(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing 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[ 82695, 3226]*) (*NotebookOutlinePosition[ 128120, 4800]*) (* CellTagsIndexPosition[ 127958, 4791]*) (*WindowFrame->Normal*) Notebook[{ Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg"], "Graphics", ShowCellBracket->False, CellMargins->{{25, 24}, {5, 7}}, ImageSize->{81, 22}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Approximation\n", StyleBox[ "Authors: Bill Davis, Horacio Porta and Jerry Uhl ", "Subtitle"], StyleBox["\[Copyright]1999", "Subtitle", FontSize->12], StyleBox["\nProducer: Bruce Carpenter\n", "Subtitle"], StyleBox["Publisher: ", "Subtitle", FontSize->12], StyleBox[ButtonBox["Math Everywhere, Inc.", ButtonData:>{ URL[ "http://www.matheverywhere.com"], None}, ButtonStyle->"MEIHyperlink"], FontSize->12], StyleBox[" Distributor: ", "Subtitle", FontSize->12], StyleBox[ButtonBox["Wolfram Research, Inc.", ButtonData:>{ URL[ "http://www.wolfram.com"], None}, ButtonStyle->"MEIHyperlink", ButtonNote->"Makers of Mathematica!"], FontSize->12] }], "PrefaceTitle", CellMargins->{{Inherited, Inherited}, {Inherited, 0}}], Cell[TextData[{ "3.04 Taylor's Formula\n", StyleBox["Samples", FontSize->16, FontSlant->"Italic"] }], "Title"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Initializations" }], "Special2"], Cell[BoxData[{ \(\(Off[General::spell];\)\), "\n", \(\(Off[General::spell1];\)\), "\n", \(\(Off[Plot::plnr];\)\), "\n", \(\(Off[ParametricPlot::ppcom];\)\), "\n", \(\(SetOptions[Limit, Analytic \[Rule] True];\)\n\), "\n", \(\(If[ MemberQ[{"\", \ "\", \ \ "\"}, \n\ \ \ \ Context[Gray]], \ Remove["\<*`Gray\>"]];\)\), "\n", \(\(<< "\";\)\), "\n", \(\(Remove["\<*`Gray\>"];\)\), "\n", \(\(<< "\";\)\), "\n", \(\(Graphics`Colors`GosiaGreen = RGBColor[0, \ 0.392187, \ 0];\)\n\), "\n", \(\(Clear[Derivative];\)\)}], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell["ThreeAxes[u,v]", "Subsubsection", CellMargins->{{19, Inherited}, {Inherited, Inherited}}, CellLabelMargins->{{27, Inherited}, {Inherited, Inherited}}], Cell[BoxData[{ \(\(Axes3D::"\" = "\";\)\), "\ \n", \(\(Axes3D[u_, v_] := Graphics3D[{{Blue, Line[{{\(-\(u\/3\)\), 0, 0}, {u, 0, 0}}]}, Text["\", {u + v, 0, 0}], {Blue, Line[{{0, \(-\(u\/3\)\), 0}, {0, u, 0}}]}, Text["\", {0, u + v, 0}], {Blue, Line[{{0, 0, \(-\(u\/3\)\)}, {0, 0, u}}]}, Text["\", {0, 0, u + v}]}];\)\), "\n", \(\(Axes3D[u_] := Axes3D[u, u\/8];\)\)}], "Input", InitializationCell->True] }, Closed]], Cell[BoxData[ \(\(CMView\ = \ {2.7, \ 1.6, \ 1.2};\)\)], "Input", InitializationCell->True] }, Closed]], Cell["Basic Problem", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "B.1) Taylor's formula for the expansion of ", Cell[BoxData[ StyleBox[\(f[x]\), FontWeight->"Bold"]]], " \n in powers of ", Cell[BoxData[ StyleBox[\((x\ - \ b)\), FontWeight->"Bold"]]] }], "Subsection", CellTags->"3.04.B1"], Cell["Before you start, run this cell:", "Special2"], Cell[BoxData[ \(\(Clear[Derivative];\)\)], "Input", AspectRatioFixed->True], Cell["B.1.a)", "Subsubsection"], Cell["Look at these:", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, b, x];\)\), "\n", \(Normal[Series[f[x], {x, b, 3}]]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{\(f[b]\), "+", RowBox[{\((\(-b\) + x)\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/2\), " ", \(\((\(-b\) + x)\)\^2\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/6\), " ", \(\((\(-b\) + x)\)\^3\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Normal[Series[f[x], {x, b, 8}]]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{\(f[b]\), "+", RowBox[{\((\(-b\) + x)\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/2\), " ", \(\((\(-b\) + x)\)\^2\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/6\), " ", \(\((\(-b\) + x)\)\^3\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/24\), " ", \(\((\(-b\) + x)\)\^4\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((4)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/120\), " ", \(\((\(-b\) + x)\)\^5\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((5)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/720\), " ", \(\((\(-b\) + x)\)\^6\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((6)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}], "+", FractionBox[ RowBox[{\(\((\(-b\) + x)\)\^7\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((7)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}], "5040"], "+", FractionBox[ RowBox[{\(\((\(-b\) + x)\)\^8\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((8)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}], "40320"]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["What's the message?", "Text"], Cell["Answer:", "Special1"], Cell["Look at some more:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, b, x];\)\), "\n", \(Normal[Series[f[x], {x, b, 2}]]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{\(f[b]\), "+", RowBox[{\((\(-b\) + x)\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/2\), " ", \(\((\(-b\) + x)\)\^2\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "b", "]"}]}]}]], "Output"] }, Open ]], Cell[TextData[{ "This has order of contact ", Cell[BoxData[ \(2\)]], " with ", Cell[BoxData[ \(f[x]\)]], " at ", Cell[BoxData[ \(x = b\)]], "." }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Normal[Series[f[x], {x, b, 6}]]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{\(f[b]\), "+", RowBox[{\((\(-b\) + x)\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/2\), " ", \(\((\(-b\) + x)\)\^2\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/6\), " ", \(\((\(-b\) + x)\)\^3\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/24\), " ", \(\((\(-b\) + x)\)\^4\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((4)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/120\), " ", \(\((\(-b\) + x)\)\^5\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((5)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/720\), " ", \(\((\(-b\) + x)\)\^6\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((6)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}]}]], "Output"] }, Open ]], Cell[TextData[{ "This has order of contact ", Cell[BoxData[ \(6\)]], " with ", Cell[BoxData[ \(f[x]\)]], " at ", Cell[BoxData[ \(x = b\)]], ".\nThe denominators are factorials.\nThe message is that the expansion of \ ", Cell[BoxData[ \(f[x]\)]], " in powers of ", Cell[BoxData[ \(\((x - \ b)\)\)]], " is\n ", Cell[BoxData[ RowBox[{\(f[b]\), " ", "+", " ", RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}], " ", \((x\ - \ b)\)}], " ", "+"}]]], " ", Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "\[Prime]", MultilineFunction->None], "[", "b", "]"}], " ", \(\((x\ - \ b)\)\^2\/\(2!\)\)}], " ", "+", " ", \(\(f\^\([3]\)\)[b]\ \((x\ - \ b)\)\^3\/\(3!\)\), " ", "+"}]]], " \n ", Cell[BoxData[ \(\(\( ... \ \(+\ \(f\^\([k]\)\)[b]\)\)\ \((x\ - \ b)\)\^k\/\(k!\)\ + \)\ ... \)]], "\nMost everyone calls this Taylor's formula.\nThis is worth memorizing." }], "SmallText"] }, Closed]], Cell["B.1.b)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Check out Taylor's formula for the coefficients of the expansion of ", Cell[BoxData[ \(f[x] = e\^x\)]], " in powers of ", Cell[BoxData[ \(\((x - \ 1)\)\)]], "." }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "The expansion of ", Cell[BoxData[ \(e\^x\)]], " in powers of ", Cell[BoxData[ \(\((x - \ 1)\)\)]], " starts out with:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(n = 6;\)\), "\n", \(\(b = 1;\)\), "\n", \(\(Clear[x, f];\)\), "\n", \(\(f[x] = E\^x;\)\n\), "\n", \(Normal[Series[f[x], {x, b, n}]]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\[ExponentialE] + \[ExponentialE]\ \((\(-1\) + x)\) + 1\/2\ \[ExponentialE]\ \((\(-1\) + x)\)\^2 + 1\/6\ \[ExponentialE]\ \((\(-1\) + x)\)\^3 + 1\/24\ \[ExponentialE]\ \((\(-1\) + x)\)\^4 + 1\/120\ \[ExponentialE]\ \((\(-1\) + x)\)\^5 + 1\/720\ \[ExponentialE]\ \((\(-1\) + x)\)\^6\)], "Output"] }, Open ]], Cell[TextData[{ "Taylor's formula for the expansion of ", Cell[BoxData[ \(f[x]\)]], " in powers of ", Cell[BoxData[ \(\((x - \ 1)\)\)]], " is\n ", Cell[BoxData[ RowBox[{\(f[b]\), " ", "+", " ", RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}], " ", \((x\ - \ b)\)}], " ", "+"}]]], " ", Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "\[Prime]", MultilineFunction->None], "[", "b", "]"}], " ", \(\((x\ - \ b)\)\^2\/\(2!\)\)}], " ", "+", " ", \(\(f\^\([3]\)\)[b]\ \((x\ - \ b)\)\^3\/\(3!\)\), " ", "+"}]]], " \n ", Cell[BoxData[ \(\(\( ... \ \(+\ \(f\^\([k]\)\)[b]\)\)\ \((x\ - \ b)\)\^k\/\(k!\)\ + \)\ ... \)]], "\nSo Taylor's formula gives you:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[a, k];\)\), "\n", \(a[k_]\ := \ \(\(\((D[f[x], {x, k}] /. x -> b)\)/\(k!\)\)\(\n\)\)\), "\n", \(Sum[a[k]\ \((x - b)\)\^k, {k, 0, n}]\)}], "Input"], Cell[BoxData[ \(\[ExponentialE] + \[ExponentialE]\ \((\(-1\) + x)\) + 1\/2\ \[ExponentialE]\ \((\(-1\) + x)\)\^2 + 1\/6\ \[ExponentialE]\ \((\(-1\) + x)\)\^3 + 1\/24\ \[ExponentialE]\ \((\(-1\) + x)\)\^4 + 1\/120\ \[ExponentialE]\ \((\(-1\) + x)\)\^5 + 1\/720\ \[ExponentialE]\ \((\(-1\) + x)\)\^6\)], "Output"] }, Open ]], Cell["\<\ Looks good. Do it again:\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(n = 12;\)\), "\n", \(Normal[Series[f[x], {x, b, n}]]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\[ExponentialE] + \[ExponentialE]\ \((\(-1\) + x)\) + 1\/2\ \[ExponentialE]\ \((\(-1\) + x)\)\^2 + 1\/6\ \[ExponentialE]\ \((\(-1\) + x)\)\^3 + 1\/24\ \[ExponentialE]\ \((\(-1\) + x)\)\^4 + 1\/120\ \[ExponentialE]\ \((\(-1\) + x)\)\^5 + 1\/720\ \[ExponentialE]\ \((\(-1\) + x)\)\^6 + \(\[ExponentialE]\ \ \((\(-1\) + x)\)\^7\)\/5040 + \(\[ExponentialE]\ \((\(-1\) + x)\)\^8\)\/40320 \ + \(\[ExponentialE]\ \((\(-1\) + x)\)\^9\)\/362880 + \(\[ExponentialE]\ \ \((\(-1\) + x)\)\^10\)\/3628800 + \(\[ExponentialE]\ \((\(-1\) + \ x)\)\^11\)\/39916800 + \(\[ExponentialE]\ \((\(-1\) + \ x)\)\^12\)\/479001600\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\n\)\(Sum[a[k]\ \((x - 1)\)\^k, {k, 0, n}]\)\)\)], "Input"], Cell[BoxData[ \(\[ExponentialE] + \[ExponentialE]\ \((\(-1\) + x)\) + 1\/2\ \[ExponentialE]\ \((\(-1\) + x)\)\^2 + 1\/6\ \[ExponentialE]\ \((\(-1\) + x)\)\^3 + 1\/24\ \[ExponentialE]\ \((\(-1\) + x)\)\^4 + 1\/120\ \[ExponentialE]\ \((\(-1\) + x)\)\^5 + 1\/720\ \[ExponentialE]\ \((\(-1\) + x)\)\^6 + \(\[ExponentialE]\ \ \((\(-1\) + x)\)\^7\)\/5040 + \(\[ExponentialE]\ \((\(-1\) + x)\)\^8\)\/40320 \ + \(\[ExponentialE]\ \((\(-1\) + x)\)\^9\)\/362880 + \(\[ExponentialE]\ \ \((\(-1\) + x)\)\^10\)\/3628800 + \(\[ExponentialE]\ \((\(-1\) + \ x)\)\^11\)\/39916800 + \(\[ExponentialE]\ \((\(-1\) + \ x)\)\^12\)\/479001600\)], "Output"] }, Open ]], Cell[TextData[{ "Play with these by going back to the beginning and changing ", Cell[BoxData[ \(f[x]\)]], ", ", Cell[BoxData[ \(b\)]], ", and ", Cell[BoxData[ \(n\)]], ". " }], "SmallText"] }, Closed]], Cell["B.1.c.i)", "Subsubsection"], Cell[CellGroupData[{ Cell["What use is Taylor's formula?", "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Many new practitioners of calculus want to jump on this formula and use it \ to slam out all expansions. But those who have been around for a while know \ this is not a good idea even if you are as fast as ", StyleBox["Mathematica", FontSlant->"Italic"], ". As a matter of fact, this formula is usually the least efficient way to \ obtain an expansion of a given function. Just think of the misery involved in \ calculating many derivatives and plugging in. " }], "SmallText"], Cell["\<\ Pity those students in the traditional course because they spend lots of their valuable time calculating many derivatives \ and plugging in.\ \>", "Special2"], Cell[TextData[{ "Just to use Taylor's formula to calculate the first ", Cell[BoxData[ \(10\)]], " terms of the expansion of\n ", Cell[BoxData[ \(f[x] = e\^x\/\(1\ - \ x\)\)]], " \nin powers of ", Cell[BoxData[ \(x\)]], ", you've got to calculate this bird's nest of derivatives:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(\(f[x_]\ = \ E\^x\/\(1 - x\);\)\n\), "\n", \(Table[D[f[x], {x, k}]/\(k!\), {k, 0, 10}]\)}], "Input"], Cell[BoxData[ \({\[ExponentialE]\^x\/\(1 - x\), \[ExponentialE]\^x\/\((1 - x)\)\^2 + \ \[ExponentialE]\^x\/\(1 - x\), 1\/2\ \((\(2\ \[ExponentialE]\^x\)\/\((1 - x)\)\^3 + \(2\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^2 + \[ExponentialE]\^x\/\(1 - x\))\), 1\/6\ \((\(6\ \[ExponentialE]\^x\)\/\((1 - x)\)\^4 + \(6\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^3 + \(3\ \[ExponentialE]\^x\)\/\((1 - x)\)\ \^2 + \[ExponentialE]\^x\/\(1 - x\))\), 1\/24\ \((\(24\ \[ExponentialE]\^x\)\/\((1 - x)\)\^5 + \(24\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^4 + \(12\ \[ExponentialE]\^x\)\/\((1 - \ x)\)\^3 + \(4\ \[ExponentialE]\^x\)\/\((1 - x)\)\^2 + \[ExponentialE]\^x\/\(1 \ - x\))\), 1\/120\ \((\(120\ \[ExponentialE]\^x\)\/\((1 - x)\)\^6 + \(120\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^5 + \(60\ \[ExponentialE]\^x\)\/\((1 - \ x)\)\^4 + \(20\ \[ExponentialE]\^x\)\/\((1 - x)\)\^3 + \(5\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^2 + \[ExponentialE]\^x\/\(1 - x\))\), 1\/720\ \((\(720\ \[ExponentialE]\^x\)\/\((1 - x)\)\^7 + \(720\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^6 + \(360\ \[ExponentialE]\^x\)\/\((1 - x)\ \)\^5 + \(120\ \[ExponentialE]\^x\)\/\((1 - x)\)\^4 + \(30\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^3 + \(6\ \[ExponentialE]\^x\)\/\((1 - x)\)\ \^2 + \[ExponentialE]\^x\/\(1 - x\))\), \(\(5040\ \[ExponentialE]\^x\)\/\((1 \ - x)\)\^8 + \(5040\ \[ExponentialE]\^x\)\/\((1 - x)\)\^7 + \(2520\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^6 + \(840\ \[ExponentialE]\^x\)\/\((1 - x)\ \)\^5 + \(210\ \[ExponentialE]\^x\)\/\((1 - x)\)\^4 + \(42\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^3 + \(7\ \[ExponentialE]\^x\)\/\((1 - x)\)\ \^2 + \[ExponentialE]\^x\/\(1 - x\)\)\/5040, \(\(1\/40320\)\((\(40320\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^9 + \(40320\ \[ExponentialE]\^x\)\/\((1 - \ x)\)\^8 + \(20160\ \[ExponentialE]\^x\)\/\((1 - x)\)\^7 + \(6720\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^6 + \(1680\ \[ExponentialE]\^x\)\/\((1 - \ x)\)\^5 + \(336\ \[ExponentialE]\^x\)\/\((1 - x)\)\^4 + \(56\ \[ExponentialE]\ \^x\)\/\((1 - x)\)\^3 + \(8\ \[ExponentialE]\^x\)\/\((1 - x)\)\^2 + \ \[ExponentialE]\^x\/\(1 - x\))\)\), \(\(1\/362880\)\((\(362880\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^10 + \(362880\ \[ExponentialE]\^x\)\/\((1 \ - x)\)\^9 + \(181440\ \[ExponentialE]\^x\)\/\((1 - x)\)\^8 + \(60480\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^7 + \(15120\ \[ExponentialE]\^x\)\/\((1 - \ x)\)\^6 + \(3024\ \[ExponentialE]\^x\)\/\((1 - x)\)\^5 + \(504\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^4 + \(72\ \[ExponentialE]\^x\)\/\((1 - \ x)\)\^3 + \(9\ \[ExponentialE]\^x\)\/\((1 - x)\)\^2 + \[ExponentialE]\^x\/\(1 \ - x\))\)\), \(\(1\/3628800\)\((\(3628800\ \[ExponentialE]\^x\)\/\((1 - \ x)\)\^11 + \(3628800\ \[ExponentialE]\^x\)\/\((1 - x)\)\^10 + \(1814400\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^9 + \(604800\ \[ExponentialE]\^x\)\/\((1 - \ x)\)\^8 + \(151200\ \[ExponentialE]\^x\)\/\((1 - x)\)\^7 + \(30240\ \ \[ExponentialE]\^x\)\/\((1 - x)\)\^6 + \(5040\ \[ExponentialE]\^x\)\/\((1 - \ x)\)\^5 + \(720\ \[ExponentialE]\^x\)\/\((1 - x)\)\^4 + \(90\ \[ExponentialE]\ \^x\)\/\((1 - x)\)\^3 + \(10\ \[ExponentialE]\^x\)\/\((1 - x)\)\^2 + \ \[ExponentialE]\^x\/\(1 - x\))\)\)}\)], "Output"] }, Open ]], Cell[TextData[{ "To get the expansion of ", Cell[BoxData[ \(f[x] = e\^x\/\(1\ - \ x\)\)]], " in powers of ", Cell[BoxData[ \(x\)]], " through the ", Cell[BoxData[ \(x\^10\)]], " term, the experienced practitioner of calculus takes:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(first = Series[E\^x, {x, 0, 10}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "+", "x", "+", \(x\^2\/2\), "+", \(x\^3\/6\), "+", \(x\^4\/24\), "+", \(x\^5\/120\), "+", \(x\^6\/720\), "+", \(x\^7\/5040\), "+", \(x\^8\/40320\), "+", \(x\^9\/362880\), "+", \(x\^10\/3628800\), "+", InterpretationBox[\(O[x]\^11\), SeriesData[ x, 0, {}, 0, 11, 1]]}], SeriesData[ x, 0, {1, 1, Rational[ 1, 2], Rational[ 1, 6], Rational[ 1, 24], Rational[ 1, 120], Rational[ 1, 720], Rational[ 1, 5040], Rational[ 1, 40320], Rational[ 1, 362880], Rational[ 1, 3628800]}, 0, 11, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(second = Series[1\/\(1 - x\), {x, 0, 10}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "+", "x", "+", \(x\^2\), "+", \(x\^3\), "+", \(x\^4\), "+", \(x\^5\), "+", \(x\^6\), "+", \(x\^7\), "+", \(x\^8\), "+", \(x\^9\), "+", \(x\^10\), "+", InterpretationBox[\(O[x]\^11\), SeriesData[ x, 0, {}, 0, 11, 1]]}], SeriesData[ x, 0, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, 0, 11, 1]]], "Output"] }, Open ]], Cell["And multiplies them together:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(first\ second\)], "Input", AspectRatioFixed->True], Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "+", \(2\ x\), "+", \(\(5\ x\^2\)\/2\), "+", \(\(8\ x\^3\)\/3\), "+", \(\(65\ x\^4\)\/24\), "+", \(\(163\ x\^5\)\/60\), "+", \(\(1957\ x\^6\)\/720\), "+", \(\(685\ x\^7\)\/252\), "+", \(\(109601\ x\^8\)\/40320\), "+", \(\(98641\ x\^9\)\/36288\), "+", \(\(9864101\ x\^10\)\/3628800\), "+", InterpretationBox[\(O[x]\^11\), SeriesData[ x, 0, {}, 0, 11, 1]]}], SeriesData[ x, 0, {1, 2, Rational[ 5, 2], Rational[ 8, 3], Rational[ 65, 24], Rational[ 163, 60], Rational[ 1957, 720], Rational[ 685, 252], Rational[ 109601, 40320], Rational[ 98641, 36288], Rational[ 9864101, 3628800]}, 0, 11, 1]]], "Output"] }, Open ]], Cell["\<\ Although Taylor's formula is not very useful for down-to-earth \ computations, it does have quite a bit of theoretical value. And since theory \ feeds new calculations and measurements, it has some practical value as well. You'll see it again in this lesson.\ \>", "SmallText"] }, Closed]], Cell["B.1.c.ii)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Does ", StyleBox["Mathematica", FontSlant->"Italic"], " use Taylor's formula to come up with expansions?" }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Only as a last resort. \nLike most experienced scientists, ", StyleBox["Mathematica", FontSlant->"Italic"], " tries to get its expansions the way you have been doing it:\nMemorize \ basic expansions like the expansions of\n ", Cell[BoxData[ \(1\/\(1\ - \ x\)\)]], ", ", Cell[BoxData[ \(Sin[x]\)]], ", ", Cell[BoxData[ \(Cos[x]\)]], ", and ", Cell[BoxData[ \(e\^x\)]], " \nin powers of ", Cell[BoxData[ \(x\)]], " and then do little adjustments to get the expansions you want." }], "SmallText"] }, Closed]], Cell["B.1.d)", "Subsubsection"], Cell[CellGroupData[{ Cell["\<\ How does Taylor's formula guarantee that it is O.K. to get \ expansions by devices other than Taylor's formula?\ \>", "Text"], Cell["Answer:", "Special1"], Cell["\<\ This is a good example of the use of Taylor's formula as a \ theoretical tool. Taylor's formula tells you that there is a definite formula for expansions. \ \ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f];\)\), "\n", \(Series[f[x], {x, 0, 7}]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ InterpretationBox[ RowBox[{\(f[0]\), "+", RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "0", "]"}], " ", "x"}], "+", RowBox[{\(1\/2\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "0", "]"}], " ", \(x\^2\)}], "+", RowBox[{\(1\/6\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "0", "]"}], " ", \(x\^3\)}], "+", RowBox[{\(1\/24\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((4)\), Derivative], MultilineFunction->None], "[", "0", "]"}], " ", \(x\^4\)}], "+", RowBox[{\(1\/120\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((5)\), Derivative], MultilineFunction->None], "[", "0", "]"}], " ", \(x\^5\)}], "+", RowBox[{\(1\/720\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((6)\), Derivative], MultilineFunction->None], "[", "0", "]"}], " ", \(x\^6\)}], "+", FractionBox[ RowBox[{ RowBox[{ SuperscriptBox["f", TagBox[\((7)\), Derivative], MultilineFunction->None], "[", "0", "]"}], " ", \(x\^7\)}], "5040"], "+", InterpretationBox[\(O[x]\^8\), SeriesData[ x, 0, {}, 0, 8, 1]]}], SeriesData[ x, 0, { f[ 0], Derivative[ 1][ f][ 0], Times[ Rational[ 1, 2], Derivative[ 2][ f][ 0]], Times[ Rational[ 1, 6], Derivative[ 3][ f][ 0]], Times[ Rational[ 1, 24], Derivative[ 4][ f][ 0]], Times[ Rational[ 1, 120], Derivative[ 5][ f][ 0]], Times[ Rational[ 1, 720], Derivative[ 6][ f][ 0]], Times[ Rational[ 1, 5040], Derivative[ 7][ f][ 0]]}, 0, 8, 1]]], "Output"] }, Open ]], Cell[TextData[{ "The fact that there is such a formula means that there is only one \ possibility for an expansion in powers of ", Cell[BoxData[ \(x\)]], ". As a result, you always get the same expansion for a given function no \ matter how you go about it.\nSo get it any way you can!" }], "SmallText"] }, Closed]] }, Closed]], Cell["Tutorial Problem", "Subsubsection"], Cell[CellGroupData[{ Cell["T.1) Taylor's formula in reverse", "Subsection", CellTags->"3.04.T1"], Cell["Before you start, run this cell:", "Special2"], Cell[BoxData[ \(\(Clear[Derivative];\)\)], "Input", AspectRatioFixed->True], Cell["T.1.a)", "Subsubsection"], Cell[TextData[{ "Here is the expansion in powers of ", Cell[BoxData[ \(x\)]], " of\n ", Cell[BoxData[ \(f[x] = 1\/\(1\ - \ x\ - \ x\^2\)\)]], "\nthrough the ", Cell[BoxData[ \(x\^12\)]], " term:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, fexpan12];\)\), "\n", \(\(f[x_] = 1\/\(1 - x - x\^2\);\)\n\), "\n", \(expan12[x_] = Normal[Series[f[x], {x, 0, 12}]]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1 + x + 2\ x\^2 + 3\ x\^3 + 5\ x\^4 + 8\ x\^5 + 13\ x\^6 + 21\ x\^7 + 34\ x\^8 + 55\ x\^9 + 89\ x\^10 + 144\ x\^11 + 233\ x\^12\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Use this partial expansion to come up with a table of the values of the \ derivatives \n ", Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "0", "]"}], ",", " ", RowBox[{ SuperscriptBox[ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "\[Prime]", MultilineFunction->None], "[", "0", "]"}], ",", " ", RowBox[{ SuperscriptBox[ SuperscriptBox[ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "\[Prime]", MultilineFunction->None], "\[Prime]", MultilineFunction->None], "[", "0", "]"}], ",", " ", "...", " ", ",", " ", \(\(f\^\((11)\)\)[0]\), ",", \(\(f\^\((12)\)\)[0]\)}], "}"}]]], "." }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Taylor's formula says that the expansion of ", Cell[BoxData[ \(f[x]\)]], " in powers of ", Cell[BoxData[ \(\((x\ - \ b)\)\)]], " is\n ", Cell[BoxData[ RowBox[{\(f[b]\), " ", "+", " ", RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}], " ", \((x\ - \ b)\)}], " ", "+"}]]], " ", Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{ RowBox[{ SuperscriptBox[ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "\[Prime]", MultilineFunction->None], "[", "b", "]"}], " ", \(\((x\ - \ b)\)\^2\)}], \(2!\)], " ", "+", " ", FractionBox[ RowBox[{ RowBox[{ SuperscriptBox["f", \((3)\), MultilineFunction->None], "[", "b", "]"}], " ", \(\((x\ - \ b)\)\^3\)}], \(3!\)], " ", "+"}]]], " \n ", Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"...", " ", RowBox[{"+", " ", FractionBox[ RowBox[{ RowBox[{ SuperscriptBox["f", \([k]\), MultilineFunction->None], "[", "b", "]"}], " ", \(\((x\ - \ b)\)\^k\)}], \(k!\)]}]}], " ", "+"}], " ", "..."}]]], "\n Here you are dealing with ", Cell[BoxData[ \(b = 0\)]], ". So the table \n", " ", Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "0", "]"}], ",", " ", RowBox[{ SuperscriptBox[ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "\[Prime]", MultilineFunction->None], "[", "0", "]"}], ",", " ", RowBox[{ SuperscriptBox[ SuperscriptBox[ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "\[Prime]", MultilineFunction->None], "\[Prime]", MultilineFunction->None], "[", "0", "]"}], ",", " ", "...", " ", ",", " ", \(\(f\^\((11)\)\)[0]\), ",", \(\(f\^\((12)\)\)[0]\)}], "}"}]]], "\n is:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[k];\)\), "\n", \(Table[\(k!\)\ Coefficient[expan12[x], x\^k], {k, 1, 12}]\)}], "Input"], Cell[BoxData[ \({1, 4, 18, 120, 960, 9360, 105840, 1370880, 19958400, 322963200, 5748019200, 111607372800}\)], "Output"] }, Open ]], Cell["Check:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[D[f[x], {x, k}] /. x -> 0, {k, 1, 12}]\)], "Input"], Cell[BoxData[ \({1, 4, 18, 120, 960, 9360, 105840, 1370880, 19958400, 322963200, 5748019200, 111607372800}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " had to grind on the last one, because along the way it had to do some \ hairy stuff. " }], "SmallText"] }, Closed]] }, Closed]], Cell["Give It a Try Problems", "Subsubsection"], Cell[CellGroupData[{ Cell["G.2) Limits, Taylor's formula and L'Hospital's rule*", "Subsection", CellTags->"3.04.G2"], Cell["Before you start, run this cell:", "Special2"], Cell[BoxData[ \(\(Clear[Derivative];\)\)], "Input", AspectRatioFixed->True], Cell["G.2.a.i)", "Subsubsection"], Cell[TextData[{ "All you know about a certain function ", Cell[BoxData[ \(f[x]\)]], " is that ", Cell[BoxData[ \(f[3]\ = \ 0\)]], " and ", Cell[BoxData[ \(\(f'\)[3]\ = \ 9\)]], ". \nCalculate \n ", Cell[BoxData[ \(\(\(lim\_\(x \[Rule] 3\)\)\(\(x\^3\ - \ 3\^3\)\/f[x]\)\)\)]], ". " }], "Text"], Cell["G.2.a.ii)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "All you know about a certain function ", Cell[BoxData[ \(f[x]\)]], " is that \n ", Cell[BoxData[ \(\(f'\)[x]\ = \ Cos[f[x]]\ + \ 4\ x\)]], " and ", Cell[BoxData[ \(f[1]\ = \ 0\)]], ".\nCalculate \n ", Cell[BoxData[ \(\(\(lim\_\(x \[Rule] 1\)\)\(f[x]\/Cos[\(\[Pi]\ x\)\/2]\)\)\)]], ". " }], "Text"], Cell["Tip:", "Special1"], Cell[TextData[{ "You have enough info to find out what ", Cell[BoxData[ \(\(f'\)[1]\)]], " is." }], "SmallText"] }, Closed]], Cell["G.2.b.i) Taylor's formula and L'Hospital's rule", "Subsubsection"], Cell[TextData[{ "Take two functions ", Cell[BoxData[ \(f[x]\)]], " and ", Cell[BoxData[ \(g[x]\)]], " such that \n\[Rule] ", Cell[BoxData[ \(f[b] = 0\)]], " and ", Cell[BoxData[ \(g[b] = 0\)]], " and \n\[Rule] ", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["g", "\[Prime]", MultilineFunction->None], "[", "b", "]"}], "\[NotEqual]", "0"}]]], " \nand look at:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, g, x, b, Derivative];\)\), "\n", \(Normal[Series[f[x], {x, b, 2}]] /. f[b] \[Rule] 0\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{\((\(-b\) + x)\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/2\), " ", \(\((\(-b\) + x)\)\^2\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "b", "]"}]}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Normal[Series[g[x], {x, b, 2}]] /. g[b] \[Rule] 0\)], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{\((\(-b\) + x)\), " ", RowBox[{ SuperscriptBox["g", "\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/2\), " ", \(\((\(-b\) + x)\)\^2\), " ", RowBox[{ SuperscriptBox["g", "\[Prime]\[Prime]", MultilineFunction->None], "[", "b", "]"}]}]}]], "Output"] }, Open ]], Cell[TextData[{ "How does this tell you that you can read off\n ", Cell[BoxData[ RowBox[{\(lim\+\(x\ \[Rule] \ b\)\ f[x]\/g[x]\), "=", FractionBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}], RowBox[{ SuperscriptBox["g", "\[Prime]", MultilineFunction->None], "[", "b", "]"}]]}]]], "?\nSome folks call this formula L'Hospital's Rule." }], "Text"], Cell["G.2.b.ii)", "Subsubsection"], Cell[TextData[{ "Take two functions ", Cell[BoxData[ \(f[x]\)]], " and ", Cell[BoxData[ \(g[x]\)]], " such that \n\[Rule] ", Cell[BoxData[ \(f[b] = 0\)]], " and ", Cell[BoxData[ \(g[b] = 0\)]], " and\n\[Rule] ", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}], "=", "0"}]]], " and ", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["g", "\[Prime]", MultilineFunction->None], "[", "b", "]"}], "=", "0"}]]], " and \n\[Rule] ", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox[ SuperscriptBox["g", "\[Prime]", MultilineFunction->None], "\[Prime]", MultilineFunction->None], "[", "b", "]"}], "\[NotEqual]", "0"}]]], " \nand look at:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, g, x, b, Derivative];\)\), "\n", \(Normal[ Series[f[x], {x, b, 3}]] /. {f[b] \[Rule] 0, \(f'\)[b] \[Rule] 0}\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{\(1\/2\), " ", \(\((\(-b\) + x)\)\^2\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/6\), " ", \(\((\(-b\) + x)\)\^3\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Normal[ Series[g[x], {x, b, 3}]] /. {g[b] \[Rule] 0, \(g'\)[b] \[Rule] 0}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{\(1\/2\), " ", \(\((\(-b\) + x)\)\^2\), " ", RowBox[{ SuperscriptBox["g", "\[Prime]\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/6\), " ", \(\((\(-b\) + x)\)\^3\), " ", RowBox[{ SuperscriptBox["g", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}]}]], "Output"] }, Open ]], Cell[TextData[{ "How does this tell you that you can read off\n ", Cell[BoxData[ RowBox[{\(lim\+\(x\ \[Rule] \ b\)\ f[x]\/g[x]\), "=", FractionBox[ RowBox[{ SuperscriptBox[ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "\[Prime]", MultilineFunction->None], "[", "b", "]"}], RowBox[{ SuperscriptBox[ SuperscriptBox["g", "\[Prime]", MultilineFunction->None], "\[Prime]", MultilineFunction->None], "[", "b", "]"}]]}]]], "?\nSome folks call this formula L'Hospital's Rule." }], "Text"], Cell["G.2.b.iii)", "Subsubsection"], Cell[TextData[{ "Take two functions ", Cell[BoxData[ \(f[x]\)]], " and ", Cell[BoxData[ \(g[x]\)]], " such that \n\[Rule] ", Cell[BoxData[ \(f[b]\ = \ 0\)]], " and ", Cell[BoxData[ \(g[b]\ = \ 0\)]], " and\n\[Rule] ", Cell[BoxData[ \(\(f'\)[b]\ = \ 0\)]], " and ", Cell[BoxData[ \(\(g'\)[b]\ = \ 0\)]], " and \n\[Rule] ", Cell[BoxData[ \(\(\(\ \)\(\(\(f'\)'\)[b]\ = \ 0\)\)\)]], " and ", Cell[BoxData[ \(\(\(g'\)'\)[b]\ = \ 0\)]], " and\n\[Rule] ", Cell[BoxData[ \(\(\(\(g'\)'\)'\)[b]\ \[NotEqual] \ 0\)]], " \nGive a formula for\n ", Cell[BoxData[ \(lim\_\(x \[Rule] b\)\ f[x]\/g[x]\)]], ",\nand explain where you got it.\nSome folks call this formula \ L'Hospital's Rule." }], "Text"], Cell["G.2.b.iv)", "Subsubsection"], Cell["\<\ Professor Lipman Bers, of Columbia University and past president of \ the American Mathematical Society, once said that L'Hospital's rule impressed \ the early practitioners of calculus, but the importance of L'Hospital's rule \ in contemporary mathematics is not overwhelming. Why is L'Hospital's rule no big deal if you know Taylor's formula?\ \>", \ "Text"] }, Closed]], Cell[CellGroupData[{ Cell["G.7) Midpoint versus trapezoidal approximation", "Subsection", CellTags->"3.04.G7"], Cell["Before you start, run this cell:", "Special2"], Cell[BoxData[ \(\(Clear[Derivative];\)\)], "Input", AspectRatioFixed->True], Cell["G.7.a)", "Subsubsection"], Cell[TextData[{ "Here are the trapezoidal and the midpoint approximations of ", Cell[BoxData[ \(f[x]\)]], " near a point ", Cell[BoxData[ \(b\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[ftrap, f, x, b, Derivative];\)\), "\n", \(ftrap[x_, b_] = f[b] + 1\/2\ \((x - b)\)\ \((\(f'\)[b] + \(f'\)[x])\)\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{\(f[b]\), "+", RowBox[{\(1\/2\), " ", \((\(-b\) + x)\), " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}], "+", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], ")"}]}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[fmidpt, f, x, b];\)\), "\n", \(fmidpt[x_, b_] = f[b] + \((x - b)\)\ \(f'\)[\(b + x\)\/2]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{\(f[b]\), "+", RowBox[{\((\(-b\) + x)\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", \(\(b + x\)\/2\), "]"}]}]}]], "Output"] }, Open ]], Cell[TextData[{ "Check them out in the case that ", Cell[BoxData[ \(f[x] = Cos[x]\)]], " and ", Cell[BoxData[ \(b = 0\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(f[x_] = Cos[x];\)\), "\n", \(\(b = 0;\)\n\), "\n", \(\(Plot[{f[x], fmidpt[x, b], ftrap[x, b]}, {x, b - 2, b + 2}, \n\t PlotStyle \[Rule] {{Thickness[0.02], Blue}, {Thickness[0.01], Red}, {Thickness[0.005], Brown}}, AspectRatio \[Rule] 1\/GoldenRatio, AxesLabel \[Rule] {"\", "\<\>"}];\)\)}], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.238095 0.253572 0.349747 [ [.02381 .24107 -6 -9 ] [.02381 .24107 6 0 ] [.2619 .24107 -6 -9 ] [.2619 .24107 6 0 ] [.7381 .24107 -3 -9 ] [.7381 .24107 3 0 ] [.97619 .24107 -3 -9 ] [.97619 .24107 3 0 ] [1.025 .25357 0 -6 ] [1.025 .25357 10 6 ] [.4875 .0787 -24 -4.5 ] [.4875 .0787 0 4.5 ] [.4875 .16614 -30 -4.5 ] [.4875 .16614 0 4.5 ] [.4875 .34101 -24 -4.5 ] [.4875 .34101 0 4.5 ] [.4875 .42845 -18 -4.5 ] [.4875 .42845 0 4.5 ] [.4875 .51588 -24 -4.5 ] [.4875 .51588 0 4.5 ] [.4875 .60332 -6 -4.5 ] [.4875 .60332 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .25357 m .02381 .25982 L s [(-2)] .02381 .24107 0 1 Mshowa .2619 .25357 m .2619 .25982 L s [(-1)] .2619 .24107 0 1 Mshowa .7381 .25357 m .7381 .25982 L s [(1)] .7381 .24107 0 1 Mshowa .97619 .25357 m .97619 .25982 L s [(2)] .97619 .24107 0 1 Mshowa .125 Mabswid .07143 .25357 m .07143 .25732 L s .11905 .25357 m .11905 .25732 L s .16667 .25357 m .16667 .25732 L s .21429 .25357 m .21429 .25732 L s .30952 .25357 m .30952 .25732 L s .35714 .25357 m .35714 .25732 L s .40476 .25357 m .40476 .25732 L s .45238 .25357 m .45238 .25732 L s .54762 .25357 m .54762 .25732 L s .59524 .25357 m .59524 .25732 L s .64286 .25357 m .64286 .25732 L s .69048 .25357 m .69048 .25732 L s .78571 .25357 m .78571 .25732 L s .83333 .25357 m .83333 .25732 L s .88095 .25357 m .88095 .25732 L s .92857 .25357 m .92857 .25732 L s .25 Mabswid 0 .25357 m 1 .25357 L s gsave 1.025 .25357 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (x) show 69.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 1.000 setlinewidth grestore .5 .0787 m .50625 .0787 L s [(-0.5)] .4875 .0787 1 0 Mshowa .5 .16614 m .50625 .16614 L s [(-0.25)] .4875 .16614 1 0 Mshowa .5 .34101 m .50625 .34101 L s [(0.25)] .4875 .34101 1 0 Mshowa .5 .42845 m .50625 .42845 L s [(0.5)] .4875 .42845 1 0 Mshowa .5 .51588 m .50625 .51588 L s [(0.75)] .4875 .51588 1 0 Mshowa .5 .60332 m .50625 .60332 L s [(1)] .4875 .60332 1 0 Mshowa .125 Mabswid .5 .09619 m .50375 .09619 L s .5 .11367 m .50375 .11367 L s .5 .13116 m .50375 .13116 L s .5 .14865 m .50375 .14865 L s .5 .18362 m .50375 .18362 L s .5 .20111 m .50375 .20111 L s .5 .2186 m .50375 .2186 L s .5 .23608 m .50375 .23608 L s .5 .27106 m .50375 .27106 L s .5 .28855 m .50375 .28855 L s .5 .30603 m .50375 .30603 L s .5 .32352 m .50375 .32352 L s .5 .3585 m .50375 .3585 L s .5 .37598 m .50375 .37598 L s .5 .39347 m .50375 .39347 L s .5 .41096 m .50375 .41096 L s .5 .44593 m .50375 .44593 L s .5 .46342 m .50375 .46342 L s .5 .48091 m .50375 .48091 L s .5 .49839 m .50375 .49839 L s .5 .53337 m .50375 .53337 L s .5 .55086 m .50375 .55086 L s .5 .56834 m .50375 .56834 L s .5 .58583 m .50375 .58583 L s .5 .06121 m .50375 .06121 L s .5 .04372 m .50375 .04372 L s .5 .02624 m .50375 .02624 L s .5 .00875 m .50375 .00875 L s .25 Mabswid .5 0 m .5 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 0 1 r .02 w .02381 .10803 m .06244 .16132 L .10458 .22215 L .14415 .28021 L .18221 .33537 L .22272 .39177 L .26171 .4423 L .30316 .49045 L .34309 .53007 L .3815 .56089 L .40095 .57349 L .42237 .58489 L .44268 .59323 L .45178 .59617 L .46172 .59881 L .4671 .59999 L .4721 .60092 L .47727 .60173 L .48196 .60232 L .48658 .60276 L .4887 .60292 L .49093 .60307 L .49332 .60318 L .49438 .60322 L .49552 .60326 L .49675 .60329 L .49789 .60331 L .49859 .60331 L .49925 .60332 L .50049 .60332 L .50163 .60331 L .50286 .60329 L .50401 .60327 L .50508 .60324 L .50754 .60314 L .51014 .603 L .51268 .60282 L .51504 .60262 L .5204 .60204 L .5293 .60067 L .53882 .59868 L .54906 .59592 L .56016 .59221 L .58032 .58361 L .62123 .55895 L .66064 .52669 L .69852 .48862 L .73886 .44159 L .77769 .39122 L .81897 .3337 L Mistroke .85873 .27599 L .89697 .21988 L .93767 .16116 L .97619 .10803 L Mfstroke 1 0 0 r .01 w .02381 .01472 m .06244 .09239 L .10458 .17454 L .14415 .24805 L .18221 .3144 L .22272 .37932 L .26171 .43538 L .30316 .48717 L .34309 .52873 L .3815 .56045 L .40095 .57327 L .42237 .58481 L .44268 .59321 L .45178 .59616 L .46172 .5988 L .4671 .59998 L .4721 .60092 L .47727 .60173 L .48196 .60232 L .48658 .60276 L .4887 .60292 L .49093 .60307 L .49332 .60318 L .49438 .60322 L .49552 .60326 L .49675 .60329 L .49789 .60331 L .49859 .60331 L .49925 .60332 L .50049 .60332 L .50163 .60331 L .50286 .60329 L .50401 .60327 L .50508 .60324 L .50754 .60314 L .51014 .603 L .51268 .60282 L .51504 .60262 L .5204 .60204 L .5293 .60067 L .53882 .59868 L .54906 .59591 L .56016 .59218 L .58032 .58351 L .60019 .57258 L .62123 .55847 L .65912 .52666 L .69946 .48415 L .73829 .43538 L .77956 .37584 L Mistroke .81932 .31182 L .85757 .24495 L .89827 .16911 L .93745 .09261 L .97619 .01472 L Mfstroke .5 .165 .165 r .005 w .02381 .28529 m .02605 .28544 L .02846 .28565 L .03279 .28613 L .03793 .2869 L .04262 .28778 L .05281 .29027 L .06244 .29333 L .08265 .30185 L .10458 .31407 L .14545 .34378 L .1848 .37884 L .22263 .41615 L .26292 .4572 L .30169 .49555 L .34292 .5326 L .38262 .56252 L .42082 .58433 L .44049 .59251 L .45141 .59609 L .46146 .59876 L .47175 .60086 L .47716 .60171 L .48287 .60241 L .48527 .60265 L .48785 .60286 L .49007 .60301 L .49249 .60314 L .49396 .60321 L .49533 .60325 L .49663 .60328 L .49729 .6033 L .49801 .60331 L .49926 .60332 L .49996 .60332 L .50061 .60332 L .5018 .60331 L .50305 .60329 L .50428 .60326 L .50562 .60322 L .50805 .60312 L .51041 .60298 L .51259 .60283 L .51754 .60237 L .52279 .60172 L .53153 .60026 L .54115 .59812 L .5606 .59211 L .58149 .58323 L .62277 .55886 L Mistroke .66253 .52801 L .70078 .49319 L .74147 .45274 L .78066 .41283 L .82229 .37217 L .86241 .33746 L .90101 .31068 L .92099 .30012 L .93117 .29572 L .94207 .29181 L .95131 .28917 L .95973 .28732 L .96378 .28662 L .96819 .28601 L .9702 .28578 L .97235 .28557 L .97619 .28529 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "For this ", Cell[BoxData[ \(f[x]\)]], "(thickest), ", Cell[BoxData[ \(fmidpt[x, b]\)]], " (red) is a slightly better approximation of ", Cell[BoxData[ \(f[x]\)]], " than ", Cell[BoxData[ \(ftrap[x, b]\)]], " (thinnest) even though both approximations have order of contact ", Cell[BoxData[ \(2\)]], " with ", Cell[BoxData[ \(f[x]\)]], " at ", Cell[BoxData[ \(x = b\)]], ". \nSee what happens for a different function and a different ", Cell[BoxData[ \(b\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f];\)\), "\n", \(\(f[x_] = x + E\^\(x/2\)\ Cos[3\ x];\)\), "\n", \(\(b = 1;\)\n\), "\n", \(\(Plot[{f[x], fmidpt[x, b], ftrap[x, b]}, {x, b - 2, b + 2}, \n\t PlotStyle \[Rule] {{Thickness[0.02], Blue}, {Thickness[0.01], Red}, {Thickness[0.005], Brown}}, AspectRatio \[Rule] 1\/GoldenRatio, AxesLabel \[Rule] {"\", "\<\>"}];\)\)}], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.261905 0.238095 0.297659 0.0269129 [ [.02381 .28516 -6 -9 ] [.02381 .28516 6 0 ] [.5 .28516 -3 -9 ] [.5 .28516 3 0 ] [.7381 .28516 -3 -9 ] [.7381 .28516 3 0 ] [.97619 .28516 -3 -9 ] [.97619 .28516 3 0 ] [1.025 .29766 0 -6 ] [1.025 .29766 10 6 ] [.2494 .02853 -18 -4.5 ] [.2494 .02853 0 4.5 ] [.2494 .16309 -12 -4.5 ] [.2494 .16309 0 4.5 ] [.2494 .43222 -6 -4.5 ] [.2494 .43222 0 4.5 ] [.2494 .56679 -12 -4.5 ] [.2494 .56679 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .29766 m .02381 .30391 L s [(-1)] .02381 .28516 0 1 Mshowa .5 .29766 m .5 .30391 L s [(1)] .5 .28516 0 1 Mshowa .7381 .29766 m .7381 .30391 L s [(2)] .7381 .28516 0 1 Mshowa .97619 .29766 m .97619 .30391 L s [(3)] .97619 .28516 0 1 Mshowa .125 Mabswid .07143 .29766 m .07143 .30141 L s .11905 .29766 m .11905 .30141 L s .16667 .29766 m .16667 .30141 L s .21429 .29766 m .21429 .30141 L s .30952 .29766 m .30952 .30141 L s .35714 .29766 m .35714 .30141 L s .40476 .29766 m .40476 .30141 L s .45238 .29766 m .45238 .30141 L s .54762 .29766 m .54762 .30141 L s .59524 .29766 m .59524 .30141 L s .64286 .29766 m .64286 .30141 L s .69048 .29766 m .69048 .30141 L s .78571 .29766 m .78571 .30141 L s .83333 .29766 m .83333 .30141 L s .88095 .29766 m .88095 .30141 L s .92857 .29766 m .92857 .30141 L s .25 Mabswid 0 .29766 m 1 .29766 L s gsave 1.025 .29766 -61 -10 Mabsadd m 1 1 Mabs scale currentpoint translate /MISOfy { /newfontname exch def /oldfontname exch def oldfontname findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end newfontname exch definefont pop } def 0 20 translate 1 -1 scale 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (x) show 69.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 1.000 setlinewidth grestore .2619 .02853 m .26815 .02853 L s [(-10)] .2494 .02853 1 0 Mshowa .2619 .16309 m .26815 .16309 L s [(-5)] .2494 .16309 1 0 Mshowa .2619 .43222 m .26815 .43222 L s [(5)] .2494 .43222 1 0 Mshowa .2619 .56679 m .26815 .56679 L s [(10)] .2494 .56679 1 0 Mshowa .125 Mabswid .2619 .05544 m .26565 .05544 L s .2619 .08236 m .26565 .08236 L s .2619 .10927 m .26565 .10927 L s .2619 .13618 m .26565 .13618 L s .2619 .19001 m .26565 .19001 L s .2619 .21692 m .26565 .21692 L s .2619 .24383 m .26565 .24383 L s .2619 .27075 m .26565 .27075 L s .2619 .32457 m .26565 .32457 L s .2619 .35148 m .26565 .35148 L s .2619 .3784 m .26565 .3784 L s .2619 .40531 m .26565 .40531 L s .2619 .45914 m .26565 .45914 L s .2619 .48605 m .26565 .48605 L s .2619 .51296 m .26565 .51296 L s .2619 .53987 m .26565 .53987 L s .2619 .00162 m .26565 .00162 L s .2619 .5937 m .26565 .5937 L s .25 Mabswid .2619 0 m .2619 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 0 1 r .02 w .02381 .25459 m .04262 .25709 L .06244 .26079 L .08255 .26565 L .10458 .27214 L .14509 .28653 L .18408 .30158 L .20396 .30886 L .22553 .31591 L .24453 .32102 L .25533 .32337 L .26546 .32515 L .27501 .32643 L .28038 .32697 L .28529 .32735 L .28797 .32751 L .29045 .32763 L .29317 .32773 L .29469 .32777 L .29537 .32778 L .29607 .3278 L .2974 .32781 L .29861 .32782 L .29992 .32782 L .30066 .32782 L .30135 .32782 L .30261 .3278 L .30375 .32778 L .30507 .32775 L .30632 .32772 L .30869 .32763 L .31091 .32752 L .31598 .32719 L .32091 .32675 L .32623 .32616 L .33584 .32477 L .34465 .32315 L .36358 .31867 L .38428 .3125 L .42484 .2985 L .44357 .29226 L .46388 .28647 L .47499 .28396 L .47987 .28304 L .48523 .28217 L .48995 .28154 L .49502 .28101 L .49782 .28079 L .50037 .28062 L .5016 .28056 L Mistroke .50294 .2805 L .50413 .28046 L .50538 .28043 L .50602 .28041 L .50672 .2804 L .50794 .28039 L .50864 .28039 L .50938 .28039 L .51007 .28039 L .51071 .2804 L .51195 .28042 L .51264 .28044 L .51329 .28046 L .51573 .28056 L .51789 .28068 L .52027 .28086 L .52509 .28135 L .53031 .28207 L .53592 .28308 L .54608 .28554 L .55567 .28861 L .56616 .29279 L .58727 .30378 L .62788 .33317 L .66698 .36749 L .70853 .40264 L .72937 .41685 L .73852 .42195 L .74857 .4266 L .75323 .42839 L .75824 .43005 L .76257 .43124 L .76729 .43229 L .77005 .43277 L .7726 .43314 L .77504 .43341 L .77629 .43352 L .77764 .43361 L .77885 .43368 L .77995 .43372 L .78101 .43375 L .78213 .43376 L .78335 .43376 L .78449 .43374 L .78575 .43369 L .78645 .43366 L .78708 .43362 L .78939 .43345 L .7907 .43332 L .79192 .43317 L Mistroke .79705 .43235 L .80148 .43136 L .80625 .43 L .8171 .42578 L .82703 .42056 L .84603 .40719 L .86676 .38814 L .90743 .34181 L .94658 .29591 L .97619 .2685 L Mfstroke 1 0 0 r .01 w .02381 .19991 m .06244 .24343 L .10458 .28581 L .14415 .318 L .16408 .33076 L .18221 .34017 L .20178 .3479 L .21254 .35106 L .22272 .35334 L .23241 .35488 L .2373 .35543 L .24012 .35567 L .24269 .35585 L .24504 .35598 L .24728 .35607 L .2485 .35611 L .24964 .35614 L .25088 .35616 L .25158 .35616 L .25221 .35617 L .25336 .35617 L .25442 .35616 L .25558 .35614 L .2568 .35612 L .25787 .35609 L .25887 .35606 L .26113 .35596 L .26373 .35581 L .26614 .35564 L .27158 .35513 L .27638 .35454 L .28148 .35378 L .30004 .34991 L .31951 .3442 L .33779 .33759 L .37648 .32106 L .41762 .30281 L .43814 .29473 L .45724 .28843 L .46672 .28588 L .47567 .28386 L .48061 .28293 L .48598 .28207 L .4905 .28148 L .49535 .28098 L .49791 .28078 L .50024 .28063 L .50136 .28057 L .50259 .28052 L .50373 .28047 L Mistroke .50481 .28044 L .50547 .28042 L .50619 .28041 L .50743 .28039 L .50814 .28039 L .50888 .28039 L .50952 .28039 L .51022 .28039 L .51147 .28041 L .51217 .28043 L .51281 .28044 L .51523 .28053 L .51653 .28059 L .51792 .28068 L .52041 .28086 L .52604 .28143 L .5309 .28211 L .53627 .28307 L .54639 .28549 L .55565 .2884 L .57652 .29749 L .59549 .30882 L .6162 .32448 L .65681 .36436 L .69591 .41193 L .73746 .46777 L .7775 .52074 L .81601 .56436 L .83593 .58186 L .84609 .58905 L .85698 .59527 L .86225 .59768 L .86724 .5996 L .8721 .60109 L .87424 .60163 L .8766 .60214 L .87864 .60251 L .88085 .60284 L .88207 .60298 L .88318 .60309 L .88424 .60317 L .88537 .60324 L .88652 .60329 L .88759 .60332 L .88887 .60332 L .89002 .6033 L .89123 .60325 L .89254 .60317 L .89378 .60306 L .89492 .60294 L Mistroke .89736 .60261 L .89959 .6022 L .9046 .60096 L .90925 .59939 L .91366 .59751 L .92353 .59192 L .93429 .58356 L .95398 .56195 L .97496 .52956 L .97619 .52736 L Mfstroke .5 .165 .165 r .005 w .02381 .26874 m .06244 .24652 L .08255 .23684 L .0932 .23259 L .10458 .22884 L .11009 .22734 L .11531 .22614 L .12038 .22516 L .12507 .22443 L .12951 .2239 L .13175 .22369 L .13422 .2235 L .13669 .22337 L .13808 .22331 L .13935 .22327 L .14052 .22325 L .14181 .22324 L .14302 .22324 L .14415 .22325 L .14544 .22328 L .14608 .22329 L .1468 .22332 L .14923 .22343 L .1506 .22351 L .15207 .22361 L .15472 .22385 L .15993 .22446 L .16467 .22521 L .17423 .22721 L .18328 .22971 L .20319 .23703 L .22161 .2456 L .26088 .26681 L .29863 .28608 L .31797 .29378 L .32794 .29693 L .33883 .29967 L .34385 .30067 L .34921 .30155 L .35427 .30221 L .35887 .30266 L .36158 .30286 L .36408 .303 L .36539 .30306 L .36682 .30311 L .36754 .30313 L .36832 .30315 L .36905 .30316 L .36972 .30317 L .37104 .30318 L Mistroke .37173 .30319 L .37247 .30318 L .37318 .30318 L .37384 .30317 L .37508 .30315 L .3763 .30312 L .37743 .30308 L .37997 .30297 L .38244 .30283 L .38469 .30267 L .38973 .30221 L .39883 .30105 L .40946 .29921 L .41951 .2971 L .45789 .2876 L .47771 .28337 L .4877 .28182 L .49339 .28116 L .49616 .28091 L .49873 .28072 L .50101 .28059 L .50344 .28048 L .5041 .28046 L .50483 .28044 L .50614 .28041 L .50741 .2804 L .50858 .28039 L .50983 .2804 L .51099 .28041 L .51226 .28044 L .51297 .28046 L .51362 .28048 L .51493 .28054 L .51636 .28061 L .51895 .28078 L .52361 .28123 L .52856 .28189 L .53747 .28356 L .54682 .28602 L .55691 .28946 L .57518 .29762 L .61629 .32239 L .63679 .33546 L .64679 .34127 L .65588 .34598 L .66471 .34985 L .67429 .35307 L .67684 .35373 L .67955 .35434 L .68205 .3548 L Mistroke .6844 .35515 L .68659 .35541 L .68783 .35551 L .68895 .35559 L .69018 .35565 L .69088 .35567 L .69152 .35569 L .6927 .3557 L .69396 .35568 L .6952 .35563 L .69591 .35559 L .69657 .35555 L .6979 .35544 L .69932 .35529 L .7019 .35491 L .70426 .35446 L .70997 .35292 L .71539 .35086 L .72505 .34566 L .73556 .33769 L .74497 .32846 L .75535 .31597 L .77636 .2835 L .81659 .19956 L .8553 .10854 L .87496 .06747 L .88535 .0492 L .89645 .03328 L .90199 .02697 L .90722 .02212 L .91191 .01875 L .91456 .01728 L .917 .01621 L .9181 .01582 L .91928 .01546 L .9203 .01521 L .92142 .01499 L .92272 .01482 L .92396 .01473 L .92516 .01472 L .92628 .01477 L .92757 .01491 L .92878 .01513 L .92986 .01538 L .93104 .01573 L .93368 .01678 L .9361 .01808 L .94082 .02156 L .94587 .02668 L .95498 .0397 L Mistroke .96054 .05012 L .9658 .06172 L .97566 .08806 L .97619 .08965 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{35, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]], Cell[TextData[{ "Again, for ", Cell[BoxData[ \(x\)]], "'s near ", Cell[BoxData[ \(b\)]], ", ", Cell[BoxData[ \(fmidpoint[x, b]\)]], " hugs the ", Cell[BoxData[ \(f[x]\)]], " curve slightly better than ", Cell[BoxData[ \(ftrap[x, b]\)]], ". \nDo a couple more for functions of your own choice and report your \ results." }], "Text"], Cell["G.7.b)", "Subsubsection"], Cell[TextData[{ "When Accurate Ann did these experiments, she exclaimed:\n\"I know why they \ came out the way they did! And I know why it is likely to happen this way no \ matter what ", Cell[BoxData[ \(f[x]\)]], " and ", Cell[BoxData[ \(b\)]], " you try. Look at this\": " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[ftrap, f, x, b];\)\), "\n", \(ftrap[x_, b_] = f[b] + 1\/2\ \((x - b)\)\ \((\(f'\)[b] + \(f'\)[x])\)\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{\(f[b]\), "+", RowBox[{\(1\/2\), " ", \((\(-b\) + x)\), " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}], "+", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], ")"}]}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[fmidpt, f, x, b];\)\), "\n", \(fmidpt[x_, b_] = f[b] + \((x - b)\)\ \(f'\)[\(b + x\)\/2]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{\(f[b]\), "+", RowBox[{\((\(-b\) + x)\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", \(\(b + x\)\/2\), "]"}]}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[{Normal[Series[fmidpt[x, b], {x, b, 3}]], Normal[Series[f[x], {x, b, 3}]], Normal[Series[ftrap[x, b], {x, b, 3}]]}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ InterpretationBox[GridBox[{ { RowBox[{\(f[b]\), "+", RowBox[{\((\(-b\) + x)\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/2\), " ", \(\((\(-b\) + x)\)\^2\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/8\), " ", \(\((\(-b\) + x)\)\^3\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}]}]}, { RowBox[{\(f[b]\), "+", RowBox[{\((\(-b\) + x)\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/2\), " ", \(\((\(-b\) + x)\)\^2\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/6\), " ", \(\((\(-b\) + x)\)\^3\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}]}]}, { RowBox[{\(f[b]\), "+", RowBox[{\((\(-b\) + x)\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/2\), " ", \(\((\(-b\) + x)\)\^2\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/4\), " ", \(\((\(-b\) + x)\)\^3\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}]}]} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ { Plus[ f[ b], Times[ Plus[ Times[ -1, b], x], Derivative[ 1][ f][ b]], Times[ Rational[ 1, 2], Power[ Plus[ Times[ -1, b], x], 2], Derivative[ 2][ f][ b]], Times[ Rational[ 1, 8], Power[ Plus[ Times[ -1, b], x], 3], Derivative[ 3][ f][ b]]], Plus[ f[ b], Times[ Plus[ Times[ -1, b], x], Derivative[ 1][ f][ b]], Times[ Rational[ 1, 2], Power[ Plus[ Times[ -1, b], x], 2], Derivative[ 2][ f][ b]], Times[ Rational[ 1, 6], Power[ Plus[ Times[ -1, b], x], 3], Derivative[ 3][ f][ b]]], Plus[ f[ b], Times[ Plus[ Times[ -1, b], x], Derivative[ 1][ f][ b]], Times[ Rational[ 1, 2], Power[ Plus[ Times[ -1, b], x], 2], Derivative[ 2][ f][ b]], Times[ Rational[ 1, 4], Power[ Plus[ Times[ -1, b], x], 3], Derivative[ 3][ f][ b]]]}]]], "Output"] }, Open ]], Cell["Then she said, \"Look at:\"", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({N[1\/6 - 1\/8], N[1\/4 - 1\/6]}\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \({0.041666666666666664`, 0.08333333333333333`}\)], "Output"] }, Open ]], Cell["\<\ And then she said, \"This explains why the experiments came out the \ way they did.\" What was she driving at?\ \>", "Text"], Cell["G.7.c)", "Subsubsection"], Cell["\<\ Not one to stop when she is ahead, Accurate Ann went on to say, \ \"Look at this\":\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[ftrap, f, x, b];\)\), "\n", \(ftrap[x_, b_] = f[b] + \ \((x - b)\)\ \((\(f'\)[b] + \(f'\)[x])\)\/2\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{\(f[b]\), "+", RowBox[{\(1\/2\), " ", \((\(-b\) + x)\), " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}], "+", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], ")"}]}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[fmidpt, f, x, b];\)\), "\n", \(fmidpt[x_, b_] = f[b] + \((x - b)\)\ \(f'\)[\(b + x\)\/2]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{\(f[b]\), "+", RowBox[{\((\(-b\) + x)\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", \(\(b + x\)\/2\), "]"}]}]}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[{Normal[Series[fmidpt[x, b], {x, b, 3}]], Normal[Series[f[x], {x, b, 3}]], Normal[Series[ftrap[x, b], {x, b, 3}]]}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ InterpretationBox[GridBox[{ { RowBox[{\(f[b]\), "+", RowBox[{\((\(-b\) + x)\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/2\), " ", \(\((\(-b\) + x)\)\^2\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/8\), " ", \(\((\(-b\) + x)\)\^3\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}]}]}, { RowBox[{\(f[b]\), "+", RowBox[{\((\(-b\) + x)\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/2\), " ", \(\((\(-b\) + x)\)\^2\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/6\), " ", \(\((\(-b\) + x)\)\^3\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}]}]}, { RowBox[{\(f[b]\), "+", RowBox[{\((\(-b\) + x)\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/2\), " ", \(\((\(-b\) + x)\)\^2\), " ", RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "b", "]"}]}], "+", RowBox[{\(1\/4\), " ", \(\((\(-b\) + x)\)\^3\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "b", "]"}]}]}]} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ { Plus[ f[ b], Times[ Plus[ Times[ -1, b], x], Derivative[ 1][ f][ b]], Times[ Rational[ 1, 2], Power[ Plus[ Times[ -1, b], x], 2], Derivative[ 2][ f][ b]], Times[ Rational[ 1, 8], Power[ Plus[ Times[ -1, b], x], 3], Derivative[ 3][ f][ b]]], Plus[ f[ b], Times[ Plus[ Times[ -1, b], x], Derivative[ 1][ f][ b]], Times[ Rational[ 1, 2], Power[ Plus[ Times[ -1, b], x], 2], Derivative[ 2][ f][ b]], Times[ Rational[ 1, 6], Power[ Plus[ Times[ -1, b], x], 3], Derivative[ 3][ f][ b]]], Plus[ f[ b], Times[ Plus[ Times[ -1, b], x], Derivative[ 1][ f][ b]], Times[ Rational[ 1, 2], Power[ Plus[ Times[ -1, b], x], 2], Derivative[ 2][ f][ b]], Times[ Rational[ 1, 4], Power[ Plus[ Times[ -1, b], x], 3], Derivative[ 3][ f][ b]]]}]]], "Output"] }, Open ]], Cell[TextData[{ "And then she said:\n\"If the third derivative ", Cell[BoxData[ \(\(f\^\((3)\)\)[b]\)]], " is positive, then:\n\[Rule] for ", Cell[BoxData[ \(x\)]], " slightly to the right of ", Cell[BoxData[ \(b\)]], ", you expect to have\n ", Cell[BoxData[ \(fmidpt[x, b] < f[x] < ftrap[x, b]\)]], " and\n\[Rule] for ", Cell[BoxData[ \(x\)]], " slightly to the left of ", Cell[BoxData[ \(b\)]], ", you expect to have\n ", Cell[BoxData[ \(ftrap[x, b] < f[x] < fmidpt[x, b]\)]], ".\nBut if the third derivative ", Cell[BoxData[ \(\(f\^\((3)\)\)[b]\)]], " is negative, then:\n\[Rule] for ", Cell[BoxData[ \(x\)]], " slightly to the right of ", Cell[BoxData[ \(b\)]], ", you expect to have\n ", Cell[BoxData[ \(ftrap[x, b] < f[x] < fmidpt[x, b]\)]], " and\n\[Rule] for ", Cell[BoxData[ \(x\)]], " slightly to the left of ", Cell[BoxData[ \(b\)]], ", you expect to have\n ", Cell[BoxData[ \(fmidpt[x, b] < f[x] < ftrap[x, b]\)]], ".\"\nWhere did she get this idea?" }], "Text"], Cell["G.7.d)", "Subsubsection"], Cell[TextData[{ "Very satisified with herself, Accurate Ann went on to say,\n\"Now I see \ why a weighted average, like ", Cell[BoxData[ \(frunge[x, b]\)]], ", of ", Cell[BoxData[ \(ftrap[x, b]\)]], " and ", Cell[BoxData[ \(fmidpt[x, b]\)]], " can be expected to be a better approximation of ", Cell[BoxData[ \(f[x]\)]], " than either ", Cell[BoxData[ \(ftrap[x, b]\)]], " or ", Cell[BoxData[ \(fmidpt[x, b]\)]], ".\"\nWhat was she driving at?" }], "Text"], Cell["G.7.e)", "Subsubsection"], Cell[TextData[{ "With an amazed Bubba staring at her, his mouth hanging half open, Accurate \ Ann seized her moment of triumph and said:\n\"Now it's clear to me for a very \ small jump size, the ", Cell[BoxData[ \(frunge[x, b]\)]], " method of estimating ", Cell[BoxData[ RowBox[{\(\[Integral]\_a\%c\), RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], \(\[DifferentialD]x\)}]}]]], " beats the ", Cell[BoxData[ \(fmidpt[x, b]\)]], " method, and the ", Cell[BoxData[ \(fmidpt[x, b]\)]], " method beats the ", Cell[BoxData[ \(ftrap[x, b]\)]], " method. And if you don't believe me, check it out for yourself.\" " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[ftrap, f, x, b, Derivative];\)\), "\n", \(\(ftrap[x_, b_]\ = \ f[b]\ + \ \((x\ - \ b)\)\ \ \((\(f'\)[b] + \(f'\)[x])\)\/2;\)\n\), "\n", \(\(Clear[fmidpt];\)\), "\n", \(\(fmidpt[x_, b_]\ = \ f[b]\ + \ \((x\ - \ b)\)\ \(f'\)[\(b + x\)\/2];\)\n\), "\n", \(\(Clear[frunge];\)\), "\n", \(\(frunge[x_, b_]\ = \ \ 1\/3\ \((2\ fmidpt[x, b] + ftrap[x, b])\);\)\n\ \ \ \ \ \), "\n", \(\(\(f'\)[x_]\ = \ Sin[x\^3];\)\n\), "\n", \(\(a\ = \ 0.0;\)\), "\n", \(\(c\ = \ 1.5;\)\n\ \ \ \ \), "\n", \(\(MathematicaEstimate\ = \ NIntegrate[\(f'\)[x], {x, a, c}];\)\n\), "\n", \(\(Clear[b, jump, EulerEstimate, TrapezoidalEstimate, \n\ \ \ \ \ \ MidpointEstimate, SimpsonEstimate];\)\n\ \ \ \ \ \ \), "\n", \(\(EulerEstimate[jump_]\ := \ \n\ \ \ \ N[ Sum[feuler[b\ + \ jump, b]\ - \ feuler[b, b], \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {b, a, c\ - \ jump, jump}]];\)\n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \), "\n", \(\(TrapezoidalEstimate[jump_]\ := \ \n\ \ \ \ N[ Sum[ftrap[b\ + \ jump, b]\ - \ ftrap[b, b], \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {b, a, c\ - \ jump, jump}]];\)\ \ \ \ \ \ \ \n\ \ \ \ \ \ \ \ \ \ \ \ \ \), "\n", \(\(MidpointEstimate[jump_]\ := \n\ \ \ \ N[ Sum[fmidpt[b\ + \ jump, b]\ - \ fmidpt[b, b], \n\ \ \ \ {b, a, c\ - \ jump, jump}]];\)\n\ \ \ \ \), "\n", \(\(RungeKuttaEstimate[jump_]\ := \ \n\ \ \ \ N[ Expand[\n\ \ \ \ \ \ Sum[ frunge[b\ + \ jump, b]\ - \ frunge[b, b], \n\ \ \ \ {b, a, c\ - \ jump, jump}]]];\)\n\n\), "\n", \(\(jump\ = \ \(c - a\)\/20;\)\n\), "\n", \(ColumnForm[\n\ \ \ \ {jump\ "\<= jump gives you these estimates:\>", \n\ \ \ \ \ \ "\<\>", \ \ \n\ \ \ \ \ TrapezoidalEstimate[ jump]\ "\<= Trapezoidal\>", \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \n\ \ \ \ \ MidpointEstimate[ jump]\ "\<= Midpoint\>", \n\ \ \ \ \ RungeKuttaEstimate[ jump]\ "\<= Runge-Kutta\>", \n\ \ \ \ \ MathematicaEstimate\ "\<= \ MathematicaEstimate\>"}]\)}], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(0.075`\ "= jump gives you these estimates:"\)}, {"\<\"\"\>"}, {\(0.5837906564284083`\ "= Trapezoidal"\)}, {\(0.5884353109354244`\ "= Midpoint"\)}, {\(0.586887092766419`\ "= Runge-Kutta"\)}, {\(0.5868834559939693`\ "= MathematicaEstimate"\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ { Times[ 0.074999999999999997, "= jump gives you these estimates:"], "", Times[ 0.58379065642840833, "= Trapezoidal"], Times[ 0.58843531093542445, "= Midpoint"], Times[ 0.58688709276641904, "= Runge-Kutta"], Times[ 0.58688345599396929, "= MathematicaEstimate"]}], Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(jump\ = \ \(c - a\)\/40;\)\n\), "\n", \(ColumnForm[\n\ \ \ \ {jump\ "\<= jump gives you these estimates:\>", \n\ \ \ \ \ \ "\<\>", \ \ \n\ \ \ \ \ TrapezoidalEstimate[ jump]\ "\<= Trapezoidal\>", \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \n\ \ \ \ \ MidpointEstimate[ jump]\ "\<= Midpoint\>", \n\ \ \ \ \ RungeKuttaEstimate[ jump]\ "\<= Runge-Kutta\>", \n\ \ \ \ \ MathematicaEstimate\ "\<= \ MathematicaEstimate\>"}]\)}], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(0.0375`\ "= jump gives you these estimates:"\)}, {"\<\"\"\>"}, {\(0.5861129836819171`\ "= Trapezoidal"\)}, {\(0.5872690319452513`\ "= Midpoint"\)}, {\(0.58688368252414`\ "= Runge-Kutta"\)}, {\(0.5868834559939693`\ "= MathematicaEstimate"\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ { Times[ 0.037499999999999999, "= jump gives you these estimates:"], "", Times[ 0.58611298368191711, "= Trapezoidal"], Times[ 0.58726903194525126, "= Midpoint"], Times[ 0.58688368252414003, "= Runge-Kutta"], Times[ 0.58688345599396929, "= MathematicaEstimate"]}], Editable->False]], "Output"] }, Open ]], Cell["Where did Accurate Ann get all this insight?", "Text"], Cell["\<\ When you have finished with this problem, please run the following \ cell:\ \>", "Special2"], Cell[BoxData[ \(\(Clear[Derivative];\)\)], "Input", AspectRatioFixed->True] }, Closed]], Cell["Literacy Problem", "Subsubsection"], Cell["L.3)", "Subsubsection"], Cell[TextData[{ "All you know about a pair of functions ", Cell[BoxData[ \(f[x]\)]], " and ", Cell[BoxData[ \(g[x]\)]], " is:\n ", Cell[BoxData[ \(f[1] = 0\)]], ", ", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "1", "]"}], "=", "6"}]]], ", and ", Cell[BoxData[ \(g[1] = 0\)]], ", and ", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["g", "\[Prime]", MultilineFunction->None], "[", "1", "]"}], "=", "2"}]]], " .\nCalculate the ", Cell[BoxData[ \(lim\+\(x\ \[Rule] \ 1\)\ f[x]\/g[x]\)]], "." }], "Text"] }, FrontEndVersion->"4.0 for Macintosh", ScreenRectangle->{{0, 640}, {0, 460}}, AutoGeneratedPackage->None, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{632, 433}, WindowMargins->{{-2, Automatic}, {Automatic, 1}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ The following notebook is copyright 1999 by Math \ Everywhere,Inc.and may not be reproduced, copied, or distributed, in whole or \ in part, through any means electronic or otherwise, without written consent \ of the copyright holder.\ \>", "Text"], Cell[CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[StyleData[All, "Working"], ScriptMinSize->9], Cell[StyleData[All, "Presentation"], ScriptMinSize->12, FontSize->18], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, PrivateFontOptions->{"FontType"->"Outline"}], Cell[StyleData[All, "TwoColumn"], PageWidth->PaperWidth, PrivateFontOptions->{"FontType"->"Outline"}] }, Closed]], Cell[CellGroupData[{ Cell["Notebook Options", "Section"], Cell["\<\ The options defined for the style below will be used at the \ Notebook level.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Notebook"], CellGrouping->Manual, ShowClosedCellArea->True, StyleMenuListing->None, Background->RGBColor[1, 0.940002, 0.900008]], Cell[StyleData["Notebook", "Printout"], Background->GrayLevel[1]], Cell[StyleData["Notebook", "TwoColumn"], PageHeaders->{{None, None, None}, {None, None, None}}, PageHeaderLines->{False, False}, PrintingOptions->{"PrintingMargins"->{{55, 25}, {40, 45}}}, Background->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["Title"], CellFrame->{{1, 1}, {1, 5}}, ShowCellBracket->False, CellMargins->{{24, 24}, {12, 5}}, CellGroupingRules->{"TitleGrouping", 0}, PageBreakBelow->False, CellFrameMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, LineSpacing->{1.1, 1}, CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontFamily->"Times", FontSize->24, FontWeight->"Bold", FontColor->GrayLevel[0], Background->RGBColor[0.925002, 0.854978, 0.774975]], Cell[StyleData["Title", "Presentation"], CellMargins->{{24, Inherited}, {60, Inherited}}, TextAlignment->Center, FontSize->24, FontColor->GrayLevel[1], Background->RGBColor[0.294118, 0.521569, 0.0941176]], Cell[StyleData["Title", "Printout"], CellMargins->{{0, Inherited}, {0, 0}}, TextAlignment->Center, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["Title", "TwoColumn"], CellFrame->{{1, 1}, {5, 0}}, CellMargins->{{0, Inherited}, {0, 0}}, TextAlignment->Center, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], ShowCellBracket->False, ShowClosedCellArea->True, CellMargins->{{6, Inherited}, {0, 0}}, CellGroupingRules->{"TitleGrouping", 10}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, FontFamily->"Times", FontSize->14], Cell[StyleData["Subtitle", "Presentation"], CellFrame->False, CellMargins->{{24, Inherited}, {6, Inherited}}], Cell[StyleData["Subtitle", "Printout"], CellMargins->{{14, Inherited}, {2, 2}}], Cell[StyleData["Subtitle", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 2}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubtitle"], CellFrame->True, ShowClosedCellArea->True, CellMargins->{{6, Inherited}, {6, Inherited}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Times", FontSize->24, FontColor->RGBColor[1, 0, 0]], Cell[StyleData["Subsubtitle", "Presentation"], CellFrame->True, CellMargins->{{24, Inherited}, {6, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Subsubtitle", "Printout"], CellMargins->{{14, Inherited}, {2, 2}}, FontColor->GrayLevel[0]], Cell[StyleData["Subsubtitle", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 2}}, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellDingbat->"\[GraySquare]", ShowCellBracket->True, ShowGroupOpenCloseIcon->True, CellMargins->{{22, Inherited}, {Inherited, 20}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontFamily->"Times", FontSize->16], Cell[StyleData["Section", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, 20}}, LineSpacing->{1.5, 0}, FontSize->18], Cell[StyleData["Section", "Printout"], CellMargins->{{14, Inherited}, {2, 10}}], Cell[StyleData["Section", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 10}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"", ShowCellBracket->True, ShowGroupOpenCloseIcon->True, CellMargins->{{19, Inherited}, {Inherited, 18}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontFamily->"Times", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0, 0.392187, 0]], Cell[StyleData["Subsection", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, 15}}], Cell[StyleData["Subsection", "Printout"], CellMargins->{{14, Inherited}, {2, 5}}, FontSize->12, FontColor->GrayLevel[0]], Cell[StyleData["Subsection", "TwoColumn"], CellFrame->{{0, 0}, {0, 1}}, CellMargins->{{14, Inherited}, {2, 10}}, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[EmptySquare]", ShowClosedCellArea->True, CellMargins->{{18, Inherited}, {Inherited, 12}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, CounterIncrements->"Subsubsection", FontFamily->"Times", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.689998, 0.0899977, 0.119997]], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, 12}}, LineSpacing->{1, 0}], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{14, Inherited}, {2, 3}}, FontSize->12, FontColor->GrayLevel[0]], Cell[StyleData["Subsubsection", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 3}}, FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["PrefaceText"], CellMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1, 1}, LimitsPositioningTokens->{}, StyleMenuListing->None, FontFamily->"Times", FontSize->10, FontWeight->"Plain"], Cell[StyleData["PrefaceText", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, Inherited}}, LineSpacing->{2, 0}, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["PrefaceText", "Printout"], CellMargins->{{0, Inherited}, {2, 2}}, LineSpacing->{1, 1}, FontSize->12, FontColor->GrayLevel[0], Background->None], Cell[StyleData["PrefaceText", "TwoColumn"], CellMargins->{{0, Inherited}, {2, 2}}, LineSpacing->{1, 1}, FontSize->12, FontColor->GrayLevel[0], Background->None] }, Closed]], Cell[StyleData["PrefaceHyperlink"], CellMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1, 1}, LimitsPositioningTokens->{}, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Times", FontSize->10, FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[CellGroupData[{ Cell[StyleData["Text"], ShowClosedCellArea->True, CellMargins->{{15, 10}, {Inherited, Inherited}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, LimitsPositioningTokens->{}, FontFamily->"Times", FontSize->16, FontColor->RGBColor[0, 0, 0.500008]], Cell[StyleData["Text", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, Inherited}}, LineSpacing->{2, 0}, FontSize->16, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Text", "Printout"], CellMargins->{{14, Inherited}, {3, 1}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 2}, FontColor->GrayLevel[0], Background->None], Cell[StyleData["Text", "TwoColumn"], CellMargins->{{14, Inherited}, {3, 1}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 2}, FontSize->14, FontColor->GrayLevel[0], Background->None] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], ShowClosedCellArea->True, CellMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, LimitsPositioningTokens->{}, FontFamily->"Times", FontSize->16], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["SmallText", "Printout"], CellMargins->{{14, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True], Cell[StyleData["SmallText", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell["\<\ The cells in this section define styles used for input and output \ to the kernel. Be careful when modifying, renaming, or removing these \ styles, because the front end associates special meanings with these style \ names.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{45, Inherited}, {Inherited, Inherited}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, GroupPageBreakWithin->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", FontWeight->"Bold"], Cell[StyleData["Input", "Presentation"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{45, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Input", "Printout"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, FontSize->9], Cell[StyleData["Input", "TwoColumn"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, LineSpacing->{1.5, 0}, FormatType->StandardForm, FontFamily->"Courier", FontSize->14, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Output", "Presentation"], CellMargins->{{45, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Output", "Printout"], CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 0}, FontSize->10, FontColor->GrayLevel[0]], Cell[StyleData["Output", "TwoColumn"], CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 0}, FontSize->12, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, StyleMenuListing->None, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Message", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Message", "Printout"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->9, FontColor->GrayLevel[0]], Cell[StyleData["Message", "TwoColumn"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->9, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Print", "Printout"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->9], Cell[StyleData["Print", "TwoColumn"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->16] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Info"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, StyleMenuListing->None], Cell[StyleData["Info", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Info", "Printout"], CellMargins->{{30, Inherited}, {Inherited, Inherited}}, FontSize->10], Cell[StyleData["Info", "TwoColumn"], CellMargins->{{30, Inherited}, {Inherited, Inherited}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{15, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, ImageMargins->{{35, Inherited}, {Inherited, 0}}, AnimationDisplayTime->0.2, StyleMenuListing->None, FontSize->14], Cell[StyleData["Graphics", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Graphics", "Printout"], CellMargins->{{30, Inherited}, {0, 0}}, CellFrameMargins->False, ImageSize->{Inherited, 150}, ImageMargins->{{45, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, FontSize->9], Cell[StyleData["Graphics", "TwoColumn"], CellMargins->{{20, Inherited}, {0, 0}}, CellFrameMargins->False, ImageSize->{Inherited, 150}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Times", FontSize->9, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["CellLabel", "Presentation"], FontSize->14], Cell[StyleData["CellLabel", "Printout"], FontColor->GrayLevel[1]], Cell[StyleData["CellLabel", "TwoColumn"], FontColor->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Special Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["PrefaceTitle"], CellFrame->{{1, 1}, {1, 5}}, ShowCellBracket->False, CellMargins->{{24, 24}, {0, 10}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, CellFrameMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, LineSpacing->{1.4, 1}, FontFamily->"Times", FontSize->24, FontWeight->"Bold", FontColor->GrayLevel[1], Background->RGBColor[0, 0.392187, 0]], Cell[StyleData["PrefaceTitle", "Presentation"], CellMargins->{{24, Inherited}, {60, Inherited}}, TextAlignment->Center, FontSize->38, FontColor->GrayLevel[1], Background->RGBColor[0.596078, 0.65098, 0.0196078]], Cell[StyleData["PrefaceTitle", "Printout"], CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["PrefaceTitle", "TwoColumn"], CellFrame->{{1, 1}, {0, 5}}, CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Special Body Text and Index", "Section"], Cell[CellGroupData[{ Cell[StyleData["Accident"], CellFrame->3, ShowCellBracket->False, CellMargins->{{24, 24}, {0, 10}}, CellFrameMargins->{{15, Inherited}, {Inherited, Inherited}}, TextAlignment->Center, LineSpacing->{1.4, 1}, FontFamily->"Times", FontSize->24, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], Cell[StyleData["Accident", "Presentation"], CellMargins->{{24, Inherited}, {60, Inherited}}, TextAlignment->Center, FontSize->36], Cell[StyleData["Accident", "Printout"], CellFrame->2, CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["Accident", "TwoColumn"], CellFrame->2, CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ContentsText"], CellMargins->{{50, 10}, {5, 5}}, FontFamily->"Times", FontSize->16], Cell[StyleData["ContentsText", "Presentation"]], Cell[StyleData["ContentsText", "Printout"], FontColor->GrayLevel[0], Background->None], Cell[StyleData["ContentsText", "TwoColumn"], FontColor->GrayLevel[0], Background->None] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special1"], CellDingbat->"\[EmptySquare]", ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, FontFamily->"Times", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.689998, 0.0899977, 0.119997]], Cell[StyleData["Special1", "Presentation"], FontSize->16], Cell[StyleData["Special1", "Printout"], FontSize->12, FontColor->GrayLevel[0]], Cell[StyleData["Special1", "TwoColumn"], FontSize->12, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special2"], CellMargins->{{6, 0}, {0, 0}}, CellGroupingRules->{"SectionGrouping", 40}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special2", "Presentation"], FontSize->12], Cell[StyleData["Special2", "Printout"], FontSize->10, FontColor->GrayLevel[0]], Cell[StyleData["Special2", "TwoColumn"], FontSize->10, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special3"], CellDingbat->"\[GraySquare]", ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special3", "Presentation"]], Cell[StyleData["Special3", "Printout"], FontColor->GrayLevel[0]], Cell[StyleData["Special3", "TwoColumn"], FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special4"], ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special4", "Presentation"], FontSize->12], Cell[StyleData["Special4", "Printout"], FontColor->GrayLevel[0]], Cell[StyleData["Special4", "TwoColumn"], FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special5"], ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special5", "Presentation"], FontSize->12], Cell[StyleData["Special5", "Printout"]], Cell[StyleData["Special5", "TwoColumn"]] }, Closed]], Cell[StyleData["IndexEntry"], ShowCellBracket->False, CellMargins->{{15, 5}, {0, 5}}, PageBreakBelow->False, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, ParagraphIndent->-40, StyleMenuListing->None, FontSize->16], Cell[StyleData["IndexSubEntry"], ShowCellBracket->False, CellMargins->{{45, 5}, {0, 0}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, ParagraphIndent->-40, StyleMenuListing->None, FontSize->16] }, Closed]], Cell[CellGroupData[{ Cell["Styles for License Agreement", "Section"], Cell[CellGroupData[{ Cell[StyleData["LicenseHeading"], ShowCellBracket->True, ShowGroupOpenCloseIcon->True, CellMargins->{{24, 24}, {-1, 2}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0.4, 0.300008, 0.6]], Cell[StyleData["LicenseHeading", "Presentation"], FontSize->12], Cell[StyleData["LicenseHeading", "Printout"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["LicenseHeading", "TwoColumn"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["LicenseText"], CellFrame->True, ShowCellBracket->False, CellMargins->{{24, 24}, {5, -1}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->12, Background->RGBColor[1, 0.537743, 0.509071]], Cell[StyleData["LicenseText", "Presentation"], FontSize->18], Cell[StyleData["LicenseText", "Printout"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["LicenseText", "TwoColumn"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Automatic Numbering", "Section"], Cell["\<\ The following styles are useful for numbered equations, figures, \ etc. They automatically give the cell a FrameLabel containing a reference to \ a particular counter, and also increment that counter.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["NumberedEquation"], CounterIncrements->"NumberedEquation"], Cell[StyleData["NumberedEquation", "Presentation"]], Cell[StyleData["NumberedEquation", "Printout"]], Cell[StyleData["NumberedEquation", "TwoColumn"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedFigure"], CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Figure ", CounterBox[ "NumberedFigure"]}]], None}}, CounterIncrements->"NumberedFigure", FormatTypeAutoConvert->False, FontFamily->"Times"], Cell[StyleData["NumberedFigure", "Presentation"], CellMargins->{{24, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["NumberedFigure", "Printout"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10], Cell[StyleData["NumberedFigure", "TwoColumn"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedTable"], CellMargins->{{6, Inherited}, {Inherited, Inherited}}, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Table ", CounterBox[ "NumberedTable"]}]], None}}, CounterIncrements->"NumberedTable", FormatTypeAutoConvert->False, FontFamily->"Times"], Cell[StyleData["NumberedTable", "Presentation"], CellMargins->{{24, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["NumberedTable", "Printout"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10], Cell[StyleData["NumberedTable", "TwoColumn"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontSize->10], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, StyleMenuListing->None, FontSize->9], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard \ ButtonFunctions, for use in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontSize->14, FontColor->RGBColor[0, 0.392187, 0], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["Hyperlink", "Printout"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}], Cell[StyleData["Hyperlink", "TwoColumn"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["MEIHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.650004, 0.680003, 0.0800031], Background->RGBColor[0, 0.392187, 0], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Come visit us!"}], Cell[StyleData["MEIHyperlink", "Printout"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}], Cell[StyleData["MEIHyperlink", "TwoColumn"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["BasicsHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.848096, 0.171878, 0.228321], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Go to Basics"}], Cell[StyleData["BasicsHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["BasicsHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["BasicsIndexHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontColor->RGBColor[0.848096, 0.171878, 0.228321], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEnd`SelectionMove[ FrontEnd`SelectedNotebook[ ], Next, CellGroup], FrontEndToken[ "SelectionCloseAllGroups"], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Link into Basics"}], Cell[StyleData["BasicsIndexHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["BasicsIndexHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["TutorialsHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.0199588, 0.346716, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Go to Tutorials"}], Cell[StyleData["TutorialsHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["TutorialsHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["TutorialsIndexHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontColor->RGBColor[0.0199588, 0.346716, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEnd`SelectionMove[ FrontEnd`SelectedNotebook[ ], Next, CellGroup], FrontEndToken[ "SelectionCloseAllGroups"], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Link into Tutorials"}], Cell[StyleData["TutorialsIndexHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["TutorialsIndexHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GiveItaTryHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.497459, 0.196094, 0.543877], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Go to GiveItaTry"}], Cell[StyleData["GiveItaTryHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["GiveItaTryHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GiveItaTryIndexHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontColor->RGBColor[0.497459, 0.196094, 0.543877], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEnd`SelectionMove[ FrontEnd`SelectedNotebook[ ], Next, CellGroup], FrontEndToken[ "SelectionCloseAllGroups"], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Link into GiveItaTry"}], Cell[StyleData["GiveItaTryIndexHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["GiveItaTryIndexHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["LiteracyHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[1, 0.433326, 0], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Go to Literacy"}], Cell[StyleData["LiteracyHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["LiteracyHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["LiteracyIndexHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontColor->RGBColor[1, 0.433326, 0], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEnd`SelectionMove[ FrontEnd`SelectedNotebook[ ], Next, CellGroup], FrontEndToken[ "SelectionCloseAllGroups"], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Link into Literacy"}], Cell[StyleData["LiteracyIndexHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["LiteracyIndexHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["PreviewHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->GrayLevel[0.250004], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Preview of Lesson"}], Cell[StyleData["PreviewHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["PreviewHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuideLink", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"]], Cell[StyleData["RefGuideLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["RefGuideLink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"]], Cell[StyleData["GettingStartedLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["GettingStartedLink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"]], Cell[StyleData["OtherInformationLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["OtherInformationLink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder \ objects in palette templates.\ \>", "Text"], Cell[StyleData["Placeholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->True, StyleMenuListing->None], Cell[StyleData["SelectionPlaceholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->PrimaryPlaceholder, StyleMenuListing->None, DrawHighlighted->True] }, Closed]], Cell[CellGroupData[{ Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles \ of most cells. If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier", FontSize->12], Cell[StyleData["InputForm"], AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{1, -5}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an \ inline cell within another.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["InlineCell"], DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, ScriptLevel->1, LimitsPositioningTokens->{}, StyleMenuListing->None, FontFamily->"Times", FontSize->16, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["InlineCell", "Presentation"], DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, ScriptLevel->1, LimitsPositioningTokens->{}, StyleMenuListing->None, FontFamily->"Times", FontSize->16, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["InlineCell", "Printout"], DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, ScriptLevel->1, LimitsPositioningTokens->{}, StyleMenuListing->None, FontColor->GrayLevel[0]], Cell[StyleData["InlineCell", "TwoColumn"], DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, ScriptLevel->1, LimitsPositioningTokens->{}, StyleMenuListing->None, FontSize->14, FontColor->GrayLevel[0]] }, Closed]] }, Closed]] }, Open ]] }], MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@0000000000000006P001T1T00000000000 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->{ "3.04.B1"->{ Cell[6964, 173, 285, 10, 86, "Subsection", CellTags->"3.04.B1"]}, "3.04.T1"->{ Cell[29934, 896, 77, 1, 60, "Subsection", CellTags->"3.04.T1"]}, "3.04.G2"->{ Cell[35074, 1072, 98, 1, 60, "Subsection", CellTags->"3.04.G2"]}, "3.04.G7"->{ Cell[42671, 1358, 92, 1, 46, "Subsection", CellTags->"3.04.G7"]} } *) (*CellTagsIndex CellTagsIndex->{ {"3.04.B1", 127585, 4775}, {"3.04.T1", 127676, 4778}, {"3.04.G2", 127766, 4781}, {"3.04.G7", 127857, 4784} } *) (*NotebookFileOutline Notebook[{ Cell[1717, 49, 2361, 33, 34, 2210, 28, "GraphicsData", "Bitmap", "Graphics"], Cell[4081, 84, 813, 23, 134, "PrefaceTitle"], Cell[4897, 109, 122, 5, 91, "Title"], Cell[CellGroupData[{ Cell[5044, 118, 104, 4, 29, "Special2"], Cell[5151, 124, 724, 16, 247, "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[5900, 144, 160, 2, 54, "Subsubsection"], Cell[6063, 148, 710, 13, 214, "Input", InitializationCell->True] }, Closed]], Cell[6788, 164, 98, 2, 23, "Input", InitializationCell->True] }, Closed]], Cell[6901, 169, 38, 0, 34, "Subsubsection"], Cell[CellGroupData[{ Cell[6964, 173, 285, 10, 86, "Subsection", CellTags->"3.04.B1"], Cell[7252, 185, 52, 0, 13, "Special2"], Cell[7307, 187, 81, 2, 23, "Input"], Cell[7391, 191, 31, 0, 42, "Subsubsection"], Cell[7425, 193, 30, 0, 34, "Text"], Cell[CellGroupData[{ Cell[7480, 197, 128, 3, 39, "Input"], Cell[7611, 202, 624, 15, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[8272, 222, 90, 2, 23, "Input"], Cell[8365, 226, 1819, 48, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[10221, 279, 35, 0, 46, "Text"], Cell[10259, 281, 27, 0, 26, "Special1"], Cell[10289, 283, 39, 0, 30, "SmallText"], Cell[CellGroupData[{ Cell[10353, 287, 128, 3, 39, "Input"], Cell[10484, 292, 398, 9, 70, "Output"] }, Open ]], Cell[10897, 304, 192, 11, 30, "SmallText"], Cell[CellGroupData[{ Cell[11114, 319, 90, 2, 23, "Input"], Cell[11207, 323, 1307, 33, 70, "Output"] }, Open ]], Cell[12529, 359, 1248, 41, 166, "SmallText"] }, Closed]], Cell[13792, 403, 31, 0, 34, "Subsubsection"], Cell[CellGroupData[{ Cell[13848, 407, 211, 8, 72, "Text"], Cell[14062, 417, 27, 0, 26, "Special1"], Cell[14092, 419, 174, 8, 30, "SmallText"], Cell[CellGroupData[{ Cell[14291, 431, 214, 6, 103, "Input"], Cell[14508, 439, 350, 6, 70, "Output"] }, Open ]], Cell[14873, 448, 1017, 31, 100, "SmallText"], Cell[CellGroupData[{ Cell[15915, 483, 199, 4, 75, "Input"], Cell[16117, 489, 350, 6, 70, "Output"] }, Open ]], Cell[16482, 498, 54, 3, 52, "SmallText"], Cell[CellGroupData[{ Cell[16561, 505, 120, 3, 39, "Input"], Cell[16684, 510, 672, 11, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17393, 526, 83, 1, 43, "Input"], Cell[17479, 529, 672, 11, 70, "Output"] }, Open ]], Cell[18166, 543, 221, 11, 30, "SmallText"] }, Closed]], Cell[18402, 557, 33, 0, 34, "Subsubsection"], Cell[CellGroupData[{ Cell[18460, 561, 45, 0, 42, "Text"], Cell[18508, 563, 27, 0, 26, "Special1"], Cell[18538, 565, 504, 9, 102, "SmallText"], Cell[19045, 576, 169, 4, 26, "Special2"], Cell[19217, 582, 337, 11, 75, "SmallText"], Cell[CellGroupData[{ Cell[19579, 597, 160, 3, 90, "Input"], Cell[19742, 602, 3169, 45, 70, "Output"] }, Open ]], Cell[22926, 650, 282, 11, 54, "SmallText"], Cell[CellGroupData[{ Cell[23233, 665, 91, 2, 23, "Input"], Cell[23327, 669, 689, 18, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[24053, 692, 100, 2, 38, "Input"], Cell[24156, 696, 407, 9, 70, "Output"] }, Open ]], Cell[24578, 708, 50, 0, 30, "SmallText"], Cell[CellGroupData[{ Cell[24653, 712, 72, 2, 23, "Input"], Cell[24728, 716, 806, 19, 70, "Output"] }, Open ]], Cell[25549, 738, 287, 5, 96, "SmallText"] }, Closed]], Cell[25851, 746, 34, 0, 34, "Subsubsection"], Cell[CellGroupData[{ Cell[25910, 750, 144, 5, 45, "Text"], Cell[26057, 757, 27, 0, 26, "Special1"], Cell[26087, 759, 579, 21, 144, "SmallText"] }, Closed]], Cell[26681, 783, 31, 0, 34, "Subsubsection"], Cell[CellGroupData[{ Cell[26737, 787, 135, 3, 64, "Text"], Cell[26875, 792, 27, 0, 26, "Special1"], Cell[26905, 794, 187, 5, 52, "SmallText"], Cell[CellGroupData[{ Cell[27117, 803, 114, 3, 39, "Input"], Cell[27234, 808, 2276, 70, 70, "Output"] }, Open ]], Cell[29525, 881, 316, 7, 96, "SmallText"] }, Closed]] }, Closed]], Cell[29868, 892, 41, 0, 34, "Subsubsection"], Cell[CellGroupData[{ Cell[29934, 896, 77, 1, 60, "Subsection", CellTags->"3.04.T1"], Cell[30014, 899, 52, 0, 13, "Special2"], Cell[30069, 901, 81, 2, 23, "Input"], Cell[30153, 905, 31, 0, 42, "Subsubsection"], Cell[30187, 907, 245, 11, 86, "Text"], Cell[CellGroupData[{ Cell[30457, 922, 198, 4, 90, "Input"], Cell[30658, 928, 168, 2, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[30863, 935, 941, 24, 72, "Text"], Cell[31807, 961, 27, 0, 26, "Special1"], Cell[31837, 963, 2390, 71, 144, "SmallText"], Cell[CellGroupData[{ Cell[34252, 1038, 121, 2, 43, "Input"], Cell[34376, 1042, 131, 2, 70, "Output"] }, Open ]], Cell[34522, 1047, 27, 0, 30, "SmallText"], Cell[CellGroupData[{ Cell[34574, 1051, 77, 1, 23, "Input"], Cell[34654, 1054, 131, 2, 70, "Output"] }, Open ]], Cell[34800, 1059, 175, 5, 33, "SmallText"] }, Closed]] }, Closed]], Cell[35002, 1068, 47, 0, 34, "Subsubsection"], Cell[CellGroupData[{ Cell[35074, 1072, 98, 1, 60, "Subsection", CellTags->"3.04.G2"], Cell[35175, 1075, 52, 0, 13, "Special2"], Cell[35230, 1077, 81, 2, 23, "Input"], Cell[35314, 1081, 33, 0, 42, "Subsubsection"], Cell[35350, 1083, 345, 14, 78, "Text"], Cell[35698, 1099, 34, 0, 42, "Subsubsection"], Cell[CellGroupData[{ Cell[35757, 1103, 378, 14, 103, "Text"], Cell[36138, 1119, 24, 0, 26, "Special1"], Cell[36165, 1121, 124, 5, 30, "SmallText"] }, Closed]], Cell[36304, 1129, 72, 0, 42, "Subsubsection"], Cell[36379, 1131, 468, 21, 96, "Text"], Cell[CellGroupData[{ Cell[36872, 1156, 161, 3, 39, "Input"], Cell[37036, 1161, 383, 9, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[37456, 1175, 108, 2, 23, "Input"], Cell[37567, 1179, 383, 9, 70, "Output"] }, Open ]], Cell[37965, 1191, 484, 12, 78, "Text"], Cell[38452, 1205, 34, 0, 42, "Subsubsection"], Cell[38489, 1207, 900, 35, 118, "Text"], Cell[CellGroupData[{ Cell[39414, 1246, 204, 5, 39, "Input"], Cell[39621, 1253, 449, 11, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[40107, 1269, 151, 4, 23, "Input"], Cell[40261, 1275, 449, 11, 70, "Output"] }, Open ]], Cell[40725, 1289, 652, 16, 78, "Text"], Cell[41380, 1307, 35, 0, 42, "Subsubsection"], Cell[41418, 1309, 806, 33, 208, "Text"], Cell[42227, 1344, 34, 0, 42, "Subsubsection"], Cell[42264, 1346, 370, 7, 96, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[42671, 1358, 92, 1, 46, "Subsection", CellTags->"3.04.G7"], Cell[42766, 1361, 52, 0, 13, "Special2"], Cell[42821, 1363, 81, 2, 23, "Input"], Cell[42905, 1367, 31, 0, 42, "Subsubsection"], Cell[42939, 1369, 184, 8, 34, "Text"], Cell[CellGroupData[{ Cell[43148, 1381, 192, 4, 56, "Input"], Cell[43343, 1387, 428, 11, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[43808, 1403, 162, 3, 56, "Input"], Cell[43973, 1408, 232, 6, 70, "Output"] }, Open ]], Cell[44220, 1417, 160, 8, 34, "Text"], Cell[CellGroupData[{ Cell[44405, 1429, 404, 8, 140, "Input"], Cell[44812, 1439, 7214, 478, 70, 7091, 474, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[52041, 1920, 573, 26, 112, "Text"], Cell[CellGroupData[{ Cell[52639, 1950, 453, 9, 158, "Input"], Cell[53095, 1961, 9921, 638, 70, 9798, 634, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]], Cell[63031, 2602, 383, 18, 60, "Text"], Cell[63417, 2622, 31, 0, 42, "Subsubsection"], Cell[63451, 2624, 309, 10, 86, "Text"], Cell[CellGroupData[{ Cell[63785, 2638, 180, 4, 56, "Input"], Cell[63968, 2644, 428, 11, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[64433, 2660, 162, 3, 56, "Input"], Cell[64598, 2665, 232, 6, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[64867, 2676, 202, 4, 55, "Input"], Cell[65072, 2682, 3838, 109, 70, "Output"] }, Open ]], Cell[68925, 2794, 43, 0, 34, "Text"], Cell[CellGroupData[{ Cell[68993, 2798, 91, 2, 38, "Input"], Cell[69087, 2802, 79, 1, 70, "Output"] }, Open ]], Cell[69181, 2806, 134, 4, 60, "Text"], Cell[69318, 2812, 31, 0, 42, "Subsubsection"], Cell[69352, 2814, 107, 3, 34, "Text"], Cell[CellGroupData[{ Cell[69484, 2821, 179, 4, 56, "Input"], Cell[69666, 2827, 428, 11, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[70131, 2843, 162, 3, 56, "Input"], Cell[70296, 2848, 232, 6, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[70565, 2859, 202, 4, 55, "Input"], Cell[70770, 2865, 3838, 109, 70, "Output"] }, Open ]], Cell[74623, 2977, 1144, 44, 320, "Text"], Cell[75770, 3023, 31, 0, 42, "Subsubsection"], Cell[75804, 3025, 518, 21, 112, "Text"], Cell[76325, 3048, 31, 0, 42, "Subsubsection"], Cell[76359, 3050, 776, 24, 138, "Text"], Cell[CellGroupData[{ Cell[77160, 3078, 2326, 45, 820, "Input"], Cell[79489, 3125, 811, 18, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[80337, 3148, 495, 8, 165, "Input"], Cell[80835, 3158, 811, 18, 70, "Output"] }, Open ]], Cell[81661, 3179, 60, 0, 34, "Text"], Cell[81724, 3181, 102, 3, 13, "Special2"], Cell[81829, 3186, 81, 2, 23, "Input"] }, Closed]], Cell[81925, 3191, 41, 0, 34, "Subsubsection"], Cell[81969, 3193, 29, 0, 42, "Subsubsection"], Cell[82001, 3195, 690, 29, 90, "Text"] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)