{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 28 "Graphing I terative Sequences" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Here's how to graph iterative sequences defined by " } {XPPEDIT 18 0 "x[n] := F(x[n-1]);" "6#>&%\"xG6#%\"nG-%\"FG6#&%\"xG6#,& %\"nG\"\"\"F0!\"\"" }{TEXT -1 26 ". Load the plots package:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Enter your function (this is the doubling function)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "F := x -> fr ac(2*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)oper atorG%&arrowGF(-%%fracG6#,$9$\"\"#F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Choose appropriate starting and ending points for evaluat ion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "start := 0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&startG\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "finish := 50;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'finishG\"#]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "S(x[0]);" "6#-%\"SG6#&%\"xG6#\"\"!" }{TEXT -1 22 " \+ creates the sequence " }{XPPEDIT 18 0 "x[0],x[1],x[2],x[3];" "6&&%\"xG 6#\"\"!&F$6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 32 ", ... by iterati ng the function " }{XPPEDIT 18 0 "F;" "6#%\"FG" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "S := seed -> seq((F@@n)(seed ),n=start..finish):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "You can se e the first 10 values of " }{XPPEDIT 18 0 "S(x[0]);" "6#-%\"SG6#&%\"xG 6#\"\"!" }{TEXT -1 19 " for some value of " }{XPPEDIT 18 0 "x[0];" "6# &%\"xG6#\"\"!" }{TEXT -1 6 " here:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "S( 1/3 )[1..10];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,# \"\"\"\"\"$#\"\"#F%F#F&F#F&F#F&F#F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "To use floating-point, rather than exact values, type " } {XPPEDIT 18 0 "evalf(x[0]);" "6#-%&evalfG6#&%\"xG6#\"\"!" }{TEXT -1 12 " instead of " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "S( evalf(1/3) )[1.. 10];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,$\"+LLLLL!#5$\"+mmmmmF%$\"*LLL L$!\"*$\"*mmmm'F*$\"*KLLL$F*$\"*kmmm'F*$\"*GLLL$F*$\"*cmmm'F*$\"*7LLL$ F*$\"*Cmmm'F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Now let's compar e the two orbits by graphing them. First the " }{XPPEDIT 18 0 "pts;" "6#%$ptsG" }{TEXT -1 58 " function generates a list of ordered pairs t o be plotted." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "pts := see d -> [seq([n, (F@@n)(seed)],n=start..finish)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Here's the first five pairs:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "pts( 1/3 )[1..5];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'7$\"\"!#\"\"\"\"\"$7$F'#\"\"#F(7$F+F&7$F(F*7$\"\"%F& " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Type a list using as many dif ferent seeds as you want:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "plot(\{ pts(1/3), pts(evalf(1/3)) \}, color=black, linestyle=[1, 2, 3, 4]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'- %'CURVESG6$7U7$$\"\"!F)$\"39LLLLLLLL!#=7$$\"\"\"F)$\"3HmmmmmmmmF,7$$\" \"#F)F*7$$\"\"$F)F07$$\"\"%F)F*7$$\"\"&F)F07$$\"\"'F)F*7$$\"\"(F)F07$$ \"\")F)F*7$$\"\"*F)F07$$\"#5F)F*7$$\"#6F)F07$$\"#7F)F*7$$\"#8F)F07$$\" #9F)F*7$$\"#:F)F07$$\"#;F)F*7$$\"#F)F07$$\"#?F) F*7$$\"#@F)F07$$\"#AF)F*7$$\"#BF)F07$$\"#CF)F*7$$\"#DF)F07$$\"#EF)F*7$ $\"#FF)F07$$\"#GF)F*7$$\"#HF)F07$$\"#IF)F*7$$\"#JF)F07$$\"#KF)F*7$$\"# LF)F07$$\"#MF)F*7$$\"#NF)F07$$\"#OF)F*7$$\"#PF)F07$$\"#QF)F*7$$\"#RF)F 07$$\"#SF)F*7$$\"#TF)F07$$\"#UF)F*7$$\"#VF)F07$$\"#WF)F*7$$\"#XF)F07$$ \"#YF)F*7$$\"#ZF)F07$$\"#[F)F*7$$\"#\\F)F07$$\"#]F)F*-%*LINESTYLEG6#F/ -F$6$7U7$F($\"3y*****HLLLL$F,7$F.$\"3d*****fmmmm'F,7$F3$\"35+++ILLLLF, 7$F6$\"3>+++gmmmmF,7$F9$\"3#)******>LLLLF,7$F<$\"3k******RmmmmF,7$F?$ \"3%)******zKLLLF,7$FB$\"3o******flmmmF,7$FE$\"3#*******>JLLLF,7$FH$\" 3&)******RimmmF,7$FK$\"3E+++![KLL$F,7$FN$\"3_+++g\\mmmF,7$FQ$\"3%***** **>*HLL$F,7$FT$\"3*)******R)fmm'F,7$FW$\"3y******z'>LL$F,7$FZ$\"3d**** **f$Rmm'F,7$Fgn$\"3D+++?(yKL$F,7$Fjn$\"3]+++SubmmF,7$F]o$\"3*)******z[ 6LLF,7$F`o$\"3y******f(Him'F,7$Fco$\"37+++?&fCL$F,7$Ffo$\"3C+++S!>\\m' F,7$Fio$\"3#*******z!Q)HLF,7$F\\p$\"3%)******fhnfmF,7$F_p$\"3C+++?BN>L F,7$Fbp$\"3\\+++SYqQmF,7$Fep$\"3()******z#4uF$F,7$Fhp$\"3u******f&=[b' F,7$F[q$\"3.+++?rj4JF,7$F^q$\"31+++SUF>iF,7$Faq$\"38+++![[&QCF,7$Fdq$ \"3E+++gp4x[F,7$Fgq$\"3^+++?R>a(*F,7$Fjq$\"3#*******RyQ3&*F,7$F]r$\"3& )******zcx;!*F,7$F`r$\"3q******f8bL!)F,7$Fcr$\"3]+++?F5ngF,7$Ffr$\"3!* ******Ra?M@F,7$Fir$\"3z******z3ToUF,7$F\\s$\"3f******f<#o`)F,7$F_s$\"3 H+++?NktqF,7$Fbs$\"3.+++SqGZTF,7$Fes$\"31+++!3uXH)F,7$Fhs$\"39+++g\"[ \"*e'F,7$F[t$\"3G+++?jHyJF,7$F^t$\"3d+++SEfcjF,7$Fat$\"3.+++!G&=8FF,7$ Fdt$\"31+++g0PEaF,7$Fgt$\"3#********>6u_)!#>7$Fjt$\"3E+++SA[0 " 0 "" {MPLTEXT 1 0 74 "plot(\{ pts(5/(7*128)), pts(3/7) \}, color=black , linestyle=[1, 2, 3, 4]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7U7$$\"\"!F)$\"3Z&G9dG9dG%!#=7$$\"\"\"F)$\" 3%4dG9dG9d)F,7$$\"\"#F)$\"3,Vr&G9dG9(F,7$$\"\"$F)F*7$$\"\"%F)F07$$\"\" &F)F57$$\"\"'F)F*7$$\"\"(F)F07$$\"\")F)F57$$\"\"*F)F*7$$\"#5F)F07$$\"# 6F)F57$$\"#7F)F*7$$\"#8F)F07$$\"#9F)F57$$\"#:F)F*7$$\"#;F)F07$$\"#F)F07$$\"#?F)F57$$\"#@F)F*7$$\"#AF)F07$$\"#BF)F57$$ \"#CF)F*7$$\"#DF)F07$$\"#EF)F57$$\"#FF)F*7$$\"#GF)F07$$\"#HF)F57$$\"#I F)F*7$$\"#JF)F07$$\"#KF)F57$$\"#LF)F*7$$\"#MF)F07$$\"#NF)F57$$\"#OF)F* 7$$\"#PF)F07$$\"#QF)F57$$\"#RF)F*7$$\"#SF)F07$$\"#TF)F57$$\"#UF)F*7$$ \"#VF)F07$$\"#WF)F57$$\"#XF)F*7$$\"#YF)F07$$\"#ZF)F57$$\"#[F)F*7$$\"# \\F)F07$$\"#]F)F5-%*LINESTYLEG6#F/-F$6$7U7$F($\"3)H9dG9d.e&!#?7$F.$\"3 gG9dG92;6!#>7$F3$\"3>dG9dG9KAF^v7$F8$\"3Q9dG9dGkWF^v7$F;$\"3xG9dG9dG*) F^v7$F>$\"3v&G9dG9dy\"F,7$FA$\"3]r&G9dG9d$F,7$FDF57$FGF*7$FJF07$FMF57$ FPF*7$FSF07$FVF57$FYF*7$FfnF07$FinF57$F\\oF*7$F_oF07$FboF57$FeoF*7$Fho F07$F[pF57$F^pF*7$FapF07$FdpF57$FgpF*7$FjpF07$F]qF57$F`qF*7$FcqF07$Ffq F57$FiqF*7$F\\rF07$F_rF57$FbrF*7$FerF07$FhrF57$F[sF*7$F^sF07$FasF57$Fd sF*7$FgsF07$FjsF57$F]tF*7$F`tF07$FctF57$FftF*7$FitF07$F\\uF57$F_uF*-Fb u6#F4-%'COLOURG6&%$RGBGF)F)F)-%+AXESLABELSG6$Q!6\"Fcz-%%VIEWG6$%(DEFAU LTGFhz" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot(\{ p ts(Pi-3) \}, color=black, linestyle=[1, 2, 3, 4]);" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$\"\"!F)$\"3;Jz *e`EfT\"!#=7$$\"\"\"F)$\"3KiezrI&=$GF,7$$\"\"#F)$\"3jC'!#>7$$\"\"(F)$\"3[)=N\\f'fQ7F,7$$\"\")F )$\"3&pPq)*=$>xCF,7$$\"\"*F)$\"3\"RvS(zjQa\\F,7$$\"#5F)$\"3#y]\"[fFx3* *F,7$$\"#6F)$\"3k:I'*=ba<)*F,7$$\"#7F)$\"3GJg#z.\"4N'*F,7$$\"#8F)$\"3b i?&e2#=q#*F,7$$\"#9F)$\"35DTq^TOS&)F,7$$\"#:F)$\"3?]#3MIG23(F,7$$\"#;F )$\"3U+l\"ogc9;%F,7$$\"#F)$\"3S.?`aGl\"H$F,7$$\"#?F)$\"3!o+k!4dI$e'F,7$$\"#@F)$\"3g8!G\"= 9hmJF,7$$\"#AF)$\"3AFgDOGALjF,7$$\"#BF)$\"3Wa?^scWmEF,7$$\"#CF)$\"3))3 T-X8*GL&F,7$$\"#DF)$\"3ux@[+p#yl'FK7$$\"#EF)$\"3aNk4!Ql:L\"F,7$$\"#FF) $\"34rG>g28jEF,7$$\"#GF)$\"3=UdQ?:EE`F,7$$\"#HF)$\"3vV[r2/BDlFK7$$\"#I F)$\"3voHa\"3Y]I\"F,7$$\"#JF)$\"3]Pf3j@45EF,7$$\"#KF)$\"3+v= " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "20 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }