(*********************************************************************** 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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Before starting, you \ may want to review the notebooks ", StyleBox["Derivatives", FontWeight->"Bold"], ", and", StyleBox[" Exercises", FontWeight->"Bold"], " (in the Notebook folder or the back of the Lecture Notes). To make the \ graphs look smoother you can increase the number of plotted points with the \ optional optional argument ", StyleBox["PlotPoints", FontFamily->"Courier", FontWeight->"Bold"], ". For example,\n", StyleBox["Plot3D[ x^2 - y^2, {x,-2,2}, {y,-2,2}, PlotPoints->30]\n", FontFamily->"Courier", FontWeight->"Bold"], "or", StyleBox[ "\nContourPlot[ x^2 - y^2, {x,-2,2}, {y,-2,2},\n PlotPoints->30]\n\n", FontFamily->"Courier", FontWeight->"Bold"], "The function ", StyleBox["Limit[]", FontFamily->"Courier", FontWeight->"Bold"], " only finds limits of functions of one variable. In problem 4, you should \ substitute different expressions for ", StyleBox["y", FontFamily->"Courier", FontWeight->"Bold"], ",\nthen take the limit as ", StyleBox["x -> 0", FontFamily->"Courier", FontWeight->"Bold"], ". For example,\n", StyleBox[ " z = Sin[x^3y^2]/(x^5 + y^5)\n z1 = z /. {y -> 2x}\n Limit[z1, x->0]\n", FontFamily->"Courier", FontWeight->"Bold"], " ", StyleBox["\n", FontFamily->"Courier", FontWeight->"Bold"], "Be 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["File", FontFamily->"Courier", FontWeight->"Bold"], " menu and close this group before printing." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Initialization"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["<True, AspectRatioFixed->True], Cell[TextData["i = {1,0,0}; j = {0,1,0}; k = {0,0,1};"], "Input", InitializationCell->True, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 1"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Let ", StyleBox["r[t_] = (3t^5 - 5t^3) i + (t^4+4t) j + (t+1)^3 k.", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["a)", FontSlant->"Italic"], " Find the point(s) where the curve defined by ", StyleBox["r[t]", FontFamily->"Courier", FontWeight->"Bold"], " is not smooth.\n", StyleBox["b)", FontSlant->"Italic"], " Plot the curve with ", StyleBox["ParametricPlot3D[]", FontFamily->"Courier", FontWeight->"Bold"], " near the singular point(s).\n", StyleBox["c)", FontSlant->"Italic"], " Find the length of the Curve from ", StyleBox["t==-1", FontFamily->"Courier", FontWeight->"Bold"], " to ", StyleBox["t==1", FontFamily->"Courier", FontWeight->"Bold"], ". (Use ", StyleBox["NIntegrate[]", FontFamily->"Courier", FontWeight->"Bold"], ")." }], "Text", CellMargins->{{18, 55}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell["", "Input", CellMargins->{{18, 126}, {Inherited, Inherited}}, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 2"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "A particle's position is given by\n", StyleBox["r[t_] = (1 + Sin[t])i + (3 t^2)j + (t + t^3)k", FontFamily->"Courier", FontWeight->"Bold"], ".\n", StyleBox["a)", FontSlant->"Italic"], " Find the acceleration, ", StyleBox["a[Pi]", FontFamily->"Courier", FontWeight->"Bold"], ", of the particle at time ", StyleBox["t == Pi", FontFamily->"Courier", FontWeight->"Bold"], ".\n", StyleBox["b)", FontSlant->"Italic"], " Find the tangential and normal components of ", StyleBox["a[Pi]", FontFamily->"Courier", FontWeight->"Bold"], ", i.e., find NUMBERS ", StyleBox["u", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["v", FontFamily->"Courier", FontWeight->"Bold"], " such that ", StyleBox["a[Pi] = u T[Pi] + v n[Pi]", FontFamily->"Courier", FontWeight->"Bold"], " without calculating ", StyleBox["T[Pi]", FontFamily->"Courier", FontWeight->"Bold"], " or ", StyleBox["n[Pi]", FontFamily->"Courier", FontWeight->"Bold"], "! (See Section 12.4 in Finney &Thomas or 2.4 in the Lecture Notes)." }], "Text", CellMargins->{{18, 18}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell["", "Input", CellMargins->{{18, 126}, {Inherited, Inherited}}, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 3"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Plot the graphs of the following functions, ", StyleBox["z = f[x,y]", FontFamily->"Courier", FontWeight->"Bold"], ", with ", StyleBox["Plot3D[]", FontFamily->"Courier", FontWeight->"Bold"], " and plot several level curves of each function with ", StyleBox["ContourPlot[]", FontFamily->"Courier", FontWeight->"Bold"], " (See Section 3.1 in the Lecture Notes and the examples in the Notebook \ directory.) Compare the graphs to the level curves.\n", StyleBox["a)", FontSlant->"Italic"], " ", StyleBox["z = x + y^2 - y x^3\n", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["b)", FontSlant->"Italic"], " ", StyleBox["z = z = (x^3 - y^3)/(x^2 + y^2)\n", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["c) ", FontSlant->"Italic"], StyleBox["z = E^(-x^2-y^2)Sin[x]", FontFamily->"Courier", FontWeight->"Bold"] }], "Text", CellMargins->{{18, 53}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell["", "Input", CellMargins->{{18, 126}, {Inherited, Inherited}}, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 4"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "By considering different paths of approach, show that the limit of the \ function\n", StyleBox["f[x_,y_] = x^4 y^2/(x^6 + x^2 y^4 + x^3y^3)", FontFamily->"Courier", FontWeight->"Bold"], "\nas ", StyleBox["x", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["y", FontFamily->"Courier", FontWeight->"Bold"], " both go to ", StyleBox["0", FontFamily->"Courier", FontWeight->"Bold"], " does not exist. Graph the function near ", StyleBox["{0,0}", FontFamily->"Courier", FontWeight->"Bold"], " and explain why this is so based on the appearance of the graph." }], "Text", CellMargins->{{18, 53}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell["", "Input", CellMargins->{{18, 126}, {Inherited, Inherited}}, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 5"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell[TextData[{ StyleBox["a)", FontSlant->"Italic"], " Find the partial derivatives of\n ", StyleBox["f[u_,v_] = u^2 Cos[u/v]", FontFamily->"Courier", FontWeight->"Bold"], " with respect to ", StyleBox["u", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["v", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["b)", FontSlant->"Italic"], " Find the second order partial derivatives of\n ", StyleBox["f[x_,y_] = x^3 y^2 - y/x^2 + E^(x y)", FontFamily->"Courier", FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Solution"], "Subsubsection", Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True], Cell["", "Input", AspectRatioFixed->True] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 832}, {0, 604}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Manual, WindowSize->{520, 509}, WindowMargins->{{68, Automatic}, {30, Automatic}}, 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. 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