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2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet \+ Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 34 "Quadratic forms and their graphs \+ " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canad a" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 78 "Quadratic forms .. determining the nature of the cri tical point at the origin " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 259 14 "quadratic form" }{TEXT -1 31 " is an expression of the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "q(x,y)=a*x^2+b*x*y+c*y^2" "6#/-%\"qG6$% \"xG%\"yG,(*&%\"aG\"\"\"*$F'\"\"#F,F,*(%\"bGF,F'F,F(F,F,*&%\"cGF,*$F(F .F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The surface " }{XPPEDIT 18 0 "z=q(x,y)" "6#/%\"zG-% \"qG6$%\"xG%\"yG" }{TEXT -1 67 ", which constitutes the graph of the q uadratic form, passes through" }{XPPEDIT 18 0 " ``(0,0,0)" "6#-%!G6%\" \"!F&F&" }{TEXT -1 49 ", and furthermore, since the partial derivative s " }{XPPEDIT 18 0 "q[x](x,y)=2*a*x+b*y" "6#/-&%\"qG6#%\"xG6$F(%\"yG,& *(\"\"#\"\"\"%\"aGF.F(F.F.*&%\"bGF.F*F.F." }{TEXT -1 5 " and " } {XPPEDIT 18 0 "q[y](x,y)=b*x+2*c*y" "6#/-&%\"qG6#%\"yG6$%\"xGF(,&*&%\" bG\"\"\"F*F.F.*(\"\"#F.%\"cGF.F(F.F." }{TEXT -1 20 " are both zero whe n " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 2 ", " }{TEXT 259 10 "th e origin" }{XPPEDIT 18 0 "``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 1 " " } {TEXT 259 22 "is a critical point of" }{TEXT -1 1 " " }{XPPEDIT 18 0 " q(x,y)" "6#-%\"qG6$%\"xG%\"yG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 87 "We shall investigate how the nature of this critical poin t depends on the coefficients " }{XPPEDIT 18 0 "a, b" "6$%\"aG%\"bG" } {TEXT -1 5 " and " }{TEXT 283 1 "c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 31 "First note that along the line " }{XPPEDIT 18 0 "y=0" " 6#/%\"yG\"\"!" }{TEXT -1 6 " (the " }{TEXT 279 1 "x" }{TEXT -1 7 " axi s) " }{XPPEDIT 18 0 "q(x,y)=``" "6#/-%\"qG6$%\"xG%\"yG%!G" }{XPPEDIT 18 0 "q(x,0) = a*x^2" "6#/-%\"qG6$%\"xG\"\"!*&%\"aG\"\"\"*$F'\"\"#F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "a<> 0" "6#0%\"aG\"\"!" }{TEXT -1 35 ", the intersection of the graph of " }{XPPEDIT 18 0 "q(x,y)" "6#-%\"qG6$%\"xG%\"yG" }{TEXT -1 16 " with the plane " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 6 " (the " } {TEXT 280 2 "xz" }{TEXT -1 31 " plane) is a parabola with the " } {TEXT 281 1 "z" }{TEXT -1 49 " axis as a line of symmetery, opening up wards if " }{TEXT 285 1 "a" }{TEXT -1 30 " is positive and downwards i f " }{TEXT 286 1 "a" }{TEXT -1 14 " is negative. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 217 193 193 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$!\"\"\"\"!$\"\"\"F*7$$!3ommm;p0k&* !#=$\"3c]R_q%=r9*F07$$!3wKL$3`0n(F07$$!3:mmm\"4m(G$)F0$\"32V'p4YMo$pF07$$!3\"QLL3i.9!zF 0$\"3Z6=$z\"z@ViF07$$!3\"ommT!R=0vF0$\"3(4CONayFj&F07$$!3u****\\P8#\\4 (F0$\"30z7`y3zL]F07$$!3+nm;/siqmF0$\"3qSopHns\\WF07$$!3[++](y$pZiF0$\" 335mBmxO.RF07$$!33LLL$yaE\"eF0$\"3m(y?Ic&pyLF07$$!3hmmm\">s%HaF0$\"3ej \"3!Go\"z%HF07$$!3Q+++]$*4)*\\F0$\"3mUqC6(*4)\\#F07$$!39+++]_&\\c%F0$ \"3Mc-XV;)Q3#F07$$!31+++]1aZTF0$\"3?U-MW$4-s\"F07$$!3umm;/#)[oPF0$\"3( 3L%\\M.:?9F07$$!3hLLL$=exJ$F0$\"3@IvIO>v+6F07$$!3*RLLLtIf$HF0$\"3&y?J4 F*o>')!#>7$$!3]++]PYx\"\\#F0$\"3he#)3W3%*3iF]q7$$!3EMLLL7i)4#F0$\"3!\\ _(*436US%F]q7$$!3c****\\P'psm\"F0$\"31!QHT/)yzFF]q7$$!3')****\\74_c7F0 $\"3KK)\\N![%)y:F]q7$$!3)3LLL3x%z#)F]q$\"3M!=Wt2u\\&o!#?7$$!3KMLL3s$QM %F]q$\"3a8,Dp@*o)=Fgr7$$!3]^omm;zr)*!#@$\"3+4t+rqAX(*!#C7$$\"3%pJL$ezw 5VF]q$\"3Q>$f!R?Fe=Fgr7$$\"3s*)***\\PQ#\\\")F]q$\"3UZsD4'35k'Fgr7$$\"3 GKLLe\"*[H7F0$\"3&e?f/fV;^\"F]q7$$\"3I*******pvxl\"F0$\"3a([5:F?#[FF]q 7$$\"3#z****\\_qn2#F0$\"3um(3N\"e(HJ%F]q7$$\"3U)***\\i&p@[#F0$\"3!RV,q tl6;'F]q7$$\"3B)****\\2'HKHF0$\"3;&Rg9Fg$)f)F]q7$$\"3ElmmmZvOLF0$\"3Cr sGPKR86F07$$\"3i******\\2goPF0$\"3+c+Hh^B?9F07$$\"3UKL$eR<*fTF0$\"3[xc ,u7\\Ix^Bqij(F07$$\"3[++]i`1h\"* F0$\"31F(*fd=^#R)F07$$\"3W++]P?Wl&*F0$\"3]:sFP\"o(\\\"*F07$F+F+-%'COLO URG6&%$RGBG$\"*++++\"!\")$F*F*F_[l-%*THICKNESSG6#\"\"#-F$6%7S7$F(F(7$F .$!3c]R_q%=r9*F07$F4$!3QAYJ&QafV)F07$F9$!3w!Q%*o>`0n(F07$F>$!32V'p4YMo $pF07$FC$!3Z6=$z\"z@ViF07$FH$!3(4CONayFj&F07$FM$!30z7`y3zL]F07$FR$!3qS opHns\\WF07$FW$!335mBmxO.RF07$Ffn$!3m(y?Ic&pyLF07$F[o$!3ej\"3!Go\"z%HF 07$F`o$!3mUqC6(*4)\\#F07$Feo$!3Mc-XV;)Q3#F07$Fjo$!3?U-MW$4-s\"F07$F_p$ !3(3L%\\M.:?9F07$Fdp$!3@IvIO>v+6F07$Fip$!3&y?J4F*o>')F]q7$F_q$!3he#)3W 3%*3iF]q7$Fdq$!3!\\_(*436US%F]q7$Fiq$!31!QHT/)yzFF]q7$F^r$!3KK)\\N![%) y:F]q7$Fcr$!3M!=Wt2u\\&oFgr7$Fir$!3a8,Dp@*o)=Fgr7$F^s$!3+4t+rqAX(*Fcs7 $Fes$!3Q>$f!R?Fe=Fgr7$Fjs$!3UZsD4'35k'Fgr7$F_t$!3&e?f/fV;^\"F]q7$Fdt$! 3a([5:F?#[FF]q7$Fit$!3um(3N\"e(HJ%F]q7$F^u$!3!RV,qtl6;'F]q7$Fcu$!3;&Rg 9Fg$)f)F]q7$Fhu$!3CrsGPKR86F07$F]v$!3+c+Hh^B?9F07$Fbv$!3[xc,u7\\Ix^Bqij(F07$F^z$!31F(*fd=^#R)F07$Fcz$!3]:sFP\"o(\\\"*F07$F+F(-Fiz 6&F[[lF_[lF_[lF\\[lF`[l-%%TEXTG6$7$$\"#&*!\"#$!\"(F^elQ\"x6\"-Fidl6$7$ F_elF\\elQ\"zFbel-Fidl6%7$$\"\"'F)$\"\"(F)Q&a~>~0Fbel-%&COLORG6&F[[l$ \"\"*F)F*F*-Fidl6%7$Fjel$F`elF)Q&a~<~0FbelFfdl-%+AXESLABELSG6$FaelQ!Fb el-%*AXESTICKSG6$F*F*-%%VIEWG6$;F(F+%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 47.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curv e 4" "Curve 5" "Curve 6" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 44 "Similarly, the intersection of the graph of \+ " }{XPPEDIT 18 0 "q(x,y)" "6#-%\"qG6$%\"xG%\"yG" }{TEXT -1 16 " with t he plane " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 6 " (the \+ " }{TEXT 282 2 "yz" }{TEXT -1 11 " plane) is " }{XPPEDIT 18 0 "q(x,y)= ``" "6#/-%\"qG6$%\"xG%\"yG%!G" }{XPPEDIT 18 0 "q(0,y)=c*y^2" "6#/-%\"q G6$\"\"!%\"yG*&%\"cG\"\"\"*$F(\"\"#F+" }{TEXT -1 27 ", which is a para bola when " }{XPPEDIT 18 0 "c<>0" "6#0%\"cG\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Consider \+ the vertical plane which makes an angle " }{XPPEDIT 18 0 "theta" "6#%& thetaG" }{TEXT -1 19 " with the positive " }{TEXT 284 1 "x" }{TEXT -1 12 " direction. " }}{PARA 0 "" 0 "" {TEXT -1 14 "Provided that " } {XPPEDIT 18 0 "cos(theta)<>0" "6#0-%$cosG6#%&thetaG\"\"!" }{TEXT -1 30 " this plane has the equation " }{XPPEDIT 18 0 "y=tan(theta)*x" "6 #/%\"yG*&-%$tanG6#%&thetaG\"\"\"%\"xGF*" }{TEXT -1 28 ". Otherwise it \+ is the plane " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 78 "We shall investigate how the shape of the curve of intersection of the plane " }{XPPEDIT 18 0 "y = tan(theta)* x;" "6#/%\"yG*&-%$tanG6#%&thetaG\"\"\"%\"xGF*" }{TEXT -1 11 " varies a s " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 9 " varies. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "To determ ine the shape of the curve of intersection as we rotate a vertical pla ne about the " }{TEXT 268 1 "z" }{TEXT -1 33 " axis, we calculate the \+ value of " }{XPPEDIT 18 0 "q(x,y)" "6#-%\"qG6$%\"xG%\"yG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "x=r*cos(theta)" "6#/%\"xG*&%\"rG\"\"\"-%$cosG6 #%&thetaGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=r*sin(theta)" "6#/% \"yG*&%\"rG\"\"\"-%$sinG6#%&thetaGF'" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z=``" "6#/%\"zG%!G" }{XPPEDIT 18 0 "q(r*cos(theta),r*sin(theta))=a*r^2*cos(theta)^2+b*r^2*cos(theta)*si n(theta) +c*r^2*sin(theta)^2" "6#/-%\"qG6$*&%\"rG\"\"\"-%$cosG6#%&thet aGF)*&F(F)-%$sinG6#F-F),(*(%\"aGF)*$F(\"\"#F)-F+6#F-F6F)**%\"bGF)*$F(F 6F)-F+6#F-F)-F06#F-F)F)*(%\"cGF)*$F(F6F)-F06#F-F6F)" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 4 "so, " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "z = k*r^2;" "6#/%\"zG*&%\"kG\"\"\"*$%\"rG\"\"#F'" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "k = a*cos(theta)^2+b*cos(theta)*si n(theta)+c*sin(theta)^2;" "6#/%\"kG,(*&%\"aG\"\"\"*$-%$cosG6#%&thetaG \"\"#F(F(*(%\"bGF(-F+6#F-F(-%$sinG6#F-F(F(*&%\"cGF(*$-F46#F-F.F(F(" } {TEXT -1 13 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "k<>0" "6#0%\"kG\"\"!" } {TEXT -1 43 " the curve is a parabola with its vertex at" }{XPPEDIT 18 0 " ``(0,0,0)" "6#-%!G6%\"\"!F&F&" }{TEXT -1 14 " and with the " } {TEXT 272 1 "z" }{TEXT -1 59 " axis as a line of symmetry. The parabol a opens upwards if " }{TEXT 273 1 "k" }{TEXT -1 36 " is positive and o pens downwards if " }{TEXT 274 1 "k" }{TEXT -1 14 " is negative. " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 233 240 240 {PLOTDATA 2 "6 --%'CURVESG6%7S7$$!\"\"\"\"!$\"\"\"F*7$$!3ommm;p0k&*!#=$\"3c]R_q%=r9*F 07$$!3wKL$3`0n(F0 7$$!3:mmm\"4m(G$)F0$\"32V'p4YMo$pF07$$!3\"QLL3i.9!zF0$\"3Z6=$z\"z@ViF0 7$$!3\"ommT!R=0vF0$\"3(4CONayFj&F07$$!3u****\\P8#\\4(F0$\"30z7`y3zL]F0 7$$!3+nm;/siqmF0$\"3qSopHns\\WF07$$!3[++](y$pZiF0$\"335mBmxO.RF07$$!33 LLL$yaE\"eF0$\"3m(y?Ic&pyLF07$$!3hmmm\">s%HaF0$\"3ej\"3!Go\"z%HF07$$!3 Q+++]$*4)*\\F0$\"3mUqC6(*4)\\#F07$$!39+++]_&\\c%F0$\"3Mc-XV;)Q3#F07$$! 31+++]1aZTF0$\"3?U-MW$4-s\"F07$$!3umm;/#)[oPF0$\"3(3L%\\M.:?9F07$$!3hL LL$=exJ$F0$\"3@IvIO>v+6F07$$!3*RLLLtIf$HF0$\"3&y?J4F*o>')!#>7$$!3]++]P Yx\"\\#F0$\"3he#)3W3%*3iF]q7$$!3EMLLL7i)4#F0$\"3!\\_(*436US%F]q7$$!3c* ***\\P'psm\"F0$\"31!QHT/)yzFF]q7$$!3')****\\74_c7F0$\"3KK)\\N![%)y:F]q 7$$!3)3LLL3x%z#)F]q$\"3M!=Wt2u\\&o!#?7$$!3KMLL3s$QM%F]q$\"3a8,Dp@*o)=F gr7$$!3]^omm;zr)*!#@$\"3+4t+rqAX(*!#C7$$\"3%pJL$ezw5VF]q$\"3Q>$f!R?Fe= Fgr7$$\"3s*)***\\PQ#\\\")F]q$\"3UZsD4'35k'Fgr7$$\"3GKLLe\"*[H7F0$\"3&e ?f/fV;^\"F]q7$$\"3I*******pvxl\"F0$\"3a([5:F?#[FF]q7$$\"3#z****\\_qn2# F0$\"3um(3N\"e(HJ%F]q7$$\"3U)***\\i&p@[#F0$\"3!RV,qtl6;'F]q7$$\"3B)*** *\\2'HKHF0$\"3;&Rg9Fg$)f)F]q7$$\"3ElmmmZvOLF0$\"3CrsGPKR86F07$$\"3i*** ***\\2goPF0$\"3+c+Hh^B?9F07$$\"3UKL$eR<*fTF0$\"3[xc,u7\\Ix^Bqij(F07$$\"3[++]i`1h\"*F0$\"31F(*fd=^#R)F 07$$\"3W++]P?Wl&*F0$\"3]:sFP\"o(\\\"*F07$F+F+-%'COLOURG6&%$RGBG$\"*+++ +\"!\")$F*F*F_[l-%*THICKNESSG6#\"\"#-F$6%7S7$F(F(7$F.$!3c]R_q%=r9*F07$ F4$!3QAYJ&QafV)F07$F9$!3w!Q%*o>`0n(F07$F>$!32V'p4YMo$pF07$FC$!3Z6=$z\" z@ViF07$FH$!3(4CONayFj&F07$FM$!30z7`y3zL]F07$FR$!3qSopHns\\WF07$FW$!33 5mBmxO.RF07$Ffn$!3m(y?Ic&pyLF07$F[o$!3ej\"3!Go\"z%HF07$F`o$!3mUqC6(*4) \\#F07$Feo$!3Mc-XV;)Q3#F07$Fjo$!3?U-MW$4-s\"F07$F_p$!3(3L%\\M.:?9F07$F dp$!3@IvIO>v+6F07$Fip$!3&y?J4F*o>')F]q7$F_q$!3he#)3W3%*3iF]q7$Fdq$!3! \\_(*436US%F]q7$Fiq$!31!QHT/)yzFF]q7$F^r$!3KK)\\N![%)y:F]q7$Fcr$!3M!=W t2u\\&oFgr7$Fir$!3a8,Dp@*o)=Fgr7$F^s$!3+4t+rqAX(*Fcs7$Fes$!3Q>$f!R?Fe= Fgr7$Fjs$!3UZsD4'35k'Fgr7$F_t$!3&e?f/fV;^\"F]q7$Fdt$!3a([5:F?#[FF]q7$F it$!3um(3N\"e(HJ%F]q7$F^u$!3!RV,qtl6;'F]q7$Fcu$!3;&Rg9Fg$)f)F]q7$Fhu$! 3CrsGPKR86F07$F]v$!3+c+Hh^B?9F07$Fbv$!3[xc,u7\\Ix^Bqij(F07$ F^z$!31F(*fd=^#R)F07$Fcz$!3]:sFP\"o(\\\"*F07$F+F(-Fiz6&F[[lF_[lF_[lF\\ [lF`[l-F$6%7S7$F(F_[l7$F.F_[l7$F4F_[l7$F9F_[l7$F>F_[l7$FCF_[l7$FHF_[l7 $FMF_[l7$FRF_[l7$FWF_[l7$FfnF_[l7$F[oF_[l7$F`oF_[l7$FeoF_[l7$FjoF_[l7$ F_pF_[l7$FdpF_[l7$FipF_[l7$F_qF_[l7$FdqF_[l7$FiqF_[l7$F^rF_[l7$FcrF_[l 7$FirF_[l7$F^sF_[l7$FesF_[l7$FjsF_[l7$F_tF_[l7$FdtF_[l7$FitF_[l7$F^uF_ [l7$FcuF_[l7$FhuF_[l7$F]vF_[l7$FbvF_[l7$FgvF_[l7$F\\wF_[l7$FawF_[l7$Ff wF_[l7$F[xF_[l7$F`xF_[l7$FexF_[l7$FjxF_[l7$F_yF_[l7$FdyF_[l7$FiyF_[l7$ F^zF_[l7$FczF_[l7$F+F_[l-%&COLORG6&F[[lF*$\"\")F)F*F`[l-%%TEXTG6$7$$\" #&)!\"#$!\"(FghlQ\"r6\"-Fbhl6$7$Fhhl$\"#&*FghlQ\"zF[il-Fbhl6%7$$\"\"'F )$\"\"(F)Q&k~>~0F[ilFhz-Fbhl6%7$Feil$FihlF)Q&k~<~0F[ilFfdl-Fbhl6%7$$F^ [lF)$F,F)Q&k~=~0F[ilF\\hl-%*AXESTICKSG6$F*F*-%+AXESLABELSG6$Q\"xF[ilQ! F[il-%%VIEWG6$;F(F+%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 24 "We know investigate how " }{TEXT 275 1 "k " }{TEXT -1 13 " varies with " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 45 ", paying particular attention to the sign of " }{TEXT 276 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 19 "Using the formulas " }{XPPEDIT 18 0 "sin(theta)*co s(theta)=sin(2*theta)/2" "6#/*&-%$sinG6#%&thetaG\"\"\"-%$cosG6#F(F)*&- F&6#*&\"\"#F)F(F)F)F1!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cos(theta )^2=(1+cos(2*theta))/2" "6#/*$-%$cosG6#%&thetaG\"\"#*&,&\"\"\"F,-F&6#* &F)F,F(F,F,F,F)!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin(theta)^2 =(1-cos(2*theta))/2" "6#/*$-%$sinG6#%&thetaG\"\"#*&,&\"\"\"F,-%$cosG6# *&F)F,F(F,!\"\"F,F)F1" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "k= 1/2" "6#/%\"kG*&\"\"\"F&\"\"#!\"\"" }{XPPEDIT 18 0 "``(a*(1+cos(2*theta))+b*sin(2*theta)+c*(1-cos(2*theta) ));" "6#-%!G6#,(*&%\"aG\"\"\",&F)F)-%$cosG6#*&\"\"#F)%&thetaGF)F)F)F)* &%\"bGF)-%$sinG6#*&F/F)F0F)F)F)*&%\"cGF),&F)F)-F,6#*&F/F)F0F)!\"\"F)F) " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` `=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{XPPEDIT 18 0 "``(a+c+(a-c)*cos(2 *theta)+b*sin(2*theta));" "6#-%!G6#,*%\"aG\"\"\"%\"cGF(*&,&F'F(F)!\"\" F(-%$cosG6#*&\"\"#F(%&thetaGF(F(F(*&%\"bGF(-%$sinG6#*&F1F(F2F(F(F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Choose " }{XPPEDIT 18 0 "rho" "6#%$rhoG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "rho*sin(phi)=a-c" "6#/*&%$rhoG\"\"\"-%$sinG6#%$phiGF&,&%\"aGF&% \"cG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "rho*cos(phi)=b" "6#/*&% $rhoG\"\"\"-%$cosG6#%$phiGF&%\"bG" }{TEXT -1 15 ", that is, let " } {XPPEDIT 18 0 "rho=sqrt((a-c)^2+b^2)" "6#/%$rhoG-%%sqrtG6#,&*$,&%\"aG \"\"\"%\"cG!\"\"\"\"#F,*$%\"bGF/F," }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi=arctan((a-c)/b)" "6#/%$phiG-%'arctanG6#*&,&%\"aG\"\"\"%\"cG!\" \"F+%\"bGF-" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "b<>0" "6#0%\"bG\"\"! " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi=Pi/2" "6#/%$phiG*&%#PiG\"\"\"\" \"#!\"\"" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "b=0" "6#/%\"bG\"\"!" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "a>c" "6#2%\"cG%\"aG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "phi=3*Pi/2" "6#/%$phiG*(\"\"$\"\"\"%#PiGF'\"\"# !\"\"" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "b=0" "6#/%\"bG\"\"!" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "a0" "6#2\"\"!,&%\"aG\"\"\"%\"cGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "rho < a+c;" "6#2%$rhoG,&%\"aG\"\"\"%\"cGF'" }{TEXT -1 7 ", then " } {TEXT 269 1 "k" }{TEXT -1 31 " is positive for all values of " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{GLPLOT2D 401 198 198 {PLOTDATA 2 "63-%'CURVESG6$7en 7$$\"\"!F)$\"\"$F)7$$\"3%)eD2LzxZo!#>$\"3a#y9$>%3T&G!#<7$$\"3)\\$px*G* f!G\"!#=$\"3N6hF9809FF27$$\"3+5@exGm]>F6$\"3b`e(fhZXa#F27$$\"3[99!=3o^ i#F6$\"3Y\"4X()*\\1kBF27$$\"35!\\D0[nkH$F6$\"3%eVItjmy<#F27$$\"37\"=Za &z%)=RF6$\"3m3s-[JI-?F27$$\"3edXa()oGjXF6$\"3U+l7rO^?=F27$$\"3W%3**Hbm (H_F6$\"3G.*oLaucj\"F27$$\"37PRr4)3T*eF6$\"3)z)45:u&yX\"F27$$\"3y\"[)y ykYxlF6$\"3yLx)f\"=&\\G\"F27$$\"3[s'ocGo$zrF6$\"3c(>8x%>gV6F27$$\"3UQ0 ;gr'p&yF6$\"3!G(flI2.%***F67$$\"3vFMt?$[t`)F6$\"3c)oC'p.xI()F67$$\"3u \"p30k@I>*F6$\"3X!y)*\\J&G5xF67$$\"3#*R7\\HeV)y*F6$\"3#py__aCg'pF67$$ \"3#G[))*)3W'\\5F2$\"3uOI#=m**>K'F67$$\"3W%)\\vYFjz5F2$\"3DS3O!f-78'F6 7$$\"30'[@XS@'46F2$\"3Myv3!\\!G!*fF67$$\"3=]&zEM0X9\"F2$\"3_Unp%zo(*)e F67$$\"3G9w$3G*Qz6F2$\"3K!zqFf6z&eF67$$\"3C\\L'pZn-@\"F2$\"3?*oUbMEr)e F67$$\"3>%3*3tc9T7F2$\"3ivo90%[,(fF67$$\"37J)p\"[R-v7F2$\"3g4W#[%zvAhF 67$$\"3Ey0DBA!*38F2$\"306m()RG0RjF67$$\"3qjE.#[AMP\"F2$\"3E='*>!z5L#pF 67$$\"3*R:_MgU2W\"F2$\"310+*3kKXw(F67$$\"3\"Qu#)**[jD]\"F2$\"3Cp%3l%\\ aK()F67$$\"3=hw\"ymX#p:F2$\"3[&y &z$pT9\"F27$$\"3CDL:iU!))p\"F2$\"3c;eT^<#eG\"F27$$\"3o(R1/.CRw\"F2$\"3 qsR8FhR]9F27$$\"32Id)H7*>J=F2$\"3+okyp/;I;F27$$\"3Gfb7wY,(*=F2$\"3#\\S lPkF2$\"3Km6B\"=y?*>F27$$\"3**oJ8:.SJ?F2$\"3mdh L4UZ\">#F27$$\"3)**p!zPD$\\4#F2$\"3mw-(G\\$\\nBF27$$\"3khO1 Y*[6MF27$$\"3[gN,?R*3v#F2$\"3[#zSZC-UT$F27$$\"3MK?$!3'*H:SA(zTO*!#=7$$\"3)\\$px*G*f !G\"F2$!3'*pU#=Zak*))F27$$\"3+5@exGm]>F2$!3)\\jT$*f^TZ)F27$$\"3[99!=3o ^i#F2$!3C*ya>2as;)F27$$\"3I_M;\"y<3'HF2$!3]brU%QP01)F27$$\"35!\\D0[nkH $F2$!3%4b8#\\63&)zF27$$\"3hNj)zrdwg$F2$!3?>HjN&[L%zF27$$\"37\"=Za&z%)= RF2$!3oSVCl:%*GzF27$$\"3Npe\\@u1TUF2$!3]/h(f)z(G%zF27$$\"3edXa()oGjXF2 $!3%Q?a6g7h)zF27$$\"3+@=F?n_'*[F2$!3-+D3LAXh!)F27$$\"3W%3**Hbm(H_F2$!3 id'3.:,w;)F27$$\"37PRr4)3T*eF2$!3y9P_=u9p%)F27$$\"3y\"[)yykYxlF2$!3Q^R L*ys$**))F27$$\"3[s'ocGo$zrF2$!3C[H()**G(GP*F27$$\"3UQ0;gr'p&yF2$!34GX tT')H+5!#<7$$\"3vFMt?$[t`)F2$!3iNR\\_nys5Fhp7$$\"3u\"p30k@I>*F2$!3o\"4 #yyC8]6Fhp7$$\"3#*R7\\HeV)y*F2$!3H+/K=@hD7Fhp7$$\"3#G[))*)3W'\\5Fhp$!3 #z%[S3:L?8Fhp7$$\"30'[@XS@'46Fhp$!35b79qHY.9Fhp7$$\"3G9w$3G*Qz6Fhp$!3` .^V7s#=]\"Fhp7$$\"3>%3*3tc9T7Fhp$!3?kV4Ry#*)e\"Fhp7$$\"3Ey0DBA!*38Fhp$ !3q8`\\9L)Go\"Fhp7$$\"3qjE.#[AMP\"Fhp$!3-H*Hc;f#pFhp7$$\"3=hw\" ymX#p:Fhp$!3Syr'*QpW)*>Fhp7$$\"3mwM!*4(4&Q;Fhp$!3_A:;#\\GH1#Fhp7$$\"3C DL:iU!))p\"Fhp$!3X#[0MU<.6#Fhp7$$\"3o(R1/.CRw\"Fhp$!3#Q4sjGC::#Fhp7$$ \"32Id)H7*>J=Fhp$!3s(Hi,&))\\#=#Fhp7$$\"3oWcb**o5k=Fhp$!3er_&*4I=$>#Fh p7$$\"3Gfb7wY,(*=Fhp$!3o\"f8;gl3?#Fhp7$$\"3/(HV?ka)G>Fhp$!3ue:2G/T0AFh p7$$\"3cM5'zg%pg>Fhp$!3Fhp$!3f;Dfe(3c?#Fh p7$$\"3**oJ8:.SJ?Fhp$!3l*zN8v&f+AFhp7$$\"3[M>YEk;j?Fhp$!3?7iE/S5$>#Fhp 7$$\"3)**p!zPD$\\4#Fhp$!3mTLO#f:G=#Fhp7$$\"3k Fhp7$$\"3)eq)=U!z`[#Fhp$!33l5i<>lb=Fhp7$$\"3#Q'HHd#HIb#Fhp$!3G)[,T%R'* pF\\^l7$FHF\\^l7$FRF\\^l7$FfnF\\^l7$F`oF\\^l7$FeoF\\^l7$FjoF\\^l7 $F_pF\\^l7$FdpF\\^l7$FjpF\\^l7$F_qF\\^l7$FdqF\\^l7$FiqF\\^l7$F^rF\\^l7 $FcrF\\^l7$FhrF\\^l7$F]sF\\^l7$FbsF\\^l7$FgsF\\^l7$F\\tF\\^l7$FatF\\^l 7$FftF\\^l7$F[uF\\^l7$F`uF\\^l7$FeuF\\^l7$F_vF\\^l7$FivF\\^l7$FcwF\\^l 7$F]xF\\^l7$FbxF\\^l7$FgxF\\^l7$F\\yF\\^l7$FayF\\^l7$FfyF\\^l7$F[zF\\^ l7$F`zF\\^l7$FezF\\^l7$FjzF\\^l7$F_[lF\\^l7$Fd[lF\\^l7$Fi[lF\\^l7$F^\\ lF\\^l7$Fc\\lF\\^l7$Fh\\lF\\^l7$F]]lF\\^l-Fb]l6&Fd]lF(F(Fe]l-F$6$7$7$$ !3!***************HF/$!3Q++++++IzF27$$\"3!***************HF/Ffal-Fb]l6 &Fd]lF)F)F)-F$6$7$7$Fdal$!3'*************4AFhp7$FialFablF[bl-F$6$7$7$$ !3G+++++++]F/F(7$$\"3;+++++++KFhpF(F[bl-F$6$7$7$F($!3++++++++DFhp7$F($ \"35+++++++?F2F[bl-%%TEXTG6&7$$\"#JF+$F+F+Q\"q6\"F[bl-%%FONTG6$%'SYMBO LG\"#5-Fgcl6&7$$!#7!\"#$!$$z!\"$Q#+rF^dlF[blF_dl-Fgcl6&7$Fgdl$!$@#Fidl Q#-rF^dlF[blF_dl-Fgcl6%7$F\\dl$\"\"#F+Q\"kF^dl-F`dl6$%*HELVETICAGFcdl- Fgcl6%7$$!#9Fidl$!#:F+Q$a+cF^dlFjel-Fgcl6%7$$!#KFidlFjdlFdflFjel-Fgcl6 %7$FhflFaelFdflFjel-%*AXESSTYLEG6#%%NONEG-%*AXESTICKSG6$F)F)-%+AXESLAB ELSG6%%!GFggl-F`dl6#%(DEFAULTG-%%VIEWG6$;Fhfl$\"#KF+;$!#EF+Fgel" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9 " "Curve 10" "Curve 11" "Curve 12" "Curve 13" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Otherwise, if " } {XPPEDIT 18 0 "abs(a+c) < rho" "6#2-%$absG6#,&%\"aG\"\"\"%\"cGF)%$rhoG " }{TEXT -1 7 ", then " }{TEXT 271 1 "k" }{TEXT -1 44 " takes both pos itive and negative values as " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 9 " varies. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 372 203 203 {PLOTDATA 2 "63-%'CURVESG6$7en7$$\"\"!F)$\"\"$F)7$$\"3%)eD 2LzxZo!#>$\"3orI>Q)>\"fM!#<7$$\"3)\\$px*G*f!G\"!#=$\"3Hp&*ya:T@QF27$$ \"3+5@exGm]>F6$\"3[5-Q3f!3=%F27$$\"3[99!=3o^i#F6$\"3gYgO6:'\\[%F27$$\" 35!\\D0[nkH$F6$\"3)3sM**Q0[s%F27$$\"37\"=Za&z%)=RF6$\"3e._4yZN()[F27$$ \"3Npe\\@u1TUF6$\"3)o9@0,**z%\\F27$$\"3edXa()oGjXF6$\"3M7.-;GD#*\\F27$ $\"3+@=F?n_'*[F6$\"3hi-X`We?]F27$$\"3W%3**Hbm(H_F6$\"32hN!)fM1J]F27$$ \"3C5lN\"oP>c&F6$\"35bF7(p&pB]F27$$\"37PRr4)3T*eF6$\"3A:i[eed)*\\F27$$ \"3Y47DWwyNiF6$\"3iCj0:GLa\\F27$$\"3y\"[)yykYxlF6$\"3CEb(Q5J;*[F27$$\" 3[s'ocGo$zrF6$\"3&)z\\GW%=st%F27$$\"3UQ0;gr'p&yF6$\"3Q(*3ml^!))\\%F27$ $\"3vFMt?$[t`)F6$\"3m-:c$3l[>%F27$$\"3u\"p30k@I>*F6$\"3Y(et!o***f%QF27 $$\"3#*R7\\HeV)y*F6$\"31QQM3:o'[$F27$$\"3#G[))*)3W'\\5F2$\"3I3m8Hn-9IF 27$$\"30'[@XS@'46F2$\"3KGlYFEo\"e#F27$$\"3G9w$3G*Qz6F2$\"3k>\"\\@'pg]? 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 27 "In the s pecial case where " }{XPPEDIT 18 0 "Delta = b^2-4*a*c;" "6#/%&DeltaG, &*$%\"bG\"\"#\"\"\"*(\"\"%F)%\"aGF)%\"cGF)!\"\"" }{XPPEDIT 18 0 "``=0 " "6#/%!G\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "k = a*cos(theta)^2+b *cos(theta)*sin(theta)+c*sin(theta)^2;" "6#/%\"kG,(*&%\"aG\"\"\"*$-%$c osG6#%&thetaG\"\"#F(F(*(%\"bGF(-F+6#F-F(-%$sinG6#F-F(F(*&%\"cGF(*$-F46 #F-F.F(F(" }{TEXT -1 2 ", " }{TEXT 261 20 "does not change sign" } {TEXT -1 4 " as " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 39 " v aries, but is zero for some value of " }{XPPEDIT 18 0 "theta" "6#%&the taG" }{TEXT -1 75 ". The origin is not a saddle point, a minimum point or a maximum point.\nIf " }{XPPEDIT 18 0 "b<>0" "6#0%\"bG\"\"!" } {TEXT -1 16 ", we again have " }{XPPEDIT 18 0 "a*c>0" "6#2\"\"!*&%\"aG \"\"\"%\"cGF'" }{TEXT -1 2 ", " }{TEXT 310 1 " " }{TEXT -1 3 "so " } {TEXT 314 1 "a" }{TEXT -1 5 " and " }{TEXT 311 1 "c" }{TEXT -1 15 " mu st have the " }{TEXT 259 9 "same sign" }{TEXT -1 6 ". \nIf " }{TEXT 312 1 "a" }{TEXT -1 5 " and " }{TEXT 308 1 "c" }{TEXT -1 31 " are posi tive, then \n " }{XPPEDIT 18 0 "k=``" "6#/%\"kG%!G" } {XPPEDIT 18 0 "a*cos(theta)^2+b*cos(theta)*sin(theta)+c*sin(theta)^2= \+ (sqrt(a)*cos(theta)+sqrt(c)*sin(theta))^2" "6#/,(*&%\"aG\"\"\"*$-%$cos G6#%&thetaG\"\"#F'F'*(%\"bGF'-F*6#F,F'-%$sinG6#F,F'F'*&%\"cGF'*$-F36#F ,F-F'F'*$,&*&-%%sqrtG6#F&F'-F*6#F,F'F'*&-F>6#F6F'-F36#F,F'F'F-" } {TEXT -1 1 " " }{TEXT 309 2 "\nk" }{TEXT -1 16 " is positive as " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 20 " varies except when \+ " }{XPPEDIT 18 0 "tan(theta)=-sqrt(a/c)" "6#/-%$tanG6#%&thetaG,$-%%sqr tG6#*&%\"aG\"\"\"%\"cG!\"\"F0" }{TEXT -1 32 ". \nThe quadratic form at tains a " }{TEXT 259 7 "minimum" }{TEXT -1 41 " value of 0 at all poin ts along the line " }{XPPEDIT 18 0 "y=-sqrt(a/c)*x" "6#/%\"yG,$*&-%%sq rtG6#*&%\"aG\"\"\"%\"cG!\"\"F,%\"xGF,F." }{TEXT -1 6 ". \nIf " }{TEXT 316 1 "a" }{TEXT -1 5 " and " }{TEXT 313 1 "c" }{TEXT -1 33 " are nega tive, then \n " }{XPPEDIT 18 0 "a*cos(theta)^2+b*cos(theta) *sin(theta)+c*sin(theta)^2 = -(sqrt(-a)*cos(theta)+sqrt(-c)*sin(theta) )^2;" "6#/,(*&%\"aG\"\"\"*$-%$cosG6#%&thetaG\"\"#F'F'*(%\"bGF'-F*6#F,F '-%$sinG6#F,F'F'*&%\"cGF'*$-F36#F,F-F'F',$*$,&*&-%%sqrtG6#,$F&!\"\"F'- F*6#F,F'F'*&-F?6#,$F6FBF'-F36#F,F'F'F-FB" }{TEXT -1 1 " " }{TEXT 315 2 "\nk" }{TEXT -1 16 " is negative as " }{XPPEDIT 18 0 "theta" "6#%&th etaG" }{TEXT -1 20 " varies except when " }{XPPEDIT 18 0 "tan(theta)=- sqrt(a/c)" "6#/-%$tanG6#%&thetaG,$-%%sqrtG6#*&%\"aG\"\"\"%\"cG!\"\"F0 " }{TEXT -1 31 ". The quadratic form attains a " }{TEXT 259 7 "maximum " }{TEXT -1 41 " value of 0 at all points along the line " }{XPPEDIT 18 0 "y = -sqrt(a/c)*x" "6#/%\"yG,$*&-%%sqrtG6#*&%\"aG\"\"\"%\"cG!\"\" F,%\"xGF,F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 82 "An alternative way of derivin g the classification criterion for the critical point" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 58 "We cons ider an alternative way to investigate the sign of " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "k = a*cos(theta)^2+b*cos(theta)*sin (theta)+c*sin(theta)^2" "6#/%\"kG,(*&%\"aG\"\"\"*$-%$cosG6#%&thetaG\" \"#F(F(*(%\"bGF(-F+6#F-F(-%$sinG6#F-F(F(*&%\"cGF(*$-F46#F-F.F(F(" } {TEXT -1 14 " ------- (i) " }}{PARA 0 "" 0 "" {TEXT -1 30 "for the cu rve of intersection " }{XPPEDIT 18 0 "z=k*r^2" "6#/%\"zG*&%\"kG\"\"\"* $%\"rG\"\"#F'" }{TEXT -1 16 " of the surface " }{XPPEDIT 18 0 "z=a*x^2 +b*x*y+c*y^2" "6#/%\"zG,(*&%\"aG\"\"\"*$%\"xG\"\"#F(F(*(%\"bGF(F*F(%\" yGF(F(*&%\"cGF(*$F.F+F(F(" }{TEXT -1 46 " with the vertical plane whic h makes an angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 19 " \+ with the positive " }{TEXT 299 1 "x" }{TEXT -1 13 " direction. " }} {PARA 0 "" 0 "" {TEXT -1 19 "Taking a factor of " }{XPPEDIT 18 0 "cos( theta)^2" "6#*$-%$cosG6#%&thetaG\"\"#" }{TEXT -1 40 " in the right han d side of (i), we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "k = cos(theta)^2*(a+b*tan(theta)+c*tan(theta)^2);" "6#/%\"kG*&-% $cosG6#%&thetaG\"\"#,(%\"aG\"\"\"*&%\"bGF--%$tanG6#F)F-F-*&%\"cGF-*$-F 16#F)F*F-F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "cos(theta)^2" "6#* $-%$cosG6#%&thetaG\"\"#" }{TEXT -1 18 " is positive when " }{XPPEDIT 18 0 "cos(theta)<>0" "6#0-%$cosG6#%&thetaG\"\"!" }{TEXT -1 14 ", the s ign of " }{TEXT 296 1 "k" }{TEXT -1 32 " is determined by the sign of \+ " }{XPPEDIT 18 0 "a+b*tan(theta)+c*tan(theta)^2" "6#,(%\"aG\"\"\"*&% \"bGF%-%$tanG6#%&thetaGF%F%*&%\"cGF%*$-F)6#F+\"\"#F%F%" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let \+ " }{XPPEDIT 18 0 "t=tan(theta)" "6#/%\"tG-%$tanG6#%&thetaG" }{TEXT -1 33 " to form the quadratic polynomial" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a + b*t+c*t^2" "6#,(%\"aG\"\"\"*&%\"bGF%%\"t GF%F%*&%\"cGF%*$F(\"\"#F%F%" }{TEXT -1 15 " ------- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 31 "The zeros of (ii) are given by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t = -b/(2*c);" "6#/%\"tG,$*&%\"bG \"\"\"*&\"\"#F(%\"cGF(!\"\"F," }{TEXT -1 1 " " }{TEXT 295 1 "+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(b^2-4*a*c)/(2*c)" "6#*&-%%sqrtG6#, &*$%\"bG\"\"#\"\"\"*(\"\"%F+%\"aGF+%\"cGF+!\"\"F+*&F*F+F/F+F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 20 "If the discriminant " }{XPPEDIT 18 0 "Delta=b^2-4*a*c" "6#/%&Delta G,&*$%\"bG\"\"#\"\"\"*(\"\"%F)%\"aGF)%\"cGF)!\"\"" }{TEXT -1 4 " is " }{TEXT 260 8 "positive" }{TEXT -1 83 ", then the quadratic polynomial \+ (ii) has two distinct real zeros, which means that " }{XPPEDIT 18 0 "a +b*t+c*t^2;" "6#,(%\"aG\"\"\"*&%\"bGF%%\"tGF%F%*&%\"cGF%*$F(\"\"#F%F% " }{TEXT -1 20 " and therefore also " }{XPPEDIT 18 0 "k=a*cos(theta)^2 +b*cos(theta)*sin(theta)+c*sin(theta)^2" "6#/%\"kG,(*&%\"aG\"\"\"*$-%$ cosG6#%&thetaG\"\"#F(F(*(%\"bGF(-F+6#F-F(-%$sinG6#F-F(F(*&%\"cGF(*$-F4 6#F-F.F(F(" }{TEXT -1 2 ", " }{TEXT 261 33 "can be both positive and n egative" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 5 " and " }{TEXT 297 1 "t" }{TEXT -1 37 " vary. \nIn this cas e the origin is a " }{TEXT 259 12 "saddle point" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 23 "On the o ther hand, if " }{XPPEDIT 18 0 "Delta = b^2-4*a*c;" "6#/%&DeltaG,&*$% \"bG\"\"#\"\"\"*(\"\"%F)%\"aGF)%\"cGF)!\"\"" }{TEXT -1 4 " is " } {TEXT 256 8 "negative" }{TEXT -1 51 ", (ii) does not have any real zer os. \nIn this case " }{XPPEDIT 18 0 "a+b*t+c*t^2;" "6#,(%\"aG\"\"\"*&% \"bGF%%\"tGF%F%*&%\"cGF%*$F(\"\"#F%F%" }{TEXT -1 20 " and therefore al so " }{XPPEDIT 18 0 "k=a*cos(theta)^2+b*cos(theta)*sin(theta)+c*sin(th eta)^2" "6#/%\"kG,(*&%\"aG\"\"\"*$-%$cosG6#%&thetaG\"\"#F(F(*(%\"bGF(- F+6#F-F(-%$sinG6#F-F(F(*&%\"cGF(*$-F46#F-F.F(F(" }{TEXT -1 2 ", " } {TEXT 261 20 "does not change sign" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 5 " and " }{TEXT 298 1 "t" }{TEXT -1 14 " vary. \nSince " }{XPPEDIT 18 0 "a*c>0" "6#2\"\"!*&%\"aG\"\"\"%\"c GF'" }{TEXT -1 2 ", " }{TEXT 304 2 " a" }{TEXT -1 5 " and " }{TEXT 305 1 "c" }{TEXT -1 15 " must have the " }{TEXT 259 9 "same sign" } {TEXT -1 7 ". \nIf " }{TEXT 306 1 "a" }{TEXT -1 5 " and " }{TEXT 300 1 "c" }{TEXT -1 20 " are positive, then " }{TEXT 302 1 "k" }{TEXT -1 23 " is always positive as " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 29 " varies, and the origin is a " }{TEXT 259 13 "minimum poi nt" }{TEXT -1 6 ". \nIf " }{TEXT 307 1 "a" }{TEXT -1 5 " and " }{TEXT 301 1 "c" }{TEXT -1 20 " are negative, then " }{TEXT 303 1 "k" }{TEXT -1 23 " is always negative as " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 29 " varies, and the origin is a " }{TEXT 259 13 "maximum poi nt" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "E xamples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Example 1 .. " }{XPPEDIT 18 0 "Delta;" "6 #%&DeltaG" }{TEXT -1 25 " positive (saddle point) " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "q(x,y)=3*x^2+7*x*y-y ^2" "6#/-%\"qG6$%\"xG%\"yG,(*&\"\"$\"\"\"*$F'\"\"#F,F,*(\"\"(F,F'F,F(F ,F,*$F(F.!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "In thi s example " }{XPPEDIT 18 0 "Delta=``" "6#/%&DeltaG%!G" }{XPPEDIT 18 0 "b^2-4*a*c = 7^2-4*``(3)*``(-1);" "6#/,&*$%\"bG\"\"#\"\"\"*(\"\"%F(%\" aGF(%\"cGF(!\"\",&*$\"\"(F'F(*(F*F(-%!G6#\"\"$F(-F36#,$F(F-F(F-" } {XPPEDIT 18 0 "`` = 61" "6#/%!G\"#h" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 38 "The critical point at the origin is a " }{TEXT 259 12 " saddle point" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 75 "The foll owing animation shows the shape of the intersection of the surface " } {XPPEDIT 18 0 "z=3*x^2+7*x*y-y^2" "6#/%\"zG,(*&\"\"$\"\"\"*$%\"xG\"\"# F(F(*(\"\"(F(F*F(%\"yGF(F(*$F.F+!\"\"" }{TEXT -1 46 " with the vertica l plane which makes an angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 19 " with the positive " }{TEXT 264 1 "x" }{TEXT -1 12 " dire ction. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "plots[animate]((3*cos(theta)^2+7*cos(theta)*sin(thet a)-\n sin(theta)^2)*r^2,theta=-2..2,s=0..Pi,frames=30,thickness=2); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "It is instructive to see \+ a 3 dimensional version of this animation. The angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 36 " for each frame is shown in degrees ." }}{PARA 0 "" 0 "" {TEXT -1 64 "(Click on the graphic and use the co ntrols in the context bar.) 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" }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 26 " negative (minimum point) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "q(x,y) = 3*x^2-2*x*y+y^2;" "6#/-% \"qG6$%\"xG%\"yG,(*&\"\"$\"\"\"*$F'\"\"#F,F,*(F.F,F'F,F(F,!\"\"*$F(F.F ," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "In this example " } {XPPEDIT 18 0 "Delta=``" "6#/%&DeltaG%!G" }{XPPEDIT 18 0 "b^2-4*a*c = \+ 2^2-4*``(3)*``(1);" "6#/,&*$%\"bG\"\"#\"\"\"*(\"\"%F(%\"aGF(%\"cGF(!\" \",&*$F'F'F(*(F*F(-%!G6#\"\"$F(-F26#F(F(F-" }{XPPEDIT 18 0 "`` = -8;" "6#/%!G,$\"\")!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Si nce " }{TEXT 266 1 "a" }{TEXT -1 23 " is positive (or since " }{TEXT 267 1 "c" }{TEXT -1 53 " is positive), the critical point at the origi n is a " }{TEXT 259 13 "minimum point" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The following animat ion shows the shape of the intersection of the surface " }{XPPEDIT 18 0 "z = 3*x^2-2*x*y+y^2;" "6#/%\"zG,(*&\"\"$\"\"\"*$%\"xG\"\"#F(F(*(F+F (F*F(%\"yGF(!\"\"*$F-F+F(" }{TEXT -1 46 " with the vertical plane whic h makes an angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 19 " \+ with the positive " }{TEXT 265 1 "x" }{TEXT -1 12 " direction. 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" } {XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 26 " negative (maximum p oint) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "q(x,y) = -x^2+x*y-2*y^2;" "6#/-%\"qG6$%\"xG%\"yG,(*$F'\"\"#!\"\"*&F' \"\"\"F(F.F.*&F+F.*$F(F+F.F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "In this example " }{XPPEDIT 18 0 "Delta=``" "6#/%&DeltaG% !G" }{XPPEDIT 18 0 "b^2-4*a*c = 1-4*``(-1)*``(-2);" "6#/,&*$%\"bG\"\"# \"\"\"*(\"\"%F(%\"aGF(%\"cGF(!\"\",&F(F(*(F*F(-%!G6#,$F(F-F(-F16#,$F'F -F(F-" }{XPPEDIT 18 0 "`` = -7;" "6#/%!G,$\"\"(!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{TEXT 318 1 "a" }{TEXT -1 23 " is negative (or since " }{TEXT 319 1 "c" }{TEXT -1 53 " is negative) , the critical point at the origin is a " }{TEXT 259 13 "maximum point " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The following animation shows the shape of the intersecti on of the surface " }{XPPEDIT 18 0 "z = -x^2+x*y-2*y^2;" "6#/%\"zG,(*$ %\"xG\"\"#!\"\"*&F'\"\"\"%\"yGF+F+*&F(F+*$F,F(F+F)" }{TEXT -1 46 " wit h the vertical plane which makes an angle " }{XPPEDIT 18 0 "theta" "6# %&thetaG" }{TEXT -1 19 " with the positive " }{TEXT 317 1 "x" }{TEXT -1 12 " direction. 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The angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 36 " for each frame is shown in degrees." }}{PARA 0 "" 0 "" {TEXT -1 64 "(Click on the graphic and use the controls in the context bar.) 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0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 " Q1 " }}{PARA 0 "" 0 "" {TEXT -1 37 "Determine whether the quadratic fo rm " }{XPPEDIT 18 0 "q(x,y)=6*x^2+x*y-2*y^2" "6#/-%\"qG6$%\"xG%\"yG,(* &\"\"'\"\"\"*$F'\"\"#F,F,*&F'F,F(F,F,*&F.F,*$F(F.F,!\"\"" }{TEXT -1 70 " has a maximum point, a minimum point or a saddle point at the ori gin." }}{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 30 "______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " } }{PARA 0 "" 0 "" {TEXT -1 37 "Determine whether the quadratic form " } {XPPEDIT 18 0 "q(x,y) = x^2+3*x*y+4*y^2;" "6#/-%\"qG6$%\"xG%\"yG,(*$F' \"\"#\"\"\"*(\"\"$F,F'F,F(F,F,*&\"\"%F,*$F(F+F,F," }{TEXT -1 70 " has \+ a maximum point, a minimum point or a saddle point at the origin." }} {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 30 "______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }} {PARA 0 "" 0 "" {TEXT -1 37 "Determine whether the quadratic form " } {XPPEDIT 18 0 "q(x,y) = -4*x^2+x*y-y^2;" "6#/-%\"qG6$%\"xG%\"yG,(*&\" \"%\"\"\"*$F'\"\"#F,!\"\"*&F'F,F(F,F,*$F(F.F/" }{TEXT -1 70 " has a ma ximum point, a minimum point or a saddle point at the origin." }} {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 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 pictures " }} {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 "p1:=plot([x ^2,-x^2],x=-1..1,color=[red,blue],thickness=2):\nt1:=plots[textplot]([ [.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`],co lor=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 := plots[text plot]([[.85,-.07,`r`],[-.07,.95,`z`]]):\nt2 := plots[textplot]([.6,.7, `k > 0`],color=red):\nt3 := plots[textplot]([.6,-.7,`k < 0`],color=blu e):\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([[[-.03, 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=blac k):\nt2 := plots[textplot]([[-.1,6,`k`],[-.14,1,`a+c`],\n [-.32,5.0 3,`a+c`],[-.32,-3.03,`a+c`]],\n font=[HELVETICA,10]):\nplots[displa y]([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]):\n p2 := 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[textplo t]([[3.1,-.2,`q`],[-.12,3.414,`+r`],\n [-.12,.586,`-r`]],font=[SYMBO L,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,10] ):\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 0 663 "p1:=plot3d([r*cos(t),r*sin(t),\n (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 (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]([-2,-2,4,'q' = tt],\n font=[SYMBOL,10],color=black);\n frms := frms,plots[dis play]([p1,p2,t1]);\nend do:\nplots[display]([frms],insequence=true,axe s=framed,\n labels=[`x`,`y`,`z`],orientation=[38,43]);" }}}{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 }