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{TEXT 277 39 "Computer Lab 3: \+ Multivariable Functions" }{TEXT 277 0 "" }}{PARA 215 "" 0 "" {TEXT 283 8 "Dr Firoz" }}{PARA 245 "" 0 "" {TEXT 226 144 "In this lab you wi ll study functions of two variables. You will find their cross sectio ns with planes and their contourplots using the command " }{TEXT 224 7 "plot3d " }{TEXT 288 4 "and " }{TEXT 224 11 "contourplot" }{TEXT 226 2 ". " }{TEXT 226 0 "" }}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{SECT 1 {PARA 229 "" 0 "" {TEXT 230 8 "Examples" }{TEXT 230 0 "" }}{SECT 0 {PARA 263 "" 0 "" {TEXT 247 41 "I. Graphs of Functions with two variab les" }{TEXT 247 0 "" }}{PARA 245 "" 0 "" {TEXT 226 5 "The " }{TEXT 224 6 "plot3d" }{TEXT 226 99 " command plots the graph of a function \+ of two variables. For instance, the graph of the function " } {XPPEDIT 18 0 "f(x, y) = x^2+y;" "6#/-%\"fG6$%\"xG%\"yG,&*$)F'\"\"#\" \"\"F-F(F-" }{TEXT 226 26 " is plotted as follows:" }{TEXT 226 0 "" }}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 21 "restart: with(plots): " }{MPLTEXT 1 256 0 "" }}}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 47 "setoptions3d(style=patchcontour, shading=zhue):" }{MPLTEXT 1 256 0 "" }}}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 43 "plot3d(x^2+y, x= -2..2, y=-2..2,axes=frame);" }{MPLTEXT 1 256 0 "" }}}{PARA 245 "" 0 "" {TEXT 226 72 "Notice that the first argument of the plot3d command is the formula for " }{XPPEDIT 18 0 "f(x, y);" "6#-%\"fG6$%\"xG%\"yG" } {TEXT 226 75 " , not an equation. In the example above, you should se e the level curves " }{XPPEDIT 18 0 "x^2+y = k;" "6#/,&*$)%\"xG\"\"#\" \"\"F)%\"yGF)%\"kG" }{TEXT 226 41 " corresponding to different choic es of " }{XPPEDIT 18 0 "z = k;" "6#/%\"zG%\"kG" }{TEXT 226 5 " . " } {TEXT 226 0 "" }}{PARA 245 "" 0 "" {TEXT 226 69 "Another way to plot t wo variable functions is to define the function " }{TEXT 248 1 "f" } {TEXT 226 23 " first and then plot " }{TEXT 248 1 "f" }{TEXT 226 2 " :" }{TEXT 226 0 "" }}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 23 "f :=(x,y) -> x^2 + y^2;" }{MPLTEXT 1 256 0 "" }}}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 65 "plot3d(f(x,y ), x=-2..2, y=-2..2,axes=normal,scaling=constrained);" }{MPLTEXT 1 256 0 "" }}}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{PARA 274 "" 0 "" {TEXT 228 56 "Or a third way is to define the expression g and plot g:" } {TEXT 228 0 "" }}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{EXCHG {PARA 262 " > " 0 "" {MPLTEXT 1 256 14 "g:= x^2 - y^2;" }{MPLTEXT 1 256 0 "" }}} {EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 39 "plot3d(g, x=-2..2, y=-2. .2,axes=boxed);" }{MPLTEXT 1 256 0 "" }}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{PARA 215 "> " 0 "" {TEXT 283 0 "" }}}{EXCHG {PARA 274 "" 0 "" {TEXT 228 66 "To plot cross sections \+ in the xz or yz planes, input: plot ( f(1," }{TEXT 245 1 "y" }{TEXT 228 4 ") , " }{TEXT 245 1 "y" }{TEXT 228 8 "=-3..3, " }{TEXT 245 1 "z" }{TEXT 228 48 "=-3..3); also try other cross-sections, such as " } {TEXT 245 1 "f" }{TEXT 228 1 "(" }{TEXT 245 1 "x" }{TEXT 228 5 ",2). " }{TEXT 228 0 "" }}{PARA 274 "" 0 "" {TEXT 228 135 "Be sure to use lim its which will enable you to see the graph. Notice that you can do thi s if you defined a function, not an expression!" }{TEXT 228 0 "" }} {PARA 215 "" 0 "" {TEXT 283 0 "" }}}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 37 " plot ( f(1,y) , y=-3..3, z=-3..3); " }{MPLTEXT 1 256 0 "" }}}{PARA 274 "" 0 "" {TEXT 228 0 "" }{TEXT 228 2 "\n" }{TEXT 228 58 "To plot horizontal planes intersecting the surface, input:" } {TEXT 228 0 "" }{TEXT 228 2 "\n" }}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 74 "plot3d ( \{ f(x,y), 2, 4, 6\}, x=-3..3 , y=-3..3 , axes=fra med,view=0..9);" }{MPLTEXT 1 256 0 "" }{MPLTEXT 1 256 2 "\n" }}} {EXCHG {PARA 274 "" 0 "" {TEXT 228 81 "To plot vertical planes interse cting the surface, input the following statements." }{TEXT 228 0 "" } {TEXT 228 2 "\n" }{TEXT 228 1 " " }{TEXT 228 0 "" }}}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 56 "a:=plot3d(f(x,y),x=-3..3,y=-3..3,view=0.. 9,axes=framed):" }{MPLTEXT 1 256 0 "" }{MPLTEXT 1 256 2 "\n" }{MPLTEXT 1 256 46 "b:=implicitplot3d(y=1,x=-3..3,y=-3..3,z=0..9):" }{MPLTEXT 1 256 0 "" }{MPLTEXT 1 256 2 "\n" }{MPLTEXT 1 256 19 "display3d(\{a,b \});" }{MPLTEXT 1 256 0 "" }}}{PARA 215 "" 0 "" {TEXT 283 0 "" }} {PARA 245 "" 0 "" {TEXT 226 49 "Note: the view= 0..9 command lets yo u set the " }{TEXT 248 1 "z" }{TEXT 226 22 " values on the graph." } {TEXT 226 0 "" }}}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{SECT 0 {PARA 209 "" 0 "" {TEXT 266 17 "II. Contourplots" }{TEXT 266 0 "" }}{PARA 274 "" 0 "" {TEXT 228 64 "First plot a 3 dimensional surface to see what \+ it looks like. " }}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 49 "plot3d(g,x=-3..3,y=-3..3,view=-3..9,a xes=normal);" }{MPLTEXT 1 256 0 "" }}{PARA 215 "> " 0 "" {TEXT 283 0 " " }}}{PARA 274 "" 0 "" {TEXT 228 47 "One way to get a two-dimensional \+ contour plot: " }{TEXT 228 0 "" }{TEXT 228 2 "\n" }{TEXT 228 24 " \+ " }{TEXT 228 0 "" }}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 43 "contourplot(g,x=-4..4,y=-4..4,contours=15);" } {MPLTEXT 1 256 0 "" }}}{EXCHG {PARA 215 "> " 0 "" {TEXT 283 0 "" }}} {EXCHG {PARA 274 "" 0 "" {TEXT 228 79 "Note that we set the number of \+ contours to 15. The default number is 10. " }{TEXT 228 9 " \+ " }{TEXT 228 0 "" }}{PARA 274 "" 0 "" {TEXT 228 74 "Here we don't \+ know the constant values corresponding to the level curves." }{TEXT 228 0 "" }}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{PARA 274 "" 0 "" {TEXT 228 67 "In the following example we specify the values of the level cu rves." }{TEXT 228 0 "" }}}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 73 "contourplot(g,x=-4..4,y=-4..4,contours=[-6,-4,-2,0,2,4,6],filled = true);" }{MPLTEXT 1 256 0 "" }}}{EXCHG {PARA 215 "> " 0 "" {TEXT 283 0 "" }}}{PARA 274 "" 0 "" {TEXT 228 138 "This plots the contours -6, - 4, -2, 0, 2, 4, 6. Note that we used the option \"filled = true\". Thi s gives a color version of the contour." }{TEXT 228 0 "" }}{PARA 274 " " 0 "" {TEXT 228 97 "The color goes from yellow to red with yellow cor responding to larger values of the level curves." }{TEXT 228 0 "" }} {EXCHG {PARA 215 "" 0 "" {TEXT 283 0 "" }}}{EXCHG {PARA 274 "" 0 "" {TEXT 228 81 "We can also generate a 3D view of the contours raised to their appropriate levels" }{TEXT 228 0 "" }}}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 85 "contourplot3d (g,x=-4..4,y=-4..4,contours=[-6,-4,-2,0,2,4,6],filled=true,axes=framed );" }{MPLTEXT 1 256 0 "" }}}{EXCHG {PARA 215 "> " 0 "" {TEXT 283 0 "" }}}{EXCHG {PARA 274 "" 0 "" {TEXT 228 201 "You can also plot together \+ the surface with its contour plot. Note that the transformation using \+ the transform command is necessary to view a two-dimensional contour p lot in 3D (here in the plane z=0)." }{TEXT 228 1 " " }{TEXT 228 0 "" } }}{EXCHG {PARA 215 "> " 0 "" {TEXT 283 0 "" }}}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 17 "with(plottools): " }{MPLTEXT 1 256 0 "" }}} {EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 72 "p := plot3d(g,x=-4..4,y= -4..4,style=patchcontour,contours=8,view=-4..4):" }{MPLTEXT 1 256 0 "" }{MPLTEXT 1 256 2 "\n" }{MPLTEXT 1 256 36 "q := contourplot(g,x=-4..4 ,y=-4..4):" }{MPLTEXT 1 256 0 "" }{MPLTEXT 1 256 2 "\n" }{MPLTEXT 1 256 33 "f := transform((x,y) -> [x,y,0]):" }{MPLTEXT 1 256 0 "" } {MPLTEXT 1 256 2 "\n" }{MPLTEXT 1 256 32 "display(\{p,f(q)\},axes=fram ed);" }{MPLTEXT 1 256 0 "" }}}{EXCHG {PARA 215 "> " 0 "" {TEXT 283 0 " " }}{PARA 215 "> " 0 "" {TEXT 283 0 "" }}}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 67 "p := plot3d(g,x=-4..4,y=-4..4,style=contour,contour s=8,view=-4..4):" }{MPLTEXT 1 256 0 "" }{MPLTEXT 1 256 2 "\n" } {MPLTEXT 1 256 49 "q := contourplot(g,x=-4..4,y=-4..4,filled= true):" }{MPLTEXT 1 256 0 "" }{MPLTEXT 1 256 2 "\n" }{MPLTEXT 1 256 33 "f := t ransform((x,y) -> [x,y,0]):" }{MPLTEXT 1 256 0 "" }{MPLTEXT 1 256 2 " \n" }{MPLTEXT 1 256 32 "display(\{p,f(q)\},axes=framed);" }{MPLTEXT 1 256 0 "" }}}{EXCHG {PARA 215 "> " 0 "" {TEXT 283 0 "" }}}{EXCHG {PARA 274 "" 0 "" {TEXT 228 0 "" }{TEXT 228 2 "\n" }{TEXT 228 148 "To plot a ll the levels and the surface together use the following statement: Th is will show the surface being sliced. Rotate the plot and observe.)" }{TEXT 228 0 "" }}}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 256 61 "plot 3d(\{g,2,4,6,8\},x=-4..4,y=-4..4,view=0..10,axes=framed);" }{MPLTEXT 1 256 0 "" }}}{EXCHG {PARA 215 "> " 0 "" {TEXT 283 0 "" }}}{PARA 215 " " 0 "" {TEXT 283 0 "" }}}{PARA 215 "" 0 "" {TEXT 283 0 "" }}}{PARA 240 "" 0 "" {TEXT 275 0 "" }}{SECT 1 {PARA 218 "" 0 "" {TEXT 250 16 "One \+ More Example" }{TEXT 250 0 "" }}{PARA 267 "" 0 "" {TEXT 221 116 "Like \+ for functions of one variable, when plotting surfaces it is important \+ to choose the appropriate domain for the " }{TEXT 221 67 "variables so that all the main features of the function are shown. " }{TEXT 221 0 "" }}{PARA 267 "" 0 "" {TEXT 221 47 "Consider, for instance, the follo wing function:" }{TEXT 221 0 "" }}{PARA 267 "" 0 "" {TEXT 221 0 "" }} {EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 0 24 "f:=3*x-x^4-4*y^2-10*x*y;" }{MPLTEXT 1 0 0 "" }}}{PARA 240 "" 0 "" {TEXT 275 0 "" }}{EXCHG {PARA 214 "" 0 "" {TEXT 275 62 "We first try to plot the function choosing a domain with both " }{TEXT 248 1 "x" }{TEXT 275 5 " and " }{TEXT 248 1 "y" }{TEXT 275 20 " in between -5 and 5" }{TEXT 236 0 "" }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 0 38 "plot3d(f,x=-5..5,y=-5..5,axes=fra med);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 214 "" 0 "" {TEXT 275 122 "The graph we obtain do es not seem to capture all the features of the function. Because we n otice the large values in the " }{TEXT 248 1 "z" }{TEXT 275 24 "-axes \+ we try the option " }{TEXT 284 5 "view " }{TEXT 275 34 "in order to li mit the range for z." }{TEXT 236 0 "" }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 0 51 "plot3d(f,x=-5..5,y=-5..5,view=-20..20,axes=normal);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 240 "" 0 "" {TEXT 275 110 "It seems the function has an absolute maximum of about 15 and a local maximum \+ of about 4 but no local minimum." }{TEXT 275 0 "" }}{PARA 240 "" 0 "" {TEXT 275 177 "We can confirm this by plotting the contourplot of the \+ function. Because the function is so steep we need to use a large numb er of level curves to capture all the main features." }{TEXT 275 0 "" }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 0 43 "contourplot(f,x=-3..3,y =-4..3,contours=60);" }{MPLTEXT 1 0 0 "" }}}{PARA 240 "" 0 "" {TEXT 275 149 "Notice how steep the function is (level curves very close to \+ each other) and how the contour shows maximums at approximately (-1.6, 2) and (1.8,-2.3)." }{TEXT 275 0 "" }}{PARA 240 "" 0 "" {TEXT 275 120 "Close to the origin we have the typical \"saddle point\" behavior as \+ it is confirmed by zooming in with the contourplot." }{TEXT 275 0 "" } }{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 296 43 "contourplot(f,x=-1..1,y =-1..1,contours=60);" }{MPLTEXT 1 296 0 "" }}}}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{PARA 240 "" 0 "" {TEXT 275 0 "" }}{SECT 1 {PARA 229 "" 0 "" {TEXT 230 38 "Exercises to practice (do not tern in)" }{TEXT 230 0 "" }}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{SECT 0 {PARA 206 "" 0 "" {TEXT 297 9 "Problem 1" }{TEXT 297 0 "" }}{PARA 214 "" 0 "" {TEXT 243 22 "Consider the function " }{XPPEDIT 2 0 "f(x, y) = x^3-x+y^3+y^2;" " 6#/-I\"fG6\"6$I\"xGF&I\"yGF&,**$)F(\"\"$\"\"\"F.F(!\"\"*$)F)F-F.F.*$)F )\"\"#F.F." }{TEXT 236 0 "" }}{PARA 247 "" 0 "" {TEXT 243 211 "Graph t he function using various domains and view points. Make sure you iden tify all the main features of the graph. Plot a contour diagram and ob serve the relation between the contours and the surface. " } {TEXT 243 0 "" }}{PARA 247 "" 0 "" {TEXT 243 0 "" }}{EXCHG {PARA 221 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{SECT 0 {PARA 206 "" 0 "" {TEXT 297 9 "Problem 2 " }{TEXT 297 0 "" }}{PARA 245 "" 0 "" {TEXT 226 22 "Consider the funct ion " }{XPPEDIT 2 0 "f(x, y) = 30*x-x^6-y^6-y^2+30*x*y;" "6#/-I\"fG6\" 6$I\"xGF&I\"yGF&,,*&\"#I\"\"\"F(F-F-*$)F(\"\"'F-!\"\"*$)F)F0F-F1*$)F) \"\"#F-F1*(F,F-F(F-F)F-F-" }{TEXT 226 0 "" }}{PARA 215 "" 0 "" {TEXT 283 110 "Graph the function using various domains and viewpoints. Make sure your plot gives a good view of all \"peaks " }{TEXT 283 115 "and valleys\". Would you say the function has a maximum value? Can you id entify any point(s) on the graph that you " }{TEXT 283 109 "might cons ider to be \"local maximum points\"? What about \"local minimum points \"? If you find any of those " }{TEXT 283 59 "points, give the approxi mate coordinates of their position." }{TEXT 283 0 "" }}{PARA 215 "" 0 "" {TEXT 283 60 "Use contourplot(s) of the function to confirm your fi ndings." }{TEXT 283 0 "" }}{PARA 240 "" 0 "" {TEXT 275 0 "" }}{PARA 240 "" 0 "" {TEXT 275 0 "" }}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{SECT 0 {PARA 206 "" 0 "" {TEXT 297 9 "Problem 3" }{TEXT 297 0 "" }}{PARA 245 "" 0 "" {TEXT 226 22 "Consider the function " }{XPPEDIT 2 0 "f(x, y) = xy/(x^2+y^2);" "6#/-I\"fG6\"6$I\"xGF&I\"yGF&*&I#xyGF&\"\"\",&*$)F(\" \"#F,F,*$)F)F0F,F,!\"\"" }{TEXT 226 0 "" }{TEXT 226 2 "\n" }{TEXT 226 98 "Graph the function using various domains and viewpoints. Also plot a contour plot of the function." }}{PARA 245 "" 0 "" {TEXT 226 49 "Co mment on the limiting behavior of the function." }{TEXT 226 0 "" }} {PARA 215 "" 0 "" {TEXT 283 0 "" }}{PARA 245 "" 0 "" {TEXT 226 28 "(a) What happens when both " }{TEXT 248 1 "x" }{TEXT 226 5 " and " } {TEXT 248 1 "y" }{TEXT 226 15 " become large? " }{TEXT 226 0 "" }} {PARA 215 "" 0 "" {TEXT 283 0 "" }}{PARA 245 "" 0 "" {TEXT 226 22 "(b) What happens as (" }{TEXT 248 1 "x" }{TEXT 226 1 "," }{TEXT 248 1 "y " }{TEXT 226 24 ") approaches the origin?" }{TEXT 226 0 "" }}{PARA 240 "" 0 "" {TEXT 275 0 "" }}{PARA 240 "" 0 "" {TEXT 275 0 "" }}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{PARA 215 "" 0 "" {TEXT 283 0 "" }}}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{PARA 215 "" 0 "" {TEXT 283 0 "" }}{PARA 220 "" 0 "" {TEXT 295 0 "" }}{PARA 239 "" 0 "" {TEXT 293 0 "" }}{PARA 230 "" 0 "" {TEXT 291 0 "" }}{PARA 259 "" 0 "" {TEXT 279 0 "" }}{PARA 216 "" 0 "" {TEXT 298 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }