$!3g*****\\QyO&eF*FN7$$!3y*****\\i:N&eF*F+7$ $!3A++++v$f$eF*FV7$$!3c*****\\Pf$=eF*F+7$$!3++++]7y+eF*F+7$$!3W+++DJ?$ y&F*F+7$$!3y********\\ildF*FN7$$!3A+++vo/[dF*F+7$$!3b******\\(o/t&F*F+ 7$$!3++++D1*Gr&F*FD7$$!3W++++DJ&p&F*FD7$$!3y*****\\PMxn&F*F+7$$!3E+++) o/[n&F*F+7$$!3A+++]i:gcF*FN7$$!3U+++19pecF*FD7$$!3b*****\\7yDk&F*F+7$$ !3+++++++DcF*F[o7$$!3W+++v=U2cF*F[o7$$!3y******\\P%)*e&F*FN7$$!3A+++Dc EsbF*FN7$$!3c*******\\(oabF*F[o7$$!3'******>\"yv^bF*FN7$$!3++++v$4r`&F *F+7$$!3?+++JXkNbF*FN7$$!3W+++]7`>bF*FN7$$!3\\+++8G)[]&F*FN7$$!3y***** \\7`>]&F*F+7$$!3!)*****zo/t[&F*FN7$$!3A++++]P%[&F*FD7$$!3c*****\\(ozma F*FI7$$!3'******zVB:Y&F*F+7$$!3]+++'G^0X&F*FI7$$!3++++](=#\\aF*F[o7$$! 3W+++D1kJaF*FD7$$!3-+++%4YsU&F*F+7$$!3y*******\\iST&F*FN7$$!3?+++vV['R &F*F<7$$!3a******\\i!*y`F*FN7$$!3c*****z$f3j`F*F[o7$$!3++++D\"G8O&F*F[ o7$$!3X+++++vV`F*FD7$$!3y*****\\(=\"yD*)\\F*FN7$$!3y*****\\P4Y(\\F*F[o7$$!3A+++]7.d\\F*FI7 $$!36+++DJXR\\F*F[o7$$!3+++++](=#\\F*FD7$$!38+++p/[<\\F*F+7$$!3*)***** \\(oH/\\F*FN7$$!3y******\\(=n)[F*FN7$$!3A+++D19p[F*F+7$$!37++++Dc^[F*F I7$$!3++++vV)R$[F*F+7$$!3*)******\\iS;[F*F+7$$!3$******H\"yv,[F*F[o7$$ !3y*****\\7G))z%F*F[o7$$!3A+++++D\"y%F*FN7$$!3*)********\\iSYF*FN7$$!3 5+++++++XF*FN7$$!3++++v=U#[%F*F+7$$!3!*******\\P%[Y%F*F[o7$$!3y*****\\ ilsW%F*F[o7$$!3A++++voHWF*F+7$$!36+++v$4@T%F*F+7$$!39+++Q4Y(R%F*FD7$$! 3++++]7`%R%F*F[o7$$!3))*****\\7`pP%F*FN7$$!3#)*****zVBSP%F*FI7$$!3y*** *****\\PfVF*F+7$$!3A+++vozTVF*FN7$$!36+++](=UK%F*F+7$$!3:+++8.d4VF*FN7 $$!3++++D1k1VF*FN7$$!3))*******\\i!*G%F*F[o7$$!3y*****\\P%[rUF*F[o7$$! 3A+++]i!RD%F*FN7$$!36+++D\"GjB%F*FN7$$!3++++++v=UF*FN7$$!3*)*****\\(=< ,UF*F[o7$$!3#)*****z=U#)>%F*FN7$$!3y******\\Pf$=%F*F+7$$!3A+++Dc,mTF*F +7$$!3E+++)=n8:%F*F[o7$$!37++++vV[TF*F[o7$$!3++++v$f38%F*FN7$$!3*)**** **\\7G8TF*FD7$$!3y*****\\7.d4%F*FN7$$!3A++++]7ySF*FN7$$!37+++voagSF*FN 7$$!3++++](oH/%F*F+7$$!3*)*****\\i!RDSF*F+7$$!3y*******\\7y+%F*F+7$$!3 5+++]ilsRF*F+7$$!3*)**********\\PRF*F+7$$!3+++++v=nQF*F+7$$!35++++](oz $F*F+7$$!3++++voHzPF*FI7$$!3!*******\\(=cEsS$F*FN7$$!3y*****\\7yDR$F*F+7$$!3#)*****zoHzP$F* FD7$$!3A++++++vLF*F[o7$$!37+++v=UdLF*FN7$$!3/+++)=#\\aLF*FD7$$!3++++]P %)RLF*F+7$$!3*)*****\\ilAK$F*FI7$$!3y*******\\(o/LF*F+7$$!3A+++v$4rG$F *F+7$$!3E+++Q4YsKF*F[o7$$!37+++]7`pKF*FN7$$!3++++DJ&>D$F*F[o7$$!3%**** **zVB!\\KF*FN7$$!3*)********\\PMKF*F+7$$!3A+++](=#*>$F*F+7$$!3+++++D1k JF*F+7$$!3))*****\\P%[YJF*F+7$$!3y******\\i!*GJF*F[o7$$!3A+++D\"G86$F* F+7$$!37+++++v$4$F*F[o7$$!3A++++vVBIF*F[o7$$!3y********\\7`HF*F[o7$$!3 A+++voaNHF*F+7$$!3E+++Q%)*3#HF*F+7$$!36+++](oz\"HF*F[o7$$!3++++D1R+HF* FN7$$!3))*******\\7G)GF*FN7$$!3y*****\\PM_'GF*F+7$$!3A+++]ilZGF*F+7$$! 36+++D\"y+$GF*FN7$$!3++++++]7GF*F+7$$!3))*****\\(=#\\z#F*F[o7$$!3y**** **\\PMxFF*F+7$$!3u*****>J&piFF*FN7$$!3A+++DcwfFF*FD7$$!36++++v=UFF*FI7 $$!3()*****>19pt#F*F+7$$!3++++v$4Ys#F*F[o7$$!3;+++cE(>s#F*F+7$$!3*)*** ***\\7.2FF*F+7$$!3#******H\"GQ#p#F*F+7$$!3y*****\\7`%*o#F*F[o7$$!3A+++ +](=n#F*FN7$$!3y*******\\i:g#F*FN7$$!3*)*********\\7`#F*FN7$$!3y***** \\(=n8DF*F+7$$!3%*******\\P4'\\#F*FN7$$!37+++Dc^yCF*F+7$$!3+++++v$4Y#F *F[o7$$!3*)*****\\PfLW#F*F[o7$$!31+++]7yDCF*F+7$$!3%******\\7.#3CF*FN7 $$!35++++]i!R#F*F+7$$!3++++vo/tBF*F[o7$$!3!*******\\(oaN#F*F[o7$$!31++ +D1*yL#F*F[o7$$!3%********\\7.K#F*F+7$$!36+++vVt-BF*F+7$$!3++++]i:&G#F *F+7$$!3))*****\\7yvE#F*F+7$$!31++++++]AF*F[o7$$!3&******\\(=UKAF*F+7$ $!36+++]P%[@#F*F+7$$!3++++DcE(>#F*F+7$$!3!********\\(oz@F*F+7$$!31+++v $4@;#F*F[o7$$!3%*******\\7`W@F*F[o7$$!36+++DJ&p7#F*F+7$$!3+++++]P4@F*F +7$$!3))*****\\(oz\"4#F*F[o7$$!31+++](=U2#F*F[o7$$!3%******\\iSm0#F*FN 7$$!36++++D1R?F*F[o7$$!3++++vV[@?F*F[o7$$!3*)******\\i!R+#F*F+7$$!3/++ +D\"Gj)>F*F[o7$$!3%**********\\(o>F*F+7$$!37+++v=<^>F*FN7$$!3++++]PfL> F*F+7$$!3/+++8`%*=>F*F+7$$!3*)*****\\i:g\">F*FN7$$!3/++++vV)*=F*F[o7$$ !3!******>\"y]&*=F*F+7$$!3%******\\Pf3)=F*F+7$$!35+++]7Gj=F*F+7$$!3+++ +DJqX=F*F+7$$!3*)********\\7G=F*FN7$$!30+++voa5=F*F+7$$!3%*******\\(oH z\"F*F+7$$!35+++D1Rv5F*F+7$$!3%******\\7`>+\"F*F+7$$!3t****** ***\\P%)*!#>$\"2%****************!#G7$$!3c+++v$fev*Fbgp$!2%*********** *****Fegp7$$!3G++++v=#\\*FbgpF+7$$!3/+++)=U#[%*FbgpF+7$$!3X********\\i S\"*FbgpFigp7$$!3-++++D1*y)FbgpFigp7$$!3c+++++]P%)FbgpFigp7$$FahlFbgpF +7$$FbcmFbgpFigp7$$!39++++++DcFbgpFcgp7$$F^`nFbgpF+7$$!3')********\\(= #\\FbgpF+7$$!3u*******\\7.d%FbgpF+7$$!3G+++++v=UFbgpFcgp7$$!3')****** \\(oH/%FbgpFigp7$$F[boFbgpF+7$$!32+++++]7GFbgpF+7$$!3/+++++D19FbgpF+7$ F+F+7$$\"3/+++++D19FbgpF+7$$\"32+++++]7GFbgpF+7$$\"3+++++]i:NFbgpF+7$$ \"3')******\\(oH/%FbgpFcgp7$$\"3G+++++v=UFbgpFigp7$$\"3u*******\\7.d%F bgpF+7$$\"3')********\\(=#\\FbgpF+7$$\"3+++++vVt_FbgpF+7$$\"39++++++Dc FbgpFigp7$$\"3++++]PM_hFbgpFcgp7$$\"3++++++DJqFbgpF+7$$\"3G+++v=#\\H(F bgpF+7$$\"3c+++++]P%)FbgpFcgp7$$\"3-++++D1*y)FbgpFcgp7$$\"3X********\\ iS\"*FbgpFcgp7$$\"3G++++v=#\\*FbgpF+7$$\"3c+++v$fev*FbgpFcgp7$$\"3t*** ******\\P%)*FbgpFigp7$$\"3%******\\7`>+\"F*F+7$$\"31+++]7`>5F*F+7$$\"3 .+++v$4r.\"F*FD7$$\"3&******>1R+/\"F*FN7$$\"3+++++voa5F*FN7$$\"3'***** *\\ilA2\"F*F[o7$$\"3%*******\\P%)*3\"F*F[o7$$\"31+++v=U26F*F+7$$\"3.++ ++++D6F*F+7$$\"3++++++D19F*F+7$$\"37+++++](o\"F*F+7$$\"3%******\\7y]q \"F*F+7$$\"31+++]ilA\"y]&*=F*F+7$$\"3/++++vV)*=F*FN7$$\"3*)* ****\\i:g\">F*F[o7$$\"3'******>JX*=>F*F+7$$\"3++++]PfL>F*F+7$$\"37+++v =<^>F*F[o7$$\"3%**********\\(o>F*F+7$$\"3/+++D\"Gj)>F*FN7$$\"3*)****** \\i!R+#F*F+7$$\"3++++vV[@?F*FN7$$\"36++++D1R?F*FN7$$\"3%******\\iSm0#F *F[o7$$\"31+++](=U2#F*FN7$$\"3))*****\\(oz\"4#F*FN7$$\"3+++++]P4@F*F+7 $$\"36+++DJ&p7#F*F+7$$\"3%*******\\7`W@F*FN7$$\"31+++v$4@;#F*FN7$$\"3! ********\\(oz@F*F+7$$\"3++++DcE(>#F*F+7$$\"36+++]P%[@#F*F+7$$\"3&***** *\\(=UKAF*F+7$$\"31++++++]AF*FN7$$\"3))*****\\7yvE#F*F+7$$\"3++++]i:&G #F*F+7$$\"36+++vVt-BF*F+7$$\"3%********\\7.K#F*F+7$$\"31+++D1*yL#F*FN7 $$\"3!*******\\(oaN#F*FN7$$\"3++++vo/tBF*FN7$$\"35++++]i!R#F*F+7$$\"3% ******\\7.#3CF*F[o7$$\"31+++]7yDCF*F+7$$\"3*)*****\\PfLW#F*FN7$$\"3+++ ++v$4Y#F*FN7$$\"37+++Dc^yCF*F+7$$\"3%*******\\P4'\\#F*F[o7$$\"3y***** \\(=n8DF*F+7$$\"3*)*********\\7`#F*F[o7$$\"3y*******\\i:g#F*F[o7$$\"3A ++++](=n#F*F[o7$$\"3y*****\\7`%*o#F*FN7$$\"3%)*****>\"GQ#p#F*F+7$$\"3* )******\\7.2FF*F+7$$\"3++++v$4Ys#F*FN7$$\"3()*****>19pt#F*F+7$$\"36+++ +v=UFF*FD7$$\"3A+++DcwfFF*FI7$$\"3u*****>J&piFF*F[o7$$\"3y******\\PMxF F*F+7$$\"3))*****\\(=#\\z#F*FN7$$\"3++++++]7GF*F+7$$\"36+++D\"y+$GF*F[ o7$$\"3A+++]ilZGF*F+7$$\"3y*****\\PM_'GF*F+7$$\"3))*******\\7G)GF*F[o7 $$\"3++++D1R+HF*F[o7$$\"36+++](oz\"HF*FN7$$\"3E+++Q%)*3#HF*F+7$$\"3A++ +voaNHF*F+7$$\"3y********\\7`HF*FN7$$\"3A++++vVBIF*FN7$$\"37+++++v$4$F *FN7$$\"3A+++D\"G86$F*F+7$$\"3y******\\i!*GJF*FN7$$\"3))*****\\P%[YJF* F+7$$\"3+++++D1kJF*F+7$$\"3A+++](=#*>$F*F+7$$\"3*)********\\PMKF*F+7$$ \"3')*****pVB!\\KF*F[o7$$\"3++++DJ&>D$F*FN7$$\"37+++]7`pKF*F[o7$$\"3E+ ++Q4YsKF*FN7$$\"3A+++v$4rG$F*F+7$$\"3y*******\\(o/LF*F+7$$\"3*)*****\\ ilAK$F*FD7$$\"3++++]P%)RLF*F+7$$\"3'******p=#\\aLF*FI7$$\"37+++v=UdLF* F[o7$$\"3A++++++vLF*FN7$$\"3#)*****zoHzP$F*FI7$$\"3y*****\\7yDR$F*F+7$ $\"3u*****>cEsS$F*F[o7$$\"3*)******\\i:5MF*FN7$$\"3++++vVtFMF*F[o7$$\" 32+++iSmIMF*FI7$$\"37++++DJXMF*F+7$$\"3A+++D1*GY$F*F+7$$\"3y******\\(o /[$F*F+7$$\"3))*****\\(o/)\\$F*FN7$$F`[qF*FN7$$\"36+++DJ?LNF*F+7$$\"31 +++i:&ya$F*FN7$$\"3A+++]7y]NF*F[o7$$\"3y*****\\Pf$oNF*FN7$$\"3))****** *\\Pfe$F*FN7$$\"3++++Dc^.OF*F+7$$\"36+++]P4@OF*F+7$$\"32+++(=Udj$F*F+7 $$\"3A+++v=nQOF*FN7$$\"3y*********\\il$F*F[o7$$\"3%******zoz\"fOF*F+7$ $\"3!******\\7GQn$F*F+7$$\"3)*******\\iS\"p$F*F[o7$$\"36+++vV)*3PF*F+7 $$\"3A++++DcEPF*F+7$$\"3y*****\\iSTu$F*FN7$$\"3!*******\\(=%F*F+7$$\"3*)*****\\(=<,UF*FN7$$\"3++++++v= UF*F[o7$$\"36+++D\"GjB%F*F[o7$$\"3A+++]i!RD%F*F[o7$$\"3y*****\\P%[rUF* FN7$$\"3))*******\\i!*G%F*FN7$$\"3++++D1k1VF*F[o7$$\"32+++7.d4VF*F[o7$ $\"36+++](=UK%F*F+7$$\"3\"******Rf$oDVF*F[o7$$\"3A+++vozTVF*F[o7$$\"3y ********\\PfVF*F+7$$\"3u*****pVBSP%F*FD7$$\"3))*****\\7`pP%F*F[o7$$\"3 ++++]7`%R%F*FN7$$\"39+++Q4Y(R%F*FI7$$\"36+++v$4@T%F*F+7$$\"3A++++voHWF *F+7$$\"3y*****\\ilsW%F*FN7$$\"3!*******\\P%[Y%F*FN7$$\"3++++v=U#[%F*F +7$$\"35+++++++XF*F[o7$$\"3*)********\\iSYF*F[o7$$\"3A+++++D\"y%F*F[o7 $$\"3y*****\\7G))z%F*FN7$$\"3&)*****>\"yv,[F*FN7$$\"3*)******\\iS;[F*F +7$$\"3++++vV)R$[F*F+7$$\"37++++Dc^[F*FD7$$\"3A+++D19p[F*F+7$$\"3y**** **\\(=n)[F*F[o7$$\"3*)*****\\(oH/\\F*F[o7$$\"38+++p/[<\\F*F+7$$\"3++++ +](=#\\F*FI7$$\"36+++DJXR\\F*FN7$$\"3A+++]7.d\\F*FD7$$\"3y*****\\P4Y( \\F*FN7$$\"3u*****>\"yD*)\\F*F[o7$$\"3))*******\\(=#*\\F*F+7$$\"3++++D cw4]F*FD7$$\"3c******\\PMF]F*FD7$$\"3A+++v=#\\/&F*F+7$$\"3u*****>c^y/& F*F[o7$$\"3y**********\\i]F*FN7$$\"3W+++D\"y+3&F*FN7$$\"3++++]il(4&F*F I7$$\"3c*****\\PM_6&F*FI7$$\"3A++++D\"G8&F*FI7$$\"3y*****\\i!R]^F*FD7$ $\"3W+++](oz;&F*FI7$$\"3e******HVAp^F*FD7$$\"3++++voa&=&F*FN7$$\"3c*** *****\\7._F*FN7$$\"3A+++DJq?_F*FI7$$\"3x*****z$4YA_F*FN7$$\"3y******\\ 7GQ_F*F+7$$\"3W+++v$feD&F*F[o7$$\"3S+++7y]q_F*FN7$$F_\\qF*F+7$$\"3++++ 7yD*G&F*F[o7$$\"3a*****\\i:5H&F*FI7$$\"3A+++]Pf3`F*FN7$$\"3y*****\\(=< E`F*FN7$$\"3X+++++vV`F*FI7$$\"3++++D\"G8O&F*FN7$$\"3c*****z$f3j`F*FN7$ $\"3a******\\i!*y`F*F[o7$$\"3?+++vV['R&F*FV7$$\"3y*******\\iST&F*F[o7$ $\"3\\+++)=#*pT&F*F[o7$$\"3R+++,-#*GaF*FD7$$\"3W+++D1kJaF*FI7$$\"3++++ ](=#\\aF*FN7$$\"3'******zVB:Y&F*F+7$$\"3c*****\\(ozmaF*FD7$$\"3A++++]P %[&F*FI7$$\"3!)*****zo/t[&F*F[o7$$\"3y*****\\7`>]&F*F+7$$\"3e******ozT .bF*F[o7$$\"3W+++]7`>bF*F[o7$$\"3++++v$4r`&F*F+7$$\"3'******>\"yv^bF*F [o7$$\"3c*******\\(oabF*FN7$$\"3A+++DcEsbF*F[o7$$\"3y******\\P%)*e&F*F [o7$$\"3W+++v=U2cF*FN7$$\"3+++++++DcF*FN7$$\"3b*****\\7yDk&F*F+7$$\"3A +++]i:gcF*F[o7$$\"3=+++(o/[n&F*F+7$$\"3y*****\\PMxn&F*F+7$$\"3W++++DJ& p&F*FI7$$\"3++++D1*Gr&F*FI7$$\"3b******\\(o/t&F*F+7$$\"3A+++vo/[dF*F+7 $$\"3y********\\ildF*F[o7$$\"3W+++DJ?$y&F*F+7$$\"3++++]7y+eF*F+7$$\"3c *****\\Pf$=eF*F+7$$\"3A++++v$f$eF*F<7$$\"3y*****\\i:N&eF*F+7$$\"3?+++c ,\"z&eF*F<7$$\"3W+++]P4reF*FN7$$\"3++++v=n))eF*F[o7$$\"3c*********\\i! fF*FN7$$\"3!)*****RfL%>fF*FN7$$\"3A+++D\"GQ#fF*FD7$$\"3y******\\iSTfF* F[o7$$\"3?+++\"y+e%fF*F[o7$$\"3W+++vV)*efF*F+7$$\"3+++++DcwfF*F+7$$\"3 '******p$4@\"*fF*FN7$$\"3a*****\\iST*fF*FN7$$\"3A+++](=<,'F*F[o7$$\"3y *****\\(oHHgF*F[o7$$\"3W++++](o/'F*F[o7$$\"3S+++PM_hgF*F[o7$$\"3++++DJ XkgF*FI7$$\"3c******\\7.#3'F*F+7$$\"3A+++v$4'*4'F*F+7$$\"3=+++7yD9hF*F [o7$$\"3y*******\\(=1k1V'F*F+7$ $\"3c******\\PfLkF*FN7$$\"3A+++v=<^kF*F[o7$$\"3y*********\\(okF*FN7$$ \"3W+++D\"Gj['F*FD7$$\"3++++]i!R]'F*FN7$$\"3b*****\\P%[@lF*F+7$$\"3B++ ++D1RlF*Fjq7$$\"3#)*****z=#*>a'F*F[o7$$\"3y*****\\iSmb'F*FI7$$\"3e**** **oa5elF*FD7$$\"3W+++](=Ud'F*F[o7$$\"3++++voz\"f'F*F[o7$$\"3c******** \\P4mF*FI7$$\"3A+++DJ&pi'F*F[o7$$\"3y******\\7`WmF*F[o7$$\"3A+++++DcmF *F+7$$\"3W+++v$4@m'F*FI7$$\"3+++++vozmF*F<7$$\"3u*****4CaMo'F*FD7$$\"3 o******[MY'p'F*F+7$$\"3c*****\\ilsp'F*FN7$$\"3A+++]P%[r'F*F<7$$\"3y*** **\\(=UKnF*FN7$$\"3W++++++]nF*FN7$$\"3++++D\"yvw'F*FN7$$\"3c******\\i: &y'F*F[o7$$\"3E+++Qf3)y'F*FN7$$\"3A+++vVt-oF*F+7$$\"3y*******\\7.#oF*F V7$$\"3X+++D1*y$oF*FV7$$\"3++++](oa&oF*F[o7$$\"3c*****\\(o/toF*F[o7$$ \"3A++++]i!*oF*F[o7$$\"3y*****\\7.#3pF*FN7$$\"3W+++]7yDpF*FN7$$\"3++++ v$fL%pF*F+7$$\"3'******>\"y+epF*F<7$$\"3c*******\\P4'pF*F[o7$$\"3A+++D c^ypF*FN7$$\"3w******\\P4'*pF*FN7$$\"3W+++v=n8qF*F+7$$Fh\\qF*F+7$$\"3U +++XDdZqF*FI7$$\"3c*****\\7G)[qF*F<7$$\"3A+++]iSmqF*F[o7$$\"3!******zo z;2(F*F+7$$\"3y*****\\P%)R3(F*F+7$$\"3++++pz;(4(F*FV7$$\"3W++++Dc,rF*F [o7$$\"3v******eL%p6(F*F[o7$$\"3++++D19>rF*FD7$$\"3c******\\(=n8(F*FV7 $$\"3T+++\"\\&[SrF*F[o7$$\"3A+++voHarF*F[o7$$\"3y********\\(=<(F*F[o7$ $\"3y*****>J&p(=(F*FI7$$\"3Y+++DJX*=(F*FN7$$\"3S+++i:5/sF*FI7$$\"3++++ ]7.2sF*FI7$$\"3c*****\\P4YA(F*Fjs7$$\"3A+++)=njA(F*F+7$$\"3A++++v=UsF* F+7$$\"3=+++)=#\\asF*F[o7$$\"3y*****\\il(fsF*FN7$$\"3W+++]PMxsF*FV7$$ \"3g******H$*fysF*FI7$$\"3++++v=#\\H(F*F+7$$\"3c**********\\7tF*FD7$$ \"3W++++]7`uF*FD7$$\"3A+++++v$f(F*FD7$$\"3y*****\\7G8h(F*FI7$$\"3W+++] i!*GwF*FD7$$\"38+++)ozTj(F*FN7$$\"3++++vV[YwF*FN7$$\"3c*******\\iSm(F* FN7$$\"3A+++D1k\"o(F*F[o7$$\"3w******\\(=#*p(F*F67$$\"3.+++IVZ+xF*F[o7 $$\"3O+++f%R;r(F*F+7$$\"3W+++voz;xF*FI7$$\"3+++++]PMxF*F<7$$\"3=+++i:N NxF*FD7$$\"3c*****\\7`>v(F*F[o7$$\"3K+++`-lcxF*F+7$$\"3A+++]7`pxF*F+7$ $\"3y*****\\P4ry(F*F+7$$\"3W++++vo/yF*F[o7$$\"3++++DcEAyF*F+7$$\"3c*** ***\\P%)RyF*FD7$$\"3F+++QMxUyF*F[o7$$\"3A+++v=UdyF*FN7$$\"3!)*****ptr9 'yF*FN7$$\"3y***********\\(yF*F+7$$\"3c*****4$\\-xyF*FN7$$\"3W+++D\"yD *yF*FN7$$\"3++++]i:5zF*FN7$$\"3c*****\\PMx#zF*FN7$$\"3A++++DJXzF*FN7$$ \"3=+++P4'*fzF*FN7$$\"3y*****\\i!*G'zF*FD7$$\"3W+++](o/)zF*F+7$$\"3)** ****\\(o/)*zF*F[o7$$\"34+++)=nQ,)F*F+7$$\"3c********\\i:!)F*FV7$$\"3A+ ++DJ?L!)F*F+7$$\"3y******\\7y]!)F*FN7$$\"3W+++v$f$o!)F*F+7$$\"3+++++v$ f3)F*FV7$$\"3[*****>J&p(3)F*FI7$$\"3c*****\\i:N5)F*F[o7$$\"3A+++]P4@\" )F*FD7$$\"3p*****>c^G7)F*FI7$$\"3y*****\\(=nQ\")F*F+7$$\"3u*****>J?L:) F*F[o7$$\"3W+++++Dc\")F*FD7$$\"3++++D\"GQ<)F*FI7$$\"3E+++0P3v\")F*F<7$ $\"3c******\\iS\">)F*FN7$$\"3A+++;3#[>)F*F+7$$\"3A+++vV)*3#)F*FD7$$\"3 =+++#o:K@)F*F[o7$$\"3y*******\\ilA)F*FN7$$\"3E+++l\"=rA)F*F[o7$$\"3x** ***zy!fQ#)F*F]y7$$\"3X+++D19W#)F*FI7$$\"3++++](=\"Gj)H)F*F+7$$\"3y*****\\7`WJ)F*F[o7$$\"3W+++]7.K $)F*F[o7$$\"3E+++ME%eM)F*FN7$$\"3++++v$4'\\$)F*FN7$$\"3c*******\\(=n$) F*Fjq7$$\"3!)*****fw%Qp$)F*F+7$$\"3g*****ztnIQ)F*Fjq7$$\"3A+++Dcw%Q)F* F[o7$$\"3y******\\PM-%)F*FN7$$\"3\\+++QMF0%)F*F[o7$$\"3R+++^9?<%)F*FD7 $$\"3W+++v=#*>%)F*F+7$$\"3u******>WCO%)F*F<7$$\"3++++++]P%)F*FV7$$\"3c *****\\7y]X)F*FN7$$\"3=+++7y+e%)F*F<7$$\"3A+++]ils%)F*F[o7$$\"3-+++%4@ TZ)F*FI7$$\"3y*****\\PM-\\)F*F<7$$\"3_+++If/$\\)F*F[o7$$\"3W++++D\"y]) F*F+7$$\"3++++D1RD&)F*FN7$$\"3b******\\(oHa)F*F+7$$\"3?+++mk;d&)F*Fjs7 $$\"3A+++voag&)F*Fjq7$$\"3y********\\7y&)F*F]y7$$\"3)******R+U'y&)F*FN 7$$\"3W+++DJq&f)F*FV7$$\"3@+++\\-6,')F*Fjq7$$\"3++++]7G8')F*FI7$$\"3b* ****\\Pf3j)F*F<7$$\"3_*****R\\Qwk)F*F[o7$$\"3A++++vV[')F*FI7$$\"3y**** *\\i:gm)F*Fjs7$$\"3]+++JY\"om)F*F[o7$$\"3W+++]Pf$o)F*F[o7$$\"3C+++M^S( p)F*FI7$$\"3++++v=<,()F*FV7$$\"3c*********\\(=()F*FD7$$\"3A+++D\"Gjt)F *FD7$$\"3y******\\i!Rv)F*FI7$$\"3/+++I=;b()F*FD7$$\"3W+++vV[r()F*FN7$$ \"3-+++l@6s()F*Fjs7$$\"3+++++D1*y)F*FD7$$\"3c*****\\iSm!))F*FV7$$\"3#) *****z=c,T&))F*F+7$$\"3W++++]Pf))F*F<7$$\"3++++DJ&p())F*F[o7$$\"3E+++0(3# y))F*F+7$$\"33+++&f-B*))F*FN7$$\"3c******\\7`%*))F*FD7$$\"3A+++v$4@\"* )F*FI7$$\"3y*******\\(oH*)F*FV7$$\"3W+++DcEZ*)F*FV7$$\"3++++]P%['*)F*F ]y7$$\"3w*****\\*>(o'*)F*F+7$$\"3c*****\\(=U#)*)F*F+7$$\"3A+++++++!*F* F+-%'COLOURG6&%$RGBGF+F+$\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!F[iv-%% VIEWG6$;$!\"*!\"\"$\"\"*Fciv%(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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Argument reduction to the interval " } {XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 189 " The first point to observe in the design of a procedure to evaluate th e arctangent function is that the problem can be reduced to that of ev aluating the arctangent function on the interval " }{XPPEDIT 18 0 "[0, 1]" "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Si nce " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 7 " i s an " }{TEXT 259 12 "odd function" }{TEXT -1 81 " we need only consid er arctan(x) for x non-negative, because we can then compute " } {XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 5 " for " } {TEXT 275 1 "x" }{TEXT -1 27 " negative from the formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x) = -arctan(-x);" " 6#/-%'arctanG6#%\"xG,$-F%6#,$F'!\"\"F," }{TEXT -1 13 " ------- (i)." } }{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 263 12 "____________" }{TEXT -1 17 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 69 "Then we can further reduce the interval over which we need to define " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 17 " to the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\" \"" }{TEXT -1 23 " by using the formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x) = Pi/2-arctan(1/x);" "6#/-%'arctanG 6#%\"xG,&*&%#PiG\"\"\"\"\"#!\"\"F+-F%6#*&F+F+F'F-F-" }{TEXT -1 14 " -- ----- (ii)." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 262 14 "_______ _______" }{TEXT -1 17 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Thus for a number " }{TEXT 276 1 " x" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "1 < x;" "6#2\"\"\"%\"xG" } {TEXT -1 10 ", compute " }{XPPEDIT 18 0 "1/x" "6#*&\"\"\"F$%\"xG!\"\" " }{TEXT -1 28 " which lies in the interval " }{XPPEDIT 18 0 "[0, 1]; " "6#7$\"\"!\"\"\"" }{TEXT -1 54 ", use the arctangent function define d on the interval " }{XPPEDIT 18 0 "[0, 1];" "6#7$\"\"!\"\"\"" }{TEXT -1 9 " to find " }{XPPEDIT 18 0 "arctan(1/x)" "6#-%'arctanG6#*&\"\"\"F '%\"xG!\"\"" }{TEXT -1 35 " and then subtract the result from " } {XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 46 "To illustrate this idea, we define a func tion " }{TEXT 0 5 "atan1" }{TEXT -1 51 " using Maple's arctangent func tion on the interval " }{XPPEDIT 18 0 "[0, 1];" "6#7$\"\"!\"\"\"" } {TEXT -1 58 ", but ensure that it gives no value outside this interval ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "atan1 := x -> if x >=0 and \+ x <=1 then arctan(x) else FAIL end if;\nplot('atan1'(x),x=-0.2..1.2,th ickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&atan1Gf*6#%\"xG6\"6$ %)operatorG%&arrowGF(@%31\"\"!9$1F0\"\"\"-%'arctanG6#F0%%FAILGF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 334 248 248 {PLOTDATA 2 "6&-%'CURVESG6$7N 7$$\"3W+++NIeJm!#@$\"3/[n&G1#eJmF*7$$\"3/+++pwgg:!#?$\"3%>X]?S11c\"F07 $$\"3!******R.d!eCF0$\"3W:ZX$3_!eCF07$$\"3+++++k]bLF0$\"3E3iR1Q\\bLF07 $$\"38+++0,e?=!#>$\"3ky.^+!z.#=F@7$$\"3I+++q&4cI$F@$\"3>9cbRjS/LF@7$$ \"3s+++!\\VhE'F@$\"3%G+0F^hzD'F@7$$\"39+++?lT6$*F@$\"3)))*G')*)[k%G*F@ 7$$\"3'******fYp$*>\"!#=$\"3epeMswm$>\"FU7$$\"31+++b/L,:FU$\"3C^iBFU7$$\"3%******>Fpvn#FU$\"3'praF $**>;EFU7$$\"3C+++([[[%HFU$\"3Cz7bRP*Q'GFU7$$\"3!******RvddD$FU$\"3M%[ (p+e_ZJFU7$$\"3.+++P^'4`$FU$\"3E0Nj$Q2VR$FU7$$\"3))*****RD6H$QFU$\"3#R :*)*\\r>gOFU7$$\"30+++i`V?TFU$\"3KVSi%Ga%3RFU7$$\"3\"******\\gO/U%FU$ \"3/!)4^gwApN#3&FU7$$\"3-+++7CkgeFU$\"3Pe#)Q_f4,`FU7$$\"3E+++\" *HWghFU$\"38o(=oCL8_&FU7$$\"3#******zORPX'FU$\"3;I`$pxi6t&FU7$$\"3*)** ***\\p=vt'FU$\"3F^dGqr\"*GfFU7$$\"3c*****RD2E0(FU$\"3)[wi\\LzC9'FU7$$ \"3Y+++Q$GdL(FU$\"30C3Q\"zFU$\"31>_:9X[$p'FU7$$\"3=++++4T6#)FU$\"3cJ\"e)\\d*\\(oFU7$ $\"3/+++?w=$\\)FU$\"3.;11vV)4/(FU7$$\"39+++!\\Dxy)FU$\"393>$4uE'4sFU7$ $\"3Q+++!4!pv!*FU$\"3WU:@6<\")ptFU7$$\"33+++]B;x$*FU$\"33e&y6ajE`(FU7$ $\"3k******\\f^n'*FU$\"3#z$RW\\`%\\o(FU7$$\"3,+++0-)f\")*FU$\"3#yUd>'* >6w(FU7$$\"3S+++gWWk**FU$\"3S`jg@A)[G)**FU$\"3iLdp&o)RXyFU7$$\"3c********40#***FU$\"3s **3*Qb0+&yFU7$%*undefinedGF]y-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!FfyFey- %*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!F_z-%%VIEWG6$;$!\"#Fdy$\"# 7Fdy%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Now we define a function " }{TEXT 0 5 "atan2" }{TEXT -1 7 " usi ng " }{TEXT 0 5 "atan1" }{TEXT -1 31 " and the formulas (i) and (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "atan2 := proc(x)\n if x < 0 then return -atan2(-x) end if; \n if x <=1 then atan1(x) else evalf(Pi/2)-atan1(1/x) end if;\nend p roc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&atan2Gf*6#%\"xG6\"F(F(C$@$2 9$\"\"!O,$-F$6#,$F,!\"\"F3@%1F,\"\"\"-%&atan1G6#F,,&-%&evalfG6#,$*&\" \"#F3%#PiGF6F6F6-F86#*&F6F6F,F3F3F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "xx := -5.3;\nevalf(atan2(xx));\nevalf(arctan(xx));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$!#`!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+D%4VQ\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+D% 4VQ\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "plot(['atan2'(x), Pi/2,-Pi/2],x=-8..8,color=[red,black$2],\n linestyle=[1,4$2] ,thickness=[2,1$2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 510 139 139 {PLOTDATA 2 "6'-%'CURVESG6&7W7$$!\")\"\"!$!35N\"[ALTkW\"!#<7$$!3OLLLLb C^wF-$!3&zkhSIN3W\"F-7$$!3?mmmOhzZtF-$!3Np*=%H@`N9F-7$$!3LLLL`b`1qF-$! 3V0$oP')H!H9F-7$$!3#HLLL(G,jmF-$!3*>NYZ$f#=U\"F-7$$!30nmm'*G7@jF-$!3Ic ,Xzo*QT\"F-7$$!3XLLLBr9/gF-$!3sxtYe'fdS\"F-7$$!3!)******pq$fn&F-$!3%eK 3Hm.kR\"F-7$$!3fLLLj<]O`F-$!31gY8Afb&Q\"F-7$$!3Q+++I]:)*\\F-$!3u-ZP\"y HLP\"F-7$$!3YmmmEQ7]YF-$!3%o#3y_U(*e8F-7$$!3HLLL`xdVVF-$!3v$)*f\\H9XM \"F-7$$!3I+++![z%)*RF-$!3\\RTI*=GdK\"F-7$$!35++++U'>l$F-$!3r!**e=gANI \"F-7$$!3/+++?D.=LF-$!33VWC?32y7F-7$$!3SLLLj0z9IF-$!3q*3OQH=0D\"F-7$$! 3!pmmma1Ul#F-$!3@B3wfJ[57F-7$$!3=nmm'eW([BF-$!3+iFenEGo6F-7$$!3S+++5(> M*>F-$!3z_3S(=He5\"F-7$$!3Unmm')p*)y;F-$!37(3`%yofL5F-7$$!3l******4d\" QL\"F-$!3q)=3SL)ou#*!#=7$$!3*)******Hn@05F-$!3+7lF\\r$\"3KpBr`:PxdF\\r7$$\"3AemmmK\"f$)*F\\r$\" 3)H='eu9ErxF\\r7$$\"3W******f0AE8F-$\"3A'4u3`esC*F\\r7$$\"3M)*****>kTh ;F-$\"3nK'o'HQ)*G5F-7$$\"3u)*****\\ct&)>F-$\"3J0Q_%[zU5\"F-7$$\"3e)*** **fo$eM#F-$\"3#)*3Tl*f$y;\"F-7$$\"3?KLL8QSpEF-$\"3![L^&4GO77F-7$$\"3p* ******f!)[,$F-$\"3y+/(y@F0D\"F-7$$\"3%fmmm\"R$zK$F-$\"3))QyVXI*)y7F-7$ $\"3s******zQ=qOF-$\"3$4!R\")fvy/8F-7$$\"3mJLLBW@#*RF-$\"3K1x4[)e`K\"F -7$$\"3.******H\"H)GVF-$\"393K:8&pPM\"F-7$$\"3mKLLL:$zl%F-$\"3!**e(Q7) =$f8F-7$$\"3E******\\7Z-]F-$\"31DC()pd\\t8F-7$$\"32nmmYRIM`F-$\"3?47BN 8[&Q\"F-7$$\"3?mmm13ltcF-$\"39_zO1[L'R\"F-7$$\"33LLL.x=5gF-$\"3pFKdQD# fS\"F-7$$\"3d******f,V>jF-$\"3ST\\jQb&QT\"F-7$$\"3?LLL8p&Qn'F-$\"3$*Q] 0LW1A9F-7$$\"33mmmE/'3*pF-$\"3p8u`YirG9F-7$$\"3Q+++!H_)GtF-$\"3Nu'p'fn =N9F-7$$\"3O+++ION_wF-$\"3p#y')=\"R&3W\"F-7$$\"\")F*$\"35N\"[ALTkW\"F- -%'COLOURG6&%$RGBG$\"*++++\"F)$F*F*Fd\\l-%*THICKNESSG6#\"\"#-%*LINESTY LEG6#\"\"\"-F$6&7S7$F($\"3c'*[zEjzq:F-7$F/Fa]l7$F4Fa]l7$F9Fa]l7$F>Fa]l 7$FCFa]l7$FHFa]l7$FMFa]l7$FRFa]l7$FWFa]l7$FfnFa]l7$F[oFa]l7$F`oFa]l7$F eoFa]l7$FjoFa]l7$F_pFa]l7$FdpFa]l7$FipFa]l7$F^qFa]l7$FcqFa]l7$FhqFa]l7 $F^rFa]l7$FcrFa]l7$F]sFa]l7$FgsFa]l7$FbtFa]l7$F\\uFa]l7$FauFa]l7$FfuFa ]l7$F[vFa]l7$F`vFa]l7$FevFa]l7$FjvFa]l7$F_wFa]l7$FdwFa]l7$FiwFa]l7$F^x Fa]l7$FcxFa]l7$FhxFa]l7$F]yFa]l7$FbyFa]l7$FgyFa]l7$F\\zFa]l7$FazFa]l7$ FfzFa]l7$F[[lFa]l7$F`[lFa]l7$Fe[lFa]l7$Fj[lFa]l-F_\\l6&Fa\\lF*F*F*-Ff \\lF[]l-Fj\\l6#\"\"%-F$6&7S7$F($!3c'*[zEjzq:F-7$F/F]al7$F4F]al7$F9F]al 7$F>F]al7$FCF]al7$FHF]al7$FMF]al7$FRF]al7$FWF]al7$FfnF]al7$F[oF]al7$F` oF]al7$FeoF]al7$FjoF]al7$F_pF]al7$FdpF]al7$FipF]al7$F^qF]al7$FcqF]al7$ FhqF]al7$F^rF]al7$FcrF]al7$F]sF]al7$FgsF]al7$FbtF]al7$F\\uF]al7$FauF]a l7$FfuF]al7$F[vF]al7$F`vF]al7$FevF]al7$FjvF]al7$F_wF]al7$FdwF]al7$FiwF ]al7$F^xF]al7$FcxF]al7$FhxF]al7$F]yF]al7$FbyF]al7$FgyF]al7$F\\zF]al7$F azF]al7$FfzF]al7$F[[lF]al7$F`[lF]al7$Fe[lF]al7$Fj[lF]alFc`lFe`lFf`l-%+ AXESLABELSG6$Q\"x6\"Q!Fcdl-%%VIEWG6$;F(Fj[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Further ar gument reduction within the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\" \"!\"\"\"" }{TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 63 "Suppose that we decide to use the \+ Maclaurin series to evaluate " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctan G6#%\"xG" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "arctan(x) = x -x^3/3 + x^5/5 - x^7/7+x^9/9 + ` . . . ` " "6#/-%'arctanG6#%\"xG,.F'\"\"\"*&F'\"\"$F+!\"\"F,*&F'\"\"&F.F,F)*&F' \"\"(F0F,F,*&F'\"\"*F2F,F)%(~.~.~.~GF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 74 "This series converges rather slowly in general, and \+ extremely slowly when " }{TEXT 278 1 "x" }{TEXT -1 48 " is close to 1. It does not converge at all for " }{TEXT 277 1 "x" }{TEXT -1 17 " gre ater than 1. " }}{PARA 0 "" 0 "" {TEXT -1 74 "It is therefore desirabl e to reduce the range further within the interval " }{XPPEDIT 18 0 "[0 ,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 18 " to some interval " }{XPPEDIT 18 0 "[0,a]" "6#7$\"\"!%\"aG" }{TEXT -1 8 ", where " }{TEXT 279 1 "a" } {TEXT -1 29 " is considerably less than 1." }}{PARA 0 "" 0 "" {TEXT -1 38 "This is possible by using the formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x)=arctan(x[0]) + arctan((x-x[0] )/(1+x*x[0]))" "6#/-%'arctanG6#%\"xG,&-F%6#&F'6#\"\"!\"\"\"-F%6#*&,&F' F.&F'6#F-!\"\"F.,&F.F.*&F'F.&F'6#F-F.F.F5F." }{TEXT -1 15 " ------- (i ii)." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 264 22 "______________ ________" }{TEXT -1 18 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "This formula comes from the add ition formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan (alpha-beta) = (tan(alpha)-tan(beta))/(1+tan(alpha)*tan(beta))" "6#/-% $tanG6#,&%&alphaG\"\"\"%%betaG!\"\"*&,&-F%6#F(F)-F%6#F*F+F),&F)F)*&-F% 6#F(F)-F%6#F*F)F)F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "y=arctan(x)" "6#/%\"yG-%'arctanG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[0]=arctan(x[0])" "6#/&%\"yG6#\"\"!-%'arcta nG6#&%\"xG6#F'" }{TEXT -1 7 ", then " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(y-y[0]) = (tan(y)-tan(y[0]))/(1+tan(y)*tan(y[0] ))" "6#/-%$tanG6#,&%\"yG\"\"\"&F(6#\"\"!!\"\"*&,&-F%6#F(F)-F%6#&F(6#F, F-F),&F)F)*&-F%6#F(F)-F%6#&F(6#F,F)F)F-" }{TEXT -1 1 " " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (x-x[0])/(1+x*x[0]);" "6#/% !G*&,&%\"xG\"\"\"&F'6#\"\"!!\"\"F(,&F(F(*&F'F(&F'6#F+F(F(F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-y[0]=arctan((x-x[0])/(1+x*x[0]))" "6# /,&%\"yG\"\"\"&F%6#\"\"!!\"\"-%'arctanG6#*&,&%\"xGF&&F06#F)F*F&,&F&F&* &F0F&&F06#F)F&F&F*" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=y[0]+arcta n((x-x[0])/(1+x*x[0]))" "6#/%\"yG,&&F$6#\"\"!\"\"\"-%'arctanG6#*&,&%\" xGF)&F/6#F(!\"\"F),&F)F)*&F/F)&F/6#F(F)F)F2F)" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Divide th e interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 20 " i nto the intervals " }{XPPEDIT 18 0 "[0,tan(Pi/12)]" "6#7$\"\"!-%$tanG6 #*&%#PiG\"\"\"\"#7!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "[tan(Pi/1 2),1]" "6#7$-%$tanG6#*&%#PiG\"\"\"\"#7!\"\"F)" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 84 "Once we have defined a (restricted) funct ion by the Maclaurin series on an interval " }{XPPEDIT 18 0 "[0,a]" "6 #7$\"\"!%\"aG" }{TEXT -1 57 ", we shall automatically have it defined \+ on the interval " }{XPPEDIT 18 0 "[-a,a]" "6#7$,$%\"aG!\"\"F%" }{TEXT -1 82 ", which is symmetrical about 0.\nIt is now sufficient to find a n approximation for " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG " }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-tan(Pi/12),tan(Pi /12)]" "6#7$,$-%$tanG6#*&%#PiG\"\"\"\"#7!\"\"F,-F&6#*&F)F*F+F," } {TEXT -1 33 " or approximately [-0.268,0.268]." }}{PARA 0 "" 0 "" {TEXT -1 20 "Formula (iii) with " }{XPPEDIT 18 0 "x[0]=1/sqrt(3)" "6# /&%\"xG6#\"\"!*&\"\"\"F)-%%sqrtG6#\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 266 1 "~" }{TEXT -1 11 " 0.577 and " }{XPPEDIT 18 0 "arctan(x[0])" "6# -%'arctanG6#&%\"xG6#\"\"!" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "arctan(1/ sqrt(3)) = Pi/6" "6#/-%'arctanG6#*&\"\"\"F(-%%sqrtG6#\"\"$!\"\"*&%#PiG F(\"\"'F-" }{TEXT -1 26 " can be used to calculate " }{XPPEDIT 18 0 "a rctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 18 " in the interval " } {XPPEDIT 18 0 "[tan(Pi/12),1]" "6#7$-%$tanG6#*&%#PiG\"\"\"\"#7!\"\"F) " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "To illustrate this idea, we define a function " }{TEXT 0 5 "atan3" }{TEXT -1 52 " using Maple's arctangent function on the inte rval " }{XPPEDIT 18 0 "[-tan(Pi/12), tan(Pi/12)];" "6#7$,$-%$tanG6#*& %#PiG\"\"\"\"#7!\"\"F,-F&6#*&F)F*F+F," }{TEXT -1 58 ", but ensure that it gives no value outside this interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "atan3 := x -> if abs (x)<=0.2679491924 then arctan(x) else FAIL end if;\nplot('atan3'(x),x= -0.4..0.4,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&atan3Gf* 6#%\"xG6\"6$%)operatorG%&arrowGF(@%1-%$absG6#9$$\"+C>\\zE!#5-%'arctanG F0%%FAILGF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7J7$$!3++++M#e)yE!#=$!31+g^FESkOe#F*$!3v*=k> 3t$GDF*7$$!3'******\\^x!*\\#F*$!3MV([7R=*[CF*7$$!3)******H\">1DBF*$!3; 63reNZ%G#F*7$$!33+++w))yr@F*$!3+W6oM%y&Q@F*7$$!3/+++S(R#**>F*$!31-9eoX At>F*7$$!36++++@)f#=F*$!3`.*QGj%31=F*7$$!3'*******fi,f;F*$!3;$RDI@USk \"F*7$$!3#******4G&R2:F*$!3CX#G*Q78'\\\"F*7$$!3')*****HF.rK\"F*$!34\"z xde$R>8F*7$$!3-+++$HsV<\"F*$!3I%*otXx,p6F*7$$!3_******R&)4n**!#>$!395t 3'[)GM**Ffo7$$!3!*******H\\[%R)Ffo$!3!e+A'p*\\[P)Ffo7$$!3S******R&y!pm Ffo$!3]7b1vv@fmFfo7$$!3;+++SO3E]Ffo$!3P$=Wm#y&=-&Ffo7$$!3&)******H3z6L Ffo$!3k:6-S3e5LFfo7$$!3\"*******z[`P2Vs\"Ffo$\"3&)3P6Gj8CqM8F*$\"3P/HGo*foK\"F*7$$\"34+++,.W2:F *$\"3S\"4k'e_<'\\\"F*7$$\"3-+++fp'Rm\"F*$\"3e#y5m\"*f)[;F*7$$\"3&***** *4%>4N=F*$\"3C]S%>6**[\"=F*7$$\"3')*****H@2h*>F*$\"3Wt*=SD7-(>F*7$$\"3 )******fc9W;#F*$\"3uKVk._`J@F*7$$\"30+++od'*GBF*$\"3eMt6zo<)G#F*7$$\"3 ;+++EcB,DF*$\"3Iq6dR&\\4X#F*7$$\"3%)*****4!Q>%e#F*$\"32]&H\"Q&p)GDF*7$ $\"3++++v>:nEF*$\"3sG \+ " 0 "" {MPLTEXT 1 0 327 "atan4 := proc(xx)\n local x,t;\n x := eva lf(xx);\n if x<0 then return -atan4(-x) end if;\n if x<=1 then \n \+ if x>=0 and x<=0.2679491924 then return atan3(x)\n else\n \+ t := evalf(1/sqrt(3)); \n return evalf(Pi/6)+atan3((x-t) /(1+x*t));\n end if;\n else return evalf(Pi/2)-atan4(1/x) end i f;\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&atan4Gf*6#%#xxG6$% \"xG%\"tG6\"F+C%>8$-%&evalfG6#9$@$2F.\"\"!O,$-F$6#,$F.!\"\"F;@%1F.\"\" \"@%31F5F.1F.$\"+C>\\zE!#5O-%&atan3G6#F.C$>8%-F06#*&F>F>-%%sqrtG6#\"\" $F;O,&-F06#,$*&\"\"'F;%#PiGF>F>F>-FH6#*&,&F.F>FLF;F>,&F>F>*&FLF>F.F>F> F;F>O,&-F06#,$*&\"\"#F;FenF>F>F>-F$6#*&F>F>F.F;F;F+F+F+" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following exam ple shows how this works in the most complicated case where the proced ure " }{TEXT 0 5 "atan4" }{TEXT -1 73 " is called 3 times, each time w ith different argument, and the procedure " }{TEXT 0 5 "atan3" }{TEXT -1 17 " is called once. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "xx := -1.2;\nprintlevel := 50:\neva lf(atan4(xx));\nprintlevel := 1:\nevalf(arctan(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$!#7!\"\"" }}{PARA 9 "" 1 "" {TEXT -1 29 "\{-- > enter atan4, args = -1.2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$! #7!\"\"" }}{PARA 9 "" 1 "" {TEXT -1 28 "\{--> enter atan4, args = 1.2 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"#7!\"\"" }}{PARA 9 "" 1 " " {TEXT -1 36 "\{--> enter atan4, args = .8333333333" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"+LLLL$)!#5" }}{PARA 9 "" 1 "" {TEXT -1 25 " \{--> enter sqrt, args = 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG\" \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"cG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"y G\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"gG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG\"\"$" }}{PARA 9 "" 1 "" {TEXT -1 39 " <-- exit sqrt (now in atan4) = 3^(1/2)\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG$\"+$p-Nx&!#5" }}{PARA 9 "" 1 "" {TEXT -1 36 "\{--> enter \+ atan3, args = .1728301293" }}{PARA 9 "" 1 "" {TEXT -1 37 "\{--> enter \+ arctan, args = .1728301293" }}{PARA 9 "" 1 "" {TEXT -1 43 "\{--> enter evalf/arctan, args = .1728301293" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"xG$\"+$H,$G%#xrG$\"+$H,$G< !#5" }}{PARA 9 "" 1 "" {TEXT -1 52 "<-- exit evalf/arctan (now in arct an) = .1711395005\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+0]R6 " 0 "" {MPLTEXT 1 0 106 "plot(['at an4'(x),Pi/2,-Pi/2],x=-8..8,color=[red,black$2],\n linestyle =[1,4$2],thickness=[2,1$2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 510 139 139 {PLOTDATA 2 "6'-%'CURVESG6&7W7$$!\")\"\"!$!35N\"[ALTkW\"!#<7$$!3OL LLLbC^wF-$!3&zkhSIN3W\"F-7$$!3?mmmOhzZtF-$!3Np*=%H@`N9F-7$$!3LLLL`b`1q F-$!3V0$oP')H!H9F-7$$!3#HLLL(G,jmF-$!3*>NYZ$f#=U\"F-7$$!30nmm'*G7@jF-$ !3Ic,Xzo*QT\"F-7$$!3XLLLBr9/gF-$!3sxtYe'fdS\"F-7$$!3!)******pq$fn&F-$! 3%eK3Hm.kR\"F-7$$!3fLLLj<]O`F-$!31gY8Afb&Q\"F-7$$!3Q+++I]:)*\\F-$!3u-Z P\"yHLP\"F-7$$!3YmmmEQ7]YF-$!3%o#3y_U(*e8F-7$$!3HLLL`xdVVF-$!3v$)*f\\H 9XM\"F-7$$!3I+++![z%)*RF-$!3\\RTI*=GdK\"F-7$$!35++++U'>l$F-$!3%4**e=gA NI\"F-7$$!3/+++?D.=LF-$!33VWC?32y7F-7$$!3SLLLj0z9IF-$!3q*3OQH=0D\"F-7$ $!3!pmmma1Ul#F-$!3@B3wfJ[57F-7$$!3=nmm'eW([BF-$!3+iFenEGo6F-7$$!3S+++5 (>M*>F-$!3,`3S(=He5\"F-7$$!3Unmm')p*)y;F-$!3M(3`%yofL5F-7$$!3l******4d \"QL\"F-$!3#)*=3SL)ou#*!#=7$$!3*)******Hn@05F-$!3+7lF\\r$\"3AoBr`:PxdF\\r7$$\"3AemmmK\"f$)*F\\r$ \"3)=='eu9ErxF\\r7$$\"3W******f0AE8F-$\"3A'4u3`esC*F\\r7$$\"3M)*****>k Th;F-$\"3nK'o'HQ)*G5F-7$$\"3u)*****\\ct&)>F-$\"3J0Q_%[zU5\"F-7$$\"3e)* ****fo$eM#F-$\"3/!4Tl*f$y;\"F-7$$\"3?KLL8QSpEF-$\"3-N8b4GO77F-7$$\"3p* ******f!)[,$F-$\"3y+/(y@F0D\"F-7$$\"3%fmmm\"R$zK$F-$\"3))QyVXI*)y7F-7$ $\"3s******zQ=qOF-$\"3$4!R\")fvy/8F-7$$\"3mJLLBW@#*RF-$\"3K1x4[)e`K\"F -7$$\"3.******H\"H)GVF-$\"393K:8&pPM\"F-7$$\"3mKLLL:$zl%F-$\"3!**e(Q7) =$f8F-7$$\"3E******\\7Z-]F-$\"31DC()pd\\t8F-7$$\"32nmmYRIM`F-$\"3?47BN 8[&Q\"F-7$$\"3?mmm13ltcF-$\"39_zO1[L'R\"F-7$$\"33LLL.x=5gF-$\"3pFKdQD# fS\"F-7$$\"3d******f,V>jF-$\"3ST\\jQb&QT\"F-7$$\"3?LLL8p&Qn'F-$\"3$*Q] 0LW1A9F-7$$\"33mmmE/'3*pF-$\"3p8u`YirG9F-7$$\"3Q+++!H_)GtF-$\"3Nu'p'fn =N9F-7$$\"3O+++ION_wF-$\"3p#y')=\"R&3W\"F-7$$\"\")F*$\"35N\"[ALTkW\"F- -%'COLOURG6&%$RGBG$\"*++++\"F)$F*F*Fd\\l-%*THICKNESSG6#\"\"#-%*LINESTY LEG6#\"\"\"-F$6&7S7$F($\"3c'*[zEjzq:F-7$F/Fa]l7$F4Fa]l7$F9Fa]l7$F>Fa]l 7$FCFa]l7$FHFa]l7$FMFa]l7$FRFa]l7$FWFa]l7$FfnFa]l7$F[oFa]l7$F`oFa]l7$F eoFa]l7$FjoFa]l7$F_pFa]l7$FdpFa]l7$FipFa]l7$F^qFa]l7$FcqFa]l7$FhqFa]l7 $F^rFa]l7$FcrFa]l7$F]sFa]l7$FgsFa]l7$FbtFa]l7$F\\uFa]l7$FauFa]l7$FfuFa ]l7$F[vFa]l7$F`vFa]l7$FevFa]l7$FjvFa]l7$F_wFa]l7$FdwFa]l7$FiwFa]l7$F^x Fa]l7$FcxFa]l7$FhxFa]l7$F]yFa]l7$FbyFa]l7$FgyFa]l7$F\\zFa]l7$FazFa]l7$ FfzFa]l7$F[[lFa]l7$F`[lFa]l7$Fe[lFa]l7$Fj[lFa]l-F_\\l6&Fa\\lF*F*F*-Ff \\lF[]l-Fj\\l6#\"\"%-F$6&7S7$F($!3c'*[zEjzq:F-7$F/F]al7$F4F]al7$F9F]al 7$F>F]al7$FCF]al7$FHF]al7$FMF]al7$FRF]al7$FWF]al7$FfnF]al7$F[oF]al7$F` oF]al7$FeoF]al7$FjoF]al7$F_pF]al7$FdpF]al7$FipF]al7$F^qF]al7$FcqF]al7$ FhqF]al7$F^rF]al7$FcrF]al7$F]sF]al7$FgsF]al7$FbtF]al7$F\\uF]al7$FauF]a l7$FfuF]al7$F[vF]al7$F`vF]al7$FevF]al7$FjvF]al7$F_wF]al7$FdwF]al7$FiwF ]al7$F^xF]al7$FcxF]al7$FhxF]al7$F]yF]al7$FbyF]al7$FgyF]al7$F\\zF]al7$F azF]al7$FfzF]al7$F[[lF]al7$F`[lF]al7$Fe[lF]al7$Fj[lF]alFc`lFe`lFf`l-%+ AXESLABELSG6$Q\"x6\"Q!Fcdl-%%VIEWG6$;F(Fj[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 82 "An arbitra ry precision procedure to evaluate the inverse tangent function: atanA P " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 6 "atanAP" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2435 "atanAP := proc(xx::realcons)\n \+ local x,z,saveDigits,eps,maxit,isneg,flag,\n even,k,sum,term,pow, pi,t;\n \n\011\011 # Increase precision for the computation by a few \+ digits\n\011\011 saveDigits := Digits;\n Digits := Digits + length(D igits)+1;\n pi := evalf(Pi); # Maple evaluates pi rapidly \n x := \+ evalf(xx);\n\n # Handle the argument reduction\n if x<0 then\n \+ isneg := true;\n x := -x;\n else isneg := false end if;\n\n \+ if x<0.267949192431122 then\n flag := 1;\n elif x>3.7320508075 6888 then\n flag := 4;\n x := 1/x;\n else\n t := eval f(root3)/3; \n if x<1 then\n flag := 2;\n x := (x-t)/(1+x*t);\n else\n flag := 3;\n x := (1-x*t )/(x+t);\n end if;\n end if;\n \n # Initialisation for Mac laurin series loop.\n eps := Float(1,-saveDigits);\n maxit := Digi ts*3;\n\n pow := x;\n z := x*x;\n sum := x;\n even := false; \+ \n for k from 3 to maxit by 2 do\n pow := pow*z;\n term := pow/k;\n if even then\n sum := sum+term;\n else\n \+ sum := sum-term;\n end if;\n if abs(term)<=eps*abs(su m) then break end if;\n even := not even;\n end do;\n \n if \+ flag=2 then sum := sum + pi/6\n elif flag=3 then sum := pi/3-sum\n \+ elif flag=4 then sum := pi/2-sum end if;\n if isneg then sum := -su m end if;\n\n # Return arctangent rounded to the original precision \n Digits := saveDigits;\n return evalf(sum)\nend proc: # of atanA P\n\n`evalf/constant/root3` := proc()\nlocal d,r;\nglobal _root3;\n \+ if Digits<=55 then evalf(1.7320508075688772935274463415058723669428052 53810380628)\n elif Digits<=length(op(1,_root3)) then evalf(_root3) \n else\n d := length(op(1,_root3));\n r := _root3;\n \+ while d " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Testing the procedure atanAP" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "plot(['atanAP'(x),Pi/2,-Pi/2],x=-8..8,color=[red,black$2],\n \+ linestyle=[1,4$2],thickness=[2,1$2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 487 198 198 {PLOTDATA 2 "6'-%'CURVESG6&7W7$$!\")\"\"!$!35N\" [ALTkW\"!#<7$$!3++++LbC^wF-$!3a\\g0/`$3W\"F-7$$!3c*****p8'zZtF-$!3%4.D %H@`N9F-7$$!3(******HbNl+(F-$!3(4lhP')H!H9F-7$$!3c*****H(G,jmF-$!3)*3! RZ$f#=U\"F-7$$!3T+++(*G7@jF-$!3!\\He%zo*QT\"F-7$$!3<+++Cr9/gF-$!3ar`[e 'fdS\"F-7$$!3!)******pq$fn&F-$!3iD$3Hm.kR\"F-7$$!3J+++k<]O`F-$!3_vs:Af b&Q\"F-7$$!3Q+++I]:)*\\F-$!3u-ZP\"yHLP\"F-7$$!3#)*****p#Q7]YF-$!3ggbz_ U(*e8F-7$$!3,+++axdVVF-$!3pSN*\\H9XM\"F-7$$!3')******z%z%)*RF-$!3FRTI* =GdK\"F-7$$!++U'>l$!\"*$!+-E_.8Fgo7$$!+?D.=LFgo$!+?32y7Fgo7$$!+k0z9IFg o$!+%H=0D\"Fgo7$$!+Zl?aEFgo$!+gJ[57Fgo7$$!+(eW([BFgo$!+oEGo6Fgo7$$!+6( >M*>Fgo$!+)=He5\"Fgo7$$!+()p*)y;Fgo$!+zofL5Fgo7$$!+6d\"QL\"Fgo$!+Q$)ou #*!#57$$!+Jn@05Fgo$!+st**zyF]r7$$!*n\"eBmFgo$!+<'Q,&eF]r7$$!+?dK\\]F]r $!+f(een%F]r7$$!*xp]Z$Fgo$!+%R@XM$F]r7$$!3))*****\\0Aqx\"!#=$!39Gh.vOm e)ps*yF[t7$$\"3#*******4+#[o\"Fes$\"3ky SBqU9p;Fes7$$\"*O9'[MFgo$\"+)>(*3K$F]r7$$\"+DD+%)\\F]r$\"+A*pOi%F]r7$$ \"*p!R>lFgo$\"+Z:PxdF]r7$$\"*E8f$)*Fgo$\"+r9ErxF]r7$$\"+f0AE8Fgo$\"+F& esC*F]r7$$\"+>kTh;Fgo$\"+HQ)*G5Fgo7$$\"+\\ct&)>Fgo$\"+%[zU5\"Fgo7$$\"* 'o$eM#F)$\"+(*f$y;\"Fgo7$$\"*\"QSpEF)$\"+4GO77Fgo7$$\"*g!)[,$F)$\"+=s_ ]7Fgo7$$\"*#R$zK$F)$\"+YI*)y7Fgo7$$\"*)Q=qOF)$\"+gvy/8Fgo7$$\"3#)***** *>W@#*RF-$\"3v24!z%)e`K\"F-7$$\"3#*******H\"H)GVF-$\"393K:8&pPM\"F-7$$ \"3%*******H:$zl%F-$\"3QB2C7)=$f8F-7$$\"3:+++]7Z-]F-$\"31DC()pd\\t8F-7 $$\"3z******\\RIM`F-$\"3!pPW`L\"[&Q\"F-7$$\"3#)******43ltcF-$\"3#HQok! [L'R\"F-7$$\"3N++++x=5gF-$\"3SMM[QD#fS\"F-7$$\"3d******f,V>jF-$\"3=T\\ jQb&QT\"F-7$$\"3e******4p&Qn'F-$\"3rV=)HVk?U\"F-7$$\"3o******H/'3*pF-$ \"3M^UgYirG9F-7$$\"3Q+++!H_)GtF-$\"3Nu'p'fn=N9F-7$$\"3O+++ION_wF-$\"3p #y')=\"R&3W\"F-7$$\"\")F*$\"35N\"[ALTkW\"F--%'COLOURG6&%$RGBG$\"*++++ \"F)$F*F*Ff\\l-%*THICKNESSG6#\"\"#-%*LINESTYLEG6#\"\"\"-F$6&7S7$F($\"3 c'*[zEjzq:F-7$$!3OLLLLbC^wF-Fc]l7$$!3?mmmOhzZtF-Fc]l7$$!3LLLL`b`1qF-Fc ]l7$$!3#HLLL(G,jmF-Fc]l7$$!30nmm'*G7@jF-Fc]l7$$!3XLLLBr9/gF-Fc]l7$FMFc ]l7$$!3fLLLj<]O`F-Fc]l7$FWFc]l7$$!3YmmmEQ7]YF-Fc]l7$$!3HLLL`xdVVF-Fc]l 7$$!3I+++![z%)*RF-Fc]l7$$!35++++U'>l$F-Fc]l7$$!3/+++?D.=LF-Fc]l7$$!3SL LLj0z9IF-Fc]l7$$!3!pmmma1Ul#F-Fc]l7$$!3=nmm'eW([BF-Fc]l7$$!3S+++5(>M*> F-Fc]l7$$!3Unmm')p*)y;F-Fc]l7$$!3l******4d\"QL\"F-Fc]l7$$!3*)******Hn@ 05F-Fc]l7$$!3qkmmm;eBmFesFc]l7$$!3Wnmmm(p]Z$FesFc]l7$$!3?\"[LLLLu*yF[t Fc]l7$$\"3c`mmmVh[MFesFc]l7$$\"3x\"******p!R>lFesFc]l7$$\"3AemmmK\"f$) *FesFc]l7$$\"3W******f0AE8F-Fc]l7$$\"3M)*****>kTh;F-Fc]l7$$\"3u)***** \\ct&)>F-Fc]l7$$\"3e)*****fo$eM#F-Fc]l7$$\"3?KLL8QSpEF-Fc]l7$$\"3p**** ***f!)[,$F-Fc]l7$$\"3%fmmm\"R$zK$F-Fc]l7$$\"3s******zQ=qOF-Fc]l7$$\"3m JLLBW@#*RF-Fc]l7$$\"3.******H\"H)GVF-Fc]l7$$\"3mKLLL:$zl%F-Fc]l7$$\"3E ******\\7Z-]F-Fc]l7$$\"32nmmYRIM`F-Fc]l7$$\"3?mmm13ltcF-Fc]l7$$\"33LLL .x=5gF-Fc]l7$FczFc]l7$$\"3?LLL8p&Qn'F-Fc]l7$$\"33mmmE/'3*pF-Fc]l7$Fb[l Fc]l7$Fg[lFc]l7$F\\\\lFc]l-Fa\\l6&Fc\\lF*F*F*-Fh\\lF]]l-F\\]l6#\"\"%-F $6&7S7$F($!3c'*[zEjzq:F-7$Ff]lFcfl7$Fi]lFcfl7$F\\^lFcfl7$F_^lFcfl7$Fb^ lFcfl7$Fe^lFcfl7$FMFcfl7$Fi^lFcfl7$FWFcfl7$F]_lFcfl7$F`_lFcfl7$Fc_lFcf l7$Ff_lFcfl7$Fi_lFcfl7$F\\`lFcfl7$F_`lFcfl7$Fb`lFcfl7$Fe`lFcfl7$Fh`lFc fl7$F[alFcfl7$F^alFcfl7$FaalFcfl7$FdalFcfl7$FgalFcfl7$FjalFcfl7$F]blFc fl7$F`blFcfl7$FcblFcfl7$FfblFcfl7$FiblFcfl7$F\\clFcfl7$F_clFcfl7$FbclF cfl7$FeclFcfl7$FhclFcfl7$F[dlFcfl7$F^dlFcfl7$FadlFcfl7$FddlFcfl7$FgdlF cfl7$FjdlFcfl7$F]elFcfl7$FczFcfl7$FaelFcfl7$FdelFcfl7$Fb[lFcfl7$Fg[lFc fl7$F\\\\lFcflFielF[flF\\fl-%+AXESLABELSG6$Q\"x6\"Q!Fiil-%%VIEWG6$;F(F \\\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Digits := 300:\nsqrt(24)/5; \nxx:= evalf(%):\natanAP(xx);\narctan(xx);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"\"\"&!\"\"\"\"'#F&F%F&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"g]lDTpxx+dzA/zCLo*omYR![$\\QoOja;SG%\\0#4 F<'RB;*3^r1$*o7![@=8**QTF6Kux2)Q'**[e%GC*Qr&oA-nd\\aN\\HBX`!*[=nD#y!)e ziACFR\"z**3#Hbyfl!)*H*zy?]xw6w:Yk\"Gf^-I-Mu\")*zDMZ#=6P4/sIh.JtL>v(!$ +$" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"g]lDTpxx+dzA/zCLo*omYR![$\\Qo Oja;SG%\\0#4F<'RB;*3^r1$*o7![@=8**QTF6Kux2)Q'**[e%GC*Qr&oA-nd\\aN\\HBX `!*[=nD#y!)eziACFR\"z**3#Hbyfl!)*H*zy?]xw6w:Yk\"Gf^-I-Mu\")*zDMZ#=6P4/ sIh.JtL>v(!$+$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Digits := 30 0:\nsqrt(3);\nxx:= evalf(%):\natanAP(xx);\narctan(xx);\nDigits := 10: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"\"$#\"\"\"\"\"#" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#$\"g]lw8(\\n3pyzPF))3\"=o`\\M:Tc&G)=QO1S!4b5Ei7T \\&ehP:y(>AD.Fw9KF,J=jJu[@`=qthk+4q%4]\"Rq#G/OJX3Rugo^O:GJ!\\bVFW]&p#Q F$*o02yT\\M4imp?woag-T'[JeOFN]7LJsl!Gw;$4hW@ahu(f'>^v>Z5!$*H" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"g]lw8(\\n3pyzPF))3\"=o`\\M:Tc&G)=QO1S!4b5 Ei7T\\&ehP:y(>AD.Fw9KF,J=jJu[@`=qthk+4q%4]\"Rq#G/OJX3Rugo^O:GJ!\\bVFW] &p#QF$*o02yT\\M4imp?woag-T'[JeOFN]7LJsl!Gw;$4hW@ahu(f'>^v>Z5!$*H" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Speed comparisons " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Maple's f unction " }{TEXT 0 6 "arctan" }{TEXT -1 16 " is faster than " }{TEXT 0 6 "atanAP" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "st := time():\nfor i to 1000 do at anAP(rand()*Float(1,-11)) end do:\ntime()-st;\nst := time():\nfor i to 1000 do arctan(rand()*Float(1,-11)) end do:\ntime()-st;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"%]8!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"$3&!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 181 "st := time():\nDigits := 50:\nfor i to 100 do atan AP(rand()*Float(1,-11)) end do:\ntime()-st;\nst := time():\nfor i to 1 00 do arctan(rand()*Float(1,-11)) end do:\ntime()-st;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%89!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$3#!\"$" }}}{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 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }