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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "Some interesting curves" }}{PARA 0 "" 0 "" {TEXT -1 58 "by Peter Stone, School of Life and Physical Sci ences, RMIT" }}{PARA 0 "" 0 "" {TEXT -1 23 "peter.stone@rmit.edu.au" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 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 ";" }}}{PARA 0 "" 0 "" {TEXT -1 71 "Acknowledgement: The equations o f the following curves are taken from: " }}{PARA 0 "" 0 "" {TEXT -1 75 " \"Mathematical Models\", by Cundy and Rollett, Oxford UP, 1961, p ages 71,72." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Lemniscate" }} {PARA 261 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y^2 = x^2-x^4;" "6#/ *$%\"yG\"\"#,&*$%\"xGF&\"\"\"*$F)\"\"%!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "pl ots[implicitplot](y^2=x^2-x^4,x=-1..1,y=-1..1,grid=[75,75]);" }}{PARA 13 "" 1 "" {GLPLOT2D 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\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%\"xG\"\"\",&!\"\"F'*$)F& \"\"#F'F,F'F'-%\"yG6#F&F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&% \"xG\"\"\",&!\"\"F'*$)F&\"\"#F'F,F'F'%\"yGF)F)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = x*(2*x^2-1)/y;" "6# /*&%#dyG\"\"\"%#dxG!\"\"*(%\"xGF&,&*&\"\"#F&*$F*F-F&F&F&F(F&%\"yGF(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "The gradient is zero whe n " }{XPPEDIT 18 0 "x = 1/sqrt(2);" "6#/%\"xG*&\"\"\"F&-%%sqrtG6#\"\"# !\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x = -1/sqrt(2);" "6#/%\"xG,$ *&\"\"\"F'-%%sqrtG6#\"\"#!\"\"F," }{TEXT -1 25 ", that is, at the poin ts:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 1 "," } {XPPEDIT 18 0 "``(1/sqrt(2),1/2);" "6#-%!G6$*&\"\"\"F'-%%sqrtG6#\"\"#! \"\"*&F'F'F+F," }{TEXT -1 1 "," }{XPPEDIT 18 0 "``(1/sqrt(2),-1/2);" " 6#-%!G6$*&\"\"\"F'-%%sqrtG6#\"\"#!\"\",$*&F'F'F+F,F," }{TEXT -1 1 "," }{XPPEDIT 18 0 "``(-1/sqrt(2),1/2);" "6#-%!G6$,$*&\"\"\"F(-%%sqrtG6#\" \"#!\"\"F-*&F(F(F,F-" }{TEXT -1 1 "," }{XPPEDIT 18 0 "``(-1/sqrt(2),-1 /2);" "6#-%!G6$,$*&\"\"\"F(-%%sqrtG6#\"\"#!\"\"F-,$*&F(F(F,F-F-" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "Notice that the formula \+ given above for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 28 " does not give a value when " }{XPPEDIT 18 0 "x = 0" "6#/ %\"xG\"\"!" }{TEXT -1 10 ", because " }{XPPEDIT 18 0 "y = 0" "6#/%\"yG \"\"!" }{TEXT -1 50 " also, and we obtain the indeterminate expression " }{XPPEDIT 18 0 "0/0" "6#*&\"\"!\"\"\"F$!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 68 "The curve can also be expressed in parame tric form by the equations:" }}{PARA 262 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "PIECEWISE([x = sin*t, ``],[y = sin*t*cos*t, ``]);" "6#- %*PIECEWISEG6$7$/%\"xG*&%$sinG\"\"\"%\"tGF+%!G7$/%\"yG**%$sinGF+%\"tGF +%$cosGF+F3F+F-" 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!G7$/*&%#dyGF*F+F,,&*&)%$cosG\"\"#F*%\"tGF*F**&)%$sinGF9F*F:F*F,F0" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 264 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = (cos^2*t-sin^2*t)/(cos*t);" "6# /*&%#dyG\"\"\"%#dxG!\"\"*&,&*&)%$cosG\"\"#F&%\"tGF&F&*&)%$sinGF.F&F/F& F(F&*&%$cosGF&%\"tGF&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The curve passes through the origi n at the parameter values " }{XPPEDIT 18 0 "t =n*Pi" "6#/%\"tG*&%\"nG \"\"\"%#PiGF'" }{TEXT -1 8 ", where " }{TEXT 263 1 "n" }{TEXT -1 12 " \+ an integer." }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 264 1 "n" } {TEXT -1 17 " is even we have " }{XPPEDIT 18 0 "dy/dx=1" "6#/*&%#dyG\" \"\"%#dxG!\"\"F&" }{TEXT -1 11 ", and when " }{TEXT 265 1 "n" }{TEXT -1 17 " is odd, we have " }{XPPEDIT 18 0 "dy/dx=-1" "6#/*&%#dyG\"\"\"% #dxG!\"\",$F&F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Dumb-bell" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y^2=x^4-x^6" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"%\"\"\"*$F)\" \"'!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plots[implicitplot](y^2=x^4-x^6,x=- 1..1,y=-1..1,grid=[75,75]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 229 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$!3(\\O)>-w!e?#F*7$Fas$!3iU7c%[nI4#F*7$Ffs$!3sOqlVuK&)>F*7$F[t$!3M_Xu \"3`s&=F*7$F`t$!3\\f*3'*QS&4,e9 F*7$Fjt$!3/C%)R%4KJO\"F*7$Fabo$!3u/??7sfZ7F*7$F_u$!34BS'f\"4;=6F*7$Fib o$!3!)*p$Q=1Rb)*Fjp7$Fdu$!37(4[)zZ!))H)Fjp7$Faco$!3nnw$=UgQK(Fjp7$Fju$ !3)fbVZF7$*='Fjp7$Fico$!3m#*[3l_(Hy%Fjp7$F_v$!3iJ2sjlh5FFjp7$Fado$!3;' fbCq6M&>Fjp7$Ffdo$!3o;2GYon$G&FfuF[eo-%*THICKNESSG6#\"\"#-%&COLORG6&%$ RGBG$\"\"%!\"\"$Fh[oFbdr$\"\"*Fbdr-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG 6$;$!\"#FbdrFg[o%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "x^4+x^2 *y^2+y^4 = x*(x^2-y^2);\nsubs(y=y(x),%);\ndiff(%,x);\nsolve(%,diff(y(x ),x));\nsubs(y(x)=y,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG \"\"%\"\"\"F)*&)F'\"\"#F))%\"yGF,F)F)*$)F.F(F)F)*&F'F),&*$F+F)F)*$F-F) !\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"%\"\"\"F)*&) F'\"\"#F))-%\"yG6#F'F,F)F)*$)F.F(F)F)*&F'F),&*$F+F)F)*$F-F)!\"\"F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,**$)%\"xG\"\"$\"\"\"\"\"%*&F'F))-%\" yG6#F'\"\"#F)F0*()F'F0F)F-F)-%%diffG6$F-F'F)F0*&)F-F(F)F3F)F*,(*$F2F)F )*$F,F)!\"\"*&F'F),&F'F0*&F-F)F3F)!\"#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,**$)%\"xG\"\"$\"\"\"\"\"%*&F(F*)-%\"yG6#F(\"\"#F*F 1*$)F(F1F*!\"$*$F-F*F*F**&F.F*,(F2F*F5F1F(F*F*!\"\"#F8F1" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&,**$)%\"xG\"\"$\"\"\"\"\"%*&F(F*)%\"yG\"\"#F *F/*$)F(F/F*!\"$*$F-F*F*F**&F.F*,(F0F*F3F/F(F*F*!\"\"#F6F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 279 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -(4*x^3+2*x *y^2-3*x^2+y^2)/(2*y*(x^2+2*y^2+x));" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&, **&\"\"%F&*$%\"xG\"\"$F&F&*(\"\"#F&F/F&%\"yGF2F&*&F0F&*$F/F2F&F(*$F3F2 F&F&*(F2F&F3F&,(*$F/F2F&*&F2F&*$F3F2F&F&F/F&F&F(F(" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can fi nd the stationary points by solving the pair of equations:" }}{PARA 285 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([4*x^3+2*x*y^2-3 *x^2+y^2 = 0, ``],[x^4+x^2*y^2+y^4 = x*(x^2-y^2), ``]);" "6#-%*PIECEWI SEG6$7$/,**&\"\"%\"\"\"*$%\"xG\"\"$F+F+*(\"\"#F+F-F+%\"yGF0F+*&F.F+*$F -F0F+!\"\"*$F1F0F+\"\"!%!G7$/,(*$F-F*F+*&F-F0F1F0F+*$F1F*F+*&F-F+,&*$F -F0F+*$F1F0F4F+F7" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "eq1 := 4*x^3+2*x*y^2-3*x^2+ y^2=0;\neq2 := x^4+x^2*y^2+y^4 = x*(x^2-y^2);\nfsolve(\{eq1,eq2\},\{x= 0.6..0.7,y=0.2..0.4\});\nfsolve(\{eq1,eq2\},\{x=-0.2..-0.1,y=0.28..0.3 \});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,**$)%\"xG\"\"$\"\"\" \"\"%*&F)F+)%\"yG\"\"#F+F0*$)F)F0F+!\"$*$F.F+F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,(*$)%\"xG\"\"%\"\"\"F+*&)F)\"\"#F+)%\"yGF.F +F+*$)F0F*F+F+*&F)F+,&*$F-F+F+*$F/F+!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG$\"+zI\")=j!#5/%\"yG$\"+Y8v')GF(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<$/%\"yG$\"+Y8v')G!#5/%\"xG$!+zI\")=8F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "There are two more stationary points obtained by reflecting the two points just found in the " }{TEXT 266 1 "x" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "We can find two of the \+ four points where the gradient is infinite by solving the pair of equa tions:" }}{PARA 286 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE( [x^2+2*y^2+x = 0, ``],[x^4+x^2*y^2+y^4 = x*(x^2-y^2), ``]);" "6#-%*PIE CEWISEG6$7$/,(*$%\"xG\"\"#\"\"\"*&F+F,*$%\"yGF+F,F,F*F,\"\"!%!G7$/,(*$ F*\"\"%F,*&F*F+F/F+F,*$F/F6F,*&F*F,,&*$F*F+F,*$F/F+!\"\"F,F1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "e q3 := x^2+2*y^2+x=0;\neq2 := x^4+x^2*y^2+y^4 = x*(x^2-y^2);\nfsolve(\{ eq3,eq2\},\{x=-0.2..-0.1,y=0.2..0.3\});\nfsolve(\{eq3,eq2\},\{x=-0.2.. -0.1,y=-0.3..-0.2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/,(*$) %\"xG\"\"#\"\"\"F+*$)%\"yGF*F+F*F)F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,(*$)%\"xG\"\"%\"\"\"F+*&)F)\"\"#F+)%\"yGF.F+F+ *$)F0F*F+F+*&F)F+,&*$F-F+F+*$F/F+!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG$!+%Q0qa\"!#5/%\"yG$\"+#GKqb#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG$!+%Q0qa\"!#5/%\"yG$!+#GKqb#F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The gradient is also infinite at the points" }{XPPEDIT 18 0 " ``(0,0)" "6#-%!G6$\"\"!F&" } {TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(1,0)" "6#-%!G6$\"\"\"\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Malte se Cross" }}{PARA 276 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x*y*(x^2 -y^2)=x^2+y^2" "6#/*(%\"xG\"\"\"%\"yGF&,&*$F%\"\"#F&*$F'F*!\"\"F&,&*$F %F*F&*$F'F*F&" }{TEXT -1 12 " . . . . 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{XPPMATH 20 "6#/*$)%\"xG\"\"#\"\"\",$*(,&F(F(*$)%\"tGF'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 4 "Let \+ " }{XPPEDIT 18 0 "f(t)=sqrt((1+t^2)/(t*(1-t^2)))" "6#/-%\"fG6#%\"tG-%% sqrtG6#*&,&\"\"\"F-*$F'\"\"#F-F-*&F'F-,&F-F-*$F'F/!\"\"F-F3" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(t)=sqrt(t*(1+t^2)/(1-t^2))" "6#/-%\"gG6 #%\"tG-%%sqrtG6#*(F'\"\"\",&F,F,*$F'\"\"#F,F,,&F,F,*$F'F/!\"\"F2" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "We can sketch the graph in pieces using the parametric eq uations:" }}{PARA 277 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE ([x = f(t), ``],[y = g(t), ``]);" "6#-%*PIECEWISEG6$7$/%\"xG-%\"fG6#% \"tG%!G7$/%\"yG-%\"gG6#F,F-" }{XPPEDIT 18 0 "PIECEWISE([x = -f(t), ``] ,[y = -g(t), ``])" "6#-%*PIECEWISEG6$7$/%\"xG,$-%\"fG6#%\"tG!\"\"%!G7$ /%\"yG,$-%\"gG6#F-F.F/" }{TEXT -1 3 " " }{XPPEDIT 18 0 "PIECEWISE([x = -g(t), ``],[y = f(t), ``]);" 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solve(%,diff(y(x),x));\nsubs(y(x)=y,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"xG\"\"\"%\"yGF&,&*$)F%\"\"#F&F&*$)F'F+F&!\"\"F&,&F)F&F,F& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"xG\"\"\"-%\"yG6#F%F&,&*$)F% \"\"#F&F&*$)F'F-F&!\"\"F&,&F+F&F.F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/,(*&-%\"yG6#%\"xG\"\"\",&*$)F)\"\"#F*F**$)F&F.F*!\"\"F*F**(F)F*-%%di ffG6$F&F)F*F+F*F**(F)F*F&F*,&F)F.*&F&F*F3F*!\"#F*F*,&F)F.F8F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(*&-%\"yG6#%\"xG\"\"\")F*\"\"#F+ \"\"$*$)F'F.F+!\"\"F*!\"#F+,(*$)F*F.F+F+*&F*F+)F'F-F+!\"$F'F2F1F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(*&%\"yG\"\"\")%\"xG\"\"#F(\"\"$* $)F'F,F(!\"\"F*!\"#F(,(*$)F*F,F(F(*&F*F()F'F+F(!\"$F'F0F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can find the stationary points by solving the pair of equations:" }}{PARA 278 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([3*y*x^2-y^3-2*x=0,``], [x*y*(x^2-y^2)=x^2+y^2,``])" "6#-%*PIECEWISEG6$7$/,(*(\"\"$\"\"\"%\"yG F+%\"xG\"\"#F+*$F,F*!\"\"*&F.F+F-F+F0\"\"!%!G7$/*(F-F+F,F+,&*$F-F.F+*$ F,F.F0F+,&*$F-F.F+*$F,F.F+F3" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "fsolve(\{x*y*(x^2-y^2)=x^2+y^2,3*y*x^2-y^3-2*x\}, \n \{x=0.8..0.9,y=-1.9..-1.8\});\nfsolve(\{x*y*( x^2-y^2)=x^2+y^2,3*y*x^2-y^3-2*x\},\n \{x=-0.9.. -0.8,y=1.8..1.9\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"yG$!+.5)[ #=!\"*/%\"xG$\"+$z " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 43 "Sketch th e following curves first by using " }{TEXT 0 12 "implicitplot" }{TEXT -1 35 ". Then try to draw the curve using " }{TEXT 0 4 "plot" }{TEXT -1 33 " in parametric form or in pieces." }}{PARA 0 "" 0 "" {TEXT -1 87 "Find the coordinates of any points on the curve where the gradient is zero or infinite." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Bow" }} {PARA 288 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^4 = y*(x^2-y^2);" " 6#/*$%\"xG\"\"%*&%\"yG\"\"\",&*$F%\"\"#F)*$F(F,!\"\"F)" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Hint " }{TEXT -1 13 ": Substitute " }{XPPEDIT 18 0 "y = t*x" "6#/%\"yG*&%\" tG\"\"\"%\"xGF'" }{TEXT -1 35 " to obtain the parametric equations" }} {PARA 289 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([x = t*(1- t^2), ``],[y = t^2*(1-t^2), ``]);" "6#-%*PIECEWISEG6$7$/%\"xG*&%\"tG\" \"\",&F+F+*$F*\"\"#!\"\"F+%!G7$/%\"yG*&F*F.,&F+F+*$F*F.F/F+F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 " for the curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "A suitable parameter range is " }{XPPEDIT 18 0 "t = -1.15" "6#/%\"tG,$-%&FloatG6$\"$:\"!\" #!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "t = 1.15" "6#/%\"tG-%&Float G6$\"$:\"!\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 79 "Can you \+ find the equations of all the tangent lines to the curve at the origin ?" }}{PARA 0 "" 0 "" {TEXT -1 39 "____________________________________ ___" }}{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 39 "_______________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Butterfly" }}{PARA 290 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2 = x^6+y^6;" "6#/*$%\"xG\"\"#,&*$F%\"\"'\"\"\"*$%\"yGF)F*" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 259 4 "Hint" }{TEXT -1 13 ": Su bstitute " }{XPPEDIT 18 0 "y = t*x" "6#/%\"yG*&%\"tG\"\"\"%\"xGF'" } {TEXT -1 46 " to obtain parametric equations for the curve." }}{PARA 291 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([x = (1+t^6)^(-1 /4),``],[ y =t*(1+t^6)^(-1/4),``])" "6#-%*PIECEWISEG6$7$/%\"xG),&\"\" \"F+*$%\"tG\"\"'F+,$*&F+F+\"\"%!\"\"F2%!G7$/%\"yG*&F-F+),&F+F+*$F-F.F+ ,$*&F+F+F1F2F2F+F3" }{TEXT -1 5 " or " }{XPPEDIT 18 0 "PIECEWISE([x = -(1+t^6)^(-1/4),``],[ y =-t*(1+t^6)^(-1/4),``])" "6#-%*PIECEWISEG6$7$ /%\"xG,$),&\"\"\"F,*$%\"tG\"\"'F,,$*&F,F,\"\"%!\"\"F3F3%!G7$/%\"yG,$*& F.F,),&F,F,*$F.F/F,,$*&F,F,F2F3F3F,F3F4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "which can be used to draw the right \+ and left \"wings\" of the butterfly respectively." }}{PARA 0 "" 0 "" {TEXT -1 30 "A suitable parameter range is " }{XPPEDIT 18 0 "t = -100 " "6#/%\"tG,$\"$+\"!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "t = 100" "6#/%\"tG\"$+\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "______ _________________________________" }}{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 39 "____________________ ___________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Bicuspidal" }}{PARA 292 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(y^2-1)^2 = (1-x^2)*(x-1)^2;" "6#/*$,&* $%\"yG\"\"#\"\"\"F)!\"\"F(*&,&F)F)*$%\"xGF(F*F)*$,&F.F)F)F*F(F)" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Hint" }{TEXT -1 12 ": Solve for " }{TEXT 267 1 "y" } {TEXT -1 11 " to obtain " }{XPPEDIT 18 0 "y = sqrt(1+sqrt(1-x^2)*(x-1) );" "6#/%\"yG-%%sqrtG6#,&\"\"\"F)*&-F&6#,&F)F)*$%\"xG\"\"#!\"\"F),&F/F )F)F1F)F)" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "y = sqrt(1-sqrt(1-x^2)*( x-1));" "6#/%\"yG-%%sqrtG6#,&\"\"\"F)*&-F&6#,&F)F)*$%\"xG\"\"#!\"\"F), &F/F)F)F1F)F1" }{TEXT -1 22 ", and their negatives." }}{PARA 0 "" 0 " " {TEXT -1 61 "The graph can then be drawn in four pieces. (Use the op tion \"" }{TEXT 268 13 "numpoints=250" }{TEXT -1 3 "\")." }}{PARA 0 " " 0 "" {TEXT -1 39 "_______________________________________" }}{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 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{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 }