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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" -1 257 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "Using Taylor series to calculate \+ " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Ne wton's approximation for " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 88 "See: \"Journey through Genius\" by William Dunham, John W iley & Sons inc., 1990, page 174." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 23 "The circle with centre " }{XPPEDIT 18 0 " ``(1/2,0)" "6#-%!G6$*&\"\"\"F'\"\"#!\"\"\"\"!" }{TEXT -1 12 " and radi us " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 14 " has \+ equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x-1/2)^ 2+(y-0)^2=1/4" "6#/,&*$,&%\"xG\"\"\"*&F(F(\"\"#!\"\"F+F*F(*$,&%\"yGF( \"\"!F+F*F(*&F(F(\"\"%F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2-x+1/4+y ^2=1/4" "6#/,**$%\"xG\"\"#\"\"\"F&!\"\"*&F(F(\"\"%F)F(*$%\"yGF'F(*&F(F (F+F)" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+y^2=x" "6#/,&*$% \"xG\"\"#\"\"\"*$%\"yGF'F(F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 12 "Solving for " }{TEXT 265 1 "y" }{TEXT -1 7 " gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 264 1 "y" }{TEXT -1 3 " = " }{TEXT 263 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(x-x^2)" "6#-%%sqrtG6#,& %\"xG\"\"\"*$F'\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 "In particular, the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y=sqrt(x-x^2)" "6#/%\"yG-%%sqrtG6#,&%\"xG\"\"\"*$F)\"\" #!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 71 "is the equation of the upper semi-circle ADE in the following picture. 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Thus t he area of the sector ACD is " }{XPPEDIT 18 0 "Pi/24" "6#*&%#PiG\"\"\" \"#C!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 29 "The area of \+ triangle BCD is " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT -1 1 " " }{TEXT 267 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "abs(BC )*`.`*abs(BD) = ``(1/2)*`.`*``(1/4)*`.`*``(sqrt(3)/4);" "6#/*(-%$absG6 #%#BCG\"\"\"%\".GF)-F&6#%#BDGF)*,-%!G6#*&F)F)\"\"#!\"\"F)F*F)-F06#*&F) F)\"\"%F4F)F*F)-F06#*&-%%sqrtG6#\"\"$F)F8F4F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "`` = sqrt(3)/32;" "6#/%!G*&-%%sqrtG6#\"\"$\"\"\"\"#K!\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "It follows that th e area of the region ABD is " }{XPPEDIT 18 0 "Pi/24-sqrt(3)/32" "6#,&* &%#PiG\"\"\"\"#C!\"\"F&*&-%%sqrtG6#\"\"$F&\"#KF(F(" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Evaluati ng the above integral using Maple gives the same result." }}{PARA 0 " " 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Int( sqrt(x-x^2),x=0..1/4);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$IntG6$*$-%%sqrtG6#,&%\"xG\"\"\"*$)F+\"\"#F,!\"\"F,/F+;\"\"!#F,\"\"% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$-%%sqrtG6#\"\"$\"\"\"#!\"\"\"# K*&#F)\"#CF)%#PiGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "For " } {XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\" \"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sqrt(x-x^2)=sqrt(x)*sqrt(1-x)" "6#/-%%sqrtG6#,&%\"xG\"\"\"*$F(\"\"#!\"\"*&-F%6#F(F)-F%6#,&F)F)F(F,F) " }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{TEXT 268 1 " x" }{TEXT -1 9 " near 0, " }{XPPEDIT 18 0 "sqrt(1-x)" "6#-%%sqrtG6#,& \"\"\"F'%\"xG!\"\"" }{TEXT -1 93 " can be approximated by a Taylor pol ynomial obtained by truncating the binomial expansion of " }{XPPEDIT 18 0 "sqrt(1-x)" "6#-%%sqrtG6#,&\"\"\"F'%\"xG!\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 58 "For example, we can obtain the degree 8 \+ Taylor polynomial " }{XPPEDIT 18 0 "p(x);" "6#-%\"pG6#%\"xG" }{TEXT -1 12 " as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "expand(taylor(sqrt(1-x),x,9)):\nconvert(%,p olynom):\np := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG f*6#%\"xG6\"6$%)operatorG%&arrowGF(,4\"\"\"F-*&#F-\"\"#F-9$F-!\"\"*&#F -\"\")F-*$)F1F0F-F-F2*&#F-\"#;F-*$)F1\"\"$F-F-F2*&#\"\"&\"$G\"F-*$)F1 \"\"%F-F-F2*&#\"\"(\"$c#F-*$)F1F@F-F-F2*&#\"#@\"%C5F-*$)F1\"\"'F-F-F2* &#\"#L\"%[?F-*$)F1FGF-F-F2*&#\"$H%\"&oF$F-*$)F1F5F-F-F2F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The funct ion " }{XPPEDIT 18 0 "f(x)=sqrt(x-x^2)" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&F '\"\"\"*$F'\"\"#!\"\"" }{TEXT -1 37 " can be approximated by the funct ion " }{XPPEDIT 18 0 "q(x) = sqrt(x)*p(x);" "6#/-%\"qG6#%\"xG*&-%%sqrt G6#F'\"\"\"-%\"pG6#F'F," }{TEXT -1 7 ", when " }{TEXT 269 1 "x" } {TEXT -1 43 " is a non-negative real number close to 0. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "expan d(sqrt(x)*p(x)):\nq := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"qGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,4*$9$#\"\"\"\"\"#F0*&#F0F1 F0*$)F.#\"\"$F1F0F0!\"\"*&#F0\"\")F0*$)F.#\"\"&F1F0F0F8*&#F0\"#;F0*$)F .#\"\"(F1F0F0F8*&#F?\"$G\"F0*$)F.#\"\"*F1F0F0F8*&#FF\"$c#F0*$)F.#\"#6F 1F0F0F8*&#\"#@\"%C5F0*$)F.#\"#8F1F0F0F8*&#\"#L\"%[?F0*$)F.#\"#:F1F0F0F 8*&#\"$H%\"&oF$F0*$)F.#\"# " 0 "" {MPLTEXT 1 0 88 "f := x -> sqrt(x-x^2);\nplot([f(x),q(x)],x=0..1,color=[red,magen ta],scaling=constrained);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6 #%\"xG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&9$\"\"\"*$)F0\"\"#F1!\"\"F (F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURV ESG6$7_o7$$\"\"!F)F(7$$\"3immTN@Ki8!#?$\"3]Y#3'G)\\%)o$!#>7$$\"3ALL$3F WYs#F-$\"3-;H1+mp7_F07$$\"3%)***\\iSmp3%F-$\"3iMDTr8')zjF07$$\"3WmmmT& )G\\aF-$\"3e(H(=Y!*yhtF07$$\"3m****\\7G$R<)F-$\"3zvY$R[aR+*F07$$\"3ILL L3x&)*3\"F0$\"3$Gml_1e#Q5!#=7$$\"3$*****\\ilyM;F0$\"3]>>tlN4o7FJ7$$\"3 emmm;arz@F0$\"3AUEHun?g9FJ7$$\"3.++D\"y%*z7$F0$\"3@V@`oHtSFJ7$$\"33++voMrU^F0$\"3AZBYfMn3AFJ7$$\"3nmmm\"z _\"4iF0$\"3'p$R!*[r@8CFJ7$$\"3[mmmT&phN)F0$\"37\\6?'f$HnFFJ7$$\"3CLLe* =)H\\5FJ$\"3Ec*\\6qHY1$FJ7$$\"3gmm\"z/3uC\"FJ$\"3DN\"ex\"yC/LFJ7$$\"3% )***\\7LRDX\"FJ$\"3\")zK)HxmN_$FJ7$$\"3]mm\"zR'ok;FJ$\"3cy$=)3y+DPFJ7$ $\"3w***\\i5`h(=FJ$\"3s>1U$pYS!RFJ7$$\"3WLLL3En$4#FJ$\"3'=pGv!*p&oSFJ7 $$\"3qmm;/RE&G#FJ$\"3y(peb6M))>%FJ7$$\"3\")*****\\K]4]#FJ$\"3DB.H]bnIV FJ7$$\"3$******\\PAvr#FJ$\"3q2E]S&G'[WFJ7$$\"3)******\\nHi#HFJ$\"3KVJT ]xm\\XFJ7$$\"3jmm\"z*ev:JFJ$\"3/R^\"f*QPJYFJ7$$\"3?LLL347TLFJ$\"33lj4* H\"z;ZFJ7$$\"3,LLLLY.KNFJ$\"3K@1fkClzZFJ7$$\"3w***\\7o7Tv$FJ$\"3WJbb4# *GU[FJ7$$\"3'GLLLQ*o]RFJ$\"3#o&4)4&\\l))[FJ7$$\"3A++D\"=lj;%FJ$\"3bj!) 4;b,I\\FJ7$$\"31++vV&RY2aFJ $\"3'eZ&>A)pL)\\FJ7$$\"39mm;zXu9cFJ$\"3+$>XQ+l?'\\FJ7$$\"3l******\\y)) GeFJ$\"3D'yI&[e\"3$\\FJ7$$\"3'*)***\\i_QQgFJ$\"339mxo/LAN%[FJ7$$\"35****\\P![hY'FJ$\"3]uOKR-@!y%FJ7$$\"3k KLL$Qx$omFJ$\"3ZG!z*G+W8ZFJ7$$\"3!)*****\\P+V)oFJ$\"3Q7&\\7+^8j%FJ7$$ \"3?mm\"zpe*zqFJ$\"37l;jg>%oa%FJ7$$\"3%)*****\\#\\'QH(FJ$\"31'H#zsvwUW FJ7$$\"3GKLe9S8&\\(FJ$\"368l$QvKHL%FJ7$$\"3R***\\i?=bq(FJ$\"3F$f*Rq&zZ ?%FJ7$$\"3\"HLL$3s?6zFJ$\"3kZVg`'z]1%FJ7$$\"3a***\\7`Wl7)FJ$\"33GulKm) =!RFJ7$$\"3#pmmm'*RRL)FJ$\"3&f:nUGPis$FJ7$$\"3Qmm;a<.Y&)FJ$\"3!>%>5%z/ ]_$FJ7$$\"3=LLe9tOc()FJ$\"3Wvq!=Cb**H$FJ7$$\"3u******\\Qk\\*)FJ$\"3KNR JiI*f1$FJ7$$\"3CLL$3dg6<*FJ$\"3?h+R!>nqv#FJ7$$\"3ImmmmxGp$*FJ$\"3z#\\% Q--\"4V#FJ7$$\"3sK$eRA5\\Z*FJ$\"3`I_883^IAFJ7$$\"3A++D\"oK0e*FJ$\"3'\\ =ofiuY+#FJ7$$\"3C+++]oi\"o*FJ$\"3a*)>puzmbg#f**FJ$\"3gEg*eZf(pjF07$$\"3[\\PMF,%G(**FJ$ \"3GQsuc9V/_F07$$\"3uu=nj+U')**FJ$\"39b0e'\\$f#o$F07$$\"\"\"F)F(-%'COL OURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7fnF'7$F+$\"3!eC3'G)\\%)o$F0F1F67$F <$\"3='H(=Y!*yhtF0F@FE7$FL$\"3y>>tlN4o7FJ7$FQ$\"3oTEHun?g9FJ7$FV$\"3KW @`oHtSFJ7$Fjn$\"3*=Ti%fMn3AFJ7$F_o$\"34JV!*[r@8CF J7$Fdo$\"3s$*y?'f$HnFFJ7$Fio$\"3Q)z47qHY1$FJ7$F^p$\"31.M2=yC/LFJ7$Fcp$ \"3cX\"\\VxmN_$FJ7$Fhp$\"3oY3\"R\"y+DPFJ7$F]q$\"3f!fA'4n//RFJ7$Fbq$\"3 #ep0X&*p&oSFJ7$Fgq$\"3[r/pDU$))>%FJ7$F\\r$\"30Sw\"e\"enIVFJ7$Far$\"3?r H6R\"H'[WFJ7$Ffr$\"3skYL))*o'\\XFJ7$F[s$\"3;'Q?H>w8j%FJ7$F`s$\"3Xy>zze z;ZFJ7$Fes$\"39XRx8/mzZFJ7$Fjs$\"3kz%f!)z.B%[FJ7$F_t$\"3o,A%4Dz'))[FJ7 $Fdt$\"3#f8#f\\p0I\\FJ7$Fit$\"3/Fze^\"R/'\\FJ7$F^u$\"3S90g/C%H)\\FJ7$F cu$\"3%y$o58#[a*\\FJ7$Fhu$\"34!\\uQ2f-+&FJ7$F]v$\"37ZH#48gd*\\FJ7$Fbv$ \"31iTS;I'R)\\FJ7$Fgv$\"3#GY(H`H%H'\\FJ7$F\\w$\"3Hk$H&)H;@$\\FJ7$Faw$ \"3i%\\))eXwG*[FJ7$Ffw$\"3!oteUn3i%[FJ7$F[x$\"3QU\\1HG9%y%FJ7$F`x$\"3u 6)40?N*=ZFJ7$Fex$\"35_yeL![\"RYFJ7$Fjx$\"3Qi%*\\Eb[dXFJ7$F_y$\"3goO%[Q UwX%FJ7$Fdy$\"3Y&Gx\\]AKN%FJ7$Fiy$\"3^***\\J7TFB%FJ7$F^z$\"3tZj&GzIK5% FJ7$Fcz$\"3ecJ,N`gaRFJ7$Fhz$\"3AB$=%Q9=)z$FJ7$F][l$\"3#y_(eF*yRi$FJ7$F b[l$\"3pT.$[dphV$FJ7$Fg[l$\"3K)ym#f\\f\\KFJ7$F\\\\l$\"3b2.Lw>C=IFJ7$Fa \\l$\"3elEok`Y%z#FJ7$F[]l$\"3j1b=(>ur`#FJ7$Fe]l$\"3/eTvF3krAFJ7$F]`l$ \"3+]PM_h!Q'>FJ-F``l6&Fb`lFc`lF(Fc`l-%(SCALINGG6#%,CONSTRAINEDG-%+AXES LABELSG6$Q\"x6\"Q!Fb[m-%%VIEWG6$;F(F]`l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 " " 0 "" {TEXT -1 44 "We wish to use this function to approximate " } {XPPEDIT 18 0 "f(x)=sqrt(x-x^2)" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&F'\"\"\" *$F'\"\"#!\"\"" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[0, 1/4]" "6#7$\"\"!*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 22 " in order to estim ate " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. 1/4);" "6#-%$IntG6$-%\"fG6#%\" xG/F);\"\"!*&\"\"\"F.\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "The absolute error graph is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(f(x)-q( x),x=0..1/4,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7Z7$$\"\"!F)F(7$$\"3WmmmT&)G\\a!#?$\"37KEh( )z&=X\"!#M7$$\"3PL$ek`o!>5!#>$\"3Z#yRbWEf-\"F07$$\"3omm\"z>)G_:F4$!3lB (4#zrZ-H!#N7$$\"3immT&QU!*3#F4$!3(psl,uKn![F<7$$\"34L$eRZXKi#F4$!3/'p5 #*GbDC\"F07$$\"3\\m;z>,_=JF4$!3i`ip/S1b\\F07$$\"3h**\\7G$[8j$F4$!3p445 ^p'R^#!#L7$$\"3Gm;z%*frhTF4$!35([]!\\T:j%)FQ7$$\"3S**\\ilFQ!p%F4$!3i)* 3%4v(42F!#K7$$\"3mLL$3_\"=M_F4$!3m78$y'oy3xFfn7$$\"3tmmTg(fJr&F4$!3-m/ c+%Gpx\"!#J7$$\"3_****\\7eP_iF4$!3*[HEBU*p1UFao7$$\"3%)****\\Pf!Qz'F4$ !3Zw33'H!z.$*Fao7$$\"3%*****\\(=ubJ(F4$!3)e;Ll!\\p))=!#I7$$\"3em;zW(*Q *y(F4$!3dhHP**zAUMFap7$$\"3+LL$3F-GN)F4$!3%3![n9r,=nFap7$$\"3_KLL$e'3I ))F4$!3GTtPi&zR9\"!#H7$$\"3Q**\\7.ad&Faq7$$\"3++v$f)[ $H4\"F_r$!3%>Sv04v)[))Faq7$$\"3OL$ek`1l9\"F_r$!3<'f#RxUC,9!#G7$$\"3ILe *[.-d>\"F_r$!3I1m()f^N)4#F\\s7$$\"3km;/Egw[7F_r$!3yZs'\\k]e=$F\\s7$$\" 3Ym\"z%*f%)QI\"F_r$!3V%\\fpcru#[F\\s7$$\"3))*\\(oza'=N\"F_r$!3Ut$f:N3j $oF\\s7$$\"3`m;zWho.9F_r$!3O%H\\'Ql*=#)*F\\s7$$\"3!*****\\i>Ad9F_r$!3d f.(RA<*39!#F7$$\"3u**\\i:jf4:F_r$!3O&R%4%e\"e!)>F[u7$$\"3!)*\\7`>r-c\" F_r$!3z!4>#*RAQs#F[u7$$\"3y**\\P4q`;;F_r$!3Y_%[Y#3#\\$QF[u7$$\"3;LL$eM %4n;F_r$!3Ur2(*pu6k^F[u7$$\"3&****\\P4v5s\"F_r$!3xPRd4gsFqF[u7$$\"3cm \"zWn*)*pF_r$!3)z'=;h+7$4#F_ w7$$\"3CLL3-=!y(>F_r$!3l#=.LcgDq#F_w7$$\"3*)*\\7G8O;.#F_r$!3N1')f:?V2N F_w7$$\"3smmm\"*\\[$3#F_r$!3gLdF_w7$$\"3ILekG=4*=#F_r$!3y0$e#HI1YsF_w7$$\"3%*****\\i4TPAF_r$!37P2 zJ0fC-qi7F]z7$$\"3emmmT>KUBF_r$!3 Gm%elpu0S\"F]z7$$\"3=$e*)fbF(oBF_r$!3[8Q3%)QQi:F]z7$$\"31+DJqJ8&R#F_r$ !3R4?F[^%3u\"F]z7$$\"31++]7nS?CF_r$!3'ojw(H]kG>F]z7$$\"31+voa-oXCF_r$! 3/QhN]*3X8#F]z7$$\"3/Dc,\">g#fCF_r$!3K7zO'3:JD#F]z7$$\"3-]PMF,%GZ#F_r$ !3mnyz[#GwP#F]z7$$\"3-v=nj+U'[#F_r$!3'ox!42$4$3DF]z7$$\"3++++++++DF_r$ !3?(oJBCHak#F]z-%+AXESLABELSG6$Q\"x6\"Q!Fi]l-%'COLOURG6&%$RGBGF(F($\"* ++++\"!\")-%%VIEWG6$;F($\"+++++D!#5%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "numapprox[infnorm] (f(x)-q(x),x=0..1/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+U#Hak#!#< " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The maximum absolute erro r over the interval " }{XPPEDIT 18 0 "[0,1/4]" "6#7$\"\"!*&\"\"\"F&\" \"%!\"\"" }{TEXT -1 19 " is approximately " }{XPPEDIT 18 0 "3*`.`*10^ (-8)" "6#*(\"\"$\"\"\"%\".GF%)\"#5,$\"\")!\"\"F%" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We set up a function " }{XPPEDIT 18 0 "r(x)" "6#-%\"rG6#%\"xG" }{TEXT -1 31 " a s the indefinite integral of " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG " }{TEXT -1 21 ", or more precisely, " }{XPPEDIT 18 0 "Int(q(t),t=0..x )" "6#-%$IntG6$-%\"qG6#%\"tG/F);\"\"!%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" } {TEXT -1 27 " approximates the function " }{XPPEDIT 18 0 "g(x)=Int(sqr t(t-t^2),t=0..x)" "6#/-%\"gG6#%\"xG-%$IntG6$-%%sqrtG6#,&%\"tG\"\"\"*$F /\"\"#!\"\"/F/;\"\"!F'" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Int(q(x),x);\nvalue(% );\nr := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,4* $%\"xG#\"\"\"\"\"#F**&F+!\"\"F(#\"\"$F+F-*&\"\")F-F(#\"\"&F+F-*&\"#;F- F(#\"\"(F+F-*(F3F*\"$G\"F-F(#\"\"*F+F-*(F7F*\"$c#F-F(#\"#6F+F-*(\"#@F* \"%C5F-F(#\"#8F+F-*(\"#LF*\"%[?F-F(#\"#:F+F-*(\"$H%F*\"&oF$F-F(#\"#F%F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,4*&#\" \"#\"\"$\"\"\"*$)9$#F0F/F1F1F1*&#F1\"\"&F1*$)F4#F8F/F1F1!\"\"*&#F1\"#G F1*$)F4#\"\"(F/F1F1F<*&#F1\"#sF1*$)F4#\"\"*F/F1F1F<*&#F8\"$/(F1*$)F4# \"#6F/F1F1F<*&#FC\"%k;F1*$)F4#\"#8F/F1F1F<*&#FC\"%gDF1*$)F4#\"#:F/F1F1 F<*&#\"#L\"&3u\"F1*$)F4#\"#F/F1F1F " 0 "" {MPLTEXT 1 0 40 "simplify(r(1/4));\narea _est1 := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"1z\\yGP,>>\"2 WH!))z5f*\\#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*area_est1G$\"+y1Jxw !#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "N ote that the evaluation does not involve the numerical evaluation of a ny square roots. " }}{PARA 0 "" 0 "" {TEXT -1 84 "The following comput ation shows the list of numbers which have to be added together." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "c := [seq(coeff(r(x),x^((2*k+1)/2))*1/2^(2*k+1),k=1..9)];\nadd(c[k ],k=1..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7+#\"\"\"\"#7#!\" \"\"$g\"#F*\"%%e$#F*\"&ko$#!\"&\"(#zT9#!\"(\"))[JO\"#F4\")!3')Q)#!#L\" +w8q\"G##!$H%\"-[sv3K;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"1z\\yGP,> >\"2WH!))z5f*\\#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "By equating this area est imate with the original expression " }{XPPEDIT 18 0 "Pi/24-sqrt(3)/32 " "6#,&*&%#PiG\"\"\"\"#C!\"\"F&*&-%%sqrtG6#\"\"$F&\"#KF(F(" }{TEXT -1 41 " for the area, we obtain an estimate for " }{XPPEDIT 18 0 "Pi" "6# %#PiG" }{TEXT -1 4 " as " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(`area estimate`+sqrt(3)/32)*`.`*24;" "6#*(,&%.area~est imateG\"\"\"*&-%%sqrtG6#\"\"$F&\"#K!\"\"F&F&%\".GF&\"#CF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "pi_est1 := evalf((area_e st1+sqrt(3)/32)*24);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(pi_est1G$\" +pEfTJ!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Comparing with Maple's value for " }{XPPEDIT 18 0 "Pi" "6#%#PiG " }{TEXT -1 47 " we see that this value is correct to 7 digits." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aEfTJ!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "A procedure which gener ates approximations for " }{XPPEDIT 18 0 "Int(sqrt(t-t^2),t = 0 .. x); " "6#-%$IntG6$-%%sqrtG6#,&%\"tG\"\"\"*$F*\"\"#!\"\"/F*;\"\"!%\"xG" } {TEXT -1 3 " : " }{TEXT 0 5 "circ " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "cir c: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 271 2 " " }{TEXT -1 15 " circ( u, n )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 14 " u - " }{TEXT -1 33 "a variable or numerical constant " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 23 9 " n - " }{TEXT -1 66 " a positive integer which specifies the number of terms. " }{TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 3 " " } {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 4 "circ" }{TEXT -1 48 " constructs finite sums of the series expansion " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sqrt(x-x^2),x = 0 .. u) = ``(2/3) *u^(3/2)-``(1/5)*u^(5/2)-``(1/28)*u^(7/2)-``(1/72)*u^(9/2)-``(5/704)*u ^(11/2)-``(7/1664)*u^(13/2)-` . . . `;" "6#/-%$IntG6$-%%sqrtG6#,&%\"xG \"\"\"*$F+\"\"#!\"\"/F+;\"\"!%\"uG,0*&-%!G6#*&F.F,\"\"$F/F,)F3*&F:F,F. F/F,F,*&-F76#*&F,F,\"\"&F/F,)F3*&FAF,F.F/F,F/*&-F76#*&F,F,\"#GF/F,)F3* &\"\"(F,F.F/F,F/*&-F76#*&F,F,\"#sF/F,)F3*&\"\"*F,F.F/F,F/*&-F76#*&FAF, \"$/(F/F,)F3*&\"#6F,F.F/F,F/*&-F76#*&FKF,\"%k;F/F,)F3*&\"#8F,F.F/F,F/% (~.~.~.~GF/" }{TEXT -1 2 " " }}{PARA 257 "" 0 "" {TEXT -1 7 "Option: " }}{PARA 0 "" 0 "" {TEXT -1 15 "info=true/false" }}{PARA 0 "" 0 "" {TEXT -1 116 "With the option \"info=true\" the numerical values of th e terms and the partial sums are printed as they are computed." }} {PARA 0 "" 0 "" {TEXT -1 84 "Note: This option only operates when the \+ first argument is a floating point number. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the pro cedure active open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the subsection. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "circ: implementation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "circ" {MPLTEXT 1 0 1244 "circ := proc(x::algebraic,n::posint)\n local sum,term,k,pr ntflg,format,d;\n prntflg := false;\n if type(x,float) then\n \+ if x<0 or x>1 then \n error \"the first argument must lie bet ween 0 and 1\"\n end if;\n if nargs=3 and args[3]=(info=true ) then\n prntflg:=true;\n end if;\n end if;\n sum := \+ 0; \n for k from 1 to n do\n if k=1 then\n term := 2 /3*x^(3/2)\n else\n term := term*(2*k-5)*(2*k-1)/((2*k-2) *(2*k+1))*x\n end if;\n sum := sum+term;\n if prntflg t hen\n printf(\" term %d -> \",k); \n format := cat(\"%\",convert(Digits,string),\".\",\n convert(Digits-1 ,string),e);\n printf(format,term);\n if Digits>32 the n printf(\"\\n \") \n else\n d := 7-(ilog10( k)+iquo(1-signum(term),2));\n printf(cat(\" \"$d));\n \+ end if; \n printf(\"sum -> \"); \n if s um<>0 then d := min(ilog10(sum),Digits-1) \n else d := 0 end i f;\n format := cat(\"%\",convert(Digits-d,string),\".\",\n \+ convert(Digits-d-1,string),f);\n printf(format,sum);\n printf(\"\\n\");\n if Digits>32 then printf(\"\\n\") \+ end if;\n end if; \n end do; \n sum;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 15 "Computation of " }{XPPEDIT 18 0 "Pi" "6#%#PiG" } {TEXT -1 14 " to 16 digits " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 66 "Newton used the method outlined \+ in the first section to calculate " }{XPPEDIT 18 0 "Pi" "6#%#PiG" } {TEXT -1 60 " to 16 decimal digits using a 20-term binomial expansion \+ of " }{XPPEDIT 18 0 "sqrt(1-x)" "6#-%%sqrtG6#,&\"\"\"F'%\"xG!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 146 "He presented the calcul ation in the treatise \"Methodus Fluxionum et Serierum Infinitarum\" w ritten in 1671, but not published until decades later. " }}{PARA 0 "" 0 "" {TEXT -1 149 "At one point he confessed, \"I am ashamed to tell y ou how many places of figures I carried these computations, having no \+ other business at the time.\" " }}{PARA 0 "" 0 "" {TEXT -1 52 "We can \+ repeat the calculation with ease using Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "First obtain the 20-term \+ binomial expansion of " }{XPPEDIT 18 0 "sqrt(1-x)" "6#-%%sqrtG6#,&\"\" \"F'%\"xG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "expand(taylor(sqrt(1-x),x,21 )):\nconvert(%,polynom):\np := unapply(%,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,L\"\"\"F-*&#F -\"\"#F-9$F-!\"\"*&#F-\"\")F-*$)F1F0F-F-F2*&#F-\"#;F-*$)F1\"\"$F-F-F2* &#\"\"&\"$G\"F-*$)F1\"\"%F-F-F2*&#\"\"(\"$c#F-*$)F1F@F-F-F2*&#\"#@\"%C 5F-*$)F1\"\"'F-F-F2*&#\"#L\"%[?F-*$)F1FGF-F-F2*&#\"$H%\"&oF$F-*$)F1F5F -F-F2*&#\"$:(\"&Ob'F-*$)F1\"\"*F-F-F2*&#\"%JC\"'W@EF-*$)F1\"#5F-F-F2*& #\"%*>%\"')GC&F-*$)F1\"#6F-F-F2*&#\"&$RH\"(/V>%F-*$)F1\"#7F-F-F2*&#\"& .?&\"(3')Q)F-*$)F1\"#8F-F-F2*&#\"'Dd=\")KWbLF-*$)F1\"#9F-F-F2*&#\"'0VL \")k)3r'F-*$)F1\"#:F-F-F2*&#\"(X[p*\"+[O[Z@F-*$)F1F:F-F-F2*&#\")N)yw\" \"+'Hn\\H%F-*$)F1\"#\"\",o$Q(fV$F-*$)F1\"#>F-F-F2*&#\"*&fJO))\"-Wp!z([FF-*$)F1\"#? F-F-F2F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Then construct the associated approximation for " } {XPPEDIT 18 0 "f(x)=`` " "6#/-%\"fG6#%\"xG%!G" }{XPPEDIT 18 0 "sqrt(x- x^2)=sqrt(x)*sqrt(1-x)" "6#/-%%sqrtG6#,&%\"xG\"\"\"*$F(\"\"#!\"\"*&-F% 6#F(F)-F%6#,&F)F)F(F,F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "expand(sqrt(x)*p(x)): \nq := unapply(%,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"qGf*6#%\"x G6\"6$%)operatorG%&arrowGF(,L*$9$#\"\"\"\"\"#F0*&#F0F1F0*$)F.#\"\"$F1F 0F0!\"\"*&#F0\"\")F0*$)F.#\"\"&F1F0F0F8*&#F0\"#;F0*$)F.#\"\"(F1F0F0F8* &#F?\"$G\"F0*$)F.#\"\"*F1F0F0F8*&#FF\"$c#F0*$)F.#\"#6F1F0F0F8*&#\"#@\" %C5F0*$)F.#\"#8F1F0F0F8*&#\"#L\"%[?F0*$)F.#\"#:F1F0F0F8*&#\"$H%\"&oF$F 0*$)F.#\"#F1F0F0F8*&#\"%JC\"'W@EF0* $)F.#FWF1F0F0F8*&#\"%*>%\"')GC&F0*$)F.#\"#BF1F0F0F8*&#\"&$RH\"(/V>%F0* $)F.#\"#DF1F0F0F8*&#\"&.?&\"(3')Q)F0*$)F.#\"#FF1F0F0F8*&#\"'Dd=\")KWbL F0*$)F.#\"#HF1F0F0F8*&#\"'0VL\")k)3r'F0*$)F.#\"#JF1F0F0F8*&#\"(X[p*\"+ [O[Z@F0*$)F.#FinF1F0F0F8*&#\")N)yw\"\"+'Hn\\H%F0*$)F.#\"#NF1F0F0F8*&# \")&RA['\",%=p)zr\"F0*$)F.#\"#PF1F0F0F8*&#\"*v'4%>\"\",o$Q(fV$F0*$)F.# \"#RF1F0F0F8*&#\"*&fJO))\"-Wp!z([FF0*$)F.#\"#TF1F0F0F8F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Integrate to obtain a function " }{XPPEDIT 18 0 "r(x)" "6#-%\"rG6#%\"xG" } {TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "g(x)=Int(sqrt(t-t^2),t= 0..x)" "6#/-%\"gG6#%\"xG-%$IntG6$-%%sqrtG6#,&%\"tG\"\"\"*$F/\"\"#!\"\" /F/;\"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "r := unapply(int(q(x),x),x); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(,L*&#\"\"#\"\"$\"\"\"*$)9$#F0F/F1F1F1*&#F1\"\"&F1*$)F4#F8F/F1F1 !\"\"*&#F1\"#GF1*$)F4#\"\"(F/F1F1F<*&#F1\"#sF1*$)F4#\"\"*F/F1F1F<*&#F8 \"$/(F1*$)F4#\"#6F/F1F1F<*&#FC\"%k;F1*$)F4#\"#8F/F1F1F<*&#FC\"%gDF1*$) F4#\"#:F/F1F1F<*&#\"#L\"&3u\"F1*$)F4#\"#F/F1F1F<*&#\"$:(\"'G\")oF1*$)F4#\"#@F/F1F1F<*&#\"%JC\"(cY,$F1*$) F4#\"#BF/F1F1F<*&#\"%*>%\"(+Ob'F1*$)F4#\"#DF/F1F1F<*&#\"&$RH\")/JicF1* $)F4#\"#FF/F1F1F<*&#\"&.?&\"*;[j@\"F1*$)F4#\"#HF/F1F1F<*&#\"'Dd=\"*'p$ 4?&F1*$)F4#\"#JF/F1F1F<*&#\"'N96\"*_()4p$F1*$)F4#F\\oF/F1F1F<*&#\"(p*Q >\"+oF>;vF1*$)F4#\"#NF/F1F1F<*&#\")N)yw\"\",w\\*oXzF1*$)F4#\"#PF/F1F1F <*&#\")lug@\"-'p\\\"p;6F1*$)F4#\"#RF/F1F1F<*&#\"*v'4%>\"\"-WljuVqF1*$) F4#\"#TF/F1F1F<*&#\"*&fJO))\".'H**\\()4fF1*$)F4#\"#VF/F1F1F " 0 "" {MPLTEXT 1 0 31 "area _est2 := evalf(r(0.25),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*area _est2G$\"5,xIZI;1Jxw!#@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 67 "We can see the individual terms which need to be a dded as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "Digits := 20:\nc := [seq(coeff(r(x),x^((2*k+ 1)/2))*1/2^(2*k+1),k=1..21)];\nc := evalf(%);\nadd(c[k],k=1..21);\nDig its := 10:" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"cG77#\"\"\"\"#7#!\" \"\"$g\"#F*\"%%e$#F*\"&ko$#!\"&\"(#zT9#!\"(\"))[JO\"#F4\")!3')Q)#!#L\" +w8q\"G##!$H%\"-[sv3K;#!$:(\".c9,4JW\"#!%JC\"/[)Quw)GD#!%*>%\"0+_bDB!* >##!&$RH\"17x=rV#)*f(#!&.?&\"2#>sof%>-`'#!'Dd=\"43I)ye2F*o6\"#!'N96\"4 %=H)ow8M0<$#!(p*Q>\"6CEsL>.\"\":)G#4IOC\"Q1i$*[:#!*&fJO))\";obO7X0HMC5Q)>&" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"cG77$\"5LLLLLLLLL$)!#@$!5++++++++ ]i!#A$!5'G9dG9dy,z#!#B$!5666666On7F!#C$!5OOOO60m!zY$!#D$!5i%QlLgRp^8&! #E$!5+](oa&o-lW$)!#F$!52Syppu\"*GY9F:$!5zvf\"3WUN&GE!#G$!5v14o3h1ea\\! #H$!5ow>3ElN'Hh*!#I$!5b$GEsj7%[4>FE$!5&*G3FLw$*enQ!#J$!5#e9QtyLQM'z!#K $!5'eQGy([B(Gm\"FM$!5I)3>jWY2Z^$!#L$!5hf@Iz2+)z](!#M$!5*Qb$y+))3()=;FU $!5%o;T&3J0m>N!#N$!5IV=nvD494x!#O$!5\"f(yDa`2#)*p\"Fgn" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5,xIZI;1Jxw!#@" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The corresponding estimate of " } {XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 10 " is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "pi_est2 : = evalf(evalf((area_est2+sqrt(3)/32)*24,20),16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(pi_est2G$\"1$z*e`EfTJ!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(Pi,16);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1$z*e`EfTJ!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 14 "The procedu re " }{TEXT 0 4 "circ" }{TEXT -1 3 " - " }{HYPERLNK 17 "circ" 1 "" "ci rc" }{TEXT -1 81 " given in the previous section may be used to calcul ate the series estimate for " }{XPPEDIT 18 0 "Int(sqrt(x-x^2),x=0..1/ 4)" "6#-%$IntG6$-%%sqrtG6#,&%\"xG\"\"\"*$F*\"\"#!\"\"/F*;\"\"!*&F+F+\" \"%F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Using an \"unknown\" variable " }{TEXT 272 1 "x" } {TEXT -1 55 " as the first argument gives the same series as before." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "circ(x,21);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,L*(\"\"#\"\"\"\" \"$!\"\"%\"xG#F'F%F&*&\"\"&F(F)#F,F%F(*&\"#GF(F)#\"\"(F%F(*&\"#sF(F)# \"\"*F%F(*(F,F&\"$/(F(F)#\"#6F%F(*(F1F&\"%k;F(F)#\"#8F%F(*(F1F&\"%gDF( F)#\"#:F%F(*(\"#LF&\"&3u\"F(F)#\"#F%F(*( \"$:(F&\"'G\")oF(F)#\"#@F%F(*(\"%JCF&\"(cY,$F(F)#\"#BF%F(*(\"%*>%F&\"( +Ob'F(F)#\"#DF%F(*(\"&$RHF&\")/JicF(F)#\"#FF%F(*(\"&.?&F&\"*;[j@\"F(F) #\"#HF%F(*(\"'Dd=F&\"*'p$4?&F(F)#\"#JF%F(*(\"'N96F&\"*_()4p$F(F)#FCF%F (*(\"(p*Q>F&\"+oF>;vF(F)#\"#NF%F(*(\")N)yw\"F&\",w\\*oXzF(F)#\"#PF%F(* (\")lug@F&\"-'p\\\"p;6F(F)#\"#RF%F(*(\"*v'4%>\"F&\"-WljuVqF(F)#\"#TF%F (*(\"*&fJO))F&\".'H**\\()4fF(F)#\"#VF%F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Using the optional argument \"" } {TEXT 262 9 "info=true" }{TEXT -1 152 "\" with a floating point number (between 0 and 1) as the frst argument causes the various terms and t he partial sums to be printed as they are computed. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "area_est2 \+ := evalf(circ(0.25,21,info=true),20);" }}{PARA 6 "" 1 "" {TEXT -1 73 " term 1 -> 8.3333333333333333333e-02 sum -> .083333333333333333 333" }}{PARA 6 "" 1 "" {TEXT -1 73 " term 2 -> -6.2500000000000000000 e-03 sum -> .077083333333333333333" }}{PARA 6 "" 1 "" {TEXT -1 73 " term 3 -> -2.7901785714285714285e-04 sum -> .07680431547619 0476190" }}{PARA 6 "" 1 "" {TEXT -1 73 " term 4 -> -2.712673611111111 1110e-05 sum -> .076777188740079365079" }}{PARA 6 "" 1 "" {TEXT -1 73 " term 5 -> -3.4679066051136363638e-06 sum -> .07677372083 3474251443" }}{PARA 6 "" 1 "" {TEXT -1 73 " term 6 -> -5.135169396033 6538465e-07 sum -> .076773207316534648078" }}{PARA 6 "" 1 "" {TEXT -1 73 " term 7 -> -8.3446502685546875002e-08 sum -> .07677 3123870031962531" }}{PARA 6 "" 1 "" {TEXT -1 73 " term 8 -> -1.446289 1746969784007e-08 sum -> .076773109407140215561" }}{PARA 6 "" 1 " " {TEXT -1 73 " term 9 -> -2.6285354244081597578e-09 sum -> .076 773106778604791153" }}{PARA 6 "" 1 "" {TEXT -1 73 " term 10 -> -4.954 5806610868090672e-10 sum -> .076773106283146725044" }}{PARA 6 "" 1 "" {TEXT -1 73 " term 11 -> -9.6129635652608197662e-11 sum -> . 076773106187017089391" }}{PARA 6 "" 1 "" {TEXT -1 73 " term 12 -> -1. 9094841263722628354e-11 sum -> .076773106167922248127" }}{PARA 6 " " 1 "" {TEXT -1 73 " term 13 -> -3.8675893763327082892e-12 sum -> .076773106164054658751" }}{PARA 6 "" 1 "" {TEXT -1 73 " term 14 -> - 7.9634383378733814575e-13 sum -> .076773106163258314917" }}{PARA 6 "" 1 "" {TEXT -1 73 " term 15 -> -1.6628723487782838586e-13 sum -> .076773106163092027682" }}{PARA 6 "" 1 "" {TEXT -1 73 " term 16 - > -3.5147074644631908828e-14 sum -> .076773106163056880607" }} {PARA 6 "" 1 "" {TEXT -1 73 " term 17 -> -7.5079800077930215958e-15 \+ sum -> .076773106163049372627" }}{PARA 6 "" 1 "" {TEXT -1 73 " ter m 18 -> -1.6188708880078355388e-15 sum -> .076773106163047753756" }}{PARA 6 "" 1 "" {TEXT -1 73 " term 19 -> -3.5196605310854116682e-16 sum -> .076773106163047401790" }}{PARA 6 "" 1 "" {TEXT -1 73 " t erm 20 -> -7.7091409257567184322e-17 sum -> .076773106163047324699 " }}{PARA 6 "" 1 "" {TEXT -1 73 " term 21 -> -1.6998207535425787590e- 17 sum -> .076773106163047307701" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%*area_est2G$\"5,xIZI;1Jxw!#@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "The corresponding estimate of " } {XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 10 " is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "pi_est2 : = evalf(evalf((area_est2+sqrt(3)/32)*24,20),16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(pi_est2G$\"1$z*e`EfTJ!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(Pi,16);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1$z*e`EfTJ!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "A procedure giving Taylor polynomials f or " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 7 " ab out " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 2 ": " }{TEXT 0 11 "arctan_poly" }{TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "arctan_poly: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 273 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 274 2 " " }{TEXT -1 22 " arctan_poly( x, n )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 14 " x - " }{TEXT -1 48 "an algebraic expression, or a numerical constant" }} {PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 23 9 " n \+ - " }{TEXT -1 90 " a positive integer which specifies the \+ number of terms of the Taylor polynomial" }{TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 3 " " }{TEXT -1 0 "" }}{PARA 257 "" 0 " " {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proced ure " }{TEXT 0 11 "arctan_poly" }{TEXT -1 66 " constructs finite parti al sums of the Maclaurin series expansion " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x)=x-x^3/3+x^5/5-x^7/7+` . . . `" "6#/ -%'arctanG6#%\"xG,,F'\"\"\"*&F'\"\"$F+!\"\"F,*&F'\"\"&F.F,F)*&F'\"\"(F 0F,F,%(~.~.~.~GF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } }{PARA 257 "" 0 "" {TEXT -1 7 "Option:" }}{PARA 0 "" 0 "" {TEXT -1 15 "info=true/false" }}{PARA 0 "" 0 "" {TEXT -1 116 "With the option \"in fo=true\" the numerical values of the terms and the partial sums are p rinted as they are computed." }}{PARA 0 "" 0 "" {TEXT -1 84 "Note: Thi s option only operates when the first argument is a floating point num ber. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 259 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure active 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 28 "arctan_poly: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "arctan_poly" {MPLTEXT 1 0 1274 "arctan_poly := \+ proc(x::algebraic,n::posint)\n local z,pow,sum,term,k,odd,prntflg,fo rmat,d;\n prntflg := false;\n if type(x,float) then\n if x<0 \+ or x>1 then \n error \"the first argument must lie between 0 a nd 1\"\n end if;\n if nargs=3 and args[3]=(info=true) then\n prntflg:=true;\n end if;\n end if;\n\n pow := x;\n \+ z := x*x;\n sum := 0;\n odd := false; \n for k from 1 to n do\n term := pow/(2*k-1);\n if odd then term := -term end if;\n \+ sum := sum+term;\n if prntflg then\n printf(\" term %d -> \",k); \n format := cat(\"%\",convert(Digits,str ing),\".\",\n convert(Digits-1,string),e);\n printf (format,term);\n if Digits>32 then printf(\"\\n \") \n \+ else\n d := 7-(ilog10(k)+iquo(1-signum(term),2));\n printf(cat(\" \"$d));\n end if; \n \+ printf(\"sum -> \");\n if sum<>0 then d := min(ilog10(sum),D igits-1) \n else d := 0 end if;\n format := cat(\"%\", convert(Digits-d,string),\".\",\n convert(Digits-d-1,string ),f);\n printf(format,sum);\n printf(\"\\n\");\n \+ if Digits>32 then printf(\"\\n\") end if;\n end if;\n odd := not odd;\n pow := pow*z;\n end do; \n sum;\nend proc:" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "arctan_poly: examples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "arctan_poly( x,11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8%\"xG\"\"\"*&#F%\"\"$F%*$) F$F(F%F%!\"\"*&#F%\"\"&F%*$)F$F.F%F%F%*&#F%\"\"(F%*$)F$F3F%F%F+*&#F%\" \"*F%*$)F$F8F%F%F%*&#F%\"#6F%*$)F$F=F%F%F+*&#F%\"#8F%*$)F$FBF%F%F%*&#F %\"#:F%*$)F$FGF%F%F+*&#F%\"#F%*$)F$FQF%F%F+*& #F%\"#@F%*$)F$FVF%F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Illustration of (delayed) evaluation using exact ari thmetic. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "arctan_poly(``(1/4),11);\neval(subs(``=(x->x),%));\ne valf(%,15);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8-%!G6##\"\"\"\"\"%F (*&#F(\"\"$F(*$)F$F,F(F(!\"\"*&#F(\"\"&F(*$)F$F2F(F(F(*&#F(\"\"(F(*$)F $F7F(F(F/*&#F(\"\"*F(*$)F$FF (*$)F$FUF(F(F/*&#F(\"#@F(*$)F$FZF(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"5$o80\\;*pgn:\"5Sm`N\\kJ&*)R'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0lo7j'y\\C!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Illustratation of evaluation using floating point ar ithmetic. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "evalf(arctan_poly(.25,11,info=true),15);" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 1 -> 2.50000000000000e-01 sum -> . 250000000000000" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 2 -> -5.2083333 3333333e-03 sum -> .244791666666667" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 3 -> 1.95312500000000e-04 sum -> .244986979166667" }} {PARA 6 "" 1 "" {TEXT -1 62 " term 4 -> -8.71930803571429e-06 su m -> .244978259858631" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 5 -> 4.23 855251736111e-07 sum -> .244978683713883" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 6 -> -2.16744162819602e-08 sum -> .2449786620 39467" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 7 -> 1.14624316875751e-09 sum -> .244978663185710" }}{PARA 6 "" 1 "" {TEXT -1 62 " term \+ 8 -> -6.20881716410317e-11 sum -> .244978663123622" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 9 -> 3.42398005373336e-12 sum -> .24497 8663127046" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 10 -> -1.91472568794 300e-13 sum -> .244978663126855" }}{PARA 6 "" 1 "" {TEXT -1 62 " \+ term 11 -> 1.08273178782491e-14 sum -> .244978663126866" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0mo7j'y\\C!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf(arctan(.25), 15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0ko7j'y\\C!#:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Using the Maclaurin series for " }{XPPEDIT 18 0 "arcta n(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 14 " to calculate " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 25 "The Maclaurin serie s for " }{XPPEDIT 18 0 "arctan(x);" "6#-%'arctanG6#%\"xG" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "Sum((-1)^n*x^(2*n+1)/(2*n+1),n = 0 .. infinity ) = x-x^3/3+x^5/5-x^7/7+` . . . `;" "6#/-%$SumG6$*(),$\"\"\"!\"\"%\"nG F*)%\"xG,&*&\"\"#F*F,F*F*F*F*F*,&*&F1F*F,F*F*F*F*F+/F,;\"\"!%)infinity G,,F.F**&F.\"\"$F:F+F+*&F.\"\"&FF+F+%(~.~.~.~GF*" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "The series converges to \+ " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 6 " for \+ " }{XPPEDIT 18 0 "-1<=x" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "``<=1 " "6#1%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "In par ticular, when " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 9 " we have " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum((-1)^n /(2*n+1),n = 0 .. infinity)=1-1/3+1/5-1/7+1/9-1/11+` . . . `" "6#/-%$S umG6$*&),$\"\"\"!\"\"%\"nGF*,&*&\"\"#F*F,F*F*F*F*F+/F,;\"\"!%)infinity G,0F*F**&F*F*\"\"$F+F+*&F*F*\"\"&F+F**&F*F*\"\"(F+F+*&F*F*\"\"*F+F**&F *F*\"#6F+F+%(~.~.~.~GF*" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "arctan(1) = Pi/4" "6#/-%'arctanG6#\"\"\"*&%#PiGF'\"\"%!\"\"" }{TEXT -1 14 " ----- -- (i). " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Sum((-1)^n/(2*n+1),n = 0 .. infinity);\nvalue(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)!\"\"%\"nG\"\"\",&F)\"\"#F *F*F(/F);\"\"!%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG# \"\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The series in (i) converges rather slowly, and so does not prov ide a good way of evaluating " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Sum((-1)^n/(2*n+1),n=0..50);\n4*value(%);\nevalf(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)!\"\"%\"nG\"\"\",&F)\" \"#F*F*F(/F);\"\"!\"#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"KGub/KD.U eDU'3)=uuln=yM\"Kv00H(RgA<*z_-.+RfrYF+6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+8')>hJ!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "To circ umvent this problem a formula such as" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "arctan(2/5)+arctan(3/7)=Pi/4" "6#/,&-%'arctanG6#* &\"\"#\"\"\"\"\"&!\"\"F*-F&6#*&\"\"$F*\"\"(F,F**&%#PiGF*\"\"%F," } {TEXT -1 14 " ------- (ii) " }}{PARA 0 "" 0 "" {TEXT -1 13 "can be use d. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Th is formula is based on the trigonometric formula: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(alpha+beta) = (tan(alpha)+tan(bet a))/(1-tan(alpha)*tan(beta));" "6#/-%$tanG6#,&%&alphaG\"\"\"%%betaGF)* &,&-F%6#F(F)-F%6#F*F)F),&F)F)*&-F%6#F(F)-F%6#F*F)!\"\"F7" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "alpha=arctan(s) " "6#/%&alphaG-%'arctanG6#%\"sG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b eta=arctan(t)" "6#/%%betaG-%'arctanG6#%\"tG" }{TEXT -1 22 " for two re al numbers " }{TEXT 275 1 "s" }{TEXT -1 5 " and " }{TEXT 276 1 "t" } {TEXT -1 6 ", then" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "tan(arctan(s)+arctan(t))=(s+t)/(1-s*t)" "6#/-%$tanG6#,&-%'arctanG6# %\"sG\"\"\"-F)6#%\"tGF,*&,&F+F,F/F,F,,&F,F,*&F+F,F/F,!\"\"F4" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "-Pi " 0 "" {MPLTEXT 1 0 45 "combine(arctan(s)+arctan(t),arctan,symbolic);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'arctanG6#*&,&%\"sG\"\"\"%\"tGF)F),&F)F)*&F(F)F*F)! \"\"F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "We can also check the formula (ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "arctan(2/5)+arctan(3/7) ;\ncombine(%,arctan);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%'arctanG6 ##\"\"#\"\"&\"\"\"-F%6##\"\"$\"\"(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$%#PiG#\"\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Construct a Taylor polynomial " }{XPPEDIT 18 0 "p(x) " "6#-%\"pG6#%\"xG" }{TEXT -1 29 " of degree 21 to approximate " } {XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 5 " for " } {TEXT 277 1 "x" }{TEXT -1 8 " near 0." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "taylor(arctan(x),x,22); \nconvert(%,polynom);\np := unapply(%,x):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+;%\"xG\"\"\"F%#!\"\"\"\"$F(#F%\"\"&F*#F'\"\"(F,#F%\"\" *F.#F'\"#6F0#F%\"#8F2#F'\"#:F4#F%\"#F8#F%\"#@F:-%\"OG6#F%\"# A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8%\"xG\"\"\"*&#F%\"\"$F%*$)F$F(F %F%!\"\"*&#F%\"\"&F%)F$F.F%F%*&#F%\"\"(F%*$)F$F2F%F%F+*&#F%\"\"*F%)F$F 7F%F%*&#F%\"#6F%*$)F$F;F%F%F+*&#F%\"#8F%)F$F@F%F%*&#F%\"#:F%*$)F$FDF%F %F+*&#F%\"#F%*$)F$FMF%F%F+*&#F%\"#@F%)F$FRF%F%" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Using " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate \+ " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 17 " we a pproximate " }{XPPEDIT 18 0 "arctan(2/5)+arctan(3/7)" "6#,&-%'arctanG 6#*&\"\"#\"\"\"\"\"&!\"\"F)-F%6#*&\"\"$F)\"\"(F+F)" }{TEXT -1 5 " by \+ " }{XPPEDIT 18 0 "p(2/5)+p(3/7)" "6#,&-%\"pG6#*&\"\"#\"\"\"\"\"&!\"\"F )-F%6#*&\"\"$F)\"\"(F+F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "Exact arithmetic can be used . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "p(2/5)+p(3/7);\nevalf(4 *%,15);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"H8(GB[X:K*G=2 -&e`*fCp3'\"HDc^`)*4YL6Bf4YCW58,v(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"0I3Ul#fTJ!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aEfTJ!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 " . . . or floating point arithmetic." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "Digits := 15:\nu1 := evalf(2/5);\n u2 := evalf(3/7);\nss := p(u1)+p(u2);\npp := 4*ss;\nDigits := 10:\neva lf(pp2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u1G$\"0+++++++%!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u2G$\"0H9dG9dG%!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ssG$\"0y?bj\")R&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ppG$\"0J3Ul#fTJ!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+aEfTJ!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Comparing with Maple's " }{XPPEDIT 18 0 "Pi" "6#%#PiG" } {TEXT -1 50 ", we see that this value is correct to 10 digits. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(Pi,15);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0z* e`EfTJ!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aEfTJ!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure \+ " }{TEXT 0 11 "arctan_poly" }{TEXT -1 53 " can be used to give some de tails of the computation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "Digits := 15:\nu1 := evalf(2/5);\n u2 := evalf(3/7);\np1 := arctan_poly(u1,11,info=true);\np2 := arctan_p oly(u2,11,info=true);\nss := p1+p2;\npp := 4*ss;\nDigits := 10:\nevalf (pp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u1G$\"0+++++++%!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u2G$\"0H9dG9dG%!#:" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 1 -> 4.00000000000000e-01 sum -> .4000000 00000000" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 2 -> -2.13333333333333 e-02 sum -> .378666666666667" }}{PARA 6 "" 1 "" {TEXT -1 62 " te rm 3 -> 2.04800000000000e-03 sum -> .380714666666667" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 4 -> -2.34057142857143e-04 sum -> .38 0480609523810" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 5 -> 2.9127111111 1111e-05 sum -> .380509736634921" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 6 -> -3.81300363636364e-06 sum -> .380505923631285" }} {PARA 6 "" 1 "" {TEXT -1 62 " term 7 -> 5.16222030769231e-07 su m -> .380506439853316" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 8 -> -7.1 5827882666667e-08 sum -> .380506368270528" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 9 -> 1.01058054023529e-08 sum -> .3805063783 76333" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 10 -> -1.44672582602105e- 09 sum -> .380506376929607" }}{PARA 6 "" 1 "" {TEXT -1 62 " term \+ 11 -> 2.09430786243048e-10 sum -> .380506377139038" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#p1G$\"0Q!RrP10Q!#:" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 1 -> 4.28571428571429e-01 sum -> .428571428571429 " }}{PARA 6 "" 1 "" {TEXT -1 62 " term 2 -> -2.62390670553936e-02 \+ sum -> .402332361516035" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 3 -> \+ 2.89165228773726e-03 sum -> .405224013803772" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 4 -> -3.79371291394102e-04 sum -> .4048446425 12378" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 5 -> 5.41958987705859e-05 sum -> .404898838411149" }}{PARA 6 "" 1 "" {TEXT -1 62 " term \+ 6 -> -8.14446716218452e-06 sum -> .404890693943987" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 7 -> 1.26578061076337e-06 sum -> .40489 1959724598" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 8 -> -2.014916074276 38e-07 sum -> .404891758232991" }}{PARA 6 "" 1 "" {TEXT -1 62 " \+ term 9 -> 3.26547022841910e-08 sum -> .404891790887693" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 10 -> -5.36645483295511e-09 sum -> . 404891785521238" }}{PARA 6 "" 1 "" {TEXT -1 62 " term 11 -> 8.9180153 2007114e-10 sum -> .404891786413040" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G$\"0SIT'y\"*[S!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ss G$\"0y?bj\")R&y!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ppG$\"0J3Ul#f TJ!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aEfTJ!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "19th century computation of " } {XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 15 " by Johann Dase" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 88 "S ee: \"Keys to Infinity\", by Clifford A. Pickover, John Wiley & Sons i nc., 1995, page 62." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 89 "In 1854, Johann Martin Zacharias Dase (1824-1861), a hu man computer, supposedly computed " }{XPPEDIT 18 0 "Pi" "6#%#PiG" } {TEXT -1 57 " correctly to 200 decimal places in less than two months. " }}{PARA 0 "" 0 "" {TEXT -1 19 "He used the formula" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(1/2)+arctan(1/5)+arctan(1/8 ) = Pi/4;" "6#/,(-%'arctanG6#*&\"\"\"F)\"\"#!\"\"F)-F&6#*&F)F)\"\"&F+F )-F&6#*&F)F)\"\")F+F)*&%#PiGF)\"\"%F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 54 "with a Maclaurin series expansion for each arctangent. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "combine(arctan(1/2)+arctan(1/5)+arctan(1/8),arctan);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Calculation of " }{XPPEDIT 18 0 "Pi" "6 #%#PiG" }{TEXT -1 13 " to 20 digits" }}{PARA 0 "" 0 "" {TEXT -1 19 "Fi rst we calculate " }{XPPEDIT 18 0 "arctan(1/2)" "6#-%'arctanG6#*&\"\" \"F'\"\"#!\"\"" }{TEXT -1 7 " using " }{TEXT 0 11 "arctan_poly" } {TEXT -1 3 " - " }{HYPERLNK 17 "arctan_poly" 1 "" "arctan_poly" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 133 "The number of terms is \+ adjusted so that the last term added is sufficiently small to indicate that the value is correct to 20 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Digits := 25:\np1 := \+ arctan_poly(.5,31,info=true);" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 1 -> 5.000000000000000000000000e-01 sum -> .500000000000000000000 0000" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 2 -> -4.166666666666666666 666667e-02 sum -> .4583333333333333333333333" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 3 -> 6.250000000000000000000000e-03 sum -> . 4645833333333333333333333" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 4 -> \+ -1.116071428571428571428571e-03 sum -> .4634672619047619047619047 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 5 -> 2.17013888888888888888888 9e-04 sum -> .4636842757936507936507936" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 6 -> -4.438920454545454545454545e-05 sum -> . 4636398865891053391053391" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 7 -> \+ 9.390024038461538461538462e-06 sum -> .4636492766131438006438006 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 8 -> -2.0345052083333333333333 33e-06 sum -> .4636472421079354673104673" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 9 -> 4.487879136029411764705882e-07 sum -> . 4636476908958490702516438" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 10 -> -1.003867701480263157894737e-07 sum -> .4636475905090789222253280 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 11 -> 2.2706531343005952380952 38e-08 sum -> .4636476132156102652312804" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 12 -> -5.183012589164402173913043e-09 sum -> . 4636476080325976760668782" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 13 -> 1.192092895507812500000000e-09 sum -> .4636476092246905715746907 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 14 -> -2.759474295156973379629 630e-10 sum -> .4636476089487431420589934" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 15 -> 6.422914307692955280172414e-11 sum -> . 4636476090129722851359230" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 16 -> -1.502133184863675025201613e-11 sum -> .4636476089979509532872862 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 17 -> 3.5277370250586307410037 88e-12 sum -> .4636476090014786903123448" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 18 -> -8.315380130495343889508929e-13 sum -> . 4636476090006471522992953" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 19 -> 1.966475030860385379275760e-13 sum -> .4636476090008437998023813 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 20 -> -4.664075393707324297000 200e-14 sum -> .4636476090007971590484442" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 21 -> 1.109139880210888095018340e-14 sum -> . 4636476090008082504472463" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 22 -> -2.643879947014326273008835e-15 sum -> .4636476090008056065672993 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 23 -> 6.3159354289786683188544 40e-16 sum -> .4636476090008062381608422" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 24 -> -1.511793054808723799725797e-16 sum -> . 4636476090008060869815367" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 25 -> 3.625218039592347887097573e-17 sum -> .4636476090008061232337171 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 26 -> -8.707631565687502277832 408e-18 sum -> .4636476090008061145260855" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 27 -> 2.094760423821050076271004e-18 sum -> . 4636476090008061166208459" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 28 -> -5.046468293750711547380145e-19 sum -> .4636476090008061161161991 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 29 -> 1.2173498077030225223943 33e-19 sum -> .4636476090008061162379341" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 30 -> -2.940209281316622193918517e-20 sum -> . 4636476090008061162085320" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 31 -> 7.109522442527897927917725e-21 sum -> .4636476090008061162156415 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G$\"::k:i613+4wkj%!#D" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Now we ca lculate " }{XPPEDIT 18 0 "arctan(1/5);" "6#-%'arctanG6#*&\"\"\"F'\"\"& !\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 144 "As before, the number of terms is adjusted so that the last term added is sufficient ly small to indicate that the value is correct to 20 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p2 := arctan_poly(.2,14,info=true);" }}{PARA 6 "" 1 "" {TEXT -1 82 " te rm 1 -> 2.000000000000000000000000e-01 sum -> .20000000000000000 00000000" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 2 -> -2.66666666666666 6666666667e-03 sum -> .1973333333333333333333333" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 3 -> 6.400000000000000000000000e-05 sum - > .1973973333333333333333333" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 4 \+ -> -1.828571428571428571428571e-06 sum -> .1973955047619047619047 619" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 5 -> 5.68888888888888888888 8889e-08 sum -> .1973955616507936507936508" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 6 -> -1.861818181818181818181818e-09 sum -> . 1973955597889754689754690" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 7 -> \+ 6.301538461538461538461538e-11 sum -> .1973955598519908535908536 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 8 -> -2.1845333333333333333333 33e-12 sum -> .1973955598498063202575203" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 9 -> 7.710117647058823529411765e-14 sum -> . 1973955598498834214339909" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 10 -> -2.759410526315789473684211e-15 sum -> .1973955598498806620234646 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 11 -> 9.9864380952380952380952 38e-17 sum -> .1973955598498807618878456" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 12 -> -3.647220869565217391304348e-18 sum -> . 1973955598498807582406247" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 13 -> 1.342177280000000000000000e-19 sum -> .1973955598498807583748424 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 14 -> -4.971026962962962962962 963e-21 sum -> .1973955598498807583698714" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G$\":9()p$e2))\\)fbR(>!#D" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Lastly we calculate " } {XPPEDIT 18 0 "arctan(1/8);" "6#-%'arctanG6#*&\"\"\"F'\"\")!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 139 "The number of terms is again adjusted so that the last term added is sufficiently small to i ndicate that the value is correct to 20 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "p3 := arctan _poly(.125,11,info=true);" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 1 -> \+ 1.250000000000000000000000e-01 sum -> .1250000000000000000000000 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 2 -> -6.5104166666666666666666 67e-04 sum -> .1243489583333333333333333" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 3 -> 6.103515625000000000000000e-06 sum -> . 1243550618489583333333333" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 4 -> \+ -6.811959402901785714285714e-08 sum -> .1243549937293643043154762 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 5 -> 8.27842288547092013888888 9e-10 sum -> .1243549945572065928625682" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 6 -> -1.058321107517589222301136e-11 sum -> . 1243549945466233817873923" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 7 -> \+ 1.399222618112197289100060e-13 sum -> .1243549945467633040492035 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 8 -> -1.8947806286936004956563 31e-15 sum -> .1243549945467614092685748" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 9 -> 2.612289469706250683349722e-17 sum -> . 1243549945467614353914695" }}{PARA 6 "" 1 "" {TEXT -1 82 " term 10 -> -3.652049423109067567182999e-19 sum -> .1243549945467614350262646 " }}{PARA 6 "" 1 "" {TEXT -1 82 " term 11 -> 5.1628674880262115905116 81e-21 sum -> .1243549945467614350314275" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G$\":vUJ]Vhna%*\\NC\"!#D" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The corresponding value o f " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 35 " is obtained by using \+ the formula: " }{XPPEDIT 18 0 "Pi = 4*(arctan(1/2)+arctan(1/5)+arctan( 1/8))" "6#/%#PiG*&\"\"%\"\"\",(-%'arctanG6#*&F'F'\"\"#!\"\"F'-F*6#*&F' F'\"\"&F.F'-F*6#*&F'F'\"\")F.F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "ss := p1+p2 +p3;\npp := 4*evalf(ss);\nevalf(%,20);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ssG$\":/%ph4$[uRj\")R&y!#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ppG$\":ixYQKz*e`EfTJ!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5&QKz*e`EfTJ!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 15 "This value for " }{XPPEDIT 18 0 "Pi" "6#% #PiG" }{TEXT -1 26 " is correct to 20 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(Pi,21) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"6YQKz*e`EfTJ!#?" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Calculation of " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 14 " to 200 digits" }}{PARA 0 "" 0 "" {TEXT -1 19 "First we calculate " }{XPPEDIT 18 0 "arctan(1/2)" "6#-%'arctanG 6#*&\"\"\"F'\"\"#!\"\"" }{TEXT -1 7 " using " }{TEXT 0 11 "arctan_poly " }{TEXT -1 3 " - " }{HYPERLNK 17 "arctan_poly" 1 "" "arctan_poly" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 143 "The number of terms is \+ adjusted so that the last term added is sufficiently small to indicate that the value is correct to at least 200 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Digits := 21 0:\np1 := arctan_poly(.5,335);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#p 1G$\"]x!***fn\"*)Q&o^0e#z)fv-D`,,)[;miL>2i0T%*p0`ra:6]i\"==e]8v$3>^A&) Hlb#*y)[)RG+1I0?()3L4\"QE?hGaq`G?S97YJiD9i613+4wkj%!$5#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Now we ca lculate " }{XPPEDIT 18 0 "arctan(1/5);" "6#-%'arctanG6#*&\"\"\"F'\"\"& !\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 154 "As before, the number of terms is adjusted so that the last term added is sufficient ly small to indicate that the value is correct to at least 200 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p2 := arctan_poly(.2,145);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%#p2G$\"]xMvn!Gp=tDLm_&='Gh+c:]k:+Z2@_+F.j@&\\lvub$RUz'))zkX/,O\"pt!= \"GE6W!)GP!GypKdyV#y*pR.T-%*)o<:5_yy.^eZMH!z%>l(\\+Pe2))\\)fbR(>!$5#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Lastly we calculate " }{XPPEDIT 18 0 "arctan(1/8);" "6#-%'arctanG6#*&\"\"\"F '\"\")!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 139 "The numb er of terms is again adjusted so that the last term added is sufficien tly small to indicate that the value is correct to 20 digits." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "p3 := arctan_poly(.125,113);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% #p3G$\"]xZ01fEt4+*)p(e#\\l-\")y=F+/4D,C5f)Q6<(oyVArY]RoR3j6\"Q/z$QMI^! zjKILN8Vf5PiencFuEAd6>T6:**3/)p<>qJdD5(Q;\\[NJ]Vhna%*\\NC\"!$5#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The corre sponding value of " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 35 " is ob tained by using the formula: " }{XPPEDIT 18 0 "Pi = 4*(arctan(1/2)+arc tan(1/5)+arctan(1/8));" "6#/%#PiG*&\"\"%\"\"\",(-%'arctanG6#*&F'F'\"\" #!\"\"F'-F*6#*&F'F'\"\"&F.F'-F*6#*&F'F'\"\")F.F'F'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "ss := p1+p2+p3;\npp := 4*evalf(ss);\nevalf(%,200);\nDigits := 10: " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#ssG$\"]xr!Qt5\"\\&fK(Qstb6\"**Q w-j%[vc-rDOz-7K?&)R8$zbXwQ_6Yte%3m:'4$[uRj\")R&y!$5#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%#ppG$\"]xG_$HW'>QI\\&*[HiW'fb5@&Q>q-TG]u6\"[G\"3%f`sJAe]&4Y%Q4ZmI#G8 l3[@)z1<@MD[.G')**3iG1k\"yI#fW\\(4#e5v$*Rpr>%)G]zKQVEYQKz*e`EfTJ!$4#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"cw?QI\\&*[HiW'fb5@&Q>q-TG]u6\"[G \"3%f`sJAe]&4Y%Q4ZmI#G8l3[@)z1<@MD[.G')**3iG1k\"yI#fW\\(4#e5v$*Rpr>%)G ]zKQVEYQKz*e`EfTJ!$*>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "This value for " }{XPPEDIT 18 0 "Pi" "6#%#PiG" } {TEXT -1 27 " is correct to 200 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(Pi,201);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#$\"dw'>QI\\&*[HiW'fb5@&Q>q-TG]u6\"[G\" 3%f`sJAe]&4Y%Q4ZmI#G8l3[@)z1<@MD[.G')**3iG1k\"yI#fW\\(4#e5v$*Rpr>%)G]z KQVEYQKz*e`EfTJ!$+#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "A procedure for con structing arctangent decompositions of " }{XPPEDIT 18 0 "Pi/4" "6#*&%# PiG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ": " }{TEXT 0 13 "arctan_decomp" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "arctan_decomp: usage" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 278 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 279 2 " " }{TEXT -1 52 " arct an_decomp( numterms, numer_size, denom_size )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 16 " \+ numterms - " }{TEXT -1 81 "a positive integer which specifies the n umber of terms in the arctangent formula " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 23 17 " numer_size - " }{TEXT -1 88 " a positive integer which specifies the maximum size of the nume rator of each fraction" }{TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 23 17 " denom_size - " }{TEXT -1 91 " a positive integer which specifies the maximum size of the denominator \+ of each fraction " }}{PARA 0 "" 0 "" {TEXT 259 5 "Notes" }{TEXT -1 66 ": numterms and denom_size must be greater than 1 with numer_size" } {XPPEDIT 18 0 " ``<=1/2" "6#1%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 11 " d enom_size" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 3 " " }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 10 "tan_decomp" }{TEXT -1 52 " constructs the fractions in formulas of the type: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(n[1]/d[1])+arctan(n[2]/d[2] )+` . . . ` = arctan(1);" "6#/,(-%'arctanG6#*&&%\"nG6#\"\"\"F,&%\"dG6# F,!\"\"F,-F&6#*&&F*6#\"\"#F,&F.6#F6F0F,%(~.~.~.~GF,-F&6#F," }{TEXT -1 4 " " }{XPPEDIT 18 0 "``(`` = Pi/4);" "6#-%!G6#/F$*&%#PiG\"\"\"\"\" %!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "n[i]" "6#&%\"nG6#%\"iG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "d[i]" "6#&%\"dG6#%\"iG" }{TEXT -1 28 " are positive integers and " }{XPPEDIT 18 0 "1/(numterms+5) <= n[i]/d[i];" "6#1*&\"\"\"F%,&%)nu mtermsGF%\"\"&F%!\"\"*&&%\"nG6#%\"iGF%&%\"dG6#F.F)" }{XPPEDIT 18 0 "`` <= 1/(numterms-1);" "6#1%!G*&\"\"\"F&,&%)numtermsGF&F&!\"\"F)" } {TEXT -1 11 " for each " }{TEXT 283 1 "i" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 7 "Option:" }}{PARA 0 "" 0 "" {TEXT -1 15 "info=true/false" }}{PARA 0 "" 0 "" {TEXT -1 94 "With the option \"info=true\" the valid sequences of fractions are pr inted as they are computed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to activat e:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure active open the subsection, place the cursor anywhere after the prompt [ > and p ress [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "arctan_decomp: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "arctan_decomp" {MPLTEXT 1 0 2383 "arctan_decomp := proc(numterms::posint,magnumer::posint,magdenom ::posint)\n local n,d,AT,i,j,k,eq,pp,den,dens,nums,trial,\n u,v ,notOK,prntflg,eqn,f1,f2;\n\n for i in [1,3] do\n if args[i]=1 \+ then\n error \"the %-1 argument must be greater than 1\",i;\n \+ end if;\n end do;\n if 2*magnumer>magdenom then\n error \+ \"the 2nd argument, %1, must be no greater than half of the 3rd argume nt, %2\",magnumer,magdenom;\n end if;\n\n if nargs=4 and args[4]=( info=true) then prntflg := true\n else prntflg := false end if;\n \+ # construct the equation to test\n AT := add(arctan(n[i]/d[i]),i=1.. numterms);\n AT := combine(AT,arctan,symbolic);\n AT := normal(op( 1,AT));\n eq := numer(AT)=denom(AT);\n f1 := 1/(numterms-1);\n f 2 := 1/(numterms+5);\n pp := [];\n den := combstruct[iterstructs] \n (Combination([seq(i,i=2..magdenom)]),size=numterms);\n # const ruct the denominators (no repetitions)\n while not combstruct[finish ed](den) do\n dens := combstruct[nextstruct](den);\n #for i \+ to numterms do d[i] := op(i,dens) end do;\n # construct the numer ators (repetitions allowed)\n for k from 0 to magnumer^numterms-1 do\n if magnumer>1 then\n nums := convert(k,base,m agnumer);\n nums := map(u->u+1,[op(nums),0$(numterms-nops(n ums))]);\n else\n nums := [1$numterms];\n e nd if;\n trial := [seq(op(i,nums)/op(i,dens),i=1..numterms)]; \n trial := sort(trial,(u,v)->evalb(u>v));\n # omit if obtained perviously\n if member(trial,pp) then next end if; \+ \n # only consider fractions f with 1/(numterms+5)< =f<=1/(numterms-1)\n notOK := false;\n for i to nops(t rial) do\n if op(i,trial)>f1 or op(i,trial)`,trial);\n # substitute in the identity\n eqn := eval(subs(\{seq(d[i]=op(i,dens),i=1..numterms),\n se q(n[i]=op(i,nums),i=1..numterms)\},eq));\n if eqn and not memb er(trial,pp) then\n if prntflg then print(op(trial)) end if ;\n pp := [op(pp),trial];\n end if;\n end do; \n end do;\n pp := map(u->[op(u)],pp);\n pp := sort(pp,(u,v)->ev alb(add(op(i,u),i=1..nops(u)) " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "arctan_decomp: examples" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "arctan_decomp(2,5,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)7$#\"\"$ \"\"(#\"\"#\"\"&7$#\"\"\"F(#F,F%7$#F)\"\"*#F(F&7$#F%F)#F,\"\"%7$#F(F%# F,F)7$#F)F&#F,\"\"'7$#F%F5#F,F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "combine(arctan(1/2)+arctan(1 /3),arctan);\ncombine(arctan(3/7)+arctan(2/5),arctan);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "arctan_decomp(3,2,15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7%#\"\"\"\"\"$#F%\"\"%#\"\"#\"\"*7%F$F$#F%\"\"( 7%#F%F*#F*\"#6F-7%F0#F%\"\"&#F%\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "combine(arctan(1/2)+arc tan(1/5)+arctan(1/8),arctan);\ncombine(arctan(1/3)+arctan(1/4)+arctan( 2/9),arctan);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 62 "Another example involving the use of the Maclaurin seri es for " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 14 " to calculate " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 33 "We look for formulas of the type " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(n[1]/d[1])+arctan(n[2]/d[2])+` . . . ` +arctan(n[k]/d[k]) = Pi/4;" "6#/,*-%'arctanG6#*&&%\"nG6#\"\"\"F,&%\"dG 6#F,!\"\"F,-F&6#*&&F*6#\"\"#F,&F.6#F6F0F,%(~.~.~.~GF,-F&6#*&&F*6#%\"kG F,&F.6#F?F0F,*&%#PiGF,\"\"%F0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "n[i];" "6#&%\"nG6#%\"iG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "d[i]" "6#&%\"dG6#%\"iG" }{TEXT -1 27 " a re positive integers for " }{XPPEDIT 18 0 "i=1,2, ` . . . `,k" "6&/%\" iG\"\"\"\"\"#%(~.~.~.~G%\"kG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 32 "We look for formulas of the type" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(1/a)+arctan(1/b)+arctan(1/c)=P i/4" "6#/,(-%'arctanG6#*&\"\"\"F)%\"aG!\"\"F)-F&6#*&F)F)%\"bGF+F)-F&6# *&F)F)%\"cGF+F)*&%#PiGF)\"\"%F+" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 280 1 "a" }{TEXT -1 2 ", " }{TEXT 281 1 "b " }{TEXT -1 5 " and " }{TEXT 282 1 "c" }{TEXT -1 24 " are positive int egers. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "unassign('a','b','c'):\narctan(1/a)+arctan(1/b)+arct an(1/c);\ncombine(%,arctan,symbolic);\nsimplify(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,(-%'arctanG6#*&\"\"\"F(%\"aG!\"\"F(-F%6#*&F(F(%\"bGF *F(-F%6#*&F(F(%\"cGF*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'arctanG6 #*&,&*&,&*&\"\"\"F+%\"aG!\"\"F+*&F+F+%\"bGF-F+F+,&F+F+*&F+F+*&F,F+F/F+ F-F-F-F+*&F+F+%\"cGF-F+F+,&F+F+*(F)F+F0F-F4F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'arctanG6#*&,**&%\"cG\"\"\"%\"bGF*F**&F)F*%\"aGF*F**& F-F*F+F*F*F*!\"\"F*,**(F)F*F-F*F+F*F*F)F/F+F/F-F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "Thus " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(1/a)+arctan(1/b)+arctan(1/c)=arctan((a*b+b*c+c* a-1)/(a*b*c-a-b-c))" "6#/,(-%'arctanG6#*&\"\"\"F)%\"aG!\"\"F)-F&6#*&F) F)%\"bGF+F)-F&6#*&F)F)%\"cGF+F)-F&6#*&,**&F*F)F/F)F)*&F/F)F3F)F)*&F3F) F*F)F)F)F+F),**(F*F)F/F)F3F)F)F*F+F/F+F3F+F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "We must h ave: " }{XPPEDIT 18 0 "a*b+b*c+c*a-1=a*b*c-a-b-c" "6#/,**&%\"aG\"\"\" %\"bGF'F'*&F(F'%\"cGF'F'*&F*F'F&F'F'F'!\"\",**(F&F'F(F'F*F'F'F&F,F(F,F *F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 38 "The following loop tests the equation " }{XPPEDIT 18 0 "a*b+b*c+c*a-1=a*b*c-a-b-c" "6#/,**&%\"aG\"\"\"%\"bGF'F'*&F(F'%\"cGF'F '*&F*F'F&F'F'F'!\"\",**(F&F'F(F'F*F'F'F&F,F(F,F*F," }{TEXT -1 42 " wit h all sets of three positive integers " }{XPPEDIT 18 0 "\{a,b,c\}" "6# <%%\"aG%\"bG%\"cG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a,b,c<=100" "6%%\"aG%\"bG1%\"cG\"$+\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "with(combstruct):\na llc := iterstructs(Combination(100),size=3):\nwhile not finished(allc) do\n s := nextstruct(allc);\n a := op(1,s);\n b := op(2,s);\n \+ c := op(3,s);\n if a*b+b*c+c*a-1=a*b*c-a-b-c then print(s) end if; \n end do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%\"\"#\"\"%\"#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<%\"\"#\"\"&\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "There appear to be just t wo possible solutions (without regard to the order): " }{XPPEDIT 18 0 "a=2,b=4,c=13" "6%/%\"aG\"\"#/%\"bG\"\"%/%\"cG\"#8" }{TEXT -1 6 " an d " }{XPPEDIT 18 0 "a=2,b=5,c=8" "6%/%\"aG\"\"#/%\"bG\"\"&/%\"cG\"\") " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "combine(arctan(1/2)+arctan(1/5)+arctan(1/8),ar ctan);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "combine(arctan(1/2)+arctan(1/4)+arctan(1/13),arctan);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The 2nd formula has no advantag e over the 1st formula which was considered in the previous section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Example 2 - using " } {TEXT 0 13 "arctan_decomp" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 13 "arctan_decomp" }{TEXT -1 3 " - " } {HYPERLNK 17 "arctan_decomp" 1 "" "arctan_decomp" }{TEXT -1 63 " can b e used to obtain the fractions in formulas of the type: " }}{PARA 0 " " 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(n[1]/d[1])+arctan(n[2]/d[2])+` . . . `+arctan(n[k]/d[k]) = P i/4;" "6#/,*-%'arctanG6#*&&%\"nG6#\"\"\"F,&%\"dG6#F,!\"\"F,-F&6#*&&F*6 #\"\"#F,&F.6#F6F0F,%(~.~.~.~GF,-F&6#*&&F*6#%\"kGF,&F.6#F?F0F,*&%#PiGF, \"\"%F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "arcta n_decomp(3,3,16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(7%#\"\"\"\"\"$#F% \"\"%#\"\"#\"\"*7%F$#F&\"#6#F%\"\"&7%F$#F*\"\"(#F&\"#;7%F$F$#F%F37%#F% F*#F*F.F77%F9F/#F%\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "combine(2*arctan(1/3)+arctan(1/7)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "arcta n_decomp(4,3,16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,7&#\"\"\"\"\"%#\" \"#\"\"*#F(\"#6#F%\"\"(7&F$F'#F%\"\"&#F%\"\")7&#\"\"$F+F/F/F17&#F(F-#F 5\"#;F*F,7&F4#F5\"#9F/#F%F)7&F7F/F8F17&F7F;F8F=7&#F%F5F*F,F,7&FAF/F,F1 7&FAF;F,F=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "A \"much\" longer calculation does not yield any significantly \+ better candidates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "arctan_decomp(4,6,20);" }{TEXT -1 0 "" }} {PARA 12 "" 1 "" {XPPMATH 20 "657&#\"\"\"\"\"%#\"\"#\"\"*#F%\"\"'#\"\" $\"#>7&F$F'#F(\"#6#F%\"\"(7&F$F'#F%\"\"&#F%\"\")7&#F-F1F5F*F,7&F$F'#F- \"#9#F%F)7&F:F5F5F77&#F(F3#F-\"#;F*F,7&FAFBF0F27&F:F7&FAF5FBF77&F AF7&#F&\"#8F*F*F,7&FIF0F*F27&FIF5F*F77&FIF7&#F%F-F*F,F27&FOF 0F2F27&FOF5F2F77&FOF" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Probably the best formula obtained from these com putations is: " }{XPPEDIT 18 0 "arctan(3/11)+2*arctan(1/5)+arctan(1/8) = Pi/4;" "6#/,(-%'arctanG6#*&\"\"$\"\"\"\"#6!\"\"F**&\"\"#F*-F&6#*&F* F*\"\"&F,F*F*-F&6#*&F*F*\"\")F,F**&%#PiGF*\"\"%F," }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "combine(arctan(1/8)+2*arctan(1/5)+arctan(3/11));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%!\"\"%#PiG\"\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Calculation of " }{XPPEDIT 18 0 "Pi" "6 #%#PiG" }{TEXT -1 21 " to 200 digits using " }{XPPEDIT 18 0 "arctan(3/ 11)+2*arctan(1/5)+arctan(1/8) = Pi/4" "6#/,(-%'arctanG6#*&\"\"$\"\"\" \"#6!\"\"F**&\"\"#F*-F&6#*&F*F*\"\"&F,F*F*-F&6#*&F*F*\"\")F,F**&%#PiGF *\"\"%F," }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 19 "First we calculate " }{XPPEDIT 18 0 "arctan(3/11);" "6#-%'arctanG6#*&\"\"$\"\"\"\"#6!\"\"" }{TEXT -1 7 " u sing " }{TEXT 0 11 "arctan_poly" }{TEXT -1 3 " - " }{HYPERLNK 17 "arct an_poly" 1 "" "arctan_poly" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 143 "The number of terms is adjusted so that the last term added is sufficiently small to indicate that the value is correct to at least \+ 200 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "Digits := 210:\np1 := arctan_poly(evalf(3/11),180); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#p1G$\"]xD(3o))>?7\">UJPpti'\\w* *f:L;>b7n1N-ysu]'Rs*zr2$[HQ$\\!4uZ*\\xTt,RW[)*fZfX-zc>z\\(=%*Q;z%[TC$H i\"o#)\\]>&4e3TUmiY1U%yND4:\\?Dm#!$5#" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "Now we calculate " }{XPPEDIT 18 0 "ar ctan(1/5);" "6#-%'arctanG6#*&\"\"\"F'\"\"&!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 154 "As before, the number of terms is adjust ed so that the last term added is sufficiently small to indicate that \+ the value is correct to at least 200 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p2 := arctan_poly( .2,145);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#p2G$\"]xMvn!Gp=tDLm_&=' Gh+c:]k:+Z2@_+F.j@&\\lvub$RUz'))zkX/,O\"pt!=\"GE6W!)GP!GypKdyV#y*pR.T- %*)o<:5_yy.^eZMH!z%>l(\\+Pe2))\\)fbR(>!$5#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Lastly we calculate " } {XPPEDIT 18 0 "arctan(1/8);" "6#-%'arctanG6#*&\"\"\"F'\"\")!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 139 "The number of terms is again adjusted so that the last term added is sufficiently small to i ndicate that the value is correct to 20 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "p3 := arctan _poly(.125,113);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#p3G$\"]xZ01fEt4 +*)p(e#\\l-\")y=F+/4D,C5f)Q6<(oyVArY]RoR3j6\"Q/z$QMI^!zjKILN8Vf5PiencF uEAd6>T6:**3/)p<>qJdD5(Q;\\[NJ]Vhna%*\\NC\"!$5#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The corresponding value o f " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 35 " is obtained by using \+ the formula: " }{XPPEDIT 18 0 "Pi = 4*(arctan(3/11)+2*arctan(1/5)+arct an(1/8));" "6#/%#PiG*&\"\"%\"\"\",(-%'arctanG6#*&\"\"$F'\"#6!\"\"F'*& \"\"#F'-F*6#*&F'F'\"\"&F/F'F'-F*6#*&F'F'\"\")F/F'F'" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ss := p1+2*p2+p3;\npp := 4*evalf(ss);\nevalf(%,200);\nDigits := \+ 10:" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#ssG$\"]xSVA26\\&fK(Qstb6\"** Qw-j%[vc-rDOz-7K?&)R8$zbXwQ_6Yte%3m:'4$[uRj\")R&y!$5#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%#ppG$\"]xO(*)GW'>QI\\&*[HiW'fb5@&Q>q-TG]u6\"[G\"3%f`sJAe]&4Y%Q4ZmI# G8l3[@)z1<@MD[.G')**3iG1k\"yI#fW\\(4#e5v$*Rpr>%)G]zKQVEYQKz*e`EfTJ!$4# " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"cw?QI\\&*[HiW'fb5@&Q>q-TG]u6\"[ G\"3%f`sJAe]&4Y%Q4ZmI#G8l3[@)z1<@MD[.G')**3iG1k\"yI#fW\\(4#e5v$*Rpr>%) G]zKQVEYQKz*e`EfTJ!$*>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "This value for " }{XPPEDIT 18 0 "Pi" "6#%#PiG" } {TEXT -1 27 " is correct to 200 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(Pi,201);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#$\"dw'>QI\\&*[HiW'fb5@&Q>q-TG]u6\"[G\" 3%f`sJAe]&4Y%Q4ZmI#G8l3[@)z1<@MD[.G')**3iG1k\"yI#fW\\(4#e5v$*Rpr>%)G]z KQVEYQKz*e`EfTJ!$+#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }} {PARA 0 "" 0 "" {TEXT -1 10 "Calculate " }{XPPEDIT 18 0 "Pi" "6#%#PiG " }{TEXT -1 32 " to 16 digits using the formula " }{XPPEDIT 18 0 "Int( sqrt(x-x^2),x=0..1/2-sqrt(2)/4)=Pi/32-1/16" "6#/-%$IntG6$-%%sqrtG6#,&% \"xG\"\"\"*$F+\"\"#!\"\"/F+;\"\"!,&*&F,F,F.F/F,*&-F(6#F.F,\"\"%F/F/,&* &%#PiGF,\"#KF/F,*&F,F,\"#;F/F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }} {PARA 0 "" 0 "" {TEXT -1 10 "Calculate " }{XPPEDIT 18 0 "Pi" "6#%#PiG " }{TEXT -1 34 " to 200 digits using the formula: " }{XPPEDIT 18 0 "ar ctan(1/4)+arctan(2/9)+arctan(1/6)+arctan(3/19) = Pi/4" "6#/,*-%'arctan G6#*&\"\"\"F)\"\"%!\"\"F)-F&6#*&\"\"#F)\"\"*F+F)-F&6#*&F)F)\"\"'F+F)-F &6#*&\"\"$F)\"#>F+F)*&%#PiGF)F*F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pictures" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 29 "code for semi-circle picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 639 "d := evalf(sqrt(3 )/4):\np1 := plot([cos(t)*.5+.5,sin(t)*.5,t=0..Pi],color=red):\ncv := \+ evalf(seq([cos(Pi*i/60)*.5+.5,sin(Pi*i/60)*.5],i=40..60)):\np2 := plot s[polygonplot]([cv,[.25,0]],\n style=patchnogrid,color=COLOR(RG B,.9,.85,.85)):\np3 := plot([[.25,0],[.25,d]],color=brown):\np4 := plo t([[.25,.03],[.22,.03],[.22,0]],color=COLOR(RGB,.4,.4,.4)):\nt1 := plo ts[textplot]([[1.09,-.02,`x`],[-0.02,.59,`y`],\n [0.28,.04,`B`],[0.23, .46,`D`],[-0.03,.04,`A`],\n [1.03,.04,`E`]],font=[HELVETICA,9]):\npl ots[display]([p1,p2,p3,p4,t1],\nxtickmarks=[.25=`1/4`,.5=`1/2`,1=`1`], ytickmarks=[0.5=`1/2`],\n scaling=constrained,view=[-.05..1.1,-.05..0. 6]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 685 "d := evalf(sqrt(3)/4):\np1 := plot([cos(t)*.5+.5,sin (t)*.5,t=0..Pi],color=red):\ncv := evalf(seq([cos(Pi*i/60)*.5+.5,sin(P i*i/60)*.5],i=40..60)):\np2 := plots[polygonplot]([cv,[.25,0]],\n \+ style=patchnogrid,color=COLOR(RGB,.9,.85,.85)):\np3 := plot([[[.25, 0],[.25,d]],\n [[.25,d],[.5,0]]],linestyle=[1,2],color=brown): \np4 := plot([[.25,.03],[.22,.03],[.22,0]],color=COLOR(RGB,.4,.4,.4)): \nt1 := plots[textplot]([[1.09,-.02,`x`],[-0.02,.59,`y`],\n [0.28,.04, `B`],[0.23,.46,`D`],[-0.03,.04,`A`],[0.52,.04,`C`]],\n font=[HELVE TICA,9]):\nplots[display]([p1,p2,p3,p4,t1],\nxtickmarks=[.25=`1/4`,.5= `1/2`,1=`1`],ytickmarks=[0.5=`1/2`],\n scaling=constrained,view=[-.05. .1.1,-.05..0.6]);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 740 "d := eval f(sqrt(3)/4):\np1 := plot([cos(t)*.5+.5,sin(t)*.5,t=0..Pi],color=red): \ncv := evalf(seq([cos(Pi*i/60)*.5+.5,sin(Pi*i/60)*.5],i=40..60)):\np2 := plots[polygonplot]([cv,[.25,0]],\n style=patchnogrid,color= COLOR(RGB,.9,.85,.85)):\np3 := plot([[[.25,0],[.25,d]],\n [[.25 ,d],[.5,0]]],linestyle=[1,2],color=brown):\np4 := plot([[.25,.03],[.28 ,.03],[.28,0]],color=brown):\nt1 := plots[textplot]([[1.09,-.02,`x`],[ -0.02,.56,`y`],\n [0.22,.04,`B`],[0.23,.46,`D`],[0.52,.04,`C`],\n \+ [-0.03,.04,`A`],[1.03,.04,`E`],\n [.25,-.03,`(1/4,0)`],[.5,-.03,`(1 /2,0)`],\n [1,-.03,`(1,0)`]],font=[HELVETICA,9],color=COLOR(RGB,.01 ,.01,.01)):\nplots[display]([p1,p2,p3,p4,t1],tickmarks=[0,0],\n scal ing=constrained,view=[-.05..1.1,-.05..0.57]);" }}}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }