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203 "" 0 "" {TEXT 295 68 "Computer Lab 4: Partial Derivatives,Th e Gradient and Lagrnage Method" }}{PARA 206 "" 0 "" {TEXT 314 0 "" }} {PARA 206 "" 0 "" {TEXT 204 162 "In this lab you will learn how to use MAPLE to find partial derivatives and plot gradients. We will use MAP LE for optimization problems using the Lagrange method." }}{PARA 206 " " 0 "" {TEXT 314 0 "" }}{SECT 1 {PARA 206 "" 0 "" {TEXT 245 8 "Example s" }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{PARA 206 "" 0 "" {TEXT 209 24 "I. Partial Derivatives " }}{PARA 206 "" 0 "" {TEXT 314 0 "" }} {PARA 202 "" 0 "" {TEXT 211 8 "Method 1" }{TEXT 235 1 " " }{TEXT 261 46 "(recommended): define the function f as an " }{TEXT 272 10 "exp ression" }{TEXT 261 72 " and use notation f. For the derivatives use notation fx,fxx,fy, etc. " }}{PARA 202 "" 0 "" {TEXT 261 121 "Find f x with the command diff(f,x) and fy with the command diff(f,y) . Fo r fxx use either diff(fx,x) or diff(f,x,x)." }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{PARA 206 "" 0 "" {TEXT 255 9 "Example 1" }{TEXT 204 1 ":" }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 8 "restart:" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 31 "f:= 3*x^2-6*x^2*y^3-4*x*cos(x);" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 14 "fx:=diff(f,x);" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 14 "fy:=diff(f,y);" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 16 "fxx:=diff(fx,x);" }}}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{PARA 206 "" 0 "" {TEXT 204 49 "you can find the value of the par tial derivative " }{TEXT 280 1 "f" }{TEXT 204 1 "_" }{TEXT 280 1 "x" } {TEXT 204 45 " at the point (2,3) using the subs command:" }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 31 "f1:=subs(x=2,y=3,fx); evalf(%);" }}}{PARA 202 "" 0 "" {TEXT 315 0 "" }}{PARA 202 "" 0 "" {TEXT 211 8 "Method 2" }{TEXT 261 18 " : \+ define f as a " }{TEXT 272 8 "function" }{TEXT 261 59 " . For the de rivatives still use notation fx,fxx,fy, etc. " }}{PARA 202 "" 0 "" {TEXT 261 71 "Find fx with the command D[1](f) and fy with the com mand D[2](f) . " }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{PARA 206 "" 0 " " {TEXT 255 15 "Example 1 again" }{TEXT 204 1 ":" }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 0 "" }}} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 39 "f1:=(x,y)-> 3*x^2-6*x^2* y^3-4*x*cos(x);" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 15 "f1x: =D[1](f1) ;" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 15 "f1y:=D[2 ](f1) ;" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 16 "f1xx:=D[1](f 1x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 49 "you can find the value o f the partial derivative " }{TEXT 280 1 "f" }{TEXT 204 1 "_" }{TEXT 280 1 "x" }{TEXT 204 68 " at the point (2,3) by evaluating the parti al derivative function:" }{TEXT 256 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 22 "f2:=f1x(2,3);evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 27 "Look at the plot commands: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "plot3d(\{sin(x*y), x + 2*y\}, x=-Pi..Pi, y=-Pi ..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot3d([sin(x*y), x + 2*y], x=-Pi..Pi, y=-Pi..Pi, color=[blue,green]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "c1:= [cos(x)-2*cos(0.4*y), sin(x)-2 *sin(0.4*y), y]:\n" }{MPLTEXT 1 0 53 "c2:= [cos(x)+2*cos(0.4*y), sin(x )+2*sin(0.4*y), y]:\n" }{MPLTEXT 1 0 53 "c3:= [cos(x)+2*sin(0.4*y), si n(x)-2*cos(0.4*y), y]:\n" }{MPLTEXT 1 0 51 "c4:= [cos(x)-2*sin(0.4*y), sin(x)+2*cos(0.4*y), y]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot3d(\{c1, c2, c3, c4\}, x=0..2*Pi, y=0..10, grid=[25,15], style =patch, color=sin(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 " plot3d([c1, c2, c3],x=0..2*Pi, y=0..10, grid=[25,15], style=patch, col or=[default,green,yellow], plotlist);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 58 "Check the following plots from our functions defined above" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot3d(f,x=-2*Pi..2*Pi, y= -2*Pi..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot3d(fx, x=-2*Pi..2*Pi, y=-2*Pi..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot3d(fy,x=-2*Pi..2*Pi, y=-2*Pi..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot3d(\{f,fx,fy\},x=-2*Pi..2*Pi, y=-2*Pi..2* Pi);" }}}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{PARA 206 "" 0 "" {TEXT 209 17 "II. The Gradient" }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{PARA 206 "" 0 "" {TEXT 204 17 "Given a function " }{XPPEDIT 18 0 "Typesetti ng:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"f \"), Typesetting:-mo(\"=\", form = \"infix\", fence = \"false\", separ ator = \"false\", lspace = \"thickmathspace\", rspace = \"thickmathspa ce\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinit y\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", \+ accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", f oreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting: -mrow(Typesetting:-mi(\"f\"), Typesetting:-mo(\"⁡\", for m = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0e m\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", ma xsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelim its = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255] \"), Typesetting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \"false\", maxs ize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", font_style_name = \"2D Comment\", s ize = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\" ), Typesetting:-mrow(Typesetting:-mi(\"x\"), Typesetting:-mo(\",\", fo rm = \"infix\", fence = \"false\", separator = \"true\", lspace = \"0e m\", rspace = \"verythickmathspace\", stretchy = \"false\", symmetric \+ = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mi(\"y\")), Typesetting:-mo(\")\", fo rm = \"postfix\", fence = \"true\", separator = \"false\", lspace = \" thinmathspace\", rspace = \"verythinmathspace\", stretchy = \"true\", \+ symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", font_sty le_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", bac kground = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mi (\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&-F,6#Q\"fF'-I#moGF$63Q\"=F'/%%f ormGQ&infixF'/%&fenceGQ&falseF'/%*separatorGF=/%'lspaceGQ/thickmathspa ceF'/%'rspaceGFB/%)stretchyGF=/%*symmetricGF=/%(maxsizeGQ)infinityF'/% (minsizeGQ\"1F'/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%0font_sty le_nameGQ+2D~CommentF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgr oundGQ.[255,255,255]F'-F#6&F1-F563Q0⁡F'F8F;F>/FAQ$0emF'/ FDFaoFEFGFIFLFOFQFSFUFXFenFhn-F#6%-F563Q\"(F'/F9Q'prefixF'/F/FAQ.thinmathspaceF'/FDF]p/FFF[pFGFIFLFOFQFSFUFXFenFhn-F#6%-F,6#Q\"xF '-F563Q\",F'F8F;/F?F[pF`o/FDQ3verythickmathspaceF'FEFGFIFLFOFQFSFUFXFe nFhn-F,6#Q\"yF'-F563Q\")F'/F9Q(postfixF'FjoF>F\\p/FDQ2verythinmathspac eF'F_pFGFIFLFOFQFSFUFXFenFhnF+F+F+" }{TEXT 204 1 " " }{TEXT 204 18 ", \+ recall that the " }{TEXT 280 8 "gradient" }{TEXT 204 4 " of " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"f\"));" "-I%mrowG6# /I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6#Q\"fF'" }{TEXT 204 1 " " }{TEXT 204 24 " is the vector function " }{XPPEDIT 18 0 "Typ esetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-m i(\"\"), Typesetting:-mrow(Typesetting:-mi(\"grad\"), Typesetting:-mo( \"⁡\", form = \"infix\", fence = \"false\", separator = \+ \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", s ymmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_styl e_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", back ground = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mo(\"(\", \+ form = \"prefix\", fence = \"true\", separator = \"false\", lspace = \+ \"thinmathspace\", rspace = \"thinmathspace\", stretchy = \"true\", sy mmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", font_style _name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", backg round = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"f\")), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true\", separato r = \"false\", lspace = \"thinmathspace\", rspace = \"verythinmathspac e\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity \", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", font_style_name = \"2D Comment\", size = \"12\", fo reground = \"[0,0,0]\", background = \"[255,255,255]\")), Typesetting: -mi(\"\")), Typesetting:-mo(\"=\", form = \"infix\", fence = \"false\" , separator = \"false\", lspace = \"thickmathspace\", rspace = \"thick mathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \" infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", font_style_name = \"2D Comment\", size = \" 12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Types etting:-mrow(Typesetting:-mo(\"[\", form = \"prefix\", fence = \"true \", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thin mathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"i nfinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"fal se\", accent = \"false\", font_style_name = \"2D Comment\", size = \"1 2\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typese tting:-mrow(Typesetting:-mi(\"f_x\"), Typesetting:-mo(\",\", form = \" infix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rs pace = \"verythickmathspace\", stretchy = \"false\", symmetric = \"fal se\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", font_style_name = \"2D Co mment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255, 255,255]\"), Typesetting:-mi(\"f_y\")), Typesetting:-mo(\"]\", form = \+ \"postfix\", fence = \"true\", separator = \"false\", lspace = \"thinm athspace\", rspace = \"verythinmathspace\", stretchy = \"true\", symme tric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", font_style_n ame = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", backgro und = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mi(\" \"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF $6#Q!F'-F#6'F+-F#6&-F,6#Q%gradF'-I#moGF$63Q0⁡F'/%%formGQ &infixF'/%&fenceGQ&falseF'/%*separatorGF?/%'lspaceGQ$0emF'/%'rspaceGFD /%)stretchyGF?/%*symmetricGF?/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/ %(largeopGF?/%.movablelimitsGF?/%'accentGF?/%0font_style_nameGQ+2D~Com mentF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255, 255]F'-F#6%-F763Q\"(F'/F;Q'prefixF'/F>Q%trueF'F@/FCQ.thinmathspaceF'/F FFgo/FHFeoFIFKFNFQFSFUFWFZFgnFjn-F#6#-F,6#Q\"fF'-F763Q\")F'/F;Q(postfi xF'FdoF@Ffo/FFQ2verythinmathspaceF'FioFIFKFNFQFSFUFWFZFgnFjnF+-F763Q\" =F'F:F=F@/FCQ/thickmathspaceF'/FFFjpFGFIFKFNFQFSFUFWFZFgnFjn-F#6%-F763 Q\"[F'FboFdoF@FfoFhoFioFIFKFNFQFSFUFWFZFgnFjn-F#6%-F,6#Q$f_xF'-F763Q\" ,F'F:F=/FAFeoFB/FFQ3verythickmathspaceF'FGFIFKFNFQFSFUFWFZFgnFjn-F,6#Q $f_yF'-F763Q\"]F'FbpFdoF@FfoFdpFioFIFKFNFQFSFUFWFZFgnFjnF+F+" }{TEXT 204 1 " " }{TEXT 204 3 ". " }{TEXT 280 5 "Maple" }{TEXT 204 51 " will compute the gradient of a function using the " }{TEXT 312 4 "grad" } {TEXT 204 31 " command, which is part of the " }{TEXT 312 6 "linalg" } {TEXT 204 67 " package. (Don't worry about the warnings generated whe n you load " }{TEXT 312 6 "linalg" }{TEXT 204 3 ".) " }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{PARA 206 "" 0 "" {TEXT 255 9 "Example 2" }{TEXT 204 72 ": Find the gradient and the gradient field for the function \+ f(x,y) = " }{TEXT 280 1 "x" }{TEXT 204 5 "^2 + " }{TEXT 280 1 "y" }} {PARA 206 "" 0 "" {TEXT 314 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 22 "restart: with(linalg):" }}{PARA 206 "> " 0 "" {MPLTEXT 1 212 19 "grad(x^2+y, [x,y]);" }}}{PARA 206 "" 0 "" {TEXT 314 0 "" }} {PARA 206 "" 0 "" {TEXT 204 13 "The argument " }{TEXT 312 5 "[x,y]" } {TEXT 204 7 " tells " }{TEXT 280 5 "Maple" }{TEXT 204 19 " that you wa nt the " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"x\"));" " -I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6#Q\"xF '" }{TEXT 204 1 " " }{TEXT 204 23 "-partial derivative of " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"f\"));" "-I%mrowG6#/I+module nameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6#Q\"fF'" }{TEXT 204 1 " " }{TEXT 204 33 " in the first component, and the " }{XPPEDIT 18 0 "Typ esetting:-mrow(Typesetting:-mi(\"y\"));" "-I%mrowG6#/I+modulenameG6\"I ,TypesettingGI(_syslibGF'6#-I#miGF$6#Q\"yF'" }{TEXT 204 1 " " }{TEXT 204 36 "-partial derivative in the second. " }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{PARA 206 "" 0 "" {TEXT 204 12 "The command " }{TEXT 312 8 "gradplot" }{TEXT 204 151 " draws the gradient field of the give n function. In other words, it plots the gradient vectors of the funct ion at each of several points in the plane. " }}{PARA 206 "" 0 "" {TEXT 204 1 " " }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 12 "with(plots):" }}{PARA 206 "> " 0 "" {MPLTEXT 1 212 36 "setoptions(thickness=2, color=blue):" }}{PARA 206 " > " 0 "" {MPLTEXT 1 212 125 "plot3d(x^2+y, x=-2..2, y=-2..2,style=patc hcontour,axes=frame); gradplot(x^2+y, x=-2..2, y=-2..2, grid=[10,10], \+ arrows=slim); " }}}{EXCHG {PARA 206 "> " 0 "" {TEXT 314 0 "" }}} {EXCHG {PARA 206 "" 0 "" {TEXT 204 121 "We can also plot the gradient \+ and the level curves together to check that the gradient is orthogonal to the level curves." }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 273 4 "p1:=" }{MPLTEXT 1 212 61 "gradplot(x^2+y, x=-2..2, y=-2..2, grid=[10, 10], arrows=slim):" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 273 39 "p 2:=contourplot(x^2+y,x=-2..2,y=-2..2):" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 273 19 "display(\{p1,p2\});" }}}{EXCHG {PARA 206 "> " 0 "" {TEXT 314 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {TEXT 314 0 "" }}}{PARA 210 "" 0 "" {TEXT 250 27 "It is important to realize " }{TEXT 284 5 "t hat " }{TEXT 263 5 "Maple" }{TEXT 284 285 " scales the gradient plot s o that the gradient vectors won't overlap each other. For example, th e vector at the point (1,1) in the plot above should be [2,1], but it \+ is clearly drawn smaller than that. The relative sizes of the vectors (and their directions!) are preserved, however." }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{PARA 206 "" 0 "" {TEXT 209 26 "III. Lagrange Multip liers" }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{PARA 206 "" 0 "" {TEXT 204 18 "Given a function " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesett ing:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"f\"), Typesetting:- mo(\"=\", form = \"infix\", fence = \"false\", separator = \"false\", \+ lspace = \"thickmathspace\", rspace = \"thickmathspace\", stretchy = \+ \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \" 1\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0 ,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesettin g:-mi(\"f\"), Typesetting:-mo(\"⁡\", form = \"infix\", f ence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \" 0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infini ty\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", \+ foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting :-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", sep arator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmathspa ce\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity \", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", font_style_name = \"2D Comment\", size = \"12\", fo reground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:- mrow(Typesetting:-mi(\"x\"), Typesetting:-mo(\",\", form = \"infix\", \+ fence = \"false\", separator = \"true\", lspace = \"0em\", rspace = \" verythickmathspace\", stretchy = \"false\", symmetric = \"false\", max size = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", font_style_name = \"2D Comment\", \+ size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255] \"), Typesetting:-mi(\"y\"), Typesetting:-mo(\",\", form = \"infix\", \+ fence = \"false\", separator = \"true\", lspace = \"0em\", rspace = \" verythickmathspace\", stretchy = \"false\", symmetric = \"false\", max size = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", font_style_name = \"2D Comment\", \+ size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255] \"), Typesetting:-mi(\"z\")), Typesetting:-mo(\")\", form = \"postfix \", fence = \"true\", separator = \"false\", lspace = \"thinmathspace \", rspace = \"verythinmathspace\", stretchy = \"true\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\")), T ypesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_s yslibGF'6%-I#miGF$6#Q!F'-F#6&-F,6#Q\"fF'-I#moGF$63Q\"=F'/%%formGQ&infi xF'/%&fenceGQ&falseF'/%*separatorGF=/%'lspaceGQ/thickmathspaceF'/%'rsp aceGFB/%)stretchyGF=/%*symmetricGF=/%(maxsizeGQ)infinityF'/%(minsizeGQ \"1F'/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%0font_style_nameGQ+ 2D~CommentF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[25 5,255,255]F'-F#6&F1-F563Q0⁡F'F8F;F>/FAQ$0emF'/FDFaoFEFGF IFLFOFQFSFUFXFenFhn-F#6%-F563Q\"(F'/F9Q'prefixF'/F/FAQ.thin mathspaceF'/FDF]p/FFF[pFGFIFLFOFQFSFUFXFenFhn-F#6'-F,6#Q\"xF'-F563Q\", F'F8F;/F?F[pF`o/FDQ3verythickmathspaceF'FEFGFIFLFOFQFSFUFXFenFhn-F,6#Q \"yF'Fep-F,6#Q\"zF'-F563Q\")F'/F9Q(postfixF'FjoF>F\\p/FDQ2verythinmath spaceF'F_pFGFIFLFOFQFSFUFXFenFhnF+F+F+" }{TEXT 204 1 " " }{TEXT 204 20 ", and a constraint " }{TEXT 280 1 "g" }{TEXT 204 3 " = " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"g\"), Typesetting:-mo(\"⁡\", form \+ = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em \", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", max size = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", font_style_name = \"2D Comment\", \+ size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255] \"), Typesetting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \"false\", maxs ize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", font_style_name = \"2D Comment\", s ize = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\" ), Typesetting:-mrow(Typesetting:-mi(\"x\"), Typesetting:-mo(\",\", fo rm = \"infix\", fence = \"false\", separator = \"true\", lspace = \"0e m\", rspace = \"verythickmathspace\", stretchy = \"false\", symmetric \+ = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mi(\"y\"), Typesetting:-mo(\",\", for m = \"infix\", fence = \"false\", separator = \"true\", lspace = \"0em \", rspace = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mi(\"z\")), Typesetting:-mo(\")\", fo rm = \"postfix\", fence = \"true\", separator = \"false\", lspace = \" thinmathspace\", rspace = \"verythinmathspace\", stretchy = \"true\", \+ symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", font_sty le_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", bac kground = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mi (\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#m iGF$6#Q!F'-F#6&-F,6#Q\"gF'-I#moGF$63Q0⁡F'/%%formGQ&infix F'/%&fenceGQ&falseF'/%*separatorGF=/%'lspaceGQ$0emF'/%'rspaceGFB/%)str etchyGF=/%*symmetricGF=/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(larg eopGF=/%.movablelimitsGF=/%'accentGF=/%0font_style_nameGQ+2D~CommentF' /%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F' -F#6%-F563Q\"(F'/F9Q'prefixF'/F/FAQ.thinmathspaceF'/FDFeo/F FFcoFGFIFLFOFQFSFUFXFenFhn-F#6'-F,6#Q\"xF'-F563Q\",F'F8F;/F?FcoF@/FDQ3 verythickmathspaceF'FEFGFIFLFOFQFSFUFXFenFhn-F,6#Q\"yF'F]p-F,6#Q\"zF'- F563Q\")F'/F9Q(postfixF'FboF>Fdo/FDQ2verythinmathspaceF'FgoFGFIFLFOFQF SFUFXFenFhnF+F+" }{TEXT 204 1 " " }{TEXT 204 68 " = k recall that abs olute maximum (minimum) can be found where the " }{TEXT 280 8 "gradien t" }{TEXT 204 4 " of " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:- mi(\"f\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#- I#miGF$6#Q\"fF'" }{TEXT 204 1 " " }{TEXT 204 37 " is a constant times \+ the gradient of " }{TEXT 280 1 "g" }{TEXT 204 3 ". " }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{PARA 206 "" 0 "" {TEXT 255 9 "Example 3" }{TEXT 204 46 ": Find the absolute maximum and minimum for " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetti ng:-mi(\"f\"), Typesetting:-mo(\"=\", form = \"infix\", fence = \"fals e\", separator = \"false\", lspace = \"thickmathspace\", rspace = \"th ickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \+ \"false\", accent = \"false\", font_style_name = \"2D Comment\", size \+ = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), T ypesetting:-mrow(Typesetting:-mi(\"f\"), Typesetting:-mo(\"&ApplyFunct ion;\", form = \"infix\", fence = \"false\", separator = \"false\", ls pace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \" false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[2 55,255,255]\"), Typesetting:-mrow(Typesetting:-mo(\"(\", form = \"pref ix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspac e\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \"fa lse\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", m ovablelimits = \"false\", accent = \"false\", font_style_name = \"2D C omment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255 ,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"x\"), Typesetting:-m o(\",\", form = \"infix\", fence = \"false\", separator = \"true\", ls pace = \"0em\", rspace = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", large op = \"false\", movablelimits = \"false\", accent = \"false\", font_st yle_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", ba ckground = \"[255,255,255]\"), Typesetting:-mi(\"y\"), Typesetting:-mo (\",\", form = \"infix\", fence = \"false\", separator = \"true\", lsp ace = \"0em\", rspace = \"verythickmathspace\", stretchy = \"false\", \+ symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", font_sty le_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", bac kground = \"[255,255,255]\"), Typesetting:-mi(\"z\")), Typesetting:-mo (\")\", form = \"postfix\", fence = \"true\", separator = \"false\", l space = \"thinmathspace\", rspace = \"verythinmathspace\", stretchy = \+ \"true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1 \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0 ,0]\", background = \"[255,255,255]\")), Typesetting:-mi(\"\")), Types etting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&-F,6#Q\"fF'-I#moGF$63 Q\"=F'/%%formGQ&infixF'/%&fenceGQ&falseF'/%*separatorGF=/%'lspaceGQ/th ickmathspaceF'/%'rspaceGFB/%)stretchyGF=/%*symmetricGF=/%(maxsizeGQ)in finityF'/%(minsizeGQ\"1F'/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/ %0font_style_nameGQ+2D~CommentF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F '/%+backgroundGQ.[255,255,255]F'-F#6&F1-F563Q0⁡F'F8F;F>/ FAQ$0emF'/FDFaoFEFGFIFLFOFQFSFUFXFenFhn-F#6%-F563Q\"(F'/F9Q'prefixF'/F /FAQ.thinmathspaceF'/FDF]p/FFF[pFGFIFLFOFQFSFUFXFenFhn-F#6' -F,6#Q\"xF'-F563Q\",F'F8F;/F?F[pF`o/FDQ3verythickmathspaceF'FEFGFIFLFO FQFSFUFXFenFhn-F,6#Q\"yF'Fep-F,6#Q\"zF'-F563Q\")F'/F9Q(postfixF'FjoF>F \\p/FDQ2verythinmathspaceF'F_pFGFIFLFOFQFSFUFXFenFhnF+F+F+" }{TEXT 204 1 " " }{TEXT 204 3 "= 2" }{TEXT 280 1 "x" }{TEXT 204 4 " + 6" } {TEXT 280 1 "y" }{TEXT 204 5 " + 10" }{TEXT 280 1 "z" }{TEXT 204 28 " \+ subject to the constraint " }{TEXT 280 1 "g" }{TEXT 204 3 " = " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"g\"), Typesetting:-mo(\"⁡\", form \+ = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em \", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", max size = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", font_style_name = \"2D Comment\", \+ size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255] \"), Typesetting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \"false\", maxs ize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", font_style_name = \"2D Comment\", s ize = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\" ), Typesetting:-mrow(Typesetting:-mi(\"x\"), Typesetting:-mo(\",\", fo rm = \"infix\", fence = \"false\", separator = \"true\", lspace = \"0e m\", rspace = \"verythickmathspace\", stretchy = \"false\", symmetric \+ = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mi(\"y\"), Typesetting:-mo(\",\", for m = \"infix\", fence = \"false\", separator = \"true\", lspace = \"0em \", rspace = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mi(\"z\")), Typesetting:-mo(\")\", fo rm = \"postfix\", fence = \"true\", separator = \"false\", lspace = \" thinmathspace\", rspace = \"verythinmathspace\", stretchy = \"true\", \+ symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", font_sty le_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", bac kground = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mi (\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#m iGF$6#Q!F'-F#6&-F,6#Q\"gF'-I#moGF$63Q0⁡F'/%%formGQ&infix F'/%&fenceGQ&falseF'/%*separatorGF=/%'lspaceGQ$0emF'/%'rspaceGFB/%)str etchyGF=/%*symmetricGF=/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(larg eopGF=/%.movablelimitsGF=/%'accentGF=/%0font_style_nameGQ+2D~CommentF' /%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F' -F#6%-F563Q\"(F'/F9Q'prefixF'/F/FAQ.thinmathspaceF'/FDFeo/F FFcoFGFIFLFOFQFSFUFXFenFhn-F#6'-F,6#Q\"xF'-F563Q\",F'F8F;/F?FcoF@/FDQ3 verythickmathspaceF'FEFGFIFLFOFQFSFUFXFenFhn-F,6#Q\"yF'F]p-F,6#Q\"zF'- F563Q\")F'/F9Q(postfixF'FboF>Fdo/FDQ2verythinmathspaceF'FgoFGFIFLFOFQF SFUFXFenFhnF+F+" }{TEXT 204 1 " " }{TEXT 204 3 " = " }{TEXT 280 1 "x" }{TEXT 204 3 "^2+" }{TEXT 280 1 "y" }{TEXT 204 3 "^2+" }{TEXT 280 1 "z " }{TEXT 204 6 "^2=35." }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 9 "restart: " }}}{EXCHG {PARA 206 " > " 0 "" {MPLTEXT 1 212 17 "f:=2*x+6*y+10*z; " }}}{EXCHG {PARA 206 "> \+ " 0 "" {MPLTEXT 1 212 15 "g:=x^2+y^2+z^2;" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 27 "eq1:=diff(f,x)=a*diff(g,x);" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 27 "eq2:=diff(f,y)=a*diff(g,y);" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 27 "eq3:=diff(f,z)=a*diff(g,z);" }} }{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 10 "eq4:=g=35;" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 40 "solve(\{eq1,eq2,eq3,eq4\},\{x,y ,z,a\}); " }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{PARA 206 "" 0 "" {TEXT 209 27 "It is important to realize " }{TEXT 204 52 "that in the \+ next step we are using the x,y,z values " }{TEXT 209 19 "we calculated above" }{TEXT 204 105 ". in a different problem you will have to sub stitute a answer, the one you got in your steps above!!!!!!" }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 52 "f1:=subs(x=1,y=3,z=5,f); f2:=subs(x=-1,y=-3,z=-5,f);" }}} {PARA 206 "" 0 "" {TEXT 314 0 "" }}{PARA 206 "" 0 "" {TEXT 204 20 "If \+ two constraints (" }{TEXT 280 1 "g" }{TEXT 204 3 " = " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesettin g:-mi(\"g\"), Typesetting:-mo(\"⁡\", form = \"infix\", f ence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \" 0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infini ty\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", \+ foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting :-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", sep arator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmathspa ce\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity \", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", font_style_name = \"2D Comment\", size = \"12\", fo reground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:- mrow(Typesetting:-mi(\"x\"), Typesetting:-mo(\",\", form = \"infix\", \+ fence = \"false\", separator = \"true\", lspace = \"0em\", rspace = \" verythickmathspace\", stretchy = \"false\", symmetric = \"false\", max size = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", font_style_name = \"2D Comment\", \+ size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255] \"), Typesetting:-mi(\"y\"), Typesetting:-mo(\",\", form = \"infix\", \+ fence = \"false\", separator = \"true\", lspace = \"0em\", rspace = \" verythickmathspace\", stretchy = \"false\", symmetric = \"false\", max size = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", font_style_name = \"2D Comment\", \+ size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255] \"), Typesetting:-mi(\"z\")), Typesetting:-mo(\")\", form = \"postfix \", fence = \"true\", separator = \"false\", lspace = \"thinmathspace \", rspace = \"verythinmathspace\", stretchy = \"true\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F' -F#6&-F,6#Q\"gF'-I#moGF$63Q0⁡F'/%%formGQ&infixF'/%&fence GQ&falseF'/%*separatorGF=/%'lspaceGQ$0emF'/%'rspaceGFB/%)stretchyGF=/% *symmetricGF=/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopGF=/%.m ovablelimitsGF=/%'accentGF=/%0font_style_nameGQ+2D~CommentF'/%%sizeGQ# 12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F'-F#6%-F563 Q\"(F'/F9Q'prefixF'/F/FAQ.thinmathspaceF'/FDFeo/FFFcoFGFIFL FOFQFSFUFXFenFhn-F#6'-F,6#Q\"xF'-F563Q\",F'F8F;/F?FcoF@/FDQ3verythickm athspaceF'FEFGFIFLFOFQFSFUFXFenFhn-F,6#Q\"yF'F]p-F,6#Q\"zF'-F563Q\")F' /F9Q(postfixF'FboF>Fdo/FDQ2verythinmathspaceF'FgoFGFIFLFOFQFSFUFXFenFh nF+F+" }{TEXT 204 1 " " }{TEXT 204 9 " = k and " }{TEXT 280 1 "h" } {TEXT 204 3 " = " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\" \"), Typesetting:-mrow(Typesetting:-mi(\"h\"), Typesetting:-mo(\"&Appl yFunction;\", form = \"infix\", fence = \"false\", separator = \"false \", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetri c = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", font_style_name \+ = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background \+ = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mo(\"(\", form = \+ \"prefix\", fence = \"true\", separator = \"false\", lspace = \"thinma thspace\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric \+ = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"x\"), Typesett ing:-mo(\",\", form = \"infix\", fence = \"false\", separator = \"true \", lspace = \"0em\", rspace = \"verythickmathspace\", stretchy = \"fa lse\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", f ont_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0] \", background = \"[255,255,255]\"), Typesetting:-mi(\"y\"), Typesetti ng:-mo(\",\", form = \"infix\", fence = \"false\", separator = \"true \", lspace = \"0em\", rspace = \"verythickmathspace\", stretchy = \"fa lse\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", f ont_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0] \", background = \"[255,255,255]\"), Typesetting:-mi(\"z\")), Typesett ing:-mo(\")\", form = \"postfix\", fence = \"true\", separator = \"fal se\", lspace = \"thinmathspace\", rspace = \"verythinmathspace\", stre tchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsiz e = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\")), Typesetting:-mi(\"\")) , Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI (_syslibGF'6%-I#miGF$6#Q!F'-F#6&-F,6#Q\"hF'-I#moGF$63Q0⁡ F'/%%formGQ&infixF'/%&fenceGQ&falseF'/%*separatorGF=/%'lspaceGQ$0emF'/ %'rspaceGFB/%)stretchyGF=/%*symmetricGF=/%(maxsizeGQ)infinityF'/%(mins izeGQ\"1F'/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%0font_style_na meGQ+2D~CommentF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundG Q.[255,255,255]F'-F#6%-F563Q\"(F'/F9Q'prefixF'/F/FAQ.thinma thspaceF'/FDFeo/FFFcoFGFIFLFOFQFSFUFXFenFhn-F#6'-F,6#Q\"xF'-F563Q\",F' F8F;/F?FcoF@/FDQ3verythickmathspaceF'FEFGFIFLFOFQFSFUFXFenFhn-F,6#Q\"y F'F]p-F,6#Q\"zF'-F563Q\")F'/F9Q(postfixF'FboF>Fdo/FDQ2verythinmathspac eF'FgoFGFIFLFOFQFSFUFXFenFhnF+F+" }{TEXT 204 1 " " }{TEXT 204 32 " = c ) are given for a function " }{XPPEDIT 18 0 "Typesetting:-mrow(Typese tting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"f\"), Typesetting :-mo(\"=\", form = \"infix\", fence = \"false\", separator = \"false\" , lspace = \"thickmathspace\", rspace = \"thickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \+ \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[ 0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetti ng:-mi(\"f\"), Typesetting:-mo(\"⁡\", form = \"infix\", \+ fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \+ \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infi nity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmat hspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infi nity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mrow(Typesetting:-mi(\"x\"), Typesetting:-mo(\",\", form = \"inf ix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rspac e = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false \", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", font_style_name = \"2D Comm ent\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,25 5,255]\"), Typesetting:-mi(\"y\"), Typesetting:-mo(\",\", form = \"inf ix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rspac e = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false \", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", font_style_name = \"2D Comm ent\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,25 5,255]\"), Typesetting:-mi(\"z\")), Typesetting:-mo(\")\", form = \"po stfix\", fence = \"true\", separator = \"false\", lspace = \"thinmaths pace\", rspace = \"verythinmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fal se\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\")), \+ Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_ syslibGF'6%-I#miGF$6#Q!F'-F#6&-F,6#Q\"fF'-I#moGF$63Q\"=F'/%%formGQ&inf ixF'/%&fenceGQ&falseF'/%*separatorGF=/%'lspaceGQ/thickmathspaceF'/%'rs paceGFB/%)stretchyGF=/%*symmetricGF=/%(maxsizeGQ)infinityF'/%(minsizeG Q\"1F'/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%0font_style_nameGQ +2D~CommentF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[2 55,255,255]F'-F#6&F1-F563Q0⁡F'F8F;F>/FAQ$0emF'/FDFaoFEFG FIFLFOFQFSFUFXFenFhn-F#6%-F563Q\"(F'/F9Q'prefixF'/F/FAQ.thi nmathspaceF'/FDF]p/FFF[pFGFIFLFOFQFSFUFXFenFhn-F#6'-F,6#Q\"xF'-F563Q\" ,F'F8F;/F?F[pF`o/FDQ3verythickmathspaceF'FEFGFIFLFOFQFSFUFXFenFhn-F,6# Q\"yF'Fep-F,6#Q\"zF'-F563Q\")F'/F9Q(postfixF'FjoF>F\\p/FDQ2verythinmat hspaceF'F_pFGFIFLFOFQFSFUFXFenFhnF+F+F+" }{TEXT 204 1 " " }{TEXT 204 65 ", then the absolute maximum (minimum) can be found where the " }{TEXT 280 8 "gradient" }{TEXT 204 4 " of " }{XPPEDIT 18 0 "Typesetti ng:-mrow(Typesetting:-mi(\"f\"));" "-I%mrowG6#/I+modulenameG6\"I,Types ettingGI(_syslibGF'6#-I#miGF$6#Q\"fF'" }{TEXT 204 1 " " }{TEXT 204 37 " is a constant times the gradient of " }{TEXT 280 2 "g " }{TEXT 204 44 "plus another constant times the gradient of " }{TEXT 280 1 "h" } {TEXT 204 3 ". " }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{PARA 206 "" 0 "" {TEXT 255 9 "Example 4" }{TEXT 204 46 ": Find the absolute maximum and minimum for " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi( \"\"), Typesetting:-mrow(Typesetting:-mi(\"f\"), Typesetting:-mo(\"=\" , form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"thickmathspace\", rspace = \"thickmathspace\", stretchy = \"false\" , symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", larg eop = \"false\", movablelimits = \"false\", accent = \"false\", font_s tyle_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", b ackground = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"f \"), Typesetting:-mo(\"⁡\", form = \"infix\", fence = \" false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", st retchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", min size = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", foregroun d = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Ty pesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \+ \"false\", lspace = \"thinmathspace\", rspace = \"thinmathspace\", str etchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsi ze = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typ esetting:-mi(\"x\"), Typesetting:-mo(\",\", form = \"infix\", fence = \+ \"false\", separator = \"true\", lspace = \"0em\", rspace = \"verythic kmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \+ \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", font_style_name = \"2D Comment\", size = \+ \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typ esetting:-mi(\"y\"), Typesetting:-mo(\",\", form = \"infix\", fence = \+ \"false\", separator = \"true\", lspace = \"0em\", rspace = \"verythic kmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \+ \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", font_style_name = \"2D Comment\", size = \+ \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typ esetting:-mi(\"z\")), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"verythinmathspace\", stretchy = \"true\", symmetric = \"false\", \+ maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", font_style_name = \"2D Comment \", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,2 55]\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\")), Typesetting:- mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I #miGF$6#Q!F'-F#6&-F,6#Q\"fF'-I#moGF$63Q\"=F'/%%formGQ&infixF'/%&fenceG Q&falseF'/%*separatorGF=/%'lspaceGQ/thickmathspaceF'/%'rspaceGFB/%)str etchyGF=/%*symmetricGF=/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(larg eopGF=/%.movablelimitsGF=/%'accentGF=/%0font_style_nameGQ+2D~CommentF' /%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F' -F#6&F1-F563Q0⁡F'F8F;F>/FAQ$0emF'/FDFaoFEFGFIFLFOFQFSFUF XFenFhn-F#6%-F563Q\"(F'/F9Q'prefixF'/F/FAQ.thinmathspaceF'/ FDF]p/FFF[pFGFIFLFOFQFSFUFXFenFhn-F#6'-F,6#Q\"xF'-F563Q\",F'F8F;/F?F[p F`o/FDQ3verythickmathspaceF'FEFGFIFLFOFQFSFUFXFenFhn-F,6#Q\"yF'Fep-F,6 #Q\"zF'-F563Q\")F'/F9Q(postfixF'FjoF>F\\p/FDQ2verythinmathspaceF'F_pFG FIFLFOFQFSFUFXFenFhnF+F+F+" }{TEXT 204 1 " " }{TEXT 204 32 "= z subje ct to the constraints " }{TEXT 280 1 "g" }{TEXT 204 3 " = " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Types etting:-mi(\"g\"), Typesetting:-mo(\"⁡\", form = \"infix \", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"i nfinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"fal se\", accent = \"false\", font_style_name = \"2D Comment\", size = \"1 2\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typese tting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true\" , separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinma thspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"inf inity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mrow(Typesetting:-mi(\"x\"), Typesetting:-mo(\",\", form = \"inf ix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rspac e = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false \", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", font_style_name = \"2D Comm ent\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,25 5,255]\"), Typesetting:-mi(\"y\"), Typesetting:-mo(\",\", form = \"inf ix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rspac e = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false \", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", font_style_name = \"2D Comm ent\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,25 5,255]\"), Typesetting:-mi(\"z\")), Typesetting:-mo(\")\", form = \"po stfix\", fence = \"true\", separator = \"false\", lspace = \"thinmaths pace\", rspace = \"verythinmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fal se\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F '-F#6&-F,6#Q\"gF'-I#moGF$63Q0⁡F'/%%formGQ&infixF'/%&fenc eGQ&falseF'/%*separatorGF=/%'lspaceGQ$0emF'/%'rspaceGFB/%)stretchyGF=/ %*symmetricGF=/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopGF=/%. movablelimitsGF=/%'accentGF=/%0font_style_nameGQ+2D~CommentF'/%%sizeGQ #12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F'-F#6%-F56 3Q\"(F'/F9Q'prefixF'/F/FAQ.thinmathspaceF'/FDFeo/FFFcoFGFIF LFOFQFSFUFXFenFhn-F#6'-F,6#Q\"xF'-F563Q\",F'F8F;/F?FcoF@/FDQ3verythick mathspaceF'FEFGFIFLFOFQFSFUFXFenFhn-F,6#Q\"yF'F]p-F,6#Q\"zF'-F563Q\")F '/F9Q(postfixF'FboF>Fdo/FDQ2verythinmathspaceF'FgoFGFIFLFOFQFSFUFXFenF hnF+F+" }{TEXT 204 1 " " }{TEXT 204 3 " = " }{TEXT 280 3 "x+y" }{TEXT 204 12 " +z =1 and " }{TEXT 280 1 "h" }{TEXT 204 3 " = " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typeset ting:-mi(\"h\"), Typesetting:-mo(\"⁡\", form = \"infix\" , fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"inf inity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mrow(Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmat hspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infi nity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Comment\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mrow(Typesetting:-mi(\"x\"), Typesetting:-mo(\",\", form = \"inf ix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rspac e = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false \", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", font_style_name = \"2D Comm ent\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,25 5,255]\"), Typesetting:-mi(\"y\"), Typesetting:-mo(\",\", form = \"inf ix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rspac e = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false \", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", font_style_name = \"2D Comm ent\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,25 5,255]\"), Typesetting:-mi(\"z\")), Typesetting:-mo(\")\", form = \"po stfix\", fence = \"true\", separator = \"false\", lspace = \"thinmaths pace\", rspace = \"verythinmathspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fal se\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F '-F#6&-F,6#Q\"hF'-I#moGF$63Q0⁡F'/%%formGQ&infixF'/%&fenc eGQ&falseF'/%*separatorGF=/%'lspaceGQ$0emF'/%'rspaceGFB/%)stretchyGF=/ %*symmetricGF=/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopGF=/%. movablelimitsGF=/%'accentGF=/%0font_style_nameGQ+2D~CommentF'/%%sizeGQ #12F'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ.[255,255,255]F'-F#6%-F56 3Q\"(F'/F9Q'prefixF'/F/FAQ.thinmathspaceF'/FDFeo/FFFcoFGFIF LFOFQFSFUFXFenFhn-F#6'-F,6#Q\"xF'-F563Q\",F'F8F;/F?FcoF@/FDQ3verythick mathspaceF'FEFGFIFLFOFQFSFUFXFenFhn-F,6#Q\"yF'F]p-F,6#Q\"zF'-F563Q\")F '/F9Q(postfixF'FboF>Fdo/FDQ2verythinmathspaceF'FgoFGFIFLFOFQFSFUFXFenF hnF+F+" }{TEXT 204 1 " " }{TEXT 204 3 " = " }{TEXT 280 1 "y" }{TEXT 204 10 "^2+x^2=2 ." }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 9 "restart: " }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 6 "f:=z; " }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 11 "h:=y^2+x^2;" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 39 " eq1:=diff(f,x)=a*diff(g,x)+b*diff(h,x);" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 39 "eq2:=diff(f,y)=a*diff(g,y)+b*diff(h,y);" }}} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 39 "eq3:=diff(f,z)=a*diff(g, z)+b*diff(h,z);" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 9 "eq4:= g=1;" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 9 "eq5:=h=2;" }}} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 46 "solve(\{eq1,eq2,eq3,eq4, eq5\},\{x,y,z,a,b\}); " }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}{PARA 206 "" 0 "" {TEXT 204 77 "note that in the next step we are using the x,y ,z values we calculated above." }}{PARA 206 "" 0 "" {TEXT 314 0 "" }}} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 212 53 "f1:=subs(x=1,y=1,z=-1,f) ; f2:=subs(x=-1,y=-1,z=3,f);;" }}}{PARA 206 "" 0 "" {TEXT 314 0 "" }} {PARA 206 "" 0 "" {TEXT 314 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 316 64 "Exercise due on Feb 27, 2008. Solve only one of the given three." }}{PARA 0 "" 0 "" {TEXT 256 20 "1. For the function " }{XPPEDIT 256 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesettin g:-mi(\"f\"), Typesetting:-mo(\"⁡\", form = \"infix\", f ence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \" 0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infini ty\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"Text\", size = \"12\", foregr ound = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mfen ced(Typesetting:-mrow(Typesetting:-mi(\"x\"), Typesetting:-mo(\",\", f orm = \"infix\", fence = \"false\", separator = \"true\", lspace = \"0 em\", rspace = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fal se\", movablelimits = \"false\", accent = \"false\", font_style_name = \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \"[25 5,255,255]\"), Typesetting:-mi(\"y\")))), Typesetting:-mo(\"=\", form \+ = \"infix\", fence = \"false\", separator = \"false\", lspace = \"thic kmathspace\", rspace = \"thickmathspace\", stretchy = \"false\", symme tric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", font_style_n ame = \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mi(\"x\"), Typesetting:-msup(Typesett ing:-mo(\"ⅇ\", form = \"prefix\", fence = \"false\", sepa rator = \"false\", lspace = \"0em\", rspace = \"verythinmathspace\", s tretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", mi nsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent \+ = \"false\", font_style_name = \"Text\", size = \"12\", foreground = \+ \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typese tting:-mo(\"−\", form = \"infix\", fence = \"false\", separator \+ = \"true\", lspace = \"0em\", rspace = \"verythickmathspace\", stretch y = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize \+ = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", font_style_name = \"Text\", size = \"12\", foreground = \"[0,0, 0]\", background = \"[255,255,255]\"), Typesetting:-msup(Typesetting:- mi(\"x\"), Typesetting:-mn(\"2\"), superscriptshift = \"0\"), Typesett ing:-mo(\"−\", form = \"infix\", fence = \"false\", separator = \+ \"true\", lspace = \"0em\", rspace = \"verythickmathspace\", stretchy \+ = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \+ \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", font_style_name = \"Text\", size = \"12\", foreground = \"[0,0,0] \", background = \"[255,255,255]\"), Typesetting:-msup(Typesetting:-mi (\"y\"), Typesetting:-mrow(Typesetting:-mn(\"2\"), Typesetting:-mo(\"& InvisibleTimes;\", form = \"infix\", fence = \"false\", separator = \" true\", lspace = \"0em\", rspace = \"verythickmathspace\", stretchy = \+ \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \" 1\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", font_style_name = \"Text\", size = \"12\", foreground = \"[0,0,0] \", background = \"[255,255,255]\"), Typesetting:-mo(\"&InvisibleTimes ;\", form = \"infix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rspace = \"verythickmathspace\", stretchy = \"false\", sym metric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_ name = \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\")), superscriptshift = \"0\"), Typesetting:-mi(\"\") ), superscriptshift = \"0\"), Typesetting:-mo(\"⁢\", fo rm = \"infix\", fence = \"false\", separator = \"true\", lspace = \"0e m\", rspace = \"verythickmathspace\", stretchy = \"false\", symmetric \+ = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255 ,255,255]\"), Typesetting:-mi(\"determine\"), Typesetting:-mo(\"&Invis ibleTimes;\", form = \"infix\", fence = \"false\", separator = \"true \", lspace = \"0em\", rspace = \"verythickmathspace\", stretchy = \"fa lse\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", f ont_style_name = \"Text\", size = \"12\", foreground = \"[0,0,0]\", ba ckground = \"[255,255,255]\"), Typesetting:-msub(Typesetting:-mi(\"f\" ), Typesetting:-mi(\"x\"), subscriptshift = \"0\", placeholder = \"fal se\"), Typesetting:-mo(\"⁢\", form = \"infix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rspace = \"verythi ckmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \+ \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", font_style_name = \"Text\", size = \"12\" , foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetti ng:-mi(\"and\"), Typesetting:-mo(\"⁢\", form = \"infix \", fence = \"false\", separator = \"true\", lspace = \"0em\", rspace \+ = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movable limits = \"false\", accent = \"false\", font_style_name = \"Text\", si ze = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\") , Typesetting:-msub(Typesetting:-mi(\"f\"), Typesetting:-mi(\"y\"), su bscriptshift = \"0\", placeholder = \"false\"), Typesetting:-mo(\"&Inv isibleTimes;\", form = \"infix\", fence = \"false\", separator = \"tru e\", lspace = \"0em\", rspace = \"verythickmathspace\", stretchy = \"f alse\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ font_style_name = \"Text\", size = \"12\", foreground = \"[0,0,0]\", b ackground = \"[255,255,255]\"), Typesetting:-mo(\"⁢\", \+ form = \"infix\", fence = \"false\", separator = \"true\", lspace = \" 0em\", rspace = \"verythickmathspace\", stretchy = \"false\", symmetri c = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", font_style_name \+ = \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \"[2 55,255,255]\"), Typesetting:-mo(\"⁢\", form = \"infix\" , fence = \"false\", separator = \"true\", lspace = \"0em\", rspace = \+ \"verythickmathspace\", stretchy = \"false\", symmetric = \"false\", m axsize = \"infinity\", minsize = \"1\", largeop = \"false\", movableli mits = \"false\", accent = \"false\", font_style_name = \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), \+ Typesetting:-mi(\"and\"), Typesetting:-mo(\"⁢\", form = \"infix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rspace = \"verythickmathspace\", stretchy = \"false\", symmetric = \" false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"Te xt\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255 ,255]\"), Typesetting:-mi(\"plot\"), Typesetting:-mo(\"&InvisibleTimes ;\", form = \"infix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rspace = \"verythickmathspace\", stretchy = \"false\", sym metric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_ name = \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typesetting:-mi(\"f\"), Typeset ting:-mo(\"⁡\", form = \"infix\", fence = \"false\", sep arator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"f alse\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ font_style_name = \"Text\", size = \"12\", foreground = \"[0,0,0]\", b ackground = \"[255,255,255]\"), Typesetting:-mfenced(Typesetting:-mrow (Typesetting:-mi(\"x\"), Typesetting:-mo(\",\", form = \"infix\", fenc e = \"false\", separator = \"true\", lspace = \"0em\", rspace = \"very thickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"Text\", size = \"1 2\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typese tting:-mi(\"y\")))), Typesetting:-mo(\",\", form = \"infix\", fence = \+ \"false\", separator = \"true\", lspace = \"0em\", rspace = \"verythic kmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \+ \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", font_style_name = \"Text\", size = \"12\" , foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetti ng:-mo(\"⁢\", form = \"infix\", fence = \"false\", sepa rator = \"true\", lspace = \"0em\", rspace = \"verythickmathspace\", s tretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", mi nsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent \+ = \"false\", font_style_name = \"Text\", size = \"12\", foreground = \+ \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-msub(Typese tting:-mi(\"f\"), Typesetting:-mrow(Typesetting:-mi(\"x\"), Typesettin g:-mo(\"⁢\", form = \"infix\", fence = \"false\", separ ator = \"true\", lspace = \"0em\", rspace = \"verythickmathspace\", st retchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", min size = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"Text\", size = \"12\", foreground = \" [0,0,0]\", background = \"[255,255,255]\")), subscriptshift = \"0\", p laceholder = \"false\"), Typesetting:-mo(\",\", form = \"infix\", fenc e = \"false\", separator = \"true\", lspace = \"0em\", rspace = \"very thickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"Text\", size = \"1 2\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typese tting:-mo(\"⁢\", form = \"infix\", fence = \"false\", s eparator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \+ \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \" 1\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", font_style_name = \"Text\", size = \"12\", foreground = \"[0,0,0] \", background = \"[255,255,255]\"), Typesetting:-msub(Typesetting:-mi (\"f\"), Typesetting:-mrow(Typesetting:-mi(\"y\"), Typesetting:-mo(\"& InvisibleTimes;\", form = \"infix\", fence = \"false\", separator = \" false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", sym metric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_ name = \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\")), subscriptshift = \"0\", placeholder = \"false\") , Typesetting:-mi(\"on\"), Typesetting:-mo(\"⁢\", form \+ = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em \", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", max size = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", font_style_name = \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Ty pesetting:-mi(\"the\"), Typesetting:-mo(\"⁢\", form = \+ \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsiz e = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits \+ = \"false\", accent = \"false\", font_style_name = \"Text\", size = \" 12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Types etting:-mi(\"same\"), Typesetting:-mo(\"⁢\", form = \"i nfix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rs pace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \+ \"false\", accent = \"false\", font_style_name = \"Text\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mi(\"window\"), Typesetting:-mo(\".\", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", \+ minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accen t = \"false\", font_style_name = \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mo(\"&Invi sibleTimes;\", form = \"infix\", fence = \"false\", separator = \"fals e\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetr ic = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"f alse\", movablelimits = \"false\", accent = \"false\", font_style_name = \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \"[ 255,255,255]\"), Typesetting:-mi(\"Remember\"), Typesetting:-mo(\"&Inv isibleTimes;\", form = \"infix\", fence = \"false\", separator = \"fal se\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmet ric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \" false\", movablelimits = \"false\", accent = \"false\", font_style_nam e = \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \" [255,255,255]\"), Typesetting:-mi(\"the\"), Typesetting:-mo(\"&Invisib leTimes;\", form = \"infix\", fence = \"false\", separator = \"false\" , lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric \+ = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255 ,255,255]\"), Typesetting:-mi(\"maple\"), Typesetting:-mo(\"&Invisible Times;\", form = \"infix\", fence = \"false\", separator = \"false\", \+ lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255 ,255,255]\"), Typesetting:-mi(\"formate\"), Typesetting:-mo(\"&Invisib leTimes;\", form = \"infix\", fence = \"false\", separator = \"false\" , lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric \+ = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255 ,255,255]\"), Typesetting:-mi(\"for\"), Typesetting:-mo(\"&InvisibleTi mes;\", form = \"infix\", fence = \"false\", separator = \"false\", ls pace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \" false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"Te xt\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255 ,255]\"), Typesetting:-mi(\"exponential\"), Typesetting:-mo(\"&Invisib leTimes;\", form = \"infix\", fence = \"false\", separator = \"false\" , lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric \+ = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"Text\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255 ,255,255]\"), Typesetting:-mi(\"functions\"), Typesetting:-mo(\".\", f orm = \"infix\", fence = \"false\", separator = \"false\", lspace = \" 0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", \+ maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", font_style_name = \"Text\", siz e = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mo(\"⁢\", form = \"infix\", fence = \"fal se\", separator = \"false\", lspace = \"0em\", rspace = \"0em\", stret chy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsiz e = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", font_style_name = \"Text\", size = \"12\", foreground = \"[0, 0,0]\", background = \"[255,255,255]\"));" "-I%mrowG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6U-I#miGF$6#Q!F'-F#6%-F,6#Q\"fF'-I#moGF$63 Q0⁡F'/%%formGQ&infixF'/%&fenceGQ&falseF'/%*separatorGF=/ %'lspaceGQ$0emF'/%'rspaceGFB/%)stretchyGF=/%*symmetricGF=/%(maxsizeGQ) infinityF'/%(minsizeGQ\"1F'/%(largeopGF=/%.movablelimitsGF=/%'accentGF 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" }{TEXT 317 31 "Plot the graph of the surface " }{XPPEDIT 2 0 "Typesetting:-mrow( Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"f\"), Types etting:-mo(\"⁡\", form = \"infix\", fence = \"false\", s eparator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \+ \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \" 1\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", font_style_name = \"2D Math\", size = \"12\", foreground = \"[0,0, 0]\", background = \"[255,255,255]\"), Typesetting:-mfenced(Typesettin g:-mrow(Typesetting:-mi(\"x\"), Typesetting:-mo(\",\", form = \"infix \", fence = \"false\", separator = \"true\", lspace = \"0em\", rspace \+ = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movable limits = \"false\", accent = \"false\", font_style_name = \"2D Math\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255] \"), Typesetting:-mi(\"y\"), Typesetting:-mo(\",\", form = \"infix\", \+ fence = \"false\", separator = \"true\", lspace = \"0em\", rspace = \" verythickmathspace\", stretchy = \"false\", symmetric = \"false\", max size = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", font_style_name = \"2D Math\", siz e = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mi(\"z\"))), Typesetting:-mi(\"\")), Typesetting:-mo(\"= \", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"thickmathspace\", rspace = \"thickmathspace\", stretchy = \"false \", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", la rgeop = \"false\", movablelimits = \"false\", accent = \"false\", font _style_name = \"2D Math\", size = \"12\", foreground = \"[0,0,0]\", ba ckground = \"[255,255,255]\"), Typesetting:-msup(Typesetting:-mi(\"x\" ), Typesetting:-mn(\"2\"), superscriptshift = \"0\"), Typesetting:-mo( \"+\", form = \"infix\", fence = \"false\", separator = \"false\" , lspace = \"mediummathspace\", rspace = \"mediummathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fal se\", font_style_name = \"2D Math\", size = \"12\", foreground = \"[0, 0,0]\", background = \"[255,255,255]\"), Typesetting:-mn(\"4\"), Types etting:-msup(Typesetting:-mi(\"y\"), Typesetting:-mn(\"2\"), superscri ptshift = \"0\"), Typesetting:-mo(\"+\", form = \"infix\", fence \+ = \"false\", separator = \"false\", lspace = \"mediummathspace\", rspa ce = \"mediummathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movable limits = \"false\", accent = \"false\", font_style_name = \"2D Math\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255] \"), Typesetting:-msup(Typesetting:-mi(\"z\"), Typesetting:-mn(\"2\"), superscriptshift = \"0\"), Typesetting:-mo(\"−\", form = \"infi x\", fence = \"false\", separator = \"true\", lspace = \"0em\", rspace = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false\" , maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movabl elimits = \"false\", accent = \"false\", font_style_name = \"2D Math\" , size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255 ]\"), Typesetting:-mn(\"18\"));" "-I%mrowG6#/I+modulenameG6\"I,Typeset tingGI(_syslibGF'6--I#miGF$6#Q!F'-F#6&-F,6#Q\"fF'-I#moGF$63Q0&ApplyFun ction;F'/%%formGQ&infixF'/%&fenceGQ&falseF'/%*separatorGF=/%'lspaceGQ$ 0emF'/%'rspaceGFB/%)stretchyGF=/%*symmetricGF=/%(maxsizeGQ)infinityF'/ %(minsizeGQ\"1F'/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%0font_st yle_nameGQ(2D~MathF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backgrou ndGQ.[255,255,255]F'-I(mfencedGF$6#-F#6'-F,6#Q\"xF'-F563Q\",F'F8F;/F?Q %trueF'F@/FDQ3verythickmathspaceF'FEFGFIFLFOFQFSFUFXFenFhn-F,6#Q\"yF'F 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\"false\", font_style_name = \"2D Math \", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,2 55]\"), Typesetting:-mo(\"⁢\", form = \"infix\", fence \+ = \"false\", separator = \"false\", lspace = \"0em\", rspace = \"0em\" , stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", acce nt = \"false\", font_style_name = \"2D Math\", size = \"12\", foregrou nd = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mn(\"1 \"))), Typesetting:-mo(\".\", form = \"infix\", fence = \"false\", sep arator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"f alse\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ font_style_name = \"2D Math\", size = \"12\", foreground = \"[0,0,0]\" , background = \"[255,255,255]\")), Typesetting:-mi(\"\"));" "-I%mrowG 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