(*********************************************************************** 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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Remember, ", StyleBox["%", FontFamily->"Courier", FontWeight->"Bold"], " stands for the previous output. For example," }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["r[t_] = t i + t^2 j + t^3 k", "Input", AspectRatioFixed->True], Cell[BoxData[ \({t, t\^2, t\^3}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["D[%,t]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({1, 2\ t, 3\ t\^2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Integrate[r[t],t]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({t\^2\/2, t\^3\/3, t\^4\/4}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["% /. t->4", "Input", AspectRatioFixed->True], Cell[BoxData[ \({8, 64\/3, 64}\)], "Output"] }, Open ]], Cell[TextData[{ "This last line shows how to substitute a value for a variable in an \ expression: \n ", StyleBox[" expression /. variable -> value", FontFamily->"Courier", FontWeight->"Bold"], "\nThe double symbol ", StyleBox["/.", FontFamily->"Courier", FontWeight->"Bold"], " applies the substitution that follows to the expression on the left. The \ substitution itself uses another double symbol ", StyleBox["->", FontFamily->"Courier", FontWeight->"Bold"], " (looks like an arrow). This example also shows that ", StyleBox["D[]", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["Integrate[]", FontFamily->"Courier", FontWeight->"Bold"], " work correctly with vector functions.\n\nYou can combine plots with the ", StyleBox["Show[]", FontFamily->"Courier", FontWeight->"Bold"], " command. First save the plots with names and list the names in ", StyleBox["Show[]", FontFamily->"Courier", FontWeight->"Bold"], ":\n", StyleBox[" p1 = Plot3D[...]\n p2 = Plot3D[...]\n Show[p1,p2]\n\n", FontFamily->"Courier", FontWeight->"Bold"], "To get a 3D line segment representing a vector from ", StyleBox["p", FontFamily->"Courier", FontWeight->"Bold"], " to ", StyleBox["q", FontFamily->"Courier", FontWeight->"Bold"], ", use ", StyleBox["v1 = Vector[p,q]", FontFamily->"Courier", FontWeight->"Bold"], " and include ", StyleBox["v1", FontFamily->"Courier", FontWeight->"Bold"], " in a ", StyleBox["Show[]", FontFamily->"Courier", FontWeight->"Bold"], " command. See the notebooks ", StyleBox["AdvancedCalculusDemo", FontWeight->"Bold"], ", and", StyleBox[" TangentsNormals", FontWeight->"Bold"], " in the Notebook folder or the back of the Lecture Notes.", StyleBox["\n\n", FontFamily->"Courier", FontWeight->"Bold"], "Sometimes ", StyleBox["Mathematica", FontSlant->"Italic"], " cannot solve an equation like ", StyleBox["f[x]==0", FontFamily->"Courier", FontWeight->"Bold"], " exactly. In such a situation you can obtain a numerical solution with ", StyleBox["NSolve[f[x]==0,x]", FontFamily->"Courier", FontWeight->"Bold"], ". To find one solution near a point ", StyleBox["a", FontFamily->"Courier", FontWeight->"Bold"], ", it is better to use ", StyleBox["FindRoot[f[x],{x,a}]", FontFamily->"Courier", FontWeight->"Bold"], " where ", StyleBox["a", FontFamily->"Courier", FontWeight->"Bold"], " is a number that tells ", StyleBox["Mathematica", FontSlant->"Italic"], " where to start looking. Plotting ", StyleBox["f[x]", FontFamily->"Courier", FontWeight->"Bold"], " can help you find good value for ", StyleBox["a", FontFamily->"Courier", FontWeight->"Bold"], ".\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", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["<{{18, 126}, {Inherited, Inherited}}, PageBreakWithin->Automatic, InitializationCell->True, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell["i = {1, 0, 0}; j = {0, 1, 0}; k = {0, 0, 1};", "Input", CellMargins->{{18, 54}, {Inherited, Inherited}}, PageBreakWithin->Automatic, InitializationCell->True, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, 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["f[t_] = (t^3-t)i + (3 - E^(2t))j + Log[t]k", FontFamily->"Courier", FontWeight->"Bold"], " and\nlet ", StyleBox["g[t_] = t^2 i - t j + Sin[Pi t]k.", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["a)", FontSlant->"Italic"], " Compute ", StyleBox["f[t].g[t]", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["Cross[f[t],g[t]]", FontFamily->"Courier", FontWeight->"Bold"], ".\n", StyleBox["b)", FontSlant->"Italic"], " Find the derivative of ", StyleBox["f[t].g[t]", FontFamily->"Courier", FontWeight->"Bold"], " at ", StyleBox["t == 1", FontFamily->"Courier", FontWeight->"Bold"], ".\n", StyleBox["c)", FontSlant->"Italic"], " Find the integral of ", StyleBox["Cross[f[t],g[t]]", FontFamily->"Courier", FontWeight->"Bold"], " from ", StyleBox["t == 1", FontFamily->"Courier", FontWeight->"Bold"], " to ", StyleBox["t == 2", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", CellMargins->{{18, 54}, {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[{ "Let ", StyleBox["r[t_] = Sin[t^2] i + Cos[t^2] j + t k", FontFamily->"Courier", FontWeight->"Bold"], "\n", StyleBox["a)", FontSlant->"Italic"], " Find a vector that is tangent to the curve at the point\n ", StyleBox["p = {P1,0,Sqrt[Pi/2]}", FontFamily->"Courier", FontWeight->"Bold"], ".\n", StyleBox["b)", FontSlant->"Italic"], " Find the equation of the line tangent to the curve ", StyleBox["r[t]", FontFamily->"Courier", FontWeight->"Bold"], " at ", StyleBox["p", FontFamily->"Courier", FontWeight->"Bold"], ".\n", StyleBox["c)", FontSlant->"Italic"], " Use ", StyleBox["ParametricPlot3D[]", FontFamily->"Courier", FontWeight->"Bold"], " to plot the curve and the tanget line near ", StyleBox["p", FontFamily->"Courier", FontWeight->"Bold"], ", then \n combine the plots with ", StyleBox["Show[]", 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 3"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Solve the initial value problem ", StyleBox["r''[t] == E^(2t) i - Cos[t]j + t k", FontFamily->"Courier", FontWeight->"Bold"], ",\n", StyleBox["r'[0] == j + k", FontFamily->"Courier", FontWeight->"Bold"], ", and ", StyleBox["r[0] == i", FontFamily->"Courier", FontWeight->"Bold"], ".\n", StyleBox["a)", FontSlant->"Italic"], " Step one: let ", StyleBox["v[t]", FontFamily->"Courier", FontWeight->"Bold"], " be integral of ", StyleBox["r''[t]", FontFamily->"Courier", FontWeight->"Bold"], " with a constant vector added to the result. Then solve the equation ", StyleBox["v[0] == j + k", FontFamily->"Courier", FontWeight->"Bold"], " for the unknown constants. This ", StyleBox["v[t]", FontFamily->"Courier", FontWeight->"Bold"], " is now ", StyleBox["r'[t]", FontFamily->"Courier", FontWeight->"Bold"], ".\n", StyleBox["b)", FontSlant->"Italic"], " Step two: let ", StyleBox["r[t]", FontFamily->"Courier", FontWeight->"Bold"], " be integral of ", StyleBox["v[t]", FontFamily->"Courier", FontWeight->"Bold"], " with another constant vector added to the result. Then solve the equation \ ", StyleBox["r[0] == i", FontFamily->"Courier", FontWeight->"Bold"], " for the unknown constants. This gives the final answer ", StyleBox["r[t]", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", CellMargins->{{18, 38}, {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", 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[{ "The position of a particle in the plane at time ", StyleBox["t", FontFamily->"Courier", FontWeight->"Bold"], " is\n", StyleBox["r[t_] = 1/(t^3+1) i - 1/(t^2+1) j.\n", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["a)", FontSlant->"Italic"], " Find the particle's speed ", StyleBox["s[t]", FontFamily->"Courier", FontWeight->"Bold"], " at any time ", StyleBox["t >= 0", FontFamily->"Courier", FontWeight->"Bold"], ". Plot ", StyleBox["s[t]", FontFamily->"Courier", FontWeight->"Bold"], " as a function of ", StyleBox["t", FontFamily->"Courier", FontWeight->"Bold"], ".\n", StyleBox["b)", FontSlant->"Italic"], " Find the particle's maximum speed for ", StyleBox["t >= 0", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", CellMargins->{{18, 37}, {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", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Problem 5"], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Let", StyleBox[" r[t_] = t i + (1+t)^3 j + (1-t)^3 k", FontFamily->"Courier", FontWeight->"Bold"], ".\na) Find the unit tangent vector ", StyleBox["T[t]", FontFamily->"Courier", FontWeight->"Bold"], ", the unit normal vector ", StyleBox["n[t]", FontFamily->"Courier", FontWeight->"Bold"], " (remember, the letter ", StyleBox["N", FontFamily->"Courier", FontWeight->"Bold"], " is reserved for the function ", StyleBox["N[]", FontFamily->"Courier", FontWeight->"Bold"], " in ", StyleBox["Mathematica", FontSlant->"Italic"], "), and the binormal vector\n", StyleBox["B[t_] = Cross[T[t],n[t]]", FontFamily->"Courier", FontWeight->"Bold"], " .\nb) Plot ", StyleBox["T[1]", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["n[1]", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["B[1]", FontFamily->"Courier", FontWeight->"Bold"], " along with the plot of ", StyleBox["r[t]", FontFamily->"Courier", FontWeight->"Bold"], " near ", StyleBox["t == 1", FontFamily->"Courier", FontWeight->"Bold"], ". 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