{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 "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{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 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 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 "Maple O utput" -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 } {PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 54 "Evaluating the sine function from its Maclaurin series" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, \+ Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 25.3.2 007" }}{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 th e Maclaurin series for " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 25 "The Maclaurin series for " }{XPPEDIT 18 0 "sin( x);" "6#-%$sinG6#%\"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#/-%$SumG6$*(),$\"\"\"!\"\"%\"nGF*)%\"xG,&*&\"\"#F*F,F*F*F*F*F*-% *factorialG6#,&*&F1F*F,F*F*F*F*F+/F,;\"\"!%)infinityG,,F.F**&F.\"\"$-F 36#F=F+F+*&F.\"\"&-F36#FAF+F**&F.\"\"(-F36#FEF+F+%(~.~.~.~GF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 55 "We can make a comparison of f inite sums approximating " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG " }{TEXT -1 24 " graphically as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "First we define a function " } {TEXT 0 2 "MS" }{TEXT -1 87 " with two arguments or input parameters. \+ The degree of the polynomial approximation to " }{XPPEDIT 18 0 "sin(x) " "6#-%$sinG6#%\"xG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "2*n+1;" "6#,&* &\"\"#\"\"\"%\"nGF&F&F&F&" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 110 "x := 'x': i := 'i': n := 'n':\nMS := (n,x) -> sum( (-1)^i*x^(2*i+1)/(2*i+1)!,i=0..n);\nMS(1,x);\nMS(2,x);\nMS(3,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MSGf*6$%\"nG%\"xG6\"6$%)operatorG%& arrowGF)-%$sumG6$*&*&)!\"\"%\"iG\"\"\")9%,&F4\"\"#F5F5F5F5-%*factorial G6#F8F3/F4;\"\"!9$F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\" \"\"*&#F%\"\"'F%*$)F$\"\"$F%F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,(%\"xG\"\"\"*&#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$\"\"&F%F%*&#F%\"%S]F%*$)F$\"\"(F%F%F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalf(MS(3,Pi/4));\nevalf(sqrt(2)/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(pk52(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5y 1rq!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The degree 3, 5 and 7 approximations " }{XPPEDIT 18 0 "y = x-x^3/ 6;" "6#/%\"yG,&%\"xG\"\"\"*&F&\"\"$\"\"'!\"\"F+" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "y = x-x^3/3!+x^5/5!;" "6#/%\"yG,(%\"xG\"\"\"*&F&\"\"$-% *factorialG6#F)!\"\"F-*&F&\"\"&-F+6#F/F-F'" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "y = x-x^3/3!+x^5/5!-x^7/7!;" "6#/%\"yG,*%\"xG\"\"\"*&F& \"\"$-%*factorialG6#F)!\"\"F-*&F&\"\"&-F+6#F/F-F'*&F&\"\"(-F+6#F3F-F- " }{TEXT -1 28 " can be plotted as follows." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = sin(x);" "6#/%\"yG-%$sinG6#%\"xG" }{TEXT -1 33 " is also plotted for comparison. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "MS := (n,x) -> sum((-1)^i*x^(2*i+1)/(2*i+1)!,i = 0.. n):\nplot([sin(x),MS(1,x),MS(2,x),MS(3,x)],x=-10..10,-1.5..1.5,\n \+ color=[black,red,green,blue],linestyle=[2,1$3],\nnumpoints=100);" } }{PARA 13 "" 1 "" {GLPLOT2D 526 299 299 {PLOTDATA 2 "6(-%'CURVESG6%7cu 7$$!#5\"\"!$\"3&))p$*)36@Sa!#=7$$!3)fllllL')y*!#<$\"3GR3j#R,)eNF-7$$!3 I$RRRHzmp*F1$\"3_?MKISj&o#F-7$$!3lIJJJ\\s/'*F1$\"31%f!y/Sx*y\"F-7$$!3( GLLLj78]*F1$\"3ql2Vvw*fk(!#>7$$!33NNNN.!zR*F1$!3)*4.kV*Quo#FC7$$!3!RNN Nb-QH*F1$!31wBUuD,18F-7$$!3#4<<i/*\\+!G%F-7$$!3'efff>#R!z )F1$!3,)R=7**Go#fF-7$$!3%=@@@,x9f)F1$!31!y!\\p2b,uF-7$$!3iVWWW'edQ)F1$ !3]y;w1P2>')F-7$$!3e====+q!=)F1$!3C'*zxBj+r%*F-7$$!3!orrr,O_2)F1$!3#Qn R0QHiv*F-7$$!3!fhhh,s(pzF1$!34uG'H!y.L**F-7$$!3`TTTTcKBzF1$!3ORJ,wm'f( **F-7$$!3Emmmm#zo(yF1$!3%G'=i1'yt***F-7$$!3!>>>>*GVIyF1$!3A!Ho=TFs***F -7$$!3_<<<=I:Jm-xF-7$$!3?ZZZZ4mypF1$!3rGToVo]2kF-7$$!3czzzz^7gnF1$! 3Cy>\">+F1f%F-7$$!3E\\\\\\\\4cnmF1$!3?b%3')4/)\\PF-7$$!3'*=>>>n*\\d'F1 $!3WIU([Hvo(GF-7$$!399999DKnkF1$!3!)eE8YR)4$=F-7$$!3G4444$['fjF1$!3&Gb N5d^)QwFC7$$!31$HHHRQVE'F1$\"3in$Q8&)zX)=FC7$$!3%onnnZG!phF1$\"3,c4Lu/ 4R6F-7$$!3y#HHHReW1'F1$\"3w'RIVXq)p@F-7$$!3s3444$)))ffF1$\"3%p,ngsWp<$ F-7$$!3sfgggGJgeF1$\"3APZYkc\"Q5%F-7$$!3g6777utgdF1$\"334E/t(H+*\\F-7$ $!3GVVVVV%Hb&F1$\"37tScc<\\qmF-7$$!3;SSSSa7i`F1$\"3W\\`:$)*F-7$$!3#>>>>Ra %\\[F1$\"3@4(oYL7i!**F-7$$!3txxxxu#H![F1$\"3!4!*pC*=/f**F-7$$!3ajjjj0S cZF1$\"3+'H\\fY;.***F-7$$!3zrrr@-:1ZF1$\"3MjE3!R0)****F-7$$!3/!)zzz)** el%F1$\"3?JG()\\\"\\S)**F-7$$!3G)yyy`\\cg%F1$\"3`m7)H_(3V**F-7$$!3k&ff f>*RbXF1$\"3s(p?(>R-x)*F-7$$!3kbbbb@d^WF1$\"3C(e4*fmzh'*F-7$$!3i::::^u ZVF1$\"3`UOxR#3DM*F-7$$!3f4444hfWTF1$\"3RtBc4'**3V)F-7$$!3_++++!R![RF1 $\"3#e%3PwOK=sF-7$$!3cUUUUezHPF1$\"3W\"p?6#3n[bF-7$$!3$4000DX\"42q!#?7$$!3/&\\\\\\Rt3.$F1$!3q))4\"*Q=$ \\5\"F-7$$!3E%[[[Ghr#HF1$!3sC$>_a=z7#F-7$$!3urrrridHGF1$!3-SNOJGypIF-7 $$!3Bfeee7*>t#F1$!3IOU&QFPC)RF-7$$!3fXXXXB)z_#F1$!3!)zj7@!H#edF-7$$!3< WVVVj_GBF1$!3c>WR@)zRE(F-7$$!34%RRRR9(>@F1$!3R;$=YN&*3`)F-7$$!3gooooOg =>F1$!3#f'=pm\"F1$!3!>%)QR>&[e**F-7$$!3IFFFF!e4h \"F1$!3e_\\.ti$>***F-7$$!3UQQQ)en*f:F1$!3o$Q^yp8%****F-7$$!3a\\\\\\\\r (*3:F1$!3]R(46.)*3)**F-7$$!3F9999tE:9F1$!3#\\+C]B(Hz)*F-7$$!3**yyyyub@ 8F1$!3qgGQIX+\"p*F-7$$!3S&eeeQ`n5\"F1$!3-fS%HB.D%*)F-7$$!3!HXWWW+j9*F- $!3,&=5BPpL#zF-7$$!3;0444\\'y4(F-$!3^Ov5!R=n^'F-7$$!3b8===ybssss]s8&F-$!3bP-<\"*)[U\"\\F-7$$!3wx'oooX([SF-$!3e=&40 tM!RRF-7$$!3eLYYY1CgHF-$!3PqM\\Z_>@4'*)F-7$$\"3j9999KA88F1$\"3Ojk[r.6q'*F-7$$ \"3&yuuu%)>ST\"F1$\"3q&eF%)=dt()*F-7$$\"32\"333[;[^\"F1$\"3ko**fv`L%)* *F-7$$\"3YNNN&[LKc\"F1$\"3Qk(4&=Sr****F-7$$\"3&)*)*)*)*[];h\"F1$\"3Q)G eu%el\"***F-7$$\"3CWWW%\\n+m\"F1$\"39N7x[(z,'**F-7$$\"3i)*)*)*)\\%[3F 1$\"3%=Tl)oq!QS*F-7$$\"36gggg-Q<@F1$\"3r3w!\\y]Ia)F-7$$\"3-%QQQ)>QFBF1 $\"3ssP%G&*R=F(F-7$$\"3r8888T[HIF1$\"38B;;7\"F-7$$\"3fDDDD\"o&HJF1$\"3? %*4(p&QU-7FC7$$\"3\"RHHH44WB$F1$!3)*Q(=u%HKo#*FC7$$\"3Aiggg+DRLF1$!3ut M$o!*GP'>F-7$$\"3r>>>>C\"eV$F1$!3)[m(f F^#p64QF-7$$\"3N00004;IPF1$!3;g\\s\\vq^bF-7$$\"33!444HS&[RF1$!3nhf$H/# z@sF-7$$\"3OOPPP\\EYTF1$!3(4&)oeOi)R%)F-7$$\"3%[UUUUv%[VF1$!3G>mb^,6X$ *F-7$$\"3?`___oH^WF1$!3ef(*Hqj3h'*F-7$$\"3c\"333G=Tb%F1$!3Q)G\"RhJ,v)* F-7$$\"3dKJJJZP,YF1$!3DTA%>WU%Q**F-7$$\"3q#====J'[YF1$!33(Gj**f\"oz**F -7$$\"3!GBBBj()ep%F1$!35YRXe&Q')***F-7$$\"3#QGGG3WJu%F1$!3QW39\"*4F&** *F-7$$\"3'*zyyGz^$z%F1$!3cmgJ:&3r'**F-7$$\"3Bvuuu<*Q%[F1$!3%o&Ri1'fO\" **F-7$$\"3[qqq?cE%*[F1$!3hfYHk)f]$)*F-7$$\"3lmmmm%RY%\\F1$!3CFo\"\\p3: t*F-7$$\"3Rgggg-iW]F1$!3Y4dS$epJX*F-7$$\"38aaaa5gW^F1$!3Wu/*Qx8/3*F-7$ $\"3U*zzzz(\\e`F1$!3OvkNklK%)zF-7$$\"3y1333CDZbF1$!3kNfsp\"*y7nF-7$$\" 31VWWWg.ldF1$!3ym)=]#=t_\\F-7$$\"3k&eee)*o:'fF1$!3AbL\"\\=/5;$F-7$$\"3 9BAAA&)zegF1$!3&pQ;\"=c3DAF-7$$\"3mgeee!Gg:'F1$!3q\\xY'e[\"o7F-7$$\"3? XXXX5egiF1$!3/Q()H#oL-E#FC7$$\"3sHKKKS8ljF1$\"3I\\i)[N.d=)FC7$$\"3Ocdd d*4,Z'F1$\"3OIr4.+Qe=F-7$$\"3+$GGG)e3vlF1$\"3_eIFD#Hx(GF-7$$\"3-JJJJ%) fqmF1$\"3_=Hq2_%zx$F-7$$\"3/zzzz46mnF1$\"3N(*o&G_APk%F-7$$\"3!puuu9r*o pF1$\"36mG%z&Q\"GL'F-7$$\"3iXWWWG%p;(F1$\"3<8)[P^v7t(F-7$$\"3k]___ia\" Q(F1$\"38EAYWpi/*)F-7$$\"3)fmmm'yBnvF1$\"3iU+,2#*p\"f*F-7$$\"3!3:::luQ n(F1$\"3#yKR8MX#Q)*F-7$$\"3iNOOO9^!y(F1$\"3y>va)yAI(**F-7$$\"3)y#GGGNj IyF1$\"3)p5AOXus***F-7$$\"39????cv!)yF1$\"3apUT#)fT'***F-7$$\"3S7777x( 3$zF1$\"3K(fW6`\\/(**F-7$$\"3m////)**4)zF1$\"3](3-(>.W>**F-7$$\"3%z$QQ QdF!3)F1$\"3!ywd(zc/X(*F-7$$\"3Crsss;bz\")F1$\"35%Re2a&ou%*F-7$$\"3c*3 44\\h#y$)F1$\"3mZC!GiYol)F-7$$\"3#osss#H$4e)F1$\"3%fY)pg-/suF-7$$\"3w` aaa7w&z)F1$\"3cqHQ[%*\\$)eF-7$$\"3?%RRRzPF**)F1$\"3QR\\[`PC(=%F-7$$\"3 ciiii$e**3*F1$\"3g=W^6z)fG$F-7$$\"3!4888$*yr=*F1$\"3ws8l'*yp`BF-7$$\"3 w<>>>h5\"H*F1$\"3;?PO(=TFL\"F-7$$\"3g/222L.&R*F1$\"3/(HGByES(HFC7$$\"3 %QXXXDM()\\*F1$!3%HV\\uW*)))Q(FC7$$\"31.---_V-'*F1$!3/3!=\"e>>>2f)y*F1$!3g&=@g7+%eNF-7$$\"#5 F*$!3&))p$*)36@SaF--%'COLOURG6&%$RGBGF*F*F*-%*LINESTYLEG6#\"\"#-F$6%7` q7$F($\"3emmmmmmm:!#:7$F/$\"3Wgup1_Jl9Fh`n7$F:$\"3i41\"32\"p!Q\"Fh`n7$ FE$\"3#>qg'Q-S*G\"Fh`n7$FO$\"3)[b2)=Uc,7Fh`n7$FY$\"3N/$)zq25=6Fh`n7$Fh n$\"3D++9jm;W5Fh`n7$F]o$\"3z!R&R[))H5(*!#;7$Fbo$\"3M3L#ywl'*)*)F^bn7$F go$\"3BL2xEFm1$)F^bn7$Fap$\"3B!o*\\:;**RwF^bn7$Feq$\"3)\\R;MF)=#3(F^bn 7$F_r$\"3-C=cH*\\j['F^bn7$Fdr$\"3d]s!=U.9#fF^bn7$Fir$\"3w4Fw3sr2aF^bn7 $F^s$\"3)oz\"f0\"zm'\\F^bn7$Fcs$\"3?i,![xpGZ%F^bn7$F]t$\"3C([)4E#\\)zS F^bn7$Fgt$\"33sEf?:)4l$F^bn7$Fau$\"37QBy&3+gH$F^bn7$F[v$\"3,3>:M5EF^bn7$Fjv$\"3%p@0(yQZ)H#F^bn7$F_w$\"3'\\stFv_L.#F ^bn7$Fdw$\"3OO))=37Dp6F^bn7$F`[l$\"3]$ykCTW(\\$*F17$Fe[l$\"3!zw5*p$f6s (F17$Fj[l$\"35n/tvhJ3jF17$F_\\l$\"3P;p+!>qz\"\\F17$Fi\\l$\"3DXb(=m60#Q F17$Fc]l$\"39U_nArb)z#F17$F]^l$\"31L6$>XD')*>F17$Fh^l$\"3-C^B2$eHD\"F1 7$Fb_l$\"3KU_j1)R]m'F-7$Fg_l$\"3?]Wzqc6Y;F-7$F\\`l$!3jp&3rZ6IC#F-7$Fa` l$!3JfE%3)yMB`F-7$Ff`l$!3M%*[l_^F:uF-7$F`al$!3AY];**=m_()F-7$Fdbl$!3M \"Qk#QHD1\"F-7$F`gl$\"3'>C(fP4 #*oHF-7$Fegl$\"3.J7a;7H?\\F-7$Fjgl$\"3v1\"Q.%[L>kF-7$F_hl$\"3,BHeCOvMy F-7$Fdhl$\"3>^RNX3,C))F-7$Fihl$\"3o%GjY\"*)od$*F-7$Fcil$\"3#H#o:*f5%e7F17$Fi]m$!3es6W;G.z>F17$Fc^m$!3C!*>EL&=l'GF17$F]_m$!3' \\QLe%Rf8QF17$Fb_m$!3*[5TyjW,#\\F17$Fg_m$!32'*Gf%>AH]Xz=6F^bn7$F_bm$!33.lgf[%F^bn7$F^hm$!3fi\">3V&Fh`n7$FE$!3a@!eH?R'>[Fh`n7$FO$!3;(*= Cb&>,E%Fh`n7$FY$!3a'*e&=$3*\\v$Fh`n7$Fhn$!36=6dgVkHLFh`n7$F]o$!35**HKF \"3)HHFh`n7$Fbo$!3yj_4p(*ocDFh`n7$Fgo$!3y`))ew]kAAFh`n7$Fap$!3DnK'4!oY :>Fh`n7$Feq$!3)[\\ZY^jJn\"Fh`n7$F_r$!3TK,[ANbH9Fh`n7$Fdr$!3/:cd#z+N@\" Fh`n7$Fir$!33.@D0H/I5Fh`n7$F^s$!3()z7%\\Qsp#))F^bn7$Fcs$!3%\\jF(3T9#H( F^bn7$F]t$!39!pQ+Co+;'F^bn7$Fgt$!3W'e'QyGK=]F^bn7$Fau$!3wU'*H7,l\\TF^b n7$F[v$!3[A:BW>)RL$F^bn7$Fev$!3sIk`s()zwEF^bn7$Fjv$!3D')[[a1N,@F^bn7$F _w$!3;I)HmK/2m\"F^bn7$Fdw$!3S'*\\(*[m?o7F^bn7$F^x$!3oUa4#37\")R*F17$Fb y$!3q*ROJJ#z3rF17$Ffz$!3-dG)H)R`Z^F17$F`[l$!3)H=/`#3O'f$F17$Fe[l$!3MI' oTKY,Z#F17$Fj[l$!3c\"Q]jN!)\\o\"F17$F_\\l$!3G*z`b\\Yr4\"F17$Fi\\l$!3ri vkoH'3r(F-7$Fc]l$!35]P3p\"ol%eF-7$F]^l$!3:4iGUkx,'F-7$Fg_l$!3gH$eY?dw&pF-7$F\\`l$!3%)eM=NFgZzF-7$ Fa`l$!3]$4i:EY&*)))F-7$Ff`l$!3]I%>mP2W#zF-7$F]dl$!3vp--:s*o^'F-7$Fgdl$!3i)4***evE9\\F-7$Fael$!3\" yr]Ak&>ALkF-7$F_hl$\"3)\\Ti ?>0j)yF-7$Fdhl$\"3?KZc5M*\\'*)F-7$Fihl$\"3/Mm/C$fJo*F-7$Fcil$\"3*e5-$3 <&>+\"F17$Fgjl$\"3))*p#*z^Mj)**F-7$Fa[m$\"3#\\40FM*z$e*F-7$Ff[m$\"3Y** HaLU**)*))F-7$F[\\m$\"3?$\\/FyjJ&zF-7$F`\\m$\"3txF9frc4qF-7$Fe\\m$\"3= N3hfh$f.'F-7$F_]m$\"3-Khu!4-\\P&F-7$Fi]m$\"3&*=Qx^H*pA&F-7$Fc^m$\"3pt! =7G3Q$fF-7$F]_m$\"3w$3AD4`Vp(F-7$Fb_m$\"3s())zJM;z4\"F17$Fg_m$\"3GAWjb -l'o\"F17$F\\`m$\"3!)Qb`T.,yCF17$Fa`m$\"3#HdiD@k5g$F17$F[am$\"3Xme[z$p l8&F17$F_bm$\"3uj<+O&RR'pF17$Fccm$\"34$y]hdQsU*F17$F]dm$\"3U!eTWO\\#[7 F^bn7$Fbdm$\"3oF4U&QuIl\"F^bn7$Fgdm$\"3=t\\^\"[Wq3#F^bn7$F\\em$\"3S!)> k$oX)*o#F^bn7$Faem$\"3QLiHMY+SLF^bn7$F[fm$\"3AQz\"[27\\4%F^bn7$Fefm$\" 3UuV4kGAX]F^bn7$F_gm$\"3wVVDh(y0;'F^bn7$Figm$\"3&=$\\tCEBJtF^bn7$F^hm$ \"3g#)Qp\")*pSv)F^bn7$Fchm$\"3*4q$y6F(Q.\"Fh`n7$Fhhm$\"3K!H1&4ArH7Fh`n 7$F]im$\"3N_m:CJD@9Fh`n7$Fgim$\"3yvE;.,()o;Fh`n7$F[[n$\"3:'*RcGc$4$>Fh `n7$Fe[n$\"36[8E%Rv3A#Fh`n7$Fj[n$\"3%=JD3qGQa#Fh`n7$F_\\n$\"3:,XU86r4H Fh`n7$Fd\\n$\"3YEA='3'*4M$Fh`n7$Fi\\n$\"3mbtTc^zyPFh`n7$Fc]n$\"31HJh0) eOD%Fh`n7$F]^n$\"3'>v7\"fMc6[Fh`n7$Fg^n$\"3M9q;BotBaFh`n7$Fa_n$\"3z[q& R1KO-'Fh`n7$Ff_n$\"3UnmmmmmmnFh`n-F[`n6&F]`nFhcoFecoFhcoFico-F$6%7`q7$ F($\"3O<.Y<.Y28!#97$F/$\"3E%*pq9y:16Fcgp7$F:$\"3y_wL#o'HI&*Fh`n7$FE$\" 37\"R[o@vo-)Fh`n7$FO$\"3rCs<`O&=s'Fh`n7$FY$\"37v\"fFwxDk#Fh`n7$Fap$\"3!)*4c@1Kn8#Fh`n7$Feq$\"3)GTC_I*GimTu!)p.ILF^bn7$Fau$\"3uF^bn7$Fev$\" 3')*\\Z2,(p+:F^bn7$Fjv$\"3=vWD`Q()G6F^bn7$F_w$\"35%f\\VSe!)p\"F17$Fj[l$\"3#>&GS8l[ \"G\"F17$F_\\l$\"3Q*yE_x+?&*)F-7$Fi\\l$\"3SG3z0GRSfF-7$Fc]l$\"3_\")o5b vObIF-7$F]^l$\"3Y)eR4]MXn'FC7$Fh^l$!3srSrz-BD@48j\"Rw$F -7$Fg_l$!3/$[.()H7&[cF-7$F\\`l$!3WEq)QUi6@(F-7$Fa`l$!3a/&=p)=.3&)F-7$F f`l$!3W\"z,:+L=R*F-7$F`al$!3%p7rLDSd*)*F-7$Fdbl$!3q`$\\J*H!)z**F-7$F^c l$!3#4`Uw*4n!p*F-7$Fccl$!3vc.'HLNC%*)F-7$Fhcl$!3G\\/'>7dL#zF-7$F]dl$!3 1SN\"e8sF-7 $F`\\m$\"3079[6Du!QF-7$F_]m$\"3?u\"G#)f$*pq\"F- 7$Fi]m$!3eQGYt?DpgFC7$Fc^m$!3wH#))zc()=D$F-7$F]_m$!3Dtdh.-D@fF-7$Fb_m$ !3gWz\\.n&z&*)F-7$Fg_m$!3%*\\^\"*GUX#G\"F17$F\\`m$!3tm)z)>x\">q\"F17$F a`m$!3E8DEVxUKAF17$F[am$!3(Q@FH%*GY#HF17$F_bm$!3i)3gMm%3_PF17$Fccm$!3* *37_=3`6\\F17$F]dm$!3wvr:F*4BW'F17$Fbdm$!3p%39X0S&Q')F17$Fgdm$!3(pJD^j t+7\"F^bn7$F\\em$!3)>lrre>&4:F^bn7$Faem$!3'fLfZk#)*p>F^bn7$F[fm$!3e$eC JQ@Gb#F^bn7$Fefm$!33?hM(ovON$F^bn7$F_gm$!3'yeg.Vk.Q%F^bn7$Figm$!3y#\\r vd!e\\bF^bn7$F^hm$!398DO#*eu&3(F^bn7$Fchm$!3W`t/+rRK*)F^bn7$Fhhm$!3M.D x#>$[R6Fh`n7$F]im$!3/aq+H6)zR\"Fh`n7$Fgim$!3UEtF&)o&REFh`n7$Fj[n$!3zelq*>$*e?$Fh`n7$F_\\n$!3 eZ%=8*4>()QFh`n7$Fd\\n$!3K0M\\.\\xl&Fh`n7$Fc]n $!3@o,BJE?2nFh`n7$F]^n$!3gIa%*[Ba2!)Fh`n7$Fg^n$!3i%*37Z,V7&*Fh`n7$Fa_n $!3UXX`l(>h5\"Fcgp7$Ff_n$!3O<.Y<.Y28Fcgp-F[`n6&F]`nFhcoFhcoFecoFico-%+ AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Ff_n;$Fh`n!\"\"$\"#:F\\[r" 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 -1 216 "Successively higher degree Taylor polynomials produce progressively better approximations over a fixed interval centred at the origin.\nThe next picture shows \+ the graphs of the degree 13, 15 and 17 Taylor polynomials." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "MS \+ := (n,x) -> sum((-1)^i*x^(2*i+1)/(2*i+1)!,i = 0..n):\nplot([sin(x),MS( 6,x),MS(7,x),MS(8,x)],x=-10..10,-1.5..1.5,\n color=[black,magen ta,coral,cyan],linestyle=[2,1$3],\nnumpoints=100);" }}{PARA 13 "" 1 " " {GLPLOT2D 529 299 299 {PLOTDATA 2 "6(-%'CURVESG6%7cu7$$!#5\"\"!$\"3& ))p$*)36@Sa!#=7$$!3)fllllL')y*!#<$\"3GR3j#R,)eNF-7$$!3I$RRRHzmp*F1$\"3 _?MKISj&o#F-7$$!3lIJJJ\\s/'*F1$\"31%f!y/Sx*y\"F-7$$!3(GLLLj78]*F1$\"3q l2Vvw*fk(!#>7$$!33NNNN.!zR*F1$!3)*4.kV*Quo#FC7$$!3!RNNNb-QH*F1$!31wBUu D,18F-7$$!3#4<<i/*\\+!G%F-7$$!3'efff>#R!z)F1$!3,)R=7**Go# fF-7$$!3%=@@@,x9f)F1$!31!y!\\p2b,uF-7$$!3iVWWW'edQ)F1$!3]y;w1P2>')F-7$ $!3e====+q!=)F1$!3C'*zxBj+r%*F-7$$!3!orrr,O_2)F1$!3#QnR0QHiv*F-7$$!3!f hhh,s(pzF1$!34uG'H!y.L**F-7$$!3`TTTTcKBzF1$!3ORJ,wm'f(**F-7$$!3Emmmm#z o(yF1$!3%G'=i1'yt***F-7$$!3!>>>>*GVIyF1$!3A!Ho=TFs***F-7$$!3_<<<=I: Jm-xF-7$$!3?ZZZZ4mypF1$!3rGToVo]2kF-7$$!3czzzz^7gnF1$!3Cy>\">+F1f%F-7$ $!3E\\\\\\\\4cnmF1$!3?b%3')4/)\\PF-7$$!3'*=>>>n*\\d'F1$!3WIU([Hvo(GF-7 $$!399999DKnkF1$!3!)eE8YR)4$=F-7$$!3G4444$['fjF1$!3&GbN5d^)QwFC7$$!31$ HHHRQVE'F1$\"3in$Q8&)zX)=FC7$$!3%onnnZG!phF1$\"3,c4Lu/4R6F-7$$!3y#HHHR eW1'F1$\"3w'RIVXq)p@F-7$$!3s3444$)))ffF1$\"3%p,ngsWp<$F-7$$!3sfgggGJge F1$\"3APZYkc\"Q5%F-7$$!3g6777utgdF1$\"334E/t(H+*\\F-7$$!3GVVVVV%Hb&F1$ \"37tScc<\\qmF-7$$!3;SSSSa7i`F1$\"3W\\`:$)*F-7$$!3#>>>>Ra%\\[F1$\"3@4(oYL 7i!**F-7$$!3txxxxu#H![F1$\"3!4!*pC*=/f**F-7$$!3ajjjj0ScZF1$\"3+'H\\fY; .***F-7$$!3zrrr@-:1ZF1$\"3MjE3!R0)****F-7$$!3/!)zzz)**el%F1$\"3?JG()\\ \"\\S)**F-7$$!3G)yyy`\\cg%F1$\"3`m7)H_(3V**F-7$$!3k&fff>*RbXF1$\"3s(p? (>R-x)*F-7$$!3kbbbb@d^WF1$\"3C(e4*fmzh'*F-7$$!3i::::^uZVF1$\"3`UOxR#3D M*F-7$$!3f4444hfWTF1$\"3RtBc4'**3V)F-7$$!3_++++!R![RF1$\"3#e%3PwOK=sF- 7$$!3cUUUUezHPF1$\"3W\"p?6#3n[bF-7$$!3$4000DX\"42q!#?7$$!3/&\\\\\\Rt3.$F1$!3q))4\"*Q=$\\5\"F-7$$!3E%[[[ Ghr#HF1$!3sC$>_a=z7#F-7$$!3urrrridHGF1$!3-SNOJGypIF-7$$!3Bfeee7*>t#F1$ !3IOU&QFPC)RF-7$$!3fXXXXB)z_#F1$!3!)zj7@!H#edF-7$$!3WR @)zRE(F-7$$!34%RRRR9(>@F1$!3R;$=YN&*3`)F-7$$!3gooooOg=>F1$!3#f'=pm\"F1$!3!>%)QR>&[e**F-7$$!3IFFFF!e4h\"F1$!3e_\\.ti$>** *F-7$$!3UQQQ)en*f:F1$!3o$Q^yp8%****F-7$$!3a\\\\\\\\r(*3:F1$!3]R(46.)*3 )**F-7$$!3F9999tE:9F1$!3#\\+C]B(Hz)*F-7$$!3**yyyyub@8F1$!3qgGQIX+\"p*F -7$$!3S&eeeQ`n5\"F1$!3-fS%HB.D%*)F-7$$!3!HXWWW+j9*F-$!3,&=5BPpL#zF-7$$ !3;0444\\'y4(F-$!3^Ov5!R=n^'F-7$$!3b8===ybss ss]s8&F-$!3bP-<\"*)[U\"\\F-7$$!3wx'oooX([SF-$!3e=&40tM!RRF-7$$!3eLYYY1 CgHF-$!3PqM\\Z_>@4'*)F-7$$\"3j9999KA88F1$\"3Ojk[r.6q'*F-7$$\"3&yuuu%)>ST\"F1$\" 3q&eF%)=dt()*F-7$$\"32\"333[;[^\"F1$\"3ko**fv`L%)**F-7$$\"3YNNN&[LKc\" F1$\"3Qk(4&=Sr****F-7$$\"3&)*)*)*)*[];h\"F1$\"3Q)Geu%el\"***F-7$$\"3CW WW%\\n+m\"F1$\"39N7x[(z,'**F-7$$\"3i)*)*)*)\\%[3F1$\"3%=Tl)oq!QS*F-7$ $\"36gggg-Q<@F1$\"3r3w!\\y]Ia)F-7$$\"3-%QQQ)>QFBF1$\"3ssP%G&*R=F(F-7$$ \"3r8888T[HIF1$\"38B;;7\"F-7$$\"3fDDDD\"o&HJF1$\"3?%*4(p&QU-7FC7$$\"3\"R HHH44WB$F1$!3)*Q(=u%HKo#*FC7$$\"3Aiggg+DRLF1$!3utM$o!*GP'>F-7$$\"3r>>> >C\"eV$F1$!3)[m(fF^#p64QF-7$$\"3N00004 ;IPF1$!3;g\\s\\vq^bF-7$$\"33!444HS&[RF1$!3nhf$H/#z@sF-7$$\"3OOPPP\\EYT F1$!3(4&)oeOi)R%)F-7$$\"3%[UUUUv%[VF1$!3G>mb^,6X$*F-7$$\"3?`___oH^WF1$ !3ef(*Hqj3h'*F-7$$\"3c\"333G=Tb%F1$!3Q)G\"RhJ,v)*F-7$$\"3dKJJJZP,YF1$! 3DTA%>WU%Q**F-7$$\"3q#====J'[YF1$!33(Gj**f\"oz**F-7$$\"3!GBBBj()ep%F1$ !35YRXe&Q')***F-7$$\"3#QGGG3WJu%F1$!3QW39\"*4F&***F-7$$\"3'*zyyGz^$z%F 1$!3cmgJ:&3r'**F-7$$\"3Bvuuu<*Q%[F1$!3%o&Ri1'fO\"**F-7$$\"3[qqq?cE%*[F 1$!3hfYHk)f]$)*F-7$$\"3lmmmm%RY%\\F1$!3CFo\"\\p3:t*F-7$$\"3Rgggg-iW]F1 $!3Y4dS$epJX*F-7$$\"38aaaa5gW^F1$!3Wu/*Qx8/3*F-7$$\"3U*zzzz(\\e`F1$!3O vkNklK%)zF-7$$\"3y1333CDZbF1$!3kNfsp\"*y7nF-7$$\"31VWWWg.ldF1$!3ym)=]# =t_\\F-7$$\"3k&eee)*o:'fF1$!3AbL\"\\=/5;$F-7$$\"39BAAA&)zegF1$!3&pQ;\" =c3DAF-7$$\"3mgeee!Gg:'F1$!3q\\xY'e[\"o7F-7$$\"3?XXXX5egiF1$!3/Q()H#oL -E#FC7$$\"3sHKKKS8ljF1$\"3I\\i)[N.d=)FC7$$\"3Ocddd*4,Z'F1$\"3OIr4.+Qe= F-7$$\"3+$GGG)e3vlF1$\"3_eIFD#Hx(GF-7$$\"3-JJJJ%)fqmF1$\"3_=Hq2_%zx$F- 7$$\"3/zzzz46mnF1$\"3N(*o&G_APk%F-7$$\"3!puuu9r*opF1$\"36mG%z&Q\"GL'F- 7$$\"3iXWWWG%p;(F1$\"3<8)[P^v7t(F-7$$\"3k]___ia\"Q(F1$\"38EAYWpi/*)F-7 $$\"3)fmmm'yBnvF1$\"3iU+,2#*p\"f*F-7$$\"3!3:::luQn(F1$\"3#yKR8MX#Q)*F- 7$$\"3iNOOO9^!y(F1$\"3y>va)yAI(**F-7$$\"3)y#GGGNjIyF1$\"3)p5AOXus***F- 7$$\"39????cv!)yF1$\"3apUT#)fT'***F-7$$\"3S7777x(3$zF1$\"3K(fW6`\\/(** F-7$$\"3m////)**4)zF1$\"3](3-(>.W>**F-7$$\"3%z$QQQdF!3)F1$\"3!ywd(zc/X (*F-7$$\"3Crsss;bz\")F1$\"35%Re2a&ou%*F-7$$\"3c*344\\h#y$)F1$\"3mZC!Gi Yol)F-7$$\"3#osss#H$4e)F1$\"3%fY)pg-/suF-7$$\"3w`aaa7w&z)F1$\"3cqHQ[%* \\$)eF-7$$\"3?%RRRzPF**)F1$\"3QR\\[`PC(=%F-7$$\"3ciiii$e**3*F1$\"3g=W^ 6z)fG$F-7$$\"3!4888$*yr=*F1$\"3ws8l'*yp`BF-7$$\"3w<>>>h5\"H*F1$\"3;?PO (=TFL\"F-7$$\"3g/222L.&R*F1$\"3/(HGByES(HFC7$$\"3%QXXXDM()\\*F1$!3%HV \\uW*)))Q(FC7$$\"31.---_V-'*F1$!3/3!=\"e>>>2f)y*F1$!3g&=@g7+%eNF-7$$\"#5F*$!3&))p$*)36@SaF--% 'COLOURG6&%$RGBGF*F*F*-%*LINESTYLEG6#\"\"#-F$6%7dq7$F($!3C6Ew+:l*[&!#: 7$$!3))GGGGoJ%*)*F1$!3k_(Rq3r.r%Fh`n7$F/$!31#o%Qp)R[.%Fh`n7$F5$!3)oF$p _CX@NFh`n7$F:$!3wVe2>gLpIFh`n7$FE$!3@p4kq]KUAFh`n7$FO$!3,oL$Hu3Ni\"Fh` n7$FY$!3u([QG.)=p6Fh`n7$Fhn$!3;Qj#pH[;d)!#;7$F]o$!3Oqp&\\/R#yhFcbn7$Fb o$!3e&>T?H\"ywVFcbn7$Fgo$!3_W'o&ynO'3$Fcbn7$Fap$!3f0h@V.DV@Fcbn7$Feq$! 3y\"f2R)3;[:Fcbn7$F_r$!3UC!pu@3)o5Fcbn7$Fdr$!3!Q%p<\\q2GtF17$Fir$!3?\" z6%Gh)[0&F17$F^s$!3%zB^uv0pc$F17$Fcs$!3)G*[r4+\"HH#F17$F]t$!39(p?%\\w# o]\"F17$Fgt$!30TF5epUJ#)F-7$Fau$!3bZKtSD)*HOF-7$F[v$\"3%[5AY&>9&4$FC7$ Fev$\"3a#4?I:DSD$F-7$Fjv$\"3uNc)Qck=m&F-7$F_w$\"3OljD!['=htF-7$Fdw$\"3 y?I'3X@(F-7$F_\\l$\"3WvdO-i$fa&F-7$Fi\\l$\"3F[2#Q#[4?QF-7$Fc]l$\"3K)GF1.z l\"=F-7$F]^l$!32IhBz0aFqFa^l7$Fh^l$!3$)He%f5#*z7#F-7$Fb_l$!3yHu63NY#)R F-7$Fg_l$!3wf)GJCP#edF-7$F\\`l$!3+\"[tjA#)RE(F-7$Fa`l$!3T/$pT%f*3`)F-7 $Ff`l$!3%o#eD]KA,%*F-7$F`al$!3#4=@'Ru9**)*F-7$Fdbl$!3)*p9uM!)*3)**F-7$ F^cl$!3?41)3`/5p*F-7$Fccl$!3!y\"*yHB.D%*)F-7$Fhcl$!3_%=7BPpL#zF-7$F]dl $!3#4e2,R=n^'F-FfdlF`el7$F[fl$!3%***ynX*RI2\"F-7$Fffl$\"3U3`ab4_i5F-F_ gl7$Fegl$\"3x5TC\"G%GB\\F-7$Fjgl$\"3I)[VZ`hIV'F-7$F_hl$\"33.KeDQI&)yF- 7$Fdhl$\"3gPIb%>@4'*)F-7$Fihl$\"3W$=R>P5,n*F-7$Fcil$\"3#\\1[%z`L%)**F- 7$Fgjl$\"3QI4%y^f`!**F-7$Fa[m$\"3>+kk+s!QS*F-7$Ff[m$\"3$3Q;[O^Ia)F-7$F [\\m$\"3M@4?SB%=F(F-7$F`\\m$\"3T.ZJB([P%eF-7$Fe\\m$\"3?CV`-U'G-%F-7$F_ ]m$\"3&*)y`t6!y6@F-7$Fi]m$\"3(zC,)o/U/7FC7$Fc^m$!3-fBz>L?j>F-7$F]_m$!3 hh(pw;,z!QF-7$Fb_m$!3i#oh*R*o*[bF-7$Fg_m$!39g2z!*4S:sF-7$F\\`m$!3(zU>f #ojE%)F-7$Fa`m$!3n!y$R)>e#=$*F-7$F[am$!3FM#z4!pl@)*F-7$F_bm$!3yuj5gNn( *)*F-7$Fccm$!3Ob3_6De]&*F-7$F]dm$!3'>&G7_Zza()F-7$Fbdm$!3W*>B64t!*Q(F- 7$Fgdm$!3(=`=%4NM>dF-7$F\\em$!3o#GR1)=\\(>$F-7$Faem$!3l7-A5(oh\"GFC7$F [fm$\"3e^7ZF>)[N$F-7$Fefm$\"3K,WTk2a\"Q)F-7$F_gm$\"3VO,'HFcr]\"F17$Fig m$\"3&eq7hFFAK#F17$F^hm$\"3EMjHMDO+NF17$Fchm$\"3;11\"yfxt4&F17$Fhhm$\" 3O,AP,XW`vF17$F]im$\"3#H(y5E6Ua5Fcbn7$Fgim$\"3P*[N+o<(Q:Fcbn7$F[[n$\"3 Gw\\y#>Qb=#Fcbn7$Fe[n$\"3=/`St))G!3$Fcbn7$Fj[n$\"3;]]t(Qe;K%Fcbn7$F_\\ n$\"3'[qEgNJ42'Fcbn7$Fd\\n$\"3I)e()R&p,Z')Fcbn7$Fi\\n$\"3e+sB^-])=\"Fh `n7$Fc]n$\"3\"RC^&Q<4<;Fh`n7$F]^n$\"3l'y_L%4[KAFh`n7$Fg^n$\"3YLRA9**ze IFh`n7$F\\_n$\"33!e)[U$\\`^$Fh`n7$Fa_n$\"3/)[.pq%eMSFh`n7$$\"3sffff`H% *)*F1$\"3SfV1$yB-r%Fh`n7$Ff_n$\"3C6Ew+:l*[&Fh`n-F[`n6&F]`n$\"*++++\"! \")$F*F*F\\do-F_`n6#\"\"\"-F$6%7dq7$F($\"3c0s0JA^d@Fh`n7$Fj`n$\"3mCs(Q DH-\"=Fh`n7$F/$\"3=0#oJe_c^\"Fh`n7$F5$\"3o@z22aJ'H\"Fh`n7$F:$\"3*z7F\" )Q(z16Fh`n7$FE$\"3\\tBeVnB/xFcbn7$FO$\"3=gX\"[w3RH&Fcbn7$FY$\"3#))G1f- $*)*f$Fcbn7$Fhn$\"3#Q)3VNr\\Sl;Fcbn7$Fbo$\"3=)o( Q#4Ai2\"Fcbn7$Fgo$\"3c3j$p%\\,^nF17$Fap$\"3KiWr%*eF))RF17$Feq$\"3-_0:V f+kBF17$F_r$\"310*4vB2B<\"F17$Fdr$\"3]R%***z\"e=^%F-7$Fir$\"3[*G6>uZqo 'FC7$F^s$!35?L-%z^$)))*FC7$Fcs$!3ls<4S6859F-7$F]t$!3gVn8'QT()z)FC7$Fgt $\"3eh2Yky[%y$FC7$Fau$\"3#*Q.TI1[C=F-7$F[v$\"3OJ$zToF4c$F-7$Fev$\"3(3Y )=us)o?&F-7$Fjv$\"3ESRySnP(y'F-7$F_w$\"3%p'*[;D5t-)F-7$Fdw$\"3kKn!f_NV 1*F-7$F^x$\"3K$RhHZCGv*F-7$Fby$\"3ApZZ#)p\"*)***F-7$Ffz$\"3CSe7)pt6))* F-7$F`[l$\"3QD))p&[&RW$*F-7$Fe[l$\"3kI!f!R.uJ%)F-7$Fj[l$\"3UOw.AMp=sF- 7$F_\\l$\"37x^->@\")[bF-7$Fi\\l$\"3[muo:SP@QF-7$Fc]l$\"3$3:p2v!4<=F-7$ F]^l$!3A3pn-rM1qFa^l7$Fh^l$!3!Rx:K@;z7#F-7$Fb_l$!33&eu*\\lV#)RF-7$Fg_l $!3Afy-F)G#edF-7$F\\`l$!3gC&fKxzRE(F-7$Fa`l$!3]\\n%[M&*3`)F-7$Ff`l$!3s -H*e6B7S*F-7$F`al$!3!43&4:u9**)*F-7$Fdbl$!3)\\Gz5.)*3)**F-7$F^cl$!3Ej' z.`/5p*F-7$Fccl$!3[-R%HB.D%*)F-7$Fhcl$!3Yz,Js$pL#zF-F\\dl7$Fgdl$!3+P-< \"*)[U\"\\F-F`elF\\jnF_jnF_glFdgl7$Fjgl$\"3x_MuM:1LkF-7$F_hl$\"3I#R\"e DQI&)yF-7$Fdhl$\"3K3g^%>@4'*)F-7$Fihl$\"3)*)e$[r.6q'*F-7$Fcil$\"35\\uc v`L%)**F-7$Fgjl$\"3I$HaU\\f`!**F-7$Fa[m$\"3B\"\\yq12QS*F-7$Ff[m$\"3+(3 <`x]Ia)F-7$F[\\m$\"3[8(3^!*R=F(F-7$F`\\m$\"3&oq,l\"3uVeF-7$Fe\\m$\"3!* f-i*)y$G-%F-7$F_]m$\"3gIn(od.<6#F-7$Fi]m$\"3n6%***R9N-7FC7$Fc^m$!3qAMB J1vj>F-7$F]_m$!3vOLK6K<4QF-7$Fb_m$!3Y*yA=3\\=b&F-7$Fg_m$!3\\:=N$ei@A(F -7$F\\`m$!3$p$ohN)32W)F-7$Fa`m$!31h)H*fF+Z$*F-7$F[am$!3M5#zLGV\"z)*F-7 $F_bm$!36X*GqeZ.+\"F17$Fccm$!3SitoE51[(*F-7$F]dm$!3*R!=]wkq7\"*F-7$Fbd m$!35&R5P)*o%[!)F-7$Fgdm$!3#H@'oYSnFoF-7$F\\em$!3Mc,$)ynKs^F-7$Faem$!3 ?LX&\\#4\"oa$F-7$F[fm$!3\"e(oSDJoH>F-7$Fefm$!3CAzD%46TS$FC7$F_gm$\"33B qFF'RF!))FC7$Figm$\"33L\")[_Or:9F-7$F^hm$\"3V%\\uqv@(Q5F-7$Fchm$!3whdI HH\"[G(FC7$Fhhm$!37oO;uMu\\\\F-7$F]im$!3&eom+gr*Q6F17$Fgim$!3#3))*y^eI RBF17$F[[n$!3Rc!QF/=z5%F17$Fe[n$!3G1EDf2nKnF17$Fj[n$!3KM#H)>Lne5Fcbn7$ F_\\n$!3ox&*==AbH;Fcbn7$Fd\\n$!3^,/\">&=Q6DFcbn7$Fi\\n$!3%R)HqqL`<'R%Fcbn7$F5$ !3o=U7Ot,\"p$Fcbn7$F:$!3y&)eJ!e!o&4$Fcbn7$FE$!3aB,CNWRy?Fcbn7$FO$!3cZ( 4-)HU!R\"Fcbn7$FY$!3'f'z&RMQ@O*F17$Fhn$!3Ey\"=NA0,c'F17$F]o$!31&3]1\\) [JYF17$Fbo$!3BY#)>6neNLF17$Fgo$!3P\"Ht'=h$Q]#F17$Fap$!39%z-4Boz%>F17$F eq$!3u)[MaBl7h\"F17$F_r$!3'4,+DiI'H8F17$Fdr$!3/.D,%)*4,5\"F17$Fir$!35d WlOC>\"**)F-7$F^s$!39i7O[GM)>(F-7$Fcs$!3iO--.%yb-&F-7$F]t$!3dB0LE:#\\8 $F-7$Fgt$!3'RPr5H%*z,*FC7$Fau$\"3qjj:n(=81\"F-7$F[v$\"3Q?g-B\"Gj8$F-7$ Fev$\"3%*[WC]Iio\\F-7$Fjv$\"3+C`h@MyfmF-7$F_w$\"3%QH)R(4'*o&zF-7$Fdw$ \"3Jq;#)*oP\"G!*F-7$F^x$\"3$ea\\&*>._t*F-7$Fby$\"3e#*psW2u*)**F-7$Ffz$ \"3sAqpc$pn()*F-7$F`[l$\"3#o%os#)GSU$*F-7$Fe[l$\"3$=oKl*p&3V)F-7$Fj[l$ \"3T#=7\\o1$=sF-7$F_\\l$\"3UoM$4.l'[bF-7$Fi\\l$\"3qmSW!H:8#QF-7$Fc]l$ \"3j6w7g*pq\"=F-7$F]^l$!3`c!**yo7r+(Fa^l7$Fh^l$!3k%*H\"Rg=z7#F-7$Fb_l$ !3+jbr*GPC)RF-7$Fg_l$!3IYewC!H#edF-7$F\\`l$!3m!yf@#)zRE(F-7$Fa`l$!3n0q ua`*3`)F-7$Ff`l$!3Tt+5uVeF-7$Fe\\m$\"3&fYF$4'QG-%F-7$F_]m$\"3!GQ72*fq6@F-7$Fi]m$\"3 g\"fX<%fU-7FC7$Fc^m$!3!)e&*H%>GP'>F-7$F]_m$!3#>ncy&[64QF-7$Fb_m$!3-7sz [)R%)F-7$Fa`m$!3v'[/$fW+X$* F-7$F[am$!35UR8[*fZ()*F-7$F_bm$!37/z:Y[s%***F-7$Fccm$!3E[$*[L,JI(*F-7$ F]dm$!3mi9yG*zy2*F-7$Fbdm$!3izZ%fHg)yzF-7$Fgdm$!39w\"ft'fG-nF-7$F\\em$ !3t!GE())>-J\\F-7$Faem$!3!3#[9\\7\"F-7$Fefm$ \"3?8)pAd1ue*FC7$F_gm$\"3v#=D$z7%e8$F-7$Figm$\"3c[+pG1(f3&F-7$F^hm$\"3 l8]SD]F.rF-7$Fchm$\"3'\\LS\"o]3N!*F-7$Fhhm$\"3e(Q%z#R!H<6F17$F]im$\"3+ /<=#)Gk?8F17$Fgim$\"3&)H:aF17$Fe[n$\"3Z/: bN*>,]#F17$Fj[n$\"3#4'4^mML)H$F17$F_\\n$\"3W!)4&*oeD]XF17$Fd\\n$\"31C; 0\"H-Ui'F17$Fi\\n$\"3=RQF*)f#[a*F17$Fc]n$\"3)Guy/byOQ\"Fcbn7$F]^n$\"3! o3fpS8p1#Fcbn7$Fb^n$\"3d!fL`k**[_#Fcbn7$Fg^n$\"3#G]Yg]9@3$Fcbn7$F\\_n$ \"3McTkx\\\"Go$Fcbn7$Fa_n$\"3%Q7ySl " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "An adaptiv e graph plotting procedure: " }{TEXT 0 5 "graph" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The procedure in this sec tion is an alternative to the standard Maple function " }{TEXT 0 4 "pl ot" }{TEXT -1 195 ", for plotting the graph of a single function. It a llows control over how the plotting is performed, in a manner which is a bit different from the control one has over the standard Maple rout ine " }{TEXT 0 4 "plot" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 120 "In particular, the precision of the \+ floating point calculations is not changed internally, whereas, the Ma ple procedure " }{TEXT 0 4 "plot" }{TEXT -1 137 " often uses hardware \+ floating point arithmetic whereby the effective precision is increased 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 268 18 "Calling Sequenc e:\n" }}{PARA 0 "" 0 "" {TEXT 269 4 " " }{TEXT -1 17 "graph( f, xrn g )\n" }{TEXT 270 1 "\n" }{TEXT -1 26 " graph( f, xrng, yrng )" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 266 11 "Parame ters:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 23 4 "f - " }{TEXT -1 83 " an expression involving a s ingle variable, say x, or a function x -> f(x)\n\n " }{TEXT 23 7 "x rng - " }{TEXT -1 79 " horizontal plotting range in the form 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 265 12 "Descrip tion:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 5 "graph" }{TEXT -1 22 " plots the graph of \+ a " }{TEXT 258 15 "single function" }{TEXT -1 26 " using an adaptive m ethod." }}{PARA 15 "" 0 "" {TEXT -1 88 "An even spacing is aimed for a long sections of the curve which are approximately linear." }}{PARA 15 "" 0 "" {TEXT -1 79 "More points are plotted along sections of the \+ curve where the graph is bending." }}{PARA 15 "" 0 "" {TEXT -1 107 "An y maximum or minimum points are located approximately by parabolic int erpolation and added to the graph.\n" }}{PARA 256 "" 0 "" {TEXT 267 8 "Options:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "adaptive=true/false" }}{PARA 0 "" 0 "" {TEXT -1 144 "Adaptive plot ting will, where necessary, sub-divide the plotting interval in an att empt to get a good graphical representation of the function. " }} {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 \"f alse\"." }}{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 width by m eans \"numpoints\" points. \nFor adaptive plotting \"numpoints\" contr ols the spacing of points along the curve, that is, in the direction o f 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\"\"\"%*numpoint sG!\"\"" }{TEXT -1 97 ". Note that, in general, the number of points p lotted 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 "\"linear ity\" controls the tolerance for the allowed deviation from a straight line." }}{PARA 0 "" 0 "" {TEXT -1 93 "Along any arc between two point s on the graph, the curve will generally deviate no more than " } {TEXT 271 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 264 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 71 "A p rocedure for evaluating the sine function from its Maclaurin series " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Althoug h the Maclaurin series converges to " }{XPPEDIT 18 0 "sin(x);" "6#-%$s inG6#%\"xG" }{TEXT -1 19 " for all values of " }{TEXT 272 1 "x" } {TEXT -1 83 ", it is only of practical use in evaluating the sine fun ction for small values of " }{TEXT 273 1 "x" }{TEXT -1 2 ".\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 657 "sine_series := proc(x::real cons)\n local z,zs,sum,term,eps,k,even,maxit;\n\n maxit := Digits* 10;\n eps := Float(1,-Digits);\n\n z := evalf(x);\n term := z;\n sum := term;\n zs := z*z;\n even := false; \n for k from 2 to maxit by 2 do\n term := term*zs/(k*(k+1));\n if even then\n sum := sum+term;\n else\n sum := sum-term;\n \+ end if;\n if abs(term)<=eps*abs(sum) then break end if;\n \+ even := not even;\n end do:\n if k > maxit then\n print(`sum \+ of`,k/2,`terms of sine series is`,sum);\n error \"reached the max imum number, %1, of iterations without convergence\",maxit;\n end if ;\n evalf(sum);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 95 "When using the default precision of 10 di gits, we can get about 10 digit accuracy in computing " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 27 " using this procedure with " }{XPPEDIT 18 0 "abs(x) <= 2;" "6#1-%$absG6#%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "xx := -1.8;\nevalf(sine_series(xx),10);\nevalf(sin(xx ),10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$!#=!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+3jZQ(*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$!+4jZQ(*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We can check the accuracy of the procedure " }{TEXT 0 13 "arcsin_series" }{TEXT -1 12 " as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot(sin(x)-'evalf [10]@sine_series'(x),x=-2..2,color=blue,numpoints=80);" }}{PARA 13 "" 1 "" {GLPLOT2D 422 258 258 {PLOTDATA 2 "6&-%'CURVESG6#7\\gq7$$!\"#\"\" !$\"\"$!#57$$!+?XM)*>!\"*$\"\"#F-7$$!+S!*o'*>F1$!\"$F-7$$!+gN.&*>F1$! \"%F-7$$!+!3yL*>F1$F)F-7$$!++Es\"*>F1$\"\"\"F-7$$!+?r1!*>F1FE7$$!+S;T) )>F1$F*F*7$$!+ghv')>F1$\"\"%F-7$$!+!o+^)>F1$!\"\"F-7$$!++_W$)>F1$!\"&F -7$$!+?(*y\")>F1FE7$$!+SU8!)>F1FM7$$!+g(y%y>F1FA7$$!+!GBo(>F1FA7$$!++y ;v>F1F+7$$!+>B^t>F1F27$$!+Qo&=(>F1FV7$$!+e8?q>F1FQ7$$!+yeao>F1F77$$!+) R!*o'>F1FA7$$!+=\\Bl>F1FQ7$$!+Q%zN'>F1FE7$$!+eR#>'>F1FM7$$!+y%o-'>F1FV 7$$!+)*Hhe>F1FM7$$!+=v&p&>F1F27$$!+Q?Ib>F1$\"\"&F-7$$!+elk`>F1F+7$$!+y 5*>&>F1F<7$$!+)fN.&>F1FQ7$$!+=,o[>F1F27$$!+QY-Z>F1FM7$$!+9UeX>F1FE7$$! +*yVT%>F1F27$$!+kLqU>F1FM7$$!+SHET>F1F+7$$!+;D#)R>F1F<7$$!+#4#QQ>F1F<7 $$!+o;%p$>F1F27$$!+V7]N>F1FE7$$!+=31M>F1F27$$!+%R?E$>F1F<7$$!+q*z6$>F1 FA7$$!+Y&R(H>F1FQ7$$!+A\"*HG>F1F+7$$!+(peo#>F1FE7$$!+s#=a#>F1FV7$$!+[y (R#>F1F27$$!+Cu`A>F1F77$$!++q4@>F1FV7$$!+wll>>F1FE7$$!+^h@=>F1FQ7$$!+E dx;>F1F77$$!+-`L:>F1F77$$!+y[*Q\">F1F77$$!+aWX7>F1FV7$$!+IS,6>F1FV7$$! +0Od4>F1F27$$!+!=L\"3>F1FQ7$$!+cFp1>F1FA7$$!+KBD0>F1FV7$$!+3>\"Q!>F1FM 7$$!+%[rB!>F1FQ7$$!+f5$4!>F1FA7$$!+_6J**=F1FM7$$!+W7p(*=F1FE7$$!+O82'* =F1F<7$$!+G9X%*=F1FV7$$!+?:$G*=F1F77$$!+7;@\"*=F1F27$$!+/t%)=F1FE7$$!+s?6$)=F1FV7$$!+l@\\\" )=F1F77$$!+eA()z=F1FV7$$!+]BDy=F1F27$$!+UCjw=F1Fjq7$$!+MD,v=F1F27$$!+E ERt=F1FV7$$!+=Fxr=F1F<7$$!+5G:q=F1F27$$!+-H`o=F1FE7$$!+%*H\"p'=F1F+7$$ !+'3$Hl=F1F<7$$!+yJnj=F1F+7$$!+rK0i=F1F+7$$!+kLVg=F1F77$$!+cM\")e=F1FM 7$$!+[N>d=F1F77$$!+SOdb=F1FE7$$!+KP&R&=F1FE7$$!+CQL_=F1FE7$$!+;Rr]=F1F 77$$!+3S4\\=F1Fjq7$$!+iLYZ=F1FV7$$!+=W$=F1FE7$$!+_vyK=F1FM7$$!+1p:J=F1FE7$$!+gi_H=F1Fjq7$$!+9c*y#=F1F27$$ !+p\\EE=F1F+7$$!+CVjC=F1F27$$!+yO+B=F1F+7$$!+KIP@=F1FM7$$!+(QU(>=F1FE7 $$!+U<6==F1FM7$$!+'4\"[;=F1F+7$$!+]/&[\"=F1FA7$$!+/)>K\"=F1F77$$!+e\"* e6=F1FE7$$!+8&e*4=F1FA7$$!+oyK3=F1F27$$!+Asp1=F1FV7$$!+wl10=F1FM7$$!+I fV.=F1FV7$$!+%G0=!=F1F27$$!+RY<+=F1FV7$$!+%*Ra)z\"F1FM7$$!+[L\"pz\"F1F +7$$!+_/H&z\"F1FM7$$!+dvm$z\"F1FV7$$!+iY/#z\"F1FV7$$!+muu\"F1FM7$$!+)zm/u\"F1F77$$!+m@'*QS>n\"F1FM7$$!+prMq;F1Faam7$$!+5#*yo;F1FM7$$!+^7 Bn;F1F+7$$!+#Htcm\"F1FQ7$$!+L`6k;F1FA7$$!+utbi;F1FQ7$$!+:%**4m\"F1F+7$ $!+c9Wf;F1FM7$$!+(\\$)yl\"F1FV7$$!+QbKc;F1Fen7$$!+zvwa;F1FE7$$!+?'4Kl \"F1Fjq7$$!+h;l^;F1F+7$$!+-P4];F1Fen7$$!+Ud`[;F1Faam7$$!+$yxpk\"F1F27$ $!+SlOX;F1FV7$$!+'HbPk\"F1F<7$$!+_S9U;F1FM7$$!+3G`S;F1F77$$!+k:#*Q;F1F A7$$!+@.JP;F1F27$$!+y!*pN;F1FV7$$!+My3M;F1Fjq7$$!+!fwCj\"F1F+7$$!+Z`'3 j\"F1FA7$$!+/TDH;F1Fjq7$$!+gGkF;F1F77$$!+;;.E;F1Fen7$$!+t.UC;F1Fjq7$$! +I\"4Gi\"F1FM7$$!+')y>@;F1F27$$!+Ume>;F1F77$$!+*Rvzh\"F1FA7$$!+cTO;;F1 F<7$$!+7Hv9;F1F27$$!+o;98;F1F+7$$!+D/`6;F1FM7$$!+#=>*4;F1Faam7$$!+QzI3 ;F1FM7$$!+%p'p1;F1FV7$$!+^a30;F1F+7$$!+3UZ.;F1FQ7$$!+kH'=g\"F1FQ7$$!+? \"Qf\"F1F77$$!+Oe?#f\"F1F27$$!+g(*f!f\"F1Fjq7$$!+%o$**)e\"F1Fen7$$!+3w Q(e\"F1Fen7$$!+K:y&e\"F1F77$$!+ca<%e\"F1F77$$!+z$pDe\"F1F27$$!+-L'4e\" F1F<7$$!+EsNz:F1Faam7$$!+]6vx:F1Fjq7$$!+u]9w:F1FA7$$!+)**QXd\"F1FQ7$$! +AH$Hd\"F1F<7$$!+YoKr:F1FM7$$!+p2sp:F1F+7$$!+#p9\"o:F1FM7$$!+;'3lc\"F1 FV7$$!+SD!\\c\"F1FM7$$!+kkHj:F1FE7$$!+)Q!ph:F1F+7$$!+7V3g:F1F27$$!+O#y %e:F1FM7$$!+f@(ob\"F1$\"\"'F-7$$!+#3m_b\"F1FA7$$!+1+m`:F1FQ7$$!+IR0_:F 1F+7$$!+ayW]:F1FQ7$$!+y<%)[:F1F<7$$!+-dBZ:F1Fen7$$!+E'Hca\"F1FE7$$!+\\ N-W:F1F27$$!+-:PU:F1F27$$!+c%>2a\"F1F^\\n7$$!+4u1R:F1Fjq7$$!+i`TP:F1F< 7$$!+:LwN:F1F<7$$!+o76M:F1Fjq7$$!+@#fC`\"F1FM7$$!+ur!3`\"F1FV7$$!+F^:H :F1F27$$!+!3.v_\"F1F+7$$!+M5&e_\"F1F+7$$!+()*)>C:F1Faan7$$!+SpaA:F1F+7 $$!+%*[*3_\"F1F77$$!+ZGC>:F1FV7$$!++3f<:F1Fjq7$$!+`(Qf^\"F1FE7$$!+1nG9 :F1FQ7$$!+fYj7:F1FE7$$!+7E)4^\"F1FA7$$!+l0L4:F1F+7$$!+=&yw]\"F1FE7$$!+ sk-1:F1Fjq7$$!+DWP/:F1FV7$$!+yBs-:F1FQ7$$!+K.2,:F1FQ7$$!+&G=%*\\\"F1FA 7$$!+Qiw(\\\"F1F27$$!+\">9h\\\"F1FA7$$!+W@Y%\\\"F1Fen7$$!+(45G\\\"F1FA 7$$!+]!e6\\\"F1FM7$$!+EHq*[\"F1FE7$$!+-yC)[\"F1FE7$$!+yEz'[\"F1FE7$$!+ bvL&[\"F1FE7$$!+KC)Q[\"F1F27$$!+3tU#[\"F1FM7$$!+%=s4[\"F1FE7$$!+gq^z9F 1FM7$$!+O>1y9F1FM7$$!+7ogw9F1FM7$$!+)o^^Z\"F1FM7$$!+llpt9F1FA7$$!+U9Cs 9F1F77$$!+=jyq9F1FA7$$!+%>J$p9F1FM7$$!+qg(yY\"F1FV7$$!+Y4Um9F1FM7$$!+A e'\\Y\"F1FV7$$!+)p5NY\"F1FE7$$!+vb0i9F1FE7$$!+_/gg9F1F+7$$!+G`9f9F1FE7 $$!+/-pd9F1FM7$$!+!3NiX\"F1FM7$$!+c*zZX\"F1FE7$$!+K[K`9F1F27$$!+3(p=X \"F1F27$$!+%e9/X\"F1FE7$$!+g%f*[9F1FA7$$!+OV]Z9F1FE7$$!+7#\\gW\"F1FV7$ $!+*3%fW9F1FM7$$!+SyJT9F1FM7$$!+!fT!Q9F1FM7$$!+mMSO9F1FE7$$!+T`wM9F1FE 7$$!+;s7L9F1FE7$$!+#4*[J9F1F27$$!+n4&)H9F1FA7$$!+UG@G9F1FM7$$!+=ZdE9F1 FV7$$!+$fO\\U\"F1F<7$$!+o%)HB9F1FM7$$!+W.m@9F1FA7$$!+>A-?9F1F27$$!+%4% Q=9F1FM7$$!+pfu;9F1FV7$$!+Wy5:9F1FE7$$!+?(pMT\"F1F+7$$!+&fJ=T\"F1FA7$$ !+qM>59F1FV7$$!+Y`b39F1FV7$$!+@s\"pS\"F1FE7$$!+'4z_S\"F1F27$$!+r4k.9F1 FV7$$!+YG+-9F1FM7$$!+AZO+9F1FM7$$!+(fE()R\"F1FV7$$!+s%)3(R\"F1FA7$$!+[ .X&R\"F1FV7$$!+BA\"QR\"F1FM7$$!+)4u@R\"F1F27$$!+'R%)))Q\"F1F27$$!+%p%f &Q\"F1F27$$!+V)\\RQ\"F1FV7$$!+#*\\I#Q\"F1FE7$$!+T,m!Q\"F1FV7$$!+!H:!z8 F1FM7$$!+R/Px8F1FM7$$!+)eDdP\"F1FV7$$!+P23u8F1FV7$$!+')eVs8F1FE7$$!+N5 zq8F1FV7$$!+%=Y\"p8F1FV7$$!+L8]n8F1FE7$$!+#[ceO\"F1F27$$!+J;@k8F1FV7$$ !+!ymDO\"F1FM7$$!+H>#4O\"F1FE7$$!+yqFf8F1F77$$!+FAjd8F1FM7$$!+wt)fN\"F 1FM7$$!+DDMa8F1FM7$$!+uwp_8F1FV7$$!+BG0^8F1FM7$$!+szS\\8F1FE7$$!+@JwZ8 F1FQ7$$!+q#=hM\"F1FM7$$!+>MZW8F1F27$$!+o&GGM\"F1FV7$$!+L8F1F A7$$!+sKhJ8F1F77$$!+a\"G+L\"F1FA7$$!+NIWG8F1FM7$$!+;z&oK\"F1FM7$$!+(zs _K\"F1FV7$$!+ywoB8F1F27$$!+gD5A8F1FE7$$!+Tu^?8F1FE7$$!+AB$*=8F1FE7$$!+ /sM<8F1FV7$$!+&3idJ\"F1FE7$$!+mp<98F1FM7$$!+Gn+68F1FM7$$!+\"\\OyI\"F1F M7$$!+s8D18F1FV7$$!+aim/8F1FE7$$!+N63.8F1FE7$$!+;g\\,8F1F27$$!+(*3\"** H\"F1FM7$$!+ydK)H\"F1FM7$$!+g1u'H\"F1FE7$$!+Tb:&H\"F1FV7$$!+/`)>H\"F1F V7$$!+m]\"))G\"F1FV7$$!+EcP(G\"F1F+7$$!+(=OfG\"F1FV7$$!+[n\\%G\"F1FV7$ $!+3t0$G\"F1FE7$$!+oyh\"G\"F1FM7$$!+G%y,G\"F1FA7$$!+))*Q(y7F1F+7$$!+\\ &*Hx7F1FV7$$!+5,'eF\"F1F27$$!+q1Uu7F1FM7$$!+I7)HF\"F1FV7$$!+!zT:F\"F1F V7$$!+]B5q7F1F27$$!+6Hmo7F1F27$$!+sMAn7F1F27$$!+KSyl7F1FA7$$!+#fWVE\"F 1F27$$!+_^!HE\"F1F27$$!+7dYh7F1F27$$!+ti-g7F1FV7$$!+Moee7F1FM7$$!+%RZr D\"F1FE7$$!+azqb7F1FM7$$!+9&oUD\"F1FE7$$!+u!HGD\"F1FV7$$!+N'*Q^7F1FM7$ $!+'>]*\\7F1F27$$!+c2^[7F1FM7$$!+;82Z7F1FM7$$!+w=jX7F1FM7$$!+OC>W7F1FM 7$$!+(*HvU7F1FV7$$!+k8/T7F1FE7$$!+J(H$R7F1FM7$$!+)4=wB\"F1FE7$$!+lk!fB \"F1FE7$$!+K[>M7F1FE7$$!+*>$[K7F1FM7$$!+m:xI7F1FE7$$!+L*f!H7F1FV7$$!++ $[tA\"F1FE7$$!+nmjD7F1FM7$$!+M]#RA\"F1FV7$$!+,M@A7F1FM7$$!+o<]?7F1FM7$ $!+N,z=7F1F27$$!+-&yq@\"F1FE7$$!+poO:7F1FM7$$!+O_l87F1FM7$$!+.O%>@\"F1 FV7$$!+q>B57F1FM7$$!+P._37F1FV7$$!+/(3o?\"F1FE7$$!+rq407F1FE7$$!+QaQ.7 F1FE7$$!+0Qn,7F1FV7$$!+s@'**>\"F1FM7$$!+R0D)>\"F1FM7$$!+1*Ql>\"F1FM7$$ !+ts#[>\"F1FM7$$!+Sc6$>\"F1FM7$$!+2SS\">\"F1FE7$$!+uBp*=\"F1FV7$$!+T2) z=\"F1FE7$$!+k2`'=\"F1FM7$$!+'y!3&=\"F1FM7$$!+33j$=\"F1FM7$$!+I3=#=\"F 1FM7$$!+_3t!=\"F1FM7$$!+v3Gz6F1FE7$$!+)*3$y<\"F1FE7$$!+?4Qw6F1FA7$$!+U 4$\\<\"F1FE7$$!+k4[t6F1FE7$$!+')4.s6F1FE7$$!+45eq6F1FV7$$!+K58p6F1FE7$ $!+a5on6F1FM7$$!+w5Bm6F1FV7$$!+)4\"yk6F1F27$$!+?6Lj6F1FV7$$!+V6)=;\"F1 FE7$$!+m6Vg6F1FM7$$!+)=\")*e6F1FM7$$!+57`d6F1FE7$$!+K73c6F1FV7$$!+a7ja 6F1FM7$$!+x7=`6F1FM7$$!++8t^6F1FV7$$!+A8G]6F1FM7$$!+W8$)[6F1FV7$$!+m8Q Z6F1FE7$$!+)QJf9\"F1FM7$$!+69[W6F1FM7$$!+M9.V6F1FM7$$!+c9eT6F1FE7$$!+( y%*)R6F1F27$$!+=\"3#Q6F1FM7$$!+]9_O6F1FM7$$!+\"yM[8\"F1FM7$$!+7\"[J8\" F1FV7$$!+W9YJ6F1FE7$$!+vZxH6F1FE7$$!+1\")3G6F1FM7$$!+oZrC6F1FM7$$!+J9M @6F1FM7$$!+iZl>6F1FM7$$!+%4oz6\"F1FE7$$!+D9G;6F1FV7$$!+cZf96F1FE7$$!+( 33H6\"F1FV7$$!+=9A66F1FE7$$!+]Z`46F1FM7$$!+\"3[y5\"F1FV7$$!+79;16F1FM7 $$!+WZZ/6F1FE7$$!+v!)y-6F1FV7$$!+195,6F1FE7$$!+QZT*4\"F1FV7$$!+p!Gx4\" F1FM7$$!++9/'4\"F1FM7$$!+KZN%4\"F1FV7$$!+j!oE4\"F1FM7$$!+%R\")44\"F1FM 7$$!+EZH*3\"F1FV7$$!+d!3w3\"F1FM7$$!+c-sj5F1FM7$$!+aC$)R5F1FM7$$!+y-GL 5F1FM7$$!+-\"Gn-\"F1FM7$$!+e+4D5F1F27$$!+9?XB5F1FE7$$!+qR\"=-\"F1FV7$$ !+Ef,\"F1FM7$$!+iwM55F1FE7$$!+='4(35F1FV7$$!+u:225F1FM7$$!+INV 05F1FE7$$!+'[&z.5F1FM7$$!+Uu:-5F1FV7$$!+)R>0+\"F1FV7$$!+SN\")))**F-FE7 $$!++JVs**F-FE7$$!+gE0c**F-FV7$$!+:AnR**F-FM7$$!+q%[*F-FM7$$!+q5go%*F-FM7$$!+:I+`%*F-FM7$$!+g\\SP%*F- FE7$$!+5p!=U*F-FE7$$!+b)3iS*F-FV7$$!++3h!R*F-FM7$$!*v7]P*F1FM7$$!+?yte $*F-FE7$$!+!*GYU$*F-FM7$$!+gz=E$*F-F27$$!+DI\"*4$*F-FE7$$!+!4QOH*F-FM7 $$!+gJOx#*F-FM7$$!+I#)3h#*F-FV7$$!++L\"[C*F-FM7$$!+q$Q&G#*F-FM7$$!+NME 7#*F-FV7$$!++&))f>*F-FM7$$!+qNrz\"*F-FM7$$!+S'QM;*F-FV7$$!+5P;Z\"*F-FM 7$$!+!y))38*F-FE7$$!+XQh9\"*F-FE7$$!+5*Q$)4*F-FV7$$!+!)R1#3*F-FV7$$!+] !*yl!*F-FE7$$!+?T^\\!*F-FV7$$!+!>RK.*F-FV7$$!+bU'p,*F-FM7$$!+?$*o+!*F- FV7$$!+!R9W)*)F-FM7$$!+lYJ>*)F-FM7$$!*%\\@a))F1FM7$$!+0SKC))F-FM7$$!+q IV%z)F-FM7$$!++w[z()F-FM7$$!+N@ak()F-FV7$$!+qmf\\()F-FV7$$!++7lM()F-FA 7$$!+Idq>()F-FM7$$!+l-w/()F-FE7$$!++[\")*o)F-FV7$$!+I$p[n)F-FM7$$!+gQ# *f')F-FE7$$!+!Ry\\k)F-FE7$$!+?H.I')F-FM7$$!+bu3:')F-FM7$$!+!*>9+')F-FV 7$$!+?l>&e)F-FV7$$!+]5Dq&)F-FM7$$!+!e0`b)F-FV7$$!+5,OS&)F-FM7$$!+XYTD& )F-FE7$$!+!=p/^)F-FE7$$!+5P_&\\)F-FM7$$!+S#y0[)F-FV7$$!+vFjl%)F-FM7$$! +5to]%)F-FM7$$!+S=uN%)F-FV7$$!+qjz?%)F-FM7$$!+04&eS)F-FE7$$!+Sa!4R)F-F M7$$!*(*ffP)F1FE7$$!+l#R)f$)F-FM7$$!+g&=PM)F-FM7$$!+]yfF$)F-FM7$$!+XrZ 6$)F-FM7$$!+IdBz#)F-FM7$$!+?V*pC)F-FM7$$!+&\\6D=)F-FM7$$!+q'G!=\")F-FM 7$$!+?I1*)zF-FM7$$!*P(4gyF1FM7$$!+!\\_L%yF-FE7$$!+0wgEyF-FE7$$!+?F')4y F-FV7$$!+Sy6$z(F-FM7$$!+bHPwxF-FM7$$!+q!G'fxF-FE7$$!+!>$)Gu(F-FM7$$!+0 $Qhs(F-FV7$$!+?MR4xF-FM7$$!+S&[Ep(F-FM7$$!+bO!fn(F-FV7$$!+q(e\"fwF-FE7 $$!+!*QTUwF-FV7$$!+0!pci(F-FE7$$!+?T#*3wF-FM7$$!+S#z@f(F-FM7$$!+5(*>Dv F-FM7$$!+v,AeuF-FM7$$!+!Hv9W(F-FM7$$!+5/tCuF-FE7$$!+Db)zS(F-FE7$$!+S1C \"R(F-FE7$$!+gd\\utF-FM7$$!+v3vdtF-FV7$$!+!*f+TtF-FM7$$!*6hUK(F1FE7$$! +SYo4tF-FE7$$!+q\"3^H(F-FM7$$!++<`!G(F-FV7$$!+I_&fE(F-FE7$$!+g(y8D(F-F M7$$!+!H-oB(F-FM7$$!+?eAAsF-FV7$$!+]$\\w?(F-FM7$$!+!)G2$>(F-FM7$$!+5k \\yrF-FM7$$!+S*>R;(F-FE7$$!+qMM\\rF-FE7$$!++qwMrF-FM7$$!+I0>?rF-FV7$$! +gSh0rF-FV7$$!+!fP54(F-FM7$$!+IeUupF-FM7$$!*29y&oF1FM7$$!+!GEfg'F-FM7$ $!*\\QSN'F1FM7$$!+XWxPjF-FM7$$!++/^@jF-FV7$$!+]jC0jF-FM7$$!+0B)*)G'F-F V7$$!+g#=FF'F-FM7$$!+5UXciF-FE7$$!+l,>SiF-FM7$$!+?h#RA'F-FM7$$!+v?m2iF -FM7$$!+I!)R\">'F-FV7$$!+&)R8vhF-FE7$$!+S*p)ehF-FM7$$!+&*egUhF-FM7$$!+ ]=MEhF-FV7$$!++y25hF-FM7$$!+bP\"Q4'F-FV7$$!+5(\\v2'F-FM7$$!+gcGhgF-FM7 $$!+:;-XgF-FM7$$!+qvvGgF-FE7$$!+DN\\7gF-FM7$$!+![Hi*fF-FM7$$!+Na'*zfF- FV7$$!+!R,P'fF-FM7$$!+XtVZfF-FM7$$!++LpR^F-FM7$$!+gqHC^F-FE7$$!+ g@!*3^F-FM7$$!+gs]$4&F-FM7$$!+bB6y]F-FV7$$!+buri]F-FM7$$!+bDKZ]F-FM7$$ !+bw#>.&F-FM7$$!+bF`;]F-FV7$$!+by8,]F-FM7$$!+aHu&)\\F-FM7$$!+`![.(\\F- FE7$$!+_J&\\&\\F-FM7$$!+^NP$*[F-FM7$$!*&RzJ[F1FM7$$!*x/[G%F1FM7$$!+:x0 RSF-FM7$$!*m5Lz$F1FM7$$!+JA^gPF-FM7$$!+-QrFPF-FM7$$!+)e98r$F-FE7$$!+u` \"\\p$F-FM7$$!+gh^yOF-FM7$$!+Xp6iOF-FV7$$!+IxrXOF-FE7$$!+;&=$HOF-FM7$$ !+-$>Hh$F-FM7$$!+)3?lf$F-FM7$$!+u37!e$F-FV7$$!+f;sjNF-FE7$$!+WCKZNF-FM 7$$!+IK#4`$F-FV7$$!+;S_9NF-FM7$$!+,[7)\\$F-FV7$$!+'eD<[$F-FV7$$!+sjKlM F-FM7$$!+Wz_KMF-FM7$$!+:&H(*R$F-FM7$$!++.L$Q$F-FM7$$!+'3JpO$F-FM7$$!+s =`]LF-FV7$$!+eE8MLF-FV7$$!+WMt$F-FE7$$!+W]PzJF-FV7$$!+=\\^kJF-FM7$$!+#za'\\JF-FM7$$!+)G9-4 $F-FM7$$!+&yt2.$F-FM7$$!+yF*=\"HF-FM7$$!*x6Iz#F1FM7$$!+GdwwFF-FV7$$!+& o>0w#F-FE7$$!+UOFWFF-FM7$$!++w-GFF-FM7$$!+e:y6FF-FM7$$!+:b`&p#F-FM7$$! +s%*GzEF-FV7$$!+IM/jEF-FE7$$!+)Q(zYEF-FM7$$!+X8bIEF-FM7$$!+-`I9EF-FV7$ $!+g#f!)f#F-FV7$$!+=K\"=e#F-FM7$$!+vrclDF-FM7$$!+K6K\\DF-FM7$$!+!4vI`# F-FM7$$!+?44oCF-FM7$$!+]n5.CF-FM7$$!+lYhqBF-FM7$$!+!eA\"QBF-FM7$$!+Ql( =K#F-FM7$$!+&\\IcI#F-FM7$$!+_WQ*G#F-FE7$$!*TQJF#F1FV7$$!+]@&yD#F-FE7$$ !+!*ecUAF-FM7$$!+I'zsA#F-FV7$$!+qL*>@#F-FM7$$!+5rq'>#F-FV7$$!+]3U\"=#F -FV7$$!+!fMh;#F-FM7$$!+I$[3:#F-FM7$$!+!H.(*3#F-FM7$$!+]#e&G?F-FM7$$!+q \"oi!>F-FM7$$!*4yRy\"F1FM7$$!+?(\\hl\"F-FM7$$!+]8KG:F-FM7$$!+eUO'\\\"F -FM7$$!+lrSk9F-FM7$$!+='G%[9F-FM7$$!+s+XK9F-FE7$$!+E:Z;9F-FM7$$!+!)H\\ +9F-FE7$$!+MW^%Q\"F-FM7$$!+))e`o8F-FV7$$!+Utb_8F-FV7$$!+&zylL\"F-FM7$$ !+[-g?8F-FM7$$!+-\"F-FV7$$!+9H$*y6 F-FM7$$!+k4Jj6F-FV7$$!+:!*oZ6F-FV7$$!+mq1K6F-FM7$$!+;^W;6F-FM7$$!+nJ#3 5\"F-FM7$$!+=7?&3\"F-FM7$$!+o#z&p5F-FM7$$!+>t&R0\"F-FE7$$!+q`LQ5F-FE7$ $!+?MrA5F-FM7$$!+]#yt(*)!#6FM7$$!)BiFxF1FM7$$!+]\\'36&FghuFM7$$!)w5%\\ #F1FM7$$\")\"*QYDF1FM7$$\")\\-,xF1FM7$$\"*y(H\"G\"F1FM7$$\"*:L5v\"F1FM 7$$\"*\"HS*G#F1FM7$$\"*BF4x#F1FM7$$\"+5*Hw-$F-FM7$$\"*fKVG$F1FM7$$\"++ 7=2MF-FM7$$\"+5)H+`$F-FM7$$\"+:TX\"f$F-FM7$$\"+?%yGl$F-FM7$$\"+s0f$o$F -FM7$$\"+DFI9PF-FM7$$\"+-)e'HPF-FE7$$\"+y[,XPF-FM7$$\"+a4PgPF-FM7$$\"* .Fdx$F1FE7$$\"+!)zx#z$F-FE7$$\"+H*G)4QF-FM7$$\"+y)zo#QF-FM7$$\"+G3$R%Q F-FM7$$\"+x<)4'QF-FE7$$\"+EF.yQF-FM7$$\"+wO3&*QF-FM7$$\"+DY87RF-FE7$$ \"+ub=HRF-FM7$$\"+ClBYRF-FM7$$\"+tuGjRF-FV7$$\"+A%Q.)RF-FM7$$\"+s$*Q(* RF-FE7$$\"+@.W9SF-FE7$$\"+q7\\JSF-FV7$$\"+?Aa[SF-FM7$$\"+:)\\\\=%F-FM7 $$\"*Td8K%F1FM7$$\"+:`fdXF-FM7$$\"*ALQz%F1FM7$$\"+y9f5[F-FV7$$\"+P(\\t #[F-FM7$$\"+'*z5W[F-FV7$$\"+ai'3'[F-FV7$$\"+7Xix[F-FE7$$\"+rFQ%*[F-FM7 $$\"+I596\\F-FM7$$\"+)G**y#\\F-FV7$$\"+YvlW\\F-FM7$$\"+0eTh\\F-FM7$$\" +kS1&F-FE7$$\"+5Osy]F-FM7$$\"+q=[&4&F-FE7$$\"+I,C7^F-FV7$ $\"+!R)**G^F-FV7$$\"+]mvX^F-FM7$$\"+0\\^i^F-FV7$$\"+gJFz^F-FM7$$\"+?9. '>&F-FM7$$\"+!o*y7_F-FM7$$\"+SzaH_F-FV7$$\"++iIY_F-FV7$$\"+bW1j_F-FM7$ $\"+5F#)z_F-FM7$$\"+q4e'H&F-FM7$$\"+I#RLJ&F-FM7$$\"*\\(4I`F1FV7$$\"+D3 PX`F-FM7$$\"+gTkg`F-FV7$$\"+!\\\"R&F-FM7$$\"+!\\PfbF-FV7$$\"+D 3ZubF-FM7$$\"+gTu*e&F-FV7$$\"+!\\<]g&F-FV7$$\"+D3H?cF-FM7$$\"+gTcNcF-F V7$$\"+!\\P3l&F-FM7$$\"+D36mcF-FM7$$\"+gTQ\"o&F-FM7$$\"+!\\dmp&F-F27$$ \"+D3$>r&F-FE7$$\"+gT?FdF-FM7$$\"+!\\xCu&F-FE7$$\"+D3vddF-FE7$$\"+gT-t dF-FE7$$\"+!\\(H)y&F-FM7$$\"+D3d.eF-FV7$$\"*;W)=eF1FM7$$\"+&=]h3'F-FM7 $$\"*@cMN'F1FM7$$\"**ft9oF1FM7$$\"*#)z+M(F1FM7$$\"+?z!Qf(F-FM7$$\"*-Ov %yF1FM7$$\"+0m#4\"zF-FM7$$\"+!>)F-FM7$$\"+!*=.7 #)F-FM7$$\"+S&zyA)F-FV7$$\"+&=FPC)F-FV7$$\"+I[df#)F-FE7$$\"+![AaF)F-FM 7$$\"+D,F\"H)F-FE7$$\"+qx62$)F-FE7$$\"+?a'HK)F-FM7$$\"+lI\")Q$)F-FM7$$ \"*rgYN)F1FE7$$\"++%RiQ)F-FE7$$\"+&4=yT)F-FE7$$\"+SugL%)F-FV7$$\"+!z'R \\%)F-FV7$$\"+Nh=l%)F-FE7$$\"+![v4[)F-FM7$$\"+D[w'\\)F-FE7$$\"+qTb7&)F -FM7$$\"+?NMG&)F-FM7$$\"+lG8W&)F-FV7$$\"+5A#*f&)F-FE7$$\"+g:rv&)F-FM7$ $\"+04]\"f)F-FV7$$\"+]-H2')F-FM7$$\"+S*o)Q')F-FM7$$\"+NwWq')F-FM7$$\"+ !)pB'o)F-FM7$$\"+Ij--()F-FE7$$\"+vc\"yr)F-FM7$$\"+?]gL()F-FV7$$\"+lVR \\()F-FE7$$\"+5P=l()F-FM7$$\"+gI(4y)F-FM7$$\"+0Cw'z)F-FE7$$\"+]*f))F1FV7$$\"+5%)3v))F-FM7$$ \"+IqD!*))F-F27$$\"+]cU0*)F-FE7$$\"+lUf?*)F-FV7$$\"+!)GwN*)F-FE7$$\"++ :$4&*)F-FE7$$\"+?,5m*)F-FM7$$\"+S(o7)*)F-FM7$$\"+gtV'**)F-FE7$$\"+vfg6 !*F-FE7$$\"+!fun-*F-FE7$$\"+5K%>/*F-FE7$$\"+I=6d!*F-FM7$$\"+]/Gs!*F-F2 7$$\"+q!\\u3*F-FM7$$\"+&o/5F1FM7$$\"+Szv05F1FM7$$\"+e2K25F1FV7$$\"+vN))35F1FV7$$\"+#RY/, \"F1FM7$$\"+5#4?,\"F1FM7$$\"+\"[g#=5F1FM7$$\"+_<^C5F1FM7$$\"+qX2E5F1FE 7$$\"+(QPw-\"F1FV7$$\"+/-?H5F1FM7$$\"+AIwI5F1F27$$\"+SeKK5F1FM7$$\"+e' ))Q.\"F1FM7$$\"+w9XN5F1FV7$$\"+$H9q.\"F1FE7$$\"+%3f'Q5F1FE7$$\"+wQIS5F 1FM7$$\"+o'[>/\"F1FM7$$\"+gMfV5F1FE7$$\"+_#Q_/\"F1FV7$$\"+WI)o/\"F1FM7 $$\"+Oy_[5F1FV7$$\"+GE<]5F1FM7$$\"+?u\"=0\"F1FM7$$\"+7AY`5F1FA7$$\"+/q 5b5F1FE7$$\"+'z^n0\"F1FE7$$\"+)e'Re5F1FV7$$\"+!QT+1\"F1FM7$$\"+shoh5F1 FE7$$\"+k4Lj5F1FM7$$\"+cd(\\1\"F1FM7$$\"+[0im5F1FM7$$\"+S`Eo5F1FV7$$\" +K,\"*p5F1FE7$$\"+C\\br5F1FM7$$\"+;(*>t5F1FV7$$\"+3X%[2\"F1FV7$$\"++$* [w5F1FM7$$\"+%))y(z5F1FM7$$\"+o%oI3\"F1FM7$$\"+gKr%3\"F1FM7$$\"+_!ej3 \"F1FV7$$\"+WG+)3\"F1FE7$$\"+Owk*3\"F1FV7$$\"+pl8\"4\"F1FV7$$\"+-bi#4 \"F1FV7$$\"+MW6%4\"F1FE7$$\"+nLg&4\"F1F27$$\"++B4(4\"F1FM7$$\"+K7e)4\" F1FE7$$\"+l,2+6F1FA7$$\"+)4f:5\"F1FM7$$\"+I[^26F1FM7$$\"+h0Z86F1FM7$$ \"+%\\f\\6\"F1FM7$$\"+E%[k6\"F1FM7$$\"+ft$z6\"F1FM7$$\"+#HE%>6F1FE7$$ \"+D_\"47\"F1FM7$$\"+eTSA6F1FM7$$\"+\"4$*Q7\"F1FE7$$\"+C?QD6F1FE7$$\"+ d4(o7\"F1FE7$$\"+!*)f$G6F1FE7$$\"+A)[)H6F1FM7$$\"+bxLJ6F1FV7$$\"+?cJM6 F1FV7$$\"+'[$HP6F1FV7$$\"+O!R*Q6F1FM7$$\"+&e%eS6F1FE7$$\"+M,BU6F1FE7$$ \"+%ovQ9\"F1FV7$$\"+M7_X6F1FM7$$\"+$ymr9\"F1FV7$$\"+KB\")[6F1FV7$$\"+# )yX]6F1FE7$$\"+KM5_6F1FE7$$\"+\")*[P:\"F1FE7$$\"+IXRb6F1FA7$$\"+!3Sq: \"F1FV7$$\"+Icoe6F1FV7$$\"+z6Lg6F1FM7$$\"+Gn(>;\"F1FV7$$\"+yAij6F1FM7$ $\"+GyEl6F1FV7$$\"+xL\"p;\"F1FV7$$\"+E*e&o6F1FM7$$\"+wW?q6F1FE7$$\"+E+ &=<\"F1FV7$$\"+vb\\t6F1FE7$$\"+C69v6F1FE7$$\"+umyw6F1FV7$$\"+CAVy6F1F2 7$$\"+tx2!=\"F1FA7$$\"+ALs\"=\"F1FV7$$\"+s)oL=\"F1FM7$$\"+r*fm=\"F1FM7 $$\"+q5&**=\"F1FM7$$\"+?s_\">\"F1FE7$$\"+qL5$>\"F1FE7$$\"+?&zY>\"F1FE7 $$\"+pcD'>\"F1FE7$$\"+==$y>\"F1FM7$$\"+ozS*>\"F1FM7$$\"+=T)4?\"F1FE7$$ \"+o-c-7F1FM7$$\"+=k8/7F1FM7$$\"+nDr07F1FM7$$\"+;()G27F1FM7$$\"+m[')37 F1FV7$$\"+;5W57F1FM7$$\"+mr,77F1FM7$$\"+;Lf87F1FE7$$\"+l%p^@\"F1FE7$$ \"+9cu;7F1FV7$$\"+k7F1FM7$$\"+kSZ@7F1FA7$$\"+9-0B7F 1FM7$$\"+jjiC7F1FM7$$\"+7D?E7F1FV7$$\"+i'yxA\"F1FM7$$\"+hK3M7F1FM7$$\" +gyQS7F1FM7$$\"+8rnY7F1FM7$$\"+mj'HD\"F1FM7$$\"+z'QXD\"F1FM7$$\"+#*46c 7F1FV7$$\"+1Lod7F1FE7$$\"+>cDf7F1FV7$$\"+Kz#3E\"F1FM7$$\"+Y-Si7F1FE7$$ \"+fD(RE\"F1FV7$$\"+s[al7F1FA7$$\"+&=-8F1FM7$$ \"+#=SQI\"F1FA7$$\"+qC[08F1FM7$$\"+dZ728F1FM7$$\"+Wqw38F1FA7$$\"+K$4/J \"F1FA7$$\"+>;078F1F77$$\"+1Rp88F1FM7$$\"+%>O`J\"F1FV7$$\"+\"[ypJ\"F1F <7$$\"+o2i=8F1F77$$\"+cIE?8F1FV7$$\"+V`!>K\"F1FE7$$\"+IwaB8F1F27$$\"+= **=D8F1F+7$$\"+0A$oK\"F1FE7$$\"+#\\u%G8F1FM7$$\"+!y;,L\"F1FV7$$\"+n!f< L\"F1FA7$$\"+a8SL8F1FM7$$\"+UO/N8F1FM7$$\"+HfoO8F1FM7$$\"+;#G$Q8F1FV7$ $\"+/0(*R8F1FQ7$$\"+\"z7;M\"F1F27$$\"+y]DV8F1F27$$\"+%onZM\"F1FV7$$\"+ \"H!GY8F1FM7$$\"+)*GzZ8F1FM7$$\"+/bI\\8F1FE7$$\"+5\"=3N\"F1FM7$$\"+<2L _8F1FE7$$\"+CL%QN\"F1FE7$$\"+IfNb8F1F27$$\"+O&ooN\"F1F+7$$\"+V6Qe8F1F+ 7$$\"+]P*)f8F1FE7$$\"+cjSh8F1FM7$$\"+j*=HO\"F1FE7$$\"+q:Vk8F1FV7$$\"+w T%fO\"F1FA7$$\"+$ycuO\"F1FA7$$\"+!Rp*o8F1FM7$$\"+'*>[q8F1F77$$\"+.Y*>P \"F1F+7$$\"+5s]t8F1FE7$$\"+;)>]P\"F1FV7$$\"+BC`w8F1F27$$\"+I]/y8F1FV7$ $\"+Owbz8F1FV7$$\"+U-2\"Q\"F1F77$$\"+\\Ge#Q\"F1FV7$$\"+ca4%Q\"F1FE7$$ \"+i!3cQ\"F1FM7$$\"+o17(Q\"F1FE7$$\"+vKj)Q\"F1FV7$$\"+#)e9!R\"F1FM7$$ \"+)[e;R\"F1FM7$$\"+/w?$R\"F1FM7$$\"+?nv%R\"F1FM7$$\"+OeI'R\"F1F27$$\" +_\\&yR\"F1F77$$\"+oSS*R\"F1F27$$\"+%=`4S\"F1F27$$\"++B]-9F1F27$$\"+;9 0/9F1FE7$$\"+K0g09F1FV7$$\"+['\\rS\"F1FA7$$\"+k()p39F1F27$$\"+!)yC59F1 FE7$$\"+'*pz69F1FA7$$\"+7hM89F1FM7$$\"+G_*[T\"F1FM7$$\"+WVW;9F1F+7$$\" +gM*zT\"F1FM7$$\"+wDa>9F1FM7$$\"+#p\"4@9F1FA7$$\"+33kA9F1F27$$\"+C**=C 9F1F27$$\"+S!RdU\"F1FM7$$\"+c\")GF9F1FM7$$\"+ss$)G9F1F27$$\"+)Q'QI9F1F 27$$\"+/b$>V\"F1F+7$$\"+?Y[L9F1FM7$$\"+OP.N9F1FA7$$\"+_GeO9F1FQ7$$\"+o >8Q9F1FM7$$\"+%3\"oR9F1F27$$\"+,-BT9F1FM7$$\"+51%HW\"F1FM7$$\"+=5lW9F1 FM7$$\"+E9OY9F1FA7$$\"+M=2[9F1FV7$$\"+UAy\\9F1FA7$$\"+]E\\^9F1FA7$$\"+ eI?`9F1F27$$\"+mM\"\\X\"F1FM7$$\"+uQic9F1FE7$$\"+#GM$e9F1FE7$$\"+!pW+Y \"F1FV7$$\"+*4b>:F1F27$$\"+99u?:F1Fen7$$\"+[+HA: F1Fen7$$\"+$oQQ_\"F1F<7$$\"+=tQD:F1F<7$$\"+_f$p_\"F1FM7$$\"+'e%[G:F1F+ 7$$\"+@K.I:F1Fen7$$\"+c=eJ:F1FA7$$\"+!\\IJ`\"F1FM7$$\"+C\"zY`\"F1F77$$ \"+fxAO:F1F<7$$\"+%Rwx`\"F1Fjq7$$\"+G]KR:F1F<7$$\"+iO(3a\"F1F27$$\"+(H ACa\"F1F77$$\"+K4(Ra\"F1FV7$$\"+m&>ba\"F1FE7$$\"+NL5Z:F1FA7$$\"+/ro[:F 1FA7$$\"+t3F]:F1FE7$$\"+UY&=b\"F1Fen7$$\"+6%QMb\"F1F<7$$\"+!=A]b\"F1F+ 7$$\"+[fgc:F1Fen7$$\"+<(*=e:F1FM7$$\"+'[t(f:F1FA7$$\"+asNh:F1FQ7$$\"+B 5%Hc\"F1FM7$$\"+#zCXc\"F1F27$$\"+h&3hc\"F1FV7$$\"+IBpn:F1FQ7$$\"+*4w#p :F1F77$$\"+o)f3d\"F1$F1F-7$$\"+POWs:F1FA7$$\"+1u-u:F1F<7$$\"+u6hv:F1FQ 7$$\"+V\\>x:F1F27$$\"+7(y(y:F1FM7$$\"+![i.e\"F1FV7$$\"+\\i%>e\"F1FV7$$ \"+=+`$e\"F1FV7$$\"+(y8^e\"F1FV7$$\"+cvp'e\"F1FA7$$\"+D8G)e\"F1FV7$$\" +%4l)*e\"F1FM7$$\"+j)[9f\"F1FE7$$\"+KE.$f\"F1Faan7$$\"++kh%f\"F1F27$$ \"+p,?'f\"F1F27$$\"+A3\"yf\"F1FQ7$$\"+u9U*f\"F1F27$$\"+E@.,;F1FE7$$\"+ yFk-;F1FA7$$\"+IMD/;F1F77$$\"+#3keg\"F1F27$$\"+MZZ2;F1FE7$$\"+'Q&34;F1 FV7$$\"+Qgp5;F1F<7$$\"+!p1Bh\"F1F77$$\"+Ut\"Rh\"F1F27$$\"+%*z_:;F1FQ7$ $\"+Y'Qrh\"F1F77$$\"+)H\\(=;F1Fjq7$$\"+]*f.i\"F1FV7$$\"+.1(>i\"F1FV7$$ \"+c7eB;F1F+7$$\"+3>>D;F1FE7$$\"+gD!oi\"F1FV7$$\"+7KTG;F1FM7$$\"+kQ-I; F1F<7$$\"+;XjJ;F1FE7$$\"+o^CL;F1FM7$$\"+?e&[j\"F1F<7$$\"+skYO;F1FQ7$$ \"+Cr2Q;F1F<7$$\"+wxoR;F1F^\\n7$$\"+G%)HT;F1F+7$$\"+!34Hk\"F1FA7$$\"+K (>Xk\"F1FQ7$$\"+%QIhk\"F1FA7$$\"+P5uZ;F1F+7$$\"+S:A\\;F1FV7$$\"+U?q];F 1F+7$$\"+XD=_;F1F77$$\"+[Im`;F1F27$$\"+]N9b;F1FE7$$\"+`Sic;F1Fen7$$\"+ cX5e;F1F77$$\"+e]ef;F1FM7$$\"+hb1h;F1FE7$$\"+kgai;F1FM7$$\"+ml-k;F1Fen 7$$\"+pq]l;F1F77$$\"+sv)pm\"F1FM7$$\"+u!o%o;F1FM7$$\"+x&[*p;F1FQ7$$\"+ !3H9n\"F1FA7$$\"+$e4Hn\"F1FQ7$$\"+'3!Ru;F1FQ7$$\"+)eqen\"F1Faan7$$\"+ \"4^tn\"F1F^\\n7$$\"+%fJ)y;F1F27$$\"+'47.o\"F1Fjq7$$\"+*f#z\"o\"F1FM7$ $\"+-JF$o\"F1$\"\")F-7$$\"+0Ov%o\"F1F+7$$\"+3TB'o\"F1FE7$$\"+5Yr(o\"F1 Faan7$$\"+8^>*o\"F1F77$$\"+;cn!p\"F1FE7$$\"+=h:#p\"F1FE7$$\"+@mj$p\"F1 FQ7$$\"+Cr6&p\"F1F77$$\"+#H&p'p\"F1FM7$$\"+gMF)p\"F1FE7$$\"+G;&)*p\"F1 F+7$$\"+'zH9q\"F1FM7$$\"+kz+.s\"F1FM7$$\"+[T_B6/=F1FE7$$\"+[sy0=F1FV7$$\"+Y DY2=F1FV7$$\"+Xy84=F1FA7$$\"+WJ\"3\"=F1F<7$$\"+U%)[7=F1FM7$$\"+TP;9=F1 FA7$$\"+S!Re\"=F1FM7$$\"+RV^<=F1Fen7$$\"+Q'*=>=F1F77$$\"+P\\'3#=F1FE7$ $\"+O-aA=F1FM7$$\"+Nb@C=F1F77$$\"+M3*e#=F1F<7$$\"+LhcF=F1FV7$$\"+K9CH= F1FA7$$\"+Jn\"4$=F1F77$$\"+I?fK=F1FV7$$\"+GtEM=F1F27$$\"+FE%f$=F1FE7$$ \"+EzhP=F1FM7$$\"+CKHR=F1FE7$$\"+B&o4%=F1F+7$$\"+AQkU=F1FA7$$\"+@\">V% =F1F+7$$\"+?W*f%=F1F77$$\"+>(pw%=F1FE7$$\"+=]M\\=F1FA7$$\"+'RB3&=F1FV7 $$\"+vB/q=F1FM7$$\"+)p?:(= F1F27$$\"+w!**H(=F1F77$$\"+buZu=F1F77$$\"+Me&f(=F1FM7$$\"+7UVx=F1FA7$$ \"+\"f7*y=F1F77$$\"+q4R!)=F1FA7$$\"+[$p=)=F1FM7$$\"+FxM$)=F1FQ7$$\"+1h #[)=F1Fjq7$$\"+%[/j)=F1FM7$$\"+jGy()=F1FM7$$\"+U7E*)=F1FE7$$\"+?'R2*=F 1F77$$\"+**z@#*=F1FE7$$\"+yjp$*=F1FM7$$\"+cZ<&*=F1FV7$$\"+NJl'*=F1F+7$ $\"+y)e$)*=F1FA7$$\"+?Y1+>F1FE7$$\"+i.x,>F1Fjq7$$\"+/hZ.>F1F<7$$\"+Y== 0>F1FM7$$\"+)e()o!>F1F<7$$\"+ILf3>F1F77$$\"+s!*H5>F1FV7$$\"+9[+7>F1FA7 $$\"+c0r8>F1FA7$$\"+)H;a\">F1Fjq7$$\"+S?7<>F1Fen7$$\"+#yF)=>F1F+7$$\"+ CN`?>F1FE7$$\"+m#RA#>F1FE7$$\"+3]%R#>F1FQ7$$\"+]2lD>F1FA7$$\"+#\\ct#>F 1Faam7$$\"+MA1H>F1FM7$$\"+wzwI>F1F27$$\"+=PZK>F1FQ7$$\"+g%zT$>F1FQ7$$ \"+-_)e$>F1F<7$$\"+X4fP>F1F27$$\"+)o'HR>F1Fjq7$$\"+IC+T>F1FE7$$\"+s\"3 F%>F1FM7$$\"+9RTW>F1FV7$$\"+c'>h%>F1FE7$$\"+)RDy%>F1F27$$\"+S6`\\>F1FV 7$$\"+#)oB^>F1F+7$$\"+I2w_>F1Fjq7$$\"+xXGa>F1FQ7$$\"+C%3e&>F1FV7$$\"+s ALd>F1FM7$$\"+?h&)e>F1FV7$$\"+n*z.'>F1F+7$$\"+9Q!>'>F1F27$$\"+iwUj>F1F aan7$$\"+5:&\\'>F1Fjq7$$\"+d`Zm>F1FA7$$\"+/#**z'>F1Fjq7$$\"+_I_p>F1Faa m7$$\"+**o/r>F1FV7$$\"+Y2ds>F1F27$$\"+%f%4u>F1F27$$\"+T%=c(>F1FA7$$\"+ )GUr(>F1FE7$$\"+Ohmy>F1F27$$\"+$)**=!)>F1FQ7$$\"+IQr\")>F1F77$$\"+ywB$ )>F1F27$$\"+D:w%)>F1F27$$\"+s`G')>F1F<7$$\"+?#4y)>F1Faan7$$\"+oIL*)>F1 F<7$$\"+:p&3*>F1FM7$$\"+i2Q#*>F1FQ7$$\"+5Y!R*>F1F<7$$\"+e%Ga*>F1Fen7$$ \"+0B&p*>F1FM7$$\"+_hZ)*>F1F27$$F3F*F7-%'COLOURG6&%$RGBGFMFM$\"*++++\" !\")-%+AXESLABELSG6$Q\"x6\"Q!F\\b_l-%%VIEWG6$;F(F`a_l%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Alternati vely we may use the graphing procedure " }{TEXT 0 5 "graph" }{TEXT -1 6 " . . " }{HYPERLNK 17 "graph" 1 "" "graph" }{TEXT -1 131 " from th e previous section, which uses 10 digit precision for evaluating funct ions, instead of using the standard Maple procedure " }{TEXT 0 4 "plot " }{TEXT -1 52 ", which increases the precision to about 15 digits.\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "graph(sin(x)-'sine_serie s'(x),x=-2..2,color=blue,numpoints=80);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7_cn7$$!\"#\"\"!$\"3)********** *****H!#F7$$!3++++]P4'*>!#<$\"3/+++++++5F-7$$!3+++++v=#*>F1$F*F*7$$!3+ +++]7G))>F1F27$$!3+++++]P%)>F1F27$$!3))*****>z>6)>F1$!33+++++++?F-7$$! 3++++](o/)>F1F+7$$!3+++++Dcw>F1$!3/+++++++5F-7$$!3/+++D\"GQ(>F1$!3)*** ************HF-7$$!3++++]ils>F1FN7$$!3++++++vo>F1$\"3K+++++++]F-7$$!3# ******H!4Bn>F1$!3K+++++++]F-7$$!3++++]P%['>F1$\"39+++++++SF-7$$!3#**** **4'oEj>F1$\"33+++++++?F-7$$!3+++++v$4'>F1F_o7$$!3'******>\"y+e>F1F27$ $!3++++]7.d>F1FV7$$!3+++++]7`>F1$!39+++++++SF-7$$!3'*******e3)G&>F1F77 $$!33+++oNt\\>F1F+7$$!3++++](=#\\>F1F+7$$!3+++++DJX>F1FN7$$!3++++]iST> F1F+7$$!3/+++)oH/%>F1Fjn7$$!3++++++]P>F1F27$$!3'******>JqX$>F1FA7$$!3+ +++]PfL>F1F77$$!3+++++voH>F1F+7$$!3'******p$4rG>F1F+7$$!3++++]7yD>F1F_ o7$$!3/+++vo/B>F1FA7$$!3+++++](=#>F1F27$$!34+++v$f$=>F1F+7$$!3++++](oz \">F1FV7$$!3+++++D19>F1FI7$$!3(********[1.\">F1F_o7$$!3++++]i:5>F1FV7$ $!3++++++D1>F1FA7$$!3/+++](oa!>F1Fjn7$$!3++++]PM->F1F27$$!35+++U&)3**= F1F27$$!3+++++vV)*=F1FN7$$!33+++Shk%*=F1Fen7$$!3++++]7`%*=F1FV7$$!3+++ ++]i!*=F1F]p7$$!3)******zzt)*)=F1FI7$$!3!******R`&f()=F1F]p7$$!3++++]( =n)=F1F77$$!3+++++D\"G)=F1Fen7$$!3++++Ya(>)=F1FI7$$!3++++]i!*y=F1FN7$$ !3+++++++v=F1F]p7$$!3++++]P4r=F1FN7$$!3+++++v=n=F1F+7$$!37+++ek`m=F1F2 7$$!35+++\"3mN'=F1F_o7$$!3++++]7Gj=F1F77$$!3+++++]Pf=F1F_o7$$!3++++](o a&=F1F+7$$!3+++++Dc^=F1F_o7$$!3/+++D\"G)[=F1F_o7$$!3++++]ilZ=F1F+7$$!3 ++++++vV=F1FI7$$!3!******z&*)4V=F1FA7$$!3++++]P%)R=F1F27$$!3+++++v$f$= F1F27$$!35+++#H#oK=F1F27$$!3++++]7.K=F1F_o7$$!3+++++]7G=F1F77$$!3++++] (=U#=F1FA7$$!3'******>1*G@=F1F_o7$$!3+++++DJ?=F1F]p7$$!3++++]iS;=F1F_o 7$$!3!******z?bd\"=F1FI7$$!3++++++]7=F1FI7$$!33+++oF()3=F1F77$$!3++++] Pf3=F1F_o7$$!3+++++vo/=F1FA7$$!3#******>[3W!=F1F77$$!3++++]7y+=F1F27$$ !3'******4uT1!=F1F_o7$$!3)******z;!>(z\"F1F+7$$!3+++++](oz\"F1F77$$!3+ +++](oHz\"F1F+7$$!3+++++D1*y\"F1F77$$!3/+++D\"Gjy\"F1F27$$!3++++]i:&y \"F1FI7$$!3'******R,n9y\"F1F_o7$$!3++++++D\"y\"F1F+7$$!3++++]PMxJ&pP;F1FA7$ $!3++++](=nj\"F1F]p7$$!3+++++D\"Gj\"F1F_o7$$!3\"******\\(=UK;F1FI7$$!3 -+++Pt&)H;F1F27$$!3++++]i!*G;F1FA7$$!3+++++++D;F1F_o7$$!3!******z&*[Vi \"F1F27$$!3++++]P4@;F1F77$$!31+++H#o2i\"F1F]p7$$!3+++++v=<;F1F27$$!3++ ++]7G8;F1F27$$!3+++++]P4;F1FA7$$!32+++RQ'og\"F1F+7$$!3++++](oag\"F1FN7 $$!3+++++Dc,;F1F+7$$!3'******>\"Gj)f\"F1Fen7$$!3++++]il(f\"F1FV7$$!3++ ++++v$f\"F1FI7$$!3/+++QMx#f\"F1FN7$$!3++++]P%)*e\"F1F27$$!3!******4[W& *e\"F1FN7$$!3'******HnPse\"F1F+7$$!3+++++v$fe\"F1F]p7$$!3++++]7.#e\"F1 F+7$$!31+++\"=Y4e\"F1FI7$$!3+++++]7y:F1F27$$!3'*******)Gkqd\"F1F27$$!3 '******zq\"eu:F1F_o7$$!3++++](=Ud\"F1F77$$!3+++++DJq:F1F+7$$!3'******p $fLp:F1FN7$$!3++++]iSm:F1F_o7$$!35+++Z*)*[c\"F1Fjn7$$!3++++++]i:F1F27$ $!3++++]Pfe:F1F]p7$$!3+++++voa:F1F]p7$$!3++++]7y]:F1FI7$$!3'*******\\i lZ:F1F+7$$!3+++++](oa\"F1F]p7$$!3++++](oHa\"F1F+7$$!31+++\"o$)=a\"F1F+ 7$$!3+++++D1R:F1F27$$!3'******REt``\"F1F_bl7$$!3++++]i:N:F1FN7$$!3++++ ++DJ:F1F_o7$$!3-+++>:F1FA7$$!3+++++]i::F1F+7$$!3++++Yzy9:F1F+7$$!3++++](=<^\"F1F27 $$!3*******>M\\2^\"F1F77$$!3+++++D\"y]\"F1F_o7$$!3++++'zFt]\"F1F_bl7$$ !3++++]i!R]\"F1F77$$!3++++++++:F1F27$$!3++++]P4'\\\"F1F27$$!3++++d]O# \\\"F1FV7$$!3+++++v=#\\\"F1F]p7$$!3++++]7G)[\"F1F_o7$$!3/+++)o/t[\"F1F I7$$!3+++++]P%[\"F1F77$$!3++++](o/[\"F1F77$$!3+++++Dcw9F1F77$$!3))**** *>H2LZ\"F1FI7$$!3++++]ils9F1FN7$$!3++++++vo9F1F+7$$!3%******4\"\\Nn9F1 F77$$!3++++]P%[Y\"F1F_o7$$!3%******z!eYk9F1FI7$$!3+++++v$4Y\"F1FI7$$!3 ++++]7.d9F1F27$$!3+++++]7`9F1F27$$!3++++](=#\\9F1F77$$!3+++++DJX9F1F77 $$!3++++]iST9F1F77$$!3++++++]P9F1F77$$!3++++]PfL9F1F77$$!3+++++voH9F1F A7$$!3++++]7yD9F1F77$$!3+++++](=U\"F1FA7$$!3++++](ozT\"F1FA7$$!3+++++D 199F1FI7$$!3++++]i:59F1F_o7$$!3++++++D19F1F_o7$$!3++++]PM-9F1F77$$!3++ +++vV)R\"F1F77$$!3++++]7`%R\"F1F27$$!3+++++]i!R\"F1F_o7$$!3++++](=nQ\" F1F27$$!3+++++D\"GQ\"F1F_o7$$!3++++]i!*y8F1F27$$!3+++++++v8F1FI7$$!3++ ++]P4r8F1FI7$$!3+++++v=n8F1F_o7$$!3++++]7Gj8F1FI7$$!3+++++]Pf8F1FI7$$! 3++++](oaN\"F1F27$$!3!******zq<[N\"F1F77$$!3+++++Dc^8F1F77$$!3++++]ilZ 8F1F+7$$!3++++++vV8F1F77$$!3'******pVtFM\"F1F27$$!3++++]P%)R8F1F27$$!3 5+++U&)eO8F1FI7$$!3+++++v$fL\"F1F77$$!3%*******=j6L8F1F77$$!3++++]7.K8 F1F_o7$$!3+++++]7G8F1Fen7$$!3.+++gZ(zK\"F1FI7$$!3++++](=UK\"F1F27$$!3- +++IO9C8F1FI7$$!3+++++DJ?8F1FI7$$!3++++]iS;8F1FI7$$!3++++++]78F1F27$$! 3++++]Pf38F1F27$$!3+++++vo/8F1FI7$$!3!******zXOSI\"F1FI7$$!3++++]7y+8F 1F77$$!3+++++](oH\"F1F27$$!3))*****>z>OH\"F1F77$$!3++++](oHH\"F1F_o7$$ !3+++++D1*G\"F1F77$$!3'******>\"G8'G\"F1F77$$!3++++]i:&G\"F1FI7$$!3+++ +++D\"G\"F1F_o7$$!3#******\\Pf3G\"F1F77$$!3++++]PMx7F1F77$$!3))*****>a )3u7F1F_o7$$!3+++++vVt7F1FI7$$!3++++]7`p7F1F27$$!3+++++]il7F1F27$$!35+ ++#zpBE\"F1F77$$!3++++](=J?$G5F1F77$$!3++++]PMF 5F1FI7$$!3+++++vVB5F1F_o7$$!3++++]7`>5F1FI7$$!3'******poa&=5F1F77$$!3+ ++++]i:5F1F77$$!35+++#zpB,\"F1FI7$$!3++++](=<,\"F1FI7$$!3+++++D\"y+\"F 1F27$$!3++++]i!R+\"F1F27$$!\"\"F*FI7$$!3++++++vV)*!#=FI7$$!3++++++](o* F[fnFI7$$!3+++++vV['*F[fnFI7$$!3+++++]P4'*F[fnF77$$!3+++++DJq&*F[fnF77 $$!3++++++DJ&*F[fnF27$$!3)********\\(=#\\*F[fnF27$$!3+++++]7`%*F[fnF77 $$!3+++++D19%*F[fnFI7$$!3+++++++v$*F[fnF77$$!3+++++v$fL*F[fnF77$$!3+++ ++](oH*F[fnF27$$!3+++++D\"yD*F[fnF77$$!3++++++v=#*F[fnF27$$!3+++++voz \"*F[fnF77$$!3+++++]iS\"*F[fnFI7$$!3+++++Dc,\"*F[fnF77$$!3++++++]i!*F[ fnF77$$!3+++++]P%)*)F[fnF77$$!3++++++D1*)F[fnF77$$!3+++++v=n))F[fnFI7$ $!3+++++]7G))F[fnFA7$$!3+++++D1*y)F[fnFI7$$!3+++++++]()F[fnFI7$$!3++++ +v$4r)F[fnFI7$$!3+++++](=n)F[fnF77$$!3+++++D\"Gj)F[fnF77$$!3)********* *\\Pf)F[fnFI7$$!3+++++voa&)F[fnFI7$$!3+++++]i:&)F[fnFI7$$!3+++++Dcw%)F [fnFI7$$!3++++++]P%)F[fnF27$$!3+++++vV)R)F[fnFI7$$!3M+++$3F>R)F[fnF77$ $!3+++++]Pf$)F[fnF77$$!3+++++DJ?$)F[fnF77$$!3++++++D\"G)F[fnF77$$!3+++ ++]7.#)F[fnF77$$!3+++++++D\")F[fnF77$$!3+++++](o/)F[fnF77$$!3++++++voz F[fnF77$$!3m*****p\"z>OzF[fnF27$$!3+++++voHzF[fnFI7$$!3)*********\\i!* yF[fnF27$$!3M+++$e9T)yF[fnF77$$!3+++++Dc^yF[fnF77$$!3++++++]7yF[fnFI7$ $!3+++++vVtxF[fnF77$$!3+++++]PMxF[fnFI7$$!3+++++DJ&p(F[fnF77$$!3m***** pTgFm(F[fnF77$$!3++++++DcwF[fnF27$$!3+++++v=F77$$Fe^oF^^pF77$$FjboF^^p$\"2%****** **********!#G7$$FbdoF^^pF77$$FffoF^^pF77$$FejoF^^pF77$$F[jlF^^pFc^p7$$ Fe^o!#?F77$F7F77$$\"3++++++]7yF`_pF77$$\"3++++++]i:F^^p$!2%*********** *****Fe^p7$$\"3++++++vVBF^^pF77$$\"3+++++++DJF^^pF77$$\"3++++++](o%F^^ pF77$$\"3+++++++]iF^^pFh_p7$$Fd_pF^^pF77$$\"3+++++++v$*F^^pF77$$\"3)** ********\\P4\"F[fnF77$$\"3+++++++]7F[fnF77$$\"3+++++D1*G\"F[fnF77$$\"3 +++++]7G8F[fnF77$$\"3+++++v=n8F[fnF77$$\"3++++++D19F[fnFI7$$\"3+++++DJ X9F[fnF77$$\"3+++++]P%[\"F[fnF77$$\"3+++++vVB:F[fnF77$$\"31+++$e*)fb\" F[fnF77$$Fg_pF[fnF27$$\"3+++++Dc,;F[fnFI7$$\"3%******p\"H23;F[fnF77$$ \"3+++++]iS;F[fnF77$$\"3+++++voz;F[fnF77$$\"3++++++v=OzF[fnFI7$$\"3++++++vozF[fnF77$$\"3+++++](o/)F[fnF77 $$\"3+++++++D\")F[fnF77$$\"3+++++]7.#)F[fnF77$$\"3++++++D\"G)F[fnF77$$ \"3+++++DJ?$)F[fnF77$$\"3+++++]Pf$)F[fnF77$$\"3M+++$3F>R)F[fnF77$$\"3+ ++++vV)R)F[fnF27$$\"3++++++]P%)F[fnFI7$$\"3+++++Dcw%)F[fnF27$$\"3+++++ ]i:&)F[fnF27$$\"3+++++voa&)F[fnF27$$\"3)**********\\Pf)F[fnF27$$\"3+++ ++D\"Gj)F[fnF77$$\"3+++++](=n)F[fnF77$$\"3+++++v$4r)F[fnF27$$\"3++++++ +]()F[fnF27$$\"3+++++D1*y)F[fnF27$$\"3+++++]7G))F[fnF_o7$$\"3+++++v=n) )F[fnF27$$\"3++++++D1*)F[fnF77$$\"3+++++]P%)*)F[fnF77$$\"3++++++]i!*F[ fnF77$$\"3+++++Dc,\"*F[fnF77$$\"3+++++]iS\"*F[fnF27$$\"3+++++voz\"*F[f nF77$$\"3++++++v=#*F[fnFI7$$\"3+++++D\"yD*F[fnF77$$\"3+++++](oH*F[fnFI 7$$\"3+++++v$fL*F[fnF77$$Fj`pF[fnF77$$\"3+++++D19%*F[fnF27$$\"3+++++]7 `%*F[fnF77$$\"3)********\\(=#\\*F[fnFI7$$\"3++++++DJ&*F[fnFI7$$\"3++++ +DJq&*F[fnF77$$\"3+++++]P4'*F[fnF77$$\"3+++++vV['*F[fnF27$$\"3++++++]( o*F[fnF27$$\"3++++++vV)*F[fnF27$$\"\"\"F*F27$$\"3++++]i!R+\"F1FI7$$\"3 +++++D\"y+\"F1FI7$$\"3++++](=<,\"F1F27$$\"35+++#zpB,\"F1F27$$\"3+++++] i:5F1F77$$\"3%******4_]f,\"F1F77$$\"3&*******[\"R(=5F1F77$$\"3++++]7`> 5F1F27$$\"3+++++vVB5F1FA7$$\"3++++]PMF5F1F27$$\"3/+++8.KG5F1F77$$\"3++ ++++DJ5F1F77$$\"3++++]i:N5F1F77$$\"3+++++D1R5F1FI7$$\"3++++](oH/\"F1FI 7$$\"3#******\\P%[Y5F1F27$$\"3+++++](o/\"F1F27$$\"3++++]7y]5F1FA7$$\"3 +++++voa5F1F27$$\"3++++]Pfe5F1F27$$\"3++++++]i5F1F77$$\"3++++]iSm5F1F7 7$$\"3+++++DJq5F1F_o7$$\"35+++UN'42\"F1F77$$\"3++++](=U2\"F1F27$$\"3%* *****4FWX2\"F1FA7$$\"3+++++]7y5F1F77$$\"3/+++)oa53\"F1FA7$$\"3++++]7.# 3\"F1F27$$\"3+++++v$f3\"F1FA7$$\"3++++]P%)*3\"F1F27$$\"33+++vVB!4\"F1F 77$$\"3)******>_a)3u7F1FA7$$\"3++++]PMx7F1F7 7$$\"3%******4FpwF\"F1F77$$\"35+++Ue.\"G\"F1F77$$\"3++++++D\"G\"F1FA7$ $\"3++++]i:&G\"F1F27$$\"3/+++8G8'G\"F1FN7$$FcapF1F77$$\"3#******>Iw)*G \"F1FA7$$\"3++++](oHH\"F1FA7$$\"3+++++](oH\"F1FI7$$\"3++++]7y+8F1F77$$ \"3!******zXOSI\"F1F27$$\"3+++++vo/8F1F27$$\"3++++]Pf38F1FI7$$\"3+++++ +]78F1FI7$$\"3++++]iS;8F1F27$$\"3+++++DJ?8F1F27$$\"3/+++)=UKK\"F1F27$$ \"3++++](=UK\"F1FI7$$\"3.+++gZ(zK\"F1F27$$FfapF1FV7$$\"3++++]7.K8F1FA7 $$\"3%*******=j6L8F1F77$$\"3+++++v$fL\"F1F77$$\"3++++]P%)R8F1FI7$$\"3/ +++QMxU8F1F_o7$$\"3++++++vV8F1F77$$\"3++++]ilZ8F1FN7$$\"3+++++Dc^8F1F7 7$$\"3/+++j!RDN\"F1FA7$$\"3++++](oaN\"F1FI7$$\"3+++++]Pf8F1F27$$\"3+++ +]7Gj8F1F27$$FiapF1FA7$$\"3++++]P4r8F1F27$$\"3+++++++v8F1F27$$\"3++++] i!*y8F1FI7$$\"3+++++D\"GQ\"F1FA7$$\"3++++](=nQ\"F1FI7$$\"3+++++]i!R\"F 1FA7$$\"3++++]7`%R\"F1FI7$$\"3+++++vV)R\"F1F77$$\"3++++]PM-9F1F77$$F\\ bpF1FA7$$\"3++++]i:59F1FA7$$\"3+++++D199F1F27$$\"3++++](ozT\"F1F_o7$$ \"3+++++](=U\"F1F_o7$$\"3++++]7yD9F1F77$$\"3+++++voH9F1F_o7$$\"3++++]P fL9F1F77$$\"3++++++]P9F1F77$$\"3++++]iST9F1F77$$F_bpF1F77$$\"3++++](=# \\9F1F77$$\"3+++++]7`9F1FI7$$\"3++++]7.d9F1FI7$$\"3#******\\(oag9F1FI7 $$\"3+++++v$4Y\"F1F27$$\"3++++]P%[Y\"F1FA7$$\"3%******4\"\\Nn9F1F77$$ \"3++++++vo9F1FN7$$\"3++++]ils9F1F+7$$\"3))*****>H2LZ\"F1F27$$\"3+++++ Dcw9F1F77$$\"3++++](o/[\"F1F77$$FbbpF1F77$$\"3/+++)o/t[\"F1F27$$\"3+++ +]7G)[\"F1FA7$$\"3+++++v=#\\\"F1Fjn7$$\"3++++d]O#\\\"F1Fen7$$\"3++++]P 4'\\\"F1FI7$$\"3++++++++:F1FI7$$\"3!******z?bK]\"F1F27$$\"3++++]i!R]\" F1F77$$\"3+++++D\"y]\"F1FA7$$\"35+++UNY3:F1F+7$$\"3++++](=<^\"F1FI7$$ \"3++++Yzy9:F1FN7$$\"3+++++]i::F1FN7$$\"3++++]7`>:F1F_o7$$FebpF1F_o7$$ \"3++++]PMF:F1F27$$\"3-+++>v:F1FI7$$\"3+++++]7y:F1FI7$$\"3 1+++\"=Y4e\"F1F27$$\"3++++]7.#e\"F1FN7$$\"3+++++v$fe\"F1Fjn7$$\"3%**** **4-jie\"F1$\"3g**************pF-7$$\"3++++]P%)*e\"F1FI7$$\"31+++O*)4 \"f\"F1Fen7$$\"3++++++v$f\"F1F27$$\"3#******HfxVf\"F1F_o7$$\"3++++]il( f\"F1Fen7$$\"3)******\\]9*)f\"F1F+7$$F]cpF1FN7$$\"3#******zi\">-;F1FA7 $$\"3++++](oag\"F1F+7$$\"32+++>e!pg\"F1FN7$$\"3+++++]P4;F1F_o7$$\"33++ +x:X5;F1FA7$$\"3++++]7G8;F1FI7$$\"3+++++v=<;F1FI7$$\"3!******zqU/i\"F1 F27$$\"3++++]P4@;F1F77$$\"3!******z&*[Vi\"F1FI7$$\"3+++++++D;F1FA7$$\" 3++++]i!*G;F1F_o7$$\"3\"******\\(=UK;F1F27$$\"3+++++D\"Gj\"F1FA7$$\"3+ +++](=nj\"F1Fjn7$$\"3'******>J&pP;F1F_o7$$FccpF1F_o7$$\"3++++]7`W;F1Fe n7$$\"3(********[\"oW;F1FV7$$\"3+++++vV[;F1F27$$\"3)******z:['\\;F1FV7 $$\"3++++]PM_;F1F_o7$$\"31+++)4:Fl\"F1$!3g**************pF-7$$\"3+++++ +Dc;F1F[ds7$$\"31+++K+xc;F1F27$$\"31+++[#[+m\"F1F_o7$$\"3++++]i:g;F1FN 7$$\"3+++++D1k;F1F+7$$\"35+++UNrk;F1F_o7$$\"3++++](ozm\"F1F77$$\"3++++ >(o)o;F1F+7$$\"3+++++](=n\"F1F27$$\"3++++]7yv;F1FN7$$\"31+++o-1w;F1F27 $$\"3++++s)o)y;F1FN7$$FfcpF1F77$$\"3++++]Pf$o\"F1F+7$$\"33+++vV)Ro\"F1 FV7$$\"3++++++](o\"F1F27$$FicpF1F27$$\"3+++++++]a)3uMN$3=F1F77$$\"3++++]Pf3=F1FA7$$\"3++++++]7=F1F27$$\"3!* *****z?bd\"=F1F27$$\"3++++]iS;=F1FA7$$\"3+++++DJ?=F1Fjn7$$\"3'******>1 *G@=F1FA7$$\"3++++](=U#=F1F_o7$$\"3(******>qR_#=F1F]p7$$\"3+++++]7G=F1 F77$$\"3$******>Di\"H=F1F27$$\"3++++]7.K=F1FA7$$\"3\"******\\7=L$=F1F+ 7$$\"3+++++v$f$=F1FI7$$\"3++++]P%)R=F1FI7$$\"3!******z&*)4V=F1F_o7$$\" 3++++++vV=F1F27$$\"3++++]ilZ=F1FN7$$\"3/+++D\"G)[=F1FA7$$\"3+++++Dc^=F 1FA7$$\"3++++](oa&=F1FN7$$\"3+++++]Pf=F1FA7$$\"3!******\\i!*G'=F1FI7$$ \"3++++]7Gj=F1F77$$\"37+++ek`m=F1FI7$$\"3+++++v=n=F1FN7$$\"3++++]P4r=F 1F+7$$F_dpF1Fjn7$$\"37+++3_Dy=F1F_]t7$$\"3++++]i!*y=F1F+7$$\"3++++Ya(> )=F1F27$$\"3+++++D\"G)=F1FV7$$\"3++++](=n)=F1F77$$\"3'******RwNp)=F1F] p7$$\"3%******>r@1*=F1F77$$\"3+++++]i!*=F1Fjn7$$\"3++++]7`%*=F1Fen7$$ \"33+++Shk%*=F1FV7$$\"3+++++vV)*=F1F+7$$\"3++++]PM->F1FI7$$\"33+++oFi- >F1F27$$\"3#******fhQ\\!>F1FI7$$\"3++++++D1>F1F_o7$$\"3++++]i:5>F1Fen7 $$\"3(********[1.\">F1FA7$$\"3&******\\9&\\8>F1F27$$\"3+++++D19>F1F27$ $\"3++++](oz\">F1Fen7$$\"34+++v$f$=>F1FN7$$\"3+++++](=#>F1FI7$$\"3&*** ***\\XwE#>F1FI7$$\"3++++]7yD>F1FA7$$\"3/+++Q4rG>F1FN7$$\"3+++++voH>F1F N7$$\"3++++]PfL>F1F77$$\"3/+++8.dM>F1Fjn7$$\"3++++++]P>F1FI7$$\"3/+++) oH/%>F1F]p7$$\"3++++]iST>F1FN7$$\"3+++++DJX>F1F+7$$\"3/+++5&o!\\>F1FV7 $$\"3++++](=#\\>F1FN7$$\"3'*******e3)G&>F1F77$$\"3+++++]7`>F1Fjn7$$\"3 ++++]7.d>F1Fen7$$\"3'******>\"y+e>F1FI7$$\"3+++++v$4'>F1FA7$$\"3++++]P %['>F1F]p7$$\"3#******H!4Bn>F1FV7$$\"3++++++vo>F1Fen7$$\"3++++]ils>F1F +7$$\"3/+++D\"GQ(>F1F+7$$\"3+++++Dcw>F1F27$$\"3++++](o/)>F1FN7$$\"3))* ****>z>6)>F1F_o7$$\"3+++++]P%)>F1FI7$$\"3++++]7G))>F1FI7$$\"3+++++v=#* >F1F77$$\"3++++]P4'*>F1FI7$$\"\"#F*FN-%'COLOURG6&%$RGBGF7F7$\"*++++\"! \")-%+AXESLABELSG6$Q\"x6\"Q!F]du-%%VIEWG6$;F(F`cu%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "grap h(sin(x)-'sine_series'(x),x=-5..5,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7^am7$$!\"&\"\"!$!31+ ++++++5!#E7$$!3++++vVB!*\\!#(F-7 $$!3+++++v$4'\\F1$\"3w*************\\%F-7$$!3++++v=<^\\F1$\"3)******** ******>\"F-7$$!3++++]iST\\F1$!3=+++++++RF-7$$!3A+++n,^K\\F1$\"3;++++++ +jF-7$$!3++++D1kJ\\F1$!3K**************)*F-7$$!3+++++](=#\\F1$!31+++++ ++8F-7$$!3t******oG7>\\F1$!3=+++++++LF-7$$!3++++v$4@\"\\F1$!3/+++++++P F-7$$!3\")*****z/H_!\\F1$\"3%)*************R$F-7$$!3++++]PM-\\F1$!3C++ +++++kF-7$$!3++++D\"yD*[F1$\"3I+++++++TF-7$$!3e*****z%y(y)[F1$\"3'**** ***********fF-7$$!3+++++D\"G)[F1$\"3=+++++++RF-7$$!3))*****Ri\"z\")[F1 $\"3)**************\\\"F-7$$!3y*****f7KJ([F1F57$$!3++++vo/t[F1$\"3))** ***********H)F-7$$!3++++]7Gj[F1$!3K+++++++]!#F7$$!3u******RA@f[F1$\"3% )*************z#F-7$$!3++++Dc^`[F1$\"3)***************HF^r7$$!3'****** >aS%\\[F1$!3-+++++++5!#D7$$!3++++++vV[F1$\"3G+++++++!)F^r7$$!31+++fp=R [F1$\"31+++++++DF-7$$!3++++vV)R$[F1$F*F*7$$!3++++](=U#[F1$!3g********* *****pF^r7$$!3q*****p#pE:[F1$\"3/+++++++JF-7$$!3++++DJX9[F1F:7$$!3v*** **H+$[2[F1F\\o7$$!3+++++vo/[F1$\"3/+++++++5F^r7$$!3++++v=#\\z%F1$!3.++ ++++?5F^s7$$!3e*****H8tKz%F1$!3Q+++++++UF-7$$!3)******>/3yy%F1F?7$$!3+ +++]i:&y%F1$!3;+++++++^F-7$$!3++++D1RvZF1$\"3g**************pF^r7$$!3+ ++++]ilZF1$\"3!***************GF-7$$!3M+++2%*4jZF1$\"35+++++++iF-7$$!3 ++++v$fev%F1$\"3%)*************>#F-7$$!3W+++D.3\\ZF1Fao7$$!3++++]P4YZF 1$\"33+++++++?F^r7$$!3++++D\"Gjt%F1$\"3-+++++++&)F-7$$!3C+++f4iNZF1F:7 $$!3+++++DcEZF1$\"3!**************H#F-7$$!3(******Hb\"==ZF1$\"31++++++ +8F-7$$!3++++voz;ZF1$\"3!**************p%F-7$$!3++++]7.2ZF1$\"39++++++ +SF^r7$$!3)******\\ilsp%F1$!3)**************>\"F-7$$!37+++3@=!p%F1$\"3 )**************R#F-7$$!3++++++](o%F1$!3%)*************z#F-7$$!3++++vVt xYF1$\"3=+++++++FF-7$$!3e******H1guYF1F:7$$!3++++](ozm%F1F_q7$$!37+++' eomm%F1$\"3Q+++++++UF-7$$!3++++DJ?eYF1F\\o7$$!3s*****\\kSpl%F1$!3%)*** **********f%F-7$$!3+++++vV[YF1Fdz7$$!3)******>j()Rk%F1$!3!************ ***GF-7$$!3++++v=nQYF1$\"37+++++++?F-7$$!3x******o)f^j%F1F^[l7$$!3++++ ]i!*GYF1F\\t7$$!3C+++^#\\'>YF1$!3)**************R#F-7$$!3++++D19>YF1$! 3i*************H%F-7$$!3+++++]P4YF1$\"31+++++++5F-7$$!3?+++$[V4f%F1Fbv 7$$!3+++++DJqXF1F\\t7$$!37+++W')z_XF1$!3C+++++++SF-7$$!3++++++DJXF1F`t 7$$!3++++](=<^%F1$!39+++++++SF^r7$$!3v*******y!y,XF1F`y7$$!3+++++v=#\\ %F1$!3)***************HF^r7$$!3#)*****pAKI[%F1$\"3g**************yF-7$ $!3++++v=U#[%F1$!3w*************\\%F-7$$!3++++]ilsWF1$\"3%************ ****)F^r7$$!3U+++>%)=lWF1$!37+++++++?F-7$$!3++++D1*GY%F1$\"3)********* *****z\"F-7$$!3+++++]7`WF1F`t7$$!3<+++[5AWWF1F]al7$$!3++++v$fLW%F1$!3Y *************p(F-7$$!3++++]PfLWF1F[y7$$!3++++D\"GQU%F1F[y7$$!3$)*****p ^fkT%F1$\"3?+++++++@F-7$$!3+++++D19WF1F]al7$$!3&)*****>Tu[S%F1$!35++++ +++]F-7$$!3++++voH/WF1$\"35+++++++]F-7$$!3++++]7`%R%F1$!33+++++++?F^r7 $$!3E+++>Bt#R%F1F^x7$$!3Q+++c7V&Q%F1$\"3=+++++++LF-7$$!3++++Dcw%Q%F1Ff w7$$!3+++++++vVF1$!31+++++++>F-7$$!3W++++*HiO%F1$\"31+++++++>F-7$$!3++ ++vVBlVF1Ffw7$$!39+++'yEgN%F1Fiz7$$!3++++](oaN%F1F^_l7$$!3++++DJqXVF1$ \"35+++++++QF-7$$!3X+++nJtUVF1F`t7$$!3')*****>kdhL%F1F_el7$$!3+++++v$f L%F1Fdel7$$!3++++v=VF1Fet7$$!3++++]iS;VF1$!3%****************)F^r7$$!3++++D1k1VF1$\"3i** ***********\\&F-7$$!3@+++2G*eI%F1Fgdl7$$!3+++++](oH%F1Fbs7$$!3A+++Nl([ H%F1$\"3K+++++++]F^r7$$!3++++v$4rG%F1F^]l7$$!3)******zNO0G%F1$\"3#**** **********p\"F-7$$!3++++]PMxUF1F\\w7$$!3++++D\"yvE%F1$!3#************* *R\"F-7$$!3K+++6bzlUF1Fet7$$!3+++++D\"yD%F1$FfpF^r7$$!39+++4QS^UF1F57$ $!3++++vo/[UF1F`u7$$!3++++]7GQUF1Fbr7$$!3P+++sx[JUF1Fc^l7$$!3++++Dc^GU F1F`hl7$$!3++++++v=UF1$!3o*************f&F-7$$!3K+++i#3i@%F1$\"3i***** ********H%F-7$$!3++++vV)*3UF1$!3?+++++++@F-7$$!3)*******\\(=#*>%F1F`\\ m7$$!3++++DJX*=%F1$!3))**************eF-7$$!3w*****pwd*)=%F1F]cl7$$!3+ ++++vozTF1Fiz7$$!3%******RK5z<%F1F[p7$$!3++++v=#*pTF1F^`l7$$!3++++'>!= mTF1$\"3)**************f$F-7$$!3++++]i:gTF1F[jl7$$!3;+++?5c\"fCTF1F\\w7$$!3++++]P4@T F1$!3#**************p\"F-7$$!3++++D\"G86%F1Fbfl7$$!3:+++P'3.6%F1$!3'** *************fF^r7$$!3;+++?%3X5%F1$!3G+++++++!)F^r7$$!3+++++Dc,TF1F^x7 $$!3++++voz\"4%F1$!3!**************p%F-7$$!3')*****f4*e\"4%F1Fbs7$$!3+ +++]7.#3%F1F_dl7$$!3C+++*GB[2%F1$F_`lF-7$$!3++++DcEsSF1F`t7$$!3++++++] iSF1F]al7$$!3/+++LT>()RF1$\"3#**************R\"F-7$$!3++++++D1RF1F\\t7 $$!3@+++?CUDQF1F^`l7$$!3+++++++]PF1F`t7$$!3++++vVBSPF1Fcjl7$$!3++++](o /t$F1FI7$$!3;+++e[-IPF1F_dl7$$!39+++x\"*eAPF1Fgv7$$!3++++DJq?PF1Fgv7$$ !3+++++v$4r$F1$!31+++++++;F-7$$!37+++\\[d3PF1Fc^l7$$!3++++v=<,PF1F]am7 $$!31+++!\\[')p$F1F`u7$$!3)*******\\iS\"p$F1F`u7$$!3++++D1k\"o$F1F[jl7 $$!3')*****>ba,o$F1Fh`m7$$!3+++++](=n$F1Fh`m7$$!3+++++D\"Gj$F1Ff_l7$$! 39+++m'R6h$F1F+7$$!3++++++v$f$F1Fcjl7$$!3++++vV)Re$F1Fbal7$$!3#)*****R W&=#e$F1$!3#**************4\"F-7$$!3++++](=Ud$F1F]am7$$!3++++DJXkNF1$! 3/+++++++5F^r7$$!3+++++voaNF1FI7$$!3'******4?$pZNF1F^x7$$!3++++v=#\\a$ F1F^]l7$$!3++++]i:NNF1F[hl7$$!3(******R)*eH`$F1F`t7$$!3++++D1RDNF1F^x7 $$!3++++,1+;NF1F\\r7$$!3+++++]i:NF1Fcjl7$$!3++++]P4'\\$F1F_z7$$!3))*** **R;+`[$F1Fdel7$$!3+++++DcwMF1Fgn7$$!3++++]7.dMF1F[il7$$!3y*****R!e0]M F1Fdel7$$!3++++++]PMF1F^x7$$!3+++++vV)R$F1F_dl7$$!3A+++o0:hLF1F^`l7$$! 3+++++]PfLF1F_z7$$!3+++++DJ?LF1F\\t7$$!3))*****pTgxG$F1F_dl7$$!3++++++ D\"G$F1Fgv7$$!3+++++v=UKF1F`t7$$!3.+++],\"y@$F1$!31++++++g5F-7$$!3++++ +]7.KF1$!3**************R6F-7$$!3++++]Pf$=$F1$F_`l!#G7$$!3+++++D1kJF1$ \"3-++++++!>\"F-7$$!3=+++&>f8;$F1$\"3#)************\\@F-7$$!3++++]7`WJ F1$\"3O++++++!*\\F^r7$$!3@+++\"4g&GJF1$\"3g*************H(F^r7$$!3++++ +++DJF1$!3K+++++++`F^r7$$!3++++vVB:JF1$\"33+++++++F-7$$!3+++++](o/$F1$!3'************ **p&F^r7$$!36+++b,CPIF1Fdel7$$!3++++]PMFIF1Fh`m7$$!3*******>LcA-$F1Fcj l7$$!3+++++D\"y+$F1Fcil7$$!3.++++#[t+$F1Fgr7$$!3++++]7G))HF1Fegm7$$!3- ++++\"\\!))HF1Fgv7$$!3++++++voHF1Fgr7$$!3++++](=#\\HF1F]am7$$!3+++++vo HHF1F_z7$$!3++++]i:5HF1F_z7$$!3+++++]i!*GF1Fegm7$$!3%******zV)R$)GF1Fe gm7$$!3++++v$f3)GF1F[il7$$!3=+++#*)*=sGF1F+7$$!3++++]P4rGF1F[jl7$$!3++ ++D\"G8'GF1Fc_m7$$!3#)*****>Xm/'GF1Ff_l7$$!3#******H+]Z&GF1F`t7$$!3+++ ++Dc^GF1F\\r7$$!3y*****>JXR%GF1F\\t7$$!3++++vozTGF1F^x7$$!3++++]7.KGF1 Fgdm7$$!3-+++W)*eHGF1F_q7$$!3++++DcEAGF1F+7$$!30+++$y17#GF1F[il7$$!3++ ++++]7GF1Fcil7$$!3;+++/PoPFF1Fgr7$$!3++++++DcEF1Fbs7$$!30+++E'[cj#F1Fe gm7$$!3++++++++DF1Fegm7$$!3++++++vVBF1Ff_l7$$!3+++++DcEAF1Fgr7$$!3++++ ++](=#F1F]am7$$!3+++++]P4@F1Ff_l7$$!3++++++DJ?F1F\\t7$$!3/+++Q4YA?F1F_ dl7$$!3++++vV[@?F1Fjy7$$!3++++](=<,#F1F_dl7$$!3<+++e[F6?F1F\\r7$$!3+++ +DJ&>+#F1Fgr7$$!3-+++5ge**>F1F_dl7$$!3+++++v=#*>F1F\\t7$$!31+++&e+t)>F 1F`u7$$!3++++]ils>F1F^`l7$$!3)******>#Qwh>F1F^`l7$$!3+++++]7`>F1Ff_l7$ $!3++++v$fL%>F1F^x7$$!3-+++pz\"4%>F1Fegm7$$!3.+++\"*y$Q$>F1F`u7$$!3+++ +]PfL>F1F\\t7$$!3++++D\"GQ#>F1F_dl7$$!3%******4_+A#>F1Ff_l7$$!3+++++D1 9>F1Fegm7$$!3++++voH/>F1Fgr7$$!3++++]7`%*=F1F[il7$$!3++++Dcw%)=F1Fgr7$ $!3+++++++v=F1Ff_l7$$!3%******4US!*z\"F1F`u7$$!3++++++v=S:(H:F1F^`l7$$!3+++++vVB:F1F_ dl7$$!3++++]i!R]\"F1F\\t7$$!3+++++]P%[\"F1F\\t7$$!3+++++DJX9F1F\\t7$$! 3/+++Q%)*3U\"F1F`u7$$!3++++++D19F1F^x7$$!3++++vV['R\"F1F\\t7$$!3%***** *4xc[R\"F1F\\t7$$!3++++](=nQ\"F1F`u7$$!3++++DJ&pP\"F1F^x7$$!3+++++v=n8 F1F^x7$$!3/+++OS:(H:F1Fgr7$$\"3++++DJ?L:F1Fgr7$$\"3+ +++](oHa\"F1F^`l7$$\"3++++vVt_:F1F^`l7$$\"3++++++]i:F1Fegm7$$\"3++++++ v=F1F^`l7$$\"3+++++D19>F1F`u 7$$\"3%******4_+A#>F1Fjy7$$\"3++++D\"GQ#>F1F^x7$$\"3++++]PfL>F1F\\t7$$ \"3-+++pz\"4%>F1F`u7$$\"3++++v$fL%>F1F_dl7$$\"3+++++]7`>F1Fjy7$$\"3)** ****>#Qwh>F1Fgr7$$\"3++++]ils>F1Fgr7$$\"3+++++v=#*>F1F\\t7$$\"3,+++ACt +?F1F`u7$$\"3++++DJ&>+#F1F^`l7$$\"3<+++e[F6?F1F[il7$$\"3++++](=<,#F1F^ x7$$\"3++++vV[@?F1Ff_l7$$\"3/+++Q4YA?F1F^x7$$\"3++++++DJ?F1F\\t7$$\"3) )*****pfIB1#F1Fgr7$$\"3+++++]P4@F1Fjy7$$\"3++++++](=#F1Fbs7$$\"3+++++D cEAF1F^`l7$$\"3++++++vVBF1Fjy7$$\"3++++)3>')Q#F1F\\t7$$\"3-+++<[NiCF1F ^`l7$$\"3++++++++DF1F`u7$$\"3++++++DcEF1F]am7$$\"3/+++P!pgt#F1Fh`m7$$ \"3++++++]7GF1F``m7$$\"3++++DcEAGF1Fc^l7$$\"3-+++W)*eHGF1Fh_m7$$\"3+++ +]7.KGF1Fc_m7$$\"3++++vozTGF1F_dl7$$\"3y*****>JXR%GF1F\\t7$$\"3A+++3ly \\GF1F[hl7$$\"3+++++Dc^GF1F[il7$$\"3#)*****>Xm/'GF1Fjy7$$\"3++++D\"G8' GF1Fgdm7$$\"3++++]P4rGF1Fjbm7$$\"3=+++#*)*=sGF1Fc^l7$$\"3++++v$f3)GF1F \\r7$$\"35+++WnK$)GF1Fgdm7$$\"3+++++]i!*GF1F`u7$$\"3++++]i:5HF1FI7$$\" 3+++++voHHF1FI7$$\"3++++](=#\\HF1Fbs7$$\"3++++++voHF1F^`l7$$\"3))***** >H2L(HF1$!3)**************z\"F-7$$\"3++++]7G))HF1F`u7$$\"3)******z:)y! *HF1Ff_l7$$\"34+++W-\\1IF1Fgn7$$\"3+++++D\"y+$F1F``m7$$\"3++++]PMFIF1F cjl7$$\"3A++++z&o.$F1F_dl7$$\"3+++++](o/$F1$\"3'**************p&F^r7$$ \"3;+++L*Ra0$F1$\"3-++++++q>F-7$$\"3++++D1kcIF1$\"3-++++++!\\\"F-7$$\" 3++++]iSmIF1Fc\\n7$$\"3;+++(fq#oIF1$\"3#)**************GF^r7$$\"3++++v =:F-7$$\"3++++ +v$f3$F1Fa^n7$$\"3%)******)[z\")3$F1$!3%**************H*F^r7$$\"3++++D Jq&4$F1$!3i**************[F^r7$$\"3#******HX$H+JF1$!3'**************4& F^r7$$\"3++++](oa5$F1Fgfp7$$\"3++++vVB:JF1Fe_n7$$F[coF1$\"3K+++++++`F^ r7$$\"3@+++\"4g&GJF1$!3g*************H(F^r7$$\"3;+++&y'pTJF1$\"3.+++++ !*R:F-7$$\"3++++]7`WJF1$!3O++++++!*\\F^r7$$\"3=+++&>f8;$F1$!3#)******* *****\\@F-7$$\"3+++++D1kJF1$!3-++++++!>\"F-7$$\"3++++]Pf$=$F1Fg`n7$$\" 3+++++]7.KF1$\"3**************R6F-7$$\"3.+++],\"y@$F1$\"31++++++g5F-7$ $\"3+++++v=UKF1Fgv7$$\"3++++++D\"G$F1F`t7$$\"3))*****pTgxG$F1F^x7$$\"3 ))******RV+5LF1Fegm7$$\"3+++++DJ?LF1F\\t7$$\"3+++++]PfLF1FI7$$\"3A+++o 0:hLF1Fgr7$$\"3+++++vV)R$F1F^x7$$\"3++++++]PMF1F_dl7$$\"3y*****R!e0]MF 1F_el7$$\"3++++]7.dMF1F\\r7$$\"3+++++DcwMF1F`y7$$\"3))*****R;+`[$F1F_e l7$$\"3++++]P4'\\$F1FI7$$\"3?+++gf<-NF1F+7$$\"3+++++]i:NF1Fh`m7$$\"3++ ++D1RDNF1F_dl7$$\"3(******R)*eH`$F1Fgv7$$\"3++++]i:NNF1F]al7$$\"3++++v =#\\a$F1Fbal7$$\"3'******4?$pZNF1F_dl7$$\"3+++++voaNF1F_z7$$\"3!****** *o)Hbb$F1F\\t7$$\"3++++DJXkNF1F`u7$$\"3++++](=Ud$F1Fbs7$$\"3#)*****RW& =#e$F1Fi^m7$$\"3++++vV)Re$F1F^]l7$$\"3++++++v$f$F1Fh`m7$$\"39+++m'R6h$ F1Fc^l7$$\"3+++++D\"Gj$F1Fjy7$$\"3'******42/wk$F1F[hl7$$\"3+++++](=n$F 1Fcjl7$$\"3')*****>ba,o$F1Fcjl7$$\"3++++D1k\"o$F1Fjbm7$$\"3)*******\\i S\"p$F1Fegm7$$\"3)******H&pi#p$F1Fbbp7$$\"3++++v=<,PF1Fbs7$$\"37+++\\[ d3PF1F+7$$\"3+++++v$4r$F1Fc_m7$$\"3++++DJq?PF1F`t7$$\"3;+++e[-IPF1F^x7 $$\"3++++](o/t$F1F_z7$$\"3++++vVBSPF1Fh`m7$$\"3+++++++]PF1Fgv7$$\"3@++ +?CUDQF1Fgr7$$\"3++++++D1RF1F\\t7$$\"3/+++LT>()RF1F[jl7$$\"3++++++]iSF 1F[hl7$$\"3++++DcEsSF1Fgv7$$\"3C+++*GB[2%F1$FhrF-7$$\"3++++]7.#3%F1F^x 7$$\"3-+++5#4B3%F1Fcjl7$$\"3s*****Ra2:4%F1F\\r7$$\"3++++voz\"4%F1Fey7$ $\"3+++++Dc,TF1F_dl7$$\"3:+++P'3.6%F1Fcjl7$$\"3++++D\"G86%F1$!35++++++ +QF-7$$\"3++++]P4@TF1Fcil7$$\"3V+++`WJATF1F]am7$$\"3++++v$f38%F1Fgdm7$ $\"3K+++va]KTF1Fcjl7$$\"3+++++]iSTF1F]gm7$$\"3-+++!\\@'[TF1$\"3!****** ********\\$F-7$$\"3++++D1R]TF1F\\o7$$\"3++++]i:gTF1Fjbm7$$\"31+++hp=jT F1F_q7$$\"3++++v=#*pTF1Fgr7$$\"3u*****>9S*pTF1FN7$$\"3+++++vozTF1Fbr7$ $\"3w*****pwd*)=%F1F`\\m7$$\"3++++DJX*=%F1$\"3))**************eF-7$$\" 3)*******\\(=#*>%F1F]cl7$$\"3++++vV)*3UF1F]cl7$$\"3K+++i#3i@%F1F^^l7$$ \"3++++++v=UF1$\"3o*************f&F-7$$\"3++++Dc^GUF1$!3i************* \\&F-7$$\"3P+++sx[JUF1F+7$$\"3++++]7GQUF1Fiz7$$\"3*******f?O*QUF1$!3I+ ++++++`F-7$$\"3++++vo/[UF1Fegm7$$\"39+++NF:_UF1F^_m7$$\"3+++++D\"yD%F1 Fh`m7$$\"3K+++6bzlUF1$!3/+++++++JF-7$$\"3++++D\"yvE%F1Fjbm7$$\"3++++]P MxUF1Fi\\l7$$\"3)******zNO0G%F1F``m7$$\"3U+++\"RNgG%F1F`t7$$\"3++++v$4 rG%F1Fbal7$$\"3%)*****fTyRH%F1Faw7$$\"3+++++](oH%F1F]am7$$\"3@+++2G*eI %F1F\\o7$$\"3++++D1k1VF1Ffhq7$$\"3++++]iS;VF1F]al7$$\"3I+++-eoCVF1Fgr7 $$\"3++++v=#F-7$$\"3'******fu!)fO%F1Ffo7$$\"3+++ ++++vVF1Fdel7$$\"3++++Dcw%Q%F1Fi]r7$$\"3)******zmUgQ%F1F`p7$$\"3++++]7 `%R%F1F^x7$$\"3++++voH/WF1Fecl7$$\"3&)*****>Tu[S%F1Fjcl7$$\"3+++++D19W F1F[hl7$$\"3++++D\"GQU%F1$!3!**************H#F-7$$\"3++++]PfLWF1Ff_r7$ $\"3++++v$fLW%F1$\"3Y*************p(F-7$$\"3<+++[5AWWF1F[hl7$$\"3w**** *\\+BDX%F1$\"3K*************\\(F-7$$\"3+++++]7`WF1Fgv7$$\"3++++D1*GY%F 1Fbbp7$$\"3U+++>%)=lWF1F^]l7$$\"3!)*****\\=xCZ%F1$!31+++++++DF-7$$\"3+ +++]ilsWF1F[hl7$$\"3++++v=U#[%F1FD7$$\"3#)*****pAKI[%F1$!3g*********** ***yF-7$$\"3+++++v=#\\%F1Fgr7$$\"3++++](=<^%F1Fjy7$$\"3++++++DJXF1Fgv7 $$\"3.+++5g+NXF1Fgs7$$\"3+++++DJqXF1F\\t7$$\"3?+++$[V4f%F1$\"3;+++++++ ^F-7$$\"3+++++]P4YF1F+7$$\"3++++D19>YF1F[\\m7$$\"3C+++^#\\'>YF1Fdz7$$ \"3++++]i!*GYF1F\\t7$$\"3++++v=nQYF1Fbal7$$\"3/+++c]>WYF1Fh_m7$$\"3+++ ++vV[YF1Fi]l7$$\"3++++DJ?eYF1Fgdl7$$\"3!******>l@'eYF1FI7$$\"3++++](oz m%F1Fh_m7$$\"3e******H1guYF1F^[l7$$\"3++++vVtxYF1F:7$$\"3++++++](o%F1F br7$$\"37+++3@=!p%F1Fi]l7$$\"3)******\\ilsp%F1FI7$$\"3I+++PY4+ZF1$!3S* ************z)F-7$$\"3++++]7.2ZF1Ff_l7$$\"3f*****H@(o4ZF1F\\r7$$\"3+++ +voz;ZF1Feam7$$\"3u*****4w+#>ZF1$!3'**************z%F-7$$\"3>+++#)f'es %F1Fgs7$$\"3+++++DcEZF1Ff_r7$$\"3C+++f4iNZF1F^[l7$$\"3++++D\"Gjt%F1$!3 -+++++++&)F-7$$\"3++++]P4YZF1F_dl7$$\"3W+++D.3\\ZF1$\"3/+++++++PF-7$$ \"3++++v$fev%F1Fi]r7$$\"3M+++2%*4jZF1$!35+++++++iF-7$$\"3+++++]ilZF1Fi \\l7$$\"3++++D1RvZF1F`t7$$\"3++++]i:&y%F1Fccr7$$\"3e*****H8tKz%F1Fi[l7 $$\"3++++v=#\\z%F1$\"3.++++++?5F^s7$$\"3+++++vo/[F1Fegm7$$\"3v*****H+$ [2[F1Fgdl7$$\"3++++DJX9[F1F^[l7$$\"3++++](=U#[F1Fgv7$$\"3++++vV)R$[F1F \\t7$$\"3++++++vV[F1F]am7$$\"3,+++.o([%[F1F^x7$$\"3c*****>01I&[F1FS7$$ \"3++++Dc^`[F1F^`l7$$\"3u******RA@f[F1Fiz7$$\"3++++]7Gj[F1F[il7$$\"3++ ++vo/t[F1$!3))*************H)F-7$$\"3G+++zWnu[F1F_dl7$$\"3+++++D\"G)[F 1FN7$$\"3e*****z%y(y)[F1$Fi`mF-7$$\"3++++D\"yD*[F1F`gl7$$\"3++++]PM-\\ F1$\"3C+++++++kF-7$$\"3\")*****z/H_!\\F1$!3%)*************R$F-7$$\"3++ ++v$4@\"\\F1Fihr7$$\"3.+++%G\"R:\\F1$\"37+++++++KF-7$$\"36+++*oh1#\\F1 Fbbp7$$\"3+++++](=#\\F1F`y7$$\"3++++D1kJ\\F1$\"3K**************)*F-7$$ \"3A+++n,^K\\F1$!3;+++++++jF-7$$\"3++++]iST\\F1Fjp7$$\"3r*****R0pL%\\F 1$\"3%**************f*F-7$$\"3++++v=<^\\F1F_z7$$\"3+++++v$4'\\F1Fh`l7$ $\"3++++DJqq\\F1$!3%**************>(F-7$$\"3')******4Bbr\\F1F^[l7$$\"3 ++++](o/)\\F1Faiq7$$\"3%******HXW9)\\F1$\"3O+++++++yF-7$$\"3++++vVB!* \\F1Fc^l7$$\"\"&F*Fc^l-%'COLOURG6&%$RGBGF\\tF\\t$\"*++++\"!\")-%+AXESL ABELSG6$Q\"x6\"Q!Fhbs-%%VIEWG6$;F(F[bs%(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 "[-Pi/2, Pi/2];" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F (F)" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 177 "The first point to observe in the design of a procedure to evaluate the sine function is that the problem can \+ be reduced to that of evaluating the sine function on the interval " } {XPPEDIT 18 0 "[-Pi/2,Pi/2]" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F(F )" }{TEXT -1 9 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 56 "We can reduce the interval over which we need to d efine " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 17 " to t he interval " }{XPPEDIT 18 0 "[-Pi/2, 3*Pi/2];" "6#7$,$*&%#PiG\"\"\"\" \"#!\"\"F)*(\"\"$F'F&F'F(F)" }{TEXT -1 17 " by periodicity: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(x+2*Pi) = sin(x);" "6#/ -%$sinG6#,&%\"xG\"\"\"*&\"\"#F)%#PiGF)F)-F%6#F(" }{TEXT -1 1 "." }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 262 10 "__________" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 " Thus we can subtract a suitable integer multiple " }{XPPEDIT 18 0 "2*k *Pi;" "6#*(\"\"#\"\"\"%\"kGF%%#PiGF%" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 6 " from " }{TEXT 274 1 "x " }{TEXT -1 37 ", or add a suitable integer multiple " }{XPPEDIT 18 0 "2*k*Pi;" "6#*(\"\"#\"\"\"%\"kGF%%#PiGF%" }{TEXT -1 4 " to " }{TEXT 275 1 "x" }{TEXT -1 57 ", to ensure that the resulting number is in th e interval " }{XPPEDIT 18 0 "[-Pi/2, 3*Pi/2]" "6#7$,$*&%#PiG\"\"\"\"\" #!\"\"F)*(\"\"$F'F&F'F(F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "The required integer is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "k = floor((x+Pi/2)/(2*Pi));" "6#/%\"kG-%&floorG6#*&,&% \"xG\"\"\"*&%#PiGF+\"\"#!\"\"F+F+*&F.F+F-F+F/" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "k = floor(x/(2*Pi)+1/4);" "6#/%\"kG-%&floorG6#,&*&% \"xG\"\"\"*&\"\"#F+%#PiGF+!\"\"F+*&F+F+\"\"%F/F+" }{TEXT -1 14 " ----- -- (i), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "floor( z)" "6#-%&floorG6#%\"zG" }{TEXT -1 23 " is the largest integer" } {XPPEDIT 18 0 "`` <= z;" "6#1%!G%\"zG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "The symmetry formula: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(Pi-x) = sin(x);" "6#/-%$sinG6#,&%#PiG\"\"\"% \"xG!\"\"-F%6#F*" }{TEXT -1 14 " ------- (ii)," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 263 10 "__________" }{TEXT -1 17 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "ca n then be used to reduce the interval over which we need to find an ap proximation for " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 17 " to the interval " }{XPPEDIT 18 0 "[-Pi/2,Pi/2]" "6#7$,$*&%#PiG \"\"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "To illustrate this idea, we def ine a function " }{TEXT 0 4 "sin1" }{TEXT -1 45 " using Maple's sine f unction on the interval " }{XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$,$*&%#P iG\"\"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 58 ", but ensure that it giv es no value outside this interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "sin1 := x -> if x <=1.57079 63267949 and x >=-1.5707963267949 then sin(x) else FAIL end if;\nplot( 'sin1'(x),x=-2..2,thickness=2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% %sin1Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(@%319$$\"/\\zEjzq:!#81$!/\\zE jzq:F2F/-%$sinG6#F/%%FAILGF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 334 248 248 {PLOTDATA 2 "6&-%'CURVESG6$7V7$$!35+++J_Pq:!#<$!25n\"R8\"***** **F*7$$!3%******R[Aac\"F*$!3sdbT4c&)****!#=7$$!35+++Q(p/c\"F*$!3Y@vb7o Y****F27$$!3!******\\Ck0b\"F*$!3GSc9%Q`z***F27$$!3\"******>ve1a\"F*$!3 9%e]m(*ea***F27$$!3$******fwZ3_\"F*$!3+)\\y!R)[*F27$$!3)******p&4`i6F*$!3Y2!*oC)3\"y\"*F27$$!3'******zV %*e3\"F*$!3-Qd9^o?Z))F27$$!3m*******p)>'***F2$!3%*H`,*QbET)F27$$!3E+++ +0\"*H\"*F2$!3b$3\"z&\\fL\"zF27$$!35++++83&H)F2$!3yb7<2L*fP(F27$$!3H++ +5k(p`(F2$!3)zXXv;'RVoF27$$!3s******pj^NmF2$!3=gY^Zu=fhF27$$!3[+++q9'= (eF2$!3q913&y1-a&F27$$!3u******z#\\N)\\F2$!3o(H(HS?\")zZF27$$!3%****** *pCC(>%F2$!3u&3_a]'3vSF27$$!36+++!GRXL$F2$!3MHi%=^'3tKF27$$!39+++I=/8D F2$!3/%\\8g(Qn'[#F27$$!31+++?a*el\"F2$!3a=4ju$Q$[;F27$$!3e++++Wn(o)!#> $!3U&y1O0]nn)F^s7$$!3')********zNu>!#?$!3A9iHFds7$$\"3`++++f`@')F ^s$\"3DKR;'4f3h)F^s7$$\"3.+++qw%)H;F2$\"3U'pn)f8kA;F27$$\"3%)******>$y *eCF2$\"3ZeqVsCFMCF27$$\"3E++++9b:LF2$\"3A]U\\U'Q^D$F27$$\"3u******\\5 a`TF2$\"3;,YlB&R^.%F27$$\"3y******>\"RV'\\F2$\"3ub9u,'HHw%F27$$\"3y*** ***\\@fkeF2$\"3sIgy\"f`T`&F27$$\"3C+++I&4Nn'F2$\"3_!>Z;Iu!*='F27$$\"3L ++++:?PvF2$\"30%R0KHgN%oF27$$\"3E+++!zM)>$)F2$\"31(4(\\0_o#R(F27$$\"3W ++++(fa<*F2$\"3t?eH]_7TzF27$$\"3Q+++gg`!)**F2$\"3#Hx3P]xTS)F27$$\"3%** ****>G2A3\"F*$\"3/Bxk@,'*H))F27$$\"3-+++$)G[k6F*$\"3%fJ6.$3%e=*F27$$\" 3++++7yh]7F*$\"3?Wzs\"[#z\"\\*F27$$\"3!******p)fdL8F*$\"355@3S&\\*>(*F 27$$\"3\"******>q7%=9F*$\"3g#GF!\\17%))*F27$$\"3%******R\")z/Y\"F*$\"3 ee-L>I@R**F27$$\"3&******f#pa-:F*$\"3=M*y`8>n(**F27$$\"3)*******zX(=_ \"F*$\"3k(Rb**pN!))**F27$$\"3#******HB-7a\"F*$\"3%>cawE@c***F27$$\"3%* ******fg'3b\"F*$\"3y#Ho!))R,)***F27$$\"33+++'))H0c\"F*$\"3)p!ef0IZ**** F27$$\"3-++++=Ol:F*$\"3Y?g&*HB&)****F27$$\"35+++8P>q:F*$\"2a7$G%=)**** **F*7$%*undefinedGFe[l-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F^\\lF]\\l-%*T HICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fg\\l-%%VIEWG6$;$!\"#F^\\l$Fb \\lF^\\l%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 25 "Now we define a function " }{TEXT 0 4 "sin2" } {TEXT -1 33 " using the formulas (i) and (ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 422 "sin2 := pro c(xx)\n local x,pi,pi2,k;\n\n pi := evalf(Pi);\n x := evalf(xx); \n\n # Reduce the range to -Pi/2<=x<=3*Pi/2\n if x>1.5*pi or x<-0. 5*pi then\n pi2 := 2*pi;\n\n # Determine the number of multi ples of 2*Pi to add or subtract.\n k := floor(x/pi2+0.25);\n \+ x := x - k*pi2;\n end if;\n\n # If x>Pi/2, replace x by Pi-x so t hat abs(x)<=Pi/2\n if x>0.5*pi then x := pi-x end if;\n sin1(x);\n end proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%sin2Gf*6#%#xxG6&%\"xG% #piG%$pi2G%\"kG6\"F-C'>8%-%&evalfG6#%#PiG>8$-F26#9$@$52,$*&$\"#:!\"\" \"\"\"F0FBFBF62F6,$*&$\"\"&FAFBF0FBFAC%>8&,$*&\"\"#FBF0FBFB>8'-%&floor G6#,&*&F6FBFJFAFB$\"#D!\"#FB>F6,&F6FB*&FOFBFJFBFA@$2,$*&$FGFAFBF0FBFBF 6>F6,&F0FBF6FA-%%sin1G6#F6F-F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot('sin2'(x),x=-3*Pi..5*Pi ,color=red,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 510 139 139 {PLOTDATA 2 "6'-%'CURVESG6#7_w7$$!3^***H>%zxC%*!#<$!3'*zMR$)z$R)=!#D7$ $!3U2dS$QAyG*F*$!37LbHm$y_O\"!#=7$$!3M:9)[#o'3:*F*$!37ccGf#))\\q#F37$$ !3DBrNm7\"R,*F*$!3Wo)ymTSS*RF37$$!3QHG$yqbp())F*$!3X3i!=3%H3_F37$$!3aR O[4riQ')F*$!3GaUW:\")QwqF37$$!3q\\W86&)H+%)F*$!3\\C:pl&QWa)F37$$!3a.3G wIFK\")F*$!3;U74\\?E:'*F37$$!3gbrUTwCkyF*$!3)4\"H3]IZ****F37$$!3cvF2@r z'z(F*$!3ETq\\7Tl$)**F37$$!3_&R=2gY$HxF*$!3\"\\(3zK5VA**F37$$!3[:SO!3' *=m(F*$!3$f\\HUC#3;)*F37$$!3WN'4+cXWf(F*$!3Mig%\\S\"4l'*F37$$!3Ov3I>Xa fuF*$!34Od85-8K#*F37$$!3H:@fyMkCtF*$!3()4)*RgYTJ')F37$$!3/f<)*)f$Q!>(F *$!3eR)=\"o5\"y(yF37$$!3#GSr$>P7cqF*$!3ksc\\xuT#)pF37$$!3eY5wRQ'=#pF*$ !3%o#HEo(\\8'fF37$$!3M!p],'Rg(y'F*$!3='4t^y&)H$[F37$$!3I85Hly7jmF*$!3$ [a0bCr'3PF37$$!3RN8Vqr!e*p*G'F*$!3_JwYfuA9l!#?7$$!3W$e@Wz63;'F*$\"3>S)o$=Jo?7F3 7$$!3M(=J\"3S#>.'F*$\"3+-e^/&ei[#F37$$!3C\"zS=AOI!fF*$\"3*=r@d:*e5PF37 $$!3E%R]bV[Tx&F*$\"3WC_'fbkL([F37$$!3*H)Hf-D&3k&F*$\"3uq!pOZ\\1*fF37$$ !3srbjplb2bF*$\"3%pi1M?^;+(F37$$!3Wg\"ymjgUP&F*$\"3UPJVWIV))yF37$$!3I[ 2s.Z'4C&F*$\"3C$4D[9i_j)F37$$!3@U3^_i43^F*$\"3\\6*GUpTsA*F37$$!3+P4I,y Av\\F*$\"3o3PZ%zill*F37$$!3S%)fpvNz3\\F*$\"3d>:C$QXx!)*F37$$!3!=.\"4]$ fB%[F*$\"3#RGsPMdc\"**F37$$!3?zg[C^#fx%F*$\"3-%f(3\"eA)z**F37$$!3gE6)) )*3\\4ZF*$\"3y&QZc+e*****F37$$!3<4@/K^:TYF*$\"3E)*H#3URY(**F37$$!3u\"4 ._O>Gd%F*$\"3mZWR<'fF!**F37$$!3IuSO)f$[/XF*$\"3oAXTFTl%y*F37$$!3(o0D:$ y9OWF*$\"3q%Q&fiU(3i*F37$$!3+Aq%yHw%*H%F*$\"3?/3M<\"f&f\"*F37$$!3/))*o Tw/G;%F*$\"32MrOKxTF&)F37$$!36H!e;/W?#RF*$\"3g;6JaC+OqF37$$!3=qq9>LG\" o$F*$\"3If5A#[1(Q^F37$$!3a&F37$$!39P&f$GYwjCF*$!31:liQn-riF37$$!3G8e`k*HEL#F*$!3R(3^bg:dB( F37$$!3V*372I&\\,AF*$!3ihkRT^:w!)F37$$!3-m$))ojg.2#F*$!3o&3!e%f8zx)F37 $$!3ulQ`h\\>K=F*$!3e1NX7ZHg'*F37$$!3Wl$zhGHSf\"F*$!3i;dQp7I(***F37$$!3 874@!yGK_\"F*$!3#)f**fv()o))**F37$$!3!)eCCu#GCX\"F*$!3&*Ga[xn-I**F37$$ !3Y0SFoxi\"Q\"F*$!3CU!4z?4;#)*F37$$!38_bIis#3J\"F*$!3ERC!fIzRm*F37$$!3 YX'o.DE#p6F*$!3!3sHUU%[/#*F37$$!3\")QH\"F3 $!3]BU,MJS)G\"F37$$\"3Kz8=K&HO.\"!#>$\"3$eEbp76O.\"Ffal7$$\"3;u0,(z!\\ Q8F3$\"3q*eCmt(\\M8F37$$\"3S5I!3k=Od#F3$\"3)R%eE@>IXDF37$$\"3iYaf%[Y(3 QF3$\"33xWBDeKgi:F*$\"3w8*)**eUm****F37$$\"3[6(zY-A*H;F* $\"3!G7>2&e_#)**F37$$\"3!>yJg9Usp\"F*$\"3Z#oF7tj,#**F37$$\"3O_QQnAckF*$\"3,z\"Q'H$orA*F37$$\"3b/U9uG;,@F*$\"3=%Q1l5&F37$$\"3%f*)H9K%4H FF*$\"3@L+bSN**3SF37$$\"3ioh'pneC'GF*$\"3OQqf&[K_v#F37$$\"3ITC]KI#e*HF *$\"3qT$RXJRDX\"F37$$\"3*RrQ!)Q(=HJF*$\"3)pJ(yPZ\\S7Ffal7$$\"3]6;2sarn KF*$!3#R$[RCr)yD\"F37$$\"3-4X5cNC1MF*$!3`h#>$3Ms:EF37$$\"3`1u8S;xWNF*$ !3r*)yl,TWBRF37$$\"3//.!eRs2)[&F*$!34-)*f)y#RRrF37$$\"3!o-GdM3(>cF*$!3/k!R\" H\"*fehF37$$\"3;fe(=XR8v&F*$!3[\\!f3'4Dr]F37$$\"3]h#>aJ*pyeF*$!3kpdN@G YNRF37$$\"3&Hmi*y\"fg+'F*$!3:7zI5c#ft#F37$$\"3Skg]U!>M8'F*$!3_G#Hm-kF*$\"3!*)y-e ;Vs=\"F37$$\"3ciZXM brF37$$\"3*[H6OM#oMtF*$\"3_<4.`8n\"o)F37$$\"39h?@g2NquF*$\"3/vkNgg(F*$\"3Ojyoi]9%p*F37$$\"3-5K6&Q`Qn(F*$\"3]2CeVs?Q)*F 37$$\"3e$f8Mf(oTxF*$\"3IEC'*ok,P**F37$$\"39xRr,=_4yF*$\"3S/2\\W#=,***F 37$$\"3!)fV,5gNxyF*$\"3]$)45<$os***F37$$\"3UryZ4vAB\")F*$\"3wn>25xrR'* F37$$\"3/$QT*3!*4p$)F*$\"3@\\jT]!\\Bq)F37$$\"3$>)[)\\D+N])F*$\"3s$p\"F37$$\"3?mlae\\#pn*F*$!33snb-k$[\\#F37$$\"3!QNwWC8\"4)*F*$ !3sLM/e*H%\\PF37$$\"35F^im$!3#\\s@#)pgr*pF37$$\"3(*)=Nkh,H.\"F^ im$!3itsThPafyF37$$\"3'QrmvVDe/\"F^im$!3C4jk!HN3f)F37$$\"3I:Sd![b$f5F^ im$!3S<7*3$ps-#*F37$$\"3w;8eBb)G2\"F^im$!3q?=E&R3kk*F37$$\"3cn\\3X0lz5 F^im$!3')[c5l9^-)*F37$$\"3?=')embT'3\"F^im$!3Dgv=U*pP\"**F37$$\"3%)oA4 )e!=$4\"F^im$!3Y?!30%[nz**F37$$\"3m>ff4c%**4\"F^im$!3QLD\"zlC*****F37$ $\"3O8eMH6Y16F^im$!3?IQ\\A*yh(**F37$$\"312d4\\m(H6\"F^im$!3=\"e-\\H(45 **F37$$\"3f+c%)o@\\>6F^im$!3dV/i0-'>!)*F37$$\"3I%\\&f)o2g7\"F^im$!333Y \"ebEAl*F37$$\"3`\"G&4G(Q!R6F^im$!3nX())*)4&oI#*F37$$\"3%*o]fn(p?:\"F^ im$!3!)e0s64i_')F37$$\"3?R$)f()eRl6F^im$!3H\\,mjj\")4zF37$$\"3E4;g2?sy 6F^im$!3M![Jur_n-(F37$$\"3Kz[gF\"[?>\"F^im$!3K\"3iqq)3>gF37$$\"3d\\\"3 wCu`?\"F^im$!3q/UV:Dp/\\F37$$\"3%)z&41-!f=7F^im$!3i,)f()zsNr$F37$$\"36 55h$z0=B\"F^im$!3Kq;S3uodCF37$$\"3PSChm:-X7F^im$!3'>g0;JR*e6F37$$\"3kq QhRtBe7F^im$\"3#pJ[l()4-g\"Ffal7$$\"3c(4lCE\"Qq7F^im$\"3%o%eU&)p4q8F37 $$\"3ICjJ&=DDG\"F^im$\"3uTgD;B**fDF37$$\"3/^v;3\"pYH\"F^im$\"3NMzCq0=7 PF37$$\"3&zx=5.8oI\"F^im$\"30w=.R2p4[F37$$\"3ous/P8t?8F^im$\"3m'=%RlN^ zfF37$$\"3Trd2V'\\YL\"F^im$\"3vp#[@e)oLqF37$$\"39oU5\\zc[8F^im$\"31'>P iLF=&zF37$$\"3)[wK^D'[i8F^im$\"34f!=tSshr)F37$$\"3?$)f'R&\\$\\P\"F^im$ \"3+zra[lOd#*F37$$\"3O,#*z_OQ(Q\"F^im$\"3c6!z$eVGb'*F37$$\"3_5e@-!3OR \"F^im$\"3=(4C\\h+&)z*F37$$\"3]>Cj^B$)*R\"F^im$\"35A4*)fsw.**F37$$\"3n G!\\5qcgS\"F^im$\"3OwEO&fw1(**F37$$\"3$yjl/0\"G79F^im$\"3?/aN\"[p*)*** F37$$\"3*3**p^]<*=9F^im$\"3W8-KI)yk)**F37$$\"3&RMu)fRbD9F^im$\"3pFu?R8 -I**F37$$\"3,(pyXT!>K9F^im$\"3!4,])pb%)H)*F37$$\"3C]IGpo#)Q9F^im$\"3;' R&yfDR'o*F37$$\"3ac&F37$$\"37\"[b1?S)H:F^im$\"3k9n@L\"p?)RF37$$\"32a OlTA\\V:F^im$\"3]#*ehI)3mp#F37$$\"3-F=l#GWrb\"F^im$\"35Za&Hin4O\"F37$$ \"3'****\\OK'zq:F^im$\"37()oz4m*[9$F--%'COLOURG6&%$RGBG$\"*++++\"!\")$ \"\"!F^inF]in-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fgin-%%VIEWG 6$;$!+izxC%*!\"*$\"+Fjzq:F\\in%(DEFAULTG" 1 2 0 1 10 2 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 64 "An arbitrary precision procedure to evalu ate the sine function: " }{TEXT 0 6 "sineAP" }{TEXT -1 2 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 22 "sineAP: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1445 "sineAP := proc(xx::rea lcons)\n local x,z,pi2,term,eps,k,even,maxit,saveDigits,\n extraDi gits,sum,pi;\n\n if xx=0 then return xx end if;\n\n # increase pre cision \n saveDigits := Digits;\n Digits := Digits+length(Digits) +1;\n x := evalf(xx);\n # argument reduction involves loss of prec ision\n # increase precision further when x has large magnitude\n \+ extraDigits := max(ilog10(x),0);\n if extraDigits>0 then\n Digi ts := Digits+extraDigits;\n x := evalf(xx)\n end if;\n \n p i := evalf(Pi); # Maple evaluates Pi rapidly\n \n # Reduce the \+ range to -Pi/2<=x<=3*Pi/2\n if x>4.7123889803846898577 or x<-1.57079 63267948966192 then\n pi2 := pi+pi;\n\n # Determine the numb er of multiples of 2*Pi to add or subtract.\n k := floor(x/pi2+0. 25);\n x := x-k*pi2;\n end if;\n \n # If x>Pi/2, replace x \+ by Pi-x so that abs(x)<=Pi/2\n if x>1.5707963267948966192 then x := \+ pi-x end if;\n\n Digits := Digits-extraDigits;\n\n # Initialisatio n for Maclaurin series loop\n eps := Float(1,-saveDigits);\n maxit := Digits*4;\n\n # calculate sin(x)\n term := x;\n sum := term; \n z := x*x;\n even := false; \n for k from 2 to maxit by 2 do\n term := term*z/(k*(k+1));\n if even then\n sum := s um+term;\n else\n sum := sum-term;\n end if;\n \+ if abs(term)<=eps*abs(sum) then break end if;\n even := not even; \n end do;\n\n Digits := saveDigits;\n evalf(sum);\nend proc: # \+ of sineAP" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Testing \+ the procedure sineAP" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot('sineAP'(x),x=-2*Pi..5*Pi,color=red, thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 499 189 189 {PLOTDATA 2 "6 '-%'CURVESG6#7eu7$$!+&H&=$G'!\"*$\"&gD\"!#77$$!+#=\\L;'F*$\"+$4&\\&>\" !#57$$!+oI^VgF*$\"+rC%QP#F37$$!+apnBfF*$\"++19=NF37$$!+S3%Q!eF*$\"+@i( >h%F37$$!+;LI&f&F*$\"+)\\A!\\jF37$$!+$zlnQ&F*$\"+wA'4\"yF37$$!+PNC_^F* $\"+Pg8[!*F37$$!+#G@x\"\\F*$\"+[G$**y*F37$$!+x?qe[F*$\"+.K:$*)*F37$$!+ rGo*z%F*$\"+sI#>'**F37$$!+mOmSZF*$\"+xH+'***F37$$!+hWk\"o%F*$\"+XUF&** *F37$$!+b_iAYF*$\"+7%R(f**F37$$!+]ggjXF*$\"+AA_*))*F37$$!+Xoe/XF*$\"+) =n[y*F37$$!+RwcXWF*$\"+U(Qhk*F37$$!+]Gh5UF*$\"+elHn()F37$$!+g!ec(RF*$ \"+(4*o1uF37$$!+%*[#yv$F*$\"+_akzdF37$$!+G<**RNF*$\"+QHVzQF37$$!+:\\@F MF*$\"+sbaF*$!+kI#GS*F37$$!+j(e*y;F*$!+W9c T**F37$$!+aBHE;F*$!+2\\g%)**F37$$!+Yfit:F*$!+m*f*****F37$$!+Q&f4_\"F*$ !+VSe()**F37$$!+IJHo9F*$!+b9^Z**F37$$!+9.'HO\"F*$!+*G(z%y*F37$$!+)\\Fw D\"F*$!+:6i8&*F37$$!+#*yY?5F*$!+u!GN_)F37$$!*'G3LyF*$!+;ZFcqF37$$!+IeT UmF3$!+V9ikhF37$$!++)[!)RAIF37$$!+Kk)H#>F3$!+9n:6>F37$$!+]7ebxF\\s$!+U(3yu(F\\s7$$\"+D =q=PF\\s$\"+,[%yr$F\\s7$$\"*\\)H>:F*$\"+>.Y8:F37$$\"+?LFhDF3$\"+f7OLDF 37$$\"+]\"[Kg$F3$\"+!3#yDNF37$$\"+!)HAXYF3$\"+*pd*zWF37$$\"*\"y>(o&F*$ \"+#*y`&Q&F37$$\"+?b@l\")F3$\"+-Mo(G(F37$$\"+BLKk5F*$\"+](3Yu)F37$$\"+ GRCu7F*$\"+ak\\j&*F37$$\"+KX;%[\"F*$\"+C\")\\i**F37$$\"+C9@X:F*$\"+Uss '***F37$$\"+;$eig\"F*$\"+'*Gr$***F37$$\"+3_In;F*$\"+9jY`**F37$$\"+*4_$ GF*$\"+kq!Q?*F37$ $\"+Rqo)=#F*$\"+c@-^\")F37$$\"+6W$[S#F*$\"+pV!*=nF37$$\"+u$3M_#F*$\"+y Uc&z&F37$$\"+PB)>k#F*$\"+G_$3z%F37$$\"+*Hc0w#F*$\"+fp#)=PF37$$\"+i-8zG F*$\"+2Sf%f#F37$$\"+:2/#*HF*$\"+t3&**[\"F37$$\"+n6&\\5$F*$\"+,)GLm$F\\ s7$$\"+?;'y@$F*$!+sX]>wF\\s7$$\"+t?xILF*$!+z,`!)=F37$$\"+&G#e[MF*$!+MQ !>-$F37$$\"+)\\#RmNF*$!+SUQ@TF37$$\"+6F?%o$F*$!+o!HP;&F37$$\"+DH,-QF*$ !+!4)[MhF37$$\"+CgQ=SF*$!+@\"=po(F37$$\"+D\"fZB%F*$!+Q\"p3)))F37$$\"+P HX^VF*$!+)pmcN*F37$$\"+]n9oWF*$!+#[3Kq*F37$$\"+dO\\EXF*$!+^;rF)*F37$$ \"+j0%[e%F*$!+Csw=**F37$$\"+qu=VYF*$!+4`1w**F37$$\"+vV`,ZF*$!+$*3T**** F37$$\"+7/9iZF*$!+Gli()**F37$$\"+[kuA[F*$!+Bz;R**F37$$\"+%[_L)[F*$!+(z=sF*$\"+[Fi\\!)F37$$\"+bpnTuF*$\"+9s *>;*F37$$\"+V>TlvF*$\"+_*=le*F37$$\"+Ip9*o(F*$\"+@\\Xk)*F37$$\"+BW,^xF *$\"+(eNq%**F37$$\"+<>)G\"yF*$\"+b_b\"***F37$$\"+5%\\Z(yF*$\"+vN%y***F 37$$\"+0phOzF*$\"+%[we'**F37$$\"+%oz*e\")F*$\"+\"Qv%Q&*F37$$\"+lCM\"Q) F*$\"+@DQT')F37$$\"+!>i(=')F*$\"+JJM:sF37$$\"+:>=c))F*$\"+p_\\%Q&F37$$ \"+)p]P'*)F*$\"+RNo[WF37$$\"+![>82*F*$\"+FbWhMF37$$\"+i#)))y\"*F*$\"+( e$>MCF37$$\"+XqX'G*F*$\"+FE!)y8F37$$\"+M\"eSS*F*$\"+EW$=2#F\\s7$$\"+C# f;_*F*$!+4!yHn*F\\s7$$\"+9.ER'*F*$!+6nTG@F37$$\"+09'ov*F*$!+,N8gKF37$$ \"+j[^n)*F*$!+N39%G%F37$$\"+A$o\"y**F*$!+beub_F37$$\"+y@))35!\")$!+hT1 jhF37$$\"+Cv%*>5Fccm$!+Kx*\\*pF37$$\"+u/3V5Fccm$!+f'>rW)F37$$\"+CM@m5F ccm$!+Y-A\\%*F37$$\"+o<_x5Fccm$!+\\c>e(*F37$$\"+6,$))3\"Fccm$!+#[!)e6Fccm$!+axx&H)F37$$\"+pig\"=\"Fccm$!+D.k=oF37$$\"+)e5W? \"Fccm$!+=CW))\\F37$$\"+V42;7Fccm$!+.PEYRF37$$\"+)HJxA\"Fccm$!+A3\\]GF 37$$\"+`;RR7Fccm$!+`a+;P(z)F37$$\"+9]_)Q\"Fccm$\"+zTO%o* F37$$\"+?8(RR\"Fccm$\"+PEp0)*F37$$\"+EwT*R\"Fccm$\"+&RUz*)*F37$$\"+KR' [S\"Fccm$\"+())R3'**F37$$\"+P-J59Fccm$\"+'e)>%***F37$$\"+Ulv:9Fccm$\"+ #e>z***F37$$\"+[G?@9Fccm$\"+M=*>(**F37$$\"+a\"\\mU\"Fccm$\"+EA\\;**F37 $$\"+ga4K9Fccm$\"+U`eJ)*F37$$\"+_IKb9Fccm$\"+(ypo9*F37$$\"+W1by9Fccm$ \"+Lh(3(zF37$$\"+7Cy+:Fccm$\"+s5CVkF37$$\"+!=9I_\"Fccm$\"+WtX)f%F37$$ \"+;(f\\`\"Fccm$\"+%4Yu]$F37$$\"+__!pa\"Fccm$\"+PXWmBF37$$\"+)y])e:Fcc m$\"+S\\r\">\"F37$$\"+Cjzq:Fccm$\"')[9$!#8-%'COLOURG6&%$RGBG$\"*++++\" Fccm$\"\"!F[anFj`n-%+AXESLABELSG6$Q\"x6\"Q!F`an-%*THICKNESSG6#\"\"#-%% VIEWG6$;$!+3`=$G'F*$\"+Fjzq:Fccm%(DEFAULTG" 1 2 0 1 10 2 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 73 "xx:= sqrt(24)/5;\n Digits := 300:\nsineAP(xx);\nevalf(sin(xx));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,$*(\"\"#\"\"\"\"\"&!\"\"\"\"'#F(F'F( " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"g]lCxc:nZS\"e4+TK3 " 0 "" {MPLTEXT 1 0 69 "xx:= 10^500: \nDigits := 300:\nsineAP(xx);\nevalf(sin(xx));\nDigits := 10:" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#$\"g]lHdyv!H4it:D-A]\"*)GTnph^zrD)3Ek? f#))H)z'H.S_6dU@XX1g,&4?BnE=B/C(*>804CNGmV:v^T<@:=sc([m$ee ]F/hVm+HF?jAc#4DCty!*fS'z.u<_RV!)38$>'\\?:RdGJFlL*4rjZ!G+z!4$p-6$!$+$ " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"g]lHdyv!H4it:D-A]\"*)GTnph^zrD) 3Ek?f#))H)z'H.S_6dU@XX1g,&4?BnE=B/C(*>804CNGmV:v^T<@:=sc([ m$ee]F/hVm+HF?jAc#4DCty!*fS'z.u<_RV!)38$>'\\?:RdGJFlL*4rjZ!G+z!4$p-6$! $+$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Speed comp arisons " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Maple's function " }{TEXT 0 3 "sin" }{TEXT -1 16 " is faster than \+ " }{TEXT 0 6 "sineAP" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "st := time():\nfor i to 5 00 do sineAP(rand()*Float(1,-11)) end do:\ntime()-st;\nst := time():\n for i to 500 do sin(rand()*Float(1,-11)) end do:\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$.)!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"$W\"!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "st := time():\nDigits := 50:\nfor i to 100 do s ineAP(rand()*Float(1,-11)) end do:\ntime()-st;\nst := time():\nfor i t o 100 do sin(rand()*Float(1,-11)) end do:\ntime()-st;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%W8!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$l\"!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 35 "Argument reduction to the interval " }{XPPEDIT 18 0 "[-Pi /6, Pi/6];" "6#7$,$*&%#PiG\"\"\"\"\"'!\"\"F)*&F&F'F(F)" }{TEXT -1 53 " can be performed by using the trigonometric formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(x) = sin(x/3)*(3-4*sin(x/3)^2) ;" "6#/-%$sinG6#%\"xG*&-F%6#*&F'\"\"\"\"\"$!\"\"F,,&F-F,*&\"\"%F,*$-F% 6#*&F'F,F-F.\"\"#F,F.F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 62 "Demonstrate how this works by using the Maple procedure . . . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "sin3 := x -> if x<=0.5235987755983 and x>=-0.5235987755983 then si n(x) else FAIL end if;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 " . . . which only has values in the interval " } {XPPEDIT 18 0 "[-Pi/6, Pi/6];" "6#7$,$*&%#PiG\"\"\"\"\"'!\"\"F)*&F&F'F (F)" }{TEXT -1 59 " together with the above formula to obtain a new pr ocedure " }{TEXT 0 4 "sin4" }{TEXT -1 23 " which gives values of " } {XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 18 " in the interv al " }{XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F) *&F&F'F(F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 26 "Check that \+ your procedure " }{TEXT 0 4 "sin4" }{TEXT -1 32 " can be plotted in th e interval " }{XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$,$*&%#PiG\"\"\"\"\"# !\"\"F)*&F&F'F(F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "___ _____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "____________________ ____________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 79 "Write a procedure to evaluate the sine function which encorporates the formula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin( x) = sin(x/3)*(3-4*sin(x/3)^2);" "6#/-%$sinG6#%\"xG*&-F%6#*&F'\"\"\"\" \"$!\"\"F,,&F-F,*&\"\"%F,*$-F%6#*&F'F,F-F.\"\"#F,F.F," }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 45 "in conjunction with the Maclaurin ser ies for " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 32 " in the manner of the procedure " }{TEXT 0 11 "sine_series" }{TEXT -1 44 " above, but use no other argument reduction." }}{PARA 0 "" 0 "" {TEXT -1 40 "Do this by inserting into the procedure " }{TEXT 0 11 "si ne_series" }{TEXT -1 23 " the two lines of code:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 260 9 "z := z/3;" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 260 18 "sum*(3-4*sum*sum);" }}{PARA 0 "" 0 "" {TEXT -1 60 "in the appropriate places, and call the resulting proc edure " }{TEXT 0 12 "sine_series2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 38 "Attempt to plot both of the functions " }{TEXT 0 11 "sine _series" }{TEXT -1 5 " and " }{TEXT 0 12 "sine_series2" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[-100,100]" "6#7$,$\"$+\"!\"\"F% " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "Explain the results. " }}{PARA 0 "" 0 "" {TEXT -1 40 "_____________________________________ ___" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 40 "________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 105 "Write an arbitrary pre cision procedure to evaluate the sine function which uses the Maclauri n series for " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 50 " together with argument reduction to the interval " }{XPPEDIT 18 0 "[-Pi/2,Pi/2]" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 32 " in the manner of the procedure " }{TEXT 0 6 "sineAP" }{TEXT -1 59 " and also incorpoprates argument reduction to the interval " } {XPPEDIT 18 0 "[-Pi/6, Pi/6]" "6#7$,$*&%#PiG\"\"\"\"\"'!\"\"F)*&F&F'F( F)" }{TEXT -1 24 " by means of the formula" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(x) = sin(x/3)*(3-4*sin(x/3)^2);" "6#/-%$s inG6#%\"xG*&-F%6#*&F'\"\"\"\"\"$!\"\"F,,&F-F,*&\"\"%F,*$-F%6#*&F'F,F-F .\"\"#F,F.F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 40 "________________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "________________________________________" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }