{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 260 "" 0 1 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 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "H eading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 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 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Discussion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 104 "First of all consider the circle center ed at the origin with radius 2. The equation for this curve is " }} {PARA 0 "" 0 "" {TEXT -1 49 " \+ " }{XPPEDIT 18 0 "x^2 + y^2 = 4" "6#/,&*$%\"xG\"\"#\"\"\"*$% \"yG\"\"#F(\"\"%" }}{PARA 0 "" 0 "" {TEXT -1 261 "If you wish plot thi s curve then you soon become aware that the set of points represented by the above equation cannot satisfy a relationship of the form y = \+ f(x) where f is a single valued function (the vertical line test fai ls). The graph of the function" }}{PARA 0 "" 0 "" {TEXT -1 48 " \+ " }{XPPEDIT 18 0 "f[1](x) = \+ sqrt(4 - x^2)" "6#/-&%\"fG6#\"\"\"6#%\"xG-%%sqrtG6#,&\"\"%\"\"\"*$F*\" \"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 37 "is the upper semi-circle of r adius 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "P1 := plot(sqrt(4-x^2),x=-2..2,scaling=constrained):%;" }}{PARA 13 " " 1 "" {GLPLOT2D 252 252 252 {PLOTDATA 2 "6&-%'CURVESG6$7_o7$$!\"#\"\" !F*7$$!1L$e9r]X*>!#:$\"1'GV9$*z`Z\"!#;7$$!1nm\"HU,\"*)>F.$\"1b^-S'y]3# F17$$!1+]PM@l$)>F.$\"1**\\c[X%>b#F17$$!1LL$e%G?y>F.$\"17\\Z=crWHF17$$! 1++voUIn>F.$\"1pQd$z\"e,OF17$$!1nmm\"p0k&>F.$\"1XE1hA.`TF17$$!1++]P&3Y $>F.$\"1\"oFHEuB2&F17$$!1LLL$Q6G\">F.$\"1p0<(4F3%eF17$$!1++v3-)[(=F.$ \"1s>(=$H'pF17$$!1nm;M!\\p$=F.$\"1(ziTJ.'4zF17$$!1++Dh9H%z\"F.$\"1#R \\y$QpM))F17$$!1LLL))Qj^4nU49F.7$$!1LL$3WDTL\"F.$\"1^t_BJ+!\\\"F.7$$ !1++]d(Q&\\7F.$\"1[#otn=;c\"F.7$$!1nmmc4`i6F.$\"1x9,jzUF;F.7$$!1LLLQW* e3\"F.$\"1zMAYO`z;F.7$$!1,+++()>'***F1$\"1Hh6?-FK=F.7$$!1LLL3k(p`(F1$\"1cgOe &\\D&=F.7$$!1nmmmj^NmF1$\"1Y&Q'>lr')=F.7$$!1ommm9'=(eF1$\"1\\i$e)4'=\" >F.7$$!1,++v#\\N)\\F1$\"18A#Qo:p$>F.7$$!1pmmmCC(>%F1$\"1$Q#R!)>Yb>F.7$ $!1*****\\FRXL$F1$\"1D#Rk?1?(>F.7$$!1+++D=/8DF1$\"1uWKO([T)>F.7$$!1mmm ;a*el\"F1$\"1Op\"yBLJ*>F.7$$!1pmm;Wn(o)!#<$\"1\"\\**o@7\")*>F.7$$!1qLL L$eV(>!#=$\"1bqZD!*****>F.7$$\"1Mmm;f`@')Feu$\"1[T'Q'39)*>F.7$$\"1)*** *\\nZ)H;F1$\"1!>y)GzM$*>F.7$$\"1lmm;$y*eCF1$\"1x!Q:+E[)>F.7$$\"1****** *R^bJ$F1$\"19BTRjKs>F.7$$\"1'*****\\5a`TF1$\"1Y1Z!*[Rc>F.7$$\"1(****\\ 7RV'\\F1$\"14v=K*3u$>F.7$$\"1'*****\\@fkeF1$\"1q%Hd4%37>F.7$$\"1JLLL&4 Nn'F1$\"16;f6gP&)=F.7$$\"1*******\\,s`(F1$\"10)*\\+/a_=F.7$$\"1lmm\"zM )>$)F1$\"1mED%yO(==F.7$$\"1*******pfa<*F1$\"1=p6Hq5xo\"F.7$$\"1LLL$)G[k6F .$\"1R+59F.7$$\"1LL$e#pa-:F.$\"1IGs'4#)* >8F.7$$\"1+++Sv&)z:F.$\"1ub#\\A(RE7F.7$$\"1LLLGUYo;F.$\"1Cg:wo#G5\"F.7 $$\"1nmm1^rZF.$\"1il>BruJeF17$$\"1+]i0j\"[$>F.$\"1dKYSLWk]F17$$\"1++v.Uac>F.$\"1 ]tQ%)=]YTF17$$\"1+D\"G:3u'>F.$\"1n)[F.$\"1e1=p$ f+%HF17$$\"1]iSwSq$)>F.$\"1&Qe.z.za#F17$$\"1+v$40O\"*)>F.$\"1u))p#es<3 #F17$$\"1](oa-oX*>F.$\"1PAj)RPIZ\"F17$$\"\"#F*F*-%'COLOURG6&%$RGBG$\"# 5!\"\"F*F*-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEW G6$;F(F_`l%(DEFAULTG" 1 2 0 1 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Whereas t he following plot gives the bottom half:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "P2 := plot(-sqrt(4-x^2),x=-2..2,scaling=constrained): %;" }}{PARA 13 "" 1 "" {GLPLOT2D 252 252 252 {PLOTDATA 2 "6&-%'CURVESG 6$7_o7$$!\"#\"\"!F*7$$!1L$e9r]X*>!#:$!1'GV9$*z`Z\"!#;7$$!1nm\"HU,\"*)> F.$!1b^-S'y]3#F17$$!1+]PM@l$)>F.$!1**\\c[X%>b#F17$$!1LL$e%G?y>F.$!17\\ Z=crWHF17$$!1++voUIn>F.$!1pQd$z\"e,OF17$$!1nmm\"p0k&>F.$!1XE1hA.`TF17$ $!1++]P&3Y$>F.$!1\"oFHEuB2&F17$$!1LLL$Q6G\">F.$!1p0<(4F3%eF17$$!1++v3- )[(=F.$!1s>(=$H'pF17$$!1nm;M!\\p$=F.$!1(ziTJ.'4zF17$$!1++Dh9H%z\"F.$ !1#R\\y$QpM))F17$$!1LLL))Qj^4nU49F.7$$!1LL$3WDTL\"F.$!1^t_BJ+!\\\"F.7$$! 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)>Yb>F.7$Fdt$!1D#Rk?1?(>F.7$Fit$!1uWKO([T)>F.7$F^u$!1Op\"yBLJ*>F.7$Fcu $!1\"\\**o@7\")*>F.7$Fiu$!1bqZD!*****>F.7$F_v$!1[T'Q'39)*>F.7$Fdv$!1!> y)GzM$*>F.7$Fiv$!1x!Q:+E[)>F.7$F^w$!19BTRjKs>F.7$Fcw$!1Y1Z!*[Rc>F.7$Fh w$!14v=K*3u$>F.7$F]x$!1q%Hd4%37>F.7$Fbx$!16;f6gP&)=F.7$Fgx$!10)*\\+/a_ =F.7$F\\y$!1mED%yO(==F.7$Fay$!1=p6Hq5xo\"F.7$F`z$!1R+59F.7$Fd[l$!1IGs'4#)*>8F.7$Fi[l$!1ub#\\A(RE7F.7$F^\\l$ !1Cg:wo#G5\"F.7$Fc\\l$!1rz`43kB(*F17$Fh\\l$!1:4a_K/A*)F17$F]]l$!1SF(Q] )p=!)F17$Fb]l$!1hzw)*=nAqF17$Fg]l$!1il>BruJeF17$F\\^l$!1dKYSLWk]F17$Fa ^l$!1]tQ%)=]YTF17$Ff^l$!1n)[ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "parameterization" {TEXT -1 16 "We can obtain a " }{TEXT 260 16 "parameterization" }{TEXT -1 87 " for the entire circle. Sinc e the following identity is true for all real values of t" }}{PARA 0 " " 0 "" {TEXT -1 41 " " } {XPPEDIT 18 0 "sin(t)^(2) + cos(t)^(2)=1" "6#/,&*$-%$sinG6#%\"tG\"\"# \"\"\"*$-%$cosG6#F)\"\"#F+\"\"\"" }}{PARA 0 "" 0 "" {TEXT -1 17 "it fo llows that " }{XPPEDIT 18 0 "x=2 sin(t)" "6#/%\"xG*&\"\"#\"\"\"-%$sin G6#%\"tGF'" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "y=2 cos(t)" "6#/%\"yG*& \"\"#\"\"\"-%$cosG6#%\"tGF'" }{TEXT -1 22 " satisfy the equation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 " \+ " }{XPPEDIT 18 0 "x^(2)+ y^(2)= 4" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F(\"\"%" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "for all values of t. Mo reover, the image of the mapping defined by the parametric equations \+ " }}{PARA 0 "" 0 "" {TEXT -1 49 " \+ " }{XPPEDIT 18 0 "x=2 cos(t), y=2 sin(t)" "6$/%\"xG*&\"\"# \"\"\"-%$cosG6#%\"tGF'/%\"yG*&\"\"#F'-%$sinG6#F+F'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 53 "generate the entire circle as t varies be tween 0 and " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot([2*cos(t),2*sin(t),t=0..2*Pi],scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 252 252 252 {PLOTDATA 2 "6&-%'CURVESG6$7S7$ $\"\"#\"\"!F*7$$\"1=AzuCF\")>!#:$\"1H`6ljbIF!#;7$$\"1BYw>/wM>F.$\"1])* =>zdm]F17$$\"12Ax&H<(\\=F.$\"1nEr\\1A1wF17$$\"1s$fP=91t\"F.$\"1.2*y=%[ -5F.7$$\"16]d1/&3e\"F.$\"1\"f/S8<^A\"F.7$$\"1qysL[^;9F.$\"1t%y,a3>T\"F .7$$\"1mQE\\6HB7F.$\"1VU!*4QE#e\"F.7$$\"1'[h7Ja@+\"F.$\"1RQvF_!3t\"F.7 $$\"1[;M*Gz-k(F1$\"12?@-JJ[=F.7$$\"1MHW_Dx]]F1$\"1xi*oit^$>F.7$$\"1Z-B XOF!p#F1$\"1?([FZB=)>F.7$$!1u%3[&R@%>\"!#=$\"1EKYV'*****>F.7$$!1'*H[=Z '\\s#F1$\"1%*Q_d%\\8)>F.7$$!17Vr+VO#H&F1$\"1%QO)4lqG>F.7$$!1e(G@'y@YvF 1$\"1cX;2L<_=F.7$$!1*\\CZ0j%35F.$\"1Hdh9q8FF.$\"1(QHKl(oV^F17$$!1&R$)Q+19)>F.$\"1'3/,,]3s#F17$ $!1_m&=Q!****>F.$\"1]W)H()=E?'Fao7$$!16Yu.wo\")>F.$!1&=)R7PE+FF17$$!15 2X$48[$>F.$!1r5bcgck]F17$$!1ImgUCl_=F.$!1>qjsqWMvF17$$!1b'>kAP[t\"F.$! 1`q!Hx\"e^**F17$$!1&)fDaN@*e\"F.$!1\\f!Q&)[U@\"F.7$$!1l#=u=8@U\"F.$!1W OTi&piS\"F.7$$!1pN-'G<(47F.$!1E:c_=F.7$$!1c\\ei6L<_F1$!1*Q%>R(\\2$>F.7$ $!1BC[]N8$e#F1$!1zOh)R[K)>F.7$$!1edteow9hFao$!1@ZQ_1****>F.7$$\"1hWE2l WvDF1$!1\\p,p$[L)>F.7$$\"1lyS\"p#35^F1$!1(p$*e#fhL>F.7$$\"1od6rnfrwF1$ !1TTVQa,Z=F.7$$\"1A&>_6g1+\"F.$!1176A&p;t\"F.7$$\"1]w*)z+(=A\"F.$!1\"z p$e9O$e\"F.7$$\"1@NHx+')>9F.$!19x67Wa39F.7$$\"1sVuyc.!e\"F.$!1heM5x;E7 F.7$$\"1G?C.v'[t\"F.$!1R'f^%R0^**F17$$\"1nJk!>+]%=F.$!1\"R9M:P*>xF17$$ \"1tX%QxP4$>F.$!1?,NzwO5_F17$$\"1;J>26R\")>F.$!1[RH7Y$>s#F17$F($\"1\\E 7Gu#3k\"!#C-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%(SCALINGG6#%,CONSTRAINEDG -%+AXESLABELSG6$%!GFe[l-%%VIEWG6$%(DEFAULTGFi[l" 1 2 0 1 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 174 "When a function is defined explici tly as a function of x or y alone it is easy to write a parameteriz ation: if a curve is described by the relation y = f(x), for " }{XPPEDIT 18 0 "a <= x,x <= b" "6$1%\"aG%\"xG1F%%\"bG" }{TEXT -1 47 ", then a parameterization for this curve is " }}{PARA 0 "" 0 "" {TEXT -1 43 " " }{XPPEDIT 18 0 "x=t, y = f(t)" "6$/%\"xG%\"tG/%\"yG-%\"fG6#F%" }{TEXT -1 7 ", f or " }{XPPEDIT 18 0 "a <= t,t <= b" "6$1%\"aG%\"tG1F%%\"bG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 44 "and, similarly if a curve is d escribed as " }{XPPEDIT 18 0 " x = g(y)" "6#/%\"xG-%\"gG6#%\"yG" } {TEXT -1 40 " then the parameterization has the form " }}{PARA 0 "" 0 "" {TEXT -1 50 " " } {XPPEDIT 18 0 "x=g(t), y= t" "6$/%\"xG-%\"gG6#%\"tG/%\"yGF(" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "For example,suppose" }} {PARA 0 "" 0 "" {TEXT -1 49 " \+ " }{XPPEDIT 18 0 "y = x^3 - 2 x " "6#/%\"yG,&*$%\"xG\"\"$\"\" \"*&\"\"#F)F'F)!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 103 "is given. Then \+ either of the following plot commands will give the same plot for any range, say, " }}{PARA 0 "" 0 "" {TEXT -1 47 " \+ " }{XPPEDIT 18 0 "-3 <= x,x <= 3" "6$1,$\" \"$!\"\"%\"xG1F'\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(x^3-2*x,x=-3..3); " }}{PARA 13 "" 1 "" {GLPLOT2D 252 252 252 {PLOTDATA 2 "6%-%'CURVESG6$ 7S7$$!\"$\"\"!$!#@F*7$$!1+++vq@pG!#:$!1o2C]F@)y\"!#97$$!1++D^NUbFF0$!1 '*4!\\vJ4a\"F37$$!1++]K3XFEF0$!16C'z&*p$)G\"F37$$!1++]F)H')\\#F0$!1xZ$ )oj?g5F37$$!1++D'3@/P#F0$!1ZuM'o2$y&)F07$$!1++Dr^b^AF0$!1iX))\\,:6pF07 $$!1++D,kZG@F0$!1(>.>y@fQ&F07$$!1++Dh\")=,?F0$!1%HSX3!*=,%F07$$!1++DO \"3V(=F0$!1R*HUH')e$GF07$$!1+++NkzViUC\"F0$\"1)Hl\"33s@cFeo7$$!1++DhkaI6F0$\"1u)\\ !4e,h\")Feo7$$!1+++]XF`**Feo$\"1MQ)*[rg/5F07$$!1++++Az2))Feo$\"1;wl_XF y5F07$$!1++]7RKvuFeo$\"1-?$o=St2\"F07$$!1-+++P'eH'Feo$\"1!4_f#zh45F07$ $!1****\\7*3=+&Feo$\"1V6u\\1E_()Feo7$$!1)***\\PFcpPFeo$\"1r=Umb[.qFeo7 $$!1)****\\7VQ[#Feo$\"1:M*y7YW\"[Feo7$$!1)***\\i6:.8Feo$\"1i^[t@<%e#Fe o7$$!1b+++v`hH!#=$\"1O-BD!\\I#fF`s7$$\"1++](QIKH\"Feo$!1@w<5A$[c#Feo7$ $\"1****\\7:xWCFeo$!1%QSqG@Mu%Feo7$$\"1,++vuY)o$Feo$!17>X5m7voFeo7$$\" 1)******4FL(\\Feo$!1uMA[Bb;()Feo7$$\"1)****\\d6.B'Feo$!16wzm;A/5F07$$ \"1++](o3lW(Feo$!1B'G)p\"*Qw5F07$$\"1*****\\A))oz)Feo$!1RJ'pJG'y5F07$$ \"1+++Ik-,5F0$!1;'HC3/(*)**Feo7$$\"1+++D-eI6F0$!1#*>;1kRg\")Feo7$$\"1+ +v=_(zC\"F0$!1ny))QB,BbFeo7$$\"1+++b*=jP\"F0$!1\"4/z3/aX\"Feo7$$\"1++v 3/3(\\\"F0$\"1dmMZFq6OFeo7$$\"1++vB4JB;F0$\"1.P.Iq-J5F07$$\"1+++DVsYw7#F0$\"1YcqV())fP&F07$$\"1++v)Q?QD#F0$\"1v% \\6&\\5TpF07$$\"1+++5jypBF0$\"1LTdy)z)o&)F07$$\"1++]Ujp-DF0$\"1q\\MG=- n5F37$$\"1+++gEd@EF0$\"12@cA))Rx7F37$$\"1++v3'>$[FF0$\"1S

N@E:F37$$ \"1++D6EjpGF0$\"1='R10c\"*y\"F37$$\"\"$F*$\"#@F*-%'COLOURG6&%$RGBG$\"# 5!\"\"F*F*-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "pl ot([t,t^3-2*t,t=-3..3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 252 252 252 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"$\"\"!$!#@F*7$$!1+++vq@pG!#:$!1o2C] F@)y\"!#97$$!1++D^NUbFF0$!1'*4!\\vJ4a\"F37$$!1++]K3XFEF0$!16C'z&*p$)G \"F37$$!1++]F)H')\\#F0$!1xZ$)oj?g5F37$$!1++D'3@/P#F0$!1ZuM'o2$y&)F07$$ !1++Dr^b^AF0$!1iX))\\,:6pF07$$!1++D,kZG@F0$!1(>.>y@fQ&F07$$!1++Dh\")=, ?F0$!1%HSX3!*=,%F07$$!1++DO\"3V(=F0$!1R*HUH')e$GF07$$!1+++NkzViUC\"F0$\"1)Hl\"33s@ cFeo7$$!1++DhkaI6F0$\"1u)\\!4e,h\")Feo7$$!1+++]XF`**Feo$\"1MQ)*[rg/5F0 7$$!1++++Az2))Feo$\"1;wl_XFy5F07$$!1++]7RKvuFeo$\"1-?$o=St2\"F07$$!1-+ ++P'eH'Feo$\"1!4_f#zh45F07$$!1****\\7*3=+&Feo$\"1V6u\\1E_()Feo7$$!1)** *\\PFcpPFeo$\"1r=Umb[.qFeo7$$!1)****\\7VQ[#Feo$\"1:M*y7YW\"[Feo7$$!1)* **\\i6:.8Feo$\"1i^[t@<%e#Feo7$$!1b+++v`hH!#=$\"1O-BD!\\I#fF`s7$$\"1++] (QIKH\"Feo$!1@w<5A$[c#Feo7$$\"1****\\7:xWCFeo$!1%QSqG@Mu%Feo7$$\"1,++v uY)o$Feo$!17>X5m7voFeo7$$\"1)******4FL(\\Feo$!1uMA[Bb;()Feo7$$\"1)**** \\d6.B'Feo$!16wzm;A/5F07$$\"1++](o3lW(Feo$!1B'G)p\"*Qw5F07$$\"1*****\\ A))oz)Feo$!1RJ'pJG'y5F07$$\"1+++Ik-,5F0$!1;'HC3/(*)**Feo7$$\"1+++D-eI6 F0$!1#*>;1kRg\")Feo7$$\"1++v=_(zC\"F0$!1ny))QB,BbFeo7$$\"1+++b*=jP\"F0 $!1\"4/z3/aX\"Feo7$$\"1++v3/3(\\\"F0$\"1dmMZFq6OFeo7$$\"1++vB4JB;F0$\" 1.P.Iq-J5F07$$\"1+++DVsYw7#F0$\"1YcqV())fP& F07$$\"1++v)Q?QD#F0$\"1v%\\6&\\5TpF07$$\"1+++5jypBF0$\"1LTdy)z)o&)F07$ $\"1++]Ujp-DF0$\"1q\\MG=-n5F37$$\"1+++gEd@EF0$\"12@cA))Rx7F37$$\"1++v3 '>$[FF0$\"1S

N@E:F37$$\"1++D6EjpGF0$\"1='R10c\"*y\"F37$$\"\"$F*$\"#@ F*-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$%!GFe[l-%%VIEWG6$%(D EFAULTGFi[l" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Similarly if " } {XPPEDIT 18 0 "x = y^2-y" "6#/%\"xG,&*$%\"yG\"\"#\"\"\"F'!\"\"" } {TEXT -1 77 " is the function you wish to plot then the following com mand does the trick." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot([t^2-t,t,t=-2..2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 252 252 252 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"'\"\"!$ !\"#F*7$$\"18Har(e;d&!#:$!1LLL$Q6G\">F07$$\"1;DH)yI8@&F0$!1nm;M!\\p$=F 07$$\"1&3\"4n^&)>[F0$!1LLL))Qj^@F+([SWF0$!1LLL=Kvl;F07$$\"1 #RR8*ycxSF0$!1nm;C2G!e\"F07$$\"1IyC)>[Tv$F0$!1LL$3yO5]\"F07$$\"17D\"*= 1]KMF0$!1++]nU)*=9F07$$\"1q?rKh,9JF0$!1LL$3WDTL\"F07$$\"1WY*R'e)3\"GF0 $!1++]d(Q&\\7F07$$\"1\")\\(==4S^#F0$!1nmmc4`i6F07$$\"1*fO&p61lAF0$!1LL LQW*e3\"F07$$\"1<))\\a(f))*>F0$!1,+++()>'***!#;7$$\"1-,Q2PaY$*Fdo$!1ommm9'=(eF do7$$\"10`j7c7nuFdo$!1,++v#\\N)\\Fdo7$$\"1zc1**o#*efFdo$!1pmmmCC(>%Fdo 7$$\"1^kuzX9$Fdo$!1+++D=/8DFdo7 $$\"1L/wzV4I>Fdo$!1mmm;a*el\"Fdo7$$\"1:nO%GJCW*!#<$!1pmm;Wn(o)Fjr7$$\" 1t<;CkDy>!#=$!1qLLL$eV(>F`s7$$!11H/,rAyyFjr$\"1Mmm;f`@')Fjr7$$!13(H1L2 UO\"Fdo$\"1)****\\nZ)H;Fdo7$$!1#)H[!)3Ka=Fdo$\"1lmm;$y*eCFdo7$$!1/eR\" Hji@#Fdo$\"1*******R^bJ$Fdo7$$!1*['fC2NGCFdo$\"1'*****\\5a`TFdo7$$!1B% *>IG()*\\#Fdo$\"1(****\\7RV'\\Fdo7$$!1QeTT![_U#Fdo$\"1'*****\\@fkeFdo7 $$!1YU=%eO*>AFdo$\"1JLLL&4Nn'Fdo7$$!1v(R[&3Ec=Fdo$\"1*******\\,s`(Fdo7 $$!1bRg&ppyR\"Fdo$\"1lmm\"zM)>$)Fdo7$$!17fn$HObc(Fjr$\"1*******pfa<*Fd o7$$!1D]T;dgU>F`s$\"1HLLeg`!)**Fdo7$$\"1]Mgz>`'*))Fjr$\"1++]#G2A3\"F07 $$\"1-)HC-v`\">Fdo$\"1LLL$)G[k6F07$$\"1[GU/5FMJFdo$\"1++]7yh]7F07$$\"1 pIZb#*[[WFdo$\"1nmm')fdL8F07$$\"1bm#e!*=[$fFdo$\"1nmm,FT=9F07$$\"1s'y \\Q.5b(Fdo$\"1LL$e#pa-:F07$$\"17&[p1B4;*Fdo$\"1+++Sv&)z:F07$$\"17Y*Gl3 `6\"F0$\"1LLLGUYo;F07$$\"16US()Hz18F0$\"1nmm1^rZF07$$\"\"#F*Fgz-%'COLOU RG6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$%!GFc[l-%%VIEWG6$%(DEFAULTGFg[ l" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Straight lines are very easy to parameterize. For example, if " }{XPPEDIT 18 0 "y = m*x+b" "6#/%\"yG ,&*&%\"mG\"\"\"%\"xGF(F(%\"bGF(" }{TEXT -1 37 " is the equation of t he line then " }{XPPEDIT 18 0 "x=t, y=mt+b" "6$/%\"xG%\"tG/%\"yG,&%# mtG\"\"\"%\"bGF*" }{TEXT -1 223 " gives a parameterization. But you can sometimes write the parametric equations directly from their geo metric description. For example, a portion of the straight line thr ough the point (1,3) parallel to the vector <" }{XPPEDIT 18 0 "2,5" " 6$\"\"#\"\"&" }{TEXT -1 28 "> has parametric equations " }{XPPEDIT 18 0 "x = 1 + 2t, y = 3 + 5t" "6$/%\"xG,&\"\"\"\"\"\"*&\"\"#F'%\"tGF' F'/%\"yG,&\"\"$F'*&\"\"&F'F*F'F'" }{TEXT -1 36 " and can be plotted b y the command:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "plot([1+2*t,3+5*t,t=-1..1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 252 252 252 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"\"\"\"!$! \"#F*7$$!1LLLLQ6G\"*!#;$!1LLLe%G?y\"!#:7$$!1mmmT.\\p$)F0$!1mmT&esBf\"F 37$$!1LLL$))Qj^(F0$!1LL$3s%3z8F37$$!1KLL$=Kvl'F0$!1LL$e/$Qk6F37$$!1omm Ts!G!eF0$!1nm;/\"=q]*F07$$!1MLL3yO5]F0$!1LL$3_>f_(F07$$!1+++vE%)*=%F0$ !1****\\(o1YZ&F07$$!1MLL3WDTLF0$!1OL$3-OJN$F07$$!1,++vvQ&\\#F0$!1,+]P* o%Q7F07$$!1mmmm&4`i\"F0$\"1oLLL3En$*!#<7$$!1KLLLQW*e)F[o$\"1lmmT!RE&GF 07$$\"1I#*******H,Q!#>$\"1)*****\\K]4]F07$$\"1(*******\\*3q)F[o$\"1(** ***\\PAvrF07$$\"1++++(=\\q\"F0$\"1,++]nHi#*F07$$\"1nmm\"fBIY#F0$\"1nm \"z*ev:6F37$$\"1LLLLO[kLF0$\"1LLL347T8F37$$\"1KLLL&Q\"GTF0$\"1LLLLY.K: F37$$\"1*****\\s]k,&F0$\"1++D\"o7Tv\"F37$$\"1JLLLvv-eF0$\"1LLL$Q*o]>F3 7$$\"1,++D2YlmF0$\"1++D\"=lj;#F37$$\"1+++v\"ep[(F0$\"1++vV&RY2MF37$$\"1mmmJy*eC\"F3$\"1mm;zXu9OF37$$\"1+++S^bJ8F 3$\"1+++]y))GQF37$$\"1+++0TN:9F3$\"1****\\i_QQSF37$$\"1++]7RV'\\\"F3$ \"1***\\7y%3TUF37$$\"1+++:#fke\"F3$\"1****\\P![hY%F37$$\"1LLL`4Nn;F3$ \"1LLL$Qx$oYF37$$\"1+++],s`$=F3$\"1mm\" zpe*z]F37$$\"1+++qfa<>F3$\"1*****\\#\\'QH&F37$$\"1LL$eg`!)*>F3$\"1KLe9 S8&\\&F37$$\"1++]#G2A3#F3$\"1++D1#=bq&F37$$\"1LLL$)G[k@F3$\"1LLL3s?6fF 37$$\"1++]7yh]AF3$\"1++DJXaEhF37$$\"1nmm')fdLBF3$\"1nmmm*RRL'F37$$\"1n mm,FT=CF3$\"1mm;a<.YlF37$$\"1LL$e#pa-DF3$\"1LLe9tOcnF37$$\"1+++Sv&)zDF 3$\"1******\\Qk\\pF37$$\"1LLLGUYoEF3$\"1LL$3dg6<(F37$$\"1nmm1^rZFF3$\" 1mmmmxGptF37$$\"1++]sI@KGF3$\"1++D\"oK0e(F37$$\"1++]2%)38HF3$\"1++v=5s #y(F37$$\"\"$F*$\"\")F*-%'COLOURG6&%$RGBG$\"#5F)F*F*-%+AXESLABELSG6$%! GFd[l-%%VIEWG6$%(DEFAULTGFh[l" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "Lissajous figure" {TEXT -1 168 "On occasion it is possible to plot curves that would be very c omplicated to express in even implicit algebraic terms. For example, \+ the following curve is known as a " }{TEXT 257 16 "Lissajous figure " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot([cos(3*t),sin(5*t),t=0..2*Pi],scaling=co nstrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 252 252 252 {PLOTDATA 2 "6&- %'CURVESG6$7_y7$$\"\"\"\"\"!F*7$$\"1'\\Vaz#HZ**!#;$\"1%pmOe%f._nu()F.$\"1,n&*zkX/uF.7$$\"1(QAvee :J)F.$\"11\"*Qg%>*>$)F.7$$\"11g:^p4#y(F.$\"1!H\"y-O7^!*F.7$$\"1nn&)4m^ !>(F.$\"1S=_T^(=e*F.7$$\"1_f7(oOB$oF.$\"1Fej*>qwy*F.7$$\"1j+kV]!pX'F.$ \"1*Rp$G5%[#**F.7$$\"1?%z\"['p^1'F.$\"1O'pF)eU#***F.7$$\"1)fmmj>\"ecF. $\"1j(zb!4&**)**F.7$$\"1rVxhIA-[F.$\"1\\&zf&>Qv(*F.7$$\"1Ii$GXay*QF.$ \"1O`Mb79(G*F.7$$\"1O)>LZwx%HF.$\"1uBa$>8J`)F.7$$\"1SE*z>\\v'>F.$\"1FN ::>-PvF.7$$\"1G3eX?)>n*!#<$\"1)=oW0CrK'F.7$$!1jHzs2^/V!#=$\"1Oz?29uP\\ F.7$$!1LNMfg1[5F.$\"1&*4.f3#eT$F.7$$!1\"Q;\"e\"pC/#F.$\"1Leg'p=zz\"F.7 $$!1m:J0-=;IF.$\"1+%[`sl\\H\"Fip7$$!1UuIJYLfRF.$!1)z:LHkDa\"F.7$$!1)[5 :7F\")z%F.$!1eB*[^h]0$F.7$$!1(>nJpJ^f&F.$!1\"))p4GXP\\%F.7$$!1JXYiqSVj F.$!1p&z#*\\bQ#eF.7$$!1>xmQjVOqF.$!1D,5F`D8qF.7$$!1$4tfcz$*o(F.$!19)eO TFf1)F.7$$!1%4(*\\(p_q#)F.$!1#p'zg**o4*)F.7$$!1\\\"RiQ_Wx)F.$!1XiH2$*o A&*F.7$$!1%=3Gk]k>*F.$!1&*4h;'[!*))*F.7$$!1...y$GWyk$y**F.7$$!11Mc\\m\"[Y*F.$!1!o388(3(***F.7$$!1 XL,s!>Da*F.$!1xlfncY)***F.7$$!1>?qv5E9'*F.$!1'yz;+)\\#)**F.7$$!1w3O\\y **z'*F.$!1+\\kN=@\\**F.7$$!1(e*oK$)oR(*F.$!12!4s\"\\m)*)*F.7$$!1#*H+U_ H$z*F.$!1_R`R\\%4$)*F.7$$!1;+]%)[FY**F.$!1=)*=Ek7\"R*F.7$$!1AM:M.$**** *F.$!1b5$HB#>\"p)F.7$$!1,_5]L.a**F.$!1?Wj,Y!Rv(F.7$$!1LuF72P4)*F.$!1Tj gg7A.mF.7$$!1vdd)yxtc*F.$!1!4_/C53F&F.7$$!1Gf'\\nb/B*F.$!1@0)4\"4M$z$F .7$$!1y^A]TP)y)F.$!1AuD#\\)3l@F.7$$!1obB:S.a#)F.$!1'QZ/6\"*zt%Fip7$$!1 eZ%omWIj(F.$\"1!yN7-&GJ7F.7$$!1K:$pF.$\"1ShOc+_+HF.7$$!1`\")zP@%Q D'F.$\"1-bQ+?Q-VF.7$$!1d*>*3g\"[_&F.$\"1J:U%Q4qg&F.7$$!18u\\6`y]ZF.$\" 19C@4u\"\\y'F.7$$!1G+'z@b!QRF.$\"1*>@W:&[4yF.7$$!1FTLN@2&)HF.$\"1okXG5 r]()F.7$$!12u7SyF,?F.$\"1wT&y5C9W*F.7$$!1v'epDo#o**Fip$\"1(*ypg3&=')*F .7$$\"1h*\\13?8z\"F_q$\"1**\\$HVb*****F.7$$\"1d'onO>m.\"F.$\"1n)z7p)e] )*F.7$$\"1$=;\">y`W?F.$\"1A!Rf=2pT*F.7$$\"1f_,C&z6.$F.$\"1q(4R9G9r)F.7 $$\"1O$eo'oF')RF.$\"1Z#[yg8Xv(F.7$$\"1o+$*G\"\\v'[F.$\"1%p1N(F.$\"1ovX4B(oH#F.7$$\"19*f;&*4zy(F.$\"1DTgzJy!G)Fip7$$\"1())* *[eljJ)F.$!1$[uhk)H!f'Fip7$$\"14A17&G&y()F.$!1#p7chi:8#F.7$$\"1I:=oYrq \"*F.$!1e5%)pE&pb$F.7$$\"14))))>!)pT&*F.$!1wp\"QM7q9&F.7$$\"1(\\.QZl^! )*F.$!17Em?&Rid'F.7$$\"1>gUx#[\"e**F.$!1FF#H3u**z(F.7$$\"19;\")>E#*)** *F.$!17/o#4x*z()F.7$$\"1)fq(\\oHX**F.$!1JMb(=,lR*F.7$$\"17Jkb$H7\")*F. $!1&4%>g$f@!)*F.7$$\"1')y?ZjDl(*F.$!1%[&Gd>Xp)*F.7$$\"1$)[TBOM9(*F.$!1 mc(RvxG#**F.7$$\"1[\"f#Qp^e'*F.$!17s#eohB'**F.7$$\"1zJWJX!yf*F.$!1BT.x #[y)**F.7$$\"1%)>49rBK&*F.$!1'fm$=\"zpF.7$$ \"1S7XI!=oK'F.$!19r#\\&*pZz&F.7$$\"1aK$\\*RM$e&F.$!1/n\"4EnDZ%F.7$$\"1 %*)HRg#*>z%F.$!1,?V$[lR/$F.7$$\"1&\\M7s\\&fRF.$!1y_N2+&Ha\"F.7$$\"1&*R 5R3W2IF.$\"1\"e*QT+qZ9Fip7$$\"1H`$\\\"HHC?F.$\"1\\?d*)fMG=F.7$$\"1([&o rFD?5F.$\"1bZPkdefMF.7$$\"11fngc&Go&!#>$\"1pLZi_z\"*\\F.7$$!1n%>Uetjg* Fip$\"19r],Hh=jF.7$$!1@vzPo'z\">F.$\"1PX^R'*Q\"[(F.7$$!1Rak1eMdGF.$\"1 [W[0!Q*\\%)F.7$$!1)='R!\\#)*pPF.$\"1*F.7$$!1byb/9[%o%F.$\"1h? 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For example " }}{PARA 0 "" 0 "" {TEXT -1 40 " \+ " }{XPPEDIT 18 0 "x= cos(2t), y = -2 sin(2t)" "6$/% \"xG-%$cosG6#*&\"\"#\"\"\"%\"tGF*/%\"yG,$*&\"\"#F*-%$sinG6#*&\"\"#F*F+ F*F*!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "is a soluti on of the initial value problem:" }}{PARA 0 "" 0 "" {TEXT -1 32 " \+ " }{XPPEDIT 18 0 "D(x)(t) = y(t), D(y)(t) = -4y(t)" "6$/--%\"DG6#%\"xG6#%\"tG-%\"yG6#F*/--F&6#F,6#F*,$*&\"\"%\" \"\"-F,6#F*F6!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "x(0) = 1, y(0) \+ = 0" "6$/-%\"xG6#\"\"!\"\"\"/-%\"yG6#F'F'" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 60 "as can easily be verified. A plot of this curve i s given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "plot([cos(2*t),-2*sin(2*t),t=0..Pi],scaling = const rained);" }}{PARA 13 "" 1 "" {GLPLOT2D 252 252 252 {PLOTDATA 2 "6&-%'C URVESG6$7S7$$\"\"\"\"\"!F*7$$\"1#4hRPij!**!#;$!1H`6ljbIFF.7$$\"18J#))4 -Qn*F.$!1])*=>zdm]F.7$$\"1N5')yke[#*F.$!1nEr\\1A1wF.7$$\"1goz=42`')F.$ !1.2*y=%[-5!#:7$$\"1b](G._U!zF.$!1\"f/S8<^A\"F@7$$\"1]$R'oTd#3(F.$!1t% y,a3>T\"F@7$$\"1H$>jubk6'F.$!1VU!*4QE#e\"F@7$$\"1GuIc:x5]F.$!1RQvF_!3t \"F@7$$\"1C3nW'R,#QF.$!12?@-JJ[=F@7$$\"1n9AwiQDDF.$!1xi*oit^$>F@7$$\"1 B^hAo8X8F.$!1?([FZB=)>F@7$$!1qB/u(p5(f!#>$!1EKYV'*****>F@7$$!1)\\T#fB[ i8F.$!1%*Q_d%\\8)>F@7$$!1crN]@=YEF.$!1%QO)4lqG>F@7$$!1zV1J*3Jx$F.$!1cX ;2L<_=F@7$$!1(\\AOF:B/&F.$!1Hdh9q8F+.2**F.$!1'3/,,]3s#F.7$$!1gKG4>&*****F.$!1]W)H ()=E?'!#=7$$!1`Is=!Q%3**F.$\"1&=)R7PE+FF.7$$!1_NDna1u'*F.$\"1r5bcgck]F .7$$!1_J.8AEj#*F.$\"1>qjsqWMvF.7$$!1v#)4Kh=u')F.$\"1`q!Hx\"e^**F.7$$!1 B*z7xng%zF.$\"1\\f!Q&)[U@\"F@7$$!1D84Pfc5rF.$\"1WOTi&piS\"F@7$$!1Wy6Ik e[gF.$\"1E:c_=F@7$$!1yCH\"el'3EF.$\"1*Q%>R(\\2$>F@7$$!177Cvnc\"H\"F .$\"1zOh)R[K)>F@7$$!1zyOHMQdIF`s$\"1@ZQ_1****>F@7$$\"1IAj`Ks(G\"F.$\"1 \\p,p$[L)>F@7$$\"1KRqX8/bDF.$\"1(p$*e#fhL>F@7$$\"1%)yb&Q)zNQF.$\"1TTVQ a,Z=F@7$$\"17w4w0I.]F.$\"1176A&p;t\"F@7$$\"1\\#)[*R]$4hF.$\"1\"zp$e9O$ e\"F@7$$\"1/wY'Q+$*4(F.$\"19x67Wa39F@7$$\"1f=s$Ry,!zF.$\"1heM5x;E7F@7$ $\"1S,@;vLu')F.$\"1R'f^%R0^**F.7$$\"1Le@`4+D#*F.$\"1\"R9M:P*>xF.7$$\"1 mGAp))oa'*F.$\"1?,NzwO5_F.7$$\"1zb'f`bp!**F.$\"1[RH7Y$>s#F.7$F($!1\\E7 Gu#3k\"!#C-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%(SCALINGG6#%,CONSTRAINEDG- %+AXESLABELSG6$%!GFf[l-%%VIEWG6$%(DEFAULTGFj[l" 1 2 0 1 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "When plotting curves in three dimensions using Maple V we need to use the command " }{TEXT 258 10 "spacecurve" }{TEXT -1 22 " w hich is part of the " }{TEXT 259 5 "plots" }{TEXT -1 57 " package. Co nsider the curve written parametrically as " }{XPPEDIT 18 0 "x = cos t , y= sin t, z=t" "6%/%\"xG*&%$cosG\"\"\"%\"tGF'/%\"yG*&%$sinGF'F(F' /%\"zGF(" }{TEXT -1 32 ". It's plot over the range [0," }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 13 "] is given by" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "plots[spacecur ve]" {MPLTEXT 1 0 41 "spacecurve([cos(t),sin(t),t],t=0..10*Pi);" }} {PARA 13 "" 1 "" {GLPLOT3D 252 252 252 {PLOTDATA 3 "6#-%'CURVESG6#7T7% $\"\"\"\"\"!F*F*7%$\"+>i89!)!#5$\"+0`5\")fF.$\"+zNT6kF.7%$\"+keFXGF.$ \"+J&yme*F.$\"+;FG#G\"!\"*7%$!+[0l`MF.$\"+>Uo%Q*F.$\"+uSUB>F:7%$!+_5)3 Q)F.$\"+3!\\`X&F.$\"+KackDF:7%$!+FRXz**F.$!+g?-2k!#6$\"+!z1d?$F:7%$!+z &fWh(F.$!+fRG#['F.$\"+[\"[o%QF:7%$!+J$4_A#F.$!+C\"z#\\(*F.$\"+1&*)z[%F :7%$\"+SM$y/%F.$!+Ei7W\"*F.$\"+k38H^F:7%$\"+Zq=8()F.$!+5b<2\\F.$\"+AAF qdF:7%$\"+P,!z\"**F.$\"+H;xy7F.$\"+!e89T'F:7%$\"+\"\\$\\$=(F.$\"+;b#o& pF.$\"+Q\\b_qF:7%$\"+O*)*ff\"F.$\"+Py\"=()*F.$\"+'H'p$p(F:7%$!+F.$\"+q.7<'*F:7%$!+t)3Is'F.$!+)**zFS(F.$\"+th#e-\"!\")7 %$!+#)>I-'*FN$!+N6z`**F.$\"+4.%**3\"Fhr7%$\"+ad#R=&F.$!+(eF9b)F.$\"+XW 0a6Fhr7%$\"+7w;p#*F.$!+`*pEv$F.$\"+\"eo\"=7Fhr7%$\"+*f[Hn*F.$\"+gfaODF .$\"+3^JF.$\"+h#R(Q:Fhr7%$!+(Rd0\\*F.$!++C3^JF.$\"+(R`Gg\"Fhr7%$ !+%Rm6s&F.$!+!pA&z\"Fhr7%$\"+6([Hn*F.$!+LbaODF.$ \"+T*4$f=Fhr7%$\"+Zu;p#*F.$\"+h.n_PF.$\"+xSUB>Fhr7%$\"+w`#R=&F.$\"+;yU ^&)F.$\"+8#Qv)>Fhr7%$!+sjI-'*FN$\"+$4\"z`**F.$\"+\\Bl^?Fhr7%$!++#4Is'F .$\"+,(zFS(F.$\"+&[md6#Fhr7%$!+^;f:)*F.$\"+see6>F.$\"+@1))z@Fhr7%$!+([ )o4!*F.$!+\"zP)QVF.$\"+dZ*RC#Fhr7%$!+!\\#QDYF.$!+zK*f'))F.$\"+$*)3\"3B Fhr7%$\"+I%**ff\"F.$!+dx\"=()*F.$\"+HIAsBFhr7%$\"+aQ\\$=(F.$!+U^#o&pF. $\"+lrLOCFhr7%$\"+1-!z\"**F.$!+$4r(y7F.$\"+,8X+DFhr7%$\"+rn=8()F.$\"+* *f<2\\F.$\"+PackDFhr7%$\"+4H$y/%F.$\"+hk7W\"*F.$\"+t&z'GEFhr7%$!+<*4_A #F.$\"+!**y#\\(*F.$\"+4Pz#p#Fhr7%$!+\")*fWh(F.$\"+'[$G#['F.$\"+Xy!pv#F hr7%$!+oRXz**F.$\"+jc,2kFN$\"+\")>-@GFhr7%$!+\"p!)3Q)F.$!+i&\\`X&F.$\" + " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "A complic ated figure can be obtained as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "spacecurve([(4+sin(3*t ))*cos(2*t),(4+sin(3*t))*sin(2*t),cos(3*t)]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "t=0..2*Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "The last example illustrates the fact \+ that when Maple V plots a graph it connects points with straight lines . We know that all points on the curve " }{XPPEDIT 18 0 "x = cos(t^3) " "6#/%\"xG-%$cosG6#*$%\"tG\"\"$" }{TEXT -1 4 ", y=" }{XPPEDIT 18 0 "s in(t^3)" "6#-%$sinG6#*$%\"tG\"\"$" }{TEXT -1 29 " lie on the unit circ le, but," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot([cos(t^3),sin(t^3),t=0..4*Pi]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Can you e xplain why the next plot is closer to what you might expect?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "p lot([cos(t^3),sin(t^3),t=0..4*Pi], numpoints=500);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "Exercises" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 79 "1. Make a Maple V plot of the con ical helix given by the parametric equations" }}{PARA 14 "" 0 "" {TEXT -1 33 " " }{XPPEDIT 18 0 "x = t* cos(3*t), y=t*sin(3*t), z=t" "6%/%\"xG*&%\"tG\"\"\"-%$cosG6#*&\"\"$F'F &F'F'/%\"yG*&F&F'-%$sinG6#*&\"\"$F'F&F'F'/%\"zGF&" }{TEXT -1 3 " " } {XPPEDIT 18 0 "-infinity < t,t < infinity" "6$2,$%)infinityG!\"\"%\"tG 2F'F%" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 111 "2. Write para metric equations for first octant portion of the curve of intersection of the sphere and cylinder" }}{PARA 14 "" 0 "" {TEXT -1 40 " \+ " }{XPPEDIT 18 0 "x^2 + y^2 + z^2 = 64 " "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F(*$%\"zG\"\"#F(\"#k" }{TEXT -1 12 ", " }{XPPEDIT 18 0 "x^2+(y-4)^2 =16" "6#/,&*$%\"xG\"\"# \"\"\"*$,&%\"yGF(\"\"%!\"\"\"\"#F(\"#;" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 35 "Make a Maple V plot of this curve. " }}{PARA 14 "" 0 "" {TEXT -1 14 "2. Show that " }{XPPEDIT 18 0 "x = sin(2t)*cos(t)" "6#/%\"xG*&-%$sinG6#*&\"\"#\"\"\"%\"tGF+F+-%$cosG6#F,F+" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "y = sin(2t)*sin(t)" "6#/%\"yG*&-%$sinG6#*&\"\"# \"\"\"%\"tGF+F+-F'6#F,F+" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "0<= t ,t <= 2*pi" "6$1\"\"!%\"tG1F%*&\"\"#\"\"\"%#piGF)" }{TEXT -1 81 " is a p arametric representation for the curve given implicitly by the equatio n " }}{PARA 14 "" 0 "" {TEXT -1 37 " \+ " }{XPPEDIT 18 0 "(x^2 + y^2)^3 = 4x^2y^2" "6#/*$,&*$%\"xG\"\"#\" \"\"*$%\"yG\"\"#F)\"\"$*(\"\"%F)*$F'\"\"#F)F+\"\"#" }{TEXT -1 1 "." }} {PARA 14 "" 0 "" {TEXT -1 186 "Plot the graph of this curve using Mapl e V by using the parametric representation and also by using the impli cit repesentation. Is it worthwhile to obtain the parametric represen tation?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{MARK "1 3 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 }