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1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 20 "Lagrange Multipliers" }} {PARA 19 "" 0 "" {TEXT -1 14 "Brian E. Blank" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 " Constrained Extrema" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Consider the path " }{XPPEDIT 18 0 " proc (t) options operator, arrow; [1-t, 2-t/2, 10-(1-t)^2-2*(2-t/2)^2] end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"7%,&\"\"\"F-F%!\"\" ,&\"\"#F-*&F%F-F0F.F.,(\"#5F-*$,&F-F-F%F.F0F.*&F0F-*$,&F0F-*&F%F-F0F.F .F0F-F.F*F*F*" }{TEXT -1 23 " \n \non the surface " }{XPPEDIT 18 0 "z = 10-x^2-2*y^2;" "6#/%\"zG,(\"#5\"\"\"*$%\"xG\"\"#!\"\"*&F*F'*$% \"yGF*F'F+" }{TEXT -1 181 ". \n\nThe surface may be imagined as a hil l. We may imagine that we are following a path over the hill. Relative to the ground our path follows the aquamarine line in the plot below. 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The point at which our path reaches a maximum height does n ot occur at the peak of the hill. It is not a critical point of the he ight function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "The maximum that we attain is a function not only of the particular surface but also on the path that we follow. We say that t he variables " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 112 " are constrained to lie o n this path. How do we use calculus to find an extremum associated wit h a constraint? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Let us examine the level curves of " }{XPPEDIT 18 0 "f; " "6#%\"fG" }{TEXT -1 3 ":\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "contourPlot := contourplot(f(x,y), x = -2.. 2, y= -2..2, contou rs=27,color=COLOR(RGB,0.75,0.7,0.7)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "projectionPlot := plot([1-t, 2-t/2, t=0..3], color=aq uamarine, thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " display(contourPlot,projectionPlot);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6@-%'CURVESG6U7$7$$!3w-++++++c!#=$!3l%*\\P4r& Ga#F*7$$!3%p*\\P4r&Gu&F*$!3_.++++++CF*7$F-7$$!3!y)*\\(=Ur&)eF*$!3w=+D \"y&G9@F*7$7$$!3a'*\\P4r&Ga'F*$!3)[.++++++)!#>F37$F97$F:$\"3U)))\\P4r& G9F>7$7$F:$\"3[l************zF>F@7$FD7$$!3h)**\\iSr&GiF*$\"3d\"**\\iSr &G9F*7$7$F.$\"3e'************R#F*FH7$FN7$$!3%)*\\(oa&G9n&F*$\"3c#\\(oa &G9Z#F*7$7$F($\"3V()\\P4r&Ga#F*FR7$F'7$$!3!\\.+Dcr&G]F*$!3ws**\\P%G9(H 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"Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26 " "Curve 27" "Curve 28" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 76 "As we ascend, passing from one height to another, \+ our path relative to the " }{XPPEDIT 18 0 "xy;" "6#%#xyG" }{TEXT -1 12 "-plane the " }{TEXT 290 7 "crosses" }{TEXT -1 18 " level curves \+ of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 173 ". At the point at wh ich we reach our maximum we no longer pass to a higher level. Our path begines to cross the level curves we have previously crossed. This su ggests that " }{TEXT 291 94 "at a constrained extremum the constraint curve should be tangent to the level curve it touches" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Suppo se that the constraint is expressed as " }{XPPEDIT 18 0 "phi(x,y) = co nstant;" "6#/-%$phiG6$%\"xG%\"yG%)constantG" }{TEXT -1 24 ". Then the \+ gradient of " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 36 " is per pendicular to the graph of " }{XPPEDIT 18 0 "phi(x,y) = constant;" "6 #/-%$phiG6$%\"xG%\"yG%)constantG" }{TEXT -1 22 " . The gradient of \+ " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 43 " is perpendicular to th e level curves of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 99 " . Th erefore the analytic condition that expresses the tangency at a constr ained extremum is that " }{TEXT 292 15 "the gradient of" }{TEXT -1 3 " " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 3 " " }{TEXT 293 20 "i s a scalar multiple" }{TEXT -1 2 " " }{XPPEDIT 18 0 "lambda;" "6#%'la mbdaG" }{TEXT -1 2 " " }{TEXT 294 22 " times the gradient of" }{TEXT -1 2 " " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The scalar mult iplier " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 16 " is cal led a " }{TEXT 295 19 "Lagrange multiplier" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 " Lagrange Multiplier Problems\n" }}{PARA 0 "" 0 "" {TEXT -1 131 "Let us illustrate these ideas by working thro ugh a classic constrained extremum problem. It is traditionally referr ed to as \nthe " }{TEXT 259 16 "Milkmaid Problem" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 8 "Example: " }}{PARA 0 "" 0 "" {TEXT 261 73 "Suppose that a farmhand lives in a h ouse that is represented by the point" }{TEXT -1 1 " " }{TEXT 260 9 "P = [0,0]" }{TEXT -1 2 ". " }{TEXT 262 40 "Suppose that the farmhand ha s to go from" }{TEXT -1 2 " " }{TEXT 263 1 "P" }{TEXT -1 2 " " } {TEXT 267 20 "to a barn located at" }{TEXT -1 1 " " }{TEXT 264 10 "Q = [10,0]" }{TEXT -1 83 ". However, the farmhand must first stop at the \+ river, represented by the equation " }{XPPEDIT 18 0 "x/16-y/5 = 1;" " 6#/,&*&%\"xG\"\"\"\"#;!\"\"F'*&%\"yGF'\"\"&F)F)F'" }{TEXT -1 2 ", " } {TEXT 269 80 "in order to fill a bucket of water to bring to the barn \+ animals. \n\nAt what point" }{TEXT -1 1 " " }{TEXT 265 1 "R" }{TEXT -1 1 " " }{TEXT 268 69 "on the river should the farmhand go in order t o minimize the distance" }{TEXT -1 1 " " }{TEXT 266 5 "PR+RQ" }{TEXT -1 2 "? 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}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ph iGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*&#\"\"\"\"#;F09$F0F0*&#F0 \"\"&F09%F0!\"\"F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn1 \+ := diff(f(x,y),x) = lambda*diff(phi(x,y),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,&*&,&*$)%\"xG\"\"#\"\"\"F-*$)%\"yGF,F-F-#!\" \"F,F+F-F-*(F,F2,*F)F-*&\"#?F-F+F-F2\"$+\"F-F.F-F1,&*&F,F-F+F-F-F6F2F- F-,$*&\"#;F2%'lambdaGF-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn2 := diff(f(x,y),y) = lambda*diff(phi(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,&*&,&*$)%\"xG\"\"#\"\"\"F-*$)%\"yGF,F-F- #!\"\"F,F0F-F-*&,*F)F-*&\"#?F-F+F-F2\"$+\"F-F.F-F1F0F-F-,$*&\"\"&F2%'l ambdaGF-F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eqn3 := phi(x ,y) = 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,&*&\"#;!\"\"%\"x G\"\"\"F+*&\"\"&F)%\"yGF+F)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "solve( \{eqn1, eqn2, eqn3\}, \{x,y,lambda\});" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<%/%\"xG#\"&!)[#\"%\"4$/%'lambdaG,$*&#\"%g<\"$x$\"\" \"-%'RootOfG6$,&*&\"$\"GF0)%#_ZG\"\"#F0F0F/!\"\"$\"+b0He6!\"*F0F0/%\"y G#!%!o(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "x0 := 24880/3091: y0 := -7680/3091: z0 := evalf(f(x0 ,y0)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$\"+b0He6!\")" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "horizontalLinePlot := plot(z 0, x=0..12,color=wheat):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "pointPlot := plot([[x0,f(x0,y0)],[x0+.01,f(x0,y0)]],color=NAVY,style= POINT,symbol=CIRCLE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "fu nctionPlot := plot(f(x,5*x/16-5),x=0..12,y= 10..17,color=plum):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "display(horizontalLinePlot,p ointPlot,functionPlot);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 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elif not type(args[3],set) then\n \+ error \"lagrangeMultiplier2d expects a set as its third argument.\" ;\n elif nops(args[3])<> 2 then\n error \"lagrangeMultiplier2d ex pects a set with two elements as its third argument.\";\n elif not ty pe(op(1,args[3]),\{name,equation\}) \n or not type(op(2,args[3]),\{ name,equation\}) then\n error \"lagrange_mult_2d expects its thi rd argument to be of the form \{var1,var2\} or \{var1 = a,var2=b\}.\"; \n else\n var_list := convert(args[3],list):\n end if:\n \n if t ype(op(1,var_list),name) then\n xx := op(1,var_list):\n else\n if no t type(lhs(op(1,var_list)),name) or not type(rhs(op(1,var_list)),real cons) then\n error \"lagrangeMultiplier2d expects each equation in its third argument to be of the form name = realconstant.\";\n else\n xx := lhs(op(1,var_list)): \n end if:\n end if: \n \n if type(op( 2,var_list),name) then\n yy := op(2,var_list):\n else\n if not typ e(lhs(op(2,var_list)),name) or not type(rhs(op(2,var_list)),realcons) then\n error \"lagrangeMultiplier2d expects each equation in its third argument to be of the form name = realconstant.\";\n else\n \+ yy := lhs(op(2,var_list)): \n end if:\n end if:\n \n if not type( args[2],equation)\n then\n error \"lagrangeMultiplier2d expect s an equation as its second argument.\";\n elif nops(lhs(args[2]))<> \+ 2\n then\n error \"lagrangeMultiplier2d expects the left side \+ of its second argument to be of the form g(x,y).\";\n elif not type(r hs(args[2]),realcons)\n then\n error \"lagrangeMultiplier2d ex pects the right side of its second argument to a real constant.\";\n \+ else \n constraintfn := lhs(args[2]):\n constraintconst := rhs(a rgs[2]):\n end if:\n \n if not has(constraintfn,xx) or \n not has( constraintfn,yy)\n then\n WARNING(\"The constraint does not involve \+ one of the stated variables.\");\n end if:\n \n if not type(args[1 ],algebraic)\n then\n error \"lagrangeMultiplier2d expects an \+ expression as its first argument.\";\n else\n expr := args[1]:\n en d if:\n \n if not has(expr,xx) or \n not has(expr,yy)\n then\n WA RNING( \"The expression to be extremized does not involve one of the s tated variables.\");\n end if:\n \n if nops(lhs(args[2]))<> 2\n \+ then\n error \"lagrangeMultiplier2d expects the left side of its \+ second argument to be of the form g(x,y).\";\n elif not type(rhs(args [2]),realcons)\n then\n error \"lagrangeMultiplier2d expects t he right side of its second argument to a real constant.\";\n else\n \+ constraint := args[2]: \n constraintfn := lhs(args[2]):\n con straintconst := rhs(args[2]):\n end if:\n \n eqn_set := \{diff(expr ,xx)=lambda*diff(constraintfn,xx),\n diff(expr,yy)=lambda *diff(constraintfn,yy),\n constraint\}:\n soln_set := fs olve(eqn_set, \{op(args[3]),lambda\});\n if not type(soln_set, set ) then\n printf(\"\\n\\nNo solution was found.\\n\\n\");\n ret urn;\n end if: \n\n return soln_set;\n end proc:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 " Descript ion of " }{TEXT 273 20 "lagrangeMultiplier2d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 20 "lagrangeMultiplier2d" } {TEXT -1 69 " - extremizes an expression in two variables subject to a constraint\n" }{TEXT 275 17 "Calling Sequence:" }{TEXT -1 60 "\n \+ lagrange_mult_2d(expr, constraint, set_of_vars)\n \n" }{TEXT 276 11 "Parameters:" }{TEXT -1 359 "\n expr - an expression to be extremized \n constraint - an equation whose right side \+ is a real constant \n set_of_vars - a set consisting of two indepe ndent variables. Optionally, either variable may be specified as the l eft side of an equation, with the right side of the equation equal to a real constant.\n\n" }{TEXT 277 12 "Descrip tion:" }}{PARA 15 "" 0 "" {TEXT -1 22 "A typical call to the " }{TEXT 278 16 "lagrange_mult_2d" }{TEXT -1 14 " function is \n" }{TEXT 279 46 "lagrange_mult_2d( f(x,y), g(x,y) = c, \{x,y\} );" }}{PARA 15 "" 0 "" {TEXT -1 167 "If a solution is found, then the set of equations tha t locate the extremum is returned. If no solution is found, then a str ing containing that information is returned." }}{PARA 15 "" 0 "" {TEXT -1 13 "If a call to " }{TEXT 280 16 "lagrange_mult_2d" }{TEXT -1 225 " either fails or returns an extremum that is not the desired \+ one, then it may be helpful to take an active role in initializing the Newton-Raphson Method of solving the equations. This is done by makin g the call in the form\n" }{TEXT 281 55 "lagrange_mult_2d( f(x,y), g(x ,y) = c, \{x = a, y = b\} );" }}{PARA 15 "" 0 "" {TEXT -1 88 "A warnin g is issued if the constraint does not depend on one of the specified \+ variables." }}{PARA 15 "" 0 "" {TEXT -1 102 "A warning is issued if th e expression to be extremized does not depend on one of the stated var iables." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "lagrangeMultiplier2d( x, x^2+y^2=1, \{x,y\});" }} {PARA 7 "" 1 "" {TEXT -1 87 "Warning, The expression to be extremized \+ does not involve one of the stated variables.\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"xG$!+++++5!\"*/%'lambdaG$!+++++]!#5/%\"yG$\"\"!F1 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "lagrangeMultiplier2d( x , x^2+y^2=1, \{x=0.2,y\});" }}{PARA 7 "" 1 "" {TEXT -1 87 "Warning, Th e expression to be extremized does not involve one of the stated varia bles.\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%'lambdaG$\"+++++]!#5/% \"xG$\"+++++5!\"*/%\"yG$\"\"!F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 20 "\nWe can now apply " }{TEXT 285 20 "lag rangeMultiplier2d" }{TEXT -1 40 " to the data of the Milkmaid probl em:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "lagrangeMultiplier2d( f(x,y), phi(x,y)=1, \{x,y\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/ %'lambdaG$\"+%=/uS&!\"*/%\"xG$\"+C]<\\!)F(/%\"yG$!+0Gj%[#F(" }}}{PARA 0 "" 0 "" {TEXT -1 97 "\nFor physical reasons we know that this extrem um is a minimum.\nThe minimum distance is therefore\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "min_val := f(8.049175024,-2.484632805);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(min_valG$\"+b0He6!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The tangential \+ nature of the constraint and the level curve of " }{TEXT 283 1 "f" } {TEXT -1 74 " corresponding to this extremal value can be seen by usi ng the function " }{HYPERLNK 17 "implicitplot" 2 "implicitplot" "" } {TEXT -1 10 " in the " }{HYPERLNK 17 "plots" 2 "plots" "" }{TEXT -1 12 " package.\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(p lots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords \+ has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "imp licitplot(\{f(x,y) = min_val, phi(x,y) = 1\}, x=-2..14,y=-5..3);" }} {PARA 13 "" 1 "" {GLPLOT2D 374 374 374 {PLOTDATA 2 "6%-%'CURVESG6co7$7 $$\"3>sw3F`C@7!#K$!\"&\"\"!7$$\"3c^Q:YQ:YM!#=$!3o#p2Bp2B*[!#<7$7$$\"3a +++++++cF1$!3=++++++D[F4F.7$7$$\"3k,++++++cF1F97$$\"3e!R:YQ:YQ(F1$!3;p 2Bp2BpZF47$7$$\"3N,+++++C5F4$!3q************zYF4F?7$FE7$$\"3\"Hp2Bp2B8 \"F4$!3kXQ:YQ:YYF47$7$$\"3;+++++++7F4$!3+++++++DYF4FK7$FQ7$$\"3'o%Q:YQ :E:F4$!3+Bp2Bp2BXF47$7$$\"3_++++++S=F4$!3#)***********\\U%F4FW7$7$$\"3 I++++++S=F4Fjn7$$\"3#3++++++#>F4$!3Y*************R%F47$7$$\"3Q,+++++[? 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7$$\"3/\\n/R!*=_5F]`lFiflFj^o7$F`_o7$$\"3[x)pvEQ%f5F]`l$!3'R(Q\\yL\">P (F17$7$$\"3#=r99G#zo5F]`lFa^mFd_o7$7$$\"3+7ZT\"G#zo5F]`lFa^m7$$\"3/\\v (\\Tz72\"F]`l$!3UUu()[2(Rw%F17$7$$\"35ZALDDpx5F]`lFfilFa`o7$7$$\"3GZAL DDpx5F]`lFfil7$$\"3[JH$p8(zx5F]`l$!3#)plkYo&)*)=F17$7$$\"3E1/G*>J'y5F] `lFejlF^ao7$Fdao7$$\"35j$\\y2T#y5F]`l$\"37Z=`2h%zG\"F17$7$$\"3_`o2$>\" zr5F]`lFh[mFhao7$7$$\"3q`o2$>\"zr5F]`lFd[m7$$\"3'RjpWIT#o5F]`l$\"3U-$= lxMz)\\F17$7$$\"3UhhRHD(p0\"F]`lF^_mFebo7$7$$\"3ChhRHD(p0\"F]`lF^_m7$$ \"3^(eZ$e)>U/\"F]`l$\"3)Gj?E32!*Q*F17$7$$\"3kL&z()z>d.\"F]`lF``mFbco7$ FhcoFh\\oFfhl-%+AXESLABELSG6$%\"xG%\"yG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Here is an automated f unction, " }{TEXT 284 1 "l" }{TEXT 332 22 "agrangeMultiplierPlot2" } {TEXT 333 1 "d" }{TEXT -1 98 " that first solves the Lagrange Multipli er problem and then illustrates the solution with a plot.\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 " Code for " }{TEXT 287 1 " " }{TEXT 286 24 "lagrangeMultiplierPlot2d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4628 "lagrangeMultiplierPlot2d : = proc()\nlocal xx, yy, constraint, constraintfn, constraintconst, exp r, eqn_set, lambda, var_list, plot1, plot2, plot3, jj, ii, soln_set, e xtreme_val, xx_, yy_;\nif nargs <> 3 and nargs <> 5 \n then \n \+ error \"lagrangeMultiplierPlot2d expects 3 or 5 arguments.\";\n elif not type(args[3],set)\n then\n error \"lagrangeMultiplierPlot 2d expects a set as its third argument.\";\n elif nops(args[3])<> 2\n then\n error \"lagrangeMultiplierPlot2d expects a set with tw o elements as its third argument.\";\n elif not type(op(1,args[3]),\{ name,equation\}) or not type(op(2,args[3]),\{name,equation\}) then\n error \"lagrangeMultiplierPlot2d expects its third argument to be of the form \{var1,var2\} or \{var1 = a,var2=b\}.\";\n else\n var_l ist := convert(args[3],list):\n end if:\n \n \n if type(op(1,var_list) ,name) then\n xx := op(1,var_list):\n else\n if not type(lhs(op(1 ,var_list)),name) or not type(rhs(op(1,var_list)),realcons) then\n \+ error \"lagrangeMultiplierPlot2d expects each equation in its third \+ argument to be of the form name = realconstant.\";\n else\n xx := lh s(op(1,var_list)): \n end if:\n end if: \n \n if type(op(2,var_list) ,name) then\n yy := op(2,var_list):\n else\n if not type(lhs(op(2, var_list)),name) or not type(rhs(op(2,var_list)),realcons) then\n \+ error \"v expects each equation in its third argument to be of the f orm name = realconstant.\";\n else\n yy := lhs(op(2,var_list)): \n end if:\n end if:\n \n if not type(args[2],equation)\n then\n error \"lagrangeMultiplierPlot2d expects an equation as its secon d argument.\";\n elif nops(lhs(args[2]))<> 2\n then\n error \+ \"lagrangeMultiplierPlot2d expects the left side of its second argumen t to be of the form g(x,y).\";\n elif not type(rhs(args[2]),realcons) \n then\n error \"lagrangeMultiplierPlot2d expects the right s ide of its second argument to a real constant.\";\n else \n constr aintfn := lhs(args[2]):\n constraintconst := rhs(args[2]):\n end i f:\n \n if not has(constraintfn,xx) or \n not has(constraintfn,yy) \n then\n WARNING(\"The constraint does not involve one of the state d variables.\");\n end if:\n \n \n if not type(args[1],algebraic) \n then\n error \"lagrangeMultiplierPlot2d expects an expressi on as its first argument.\";\n else\n expr := args[1]:\n end if:\n \+ \n if not has(expr,xx) or \n not has(expr,yy)\n then\n WARNING( \+ \"The expression to be extremized does not involve one of the stated v ariables.\");\n end if:\n \n if nops(lhs(args[2]))<> 2\n then\n error \"lagrangeMultiplierPlot2d expects the left side of its sec ond argument to be of the form g(x,y).\";\n elif not type(rhs(args[2] ),realcons)\n then\n error \"lagrangeMultiplierPlot2d expects \+ the right side of its second argument to a real constant.\";\n else\n constraint := args[2]: \n constraintfn := lhs(args[2]):\n co nstraintconst := rhs(args[2]):\n end if:\n \n eqn_set := \{diff(exp r,xx)=lambda*diff(constraintfn,xx),\n diff(expr,yy)=lambd a*diff(constraintfn,yy),\n constraint\}:\n soln_set := f solve(eqn_set, \{op(args[3]),lambda\});\n if not type(soln_set, se t) then\n printf(\"\\n\\nNo solution was found.\\n\\n\");\n re turn;\n end if: \n for jj from 1 to 3 do\n if has(op(jj,soln_s et),xx) then xx_ := rhs(op(jj,soln_set)) end if:\n if has(op(jj,soln_ set),yy) then yy_ := rhs(op(jj,soln_set)) end if: end do:\n\nextreme _val := subs( \{xx=xx_,yy=yy_\}, expr):\n\nprintf(\"\\nThe extreme val ue is %f. It occurs at %s = %f, %s = %f.\\n\\n\", extreme_val, xx,xx_, yy,yy_);\n\nif nargs = 3 then\n\nplot1 := plots[implicitplot](constrai nt,xx=xx_-2..xx_+2,yy=yy_-2..yy_+2,thickness=3,color=navy):\nplot2 := \+ plots[implicitplot](expr=extreme_val,xx=xx_-2..xx_+2,yy=yy_-2..yy_+2,c olor=blue,thickness=2):\nplot3 := plots[contourplot](expr,xx=xx_-2..xx _+2,yy=yy_-2..yy_+2,color=wheat): \nreturn plots[display](\{plot1,plot 2,plot3\});\nend if:\n\nif not type(args[4],equation) or not type(args [5],equation)\nthen\nerror \"The fourth and fifth arguments of plotlm2 d must be equations.\";\nelif not type(rhs(args[4]),range) or not type (rhs(args[5]),range)\nthen\nerror \"The fourth and fifth arguments of \+ plotlm2d must be equations of the form var = a..b.\";\nelif \{lhs(args [4]),lhs(args[5])\}<>\{xx,yy\} then\nerror \"The variables in the four th and fifth arguments of plotlm2d must match those of the third argum ent.\"\nelse\nplot1 := plots[implicitplot](constraint,args[4],args[5], thickness=3,color=navy):\nplot2 := plots[implicitplot](expr=extreme_va l,args[4],args[5],color=blue, thickness=2):\nplot3 := plots[contourplo t](expr,args[4],args[5],color=wheat): \nreturn plots[display](\{plot1, plot2,plot3\}); \nend if\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 " Description of " }{TEXT 297 1 " " }{TEXT 317 23 "lagrangeMultiplierPlot2" }{TEXT 296 1 "d" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 298 24 "lagrangeMultipli erPlot2d" }{TEXT -1 10 " - plots \n" }{TEXT 299 17 "Calling Sequence: " }{TEXT -1 144 "\n lagrangeMultiplierPlot2d(expr, constraint, set _of_vars)\n lagrangeMultiplierPlot2d(expr, constraint, set_of_vars , range1, range2)\n \n" }{TEXT 300 11 "Parameters:" }{TEXT -1 587 " \n expr - an expression to be extremized \n constra int - an equation whose right side is a real constant \n set_of _vars - a set consisting of two independent variables, say x and y. Op tionally, either variable may \n be specified as the left side of an equation, with the right side of the equation \+ equal to a \n real constant.\n range1 \+ - (optional) an equation that specifies a plotting range such as x = a..b \n range2 - (optional) an equation that specifies a plotting range such as y = a..b \n" }{TEXT 301 12 "Description:" }} {PARA 15 "" 0 "" {TEXT -1 23 "A typical call to the " }{TEXT 302 8 "p lotlm2d" }{TEXT -1 16 " function is \n" }{TEXT 303 38 "plotlm2d( f(x ,y), g(x,y) = c, \{x,y\} );" }}{PARA 15 "" 0 "" {TEXT -1 5 "When " } {TEXT 306 8 "plotlm2d" }{TEXT -1 409 " finds an extreme value, it pri nts the relevant information. Then it plots the constraint (in navy) i n the plane of the specified independent variables. To that plot it ad ds the level curve (in blue) of the function that corresponds to its e xtreme value. These two plotted curves should be tangent to each othe r. Finally, various other level curves (in wheat) of the function that was extremeized are added." }}{PARA 15 "" 0 "" {TEXT -1 13 "If a call to " }{TEXT 304 8 "plotlm2d" }{TEXT -1 225 " either fails or returns an extremum that is not the desired one, then it may be helpful to ta ke an active role in initializing the Newton-Raphson Method of solving the equations. This is done by making the call in the form\n" }{TEXT 307 47 "plotlm2d( f(x,y), g(x,y) = c, \{x = a, y = b\} );" }}{PARA 15 "" 0 "" {TEXT -1 75 "The typical calls that have been discussed are th e three argument form of " }{TEXT 308 8 "plotlm2d" }{TEXT -1 88 ". I f the default plot window needs to be adjusted, then use the five argu ment form of " }{TEXT 309 8 "plotlm2d" }{TEXT -1 2 ". " }}{PARA 15 " " 0 "" {TEXT -1 37 "A typical five argument call to the " }{TEXT 310 8 "plotlm2d" }{TEXT -1 16 " function is \n" }{TEXT 311 57 "plotlm2d( f(x,y), g(x,y) = c, \{x,y\}, x = a..b, y = c..d);" }{TEXT -1 4 "\nor \n" }{TEXT 312 61 "plotlm2d( f(x,y), g(x,y) = c, \{x=p,y=q\}, x = a..b , y = c..d);" }{TEXT -1 75 "\nIn this case the viewing window is as sp ecified by the last two arguments." }}{PARA 15 "" 0 "" {TEXT -1 88 "A \+ warning is issued if the constraint does not depend on one of the spec ified variables." }}{PARA 15 "" 0 "" {TEXT -1 102 "A warning is issued if the expression to be extremized does not depend on one of the stat ed variables." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "lagrangeMultiplierPlot2d( x, x^2+y^ 2=1, \{x,y\});" }}{PARA 7 "" 1 "" {TEXT -1 87 "Warning, The expression to be extremized does not involve one of the stated variables.\n" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 73 "The extre me value is -1.000000. 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36++++++?@F*F\\]n7$FiavF^av7$7$F(FehlFfav7$7$FOFcflF^bv-F_p6&Fap$\")! \\DP\"FdpFcbv$\")viobFdp-Fcfu6#\"\"$-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG 6#%(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" "Curve 9" "Curve 10" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 18 "\nWe can now apply " }{TEXT 313 1 " " } {TEXT 314 24 "lagrangeMultiplierPlot2d" }{TEXT -1 40 " to the data o f the Milkmaid problem:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "lagrangeMultiplierPlot2d( f(x,y), phi(x,y)=1, \{x,y\} );" }}{PARA 6 " " 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 73 "The extreme value \+ is 11.582906. 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} {TEXT 316 10 "z = f(x,y)" }{TEXT -1 409 ". We give the plot structure a name in order to display it later. Before displaying this surface, we will plot two relevant curves on it. In order to better see these \+ curves, we plot the surface with a light color by specifying the red, \+ blue, and green components to be just slightly less than 1 - when thes e three components are 1, then white results and when these three comp onents are 0, black results.\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "surfacePlot := plot3d( f(x,y), x= -2..15, y=-6..6, color=COLOR (RGB,.85,.85,.90),style=WIREFRAME):" }}}{PARA 0 "" 0 "" {TEXT -1 135 " \n\nNext we add the curve on the surface that lies over the constraint . We will plot this curve in navy. Again we will defer viewing it.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "curveOverConstraint := sp acecurve([x, 5*(x/16-1), f(x,5*(x/16-1)), x=-2..15], thickness=3, colo r=navy ):" }}}{PARA 0 "" 0 "" {TEXT -1 171 "\nFinally, we plot the con tour of the extremized function that corresponds to its minimum value \+ relative to the constraint. We specify blue for the color of this cont our.\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "contourAtMin := plot3d( f(x,y), x= -2..15, y=-6..6, style=CONTOUR, contours = [min_va l], thickness=3, color=blue):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 68 "\nFinally we display the superposition of our three plot structures:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display( \{surfacePlot, curveOverConstraint, contourAtMin\} );" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%%GRIDG6(;$! \"#\"\"!$\"#:F);$!\"'F)$\"\"'F)X,%)anythingG6\"6\"[gl'!%\"!!#\\bm\":\" :4033BDAFC3688E5340330D7FA16C6D564032629A292A367D4031BD8D1DDB651340311 F0A04532AD4403087F80FD86EB4402FF321EB203A07402EEB1DF05D3100402DFCE74D0 3AB2C402D2FD05F129838402C8E29C95D3250402C24D8976ED791402C0000000000004 02C24D8976ED791402C8E29C95D3250402D2FD05F129838402DFCE74D03AB2C402EEB1 DF05D3100402FF321EB203A07403087F80FD86EB440311F0A04532AD44031BD8D1DDB6 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