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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[ 156086, 5779]*) (*NotebookOutlinePosition[ 157292, 5817]*) (* CellTagsIndexPosition[ 157248, 5813]*) (*WindowFrame->Normal*) Notebook[{ Cell["Math 225: Calculus III Solution 3", "Subsection", CellMargins->{{18, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Initialization", "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[BoxData[ \(<< "\"\)], "Input", CellMargins->{{18, 126}, {Inherited, Inherited}}, PageBreakWithin->Automatic, InitializationCell->True, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[BoxData[ \(i = {1, 0, 0}; 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To find this point we simply solve ", Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{ SuperscriptBox["s", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", "0"}], "Input"]]], StyleBox[".", FontFamily->"Courier", FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ SuperscriptBox["s", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "]"}]], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(\(-\(\(32\ t\^3\)\/\((1 + t\^2)\)\^5\)\) + \(8\ t\)\/\((1 + t\^2)\)\^4 - \(108\ t\^6\)\/\((1 + t\^3)\)\^5 + \(36\ t\^3\)\/\((1 + t\^3)\)\^4\)\/\(2\ \@\(\(4\ t\^2\)\/\((1 + t\^2)\)\^4 + \(9\ t\^4\)\/\((1 + t\^3)\)\^4\)\)\)], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "The maximum seems to be near ", StyleBox["t==1", "Input", FontFamily->"Courier", FontWeight->"Bold"], " so we use this as a starting value in ", StyleBox["FindRoot[]", "Input"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"maxpt", "=", " ", RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ SuperscriptBox["s", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], ",", \({t, 1}\)}], "]"}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ \({t \[Rule] 0.746855147837759769`}\)], "Output"] }, Open ]], Cell[TextData[{ "We can also use ", StyleBox["NSolve[]", FontFamily->"Courier", FontWeight->"Bold"], " to find all the roots, including complex numbers. This takes longer." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NSolve", "[", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["s", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "==", "0"}], ",", "t"}], "]"}]], "Input", AspectRatioFixed->True], Cell[BoxData[ \({{t \[Rule] \(-1.58162908131935076`\) - 1.1661387728295296`\ I}, { t \[Rule] \(-1.58162908131935076`\) + 1.1661387728295296`\ I}, { t \[Rule] \(-0.681206743210020349`\) - 0.599794106069625954`\ I}, { t \[Rule] \(-0.681206743210020349`\) + 0.599794106069625954`\ I}, { t \[Rule] \(-0.345867790533920782`\) - 0.250430642833388539`\ I}, { t \[Rule] \(-0.345867790533920782`\) + 0.250430642833388539`\ I}, { t \[Rule] \(0.238278961499897734`\[InvisibleSpace]\) - 0.779832634326388962`\ I}, { t \[Rule] \(0.238278961499897734`\[InvisibleSpace]\) + 0.779832634326388962`\ I}, { t \[Rule] \(0.262336503045808644`\[InvisibleSpace]\) - 0.362412188757479469`\ I}, { t \[Rule] \(0.262336503045808644`\[InvisibleSpace]\) + 0.362412188757479469`\ I}, { t \[Rule] \(0.276915576256147577`\[InvisibleSpace]\) - 0.953489038926348086`\ I}, { t \[Rule] \(0.276915576256147577`\[InvisibleSpace]\) + 0.953489038926348086`\ I}, { t \[Rule] \(0.347539928010534904`\[InvisibleSpace]\) - 1.14950494148227133`\ I}, { t \[Rule] \(0.347539928010534904`\[InvisibleSpace]\) + 1.14950494148227133`\ I}, {t \[Rule] 0.746855147837857114`}, { t \[Rule] \(1.11020507233197451`\[InvisibleSpace]\) - 1.68346871663068098`\ I}, { t \[Rule] \(1.11020507233197451`\[InvisibleSpace]\) + 1.68346871663068098`\ I}}\)], "Output"] }, Open ]], Cell[TextData[{ "The maximum of ", StyleBox["s[t]", FontFamily->"Courier", FontWeight->"Bold"], " is thus:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(N[s[t] /. maxpt]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1.03645333860124599`\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Problem\ 5\)], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Let", StyleBox[" ", FontFamily->"Courier", FontWeight->"Bold"], Cell[BoxData[ \(r[t_] = t\ i + \((1 + t)\)\^3\ j + \((1 - t)\)\^3\ k\)], "Input"], ".\na) Find the unit tangent vector ", StyleBox["T[t]", "Input"], ", the unit normal vector ", StyleBox["n[t]", "Input"], " (remember, the letter ", StyleBox["N", "Input"], " is reserved for the function ", StyleBox["N[]", "Input"], " 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]", "Input"], ", ", StyleBox["n[1]", "Input"], " and ", StyleBox["B[1]", "Input"], " along with the plot of ", StyleBox["r[t]", "Input"], " near ", StyleBox["t==1", "Input"], ". " }], "Text", CellMargins->{{18, 4}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[BoxData[ \(Solution\)], "Subsubsection", CellMargins->{{Inherited, 126}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(r[t_] = t\ i + \((1 + t)\)\^3\ j + \((1 - t)\)\^3\ k\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({t, \((1 + t)\)\^3, \((1 - t)\)\^3}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["a) ", FontSlant->"Italic"], "The unit tangent vector is given by" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(T[t_]\), "=", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["r", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], \(Norm[\(r'\)[t]]\)], " ", "//", "Simplify"}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ \({1\/\@\(19 + 108\ t\^2 + 18\ t\^4\), \(3\ \((1 + t)\)\^2\)\/\@\(19 + 108\ t\^2 + 18\ t\^4\), \(-\(\(3\ \((\(-1\) + t)\)\^2\)\/\@\(19 + 108\ t\^2 + 18\ t\^4\)\)\)} \)], "Output"] }, Open ]], Cell["The unit normal vector is given by", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(n[t_]\), "=", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["T", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], RowBox[{"Norm", "[", RowBox[{ SuperscriptBox["T", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "]"}]], "//", "Simplify"}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(- \(\(3\ \@2\ t\ \((3 + t\^2) \)\)\/\(\@\(\(19 - 35\ t\^2 + 18\ t\^4\)\/\((19 + 108\ t\^2 + 18\ t\^4)\)\^2\)\ \((19 + 108\ t\^2 + 18\ t\^4)\)\^\(3/2\)\)\)\), \(-\(\(\(-19\) + 35\ t - 36\ t\^3 + 18\ t\^4\)\/\(\@2\ \@\(\(19 - 35\ t\^2 + 18\ t\^4\)\/\((19 + 108\ t\^2 + 18\ t\^4)\)\^2\)\ \((19 + 108\ t\^2 + 18\ t\^4)\)\^\(3/2\)\)\)\), \(-\(\(\(-19\) - 35\ t + 36\ t\^3 + 18\ t\^4\)\/\(\@2\ \@\(\(19 - 35\ t\^2 + 18\ t\^4\)\/\((19 + 108\ t\^2 + 18\ t\^4)\)\^2\)\ \((19 + 108\ t\^2 + 18\ t\^4)\)\^\(3/2\)\)\)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(B[t_] = T[t]\[Cross]n[t]\ // Simplify\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({\(3\ \@2\ \@\(\(19 - 35\ t\^2 + 18\ t\^4\)\/\((19 + 108\ t\^2 + 18\ t\^4)\)\^2 \)\ \((19 + 89\ t\^2 - 90\ t\^4 - 18\ t\^6)\)\)\/\(19 - 35\ t\^2 + 18\ t\^4\), \(\@\(\(19 - 35\ t\^2 + 18\ t\^4\)\/\((19 + 108\ t\^2 + 18\ t\^4)\)\^2 \)\ \((\(-19\) + 19\ t - 108\ t\^2 + 108\ t\^3 - 18\ t\^4 + 18\ t\^5)\)\)\/\(\@2\ \((19 - 35\ t\^2 + 18\ t\^4)\)\), \(\@\(\(19 - 35\ t\^2 + 18\ t\^4\)\/\((19 + 108\ t\^2 + 18\ t\^4)\)\^2 \)\ \((19 + 19\ t + 108\ t\^2 + 108\ t\^3 + 18\ t\^4 + 18\ t\^5)\)\)\/\(\@2\ \((19 - 35\ t\^2 + 18\ t\^4)\)\)}\)], "Output"] }, Open ]], Cell[TextData[{ "To create the graphics for displaying these vactors at the point ", 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