(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 9741, 334]*) (*NotebookOutlinePosition[ 10684, 365]*) (* CellTagsIndexPosition[ 10640, 361]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ \(Off[General::spell]\)], "Input", Background->RGBColor[1, 1, 0]], Cell[TextData[{ StyleBox["Math 2443: Calculus & Analytic Geometry IV", FontSize->12, FontColor->GrayLevel[0]], "\n", StyleBox["Construction of a Tangent Plane", FontSize->24], "\n", StyleBox[ "by TJ Murphy with adaptations of work by Brad Kline at USAFA and Paul \ Goodey at OU", FontSize->12, FontColor->GrayLevel[0]] }], "Text", CellFrame->{{0, 0}, {0, 3}}, FontSize->24, FontColor->RGBColor[1, 0, 1]], Cell["\<\ Shows the construction of a tangent plane to a surface at a \ point.\ \>", "Text", CellFrame->{{0, 0}, {0, 3}}], Cell[CellGroupData[{ Cell["Inputing the Surface and the Point", "Text", CellFrame->{{0, 0}, {0, 3}}, FontSize->18, Background->RGBColor[1, 0, 1]], Cell[CellGroupData[{ Cell["\<\ The following command allows us to change the viewpoint of the \ graphics objects and only type the change in one spot.\ \>", "Text"], Cell[BoxData[ \(\(viewpoint = \n\t\ \ \ ViewPoint -> {3.098, \ 1.116, \ 0.779}; \)\)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "We start with a surface, in this example the elliptic paraboloid ", Cell[BoxData[ RowBox[{\(f \((x, y)\)\), "=", FormBox[\(\(-x\^2\) - y\^2\), "TraditionalForm"]}]]], "." }], "Text"], Cell[BoxData[ \(\(paraboloid = Plot3D[\(-x^2\) - y^2, \n\t\t{x, \(-5\), 5}, {y, \(-5\), 5}, \n\t\t BoxRatios -> {1, 1, 1}, \n\t\tBoxed -> False, Axes -> False, \n\t\t PlotPoints -> 40, Mesh -> False]; \)\)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We are interested in the tangent plane to the surface S=f(x,y) at a \ particular point. In this example the point is (2,-1,-5).\ \>", "Text"], Cell[BoxData[ \(\(point = Graphics3D[{PointSize[ .025], \n\t\t\tPoint[{2, \(-1\), \(-5\)}]}]; \)\)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ The following command shows the surface and the point (2,-1,-5) \ with our chosed viewpoint. Note that many planes go through this point, but \ we want the one that emulates the surface most closely at this point.\ \>", "Text"], Cell[BoxData[ \(\(Show[paraboloid, point, viewpoint]; \)\)], "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Useful Vertical Traces", "Text", CellFrame->{{0, 0}, {0, 3}}, FontSize->18, Background->RGBColor[1, 0, 1]], Cell[CellGroupData[{ Cell["\<\ To begin constructing the tangent plane to the surface at the \ point, we first intersect the surface with the plane y=-1, calling this plane \ \"yplane\". (Note: the first graph of this plane is in the default \ viewpoint.)\ \>", "Text"], Cell[BoxData[ \(\(yplane = ParametricPlot3D[{u, \(-1\), w}, \n\t\t{u, \(-5\), 6}, {w, \(-50\), 5}, \n\t\tBoxRatios -> {1, 1, 1}, \n\t\tBoxed -> False, Axes -> False, PlotPoints -> 2]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(\(Show[paraboloid, point, yplane, \n\t\ \ viewpoint]; \)\)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ This intersection results in a curve called the trace of S in the \ plane y=-1. We call this curve \"ycurve\". (Note: the first graph of this \ curve is in the default viewpoint.)\ \>", "Text"], Cell[BoxData[ \(\(ycurve = \n\t ParametricPlot3D[{u, \(-1\), \(-u^2\) - 1}, \n\t\t{u, \(-5\), 5}, BoxRatios -> {1, 1, 1}, \n\t\tBoxed -> False, Axes -> False]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(\(Show[ycurve, point, viewpoint]; \)\)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We also intersect S with the plane x=2, calling this plane \"xplane\ \". (Note: the first graph of this plane is in the default viewpoint.)\ \>", "Text"], Cell[BoxData[ \(\(xplane = ParametricPlot3D[{2, v, w}, \n\t\t{v, \(-5\), 5}, {w, \(-50\), 5}, \n \t\tBoxRatios -> {1, 1, 1}, \n\t\tBoxed -> False, Axes -> False, PlotPoints -> 2]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(\(Show[paraboloid, point, yplane, xplane, viewpoint]; \)\)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ The curve resulting from this intersection we call \"xcurve\" and \ here we show both curves and emphasize that they intersect at (2,-1,-5). \ (Note: the first graph of xcurve is in the default viewpoint.)\ \>", "Text"], Cell[BoxData[ \(\(xcurve = \n\t ParametricPlot3D[{2, v, \(-4\) - v^2}, \n\t\t{v, \(-5\), 5}, BoxRatios -> {1, 1, 1}, \n\t\tBoxed -> False, Axes -> False]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(\(Show[ycurve, xcurve, point, viewpoint]; \)\)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Their Tangent Lines Uniquely Determine the Plane We Seek\ \>", "Text", CellFrame->{{0, 0}, {0, 3}}, FontSize->18, Background->RGBColor[1, 0, 1]], Cell[CellGroupData[{ Cell["\<\ Each of the curves \"ycurve\" and \"xcurve\" has a tangent line at \ the point (2,-1,-5). We call these lines \"ytangent\" and \"xtangent\" \ respectively.\ \>", "Text"], Cell[BoxData[ \(\(ytangent = \n\t Graphics3D[{Thickness[ .01], \n\t\t\t Line[{{0, \(-1\), 3}, {7, \(-1\), \(-25\)}}]}]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(\(xtangent = \n\t Graphics3D[{Thickness[ .01], \n\t\t\t Line[{{2, \(-5\), \(-13\)}, {2, 5, 7}}]}]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(\(Show[ycurve, xcurve, point, ytangent, \n\t\ \ xtangent, viewpoint]; \)\)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ The tangent lines \"ytangent\" and \"xtangent\" uniquely determine \ a plane. As it turns out, this is the plane we were seeking. (Note: the first \ graph of this plane is in the default viewpoint.)\ \>", "Text"], Cell[BoxData[ \(\(tanplane = \n\t ParametricPlot3D[{u, v, \(-4\) u + 2 v + 5}, \n \t\t{u, 0, 7}, {v, \(-5\), 5}, \n\t\tBoxRatios -> {1, 1, 1}, \n\t\t Boxed -> False, Axes -> False, PlotPoints -> 2]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(\(Show[ycurve, xcurve, point, ytangent, \n\t\ \ xtangent, tanplane, viewpoint]; \)\)], "Input", Background->RGBColor[1, 1, 0]], Cell[BoxData[ \(\(Show[paraboloid, ycurve, xcurve, point, \n\tytangent, xtangent, tanplane, viewpoint]; \)\)], "Input", Background->RGBColor[1, 1, 0]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["An Equation for the Tangent Plane", "Text", CellFrame->{{0, 0}, {0, 3}}, FontSize->18, Background->RGBColor[1, 0, 1]], Cell[TextData[{ "In general the equation for a plane containing the point (a,b,c) is\n\nA(x \ - a) + B(y - b) + C(z - c) = 0\n\nwhere A, B, and C are the components of the \ normal vector to the plane. In order to calculate these coefficients for our \ tangent plane, we note that the plane contains the two tangent lines from \ above. One of those tangent lines has slope ", Cell[BoxData[ \(TraditionalForm\`\(-A\)\/C = \(f\_x\)(a, b)\)]], " and the other tangent line has slope ", Cell[BoxData[ \(TraditionalForm\`\(-B\)\/C = \(f\_y\)(a, b)\)]], ". Some algebra gives us that the equation for the tangent plane to f(x,y) \ at (a,b,f(a,b)) is\n\n", Cell[BoxData[ \(TraditionalForm \`z - f(a, b) = \(\(f\_x\)(a, b)\) \((x\ - a)\) + \(\(f\_y\)(a, b)\) \((y - b)\)\)]], "." }], "Text"] }, Closed]], Cell[TextData[{ "This work is part of the Multivariable Calculus with ", StyleBox["Mathematica", FontSlant->"Italic"], " Project at the University of Oklahoma " }], "Text", CellFrame->{{0, 0}, {0, 3}}, FontSize->10] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 832}, {0, 604}}, WindowToolbars->"RulerBar", CellGrouping->Manual, WindowSize->{520, 509}, WindowMargins->{{75, Automatic}, {Automatic, 5}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, MacintoshSystemPageSetup->"\<\ 00l0005X0FP000003^P;H?oCokH?B`^Z;085:0?l0@00005X0]0000003^PF`003 1@00I00200000@0200000BL?004000]OMc0000000000030001000000000@0?oC of`?BaMD0200000000400000000001T1\>" ] (*********************************************************************** Cached data follows. 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