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2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cst yle607" -1 1051 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_ps tyle45" -1 244 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 } 0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle608" -1 1052 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle46" -1 245 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {PARA 226 "" 0 "" {TEXT 989 36 " Worksheet 5, Math 152 Fal l 2005" }{TEXT 989 0 "" }}{PARA 227 "" 0 "" {TEXT 990 0 "" }}{PARA 227 "" 0 "" {TEXT 990 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{SECT 0 {PARA 229 "" 0 "" {TEXT 992 47 " CURVES DEFINED BY PARAMETRIC & POLA R EQUATIONS" }{TEXT 992 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }} {PARA 230 "" 0 "" {TEXT 993 1 " " }{TEXT 994 5 "Goals" }{TEXT 995 78 " : In this worksheet, curves are defined by parametric and polar equat ions and" }{TEXT 996 0 "" }}{PARA 231 "" 0 "" {TEXT 997 47 " \+ then plotted and analyzed using " }{TEXT 998 5 "Maple" }{TEXT 997 3 ". " }{TEXT 997 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }} {PARA 232 "" 0 "" {TEXT 999 1 " " }{TEXT 1000 41 " I Parametric Equa tions for Curves and " }{TEXT 1001 5 "Maple" }{TEXT 1000 1 "." }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 49 " The most flexible way of \+ describing the curve " }{TEXT 1001 1 "C" }{TEXT 999 10 " in the " } {TEXT 1002 4 "xy -" }{TEXT 999 36 " plane, is with a pair of functions :" }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 22 " " }{TEXT 1001 1 "C" }{TEXT 1000 1 " " }{TEXT 1002 1 " " }{TEXT 1000 2 ": " }{TEXT 1002 2 " " } {TEXT 999 1 " " }{TEXT 1002 1 "x" }{TEXT 999 7 " = " }{TEXT 1002 2 "f " }{TEXT 999 2 "( " }{TEXT 1002 2 "t " }{TEXT 999 7 ") , " } {TEXT 1002 1 "y" }{TEXT 999 7 " = " }{TEXT 1002 1 "g" }{TEXT 999 3 " ( " }{TEXT 1002 1 "t" }{TEXT 999 20 " ) for " }{TEXT 1002 1 "t" }{TEXT 999 19 " in the interval " }{TEXT 1002 1 "I" } {TEXT 999 2 " ." }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }} {PARA 232 "" 0 "" {TEXT 999 35 " The ability to plot curves using " } {TEXT 1002 6 " Maple" }{TEXT 999 55 " is particularly helpful when th e given curve is very " }{TEXT 999 0 "" }}{PARA 233 "" 0 "" {TEXT 1003 14 " complicated." }{TEXT 1003 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 1 " " }{TEXT 1004 1 " " }{TEXT 1005 9 "Example 1" }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 35 " \+ Consider the graph of the curve " }{TEXT 1001 1 "C" }{TEXT 999 15 " \+ described by:" }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 1002 18 " \+ x " }{TEXT 999 6 "= cos " }{TEXT 1002 1 "t" }{TEXT 999 5 " , " }{TEXT 1002 2 " y" }{TEXT 999 9 " = sin ( " }{TEXT 1002 3 "t + " }{TEXT 999 6 " sin 5" }{TEXT 1002 2 "t " }{TEXT 999 2 ") " }{TEXT 1002 1 " " }{TEXT 999 17 " for 0 " }{TEXT 1006 1 "<" }{TEXT 999 1 " " }{TEXT 1002 2 " t" }{TEXT 999 7 " < 2 " }{XPPEDIT 18 0 "Pi :" "6#%#PiG" }{TEXT 999 5 " ." }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 50 " First, let's define the parametric functions in " } {TEXT 1002 5 "Maple" }{TEXT 999 1 "." }{TEXT 999 0 "" }}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 13 "f:=t->cos(t);" }{MPLTEXT 1 1007 0 "" }}}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 22 "g:=t->sin(t+sin(5*t ));" }{MPLTEXT 1 1007 0 "" }}}{PARA 228 "" 0 "" {TEXT 991 0 "" }} {PARA 232 "" 0 "" {TEXT 999 33 " Now we utilize the following " } {TEXT 1008 5 " plot" }{TEXT 999 41 " command to plot the parametric c urve, " }{TEXT 1001 3 "C ." }{TEXT 999 0 "" }}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 74 "plot([f(t),g(t),t=0..2*Pi],color=magenta,thick ness=2,scaling=constrained);" }{MPLTEXT 1 1007 0 "" }}}{EXCHG {PARA 228 "" 0 "" {TEXT 991 0 "" }}}{PARA 232 "" 0 "" {TEXT 999 71 " In o rder to simply animate the trace of this plot as the parameter " } {TEXT 1002 2 " t" }{TEXT 999 15 " ranges from " }{XPPEDIT 18 0 "0;" "6#\"\"!" }{TEXT 999 7 " to " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\" \"%#PiGF%" }{TEXT 999 5 " , " }{TEXT 999 0 "" }}{PARA 233 "" 0 "" {TEXT 1003 49 " we execute the following animation command: " } {TEXT 1003 0 "" }}{PARA 232 "" 0 "" {TEXT 999 1 " " }{TEXT 1009 2 " " }{TEXT 1010 77 "After executing this command, click on the displayed \+ graph, and then use the " }{TEXT 1011 5 "PLAY " }{TEXT 999 0 "" }} {PARA 235 "" 0 "" {TEXT 1012 32 " button to view the animation. " } {TEXT 1012 0 "" }}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 12 "with( plots):" }{MPLTEXT 1 1007 0 "" }}}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 99 "animate([f(d*t),g(d*t),t=0..2*Pi],d=0..1,frames=100,color=m agenta,thickness=2,scaling=constrained);" }{MPLTEXT 1 1007 0 "" }}} {EXCHG {PARA 228 "" 0 "" {TEXT 991 0 "" }}}{PARA 232 "" 0 "" {TEXT 999 3 " " }{TEXT 1013 1 " " }{TEXT 1014 7 "Remark:" }{TEXT 1000 1 " \+ " }{TEXT 999 25 "Increasing the value of " }{TEXT 1008 6 "frames" } {TEXT 999 52 " in the above animation command should improve the " } {TEXT 999 0 "" }}{PARA 233 "" 0 "" {TEXT 1003 102 " \+ accuracy of the plot. However, the amount of memory required to gener ate the plot " }{TEXT 1003 0 "" }}{PARA 232 "" 0 "" {TEXT 999 46 " \+ increases as the value of " }{TEXT 1008 7 " frames" } {TEXT 999 13 " increases. " }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 236 "" 0 "" {TEXT 1015 2 " " }{TEXT 1015 51 "Now recall the following lemma discu ssed in class: " }{TEXT 1015 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" } }{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 230 "" 0 "" {TEXT 1016 2 " \+ " }{TEXT 1017 5 "Lemma" }{TEXT 1016 4 " " }{TEXT 1018 148 " \+ \+ \+ " }{TEXT 996 0 "" }}{PARA 232 "" 0 "" {TEXT 1001 19 " If " }{TEXT 999 23 "a simple closed curve " }{TEXT 1001 1 "C" }{TEXT 999 32 " is described parametrically by" }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 43 " " } {TEXT 1001 1 "C" }{TEXT 1002 1 " " }{TEXT 999 1 ":" }{TEXT 1001 1 " " }{TEXT 999 4 " " }{TEXT 1002 2 "x " }{TEXT 999 4 " = " }{TEXT 1002 2 "f " }{TEXT 999 3 " ( " }{TEXT 1002 1 "t" }{TEXT 999 8 " ) , \+ " }{TEXT 1002 1 "y" }{TEXT 999 5 " = " }{TEXT 1002 1 "g" }{TEXT 999 3 " ( " }{TEXT 1002 1 "t" }{TEXT 999 19 " ) for " } {TEXT 1002 1 "a" }{TEXT 999 2 " " }{TEXT 1006 1 "<" }{TEXT 999 2 " " }{TEXT 1002 1 "t" }{TEXT 999 2 " " }{TEXT 1006 1 "<" }{TEXT 999 2 " \+ " }{TEXT 1002 1 "b" }{TEXT 999 5 " , " }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 29 " where " }{TEXT 1002 3 "f \+ '" }{TEXT 999 3 " ( " }{TEXT 1002 2 "t " }{TEXT 999 5 ") , " }{TEXT 1002 1 "g" }{TEXT 999 3 " ( " }{TEXT 1002 1 "t" }{TEXT 999 29 " ) are both continuous on [" }{TEXT 1002 2 " a" }{TEXT 999 5 " , " } {TEXT 1002 1 "b" }{TEXT 999 4 " ] ," }{TEXT 1001 1 " " }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 1001 29 " R" } {TEXT 999 36 " is the bounded region interior to " }{TEXT 1001 1 "C" }{TEXT 999 7 " , and:" }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 18 " " }{TEXT 1019 5 "( i )" }{TEXT 1000 3 " " }{TEXT 999 1 " " }{TEXT 1001 1 "C" }{TEXT 999 29 " has positive orientation , " }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 35 " ( i.e. " }{TEXT 1001 1 "R" }{TEXT 999 19 " is always on the " }{TEXT 1002 15 "Right H and Side" }{TEXT 999 5 " of " }{XPPEDIT 18 0 "P[t];" "6#&%\"PG6#%\"tG " }{TEXT 999 9 " = ( " }{TEXT 1002 1 "f" }{TEXT 999 3 " ( " } {TEXT 1002 1 "t" }{TEXT 999 5 " ) , " }{TEXT 1002 1 "g" }{TEXT 999 3 " ( " }{TEXT 1002 1 "t" }{TEXT 999 4 " ) )" }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 87 " \+ as " }{TEXT 1002 1 "t" }{TEXT 999 15 " ranges from " }{TEXT 1002 1 "a" }{TEXT 999 6 " to " } {TEXT 1002 1 "b" }{TEXT 999 3 " ) " }{TEXT 999 0 "" }}{PARA 237 "" 0 " " {TEXT 1020 0 "" }}{PARA 232 "" 0 "" {TEXT 999 18 " \+ " }{TEXT 1001 6 " then " }{TEXT 999 20 "the area of region " }{TEXT 1001 1 "R" }{TEXT 999 17 " is given by : " }{TEXT 1004 2 " " } {TEXT 1021 6 "Area (" }{TEXT 1022 1 " " }{TEXT 1021 4 "R ) " }{TEXT 1022 1 " " }{TEXT 1004 4 " = " }{XPPEDIT 300 0 "Int(g(t)*` f ' ( t ) \+ `,t = a .. b):" "6#-%$IntG6$*&-%\"gG6#%\"tG\"\"\"%,~f~'~(~t~)~GF+/F*;% \"aG%\"bG" }{TEXT 1002 2 " " }{TEXT 1023 4 " ." }{TEXT 999 0 "" }} {PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 16 " \+ " }{TEXT 1019 7 "( i i )" }{TEXT 1000 3 " " }{TEXT 999 1 " " }{TEXT 1001 1 "C" }{TEXT 999 28 " has negative orientation ," } {TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 35 " \+ ( i.e. " }{TEXT 1001 1 "R" }{TEXT 999 18 " is always on the" } {TEXT 1002 15 " Left Hand Side" }{TEXT 999 5 " of " }{XPPEDIT 18 0 "P [t];" "6#&%\"PG6#%\"tG" }{TEXT 999 9 " = ( " }{TEXT 1002 1 "f" } {TEXT 999 3 " ( " }{TEXT 1002 1 "t" }{TEXT 999 5 " ) , " }{TEXT 1002 1 "g" }{TEXT 999 3 " ( " }{TEXT 1002 1 "t" }{TEXT 999 4 " ) )" }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 88 " \+ as " }{TEXT 1002 1 "t" }{TEXT 999 15 " ranges from " }{TEXT 1002 1 "a" }{TEXT 999 6 " to " }{TEXT 1002 1 "b" }{TEXT 999 3 " ) " }{TEXT 999 0 "" }} {PARA 237 "" 0 "" {TEXT 1020 0 "" }}{PARA 232 "" 0 "" {TEXT 999 20 " \+ " }{TEXT 1001 4 "then" }{TEXT 999 22 " the area of \+ region " }{TEXT 1001 1 "R" }{TEXT 999 19 " is given by : " } {TEXT 1021 10 "Area ( R )" }{TEXT 1004 5 " = " }{XPPEDIT 302 0 "-Int (g(t)*` f ' ( t ) `,t = a .. b):" "6#,$-%$IntG6$*&-%\"gG6#%\"tG\"\"\"% ,~f~'~(~t~)~GF,/F+;%\"aG%\"bG!\"\"" }{TEXT 1002 5 " ." }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 17 " " }{TEXT 1024 148 " \+ \+ " }{TEXT 999 0 "" }}{PARA 233 "" 0 "" {TEXT 1003 6 " \+ " }{TEXT 1003 0 "" }}{PARA 233 "" 0 "" {TEXT 1003 5 " " }{TEXT 1003 0 "" }}{PARA 238 "" 0 "" {TEXT 1025 1 " " }{TEXT 1026 1 " " } {TEXT 1027 7 "Example" }{TEXT 1026 1 " " }{TEXT 1027 1 "2" }{TEXT 1025 0 "" }}{PARA 233 "" 0 "" {TEXT 1003 25 " Consider the equation : " }{TEXT 1003 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 1028 40 " " }{TEXT 1029 3 " " }{TEXT 1008 5 "( * )" }{TEXT 1029 1 " " }{TEXT 1028 1 " " }{TEXT 999 5 " " }{TEXT 1000 2 " " }{XPPEDIT 837 0 "x^3+y^3=6*x* y" "6#/,&*$%\"xG\"\"$\"\"\"*$%\"yGF'F(*(\"\"'F(F&F(F*F(" }{TEXT 1002 2 " " }{TEXT 1023 2 " " }{TEXT 1002 1 "." }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 26 " It's g raph is called a " }{TEXT 1028 1 " " }{TEXT 1005 19 "folium of Descart es" }{TEXT 1030 1 " " }{TEXT 999 1 "." }{TEXT 999 0 "" }}{PARA 233 "" 0 "" {TEXT 1003 94 " It can be shown (see exercises) that this curve \+ has the following parametric representation:" }{TEXT 1003 0 "" }} {PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 15 " \+ " }{TEXT 1029 11 " " }{TEXT 1008 8 " ( ** ) " } {TEXT 1029 2 " " }{TEXT 999 3 " " }{TEXT 1004 1 " " }{TEXT 1005 1 " x" }{TEXT 1011 2 " " }{TEXT 1031 3 "= " }{TEXT 1022 1 " " }{XPPEDIT 298 0 "6t/(1+t^3):" "6#*(\"\"'\"\"\"%\"tGF%,&F%F%*$F&\"\"$F%!\"\"" } {TEXT 1000 11 " , " }{TEXT 1022 1 " " }{TEXT 1031 1 " " } {TEXT 1005 2 "y " }{TEXT 1031 4 " = " }{TEXT 1022 2 " " }{XPPEDIT 289 0 "6*t^2/(1+t^3)" "6#*(\"\"'\"\"\"*$%\"tG\"\"#F%,&F%F%*$F'\"\"$F%! \"\"" }{TEXT 1002 1 " " }{TEXT 1001 1 " " }{TEXT 1021 1 " " }{TEXT 1001 11 " , for " }{XPPEDIT 18 0 "-infinity:" "6#,$%)infinityG!\" \"" }{TEXT 999 1 " " }{TEXT 1001 13 " < t < + " }{XPPEDIT 18 0 "in finity:" "6#%)infinityG" }{TEXT 999 1 " " }{TEXT 1001 10 " , " }{XPPEDIT 18 0 "t<>-1" "6#0%\"tG,$\"\"\"!\"\"" }{TEXT 999 1 " " } {TEXT 1001 5 " ." }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 " " }}{PARA 233 "" 0 "" {TEXT 1003 29 " We plot this folium below." } {TEXT 1003 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 8 "restart:" }{MPLTEXT 1 1007 0 "" }}} {EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 18 "f:=t->6*t/(1+t^3);" } {MPLTEXT 1 1007 0 "" }}{PARA 234 "> " 0 "" {MPLTEXT 1 1007 20 "g:=t->6 *t^2/(1+t^3);" }{MPLTEXT 1 1007 0 "" }}}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 134 "plot([[f(t),g(t),t=0..100],[f(t),g(t),t=-1/2..0], [f(t),g(t),t=-100..-2]], color=[magenta,green,blue],thickness=3,scalin g=constrained);" }{MPLTEXT 1 1007 0 "" }}}{EXCHG {PARA 228 "" 0 "" {TEXT 991 0 "" }}}{PARA 232 "" 0 "" {TEXT 999 29 " Convince yourself \+ that as " }{TEXT 1002 1 "t" }{TEXT 999 37 " ranges over the interval ( 0 , + " }{XPPEDIT 18 0 "infinity:" "6#%)infinityG" }{TEXT 999 45 " ) , the portion of the folium in the first" }{TEXT 999 0 "" }} {PARA 232 "" 0 "" {TEXT 999 29 " quadrant is traced out, as " }{TEXT 1002 2 " t" }{TEXT 999 31 " ranges over the interval ( " }{XPPEDIT 18 0 "-1;" "6#,$\"\"\"!\"\"" }{TEXT 999 45 " , 0 ) the portion in the second quadrant is" }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 19 " traced, and as " }{TEXT 1002 2 "t " }{TEXT 999 29 " ranges over t he interval ( " }{XPPEDIT 18 0 "-infinity" "6#,$%)infinityG!\"\"" } {TEXT 999 7 " , " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT 999 52 " ) the portion in the fourth quadrant is traced. " }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 98 " Further note that the loo p in the folium is traced-out in the \"counter-clockwise\" direction a s " }{TEXT 1002 1 "t" }{TEXT 999 9 " ranges " }{TEXT 999 0 "" }} {PARA 232 "" 0 "" {TEXT 999 29 " over the interval [ 0 , + " } {XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT 999 8 " ) . " } {TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 6 " From" }{TEXT 1000 1 " " }{TEXT 999 5 " part" }{TEXT 1000 1 " " }{TEXT 999 25 "( ii ) of the lemma above" }{TEXT 1000 1 "," }{TEXT 999 59 " it follows that the area enclosed by the loop is give n by:" }{TEXT 999 0 "" }}{PARA 233 "" 0 "" {TEXT 1003 21 " \+ " }{TEXT 1003 0 "" }}{PARA 232 "" 0 "" {TEXT 999 44 " \+ " }{TEXT 1032 1 " " }{TEXT 999 4 " " }{TEXT 1001 1 "A" }{TEXT 999 4 " = " }{XPPEDIT 336 0 "-Int(g(t) *` f ' ( t ) `,t = 0 .. infinity);" "6#,$-%$IntG6$*&-%\"gG6#%\"tG\"\" \"%,~f~'~(~t~)~GF,/F+;\"\"!%)infinityG!\"\"" }{TEXT 1002 16 " . \+ " }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 34 " We now compute this integral in " }{TEXT 1002 5 "Maple" }{TEXT 999 1 "." }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 19 "fprime :=t->D(f)(t);" }{MPLTEXT 1 1007 0 "" }{MPLTEXT 1 1007 41 "\nA:=-1*Int( g(t)*fprime(t),t=0..infinity);" }{MPLTEXT 1 1007 0 "" }{MPLTEXT 1 1007 26 "\nArea:=simplify(value(A));" }{MPLTEXT 1 1007 0 "" }}}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 230 "" 0 "" {TEXT 996 24 " II Po lar Curves and " }{TEXT 1016 5 "Maple" }{TEXT 996 2 " ." }{TEXT 996 0 "" }}{PARA 232 "" 0 "" {TEXT 1002 8 " Maple " }{TEXT 999 73 "may also be used to plot curves described in polar coordinates using the " } {TEXT 1008 9 "polarplot" }{TEXT 1029 1 " " }{TEXT 999 8 "command " } {TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 24 " which is found in t he " }{TEXT 1008 11 "with(plots)" }{TEXT 999 21 " package. If we enter " }{TEXT 999 0 "" }}{PARA 230 "" 0 "" {TEXT 996 1 " " }{TEXT 1033 1 "[ " }{TEXT 996 3 " > " }{TEXT 1034 45 "polarplot ( R ( theta ) , theta = a . . b ) ;" }{TEXT 996 0 "" }}{PARA 232 "" 0 "" {TEXT 999 7 " then \+ " }{TEXT 1002 5 "Maple" }{TEXT 999 27 " will return the graph of " } {TEXT 1002 1 "r" }{TEXT 999 5 " = " }{TEXT 1002 1 "R" }{TEXT 999 3 " ( " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT 999 29 " ) , where t he polar angle " }{TEXT 1002 1 " " }{XPPEDIT 18 0 "theta " "6#%&thetaG " }{TEXT 999 23 " is restricted to: " }{TEXT 1002 1 "a" }{TEXT 999 1 " " }{TEXT 1006 1 "<" }{TEXT 999 1 " " }{XPPEDIT 18 0 "theta;" " 6#%&thetaG" }{TEXT 999 2 " " }{TEXT 1006 2 " <" }{TEXT 999 1 " " } {TEXT 1002 1 "b" }{TEXT 999 2 " ," }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 7 " and " }{TEXT 1002 1 "r" }{TEXT 999 29 " denotes the \+ polar distance." }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }} {PARA 239 "" 0 "" {TEXT 1035 9 " Example" }{TEXT 1036 1 " " }{TEXT 1035 1 "3" }{TEXT 1036 0 "" }}{PARA 232 "" 0 "" {TEXT 999 37 " In ord er to graph the polar curve " }{TEXT 1002 1 "r" }{TEXT 999 7 " = \+ " }{TEXT 1002 1 "F" }{TEXT 999 2 " (" }{TEXT 1002 1 " " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT 999 5 " ) " }{TEXT 1002 1 "G" }{TEXT 999 3 " ( " }{XPPEDIT 18 0 "theta " "6#%&thetaG" }{TEXT 999 13 " ) \+ where " }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 18 " " }{TEXT 1011 2 " " } {TEXT 1001 1 "F" }{TEXT 1000 3 " ( " }{TEXT 1001 1 " " }{XPPEDIT 597 0 "theta" "6#%&thetaG" }{TEXT 1006 2 " " }{TEXT 1037 5 " ) =" } {TEXT 1037 2 " " }{XPPEDIT 596 0 "100/(100+(theta-Pi/2)^8);" "6#*&\"$ +\"\"\"\",&F$F%*$,&%&thetaGF%*&%#PiGF%\"\"#!\"\"F-\"\")F%F-" }{TEXT 1002 1 " " }{TEXT 1038 1 " " }{TEXT 1039 1 " " }{TEXT 1038 13 " \+ and " }{TEXT 1040 1 " " }{TEXT 1041 2 " " }{TEXT 1042 4 "G ( " } {XPPEDIT 595 0 "theta" "6#%&thetaG" }{TEXT 1001 1 " " }{TEXT 1042 1 " \+ " }{TEXT 1042 18 " ) = 2 - sin 7 " }{XPPEDIT 594 0 "theta" "6#%&the taG" }{TEXT 1002 1 " " }{TEXT 1042 1 " " }{TEXT 1042 4 " - " }{TEXT 1041 1 " " }{XPPEDIT 287 0 "cos(30*theta)/2;" "6#*&-%$cosG6#*&\"#I\"\" \"%&thetaGF)F)\"\"#!\"\"" }{TEXT 1002 1 " " }{TEXT 1040 1 " " }{TEXT 1041 2 " " }{TEXT 1040 2 " ," }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 233 "" 0 "" {TEXT 1003 25 " we enter the follo wing:" }{TEXT 1003 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 35 "F:=theta->100/(100+(theta-Pi/2 )^8);" }{MPLTEXT 1 1007 0 "" }}{PARA 234 "> " 0 "" {MPLTEXT 1 1007 43 "G:=theta->(2-sin(7*theta)-cos(30*theta)/2);" }{MPLTEXT 1 1007 0 "" }} {PARA 234 "> " 0 "" {MPLTEXT 1 1007 28 "R:=theta->F(theta)*G(theta);" }{MPLTEXT 1 1007 0 "" }}}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 12 "with(plots):" }{MPLTEXT 1 1007 0 "" }}}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 99 "polarplot(R(theta),theta=-Pi/2..3*Pi/2,numpoints=2 000,color=brown,thickness=2,scaling=constrained);" }{MPLTEXT 1 1007 0 "" }}}{PARA 233 "" 0 "" {TEXT 1003 2 " " }{TEXT 1003 0 "" }}{PARA 232 "" 0 "" {TEXT 999 69 " One way to color-in the graph of this mapl eleaf is to replace the " }{TEXT 1008 6 "CURVES" }{TEXT 999 17 " obje ct name by " }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 1 " " } {TEXT 1008 9 " POLYGONS" }{TEXT 999 14 ". Note that " }{TEXT 1002 5 "Maple" }{TEXT 999 29 " can shade polygonal regions." }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 109 "mapleleaf:=polarplot(R(theta),theta=-Pi/2..3*Pi/2 ,numpoints=2000,axes=box, color=orange,scaling=constrained):" } {MPLTEXT 1 1007 0 "" }}{PARA 234 "> " 0 "" {MPLTEXT 1 1007 43 "MAPLELE AF:=subs(CURVES=POLYGONS,mapleleaf):" }{MPLTEXT 1 1007 0 "" }}}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 19 "display(MAPLELEAF);" } {MPLTEXT 1 1007 0 "" }}}{EXCHG {PARA 228 "" 0 "" {TEXT 991 0 "" }}} {PARA 233 "" 0 "" {TEXT 1003 2 " " }{TEXT 1003 0 "" }}{PARA 232 "" 0 "" {TEXT 999 50 " Now suppose that we wish to color-in the limacon " } {TEXT 1002 1 " " }{TEXT 1021 1 " " }{TEXT 1005 2 " r" }{TEXT 1031 15 " = 1 + 2 sin " }{XPPEDIT 286 0 "theta;" "6#%&thetaG" }{TEXT 1002 2 " " }{TEXT 1021 1 " " }{TEXT 1002 32 " as shown in the figure below. " }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 237 "" 0 "" {TEXT 1020 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 233 "" 0 "" {TEXT 1003 2 " " }{TEXT 1003 0 "" }}{PARA 233 "" 0 "" {TEXT 1003 100 " We may accomplish this by converting the two looped regions in to \"polygons\" for shading purposes." }{TEXT 1003 0 "" }}{PARA 232 "" 0 "" {TEXT 999 71 " The color of the larger outer region is skyblue , and is defined in " }{TEXT 1002 5 "Maple" }{TEXT 999 23 " as the f ollowing hue:" }{TEXT 999 0 "" }}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 52 "macro(skyblue = COLOR(RGB, 0.1960, 0.6000, 0.8000));" } {MPLTEXT 1 1007 0 "" }}}{PARA 233 "" 0 "" {TEXT 1003 67 " Now we colo r the entire region bounded by the limacon in skyblue." }{TEXT 1003 0 "" }}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 25 "r:=theta->1+2*sin( theta);" }{MPLTEXT 1 1007 0 "" }}}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 83 "limacon:=polarplot(r(theta),theta=-Pi/6..7*Pi/6,color=skybl ue,scaling=constrained):" }{MPLTEXT 1 1007 0 "" }}}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 47 "ShadedOuterLoop:=subs(CURVES=POLYGONS,lima con):" }{MPLTEXT 1 1007 0 "" }}}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 24 "display(ShadedOuterLoop)" }{MPLTEXT 1 1007 1 ";" }{MPLTEXT 1 1007 0 "" }}}{PARA 233 "" 0 "" {TEXT 1003 1 " " }{TEXT 1003 0 "" }} {PARA 233 "" 0 "" {TEXT 1003 59 " The inner-loop may be shaded turqu oise with the command:" }{TEXT 1003 0 "" }}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 89 "innerloop:=polarplot(r(theta),theta=7*Pi/6..11*Pi/ 6,color=turquoise,scaling=constrained):" }{MPLTEXT 1 1007 0 "" }}} {EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 49 "ShadedInnerLoop:=subs(C URVES=POLYGONS,innerloop):" }{MPLTEXT 1 1007 0 "" }}}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 25 "display(ShadedInnerLoop);" }{MPLTEXT 1 1007 0 "" }}}{PARA 233 "" 0 "" {TEXT 1003 1 " " }{TEXT 1003 0 "" }} {PARA 232 "" 0 "" {TEXT 999 70 " Now we display both of these col ored regions together. First, we" }{TEXT 1002 1 " " }{TEXT 1043 4 "lis t" }{TEXT 999 35 " the inner turquoise region in the " }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 6 " " }{TEXT 1008 8 "display " } {TEXT 999 46 " command, and then the outer skyblue section. " }{TEXT 999 0 "" }}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 43 "display([Sha dedInnerLoop,ShadedOuterLoop]);" }{MPLTEXT 1 1007 0 "" }}}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 3 " (" }{TEXT 1002 1 " " }{TEXT 1021 4 "Note" }{TEXT 1022 1 ":" }{TEXT 1004 1 " " } {TEXT 999 102 "The order of the entries in the display list is signifi cant. If we transpose the order of the colored " }{TEXT 999 0 "" }} {PARA 233 "" 0 "" {TEXT 1003 105 " regions, then the in ner region will be shaded first, and thereafter, covered in skyblue.)" }{TEXT 1003 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 1007 43 "display([ShadedOuterLoop,ShadedInner Loop]);" }{MPLTEXT 1 1007 0 "" }}}{EXCHG {PARA 228 "" 0 "" {TEXT 991 0 "" }}}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 240 "" 0 "" {TEXT 1044 0 "" }}{PARA 230 "" 0 "" {TEXT 1045 154 " \+ \+ " }{TEXT 1046 6 " " }{TEXT 1047 1 " " }{TEXT 996 0 "" }}{PARA 241 "" 0 "" {TEXT 1048 11 " Exercises" }{TEXT 1048 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 1000 6 " 1. " }{TEXT 999 5 "Let " }{TEXT 1001 1 "C" }{TEXT 999 22 " be the described by:" } {TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 9 " " }{TEXT 1002 1 "x" }{TEXT 999 4 " = " }{TEXT 1002 3 " f " }{TEXT 999 2 "( " }{TEXT 1002 1 "t" }{TEXT 999 6 " ) = " }{XPPEDIT 18 0 "2*cos(t)+sin(2*t);" "6#,&*&\"\"#\"\"\"-%$cosG6#%\"tGF &F&-%$sinG6#*&F%F&F*F&F&" }{TEXT 999 16 " and " }{TEXT 1002 1 "y" }{TEXT 999 5 " = " }{TEXT 1002 1 "g" }{TEXT 999 3 " ( " } {TEXT 1002 1 "t" }{TEXT 999 6 " ) = " }{XPPEDIT 18 0 " 4*sin(t)" "6#* &\"\"%\"\"\"-%$sinG6#%\"tGF%" }{TEXT 999 13 " for 0 " }{TEXT 1006 1 "<" }{TEXT 999 2 " " }{TEXT 1002 1 "t" }{TEXT 999 2 " " } {TEXT 1006 1 "<" }{TEXT 999 2 " " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\" \"\"%#PiGF%" }{TEXT 999 7 " . " }{TEXT 999 0 "" }}{PARA 228 "" 0 " " {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 8 " " }{TEXT 1000 6 "( a ) " }{TEXT 999 23 " Define the functions " }{TEXT 1002 3 "f " }{TEXT 999 4 "and " }{TEXT 1002 2 " g" }{TEXT 999 6 " in " } {TEXT 1002 6 "Maple " }{TEXT 999 25 "and then plot the curve " } {TEXT 1001 1 "C" }{TEXT 999 5 " as " }{TEXT 1002 2 "t " }{TEXT 999 16 "ranges over the " }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 34 " interval [ 0 , " }{XPPEDIT 18 0 "2*Pi;" "6#*&\" \"#\"\"\"%#PiGF%" }{TEXT 999 8 " ] . " }{TEXT 1002 14 "See example \+ 1 " }{TEXT 999 1 "." }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 6 " " }{TEXT 1000 8 " ( b ) " }{TEXT 999 21 " Animate the plot of " }{TEXT 1001 2 " C" }{TEXT 999 19 " as in example 1 ." }{TEXT 999 0 " " }}{PARA 232 "" 0 "" {TEXT 999 5 " " }{TEXT 1000 9 " ( c ) " } {TEXT 999 71 " What are the Cartesian coordinates of the point at whic h the graph of " }{TEXT 1001 2 " C" }{TEXT 999 17 " is not smooth, " }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 1002 20 " i. e" }{TEXT 999 17 ". at which point " }{TEXT 1000 1 " " }{TEXT 1001 1 " P" }{TEXT 999 7 " = ( " }{TEXT 1002 1 "x" }{TEXT 999 3 " , " }{TEXT 1002 1 "y" }{TEXT 999 22 " ) does the graph of " }{TEXT 1001 1 "C" } {TEXT 999 16 " have a corner?" }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 2 " " }{TEXT 1000 10 "2. ( a ) " }{TEXT 999 22 " Derive an alytically, " }{TEXT 1002 4 "i.e." }{TEXT 999 17 ", without using " } {TEXT 1002 5 "Maple" }{TEXT 999 33 " , the parametric representation " }{TEXT 1008 6 "( ** )" }{TEXT 1049 1 " " }{TEXT 999 8 " for the" } {TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 16 " " } {TEXT 1002 19 "folium of Descartes" }{TEXT 999 2 " ," }{TEXT 1029 1 " \+ " }{TEXT 1008 5 "( * )" }{TEXT 1049 1 " " }{TEXT 999 45 ", given in th e text above, that is Example 2." }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 16 " ( " }{TEXT 1002 5 "Hint " }{TEXT 999 1 ":" }{TEXT 1002 2 " " }{TEXT 999 3 "Let" }{TEXT 1002 5 " P " }{TEXT 999 40 "be the point of intersection of the line" }{TEXT 1002 13 " y \+ = t x " }{TEXT 999 9 "with the " }{TEXT 999 0 "" }}{PARA 232 "" 0 " " {TEXT 999 33 " folium" }{TEXT 1021 1 " " } {TEXT 1008 5 "( * )" }{TEXT 999 34 ", then solve for the coordinates \+ " }{TEXT 1002 3 "x " }{TEXT 999 5 "and " }{TEXT 1002 1 "y" }{TEXT 999 6 " of " }{TEXT 1002 1 "P" }{TEXT 999 14 " in terms of " } {TEXT 1002 4 " t ." }{TEXT 999 1 ")" }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 237 "" 0 "" {TEXT 1020 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 26 " " }{TEXT 1002 25 "Note tha t the line y = " }{TEXT 999 3 " - " }{TEXT 1002 1 "2" }{TEXT 999 4 " - " }{TEXT 1002 45 " x is an oblique asymptote for the folium." } {TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 4 " " }{TEXT 1000 10 " ( b ) " }{TEXT 999 1 " " }{TEXT 1002 20 "Without using Maple" } {TEXT 999 19 ", verify that the " }{TEXT 1002 19 "folium of Descartes " }{TEXT 999 2 " ," }{TEXT 1029 1 " " }{TEXT 1008 5 "( * )" }{TEXT 999 23 " , can be written as " }{TEXT 999 0 "" }}{PARA 233 "" 0 "" {TEXT 1003 37 " the polar equation :" }{TEXT 1003 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 62 " " } {XPPEDIT 18 0 "r=6*sec(theta)*tan(theta)/(1+[tan(theta)]^3:" "6#/%\"rG **\"\"'\"\"\"-%$secG6#%&thetaGF'-%$tanGF*F',&F'F'*$7#F,\"\"$F'!\"\"" } {TEXT 999 6 " ." }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 " " }}{PARA 232 "" 0 "" {TEXT 999 8 " " }{TEXT 1000 7 "( c ) " } {TEXT 999 6 " Use " }{TEXT 1002 7 "Maple's" }{TEXT 999 2 " " }{TEXT 1008 10 "polarplot " }{TEXT 999 31 " command to sketch the folium " } {XPPEDIT 18 0 "theta = [-Pi/8, 5*Pi/8];" "6#/%&thetaG7$,$*&%#PiG\"\"\" \"\")!\"\"F+*(\"\"&F)F(F)F*F+" }{TEXT 999 2 " " }{TEXT 1008 7 " ( * \+ )" }{TEXT 999 2 " ." }{TEXT 999 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 " " }}{PARA 233 "" 0 "" {TEXT 1003 25 " " } {TEXT 1003 0 "" }}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 232 "" 0 "" {TEXT 999 2 " " }{TEXT 1000 3 "3. " }{TEXT 999 72 " In parts ( a ) \+ and ( b ), first plot the given parametric curve with " }{TEXT 1002 5 "Maple" }{TEXT 999 25 " and then find the area " }{TEXT 999 0 "" }} {PARA 233 "" 0 "" {TEXT 1003 20 " it encloses." }{TEXT 1003 0 " " }}{PARA 232 "" 0 "" {TEXT 999 4 " " }{TEXT 1000 9 " ( a ) " } {TEXT 999 2 " " }{TEXT 1002 1 "x" }{TEXT 999 7 " = 2 " }{TEXT 1002 3 "cos" }{TEXT 999 3 " ( " }{XPPEDIT 18 0 "3*t/2:" "6#*(\"\"$\"\"\"%\" tGF%\"\"#!\"\"" }{TEXT 999 7 " ) - " }{TEXT 1002 3 "cos" }{TEXT 999 2 " " }{TEXT 1002 1 "t" }{TEXT 999 7 " , " }{TEXT 1002 1 "y" } {TEXT 999 7 " = 2 " }{TEXT 1002 4 "sin " }{TEXT 999 3 "( " } {XPPEDIT 18 0 "t/2" "6#*&%\"tG\"\"\"\"\"#!\"\"" }{TEXT 999 9 " ) - " }{TEXT 1002 3 "sin" }{TEXT 999 1 " " }{TEXT 1002 1 "t" }{TEXT 999 13 " for 0 " }{TEXT 1006 1 "<" }{TEXT 999 2 " " }{TEXT 1002 1 " t" }{TEXT 999 6 " < " }{XPPEDIT 18 0 "4*Pi:" "6#*&\"\"%\"\"\"%#PiGF %" }{TEXT 999 4 " ." }{TEXT 999 0 "" }}{PARA 232 "" 0 "" {TEXT 999 6 " " }{TEXT 1000 6 "( b ) " }{TEXT 999 3 " " }{TEXT 1002 1 "x" }{TEXT 999 7 " = 2 " }{TEXT 1002 5 "cos t" }{TEXT 999 5 " + " } {TEXT 1002 3 "cos" }{TEXT 999 4 " ( 3" }{TEXT 1002 1 "t" }{TEXT 999 7 " ) , " }{TEXT 1002 1 "y" }{TEXT 999 7 " = 2 " }{TEXT 1002 5 "sin \+ t" }{TEXT 999 5 " - " }{TEXT 1002 4 "sin " }{TEXT 999 3 "( 2" } {TEXT 1002 2 " t" }{TEXT 999 13 " ) for 0 " }{TEXT 1006 1 "<" } {TEXT 999 2 " " }{TEXT 1002 1 "t" }{TEXT 999 5 " < " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT 999 7 " . " }{TEXT 999 0 "" }}}{PARA 228 "" 0 "" {TEXT 991 0 "" }}{PARA 242 "" 0 "" {TEXT 1050 0 "" }}{PARA 243 "" 0 "" {TEXT 1051 0 "" }}{PARA 244 "" 0 "" {TEXT 1052 0 "" }}{PARA 245 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }