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{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 209 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 210 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 211 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 212 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 211 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 212 "Ti mes" 1 16 102 102 255 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 213 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 213 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 214 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 215 "Times" 1 16 102 102 255 1 1 2 2 2 2 2 0 0 0 1 } {PSTYLE "" -1 214 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 216 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 217 "Times" 1 16 102 102 255 1 1 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 215 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {CSTYLE "" -1 218 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 219 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 220 "T imes" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 221 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }} {SECT 0 {PARA 208 "" 0 "" {TEXT 200 0 "" }{TEXT 200 0 "" }{TEXT 200 9 "Maple Lab" }}{PARA 209 "" 0 "" {TEXT 200 6 "Week 5" }}{PARA 210 "" 0 "" {TEXT 200 33 "Gradient; finding critical points" }}{PARA 211 "" 0 " " {TEXT 200 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 200 11 "Start with:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{PARA 0 " " 0 "" {TEXT 200 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 222 18 "Plotting gradients" }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 200 260 "Using the \"gradplot\" command result in a two dimensio nal vector field with the gradient vector, evaluated at (x, y), locate d at the point (x, y). The syntax is very similar to the rest of the \+ plotting functions we have discussed thus far, and is as follows:" }} {PARA 0 "" 0 "" {TEXT 200 0 "" }}{PARA 212 "" 0 "" {TEXT 211 10 "gradp lot( " }{TEXT 212 10 "expression" }{TEXT 213 31 " , x = a..b, y = c..d , options)" }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}{PARA 0 "" 0 "" {TEXT 200 31 "For example, take the function " }{XPPEDIT 2 0 "Typesetting:-m row(Typesetting:-mi(\"z\", italic = \"true\", mathvariant = \"italic\" ), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", for m = \"infix\", lspace = \"thickmathspace\", rspace = \"thickmathspace \", minsize = \"1\", maxsize = \"infinity\"), Typesetting:-mn(\"2\", m athvariant = \"normal\"), Typesetting:-msup(Typesetting:-mi(\"x\", ita lic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetti ng:-mn(\"2\", mathvariant = \"normal\")), superscriptshift = \"0\"), T ypesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", sep arator = \"false\", stretchy = \"false\", symmetric = \"false\", large op = \"false\", movablelimits = \"false\", accent = \"false\", form = \+ \"infix\", lspace = \"mediummathspace\", rspace = \"mediummathspace\", minsize = \"1\", maxsize = \"infinity\"), Typesetting:-mn(\"3\", math variant = \"normal\"), Typesetting:-msup(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting: -mn(\"2\", mathvariant = \"normal\")), superscriptshift = \"0\"), Type setting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sysl ibGF'6*-I#miGF$6%Q\"zF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#m oGF$60Q\"=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyG F=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ &infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(ma xsizeGQ)infinityF'-I#mnGF$6$Q\"2F'F9-I%msupGF$6%-F,6%Q\"xF'F/F2-F#6#FX /%1superscriptshiftGQ\"0F'-F660Q\"+F'F9F;F>F@FBFDFFFHFJ/FNQ0mediummath spaceF'/FQFeoFRFU-FY6$Q\"3F'F9-Fgn6%-F,6%Q\"yF'F/F2F\\oF^o-F,6#Q!F'" } {TEXT 200 65 " and look at the gradient field that is produced by \"gr adplot\"." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "gradplot(2*x^2 +3*y^2,x=-2..2,y=-2..2,arrows=thick);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 107 "The options available with \"gradplot\" can be used to make the plot a bit nicer. Some of the options are:" }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}{PARA 213 "" 0 "" {TEXT 214 9 "arrows = " }{TEXT 215 25 "[line, slim, thin, \+ thick]" }}{PARA 214 "" 0 "" {TEXT 216 12 "thickness = " }{TEXT 217 1 " n" }}{PARA 215 "" 0 "" {TEXT 200 0 "" }}{PARA 0 "" 0 "" {TEXT 200 6 "w here " }{TEXT 218 2 "n " }{TEXT 200 92 "is a number greater than zero. The default thickness is 1, while the default arrow type is " }{TEXT 219 4 "thin" }{TEXT 200 3 ". " }{TEXT 200 0 "" }}}{PARA 0 "" 0 "" {TEXT 200 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 223 53 "Displaying the \+ contourplot and gradient plot together" }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 200 43 "As an example, we consider t hree functions " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"z \", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"= \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", form = \"infix\", lspace \+ = \"thickmathspace\", rspace = \"thickmathspace\", minsize = \"1\", ma xsize = \"infinity\"), Typesetting:-mn(\"2\", mathvariant = \"normal\" ), Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", mathvar iant = \"italic\"), Typesetting:-mrow(Typesetting:-mn(\"2\", mathvaria nt = \"normal\")), superscriptshift = \"0\"), Typesetting:-mo(\"+\", m athvariant = \"normal\", fence = \"false\", separator = \"false\", str etchy = \"false\", symmetric = \"false\", largeop = \"false\", movable limits = \"false\", accent = \"false\", form = \"infix\", lspace = \"m ediummathspace\", rspace = \"mediummathspace\", minsize = \"1\", maxsi ze = \"infinity\"), Typesetting:-mn(\"3\", mathvariant = \"normal\"), \+ Typesetting:-msup(Typesetting:-mi(\"y\", italic = \"true\", mathvarian t = \"italic\"), Typesetting:-mrow(Typesetting:-mn(\"2\", mathvariant \+ = \"normal\")), superscriptshift = \"0\"), Typesetting:-mo(\",\", math variant = \"normal\", fence = \"false\", separator = \"true\", stretch y = \"false\", symmetric = \"false\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", form = \"infix\", lspace = \"0em\" , rspace = \"verythickmathspace\", minsize = \"1\", maxsize = \"infini ty\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \"fals e\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , form = \"\", lspace = \"0em\", rspace = \"0em\", minsize = \"1\", ma xsize = \"infinity\"), Typesetting:-mi(\"z\", italic = \"true\", mathv ariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symme tric = \"false\", largeop = \"false\", movablelimits = \"false\", acce nt = \"false\", form = \"infix\", lspace = \"thickmathspace\", rspace \+ = \"thickmathspace\", minsize = \"1\", maxsize = \"infinity\"), Typese tting:-mn(\"1\", mathvariant = \"normal\"), Typesetting:-mo(\"&uminus0 ;\", mathvariant = \"normal\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", form = \"infix\", lspac e = \"mediummathspace\", rspace = \"mediummathspace\", minsize = \"1\" , maxsize = \"infinity\"), Typesetting:-mn(\"2\", mathvariant = \"norm al\"), Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", mat hvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mn(\"2\", mathv ariant = \"normal\")), superscriptshift = \"0\"), Typesetting:-mo(\"&u minus0;\", mathvariant = \"normal\", fence = \"false\", separator = \" false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", form = \"infix\", \+ lspace = \"mediummathspace\", rspace = \"mediummathspace\", minsize = \+ \"1\", maxsize = \"infinity\"), Typesetting:-mn(\"3\", mathvariant = \+ \"normal\"), Typesetting:-msup(Typesetting:-mi(\"y\", italic = \"true \", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mn(\"2\" , mathvariant = \"normal\")), superscriptshift = \"0\"), Typesetting:- mo(\",\", mathvariant = \"normal\", fence = \"false\", separator = \"t rue\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", form = \"infix\", l space = \"0em\", rspace = \"verythickmathspace\", minsize = \"1\", max size = \"infinity\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symme tric = \"false\", largeop = \"false\", movablelimits = \"false\", acce nt = \"false\", form = \"\", lspace = \"0em\", rspace = \"0em\", minsi ze = \"1\", maxsize = \"infinity\"));" "-I%mrowG6#/I+modulenameG6\"I,T ypesettingGI(_syslibGF'66-I#miGF$6%Q\"zF'/%'italicGQ%trueF'/%,mathvari antGQ'italicF'-I#moGF$60Q\"=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separa torGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%' accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(m insizeGQ\"1F'/%(maxsizeGQ)infinityF'-I#mnGF$6$Q\"2F'F9-I%msupGF$6%-F,6 %Q\"xF'F/F2-F#6#FX/%1superscriptshiftGQ\"0F'-F660Q\"+F'F9F;F>F@FBFDFFF HFJ/FNQ0mediummathspaceF'/FQFeoFRFU-FY6$Q\"3F'F9-Fgn6%-F,6%Q\"yF'F/F2F \\oF^o-F660Q\",F'F9F;/F?F1F@FBFDFFFHFJ/FNQ$0emF'/FQQ3verythickmathspac eF'FRFU-F660Q\"~F'F9F;F>F@FBFDFFFH/FKQ!F'Fcp/FQFdpFRFUF+F5-FY6$FTF9-F6 60Q*&uminus0;F'F9F;F>F@FBFDFFFHFJFdoFfoFRFUFXFfnF_qFgoFjoF_pFgp" } {TEXT 200 4 "and " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\" z\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"= \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", form = \"infix\", lspace \+ = \"thickmathspace\", rspace = \"thickmathspace\", minsize = \"1\", ma xsize = \"infinity\"), Typesetting:-mn(\"2\", mathvariant = \"normal\" ), Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", mathvar iant = \"italic\"), Typesetting:-mrow(Typesetting:-mn(\"2\", mathvaria nt = \"normal\")), superscriptshift = \"0\"), Typesetting:-mo(\"&uminu s0;\", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", form = \"infix\", lspa ce = \"mediummathspace\", rspace = \"mediummathspace\", minsize = \"1 \", maxsize = \"infinity\"), Typesetting:-mn(\"3\", mathvariant = \"no rmal\"), Typesetting:-msup(Typesetting:-mi(\"y\", italic = \"true\", m athvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mn(\"2\", mat hvariant = \"normal\")), superscriptshift = \"0\"), Typesetting:-mo(\" .\", mathvariant = \"normal\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", form = \"infix\", lspac e = \"0em\", rspace = \"0em\", minsize = \"1\", maxsize = \"infinity\" ));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6*-I#miGF$6 %Q\"zF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q\"=F'/F3Q 'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF =/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspac eGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinity F'-I#mnGF$6$Q\"2F'F9-I%msupGF$6%-F,6%Q\"xF'F/F2-F#6#FX/%1superscriptsh iftGQ\"0F'-F660Q*&uminus0;F'F9F;F>F@FBFDFFFHFJ/FNQ0mediummathspaceF'/F QFeoFRFU-FY6$Q\"3F'F9-Fgn6%-F,6%Q\"yF'F/F2F\\oF^o-F660Q\".F'F9F;F>F@FB FDFFFHFJ/FNQ$0emF'/FQFcpFRFU" }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plot1:=contourplot(2*x^2+3*y ^2,x=-2..2,y=-2..2,color=blue,thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "plot2:=" }{MPLTEXT 1 0 51 "contourplot(2*x^2+3*y^2, x=-2..2,y=-2..2,color=red);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "display([plot1,plot2],insequence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 122 "If you click on the graph above and hit the play butto n in the menu bar, you'll see that the two contour plots are exactly" }}{PARA 0 "" 0 "" {TEXT 200 123 "the same for the first two functions; even though the point (0,0) is the point of local minimum for the fir st function, and" }}{PARA 0 "" 0 "" {TEXT 200 42 "the point of local m aximum for the second." }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}{PARA 0 "" 0 "" {TEXT 200 99 "Plotting the gradient helps to decide which functio n has local maximum and which has local minimum." }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "grad1:=" } {MPLTEXT 1 0 50 "gradplot(2*x^2+3*y^2,x=-2..2,y=-2..2,arrows=slim);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "grad2:=" }{MPLTEXT 1 0 52 " gradplot(1-2*x^2-3*y^2,x=-2..2,y=-2..2,arrows=slim);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "display(\{plot1,grad1\});" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "display(\{plot2,grad 2\});" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 130 "From the pictures, you see \+ that the gradient loves local maximum, hates local minimum, and (to co mplete the story) has a love-hate" }}{PARA 0 "" 0 "" {TEXT 200 32 "rel ationship with saddle points:" }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "plot3:=" }{MPLTEXT 1 0 65 "co ntourplot(2*x^2-3*y^2,x=-2..2,y=-2..2,color=green,thickness=2):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "grad3:=" }{MPLTEXT 1 0 50 "gr adplot(2*x^2-3*y^2,x=-2..2,y=-2..2,arrows=slim):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "display(\{plot3,grad3\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 222 17 "Solving equations" }}{EXCHG {PARA 0 "" 0 "" {TEXT 200 120 "This comes handy if you want to find critical points; or just solve an equation that might not have a \"nice\" solution." }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}{PARA 0 "" 0 "" {TEXT 200 12 "For example," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "solve(2-exp(x)=2*x,x);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 200 35 "finds the solution of the equat ion " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mn(\"2\", mathvari ant = \"normal\"), Typesetting:-mo(\"&uminus0;\", mathvariant = \"norm al\", fence = \"false\", separator = \"false\", stretchy = \"false\", \+ symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", form = \"infix\", lspace = \"mediummathspace\", r space = \"mediummathspace\", minsize = \"1\", maxsize = \"infinity\"), Typesetting:-msup(Typesetting:-mi(\"e\", italic = \"true\", mathvaria nt = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \" true\", mathvariant = \"italic\")), superscriptshift = \"0\"), Typeset ting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", form = \"inf ix\", lspace = \"thickmathspace\", rspace = \"thickmathspace\", minsiz e = \"1\", maxsize = \"infinity\"), Typesetting:-mn(\"2\", mathvariant = \"normal\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant \+ = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF '6(-I#mnGF$6$Q\"2F'/%,mathvariantGQ'normalF'-I#moGF$60Q*&uminus0;F'F// %&fenceGQ&falseF'/%*separatorGF8/%)stretchyGF8/%*symmetricGF8/%(largeo pGF8/%.movablelimitsGF8/%'accentGF8/%%formGQ&infixF'/%'lspaceGQ0medium mathspaceF'/%'rspaceGFJ/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I%msup GF$6%-I#miGF$6%Q\"eF'/%'italicGQ%trueF'/F0Q'italicF'-F#6#-FW6%Q\"xF'FZ Fgn/%1superscriptshiftGQ\"0F'-F360Q\"=F'F/F6F9F;F=F?FAFCFE/FIQ/thickma thspaceF'/FLFeoFMFPF+F[o" }{TEXT 200 2 ". " }{TEXT 220 34 "Remember th at Maple does not know " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting: -msup(Typesetting:-mi(\"e\", italic = \"true\", mathvariant = \"italic \"), Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathv ariant = \"italic\")), superscriptshift = \"0\"));" "-I%mrowG6#/I+modu lenameG6\"I,TypesettingGI(_syslibGF'6#-I%msupGF$6%-I#miGF$6%Q\"eF'/%'i talicGQ%trueF'/%,mathvariantGQ'italicF'-F#6#-F/6%Q\"xF'F2F5/%1superscr iptshiftGQ\"0F'" }{TEXT 200 2 ", " }{TEXT 221 25 "you need to enter ex p(x)." }}{PARA 0 "" 0 "" {TEXT 200 33 "If you want a numeric answer, d o:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 43 "Maple can also solve systems of equation s. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }