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0 1 }{CSTYLE "2D Outpu t" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Dictiona ry Hyperlink" -1 45 "Times" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 } {CSTYLE "Help Emphasized" -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 Underlined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle3" -1 215 "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" -1 216 "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 217 30 "A first look at Tangent Planes " }}{PARA 19 "" 0 "" {TEXT 218 36 "\302\251 Mike May, S.J.2006 - maymk @slu.edu" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 201 327 "A first naive way to look at tangent planes is to t ake the tangent lines to the two cross section curves at a point and u se those lines to define a plane. (The naive approach works if the fu nction is differentiable, but that is a detail we will look at later.) This worksheet is intended as a demonstration of this technique." } {TEXT 201 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 219 63 "Constructing a \+ plane defined by tangent lines to cross sections" }{TEXT 219 0 "" }} {PARA 211 "" 0 "" {TEXT 201 86 "We start by defining a function that w e will work with. Let z=f(x,y) =x^2-3*x*y+y^2." }}{PARA 211 "" 0 "" {TEXT 201 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 201 218 "Notice that the graph goes up in one \+ direction and down in another. We are at a saddle, so tangent planes \+ will tend to cut the surface, much as the tangent line to a cubic curv e cuts the curve at the inflection point." }}{PARA 211 "" 0 "" {TEXT 201 53 "To find the tangent plane we consider cross sections." }} {PARA 211 "" 0 "" {TEXT 201 95 "We will start with the point (-1,3). W e 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(x0, y0);" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 201 92 "Next compute the functions in one variable that we obtain by holdi ng either x or y constant." }{TEXT 201 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 201 121 "Now compute the derivatives of the fu nctions in one variable and substitute in the point to find an x-slope and y-slope." }{TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Dfx := diff(fx,x); mx := subs(x=x0,Dfx);\n" }{MPLTEXT 1 0 40 "Df y := diff(fy,y); my := subs(y=y0,Dfy);" }{MPLTEXT 1 0 0 "" }}}{PARA 211 "" 0 "" {TEXT 201 72 "We see that the x slope is -11, the y-slope \+ is 9, and the z value is 19." }}{PARA 211 "" 0 "" {TEXT 201 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, vi ew=-10..20, axes=normal, color=[blue,red]);" }{MPLTEXT 1 0 0 "" }}} {PARA 211 "" 0 "" {TEXT 201 125 "The two surfaces look like good appro ximations to each other. Let us clean up the picture with some more m agical Maple code." }}{PARA 211 "" 0 "" {TEXT 201 105 "Since we want t o plot the cross-sections of the graph, we use the command spacecurve \+ in the plots package" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with (plots):\n" }{MPLTEXT 1 0 39 "s1 := plot3d(f(x,y), x=-3..1, y=1..5, \n " }{MPLTEXT 1 0 29 " 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 20 ", x=-3..1, y=1..5, \n" }{MPLTEXT 1 0 28 " v iew=0..40, color=red):\n" }{MPLTEXT 1 0 43 "c1 := spacecurve([x,y0,f(x ,y0)], x=-3..1, \n" }{MPLTEXT 1 0 32 " color=green, thickness =3): \n" }{MPLTEXT 1 0 42 "c2 := spacecurve([x0,y,f(x0,y)], y=1..5, \n" } {MPLTEXT 1 0 33 " color=yellow, thickness =3):\n" }{MPLTEXT 1 0 39 "display3d(\{s1,s2, c1, c2\}, axes=boxed);" }{MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT 201 130 "Thus we see that we have constructed a \+ plot that contains the appropriate point of the surface and contains t he two tangent lines." }{TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 210 "" 0 "" {TEXT 206 21 "An autom ated approach" }{TEXT 206 0 "" }}{PARA 0 "" 0 "" {TEXT 201 80 "We can \+ set up a block of code that does the work of the example all in one st ep." }{TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f := \+ (x,y) -> x^2-3*x*y+y^2;\n" }{MPLTEXT 1 0 19 "x0:=-1; y0:=3;\n" } {MPLTEXT 1 0 12 "width := 5:\n" }{MPLTEXT 1 0 14 "z0:=f(x0,y0);\n" } {MPLTEXT 1 0 14 "fx:=f(x,y0); \n" }{MPLTEXT 1 0 19 "Dfx := diff(fx,x); \n" }{MPLTEXT 1 0 25 "xslope:= subs(x=x0,Dfx);\n" }{MPLTEXT 1 0 13 "fy :=f(x0,y);\n" }{MPLTEXT 1 0 19 "Dfy := diff(fy,y);\n" }{MPLTEXT 1 0 25 "yslope:= subs(y=y0,Dfy);\n" }{MPLTEXT 1 0 72 "fsurface := plot3d(f (x,y), x=x0-width..x0+width, y=y0-width..y0+width, \n" }{MPLTEXT 1 0 16 " color=red):\n" }{MPLTEXT 1 0 75 "ftanplane := plot3d(z0+xslope *(x-x0)+yslope*(y-y0), x=x0-width..x0+width, \n" }{MPLTEXT 1 0 39 " \+ y=y0-width..y0+width, color=blue):\n" }{MPLTEXT 1 0 60 "xcurve := spa cecurve([x,y0,f(x,y0)], x=x0-width..x0+width, \n" }{MPLTEXT 1 0 32 " \+ color=green, thickness =3):\n" }{MPLTEXT 1 0 60 "ycurve := spacecurv e([x0,y,f(x0,y)], y=y0-width..y0+width, \n" }{MPLTEXT 1 0 33 " colo r=yellow, thickness =3):\n" }{MPLTEXT 1 0 60 "display3d(\{fsurface,fta nplane, xcurve, ycurve\}, axes=boxed);" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 201 166 "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 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 219 9 "Exercises" }{TEXT 219 0 "" }}{PARA 0 "" 0 "" {TEXT 220 50 "1) Modify the code above to examine the function " }{XPPEDIT 220 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesett ing:-mi(\"\"), Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x \", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mn(\"2 \", mathvariant = \"normal\"), superscriptshift = \"0\")), Typesetting :-mo(\"+\", mathvariant = \"normal\", fence = \"false\", separator = \+ \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.222 2222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mn (\"2\", mathvariant = \"normal\"), Typesetting:-mo(\"⁢ \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspac e = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant \+ = \"italic\"), Typesetting:-mo(\"⁢\", mathvariant = \"n ormal\", fence = \"false\", separator = \"false\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Type setting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\")), Type setting:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", separa tor = \"false\", stretchy = \"false\", symmetric = \"false\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesett ing:-mn(\"6\", mathvariant = \"normal\"), Typesetting:-mo(\"&Invisible Times;\", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetting:-msup(Typesetting: -mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Typesetting:- mn(\"2\", mathvariant = \"normal\"), superscriptshift = \"0\")), Types etting:-mi(\"\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "- I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F #6)F+-F#6#-I%msupGF$6%-F,6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ'it alicF'-I#mnGF$6$Q\"2F'/F=Q'normalF'/%1superscriptshiftGQ\"0F'-I#moGF$6 -Q\"+F'FC/%&fenceGQ&falseF'/%*separatorGFN/%)stretchyGFN/%*symmetricGF N/%(largeopGFN/%.movablelimitsGFN/%'accentGFN/%'lspaceGQ,0.2222222emF' /%'rspaceGFgn-F#6'F?-FI6-Q1⁢F'FCFLFOFQFSFUFWFY/FfnQ&0.0 emF'/FinF`oF6F\\o-F,6%Q\"yF'F9F " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 201 50 "2) Modify the code above to examine the function " } {XPPEDIT 201 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:- mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mn(\"3\", m athvariant = \"normal\"), Typesetting:-mo(\"⁢\", mathva riant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0e m\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic \")), Typesetting:-mo(\"−\", mathvariant = \"normal\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesettin g:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", ma thvariant = \"italic\"), Typesetting:-mn(\"3\", mathvariant = \"normal \"), superscriptshift = \"0\")), Typesetting:-mo(\"+\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222 222em\"), Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"y\", i talic = \"true\", mathvariant = \"italic\"), Typesetting:-mn(\"3\", ma thvariant = \"normal\"), superscriptshift = \"0\")), Typesetting:-mo( \"−\", mathvariant = \"normal\", fence = \"false\", separator = \+ \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.222 2222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mn (\"3\", mathvariant = \"normal\"), Typesetting:-mo(\"⁢ \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspac e = \"0.0em\"), Typesetting:-mi(\"y\", italic = \"true\", mathvariant \+ = \"italic\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%m rowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6+ F+-F#6%-I#mnGF$6$Q\"3F'/%,mathvariantGQ'normalF'-I#moGF$6-Q1&Invisible Times;F'F7/%&fenceGQ&falseF'/%*separatorGF@/%)stretchyGF@/%*symmetricG F@/%(largeopGF@/%.movablelimitsGF@/%'accentGF@/%'lspaceGQ&0.0emF'/%'rs paceGFO-F,6%Q\"xF'/%'italicGQ%trueF'/F8Q'italicF'-F;6-Q(−F'F7F>F AFCFEFGFIFK/FNQ,0.2222222emF'/FQFhn-F#6#-I%msupGF$6%FRF3/%1superscript shiftGQ\"0F'-F;6-Q\"+F'F7F>FAFCFEFGFIFKFgnFin-F#6#-F]o6%-F,6%Q\"yF'FUF XF3F_oFZ-F#6%F3F:FioF+F+" }{TEXT 201 74 " at the points \{(-1, -1), (- 1, 1), (1, -1), (1,1)\}. Explain what you find" }{TEXT 205 1 "." } {TEXT 201 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 215 0 "" }}}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 215 0 "" }}}{PARA 205 "" 0 "" {TEXT 221 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }