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}{CSTYLE "Dictionary Hyperlink " -1 45 "Times" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help Emp hasized" -1 22 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "LaTeX" -1 32 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help \+ Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Prompt " -1 1 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Under lined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help U nderlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 } {CSTYLE "_cstyle3" -1 214 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle256" -1 215 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }} {SECT 0 {PARA 18 "" 0 "" {TEXT 216 30 "A first look at Tangent Planes" }}{PARA 19 "" 0 "" {TEXT 217 39 "\251 Mike May, S.J.2006 - maymk@slu. edu" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 200 323 "A first naive way to look at tangent planes is to take \+ the tangent lines to the two cross section curves at a point and turn \+ those lines into a plane. (The naive approach works if the function i s differentiable, but that is a detail we will look at later.) This w orksheet is intended as a demonstration of that technique." }{TEXT 200 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 218 63 "Constructing a plane \+ defined by tangent lines to cross sections" }{TEXT 218 0 "" }}{PARA 211 "" 0 "" {TEXT 200 86 "We start by defining a function that we will work with. Let z=f(x,y) =x^2-3*x*y+y^2." }}{PARA 211 "" 0 "" {TEXT 200 47 "It is first of all useful to look at the graph." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f:= (x,y) -> x^2-3*x*y+y^2;" } {MPLTEXT 1 0 15 "plot3d(f(x,y), " }{MPLTEXT 1 0 36 "x=-10..10, y=-10.. 10, view=-100..100" }{MPLTEXT 1 0 2 ");" }{MPLTEXT 1 0 0 "" }}}{PARA 211 "" 0 "" {TEXT 200 218 "Notice that the graph goes up in one direct ion and down in another. We are at a saddle, so tangent planes will t end to cut the surface, much as the tangent line to a cubic curve cuts the curve at the inflection point." }}{PARA 211 "" 0 "" {TEXT 200 55 "To find the tangent plane we considered cross sections." }}{PARA 211 "" 0 "" {TEXT 200 95 "We will start with the point (-1,3). We start by finding the z-value at the point on the graph." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "x0 := -1; y0 := 3;" }{MPLTEXT 1 0 16 "z0 := f(x 0, y0);" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 200 92 "Next comp ute the functions in one variable that we obtain by holding either x o r y constant." }{TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "fx := f(x,y0); fy := f(x0,y);" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 200 121 "Now compute the derivatives of the functions in o ne variable and substitute in the point to find an x-slope and y-slope ." }{TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Dfx := \+ diff(fx,x); mx := subs(x=x0,Dfx);\n" }{MPLTEXT 1 0 40 "Dfy := diff(fy, y); my := subs(y=y0,Dfy);" }{MPLTEXT 1 0 0 "" }}}{PARA 211 "" 0 "" {TEXT 200 72 "We see that the x slope is -11, the y-slope is 9, and th e z value is 19." }}{PARA 211 "" 0 "" {TEXT 200 63 "Let's try plotting the surface and the plane we have obtained. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "plot3d([f(x,y), z" }{MPLTEXT 1 0 21 "0+mx*(x-x0)+ my*(y-y0)" }{MPLTEXT 1 0 68 " ], x=-3..2, y=-0..4, view=-10..20, axes =normal, color=[blue,red]);" }{MPLTEXT 1 0 0 "" }}}{PARA 211 "" 0 "" {TEXT 200 125 "The two surfaces look like good approximations to each \+ other. Let us clean up the picture with some more magical Maple code. " }}{PARA 211 "" 0 "" {TEXT 200 105 "Since we want to plot the cross-s ections of the graph, we use the command spacecurve in the plots packa ge" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(plots):\n" } {MPLTEXT 1 0 40 "s1 := plot3d(f(x,y), x=-3..1, y=1..5, \n" }{MPLTEXT 1 0 30 " view=0..40, color=blue):\n" }{MPLTEXT 1 0 13 "s2 := plot3d (" }{MPLTEXT 1 0 1 "z" }{MPLTEXT 1 0 21 "0+mx*(x-x0)+my*(y-y0)" } {MPLTEXT 1 0 21 ", x=-3..1, y=1..5, \n" }{MPLTEXT 1 0 29 " view=0.. 40, color=red):\n" }{MPLTEXT 1 0 44 "c1 := spacecurve([x,y0,f(x,y0)], \+ x=-3..1, \n" }{MPLTEXT 1 0 33 " color=green, thickness =3):\n" } {MPLTEXT 1 0 43 "c2 := spacecurve([x0,y,f(x0,y)], y=1..5, \n" } {MPLTEXT 1 0 34 " color=yellow, thickness =3):\n" }{MPLTEXT 1 0 41 "display3d(\{s1,s2, c1, c2\}, axes=boxed);" }{MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT 200 131 "Thus we see that we have constructed a \+ point that contains the appropriate point of the surface and contains \+ the two tangent lines." }{TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 210 "" 0 "" {TEXT 205 21 "An autom ated approach" }{TEXT 205 0 "" }}{PARA 0 "" 0 "" {TEXT 200 80 "We can \+ set up a block of code that does the work of the example all in one st ep." }{TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := \+ (x,y) -> x^2-3*x*y+y^2;\n" }{MPLTEXT 1 0 20 "x0:=-1; y0:=3;\n" } {MPLTEXT 1 0 13 "width := 5:\n" }{MPLTEXT 1 0 15 "z0:=f(x0,y0);\n" } {MPLTEXT 1 0 15 "fx:=f(x,y0); \n" }{MPLTEXT 1 0 20 "Dfx := diff(fx,x); \n" }{MPLTEXT 1 0 26 "xslope:= subs(x=x0,Dfx);\n" }{MPLTEXT 1 0 14 "fy :=f(x0,y);\n" }{MPLTEXT 1 0 20 "Dfy := diff(fy,y);\n" }{MPLTEXT 1 0 26 "yslope:= subs(y=y0,Dfy);\n" }{MPLTEXT 1 0 73 "fsurface := plot3d(f (x,y), x=x0-width..x0+width, y=y0-width..y0+width, \n" }{MPLTEXT 1 0 17 " color=red):\n" }{MPLTEXT 1 0 76 "ftanplane := plot3d(z0+xslope *(x-x0)+yslope*(y-y0), x=x0-width..x0+width, \n" }{MPLTEXT 1 0 40 " \+ y=y0-width..y0+width, color=blue):\n" }{MPLTEXT 1 0 61 "xcurve := spa cecurve([x,y0,f(x,y0)], x=x0-width..x0+width, \n" }{MPLTEXT 1 0 33 " \+ color=green, thickness =3):\n" }{MPLTEXT 1 0 61 "ycurve := spacecurv e([x0,y,f(x0,y)], y=y0-width..y0+width, \n" }{MPLTEXT 1 0 34 " colo r=yellow, thickness =3):\n" }{MPLTEXT 1 0 62 "display3d(\{fsurface,fta nplane, xcurve, ycurve\}, axes=boxed);" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 200 166 "The advantage of this set-up is that we can con sider a different example by modifying the first 2 lines of the block \+ of code above and re-executing the block of code." }{TEXT 200 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 218 9 "Exercises" }{TEXT 218 0 "" }}{PARA 0 "" 0 "" {TEXT 219 50 "1) Modify the code above to examine the function " }{XPPEDIT 219 0 "x^2+2*x*y+6*y^2;" "6#,(*$)%\"xG\"\"#\"\"\"F(*(F'F(F&F(%\"yGF(F(*&\" \"'F(*$)F*F'F(F(F(" }{TEXT 219 173 " at two points of your choosing. \+ Verify that the plane obtained is tangent to the surface. (You may wa nt to reduce the width variable to zoom in on the point of tangency.)" }{TEXT 219 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT 200 50 "2) Modify the code above to examine the function " }{XPPEDIT 200 0 "3*x-x^3+y^3-3*y;" "6#,**&\"\"$\"\"\"%\"xG F&F&*$)F'F%F&!\"\"*$)%\"yGF%F&F&*&F%F&F-F&F*" }{TEXT 200 76 " at the p oints \{(-1, -1), (-1, 1), (1, -1), (1,1)\}. Explain what you find" } {TEXT 204 1 "." }{TEXT 200 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 214 0 "" }}}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 214 0 "" }}} {PARA 205 "" 0 "" {TEXT 220 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }