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}{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Graphs of implicit functions" }} {PARA 0 "" 0 "" {TEXT -1 58 "by Peter Stone, School of Life and Physic al Sciences, RMIT" }}{PARA 0 "" 0 "" {TEXT -1 23 "peter.stone@rmit.edu .au" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Ve rsion: 16.8.2004" }}{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 34 "Graphs of some impli cit functions " }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 12 "implicitplot" }{TEXT -1 14 " in the Maple " }{TEXT 0 5 "plots" } {TEXT -1 82 " package cab be used to plot the graph of an equation inv olving the two variables " }{TEXT 295 1 "x" }{TEXT -1 5 " and " } {TEXT 296 1 "y" }{TEXT -1 21 " without solving for " }{TEXT 271 1 "y" }{TEXT -1 13 " in terms of " }{TEXT 272 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "Here are some examples." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F( F(" }{TEXT -1 13 " ------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 73 "The graph is a circle with its centre at \+ the origin, and having radius 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "plots[implicitplot](x^2+y^2= 1,x=-1.1..1.1,y=-1.1..1.1,\nscaling=constrained);" }}{PARA 13 "" 1 "" 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be drawn in two pieces using " }{TEXT 0 4 "p lot" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot([sqrt(1-x^2),-sqrt(1-x^2)],x=-1..1,t hickness=2,\nscaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6$7_o7$$!\"\"\"\"!$F*F*7$$!3-n;HdNvs* *!#=$\"3aIk@d'**oP(!#>7$$!3/MLe9r]X**F/$\"3^xD,?$RD/\"F/7$$!3/,](=ng#= **F/$\"3W*\\#GuA(fF\"F/7$$!3%pmm\"HU,\"*)*F/$\"3=cuB4yNs9F/7$$!3()*** \\PM@l$)*F/$\"3gMpy'*3z+=F/7$$!3!RLL$e%G?y*F/$\"3iA8`Ih^w?F/7$$!3u**** \\(oUIn*F/$\"3nSQYJr=ODF/7$$!3ommm;p0k&*F/$\"3Y%G&e[NT?HF/7$$!3E++vV5S u$*F/$\"3'eGkq$fY\"[$F/7$$!3wKL$3(eMbF/7$$!3\"QLL3i.9!zF/$\"3^7**H-%f#HhF/7$$!3\"ommT !R=0vF/$\"3^qi^Nc\\3mF/7$$!3u****\\P8#\\4(F/$\"3ifl'faLr/(F/7$$!3+nm;/ siqmF/$\"37dnjs%HaF/$\"3e&R<6BowR)F/7$$!3Q+++]$*4)* \\F/$\"3]Y1e+6Nh')F/7$$!39+++]_&\\c%F/$\"3S:_+\"3ds*))F/7$$!31+++]1aZT F/$\"3i'GE3]N$*4*F/7$$!3umm;/#)[oPF/$\"33y-$=zZFE*F/7$$!3hLLL$=exJ$F/$ \"3:IF>)f#eL%*F/7$$!3*RLLLtIf$HF/$\"3kU7=H\\If&*F/7$$!3]++]PYx\"\\#F/$ \"3)G16\">%yXo*F/7$$!3EMLLL7i)4#F/$\"3i8>'>!*4tx*F/7$$!3c****\\P'psm\" F/$\"35Fh>K5.g)*F/7$$!3')****\\74_c7F/$\"3JpBi\"oV2#**F/7$$!3)3LLL3x%z #)F2$\"3%4o%3*=mc'**F/7$$!3KMLL3s$QM%F2$\"3#oX(\\%3h0***F/7$$!3]^omm;z r)*!#@$\"2sF&QF^******!#<7$$\"3%pJL$ezw5VF2$\"3)*Q2K>Vq!***F/7$$\"3s*) ***\\PQ#\\\")F2$\"3s^4RW'Rn'**F/7$$\"3GKLLe\"*[H7F/$\"3+'Q!p2+8C**F/7$ $\"3I*******pvxl\"F/$\"3]s:1(pJ;')*F/7$$\"3#z****\\_qn2#F/$\"3;GKN_W(> y*F/7$$\"3U)***\\i&p@[#F/$\"3!QaP4mWqo*F/7$$\"3B)****\\2'HKHF/$\"3+\\t ky/Ug&*F/7$$\"3ElmmmZvOLF/$\"3%e0ez0!)oU*F/7$$\"3i******\\2goPF/$\"3wC !*\\-?qi#*F/7$$\"3UKL$eR<*fTF/$\"3CILE@Ro$4*F/7$$\"3m******\\)Hxe%F/$ \"3-\"f%eX^`&)))F/7$$\"3ckm;H!o-*\\F/$\"35DIn2b'em)F/7$$\"3y)***\\7k.6 aF/$\"3a'=*zS\"f&4%)F/7$$\"3#emmmT9C#eF/$\"3H&p3sIf,8)F/7$$\"33****\\i !*3`iF/$\"3;c[JlKx.yF/7$$\"3%QLLL$*zym'F/$\"3o4V`oXZ_uF/7$$\"3wKLL3N1# 4(F/$\"3!Q)Q?)e4+0(F/7$$\"3Nmm;HYt7vF/$\"3+_Th$[5**f'F/7$$\"3Y*******p (G**yF/$\"3_pyiCh)>8'F/7$$\"3]mmmT6KU$)F/$\"3I@,y!QMT^&F/7$$\"3fKLLLbd Q()F/$\"3e&)*oZS?='[F/7$$\"3amm\"zW?)\\*)F/$\"3ud/FE;-hWF/7$$\"3[++]i` 1h\"*F/$\"3$*pj$>D\\$4SF/7$$\"3Y++++PDj$*F/$\"3v!)RQ\\fL6NF/7$$\"3W++] P?Wl&*F/$\"3:\"G)fhN(e\"HF/7$$\"3K+]7G:3u'*F/$\"3fG;Bq;AKDF/7$$\"3A++v =5s#y*F/$\"37vO>U4Dt?F/7$$\"3;+D1k2/P)*F/$\"3mLW(eL^zz\"F/7$$\"35+]P40 O\"*)*F/$\"31H.f%oH+Z\"F/7$$\"3k]7.#Q?&=**F/$\"3G#>z^*=&RF\"F/7$$\"31+ voa-oX**F/$\"3(pV\\8H')3/\"F/7$$\"3[\\PMF,%G(**F/$\"3?&=hJ*p=ltF27$$\" \"\"F*F+-%'COLOURG6&%$RGBG$\"#5F)F+F+-F$6$7_oF'7$F-$!3aIk@d'**oP(F27$F 4$!3^xD,?$RD/\"F/7$F9$!3W*\\#GuA(fF\"F/7$F>$!3=cuB4yNs9F/7$FC$!3gMpy'* 3z+=F/7$FH$!3iA8`Ih^w?F/7$FM$!3nSQYJr=ODF/7$FR$!3Y%G&e[NT?HF/7$FW$!3'e Gkq$fY\"[$F/7$Ffn$!3u)R\"3d;![&RF/7$F[o$!36'pC*=pM(eMbF/7$Fjo$!3^7**H-%f#HhF/7$F_p$!3^qi^Nc\\3mF/7$Fdp $!3ifl'faLr/(F/7$Fip$!37dnj)f#eL%*F/7$Ffs$!3k U7=H\\If&*F/7$F[t$!3)G16\">%yXo*F/7$F`t$!3i8>'>!*4tx*F/7$Fet$!35Fh>K5. g)*F/7$Fjt$!3JpBi\"oV2#**F/7$F_u$!3%4o%3*=mc'**F/7$Fdu$!3#oX(\\%3h0*** F/7$Fiu$!2sF&QF^******F^v7$F`v$!3)*Q2K>Vq!***F/7$Fev$!3s^4RW'Rn'**F/7$ Fjv$!3+'Q!p2+8C**F/7$F_w$!3]s:1(pJ;')*F/7$Fdw$!3;GKN_W(>y*F/7$Fiw$!3!Q aP4mWqo*F/7$F^x$!3+\\tky/Ug&*F/7$Fcx$!3%e0ez0!)oU*F/7$Fhx$!3wC!*\\-?qi #*F/7$F]y$!3CILE@Ro$4*F/7$Fby$!3-\"f%eX^`&)))F/7$Fgy$!35DIn2b'em)F/7$F \\z$!3a'=*zS\"f&4%)F/7$Faz$!3H&p3sIf,8)F/7$Ffz$!3;c[JlKx.yF/7$F[[l$!3o 4V`oXZ_uF/7$F`[l$!3!Q)Q?)e4+0(F/7$Fe[l$!3+_Th$[5**f'F/7$Fj[l$!3_pyiCh) >8'F/7$F_\\l$!3I@,y!QMT^&F/7$Fd\\l$!3e&)*oZS?='[F/7$Fi\\l$!3ud/FE;-hWF /7$F^]l$!3$*pj$>D\\$4SF/7$Fc]l$!3v!)RQ\\fL6NF/7$Fh]l$!3:\"G)fhN(e\"HF/ 7$F]^l$!3fG;Bq;AKDF/7$Fb^l$!37vO>U4Dt?F/7$Fg^l$!3mLW(eL^zz\"F/7$F\\_l$ !31H.f%oH+Z\"F/7$Fa_l$!3G#>z^*=&RF\"F/7$Ff_l$!3(pV\\8H')3/\"F/7$F[`l$! 3?&=hJ*p=ltF2F_`l-Fc`l6&Fe`lF+Ff`lF+-%(SCALINGG6#%,CONSTRAINEDG-%+AXES LABELSG6$Q\"x6\"Q!6\"-%*THICKNESSG6#\"\"#-%%VIEWG6$;F(F``l%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We tend to get better looking graphs with " }{TEXT 0 4 "plot" }{TEXT -1 11 " than with " }{TEXT 0 12 "implicitplot" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT 0 12 "implicitplot" }{TEXT -1 89 " works by find ing where equally spaced vertical and horizontal grid lines meet the c urve." }}{PARA 0 "" 0 "" {TEXT -1 141 "The default grid uses 25 horizo ntal and 25 vertical lines. You can increase the number of grid lines, but this doesn't always help very much." }}{PARA 0 "" 0 "" {TEXT -1 95 "As an illustration, the following commands draw the grid along wit h the \"curve\" obtained using " }{TEXT 0 12 "implicitplot" }{TEXT -1 20 " with a 5 by 5 grid." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 340 "curve := plots[implicitplot](x^2+y ^2=1,x=-1..1,y=-1..1,grid=[5,5]):\nvert := plot([[[-1,-1],[-1,1]],[[-. 5,-1],[-.5,1]],[[.5,-1],[.5,1]],\n[[1,-1],[1,1]]],color=blue):\nhoriz \+ := plot([[[-1,-1],[1,-1]],[[-1,-.5],[1,-.5]],[[-1,.5],[1,.5]],\n[[-1,1 ],[1,1]]],color=black):\nplots[display](\{curve,vert,horiz\},tickmarks =[3,3],\n scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6/-%'CURVESG657$7$$!3++++++++]!#=$!3fKLLLLLL$)F*7 $F+F(7$7$$!3qLLLLLLL$)F*F(7$$!\"\"\"\"!$F5F57$F27$$!3+++++++]()F*$\"3+ ++++++]PF*7$7$F+$\"3++++++++]F*F87$7$F0F?7$$!3SnmmmmmmmF*$\"3Snmmmmmmm F*7$7$F($\"3qLLLLLLL$)F*FC7$7$F6F37$F(F07$7$F($\"3fKLLLLLL$)F*7$$!3+++ ++++]PF*$\"3+++++++]()F*7$7$F6$\"\"\"F5FS7$FM7$F;F97$7$F?F+Fgn7$7$F?FJ FY7$7$F?F07$FFFD7$7$FJF(F^o7$7$FQF(7$FVFT7$7$FZF6Fco7$Feo7$FJF?7$7$FQF ?7$F?FQ-%'COLOURG6&%$RGBGFenF5F5-F$6$7$7$F3F37$FZF3-F\\p6&F^pF5F5F5-F$ 6$7$7$F3F(7$FZF(Fdp-F$6$7$7$F3F?7$FZF?Fdp-F$6$7$7$F3FZ7$FZFZFdp-F$6$7$ FbpFcq-F\\p6&F^pF6F6$\"*++++\"!\")-F$6$7$7$F(F37$F(FZFhq-F$6$7$7$F?F37 $F?FZFhq-F$6$7$FcpFdqFhq-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%%\" xG%\"yG%(DEFAULTG-%*AXESTICKSG6$\"\"$Fgs-%%VIEWG6$FcsFcs" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "On the ot her hand " }{TEXT 0 4 "plot" }{TEXT -1 140 " does not use equally spac ed points, but decides how to space the points so that more points are used when the function is changing rapidly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y^2-2*x*y-1 = 0;" "6#/,(*$%\"yG\"\"#\"\"\"*(F'F( %\"xGF(F&F(!\"\"F(F+\"\"!" }{TEXT -1 14 " ------- (ii)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plots[i mplicitplot](y^2-2*x*y-1=0,x=-2..2,y=-4..4);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6$-%'CURVESG6]t7$7$$!3+++++++v=!#<$ !\"%\"\"!7$$!3gmmmm\"H#e=F*$!3'pmmmmTN'RF*7$7$$!33++++++S=F*$!3;++++++ IRF*F.7$7$$!\"#F-$\"3x8Cp#F?Fbo7$7$$! 3%*************z;F*$!3.$yM/8RR<_cpe\\$F*7$7$ $!3Ww/>w/>J:F*$!3K++++++gLF*Fap7$Fgp7$$!34++++](e_\"F*$!3i+++++D[LF*7$ 7$$!3C++++++?:F*$!3$z/>w/>wL$F*F]q7$Fho7$$!3iYQ:YQly:F*$\"3aAp2Bp2tFF? 7$7$Fdq$\"3+R<_cp3EHF?Fiq7$Fcq7$$!3QQ_4Q_fX9F*$!3?C&4Q_4))=$F*7$7$$!3K ++++++g8F*$!3CEyM/8RZIF*Fcr7$7$$!3Y++++++?:F*$\"3XQ<_cp3EHF?7$$!3vLLLL $3-V\"F*$\"3almmmm;/IF?7$7$Fjr$\"3f.>w/>w/KF?Fds7$Fir7$$!37#====o!e8F* $!3)ojjjjjQ/$F*7$7$$!39u%*y:j_b8F*$!3[++++++SIF*F^t7$Fdt7$$!3EPZ*y:jxF \"F*$!3=F0@%otW)GF*7$7$$!3Q+++++++7F*$!3yH9dG9deFF*Fjt7$7$Fjr$\"3:/>w/ >w/KF?7$$!35POOOO'QG\"F*$\"3!fssssssF$F?7$7$Fau$\"3k$*y:j_5UNF?Fiu7$F` u7$$!3Y++++]()*=\"F*$!3_+++++DSFF*7$7$$!3gN#)eqk5F*$!3lyxxxxFSCF*7$7$$!3^smmm mm;**F?$!3y+++++++CF*Fhx7$7$$!3Srmmmmm;**F?$!3M+++++++CF*7$$!3gRLLLLLe $*F?$!3WMLLLLL)G#F*7$7$$!3e0++++++))F?$!3H*eqkF*F`\\l7$F[[l7$$!312+++]7`')F?$\"3())********\\i]%F?7$7$$!3E &ommmmm,)F?$\"3;$************z%F?F\\]l7$Fb]l7$$!3]UOOOOO6tF?$\"3#)psss ssA]F?7$7$Fg\\l$\"3_Ww6%HN#)3&F?Fh]l7$Ff\\l7$$!3d/++++]7nF?$!3#3+++++v &=F*7$7$$!3)\\\"44444ffF?$!3!4++++++w\"F*Fb^l7$Fh^l7$$!3UfaaaaazdF?$!3 y444444CF?$!36YXXXXX+7F*7$7$$!3n#H9dG9d8\"F?$!3x++++++? 6F*Fhdl7$F^el7$$!3O\\++++vo%*!#>$!3S++++]i!4\"F*7$7$$!3)[.++++++)Fgel$ !3gFFFFFFx5F*Fdel7$Fadl7$$!3Ew&G9dGkM#F?$\"3ARr&G9dG*yF?7$7$$!3a2+++++ ]AF?$\"3w$*************zF?Fafl7$Fgfl7$$!3O/++++]i6F?$\"3)\\********** \\s)F?7$7$F\\fl$\"3(=h%Q:YQ:\"*F?F]gl7$F[fl7$$\"3@K6F*Fi jl7$7$$\"3:\"G9dG9d8\"F?Fb[m7$$\"3yosssss(*>F?$\"3+XXXXXX+7F*7$7$Fdjl$ \"3uXQ:YQ:c7F*Fh[m7$Fcjl7$$\"3o%*******\\7`MF?$!3m-++++D1pF?7$7$$\"3)o *************RF?$!3ubh%Q:YQl'F?Fb\\m7$F^\\m7$$\"3E'*********\\(*HF?$\" 3c**********\\?8F*7$7$$\"3?txxxxxFPF?$\"3]************R9F*F^]m7$7$$\"3 vtxxxxxFPF?Fg]m7$$\"3mF#p2Bpd!RF?$\"3C`h%Q:Y)e9F*7$7$Fi\\m$\"37mmmmmms 9F*F]^m7$Fh\\m7$$\"3SRWWWWWpYF?$!3]\"*))))))))))QhF?7$7$$\"35'******** ****f&F?$!3%)ommmmmmdF?Fg^m7$Fc^m7$$\"3-kmmmm\"z*[F?$\"39mmmmmT!e\"F*7 $7$F^_m$\"3ko2Bp2B*p\"F*Fc_m7$7$$\"3?(************f&F?F`_m7$$\"3-&**** *****\\ifF?$!3C.+++++DbF?7$7$$\"3_(************>(F?$!3'*[w6%HN#)3&F?F` `m7$7$F^`m$\"3')o2Bp2B*p\"F*7$$\"3m^aaaaazdF?$\"3n344444C%>F*F[bm7$Ff`m7$$\"3SJOOOOO6tF ?$!3PvsssssA]F?7$7$$\"3^Zmmmmm;!)F?$!3[1++++++[F?Febm7$7$F\\cm$!3/2+++ +++[F?7$$\"3'[*******\\7`')F?$!34,++++D1XF?7$7$$\"3!y************z)F?$ !3emmmmmm'[%F?Fdcm7$Fabm7$$\"3%=#p2Bp2)f(F?$\"3w9YQ:YQ+?F*7$7$$\"3%R%Q :YQ:'*zF?$\"3i************z?F*F`dm7$7$$\"3/XQ:YQ:'*zF?Fidm7$$\"3G(**** **\\PMZ)F?$\"3I*******\\7`9#F*7$7$F[dm$\"3'zeqk5F*$\"3KxxxxxFSCF*7$7$F`fm$\"3E% ot%*y:jZ#F*Fagm7$F_fm7$$\"3a*********\\-9\"F*$!3a,+++++0OF?7$7$$\"3s** ***********>\"F*$!3J&*y:j_5UNF?F[hm7$Fggm7$$\"3EFF*Fghm7$7$$\"3#\\B)eqkw/>w/KF?F`jm7$F\\jm7$$\"3QOZ*y:jxF\"F*$\"3SD0@%otW)GF*7$7$$\" 3Ct%*y:j_b8F*$\"3:************RIF*F\\[n7$Fb[n7$$\"3Y\"====o!e8F*$\"3bN OOOO'Q/$F*7$7$Fgjm$\"3MDyM/8RZIF*Fh[n7$Ffjm7$$\"3kKLLL$3-V\"F*$!3wnmmm m;/IF?7$7$$\"3d************>:F*$!36S<_cp3EHF?Fb\\n7$F^\\n7$$\"3sP_4Q_f X9F*$\"3'G_4Q_4))=$F*7$7$Fi\\n$\"3/Z!>w/>wL$F*F^]n7$7$Fi\\n$!3cR<_cp3E HF?7$$\"3IXQ:YQly:F*$!3wCp2Bp2tFF?7$7$$\"3E************z;F*$!31,+++++# p#F?F[^n7$Fd]n7$$\"3V********\\(e_\"F*$\"3G*********\\#[LF*7$7$$\"3bv/ >w/>J:F*$\"3)*)***********fLF*Fg^n7$F]_n7$$\"3$RI\"R<_17;F*$\"37!R<_cp e\\$F*7$7$Fb^n$\"3p\"yM/8RF*$!3)[Ts^l*o?BF?Fabn7$7$Fd`n$\"3$))******** ***HRF*7$$\"3;mmmm\"H#e=F*$\"3?lmmm;ajRF*7$7$$\"3M***********\\(=F*$\" 3m)*************RF*F`cn-%'COLOURG6&%$RGBG\"\"\"F-F--%+AXESLABELSG6$%\" xG%\"yG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "If we substitute a valu e for " }{TEXT 285 1 "x" }{TEXT -1 54 " in equation (ii), we obtain a \+ quadratic equation for " }{TEXT 286 1 "y" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 18 "For example, when " }{XPPEDIT 18 0 "x = 0" "6#/%\" xG\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y^2-1=0" "6#/,&*$%\"yG\"\"# \"\"\"F(!\"\"\"\"!" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "x = 1" "6#/%\" xG\"\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "``-1" "6#,&%!G\"\"\"F%!\" \"" }{TEXT -1 70 ".\nIn general, by completing the square, equation (i i) is equivalent to" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y^2-2*x*y+x^2=1+x^2" "6#/,(*$%\"yG\"\"#\"\"\"*(F'F(%\"xGF(F&F(! \"\"*$F*F'F(,&F(F(*$F*F'F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 11 "which gives" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(y-x)^2=1+x^2" "6#/*$,&%\"yG\"\"\"%\"xG!\"\"\"\"#,&F'F'*$F(F*F'" } {TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "y-x= sqrt(1+x^2)" "6#/,&%\"yG\"\"\"% \"xG!\"\"-%%sqrtG6#,&F&F&*$F'\"\"#F&" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "y-x= -sqrt(1+x^2)" "6#/,&%\"yG\"\"\"%\"xG!\"\",$-%%sqrtG6#,&F&F&*$F '\"\"#F&F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y= x+sqrt(1+x^2)" "6#/%\"yG,&%\"xG\"\"\"-%%sqrtG6#,&F'F '*$F&\"\"#F'F'" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "y= x-sqrt(1+x^2)" " 6#/%\"yG,&%\"xG\"\"\"-%%sqrtG6#,&F'F'*$F&\"\"#F'!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "Ag ain, the graph can be drawn in two pieces using " }{TEXT 0 4 "plot" } {TEXT -1 2 ". 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-%%sqrtG6#,&\"\"\"F'*$%\"xG\"\"#F'" }{TEXT -1 1 " " }{TEXT 262 1 "~" } {TEXT -1 1 " " }{TEXT 264 1 "x" }{TEXT -1 11 ", and when " }{TEXT 268 1 "x" }{TEXT -1 23 " is large and negative " }{XPPEDIT 18 0 "sqrt(1+x^ 2" "6#-%%sqrtG6#,&\"\"\"F'*$%\"xG\"\"#F'" }{TEXT -1 1 " " }{TEXT 263 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-x" "6#,$%\"xG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "I ndeed, as " }{XPPEDIT 18 0 "x -> infinity" "6#f*6#%\"xG7\"6$%)operator G%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x+sqrt(1+x^2)" "6#,&%\"xG\"\"\"-%%sqrt G6#,&F%F%*$F$\"\"#F%F%" }{TEXT -1 4 " -> " }{XPPEDIT 18 0 "2*x" "6#*& \"\"#\"\"\"%\"xGF%" }{TEXT -1 7 " while " }{XPPEDIT 18 0 "x-sqrt(1+x^2 );" "6#,&%\"xG\"\"\"-%%sqrtG6#,&F%F%*$F$\"\"#F%!\"\"" }{TEXT -1 7 " -> 0, " }}{PARA 0 "" 0 "" {TEXT -1 7 "and as " }{XPPEDIT 18 0 "x -> -inf inity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\",$%)infinityG!\"\"F*F*F* " }{TEXT -1 1 "," 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= 0" "6#/%\"xG\"\"!" }{TEXT -1 2 ", " } {TEXT 276 1 "y" }{TEXT -1 15 " can only be 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "xx := 0;\nfs olve(y^3-2*xx*y-1=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "Later on we shall be abl e to show that the left most point on the lower branch of the graph, w here the tangent line is vertical, has coordinates " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 18 ".944940787421155, " }{XPPEDIT 18 0 "y = -0" "6#/%\"yG,$\"\"!!\"\"" }{TEXT -1 32 ".793700525984100, appr oximately." }}{PARA 0 "" 0 "" {TEXT -1 19 "We can ensure that " } {TEXT 0 6 "fsolve" }{TEXT -1 55 " gives a single solution in one of th e three intervals:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 0" "6#/%\"yG\"\"!" }{TEXT -1 5 " to " }{XPPEDIT 18 0 "y = infini ty;" "6#/%\"yG%)infinityG" }{TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = -0" "6#/%\"yG,$\"\"!!\"\"" }{TEXT -1 20 " .793700525984100 to " }{XPPEDIT 18 0 "y = 0" "6#/%\"yG\"\"!" }{TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "y=-infin ity" "6#/%\"yG,$%)infinityG!\"\"" }{TEXT -1 6 " to " }{XPPEDIT 18 0 "y = -0" "6#/%\"yG,$\"\"!!\"\"" }{TEXT -1 17 ".793700525984100," }} {PARA 0 "" 0 "" {TEXT -1 67 "by specifying ranges for the solution as \+ in the following examples." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "xx := 1.5;\nfsolve(y^3-2*xx*y-1=0, y=0..infinity);\nfsolve(y^3-2*xx*y-1=0,y=-.793700525984100..0);\nfsolv e(y^3-2*xx*y-1=0,y=-infinity..-.793700525984100);\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#xxG$\"#:!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+U_Qz=!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+`N'HZ$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+')))3K:!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "xx := 0.945;\nfsolve (y^3-2*xx*y-1=0,y=0..infinity);\nfsolve(y^3-2*xx*y-1=0,y=-.79370052598 4100..0);\nfsolve(y^3-2*xx*y-1=0,y=-infinity..-.793700525984100);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"$X*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4UV(e\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+( 3UV(y!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+++++!)!#5" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 31 ": The last solution is exactly " }{XPPEDIT 18 0 "-4/5" "6#,$*&\"\" %\"\"\"\"\"&!\"\"F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 46 "Th is can be checked by using exact arithmetic." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "subs(y=-4/5,y^3-2* x*y-1=0);\nsolve(%,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "We can draw the graph of equation (iii) in three pie ces by defining three procedures as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 383 "hh := 0.944940787 421155:\nkk := -0.793700525984100:\n# procedure for upper part\nf := p roc(x) fsolve(y^3-2*x*y-1=0,y=0..infinity) end proc:\n# procedure for \+ middle part\ng := proc(x)\n if x>=hh then fsolve(y^3-2*x*y-1=0,y=kk. .0)\n else FAIL end if\nend proc:\n# procedure for lower part\nh := \+ proc(x)\n if x>=hh then fsolve(y^3-2*x*y-1=0,y=-infinity..kk)\n el se FAIL end if\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "xx := 0.945;\nf(xx);\ng(xx); \nh(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"$X*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4UV(e\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+(3UV(y!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+++++ !)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "plot(['f(x)','g(x)','h(x)'],x=-2.5..2.5,y,color=[red, green,blue],\n thickness=2,numpoints=200);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6$7dw7$$!#D!\"\"$\"+X@P 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "We can again use the Maple facility for plotting the graph of a pa rametric function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 111 "plot([y^2/2-1/(2*y),y,y=-4..4],discont=true ,\n color=red,view=[-2.5..2.5,-2.2..2.2],labels=[x,y],thickness=2); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$ 7gn7$$\"3+++++++D\")!#<$!\"%\"\"!7$$\"3Ejwd==$Gy(F*$!3_5x]$Q6G\"RF*7$$ \"3=\"[ic\"3S\"\\(F*$!3!)oF\\M!\\p$QF*7$$\"3sM!=Fo`1<(F*$!3al+$)))Qj^P F*7$$\"3A&*RLZ4FboF*$!3#p#=+>KvlOF*7$$\"3cPXY)))e)[lF*$!310h+D2G!e$F*7 $$\"3/F3j6TWriF*$!3xf7$=yO5]$F*7$$\"3w(eq'y*o4*fF*$!3JJ?moU)*=MF*7$$\" 3!R?OV&F*$!3#G#4+f(Q&\\KF*7$$\"3 Mo'[.D-*e^F*$!3e/;Me4`iJF*7$$\"34k%G_#)*RB\\F*$!3CW:;SW*e3$F*7$$\"3;\" *f\"eZZbm%F*$!3_g2+s)>'**HF*7$$\"3S&=j^J.WT%F*$!3)y,u@0\"*H\"HF*7$$\"3 '\\8EOHn(zTF*$!3k$)4MK\"3&HGF*7$$\"3AEqB$R**H(RF*$!39QfKVwp`FF*7$$\"3@ NOA\\H(\\t$F*$!3]j&R$R;bjEF*7$$\"3%)Qh]Ui-SNF*$!3_%H#\\\\h=(e#F*7$$\"3 (o#>gr.-@LF*$!3Q!H.0$\\N)\\#F*7$$\"3&edm\"Qy;MJF*$!3X=s#)\\Us>CF*7$$\" 3S?xBj#yn$HF*$!3%>4L3$RXLBF*7$$\"3M;CHd(yiv#F*$!3!>R(*f=/8D#F*7$$\"3'* p`C]IxvDF*$!3k([N`a*el@F*7$$\"3&f\"RRs(>rT#F*$!378H*zuwo3#F*7$$\"3u\"H 7=7/PD#F*$!3'[QHBOu>+#F*7$$\"3!3m*p\\#[D4#F*$!31kd+XYy8>F*7$$\"3A.<]RJ \\f>F*$!3cpf#oB:q$=F*7$$\"3r:#QeK$[B=F*$!3#*G^#G<-Tv\"F*7$$\"39]+\"F*7$$\"3'[bE\"*eq&f'*!#=$!3cXTkOs#z<*F_w7$$\"3 K$H]W7iZZ*F_w$!3MEj&*H7#Q\\(F_w7$$ \"30O03TcLB(*F_w$!3'R&[++-CkmF_w7$$\"3S>ln[!Q)G5F*$!3v(e,<0te\"eF_w7$$ \"3%[ip;q\\)G6F*$!3q0wr63`u\\F_w7$$\"3c;OGbBLy7F*$!3\\^rfrYU,UF_w7$$\" 3k=ZfQ34j:F*$!3*4&f.!zd`J$F_w7$$\"3qYVe!=5P,#F*$!3ONwG3!\\G_#F_w7$$\"3 //O/Ig/%*HF*$!3KjUk^$pyn\"F_w7$$\"3IRnBl\">V$RF*$!3U;IXxE\\t7F_w7$$\"3 5'\\sox[nv&F*$!3A&p7$$\"3_Tu92NvswF*$!3IrKYC-P=lFiz7$$\"3)*f P=2()o]6!#;$!3SZ)3jT!eXVFiz7$$\"3PNL4L(yT`\"Fb[l$!3WN;B7b=fKFiz7$$\"3` xO8b<@,BFb[l$!3%QUa\"31zs@Fiz7$$\"3?T3-oPEoIFb[l$!3'y\"e6cJfH;Fiz7$$\" 3FJk!o]!Q-YFb[l$!3)=@xSq&R'3\"Fiz7$$\"3h^OC(G,l8'Fb[l$!3#*)3z0ypz9)!#? 7$$\"3Yd1>3Ru/#*Fb[l$!3.fgQ?D)>V&F^]l7$$\"3%3(y=7e%4%=!#:$!3eHI>g_*fr# F^]l7$$\"3E************\\i!#5$!3]+++++++!)!#E7gn7$$!3E************\\iF ]^l$\"3]+++++++!)F`^l7$$!3Lc15Bl4N=Fg]l$\"3KWSQlAlCFF^]l7$$!3s8Ap'y%\\ v\"*Fb[l$\"3_)3o2`'H\\aF^]l7$$!3jwz.nr*p6'Fb[l$\"39L@:'zSR<)F^]l7$$!3) =&3Nkbu(e%Fb[l$\"3yH:Fb[l$\"39`3Y=RdpKF iz7$$!3-#)4.%[Vo9\"Fb[l$\"3)4Z9YiJ%fVFiz7$$!3CvH`g\">Tk(F*$\"3o1<#p.Z \"RlFiz7$$!3G[RKM@*3t&F*$\"3QU*G#\\C')=()Fiz7$$!3Wm)*\\?EM))RF*$\"3!Rg (***)z>^7F_w7$$!3!*p.\"f)GB`IF*$\"3%QJs]t40j\"F_w7$$!3[y@F*$\"3 CW$*p\">hO[#F_w7$$!3IZI;D-/S9F*$\"3.6F*$\"3; Z*Q*HG>(>%F_w7$$!3a+-icq&fx)F_w$\"3q,uohAj*)\\F_w7$$!3\"e^GcSAx\"pF_w$ \"3i%ozLRd,\"eF_w7$$!3GD#)fZq(>H&F_w$\"3av\"\\$ecuemF_w7$$!37ynDk(3m%Q F_w$\"3-u2***[7Y](F_w7$$!3eCR%o%yfjCF_w$\"3/`Re'\\!pu$)F_w7$$!3)\\b5F! 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We need to solve the equation " } {XPPEDIT 18 0 "dx/dy=0" "6#/*&%#dxG\"\"\"%#dyG!\"\"\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "g := y -> y^2/2-1/(2*y):\n'g(x)'=g(x);\nDiff(g(y),y) =0;\nvalue(%);\nhh := evalf(fsolve(%),18):\nkk := evalf(g(hh),18):\nx= evalf(kk,15);\ny=evalf(hh,15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"gG6#%\"xG,&*&\"\"#!\"\"F'F*\"\"\"*&F,F,*&F*F,F'F,F+F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%%DiffG6$,&*&\"\"#!\"\"%\"yGF)\"\"\"*&F,F,*&F) F,F+F,F*F*F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"yG\"\"\"*&F &F&*&\"\"#F&)F%F)F&!\"\"F&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"xG$\"0b6U(yS\\%*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG$!0+T)f _+Pz!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^4-y^3 = 2;" "6# /,&*$%\"xG\"\"%\"\"\"*$%\"yG\"\"$!\"\"\"\"#" }{TEXT -1 14 " ------- ( iv)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plots[implicitplot](x^4-y^3=2,x=-3..3,y=-1.5..4,grid= [50,50]);" }}{PARA 13 "" 1 "" {GLPLOT2D 486 265 265 {PLOTDATA 2 "6$-%' CURVESG6bz7$7$$!3EEj\"3/-^v#!#<$\"3%>.:Lkaj\"QF*7$$!3))o>BKHG!z#F*$\"3 a73/-^v()QF*7$F-7$$!3>i7G8[cAGF*$\"3#zL+o(ff\\RF*7$7$$!3qwV:$[w)[GF*$ \"3W'*************RF*F37$7$$!3R*[C71`Ej#F*$\"3iL#oRB\"G$e$F*7$$!3aq:Ci =csEF*$\"3]YC71`EjOF*7$FD7$$!3V/l3)pR.r#F*$\"3W$Hzl0yWt$F*7$7$$!3Lm1&z %p5KFF*$\"3_H;3/-^vPF*FJ7$FPF'7$7$$!3__Ej\"3/-^#F*$\"39B&)*z=bDN$F*7$$ !3mM[\"\\w+Pb#F*$\"3czS?5bxQMF*7$Ffn7$$!3mte**HY5'f#F*$\"3wJq`zw^9dG9dG9#F*$\"3'QVW!3RrrEF*7$$!3C#\\DS^V4>#F*$\"3yy*[C71`w#F*7$7$Fhs $\"3My*[C71`w#F*7$$!3W?$=.*G:TAF*$\"3CD@F*$\"3VB%z;>@;j#F*7$7$$!3J\"\\6+3#[J@F*$\"3'[z*[C71`EF *F^v7$Fdv7$Fcs$\"3ULW/3RrrEF*7$7$$!3QoMn$=fz*=F*$\"3u,!Gf%)=;A#F*7$$!3 'Q%\\JGTF*$\"3ZWAhIlK;BF*7$7$$!3kV\\JGTF*Ffw7$$!3g6&ea^*>(*>F*$ \"3&R`[Z$oH2CF*7$7$$!3LRcqoC84?F*$\"3%zUr&G9dGCF*F\\x7$FbxF`u7$7$$!3uJ ;3/-^vF*7$$!3#Q!QVB*)\\C=F*$\"3axQpMn$=4#F*7$F^y7$$! 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-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The equation (v) can be solved for " }{TEXT 288 1 "x" }{TEXT -1 13 " in terms of " }{TEXT 289 1 "y" }{TEXT -1 9 " to give " }{XPPEDIT 18 0 "y= 1/x" "6#/%\"yG*& \"\"\"F&%\"xG!\"\"" }{TEXT -1 5 " or " }{XPPEDIT 18 0 "y=-1/x" "6#/% \"yG,$*&\"\"\"F'%\"xG!\"\"F)" }{TEXT -1 57 ". This enables the graph t o be drawn in two pieces using " }{TEXT 0 4 "plot" }{TEXT -1 2 ". 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" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The \+ curve meets the line " }{XPPEDIT 18 0 "x+y=k" "6#/,&%\"xG\"\"\"%\"yGF& %\"kG" }{TEXT -1 7 " where " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x^3+(-x+k)^3-3*x*(-x+k) = 0" "6#/,(*$%\"xG\"\"$\"\"\"*$ ,&F&!\"\"%\"kGF(F'F(*(F'F(F&F(,&F&F+F,F(F(F+\"\"!" }{TEXT -1 14 ", --- ---- (v) " }}{PARA 0 "" 0 "" {TEXT -1 32 "as can be seen by substituti ng " }{XPPEDIT 18 0 "y = k - x" "6#/%\"yG,&%\"kG\"\"\"%\"xG!\"\"" } {TEXT -1 6 " in " }{XPPEDIT 18 0 "x^3+y^3-3*x*y=0" "6#/,(*$%\"xG\"\" $\"\"\"*$%\"yGF'F(*(F'F(F&F(F*F(!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The quadratic e quation (v) can be solved for " }{TEXT 293 1 "x" }{TEXT -1 13 " in ter ms of " }{TEXT 294 1 "k" }{TEXT -1 9 " giving: " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = (3+3*k+sqrt(9+6*k-3*k^2))*k/(2*(3*k +3));" "6#/%\"xG*(,(\"\"$\"\"\"*&F'F(%\"kGF(F(-%%sqrtG6#,(\"\"*F(*&\" \"'F(%\"kGF(F(*&F'F(*$)F2\"\"#F(F(!\"\"F(F(F*F(*&F6F(,&*&F'F(F*F(F(F'F (F(F7" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "x=(3+3*k-sqrt(9+6*k-3*k^2 ))*k/(2*(3*k+3))" "6#/%\"xG*(,(\"\"$\"\"\"*&F'F(%\"kGF(F(-%%sqrtG6#,( \"\"*F(*&\"\"'F(F*F(F(*&F'F(*$F*\"\"#F(!\"\"F5F(F*F(*&F4F(,&*&F'F(F*F( F(F'F(F(F5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "su bs(y=k-x,x^3+y^3-3*x*y=0);\nsimplify(%);\ncollect(%,x);\nop(map(_U->x= _U,[solve(%,x)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"$ \"\"\"F)*$),&%\"kGF)F'!\"\"F(F)F)*(F(F)F'F)F,F)F.\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*$)%\"kG\"\"$\"\"\"F)*(F(F))F'\"\"#F)%\"xGF)!\" \"*(F(F)F'F))F-F,F)F)*(F(F)F'F)F-F)F.*&F(F)F0F)F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,&*&\"\"$\"\"\"%\"kGF)F)F(F)F))%\"xG\"\"#F)F) *&,&*&F(F))F*F-F)!\"\"*&F(F)F*F)F2F)F,F)F)*$)F*F(F)F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"xG,$**\"\"#!\"\",&*&\"\"$\"\"\"%\"kGF,F, F+F,F(,(*&F+F,F-F,F,F+F,*$,(*&F+F,)F-F'F,F(*&\"\"'F,F-F,F,\"\"*F,#F,F' F,F,F-F,F,/F$,$**F'F(F)F(,(*&F+F,F-F,F,F+F,F0F(F,F-F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The discriminant " }{XPPEDIT 18 0 "-3* (k+1)*(k-3)" "6#,$*(\"\"$\"\"\",&%\"kGF&F&F&F&,&F(F&F%!\"\"F&F*" } {TEXT -1 17 " is positive for " }{TEXT 279 1 "k" }{TEXT -1 9 " between " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 17 " and 3 exclusiv e." }}{PARA 0 "" 0 "" {TEXT -1 25 "This means that the line " } {XPPEDIT 18 0 "x+y=k" "6#/,&%\"xG\"\"\"%\"yGF&%\"kG" }{TEXT -1 45 " me ets the curve at two distinct points when " }{XPPEDIT 18 0 "-1 < k;" " 6#2,$\"\"\"!\"\"%\"kG" }{XPPEDIT 18 0 "``<3" "6#2%!G\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "For example, taking " }{XPPEDIT 18 0 "k=2" "6#/%\"kG\"\"#" }{TEXT -1 22 " we see that the line " } {XPPEDIT 18 0 "x+y=2" "6#/,&%\"xG\"\"\"%\"yGF&\"\"#" }{TEXT -1 30 " me ets the curve at the points" }{XPPEDIT 18 0 "``(2/3,4/3)" "6#-%!G6$*& \"\"#\"\"\"\"\"$!\"\"*&\"\"%F(F)F*" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(4/3,2/3)" "6#-%!G6$*&\"\"%\"\"\"\"\"$!\"\"*&\"\"#F(F)F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 263 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit((3+3 *k+sqrt(9+6*k-3*k^2))*k/(2*(3*k+3)),k=-1,right)=-infinity" "6#/-%&Limi tG6%*(,(\"\"$\"\"\"*&F)F*%\"kGF*F*-%%sqrtG6#,(\"\"*F**&\"\"'F*F,F*F**& F)F**$F,\"\"#F*!\"\"F*F*F,F**&F6F*,&*&F)F*F,F*F*F)F*F*F7/F,,$F*F7%&rig htG,$%)infinityGF7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 29 "This can be seen as follows. " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(3+3*k+sqrt(9+6*k-3*k^2))*k /(2*(3*k+3))=``((3+3*k+sqrt(9+6*k-3*k^2))*k/(2*(3*k+3)))*``((3+3*k-sqr t(9+6*k-3*k^2))/(3+3*k-sqrt(9+6*k-3*k^2)))" "6#/*(,(\"\"$\"\"\"*&F&F'% \"kGF'F'-%%sqrtG6#,(\"\"*F'*&\"\"'F'F)F'F'*&F&F'*$F)\"\"#F'!\"\"F'F'F) F'*&F3F',&*&F&F'F)F'F'F&F'F'F4*&-%!G6#*(,(F&F'*&F&F'F)F'F'-F+6#,(F.F'* &F0F'F)F'F'*&F&F'*$F)F3F'F4F'F'F)F'*&F3F',&*&F&F'F)F'F'F&F'F'F4F'-F:6# *&,(F&F'*&F&F'F)F'F'-F+6#,(F.F'*&F0F'F)F'F'*&F&F'*$F)F3F'F4F4F',(F&F'* &F&F'F)F'F'-F+6#,(F.F'*&F0F'F)F'F'*&F&F'*$F)F3F'F4F4F4F'" }{TEXT -1 1 " " }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (9+18*k+9* k^2-9-6*k+3*k^2)*k/(2*(3*k+3)*(3+3*k-sqrt(9+6*k-3*k^2)));" "6#/%!G*(,. \"\"*\"\"\"*&\"#=F(%\"kGF(F(*&F'F(*$F+\"\"#F(F(F'!\"\"*&\"\"'F(F+F(F/* &\"\"$F(*$F+F.F(F(F(F+F(*(F.F(,&*&F3F(F+F(F(F3F(F(,(F3F(*&F3F(F+F(F(-% %sqrtG6#,(F'F(*&F1F(F+F(F(*&F3F(*$F+F.F(F/F/F(F/" }{TEXT -1 1 " " }} {PARA 261 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (12*k^2+12*k^3)/ (2*(3*k+3)*(3+3*k-sqrt(9+6*k-3*k^2)));" "6#/%!G*&,&*&\"#7\"\"\"*$)%\"k G\"\"#F)F)F)*&F(F)*$)F,\"\"$F)F)F)F)*(F-F),&*&F1F)%\"kGF)F)F1F)F),(F1F )*&F1F)F5F)F)-%%sqrtG6#,(\"\"*F)*&\"\"'F)F5F)F)*&F1F)*$F5F-F)!\"\"FAF) FA" }{TEXT -1 1 " " }}{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*k^2/(3+3*k-sqrt(9+6*k-3*k^2))" "6#/%!G*(\"\"#\"\"\"*$%\"kGF&F',( \"\"$F'*&F+F'F)F'F'-%%sqrtG6#,(\"\"*F'*&\"\"'F'F)F'F'*&F+F'*$F)F&F'!\" \"F6F6" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "The result follows since the expression " }{XPPEDIT 18 0 "3+3*k-sqrt(9+6*k-3*k^2)" "6#,(\"\"$\"\"\"*&F$F%%\"kGF%F%-%%sqrtG 6#,(\"\"*F%*&\"\"'F%F'F%F%*&F$F%*$F'\"\"#F%!\"\"F2" }{TEXT -1 17 " is \+ negative for " }{XPPEDIT 18 0 "-1-1" "6#f*6#%\"kG7\"6$%)operatorG%&arrowG6\",$\"\" \"!\"\"F*F*F*" }{TEXT -1 17 " from the right. " }}{PARA 0 "" 0 "" {TEXT -1 43 "This can be seen from the following graph. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot( 3+3*k-sqrt(9+6*k-3*k^2),k=-1..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 469 234 234 {PLOTDATA 2 "6%-%'CURVESG6$7V7$$!\"\"\"\"!$F*F*7$$!3u****\\(oU In*!#=$!3)z)\\e;2Dd_F/7$$!3[*****\\P&3Y$*F/$!3>waRjg'Q#oF/7$$!3I**\\il :gh!*F/$!3GUX[VRFrwF/7$$!3C++Dcx6x()F/$!3gI\\y0Rhe#)F/7$$!3%)**\\Pff=d %)F/$!3X&fR@C/Jr)F/7$$!3b++]iTDP\")F/$!3e)[pht$R5!*F/7$$!3A****\\P\"\\ J\\(F/$!3Q`2RZfQr#*F/7$$!3g***\\7V0@&oF/$!305hYn/g6#*F/7$$!3w++DcexdiF /$!3;))GaW6\"*[*)F/7$$!3j***\\i+#QUcF/$!3kf[#QsmH^)F/7$$!3$****\\i!3%f +&F/$!3r6K)y4G!>zF/7$$!3;++D\"oS:P%F/$!34j^9ici0sF/7$$!3h*****\\<#)*=P F/$!3mD@d5_YjjF/7$$!3#*****\\(G3U9$F/$!39C2M%\\1=a&F/7$$!3Y*****\\-\\r \\#F/$!3gL#*=#\\Mq`%F/7$$!3?+++vGVZ=F/$!3?y>J(QZ8X$F/7$$!3_*****\\(4J@ 7F/$!3A3JV6XxQBF/7$$!3;,+]iIKFl!#>$!3WRPj=XTw7F/7$$\"3(R,++]siL#!#?$\" 3-sF9]4=wYFeq7$$\"3K,+++!R5'fF_q$\"3Wp,a3WW:7F/7$$\"3!)***\\P/QBE\"F/$ \"3*=I^kU'yEEF/7$$\"39******\\\"o?&=F/$\"3$)=zE4*\\-#RF/7$$\"3k++vVb4* \\#F/$\"3EcMYN*Q[Q&F/7$$\"3w++DJ'=_6$F/$\"3ST9Ku$G=#oF/7$$\"3#4++vVy!e PF/$\"3M'p#=R!>NO)F/7$$\"3'4+](=WU[VF/$\"3I(y*GrI3;)*F/7$$\"3s****\\7B >&)\\F/$\"3%4N\"f$**=@9\"!#<7$$\"3w***\\P>:mk&F/$\"3c]9q#=w5f$4=F`t7$$\"36****\\(yb^6)F/$\"3%> kzL!G'e)>F`t7$$\"3')***\\PMaKs)F/$\"3]bV)\\MS*f@F`t7$$\"3a****\\7TW)R* F/$\"3%G\"H+9*)*pN#F`t7$$\"3z*****\\@80+\"F`t$\"3p/L\"R9Qu`#F`t7$$\"31 ++]7,Hl5F`t$\"3iYcUg]hLFF`t7$$\"3()**\\P4w)R7\"F`t$\"3u6\"oQHCX\"HF`t7 $$\"3;++]x%f\")=\"F`t$\"3ee[_n7u:JF`t7$$\"3!)**\\P/-a[7F`t$\"3-Mz5L7P3 LF`t7$$\"3/+](=Yb;J\"F`t$\"3OP#*z!Q\"=8NF`t7$$\"3')****\\i@Ot8F`t$\"3X >\"[&3=)or$F`t7$$\"3')**\\PfL'zV\"F`t$\"3=$fA`xlQ$RF`t7$$\"3>+++!*>=+: F`t$\"3kyA&zSDl9%F`t7$$\"3-++DE&4Qc\"F`t$\"3CZnDPE#yO%F`t7$$\"3=+]P%>5 pi\"F`t$\"3G'GJg@47f%F`t7$$\"39+++bJ*[o\"F`t$\"3*[\"f>!yE+![F`t7$$\"33 ++Dr\"[8v\"F`t$\"3io\")=_>oV]F`t7$$\"3++++Ijy5=F`t$\"3'[mAGswcE&F`t7$$ \"31+]P/)fT(=F`t$\"3*3yo6&yy1bF`t7$$\"31+]i0j\"[$>F`t$\"3`2i$*[)Q?u&F` t7$$\"\"#F*$\"\"'F*-%'COLOURG6&%$RGBG$\"#5F)F+F+-%+AXESLABELSG6$Q\"k6 \"Q!Fc\\l-%%VIEWG6$;F(Fe[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Alternatively, one can check that " }{XPPEDIT 18 0 "3+3*k -sqrt(9+6*k-3*k^2)" "6#,(\"\"$\"\"\"*&F$F%%\"kGF%F%-%%sqrtG6#,(\"\"*F% *&\"\"'F%F'F%F%*&F$F%*$F'\"\"#F%!\"\"F2" }{TEXT -1 25 " is equal to 0 \+ only when " }{XPPEDIT 18 0 "k=-1" "6#/%\"kG,$\"\"\"!\"\"" }{TEXT -1 10 " and when " }{XPPEDIT 18 0 "k=0" "6#/%\"kG\"\"!" }{TEXT -1 61 ". T hen, by continuity, it cannot change sign in the interval " }{XPPEDIT 18 0 "-1 " 0 "" {MPLTEXT 1 0 147 "Limit((3+3*k+sqr t(9+6*k-3*k^2))*k/(2*(3*k+3)),k=-1,'right');\n``=value(%);\nLimit((3+3 *k-sqrt(9+6*k-3*k^2))*k/(2*(3*k+3)),k=-1,'right');\n``=value(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6%*(,(\"\"$\"\"\"*&F(F)%\"kGF )F)*$,(\"\"*F)*&\"\"'F)F+F)F)*&F(F))F+\"\"#F)!\"\"#F)F3F)F)F+F),&*&F0F )F+F)F)F0F)F4/F+F4%&rightG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$%) infinityG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6%*(,(\"\"$ \"\"\"*&F(F)%\"kGF)F)*$,(\"\"*F)*&\"\"'F)F+F)F)*&F(F))F+\"\"#F)!\"\"#F )F3F4F)F+F),&*&F0F)F+F)F)F0F)F4/F+F4%&rightG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "It follows that the line " }{XPPEDIT 18 0 "x+y=-1" "6#/,&%\"xG\"\" \"%\"yGF&,$F&!\"\"" }{TEXT -1 39 " is an oblique asymptote for the cur ve " }{XPPEDIT 18 0 "x^3+y^3 = 3*x*y" "6#/,&*$%\"xG\"\"$\"\"\"*$%\"yGF 'F(*(F'F(F&F(F*F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "Als o, the line " }{XPPEDIT 18 0 "x+y=3" "6#/,&%\"xG\"\"\"%\"yGF&\"\"$" } {TEXT -1 44 " is a tangent line to the curve at the point" }{XPPEDIT 18 0 "``(3/2,3/2);" "6#-%!G6$*&\"\"$\"\"\"\"\"#!\"\"*&F'F(F)F*" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 298 "p1 := plots[implicitplot](x^3+y^3=3*x*y,x=-3..3 ,y=-3..3,grid=[50,50]):\np2 := plot([-x-1,-x+2,-x+3],x=-3..3,color=bla ck,linestyle=2):\np3 := plot([[[2/3,4/3],[4/3,2/3],[3/2,3/2]]$3],style =point,color=black,\n symbol=[circle,diamond,cross]):\np lots[display]([p1,p2,p3],view=[-3..3,-3..3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6+-%'CURVESG6bw7$7$$!37j\"3/-^v(G!# <$\"3zUr&GR8!G>F*7$$!3;?A@#[T8(HF*$\"3P1`Ej\"3/-#F*7$7$$!3i?A@#[T8(HF* F07$$!\"$\"\"!$\"3g'p#)yHki/#F*7$7$$!3EEj\"3/-^v#F*$\"3&4$fIC0M5=F*7$$ !3$z)>))y$*4WGF*$\"3[pMn$=fz*=F*7$7$$!3\\()>))y$*4WGF*FEF'7$7$$!3R*[C7 1`Ej#F*$\"3=:ScJ8N$p\"F*7$$!3\"=\\x#G+;;FF*$\"3'Gj\"3/-^v,R(e#F*$\" 3?'z*[C71`;F*7$FjnFL7$7$$!3k:3/-^v(Q#F*$\"3%yX%os,(>Y\"F*7$$!3S^nrncid CF*$\"3bfz*[C71`\"F*7$FfoFen7$7$$!3yy*[C71`E#F*$\"3%H>$Q9B$zM\"F*7$$!3 \"o@q,HkmK#F*$\"3!H71`Ej\"39F*7$FbpFao7$7$$!3YTr&G9dG9#F*$\"3**R03jFIN 7F*7$$!3kWdDWdD%>#F*$\"3E'G9dG9dG\"F*7$F^qF]p7$7$$!3.0`Ej\"3/-#F*$\"3e PB_\"[zV7\"F*7$$!3OYy%Hq!4g?F*$\"3g\\C71`Ej6F*7$Fjq7$$!3\">9dG9dG9#F*F \\q7$7$$!3QoMn$=fz*=F*$\"3g)RVP6Hb,\"F*7$$!3NBhIl#)yB>F*$\"3'HhIlK;3/ \"F*7$FirFeq7$7$$!3uJ;3/-^vW.*)>4*!#=7$$!3o9VjTQ)[y\"F*$\"3 '3w(QpMn$=*Fes7$7$Fgs$\"3'>w(QpMn$=*FesFdr7$F`s7$$!3%==Q$))z#Gr\"F*$\" 35kK&>J^ob)Fes7$7$$!35&z*[C71`;F*$\"3Q-m^]W3^!)FesF`t7$7$$!3Wez*[C71` \"F*$\"3H/ADbO7PqFes7$$!3@P/yr!Q/k\"F*$\"3Q%RpMn$=fzFes7$7$$!3WP/yr!Q/ k\"F*Fdu7$Fgt$\"3[.m^]W3^!)Fes7$7$$!3z@hIlK;39F*$\"3A#=o/v$G%Fes7$7$$! 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" }}{PARA 15 "" 0 "" {TEXT -1 32 "In some cases, we can obtain an " }{TEXT 260 8 "expl icit" }{TEXT -1 65 " formula for a function whose graph gives a sectio n of the curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "6 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }