(*********************************************************************** 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 12372, 476]*) (*NotebookOutlinePosition[ 13523, 512]*) (* CellTagsIndexPosition[ 13479, 508]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData["Math 225: Calculus III Assignment 5"], "Subsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData["Name:"], "Subsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData["Section:"], "Subsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[ "\nI affirm that the solutions presented in this assignment are entirely my \ own work"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Signature:"], "Subsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Instructions"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This assignment contains problems involving functions of several \ variables: the chain rule, tangent planes, the gradient of a function, local \ maxima/minima & saddle points on the graph of a function, see Chapter 13 in \ Finney & Thomas or Chapter 3 in the Lecture Notes. Before starting this \ assignment you may want to review the notebooks ", StyleBox["Derivatives", FontWeight->"Bold"], ",", StyleBox[" TangentsNormals", FontWeight->"Bold"], ", and", StyleBox[" AdvancedCalculusDemo", FontWeight->"Bold"], " (in the Notebook folder or the back of the Lecture Notes). The functions \ ", StyleBox["Grad[]", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["CriticalPoints[]", FontFamily->"Courier", FontWeight->"Bold"], ", and", StyleBox[" Discriminant[]", FontFamily->"Courier", FontWeight->"Bold"], " are used for the first time in this assignment.\n\nRecall that to \ substitute a value into an expression use: ", StyleBox["expr /. var -> value.\n", FontFamily->"Courier", FontWeight->"Bold"], "For example,\n", StyleBox[" x^2 - y^2 /. x->2", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox[" ", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["4 - y^2", FontFamily->"Courier"], "\nIf an expression has already been defined as a function, then just use \ the function notation to get values. For example\n", StyleBox[" f[x_,y_] = x^2 - y^2;\n f[2,y]\n ", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["4 - y^2", FontFamily->"Courier"], StyleBox["\n", FontFamily->"Courier", FontWeight->"Bold"], "\nTo plot the level curve ", StyleBox["f[x,y] == c", FontFamily->"Courier", FontWeight->"Bold"], ", use\n", StyleBox[ "ContourPlot[f[x,y],{x,minx,maxx},{y,miny,maxy},\n Contours->{c}, \ ContourShading->False]", FontFamily->"Courier", FontWeight->"Bold"], "\nTo plot a level surface ", StyleBox["f[x,y,z] == c", FontFamily->"Courier", FontWeight->"Bold"], ", use", StyleBox[ "\nContourPlot3D[f[x,y,z],{x,minx,maxx},{y,miny,maxy},\n {z,minz,maxz}, \ Contours->{c}]", FontFamily->"Courier", FontWeight->"Bold"], "\nTo get a line segment from ", StyleBox["p", FontFamily->"Courier", FontWeight->"Bold"], " to ", StyleBox["q", FontFamily->"Courier", FontWeight->"Bold"], " representing a 3D vector use ", StyleBox["Vector[p,q]", FontFamily->"Courier", FontWeight->"Bold"], " and combine it with other plots with the ", StyleBox["Show[]", FontFamily->"Courier", FontWeight->"Bold"], " command.\n\nBe sure to type in comments explaining what you are doing.\n\n\ Remember to uncheck ", StyleBox["Show In/Out Names", FontFamily->"Courier", FontWeight->"Bold"], " in the ", StyleBox["Kernel", FontFamily->"Courier", FontWeight->"Bold"], " menu and close this group before printing." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Initialization"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData["<Automatic, InitializationCell->True, AspectRatioFixed->True], Cell[TextData["Needs[\"Graphics`ContourPlot3D`\"]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData["i = {1, 0, 0}; j = {0, 1, 0}; k = {0, 0, 1};"], "Input", PageBreakWithin->Automatic, InitializationCell->True, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 1"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Let\n", StyleBox[" w[x_,y_] = Tan[x^2 + y^2]", FontFamily->"Courier", FontWeight->"Bold"], "\nand\n", StyleBox[" x[r_,s_] = E^(-r) + E^s; y[r_,s_] = r/s", FontFamily->"Courier", FontWeight->"Bold"], "\n\nFind the derivatives of ", StyleBox["w", FontFamily->"Courier", FontWeight->"Bold"], " with respect to ", StyleBox["r", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["s ", FontFamily->"Courier", FontWeight->"Bold"], "when ", StyleBox["r == 2", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["s == 1", FontFamily->"Courier", FontWeight->"Bold"], ".\n(The dependencies between the variables must be made explicit: use\n", StyleBox["D[w[x[r,s],y[r,s]],r]", FontFamily->"Courier", FontWeight->"Bold"], ", and ", StyleBox["D[w[x[r,s],y[r,s]],s]", FontFamily->"Courier", FontWeight->"Bold"], ") ." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 2"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Let ", StyleBox["f[x_,y_] = x^3 y - x^2 y^2 - y^4.", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["a) ", FontSlant->"Italic"], "Find the gradient of ", StyleBox["f[x,y]", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["Grad[f[x,y],{x,y}]", FontFamily->"Courier", FontWeight->"Bold"], ".\n", StyleBox["b)", FontSlant->"Italic"], " Find the direction in which ", StyleBox["f[x_,y_]", FontFamily->"Courier", FontWeight->"Bold"], " increases most rapidly at the point ", StyleBox["{1,2}", FontFamily->"Courier", FontWeight->"Bold"], " and find the derivative of ", StyleBox["f[x,y]", FontFamily->"Courier", FontWeight->"Bold"], " in that direction.\n", StyleBox["c)", FontSlant->"Italic"], " Plot the level curve of ", StyleBox["f[x,y]", FontFamily->"Courier", FontWeight->"Bold"], " through the point ", StyleBox["{1,2}", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 3"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Let ", StyleBox["f[x_,y_,z_] = y^3 + z^2 - x^2", FontFamily->"Courier", FontWeight->"Bold"], ". \n", StyleBox["a)", FontSlant->"Italic"], " Plot the level surface ", StyleBox["f[x,y,z] == 1", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["b)", FontSlant->"Italic"], " Find ", StyleBox["Grad[f[x,y,z]]", FontFamily->"Courier", FontWeight->"Bold"], " at the point ", StyleBox["{1,1,1}", FontFamily->"Courier", FontWeight->"Bold"], " and plot this vector with the level surface from part ", StyleBox["a)", FontSlant->"Italic"], ". (", StyleBox["{1,1,1}", FontFamily->"Courier", FontWeight->"Bold"], " is on this level surface)." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 4"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ StyleBox["a)", FontSlant->"Italic"], " Find the equation for the plane tangent to the level surface ", StyleBox["z - x^2 + y^2 == 1", FontFamily->"Courier", FontWeight->"Bold"], " at the point ", StyleBox["{3,2,6}", FontFamily->"Courier", FontWeight->"Bold"], ".\n", StyleBox["b)", FontSlant->"Italic"], " Graph the surface together with this tangent plane. (Suggestion: choose a \ domain for the tangent plane smaller than the domain for the surface so that \ the point of tangency is easy to see.)" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 5"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Let ", StyleBox["f[x_,y_] = 3 x y - x^3 - y^2", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["a)", FontSlant->"Italic"], " Test the function ", StyleBox["f[x,y]", FontFamily->"Courier", FontWeight->"Bold"], " for local maxima, local minima, and saddle points.\n", StyleBox["b) ", FontSlant->"Italic"], "Find the value of ", StyleBox["f[x,y]", FontFamily->"Courier", FontWeight->"Bold"], " at the points found in ", StyleBox["a)", FontSlant->"Italic"], ".\n", StyleBox["c) ", FontSlant->"Italic"], "Produce a graph of ", StyleBox["f[x,y]", FontFamily->"Courier", FontWeight->"Bold"], "that includes the points found in ", StyleBox["a)", FontSlant->"Italic"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", PageBreakWithin->Automatic, AspectRatioFixed->True] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 832}, {0, 604}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Manual, WindowSize->{520, 509}, WindowMargins->{{84, Automatic}, {Automatic, 1}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@00000000000000060801T1T00000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1709, 49, 148, 3, 44, "Subsection", PageBreakWithin->Automatic], Cell[1860, 54, 115, 3, 44, "Subsection", PageBreakWithin->Automatic], Cell[1978, 59, 118, 3, 44, "Subsection", PageBreakWithin->Automatic], Cell[CellGroupData[{ Cell[2121, 66, 160, 4, 46, "Text"], Cell[2284, 72, 120, 3, 44, "Subsection", PageBreakWithin->Automatic] }, Open ]], Cell[CellGroupData[{ Cell[2441, 80, 92, 2, 44, "Subsection"], Cell[2536, 84, 3065, 100, 528, "Text"], Cell[CellGroupData[{ Cell[5626, 188, 127, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[5756, 193, 130, 3, 27, "Input", PageBreakWithin->Automatic, InitializationCell->True], Cell[5889, 198, 115, 2, 27, "Input", InitializationCell->True], Cell[6007, 202, 155, 3, 27, "Input", PageBreakWithin->Automatic, InitializationCell->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[6211, 211, 122, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[6336, 216, 1020, 40, 146, "Text"], Cell[7359, 258, 121, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[7483, 263, 73, 2, 27, "Input", PageBreakWithin->Automatic] }, Open ]], Cell[CellGroupData[{ Cell[7593, 270, 122, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[7718, 275, 1092, 45, 99, "Text"], Cell[8813, 322, 121, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[8937, 327, 73, 2, 27, "Input", PageBreakWithin->Automatic] }, Open ]], Cell[CellGroupData[{ Cell[9047, 334, 122, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[9172, 339, 801, 33, 82, "Text"], Cell[9976, 374, 121, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[10100, 379, 73, 2, 27, "Input", PageBreakWithin->Automatic] }, Open ]], Cell[CellGroupData[{ Cell[10210, 386, 122, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[10335, 391, 616, 19, 81, "Text"], Cell[10954, 412, 121, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[11078, 417, 73, 2, 27, "Input", PageBreakWithin->Automatic] }, Open ]], Cell[CellGroupData[{ Cell[11188, 424, 122, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[11313, 429, 843, 35, 82, "Text"], Cell[12159, 466, 121, 3, 41, "Subsubsection", PageBreakWithin->Automatic], Cell[12283, 471, 73, 2, 27, "Input", PageBreakWithin->Automatic] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)