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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "T he graph of such a function is a " }{TEXT 259 17 "quadratic surface" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "The intersection of the \+ surface " }{XPPEDIT 18 0 "s(x,y)" "6#-%\"sG6$%\"xG%\"yG" }{TEXT -1 30 " with a plane parallel to the " }{TEXT 267 1 "y" }{TEXT -1 1 "," } {TEXT 268 1 "z" }{TEXT -1 26 " plane or parallel to the " }{TEXT 269 1 "x" }{TEXT -1 1 "," }{TEXT 270 1 "z" }{TEXT -1 102 " plane, is a par abola with its axis vertical, that is, the standard graph of a quadrat ic function. 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00C022000000000000C022A00000000000C023800000000000C024A00000000000C026 000000000000C027A00000000000C029800000000000C02BA00000000000C02E000000 000000-%&COLORG6&%$RGBG$\"\")!\"\"$\"\"'F6$\"\"\"F)-%+AXESLABELSG6%%\" xG%\"yGQ!F/-%*AXESSTYLEG6#%$BOXG-%+LIGHTMODELG6#%(LIGHT_4G" 1 2 0 1 10 0 2 1 6 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The fir st order partial derivatives of " }{XPPEDIT 18 0 "s(x,y) = c+a*x+b*y+p *x^2+q*x*y+r*y^2" "6#/-%\"sG6$%\"xG%\"yG,.%\"cG\"\"\"*&%\"aGF+F'F+F+*& %\"bGF+F(F+F+*&%\"pGF+*$F'\"\"#F+F+*(%\"qGF+F'F+F(F+F+*&%\"rGF+*$F(F3F +F+" }{TEXT -1 11 " are . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "s := (x,y) -> c+a*x+b*y+p*x ^2+q*x*y+r*y^2:\n's(x,y)'=s(x,y);\nDiff('s(x,y)',x)=diff(s(x,y),x);\nD iff('s(x,y)',y)=diff(s(x,y),y); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"sG6$%\"xG%\"yG,.%\"cG\"\"\"*&%\"aGF+F'F+F+*&%\"bGF+F(F+F+*&%\"pGF+) F'\"\"#F+F+*(%\"qGF+F'F+F(F+F+*&%\"rGF+)F(F3F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"sG6$%\"xG%\"yGF*,(%\"aG\"\"\"*(\"\"#F.% \"pGF.F*F.F.*&%\"qGF.F+F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%Dif fG6$-%\"sG6$%\"xG%\"yGF+,(%\"bG\"\"\"*&%\"qGF.F*F.F.*(\"\"#F.%\"rGF.F+ F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 " . . . and the 2nd order partial derivatives are . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "Diff(' s(x,y)',x$2)=diff(s(x,y),x$2);\nDiff('s(x,y)',x,y)=diff(s(x,y),x,y);\n Diff('s(x,y)',y$2)=diff(s(x,y),y$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%%DiffG6$-%\"sG6$%\"xG%\"yG-%\"$G6$F*\"\"#,$%\"pGF/" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%%DiffG6%-%\"sG6$%\"xG%\"yGF*F+%\"qG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"sG6$%\"xG%\"yG-%\"$G6$F+\"\"# ,$%\"rGF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "For a general func tion of 2 variables " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" } {TEXT -1 75 ", which has (continuous) 1st and 2nd order partial deriva tives at the point" }{XPPEDIT 18 0 "``(0, 0);" "6#-%!G6$\"\"!F&" } {TEXT -1 40 ", we can construct a quadratic function " }{XPPEDIT 18 0 "s(x,y);" "6#-%\"sG6$%\"xG%\"yG" }{TEXT -1 24 " with the same value as " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 13 " at th e point" }{XPPEDIT 18 0 "``(0,0);" "6#-%!G6$\"\"!F&" }{TEXT -1 61 ", a nd such that the 1st and 2nd order partial derivatives of " }{XPPEDIT 18 0 "s(x,y)" "6#-%\"sG6$%\"xG%\"yG" }{TEXT -1 16 " match those of " } {XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 13 " at the po int" }{XPPEDIT 18 0 "``(0,0);" "6#-%!G6$\"\"!F&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 60 ": The continuity of the partial derivatives guarantees th at " }{XPPEDIT 18 0 "g[xy](0,0)=g[yx](0,0)" "6#/-&%\"gG6#%#xyG6$\"\"!F *-&F&6#%#yxG6$F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "Th e required values are:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c=g(0,0)" "6#/%\"cG-%\"gG6$\"\"!F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a=g[x](0,0)" "6#/%\"aG-&%\"gG6#%\"xG6$\"\"!F+" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b=g[y](0,0)" "6#/%\"bG-&%\"gG6#%\"yG6$\"\"!F+" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "p=1/2" "6 #/%\"pG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "g[xx](0,0 )" "6#-&%\"gG6#%#xxG6$\"\"!F)" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "q=g[xy ](0,0)" "6#/%\"qG-&%\"gG6#%#xyG6$\"\"!F+" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "r=1/2" "6#/%\"rG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "g[yy](0,0)" "6#-&%\"gG6#%#yyG6$\"\"!F)" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s(x,y)=g(0,0)+g[x](0,0)*x+g[y](0,0)*y+1 /2" "6#/-%\"sG6$%\"xG%\"yG,*-%\"gG6$\"\"!F-\"\"\"*&-&F+6#F'6$F-F-F.F'F .F.*&-&F+6#F(6$F-F-F.F(F.F.*&F.F.\"\"#!\"\"F." }{XPPEDIT 18 0 "``(g[xx ](0,0)*x^2+2*g[xy](0,0)*x*y+g[yy](0,0)*y^2)" "6#-%!G6#,(*&-&%\"gG6#%#x xG6$\"\"!F.\"\"\"*$%\"xG\"\"#F/F/**F2F/-&F*6#%#xyG6$F.F.F/F1F/%\"yGF/F /*&-&F*6#%#yyG6$F.F.F/*$F9F2F/F/" }{TEXT -1 2 ". " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{TEXT 264 38 "______________________________________ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "By translation from" }{XPPEDIT 18 0 "``(0,0)" "6#-%!G6$\" \"!F&" }{TEXT -1 11 " to a point" }{XPPEDIT 18 0 "``(a,b);" "6#-%!G6$% \"aG%\"bG" }{TEXT -1 46 " we obtain the following more general result. " }}{PARA 0 "" 0 "" {TEXT -1 30 "For a function of 2 variables " } {XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 75 ", which ha s (continuous) 1st and 2nd order partial derivatives at the point" } {XPPEDIT 18 0 "``(a,b);" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 25 ", the quad ratic function " }{XPPEDIT 18 0 "s(x,y);" "6#-%\"sG6$%\"xG%\"yG" } {TEXT -1 65 " with the same value and partial derivatives of order 1 a nd 2 as " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(a, b);" "6#-%!G6$%\"aG%\"bG" } {TEXT -1 5 " is: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " " }{XPPEDIT 18 0 "s(x,y) = g(a,b)+ g[x](a,b)*(x-a)+g[y](a,b)*(y-b)+``;" "6#/-%\"sG6$%\"xG%\"yG,*-%\"gG6$% \"aG%\"bG\"\"\"*&-&F+6#F'6$F-F.F/,&F'F/F-!\"\"F/F/*&-&F+6#F(6$F-F.F/,& F(F/F.F6F/F/%!GF/" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " 1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[g[xx](a ,b)*(x-a)^2+2*g[xy](a,b)*(x-a)*(y-b)+g[yy](a, b)*(y-b)^2];" "6#7#,(*&- &%\"gG6#%#xxG6$%\"aG%\"bG\"\"\"*$,&%\"xGF.F,!\"\"\"\"#F.F.**F3F.-&F(6# %#xyG6$F,F-F.,&F1F.F,F2F.,&%\"yGF.F-F2F.F.*&-&F(6#%#yyG6$F,F-F.*$,&F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Quadratic \+ approximation examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 30 "For a function of 2 variables " } {XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 75 ", which ha s (continuous) 1st and 2nd order partial derivatives at the point" } {XPPEDIT 18 0 "``(a,b);" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 25 ", the quad ratic function " }{XPPEDIT 18 0 "s(x,y);" "6#-%\"sG6$%\"xG%\"yG" } {TEXT -1 65 " with the same value and partial derivatives of order 1 a nd 2 as " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(a, b);" "6#-%!G6$%\"aG%\"bG" } {TEXT -1 5 " is: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " " }{XPPEDIT 18 0 "s(x,y) = g(a,b)+ g[x](a,b)*(x-a)+g[y](a,b)*(y-b)+``;" "6#/-%\"sG6$%\"xG%\"yG,*-%\"gG6$% \"aG%\"bG\"\"\"*&-&F+6#F'6$F-F.F/,&F'F/F-!\"\"F/F/*&-&F+6#F(6$F-F.F/,& F(F/F.F6F/F/%!GF/" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " 1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[g[xx](a ,b)*(x-a)^2+2*g[xy](a,b)*(x-a)*(y-b)+g[yy](a, b)*(y-b)^2];" "6#7#,(*&- &%\"gG6#%#xxG6$%\"aG%\"bG\"\"\"*$,&%\"xGF.F,!\"\"\"\"#F.F.**F3F.-&F(6# %#xyG6$F,F-F.,&F1F.F,F2F.,&%\"yGF.F-F2F.F.*&-&F(6#%#yyG6$F,F-F.*$,&F " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "g(x ,y)=3/(1+x^2+x*y+2*y^2)" "6#/-%\"gG6$%\"xG%\"yG*&\"\"$\"\"\",*F+F+*$F' \"\"#F+*&F'F+F(F+F+*&F.F+*$F(F.F+F+!\"\"" }{TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "g := (x,y) -> 3/(1+x^2+x*y+2*y^2);\ng_x := D[1](g);\ng_y := D[2](g);\ng_xx := D[1,1](g);\ng_xy := D[1,2](g);\ng_yy := D[2,2](g); " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$ *&\"\"\"F/,*F/F/*$)9$\"\"#F/F/*&F3F/9%F/F/*&F4F/)F6F4F/F/!\"\"\"\"$F)F )F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$g_xGf*6$%\"xG%\"yG6\"6$%)ope ratorG%&arrowGF),$*&,*\"\"\"F0*$)9$\"\"#F0F0*&F3F09%F0F0*&F4F0)F6F4F0F 0!\"#,&F3F4F6F0F0!\"$F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$g_yG f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*&,*\"\"\"F0*$)9$\"\"#F0F0*& F3F09%F0F0*&F4F0)F6F4F0F0!\"#,&F3F0*&\"\"%F0F6F0F0F0!\"$F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%g_xxGf*6$%\"xG%\"yG6\"6$%)operatorG %&arrowGF),&*&,*\"\"\"F0*$)9$\"\"#F0F0*&F3F09%F0F0*&F4F0)F6F4F0F0!\"$, &F3F4F6F0F4\"\"'*&F;F0F/!\"#!\"\"F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%g_xyGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*(,*\"\"\"F0*$ )9$\"\"#F0F0*&F3F09%F0F0*&F4F0)F6F4F0F0!\"$,&F3F0*&\"\"%F0F6F0F0F0,&F3 F4F6F0F0\"\"'*&\"\"$F0F/!\"#!\"\"F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%g_yyGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*&,*\"\"\"F0*$ )9$\"\"#F0F0*&F3F09%F0F0*&F4F0)F6F4F0F0!\"$,&F3F0*&\"\"%F0F6F0F0F4\"\" '*&\"#7F0F/!\"#!\"\"F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "We find a quadratic approximation for " } {XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 15 " at the or igin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "a := 0;\nb := 0;\ng(a,b)+g_x(a,b)*(x-a)+g_y(a,b)*(y- b)+\n (g_xx(a,b)*(x-a)^2+2*g_xy(a,b)*(x-a)*(y-b)+g_yy(a,b)*(y-b)^2)/ 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"\"$ \"\"\"*&F$F%)%\"xG\"\"#F%!\"\"*(F$F%F(F%%\"yGF%F**&\"\"'F%)F,F)F%F*" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "This re sult can be obtained more easily using the Maple procedure " }{TEXT 0 7 "mtaylor" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "g := (x,y) -> 3/(1+x^2+x*y+2*y^2): \nmtaylor(g(x,y),[x,y],3);\ns := unapply(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"\"$\"\"\"*&F$F%)%\"xG\"\"#F%!\"\"*(F$F%F(F%%\"yGF%F **&\"\"'F%)F,F)F%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGf*6$%\"xG %\"yG6\"6$%)operatorG%&arrowGF),*\"\"$\"\"\"*&F.F/)9$\"\"#F/!\"\"*(F.F /F2F/9%F/F4*&\"\"'F/)F6F3F/F4F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "p1 := plot3d(g(x,y),x=-0. 5..0.5,y=-0.5..0.5,color=COLOR(RGB,1,.8,.6)):\np2 := plot3d(s(x,y),x=- 0.5..0.5,y=-0.5..0.5,color=magenta,\n grid=[50,50],style=wireframe, gridstyle=triangular):\nplots[display]([p1,p2],axes=boxed,lightmodel=l ight4);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6(-%%GR IDG6&;$!\"&!\"\"$\"\"&F)F&X,%)anythingG6\"6\"[gl'!%\"!!#\\bm\":\":3FF8 0000000000003FF945BBABC239183FFA95A95A95A95B3FFBED61BED61BED3FFD49C341 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LTG-%*AXESSTYLEG6#%$BOXG-%*GRIDSTYLEG6#%+TRIANGULARG" 1 2 0 1 10 0 1 1 6 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {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 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x,y)=x*y*exp(-x^2-y^2)" "6#/-% \"gG6$%\"xG%\"yG*(F'\"\"\"F(F*-%$expG6#,&*$F'\"\"#!\"\"*$F(F0F1F*" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "g := (x,y) -> x*y*exp(-x^2-y^2);\ng_x := D[1](g );\ng_y := D[2](g);\ng_xx := D[1,1](g);\ng_xy := D[1,2](g);\ng_yy := D [2,2](g); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"xG%\"yG6\"6 $%)operatorG%&arrowGF)*(9$\"\"\"9%F/-%$expG6#,&*$)F.\"\"#F/!\"\"*$)F0F 7F/F8F/F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$g_xGf*6$%\"xG%\"yG 6\"6$%)operatorG%&arrowGF),&*&9%\"\"\"-%$expG6#,&*$)9$\"\"#F0!\"\"*$)F /F8F0F9F0F0**F8F0F6F0F/F0F1F0F9F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$g_yGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*&9$\"\"\"-%$expG6 #,&*$)F/\"\"#F0!\"\"*$)9%F7F0F8F0F0**F7F0F/F0F:F0F1F0F8F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%g_xxGf*6$%\"xG%\"yG6\"6$%)operatorG %&arrowGF),&*(9$\"\"\"9%F0-%$expG6#,&*$)F/\"\"#F0!\"\"*$)F1F8F0F9F0!\" '**\"\"%F0)F/\"\"$F0F1F0F2F0F0F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%g_xyGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),*-%$expG6#,&*$)9$\" \"#\"\"\"!\"\"*$)9%F5F6F7F6*(F5F6F3F6F.F6F7*(F5F6F9F6F.F6F7**\"\"%F6F3 F6F9F6F.F6F6F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%g_yyGf*6$%\"x G%\"yG6\"6$%)operatorG%&arrowGF),&*(9$\"\"\"9%F0-%$expG6#,&*$)F/\"\"#F 0!\"\"*$)F1F8F0F9F0!\"'**\"\"%F0F/F0)F1\"\"$F0F2F0F0F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "p lot3d(g(x,y),x=-2..2,y=-2..2,color=COLOR(RGB,1,.8,.6),\n grid=[30,3 0],lightmodel=light4,axes=boxed);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6&-%%GRIDG6&;$!\"#\"\"!$\"\"#F)F&X,%)anythingG6\" 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proximation for " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" } {TEXT -1 14 " at the point " }{XPPEDIT 18 0 "``(1/sqrt(2),-1/sqrt(2)) " "6#-%!G6$*&\"\"\"F'-%%sqrtG6#\"\"#!\"\",$*&F'F'-F)6#F+F,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "a := 1/sqrt(2);\nb := -1/sqrt(2);\ng(a,b)+g_x(a,b)*( x-a)+g_y(a,b)*(y-b)+\n (g_xx(a,b)*(x-a)^2+2*g_xy(a,b)*(x-a)*(y-b)+g_y y(a,b)*(y-b)^2)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG,$*$-%%sqr tG6#\"\"#\"\"\"#F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG,$*$-%%s qrtG6#\"\"#\"\"\"#!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG 6#!\"\"#F'\"\"#*&F$\"\"\"),&%\"xGF+*&#F+F)F+*$-%%sqrtG6#F)F+F+F'F)F+F+ *&F$F+),&%\"yGF+*&#F+F)F+F2F+F+F)F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 66 "This result can be obtained more easi ly using the Maple procedure " }{TEXT 0 7 "mtaylor" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "g := (x,y) -> x*y*exp(-x^2-y^2):\nmtaylor(g(x,y),[x=1/sqrt(2),y= -1/sqrt(2)],3);\ns := unapply(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#!\"\"#F'\"\"#*&F$\"\"\"),&%\"xGF+*&#F+F)F+*$-%%sqrtG6#F) F+F+F'F)F+F+*&F$F+),&%\"yGF+*&#F+F)F+F2F+F+F)F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(-%$expG 6#!\"\"#F1\"\"#*&F.\"\"\"),&9$F5*&#F5F3F5*$-%%sqrtG6#F3F5F5F1F3F5F5*&F .F5),&9%F5*&#F5F3F5F " 0 "" {MPLTEXT 1 0 227 "p1 := plot3d(g(x,y),x=. 5..1,y=-.5..-1,color=COLOR(RGB,1,.8,.6)):\np2 := plot3d(s(x,y),x=.5..1 ,y=-.5..-1,color=magenta,\n grid=[40,40],style=wireframe,gridstyle= triangular):\nplots[display]([p1,p2],axes=boxed,lightmodel=light4);" } }{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6(-%%GRIDG6&;$\" \"&!\"\"$\"\"\"\"\"!;$F)F,$!\"&F)X,%)anythingG6\"6\"[gl'!%\"!!#\\bm\": \":BFC25618372A5850BFC2B5C7AD336291BFC310F6ED97B45BBFC3673882D46CB7BFC 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$*&\"\"(\"\"\"\"#5!\"\",$*&\"\"*F(F)F*F*" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "a := 7/10;\nb := -9/10;\ng(a,b)+g_x(a,b)*(x-a)+g_y(a,b)*(y-b)+\n (g_xx(a, b)*(x-a)^2+2*g_xy(a,b)*(x-a)*(y-b)+g_yy(a,b)*(y-b)^2)/2;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"aG#\"\"(\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG#!\"*\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.#\"&Z[\"\"% +S\"\"\"*&#\"#8\"$](F'%\"xGF'F'*&#\"\"*\"$+&F'%\"yGF'!\"\"*&#\"$\\\"\" #]F'),&F,F'#\"\"(\"#5F2\"\"#F'F'*(#\"#O\"\"&F'F8F',&F1F'#F/F;F'F'F'*&# \"$j\"\"$+\"F')FAF " 0 "" {MPLTEXT 1 0 105 "g := ( x,y) -> x^4/3+y^4/2-4*x*y^2+2*x^2+2*y^2+3:\nmtaylor(g(x,y),[x=7/10,y=- 9/10],3):\ns := unapply(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"sGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),.#\"&Z[\"\"%+S\"\"\"*&#\" #8\"$](F19$F1F1*&#\"\"*\"$+&F19%F1!\"\"*&#\"$\\\"\"#]F1),&F6F1#\"\"(\" #5F<\"\"#F1F1*(#\"#O\"\"&F1FBF1,&F;F1#F9FEF1F1F1*&#\"$j\"\"$+\"F1)FKFF F1F1F)F)F)" }}}{PARA 0 "" 0 "" 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}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 42 "Find t he 2nd degree Taylor polynomial for " }{XPPEDIT 18 0 "g(x,y)=x^2*ln(y) " "6#/-%\"gG6$%\"xG%\"yG*&F'\"\"#-%#lnG6#F(\"\"\"" }{TEXT -1 13 " at t he point" }{XPPEDIT 18 0 " ``(-2,1)" "6#-%!G6$,$\"\"#!\"\"\"\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 77 "Plot 3 dimensional grap hs to compare the Taylor polynomial with the function " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "______________________________" }}{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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 30 "__ ____________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 42 "Find the 2nd degree Taylor polynomial for " }{XPPEDIT 18 0 "g(x,y) = (x-y)/(x+y);" "6#/-%\"gG6$%\"xG%\"yG*&,&F'\"\"\"F(!\"\"F+, &F'F+F(F+F," }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(1, 2);" " 6#-%!G6$\"\"\"\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 81 "P lot three dimensional graphs to compare the Taylor polynomial with the function " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "______________________________" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 30 "______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 42 "Find the 2nd degree Taylor polyno mial for " }{XPPEDIT 18 0 "g(x,y) = cos(x^2*y);" "6#/-%\"gG6$%\"xG%\"y G-%$cosG6#*&F'\"\"#F(\"\"\"" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(2, 0);" "6#-%!G6$\"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 81 "Plot three dimensional graphs to compare the Taylor po lynomial with the function " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG% \"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "_________________ _____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 30 "______________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 42 "Find the 2nd de gree Taylor polynomial for " }{XPPEDIT 18 0 "g(x,y) = arctan(y/x);" "6 #/-%\"gG6$%\"xG%\"yG-%'arctanG6#*&F(\"\"\"F'!\"\"" }{TEXT -1 13 " at t he point" }{XPPEDIT 18 0 "``(1,1);" "6#-%!G6$\"\"\"F&" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 81 "Plot three dimensional graphs to comp are the Taylor polynomial with the function " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 " ______________________________" }}{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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 30 "____________ __________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 42 "Find the 2nd degree Taylor polynomial for " }{XPPEDIT 18 0 "g(x,y) = sin(x)*sin(y);" "6#/-%\"gG6$%\"xG%\"yG*&-%$sinG6#F'\"\"\"-F+6#F(F- " }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(0, 0);" "6#-%!G6$\" \"!F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 81 "Plot three dime nsional graphs to compare the Taylor polynomial with the function " } {XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "______________________________" }}{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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 30 "______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 18 "Code for pi ctures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 30 "parabolic intersection curves " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 275 "p 1:=plot([x^2,-x^2],x=-1..1,color=[red,blue],thickness=2):\nt1:=plots[t extplot]([[.95,-.07,`x`],[-.07,.95,`z`]]):\nt2 := plots[textplot]([.6, .7,`a > 0`],color=COLOR(RGB,.9,0,0)):\nt3 := plots[textplot]([.6,-.7,` a < 0`],color=blue):\nplots[display]([p1,t1,t2,t3],tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 358 "p1 := plot([x^2,-x^2,0],x=- 1..1,\n color=[red,blue,COLOR(RGB,0,.8,0)],thickness=2):\nt1 := plo ts[textplot]([[.85,-.07,`r`],[-.07,.95,`z`]]):\nt2 := plots[textplot]( [.6,.7,`k > 0`],color=red):\nt3 := plots[textplot]([.6,-.7,`k < 0`],co lor=blue):\nt4 := plots[textplot]([-.8,.1,`k = 0`],color=COLOR(RGB,0,. 8,0)):\nplots[display]([p1,t1,t2,t3,t4],tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 30 "variation of scaling constant \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 526 "p1 := plot([3*cos(t)^2+7*cos(t)*sin(t)-sin(t)^2,1],t=0..Pi,\n labels=[``,``],tickmarks=[0,0],color=[red,blue]):\np2 := plot([[[-.0 3,5.03],[.03,5.03]],[[-.03,-3.03],[.03,-3.03]],\n [[0,0],[3.2,0]],[[ 0,-3.5],[0,6]]],\n color=black):\nt1 := plots[textplot]([[3.1,-.4,` q`],[-.12,5.03,`+r`],\n [-.12,-3.03,`-r`]],font=[SYMBOL,10],color=bl ack):\nt2 := plots[textplot]([[-.1,6,`k`],[-.14,1,`a+c`],\n [-.32,5 .03,`a+c`],[-.32,-3.03,`a+c`]],\n font=[HELVETICA,10]):\nplots[disp lay]([p1,p2,t1,t2],view=[-.32..3.2,-3.5..6],axes=none);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 523 "p1 := plot([3*cos(t)^2-2*cos(t)*sin(t)+sin (t)^2,2],t=0..Pi,\n labels=[``,``],tickmarks=[0,0],color=[red,blue]): \np2 := plot([[[-.03,3.414],[.03,3.414]],[[-.03,.586],[.03,.586]],\n \+ [[-.05,0],[3.2,0]],[[0,-.2],[0,4]]],color=black):\nt1 := plots[textp lot]([[3.1,-.2,`q`],[-.12,3.414,`+r`],\n [-.12,.586,`-r`]],font=[SYM BOL,10],color=black):\nt2 := plots[textplot]([[-.1,4,`k`],[-.14,2,`a+c `],\n [-.32,3.414,`a+c`],[-.32,.586,`a+c`]],\n font=[HELVETICA,1 0]):\nplots[display]([p1,p2,t1,t2],view=[-.32..3.2,-.2..4],axes=none); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 538 "p1 := plot([-cos(t)^2+cos(t )*sin(t)-2*sin(t)^2,-1.5],t=0..Pi,\n labels=[``,``],tickmarks=[0,0],c olor=[red,blue]):\np2 := plot([[[-.03,-.793],[.03,-.793]],[[-.03,-2.21 ],[.03,-2.21]],\n [[-.05,0],[3.2,0]],[[0,-2.5],[0,.2]]],color=black ):\nt1 := plots[textplot]([[3.1,-.1,`q`],[-.12,-.793,`+r`],\n [-.12, -2.21,`-r`]],font=[SYMBOL,10],color=black):\nt2 := plots[textplot]([[- .1,.2,`k`],[-.14,-1.5,`a+c`],\n [-.32,-.793,`a+c`],[-.32,-2.21,`a+c `]],\n font=[HELVETICA,10]):\nplots[display]([p1,p2,t1,t2],view=[-. 32..3.2,-2.6..0.2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "aa := -1;\nbb := 1;\ncc := - 2;\nsqrt((aa-cc)^2+bb^2);\nevalf(%)/2;\n-1.5-%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#aaG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bbG\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ccG!\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*$-%%sqrtG6#\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5y1rq!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+\"y1r?#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 11 "animations " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 672 "p1:=plot3d([r*cos(t),r*sin(t),\n (3*cos(t)^2+7*cos(t)*sin( t)-sin(t)^2)*r^2],\n r=-2..2,t=0..Pi,color=COLOR(RGB,.7,.7,.7), style=wireframe,\n gridstyle=triangular,grid=[30,30]):\nd := eval f(Pi/30):\nfrms := NULL:\nfor i from 0 to 30 do\n di := d*i;\n tt \+ := round(evalf(di*180/Pi));\n p2:=plots[spacecurve]([r*cos(di),r*sin (di),\n (3*cos(di)^2+7*cos(di)*sin(di)-sin(di)^2)*r^2],r=-2..2,\n color=red,thickness=3);\n t1 := plots[textplot3d]([-1.5,1.5,15 ,'q' = tt],\n font=[SYMBOL,10],color=black);\n frms := frms,plo ts[display]([p1,p2,t1]);\nend do:\nplots[display]([frms],insequence=tr ue,axes=framed,\n labels=[`x`,`y`,`z`],orientation=[-50,50]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 667 "p1:=plot3d([r*cos(t),r*sin(t),\n (3*cos(t)^2-2*cos(t)*sin( t)+sin(t)^2)*r^2],\n r=-2..2,t=0..Pi,color=COLOR(RGB,.7,.7,.7), style=wireframe,\n gridstyle=triangular,grid=[30,30]):\nd := eval f(Pi/30):\nfrms := NULL:\nfor i from 0 to 30 do\n di := d*i;\n tt \+ := round(evalf(di*180/Pi));\n p2:=plots[spacecurve]([r*cos(di),r*sin (di),\n (3*cos(di)^2-2*cos(di)*sin(di)+sin(di)^2)*r^2],r=-2..2,\n color=red,thickness=3);\n t1 := plots[textplot3d]([0,2,12,'q' \+ = tt],\n font=[SYMBOL,10],color=black);\n frms := frms,plots[di splay]([p1,p2,t1]);\nend do:\nplots[display]([frms],insequence=true,ax es=framed,\n labels=[`x`,`y`,`z`],orientation=[-60,60]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 665 "p 1:=plot3d([r*cos(t),r*sin(t),\n (-cos(t)^2+cos(t)*sin(t)-2*sin( t)^2)*r^2],\n r=-2..2,t=0..Pi,color=COLOR(RGB,.7,.7,.7),style=w ireframe,\n gridstyle=triangular,grid=[30,30]):\nd := evalf(Pi/30 ):\nfrms := NULL:\nfor i from 0 to 30 do\n di := d*i;\n tt := roun d(evalf(di*180/Pi));\n p2:=plots[spacecurve]([r*cos(di),r*sin(di),\n (-cos(di)^2+cos(di)*sin(di)-2*sin(di)^2)*r^2],r=-2..2,\n co lor=red,thickness=3);\n t1 := plots[textplot3d]([0,2,-1,'q' = tt],\n font=[SYMBOL,10],color=black);\n frms := frms,plots[display]([ p1,p2,t1]);\nend do:\nplots[display]([frms],insequence=true,axes=frame d,\n labels=[`x`,`y`,`z`],orientation=[-30,25]);" }{TEXT -1 0 "" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 \+ 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }