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Before you begin, you may want to review the notebooks ", StyleBox["MaxMin", FontWeight->"Bold"], " and ", StyleBox["AdvancedCalculusDemo", FontWeight->"Bold"], " in the Notebook folder or the back of the Lecture Notes. There are also \ many examples of how to do these problems in the Quizzes folder.\n\nBe sure \ to type in comments explaining what you are doing.\n\nRemember 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[{ "Find the absolute maximum and minimum values of\n\t", StyleBox["f[x_,y_] = x^4 - x^2 y + x^3 + y^2", FontFamily->"Courier", FontWeight->"Bold"], "\non the rectangular region bounded by the lines ", StyleBox["x == -3/2", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["x == 1", FontFamily->"Courier", FontWeight->"Bold"], ",\n", StyleBox["y == -1/2", FontFamily->"Courier", FontWeight->"Bold"], ", and ", StyleBox["y == 1", 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[{ "Find the point(s) on the curve ", StyleBox["x y - 2x - y == 1", FontFamily->"Courier", FontWeight->"Bold"], " nearest to the point ", StyleBox["{1,2}", FontFamily->"Courier", FontWeight->"Bold"], " by using Lagrange multipliers to find the minimum of the squared-distance \ function\n", StyleBox["f[x_,y_] = (x-1)^2 + (y-2)^2", FontFamily->"Courier", FontWeight->"Bold"], " subject to the above constraint." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 3"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Find the extreme values of\n\t", StyleBox["f[x_,y_] = x^3 + 3 x y - y^2", FontFamily->"Courier", FontWeight->"Bold"], "\non the circular disk ", StyleBox["x^2 + y^2 <= 1", 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 4"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ "Suppose that the temperature ", StyleBox["T", FontFamily->"Courier", FontWeight->"Bold"], " in degrees at the point ", StyleBox["{x,y,z}", FontFamily->"Courier", FontWeight->"Bold"], " on the sphere\n", StyleBox["x^2 + y^2 + z^2 == 1", FontFamily->"Courier", FontWeight->"Bold"], " is ", StyleBox["T[x_,y_,z_] = 10(1-z^2) + x^2 y^2", FontFamily->"Courier", FontWeight->"Bold"], ".\nLocate where the highest and lowest temperatures on the sphere occur \ and what the temperatures are at those points. (Use Lagrange multipliers and \ be sure to ignore any complex numbers that arise.)" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 5"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[ "Your firm has been asked to design a storage tank for liquid petroleum gas. \ The customer's specifications call for a cylindrical tank that holds 8000 m^3 \ of gas. The customer also wants to use the smallest amount of material \ possible in building the tank. 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