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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 347155, 16869]*) (*NotebookOutlinePosition[ 392709, 18447]*) (* CellTagsIndexPosition[ 392513, 18437]*) (*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[{ "Accumulation\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[{ "2.02 Breaking the Code of the Integral:\nThe Fundamental Formula\n", StyleBox[" BASICS", 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", \(\(Off[ParametricPlot3D::ppcom];\)\), "\n", \(\(SetOptions[Limit, Analytic \[Rule] True];\)\), "\n", \(\(Needs["\"];\)\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", \(\($Pre\ = \ Chop;\)\)}], "Input", InitializationCell->True] }, Closed]], Cell["Basic Problem", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "B.2) The fundamental formula \n ", Cell[BoxData[ \(f[t] - f[a] = \[Integral]\_a\%t\( f'\)[x] \[DifferentialD]x\)]] }], "Subsection", CellTags->"2.02.B2"], Cell["B.2.a)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "The fundamental formula of calculus completely breaks the code of the \ integral by telling you that\n ", Cell[BoxData[ \(f[t] - f[a] = \[Integral]\_a\%t\( f'\)[x] \[DifferentialD]x\)]], ".\nIt's so important that it is programmed into ", StyleBox["Mathematica", FontSlant->"Italic"], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(Clear[f, t, a, Derivative]\), ";", "\n", RowBox[{\(\[Integral]\_a\%t\), RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], \(\[DifferentialD]x\)}]}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-f[a]\) + f[t]\)], "Output"] }, Open ]], Cell["Where does this beauty come from?", "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "The Race Track Principle strikes again.\nFix a point and go to the race \ track and run a race between two functions:\n ", Cell[BoxData[ \(Newton[t] = f[t] - f[a]\)]], "\nand\n ", Cell[BoxData[ \(Leibniz[t] = \[Integral]\_a\%t\( f'\)[x] \[DifferentialD]x\)]], ".\nWhen ", Cell[BoxData[ \(t = a\)]], ", you get\n ", Cell[BoxData[ \(Newton[a] = \(f[a] - f[a] = 0\)\)]], "\nand\n ", Cell[BoxData[ \(Leibniz[a] = \(\[Integral]\_a\%a\( f'\)[x] \[DifferentialD]x = 0\)\)]], "\nbecause ", Cell[BoxData[ \(\[Integral]\_a\%a\( f'\)[x] \[DifferentialD]x\)]], " measures the area of a (degenerate) rectangle whose height is ", Cell[BoxData[ \(\(f'\)[a]\)]], " and whose base is ", Cell[BoxData[ \(0\)]], ".\nThis means that if you start the race at ", Cell[BoxData[ \(t = a\)]], ", the two functions \n ", Cell[BoxData[ \(Leibniz[t]\)]], " and ", Cell[BoxData[ \(Newton[t]\)]], " \nstart their race together. If you can see why \n ", Cell[BoxData[ \(\(Newton'\)[t]\)]], " ", Cell[BoxData[ \( = \)]], " ", Cell[BoxData[ \(\(Leibniz'\)[t]\)]], " \nfor ", Cell[BoxData[ \(t > a\)]], ", then the Race Track Principle will step in to tell you that they remain \ tied throughout their race; that is, \n ", Cell[BoxData[ \(Newton[t] = \ Leibniz[t]\)]], " \nfor all ", Cell[BoxData[ \(t\)]], "'s with ", Cell[BoxData[ \(t > a\)]], ". \nTo see why ", Cell[BoxData[ \(\(Newton'\)[t] = \(Leibniz'\)[t]\)]], " for ", Cell[BoxData[ \(t > a\)]], ", note \n ", Cell[BoxData[ \(Newton[t] = f[t] - f[a]\)]], ";\nso\n ", Cell[BoxData[ \(\(Newton'\)[t] = \(\(f'\)[t] - 0 = \(f'\)[t]\)\)]], " (because ", Cell[BoxData[ \(a\)]], " is a constant)\nand\n ", Cell[BoxData[ \(Leibniz[t] = \[Integral]\_a\%t\( f'\)[x] \[DifferentialD]x\)]], ";\nso\n ", Cell[BoxData[ \(\(Leibniz'\)[t] = \(f'\)[t]\)]], " (See B.1) above).\nThe upshot: \n ", Cell[BoxData[ \(\(Newton'\)[t] = \(Leibniz'\)[t]\)]], ".\nNow you know that \n ", Cell[BoxData[ \(Newton[t] = Leibniz[t]\)]], ". \nThis is the same as \n ", Cell[BoxData[ \(f[t] - f[a] = \[Integral]\_a\%t\( f'\)[x] \[DifferentialD]x\)]], ".\nAnd you're out of here." }], "SmallText"] }, Closed]], Cell["B.2.b) Putting the fundamental formula to work", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Use the fundamental formula to calculate the exact value of \n ", Cell[BoxData[ \(\[Integral]\_0\%\(\[Pi]\/2\)Sin[x] \[DifferentialD]x\)]], " \nand explain what the calculation measures. " }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "The fundamental formula says\n ", Cell[BoxData[ \(f[t] - f[a] = \[Integral]\_a\%t\( f'\)[x] \[DifferentialD]x\)]], ". \nTo use the fundamental formula to calculate \n ", Cell[BoxData[ \(\[Integral]\_0\%\(\[Pi]\/2\)Sin[x] \[DifferentialD]x\)]], ", \nlook for a function ", Cell[BoxData[ \(f[x]\)]], " with \n ", Cell[BoxData[ \(\(f'\)[x] = Sin[x]\)]], ". \nA moment's reflection tells you that\n ", Cell[BoxData[ \(f[x] = \(-Cos[x]\)\)]], " \nwill do: " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(Clear[x, f]\), ";", "\n", \(f[x_] = \(-Cos[x]\)\), ";", "\n", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Sin[x]\)], "Output"] }, Open ]], Cell[TextData[{ "Good. \nThe fundamental formula, \n ", Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"f", "[", StyleBox[\(\[Pi]\/2\), FontSize->16], "]"}], "-", \(f[0]\)}], "=", \(\[Integral]\_0\%\(\[Pi]\/2\)\(f'\)[x] \[DifferentialD]x = \[Integral]\_0\%\(\[Pi]\/2\)Sin[x] \[DifferentialD]x\)}]]], ",\ntells you that ", Cell[BoxData[ \(\[Integral]\_0\%\(\[Pi]\/2\)Sin[x] \[DifferentialD]x\)]], " is given by:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(f[\[Pi]\/2] - f[0]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell["Check:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_0\%\(\[Pi]\/2\)Sin[x] \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[TextData[{ "Nailed it.\nThe calculation \n ", Cell[BoxData[ \(\[Integral]\_0\%\(\[Pi]\/2\)Sin[x] \[DifferentialD]x = 1\)]], " \nmeans that the shaded area below measures out to exactly one square \ unit:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(FilledPlot[Sin[x], {x, 0, \[Pi]\/2}, \n\t AspectRatio \[Rule] Automatic, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] Blue, PlotLabel \[Rule] 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That shaded area measures out to exactly one square unit; no more, no less.\ \ \>", "SmallText"] }, Closed]], Cell["B.2.c)", "Subsubsection"], Cell[CellGroupData[{ Cell["\<\ What calculational advantage does the fundamental formula give you? \ \ \>", "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Without resorting to approximations, you can calculate the exact value of \ an integral\n ", Cell[BoxData[ \(\[Integral]\_a\%t g[x] \[DifferentialD]x\)]], " \nif you can come up with a function ", Cell[BoxData[ \(f[x]\)]], " with \n ", Cell[BoxData[ \(\(f'\)[x] = g[x]\)]], ". \nOnce you have your hands on ", Cell[BoxData[ \(f[x]\)]], ", you just write\n ", Cell[BoxData[ \(\[Integral]\_a\%t g[x] \[DifferentialD]x = f[t] - f[a]\)]], " \nand you're done." }], "SmallText"] }, Closed]], Cell["B.2.d)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "What calculational advantage does the fundamental formula give ", StyleBox["Mathematica", FontSlant->"Italic"], "? " }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " uses the fundamental formula just the way you use it. \nWhen you ask ", StyleBox["Mathematica", FontSlant->"Italic"], " to calculate the exact value of\n ", Cell[BoxData[ \(\[Integral]\_a\%t g[x] \[DifferentialD]x\)]], ", \n", StyleBox["Mathematica", FontSlant->"Italic"], " looks for a function ", Cell[BoxData[ \(f[x]\)]], " with \n ", Cell[BoxData[ \(\(f'\)[x] = g[x]\)]], ". \nOnce ", StyleBox["Mathematica", FontSlant->"Italic"], " has its hands on ", Cell[BoxData[ \(f[x]\)]], ", ", StyleBox["Mathematica", FontSlant->"Italic"], " reports \n ", Cell[BoxData[ \(\[Integral]\_a\%t g[x] \[DifferentialD]x = f[t] - f[a]\)]], ". \nWatch ", StyleBox["Mathematica", FontSlant->"Italic"], " use the fundamental formula:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, g, t];\)\), "\n", \(\(g[x_] = x\ E\^\(-x\);\)\), "\n", \(\[Integral]\_2\%t g[x] \[DifferentialD]x\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(3\/\[ExponentialE]\^2 - \[ExponentialE]\^\(-t\)\ \((1 + t)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, g, t];\)\), "\n", \(\(g[x_] = Sin[x\/3];\)\), "\n", \(\[Integral]\_\[Pi]\%t g[x] \[DifferentialD]x\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(3\/2 - 3\ Cos[t\/3]\)], "Output"] }, Open ]], Cell[TextData[{ "The fundamental formula is directly programmed into ", StyleBox["Mathematica", FontSlant->"Italic"], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{\(Clear[f, x, a, t, Derivative];\), "\n", RowBox[{\(\[Integral]\_a\%t\), RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], \(\[DifferentialD]x\)}]}]}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-f[a]\) + f[t]\)], "Output"] }, Open ]], Cell["The fundamental formula is a big deal.", "SmallText"] }, Closed]] }, Closed]], Cell["Tutorial Problems ", "Subsubsection"], Cell[CellGroupData[{ Cell["\<\ T.1) Getting the feel of the fundamental formula by using it to \ calculate integrals\ \>", "Subsection", CellTags->"2.02.T1"], Cell[TextData[{ "Newton and Leibniz are called the founders of calculus because they \ cracked the code of the integral by writing down the fundamental formula of \ calculus\n ", Cell[BoxData[ \(f[t] - f[a] = \[Integral]\_a\%t\( f'\)[x] \[DifferentialD]x\)]], "\nand, in one line of writing, changed the character of science.\nThe \ fundamental formula is so important that ", StyleBox["Mathematica", FontSlant->"Italic"], " knows it cold:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(Clear[f, x, t, a, Derivative]\), ";", "\n", RowBox[{\(\[Integral]\_a\%t\), RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], \(\[DifferentialD]x\)}]}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-f[a]\) + f[t]\)], "Output"] }, Open ]], Cell[TextData[{ "The fundamental formula of calculus \n ", Cell[BoxData[ \(f[t] - f[a] = \[Integral]\_a\%t\( f'\)[x] \[DifferentialD]x\)]], " \nis your calculating jewel because it gives you a powerful, systematic \ way of calculating the exact value of integrals without resort to \ approximations. \nAnd using it is a real pleasure. \nTo attempt to calculate \ \n ", Cell[BoxData[ \(\[Integral]\_a\%t g[x] \[DifferentialD]x\)]], " \nfor a given function ", Cell[BoxData[ \(g[x]\)]], ", you find another function ", Cell[BoxData[ \(f[x]\)]], " such that\n ", Cell[BoxData[ \(\(f'\)[x] = g[x]\)]], "\nand then you just calculate\n ", Cell[BoxData[ \(\[Integral]\_a\%t g[ x] \[DifferentialD]x = \(\[Integral]\_a\%t\( f'\)[ x] \[DifferentialD]x = f[t] - f[a]\)\)]], ".\nIn principle, nothing could be simpler, and you really don't have to \ depend on the machine to do the simple cases.\nTo try it out, look at ", StyleBox["Mathematica", FontSlant->"Italic"], "'s calculation of \n ", Cell[BoxData[ \(\[Integral]\_3\%9\( x\^2\) \[DifferentialD]x\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x];\)\), "\n", \(\[Integral]\_3\%9\( x\^2\) \[DifferentialD]x\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(234\)], "Output"] }, Open ]], Cell["T.1.a.i)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Explain how ", StyleBox["Mathematica", FontSlant->"Italic"], " exploited the fundamental formula to do the calculation." }], "Text"], Cell["Answer:", "Subsubsection"], Cell[TextData[{ "The fundamental formula says\n ", Cell[BoxData[ \(f[t] - f[a] = \[Integral]\_a\%t\( f'\)[x] \[DifferentialD]x\)]], ". \nTo use the fundamental formula to calculate \n ", Cell[BoxData[ \(\[Integral]\_3\%9\( x\^2\) \[DifferentialD]x\)]], ", \n", StyleBox["Mathematica", FontSlant->"Italic"], " looked for a function ", Cell[BoxData[ \(f[x]\)]], " with \n ", Cell[BoxData[ \(\(f'\)[x] = x\^2\)]], ". \nIt found ", Cell[BoxData[ RowBox[{\(f[x]\), "=", StyleBox[\(x\^3\/3\), FontSize->16]}]]], ": " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(Clear[x, f]\), ";", "\n", \(f[x_] = x\^3\/3\), ";", "\n", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ \(x\^2\)], "Output"] }, Open ]], Cell[TextData[{ "Good. \nNow using the fundamental formula, you get \n ", Cell[BoxData[ \(f[9] - f[3] = \(\[Integral]\_3\%9\( f'\)[ x] \[DifferentialD]x = \[Integral]\_3\%9\( x\^2\) \[DifferentialD]x\)\)]], ".\n", StyleBox["Mathematica", FontSlant->"Italic"], " proclaims that ", Cell[BoxData[ \(\[Integral]\_3\%9\( x\^2\) \[DifferentialD]x\)]], " is given by:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(f[9] - f[3]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(234\)], "Output"] }, Open ]], Cell["This is right on target:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_3\%9\( x\^2\) \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(234\)], "Output"] }, Open ]], Cell["Beautiful and simple.", "SmallText"], Cell["In mathematics, most beautiful ideas are simple.", "Special2"] }, Closed]], Cell["T.1.a.ii)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Use the fundamental formula to calculate \n ", Cell[BoxData[ \(\[Integral]\_0\%\(\[Pi]\/2\)Cos[5 x] \[DifferentialD]x\)]], " \nand explain what the calculation measures. " }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "The fundamental formula says\n ", Cell[BoxData[ \(f[t] - f[a] = \[Integral]\_a\%t\( f'\)[x] \[DifferentialD]x\)]], ". \nTo use the fundamental formula to calculate \n ", Cell[BoxData[ \(\[Integral]\_0\%\(\[Pi]\/2\)Cos[5 x] \[DifferentialD]x\)]], ", \nlook for a function ", Cell[BoxData[ \(f[x]\)]], " with \n ", Cell[BoxData[ \(\(f'\)[x] = Cos[5 x]\)]], ". \nA moment's reflection tells you that\n ", Cell[BoxData[ RowBox[{\(f[x]\), "=", StyleBox[\(Sin[5 x]\/5\), FontSize->16]}]]], " \nwill do: " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(Clear[x, f]\), ";", "\n", \(f[x_] = 1\/5\ Sin[5\ x]\), ";", "\n", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Cos[5\ x]\)], "Output"] }, Open ]], Cell[TextData[{ "Good. \nThe fundamental formula, \n ", Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"f", "[", StyleBox[\(\[Pi]\/2\), 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1\)\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(x\^\(1 + n\)\/\(1 + n\)\)], "Output"] }, Open ]], Cell["Look at:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]], "Input", AspectRatioFixed->True], Cell[BoxData[ \(x\^n\)], "Output"] }, Open ]], Cell[TextData[{ "Excellent: ", Cell[BoxData[ \(\(f'\)[x] = x\^n\)]], ".\nThe fundamental formula says that\n ", Cell[BoxData[ \(\[Integral]\_a\%t\( x\^n\) \[DifferentialD]x = \(\[Integral]\_a\%t\( f'\)[ x] \[DifferentialD]x = f[t] - f[a]\)\)]], " \n ", Cell[BoxData[ RowBox[{"=", StyleBox[\(t\^\(n + 1\)\/\(n + 1\) - a\^\(n + 1\)\/\(n + 1\)\), FontSize->18]}]]], ".\nSome folks like to use the compact notation\n ", Cell[BoxData[ RowBox[{ StyleBox[\(x\^\(n + 1\)\/\(n + 1\)\), FontSize->16], SubsuperscriptBox[ StyleBox["|", FontSize->24], "a", "t"]}]]], " to stand for ", Cell[BoxData[ StyleBox[\(t\^\(n + 1\)\/\(n + 1\) - a\^\(n + 1\)\/\(n + 1\)\), 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You can also get this from a differential equation:\ \>", "SmallText"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[a, x, t, f];\)\), "\n", \(Simplify[ DSolve[{\(f'\)[t]\ == \ t\^n, \n\ \ \ \ \ \ \ \ f[a]\ == \ 0}, f[t], t]]\)}], "Input"], Cell[BoxData[ \({{f[ t] \[Rule] \(\(-a\^\(1 + n\)\) + t\^\(1 + n\)\)\/\(1 + n\)}}\)], \ "Output"] }, Open ]], Cell["T.1.b.ii)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ RowBox[{\(\[Integral]\_1\%t\), RowBox[{ StyleBox[\(1\/x\), FontSize->16], \(\[DifferentialD]x\)}]}]]] }], "Text"], Cell["Answer:", "Special1"], Cell["Try:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(f[x_] = Log[x]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Log[x]\)], "Output"] }, Open ]], Cell["Look at:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(f'\)[x]\)], "Input"], Cell[BoxData[ \(1\/x\)], "Output"] }, Open ]], Cell[TextData[{ "Fine: ", Cell[BoxData[ RowBox[{\(\(f'\)[x]\), "=", StyleBox[\(1\/x\), FontSize->16]}]]], ".\nThe fundamental formula says that\n ", Cell[BoxData[ RowBox[{ RowBox[{\(\[Integral]\_1\%t\), RowBox[{ StyleBox[\(1\/x\), FontSize->16], \(\[DifferentialD]x\)}]}], "=", \(\[Integral]\_1\%t\( f'\)[x] \[DifferentialD]x\)}]]], " \n ", Cell[BoxData[ \(\(\(=\)\(f[t] - f[1]\)\)\)]], " \n ", Cell[BoxData[ \(\(\(=\)\(Log[t] - Log[1]\)\)\)]], " \n ", Cell[BoxData[ \(\(\(=\)\(Log[t]\)\)\)]], ".\nOr in the compact notation,\n ", Cell[BoxData[ RowBox[{ RowBox[{\(\[Integral]\_1\%t\), RowBox[{ StyleBox[\(1\/x\), FontSize->16], \(\[DifferentialD]x\)}]}], "=", RowBox[{ RowBox[{\(Log[x]\), SubsuperscriptBox[ StyleBox["|", FontSize->24], "1", "t"]}], "=", \(Log[t]\)}]}]]], ".\n (Remember ", Cell[BoxData[ \(Log[1] = 0\)]], ".)\nCheck:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_1\%t\( 1\/x\) \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Log[t]\)], "Output"] }, Open ]], Cell["Nailed it.", "SmallText"] }, Closed]], Cell["T.1.b.iii)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ \(\[Integral]\_1\%3 x\^3 - 5 x\^2 + 4 x + 2 \[DifferentialD]x\)]] }], "Text"], Cell["Answer:", "Special1"], Cell["Try:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(f[x_] = x\^4\/4 - \(5\ x\^3\)\/3 + \(4\ x\^2\)\/2 + 2\ x\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(2\ x + 2\ x\^2 - \(5\ x\^3\)\/3 + x\^4\/4\)], "Output"] }, Open ]], Cell["Look at:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(f'\)[x]\)], "Input"], Cell[BoxData[ \(2 + 4\ x - 5\ x\^2 + x\^3\)], "Output"] }, Open ]], Cell[TextData[{ "Just as needed: ", Cell[BoxData[ \(\(f'\)[x] = x\^3 - 5 x\^2 + 4 x + 2\)]], ".\nThe fundamental formula tells you that\n ", Cell[BoxData[ RowBox[{\(\[Integral]\_1\%3\((x\^3 - 5 x\^2 + 4 x + 2)\) \[DifferentialD]x\), "=", RowBox[{ RowBox[{\(\[Integral]\_1\%3\), RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], \(\[DifferentialD]x\)}]}], "=", RowBox[{\(f[x]\), SubsuperscriptBox[ StyleBox["|", FontSize->24], "1", "3"]}]}]}]]], ",\nwhich is given by:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(f[3] - f[1]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-\(10\/3\)\)\)], "Output"] }, Open ]], Cell["Check:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_1\%3\((x\^3 - 5\ x\^2 + 4\ x + 2)\) \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-\(10\/3\)\)\)], "Output"] }, Open ]], Cell["\<\ Got it. Not bad for beginners.\ \>", "SmallText"] }, Closed]], Cell["T.1.b.iv)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ \(\[Integral]\_0\%t Cos[x\^3] \(x\^2\) \[DifferentialD]x\)]], " " }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "You want \n ", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "=", \(Cos[x\^3]\ x\^2\)}]]], ".\n Try: " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, y, x, t];\)\), "\n", \(f[x_] = Sin[x\^3]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Sin[x\^3]\)], "Output"] }, Open ]], Cell["Look at:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(D[f[x], x]\)], "Input"], Cell[BoxData[ \(3\ x\^2\ Cos[x\^3]\)], "Output"] }, Open ]], Cell[TextData[{ "This carries an extra constant ", Cell[BoxData[ \(3\)]], "; kill the constant:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, y, x, t];\)\), "\n", \(f[x_] = Sin[x\^3]\/3\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Sin[x\^3]\/3\)], "Output"] }, Open ]], Cell["Look at:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(D[f[x], x]\)], "Input"], Cell[BoxData[ \(x\^2\ Cos[x\^3]\)], "Output"] }, Open ]], Cell[TextData[{ "Right on the money; \n ", Cell[BoxData[ \(\(f'\)[x] = Cos[x\^3] x\^2\)]], "\nThe fundamental formula tells you that\n ", Cell[BoxData[ \(\[Integral]\_0\%t Cos[x\^3] \(x\^2\) \[DifferentialD]x\)]], "\n ", Cell[BoxData[ \(\(\(=\)\(\[Integral]\_0\%t\( f'\)[x] \[DifferentialD]x\)\)\)]], "\n ", Cell[BoxData[ RowBox[{"=", RowBox[{\(f[x]\), SubsuperscriptBox[ StyleBox["|", FontSize->24], "0", "t"]}]}]]], ",\nwhich is:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(f[t] - f[0]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Sin[t\^3]\/3\)], "Output"] }, Open ]], Cell["Check:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_0\%t\( Cos[x\^3]\ x\^2\) \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Sin[t\^3]\/3\)], "Output"] }, Open ]], Cell["\<\ On the bull's eye. You can also get this from a differential equation:\ \>", "SmallText"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[a, x, t, f];\)\), "\n", \(DSolve[{\(f'\)[t]\ == \ Cos[t\^3]\ t\^2, \n\ \ \ \ \ \ \ \ f[0]\ == \ 0}, f[t], t]\)}], "Input"], Cell[BoxData[ \({{f[t] \[Rule] Sin[t\^3]\/3}}\)], "Output"] }, Open ]], Cell["T.1.b.v)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ \(\[Integral]\_0\%\(\[Pi]\/2\)Sin[7 x] \[DifferentialD]x\)]], " " }], "Text"], Cell["Answer:", "Special1"], Cell["Try:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(f[x_] = \(-Cos[7\ x]\)\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-Cos[7\ x]\)\)], "Output"] }, Open ]], Cell["Look at:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(f'\)[x]\)], "Input"], Cell[BoxData[ \(7\ Sin[7\ x]\)], "Output"] }, Open ]], Cell[TextData[{ "But this ", Cell[BoxData[ \(f[x]\)]], " doesn't work because it carries an extra factor out front. Adjust ", Cell[BoxData[ \(f[x]\)]], " to kill the extra factor:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(f[x_] = \(-\(1\/7\)\)\ Cos[7\ x]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-\(1\/7\)\)\ Cos[7\ x]\)], "Output"] }, Open ]], Cell["Look at:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(f'\)[x]\)], "Input"], Cell[BoxData[ \(Sin[7\ x]\)], "Output"] }, Open ]], Cell[TextData[{ "Good.\nThe fundamental formula tells you that\n ", Cell[BoxData[ RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)Sin[7 x] \[DifferentialD]x\), "=", RowBox[{\(\[Integral]\_0\%\(\[Pi]\/2\)\(f'\)[x] \[DifferentialD]x\), "=", RowBox[{\(f[x]\), SubsuperscriptBox[ StyleBox["|", FontSize->24], "0", \(\[Pi]\/2\)]}]}]}]]], "\n ", Cell[BoxData[ RowBox[{"=", RowBox[{ StyleBox[\(\(-Cos[7 x]\)\/7\), FontSize->16], SubsuperscriptBox[ StyleBox["|", FontSize->24], "0", \(\[Pi]\/2\)]}]}]]], ",\nwhich is given by: " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(f[\[Pi]\/2] - f[0]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\/7\)], "Output"] }, Open ]], Cell["Check:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_0\%\(\[Pi]\/2\)Sin[7\ x] \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\/7\)], "Output"] }, Open ]], Cell["Got it.", "SmallText"] }, Closed]], Cell["T.1.b.vi)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ \(\[Integral]\_0.2\%0.9 3.1 \( E\^\(\(-10\) x\)\) \[DifferentialD]x\)]] }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "You want\n ", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "=", \(3.1\ E\^\(\(-10\)\ x\)\)}]]], ".\nTry:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, K, x];\)\), "\n", \(f[x_] = 3.1\ E\^\(\(-10\)\ x\)\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(3.1`\ \[ExponentialE]\^\(\(-10\)\ x\)\)], "Output"] }, Open ]], Cell["Look at:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(f'\)[x]\)], "Input"], Cell[BoxData[ \(\(-31.`\)\ \[ExponentialE]\^\(\(-10\)\ x\)\)], "Output"] }, Open ]], Cell[TextData[{ "But this ", Cell[BoxData[ \(f[x]\)]], " doesn't work because it carries an extra multiplicative factor of ", Cell[BoxData[ \(\(-10\)\)]], " out front. Adjust ", Cell[BoxData[ \(f[x]\)]], " to kill the extra factor:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(f[x_] = \(-\(1\/10\)\)\ 3.1\ E\^\(\(-10\)\ x\)\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-0.31`\)\ \[ExponentialE]\^\(\(-10\)\ x\)\)], "Output"] }, Open ]], Cell["Look at:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(f'\)[x]\)], "Input"], Cell[BoxData[ \(3.1`\ \[ExponentialE]\^\(\(-10\)\ x\)\)], "Output"] }, Open ]], Cell[TextData[{ "OK,\n ", Cell[BoxData[ \(\(f'\)[x] = 3.1 E\^\(\(-10\) x\)\)]], ".\nThe fundamental formula tells you that\n ", Cell[BoxData[ RowBox[{\(\[Integral]\_0.2\%0.9 3.1 \( E\^\(\(-10\) x\)\) \[DifferentialD]x\), "=", RowBox[{\(\[Integral]\_0.2\%0.9\( f'\)[x] \[DifferentialD]x\), "=", RowBox[{\(f[x]\), SubsuperscriptBox[ StyleBox["|", FontSize->24], "0.2", "0.9"]}]}]}]]], " \nwhich is given by:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(f[0.9] - f[0.2]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.04191568076408307`\)], "Output"] }, Open ]], Cell["Check:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_0.2\%0.9\( 3.1\ E\^\(\(-10\)\ x\)\) \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.041915680764083064`\)], "Output"] }, Open ]], Cell["Getting good at this.", "SmallText"] }, Closed]], Cell["T.1.b.vii)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ StyleBox[\(\[Integral]\_0\%1\( 1\/\(1 + 9 x\^2\)\) \[DifferentialD]x\), FontSize->16]]] }], "Text"], Cell["Answer:", "Special1"], Cell["Try:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(f[x_] = ArcTan[9\ x]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(ArcTan[9\ x]\)], "Output"] }, Open ]], Cell["Look at:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(f'\)[x]\)], "Input"], Cell[BoxData[ \(9\/\(1 + 81\ x\^2\)\)], "Output"] }, Open ]], Cell[TextData[{ "Ugh. You wanted\n ", Cell[BoxData[ RowBox[{\(\(f'\)[x]\), "=", StyleBox[\(1\/\(1 + 9 x\^2\)\), FontSize->16]}]]], ". \n This is way off. \nEvidently the ", Cell[BoxData[ \(9\)]], " was squared in the denominator.\nFix this by trying:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(f[x_] = ArcTan[3\ x]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(ArcTan[3\ x]\)], "Output"] }, Open ]], Cell["Look at:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(f'\)[x]\)], "Input"], Cell[BoxData[ \(3\/\(1 + 9\ x\^2\)\)], "Output"] }, Open ]], Cell[TextData[{ "You wanted ", Cell[BoxData[ RowBox[{\(\(f'\)[x]\), "=", StyleBox[\(1\/\(1 + 9 x\^2\)\), FontSize->16]}]]], ". But this ", Cell[BoxData[ \(f[x]\)]], " doesn't work because it carries an extra factor on top. Adjust ", Cell[BoxData[ \(f[x]\)]], " to kill the extra factor:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(f[x_] = 1\/3\ ArcTan[3\ x]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\/3\ ArcTan[3\ x]\)], "Output"] }, Open ]], Cell["Look at:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(f'\)[x]\)], "Input"], Cell[BoxData[ \(1\/\(1 + 9\ x\^2\)\)], "Output"] }, Open ]], Cell[TextData[{ "You wanted ", Cell[BoxData[ RowBox[{\(\(f'\)[x]\), "=", StyleBox[\(1\/\(1 + 9 x\^2\)\), FontSize->16]}]]], " and you got it." }], "SmallText"], Cell[TextData[{ "The fundamental formula tells you that\n ", Cell[BoxData[ RowBox[{ StyleBox[\(\[Integral]\_0\%1\( 1\/\(1 + 9 x\^2\)\) \[DifferentialD]x\), FontSize->16], "=", RowBox[{\(\[Integral]\_0\%1\( f'\)[x] \[DifferentialD]x\), "=", RowBox[{\(f[x]\), SubsuperscriptBox[ StyleBox["|", FontSize->24], "0", "1"]}]}]}]]], "\nwhich is given by: " }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(f[1] - f[0]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(ArcTan[3]\/3\)], "Output"] }, Open ]], Cell["If you want a decimal, then use:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(N[f[1] - f[0]]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.4163485907994181`\)], "Output"] }, Open ]], Cell["Check:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(NIntegrate[1\/\(1 + 9\ x\^2\), {x, 0, 1}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.4163485907994179`\)], "Output"] }, Open ]], Cell["Right on.", "SmallText"] }, Closed]], Cell["T.1.b.viii)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ " ", Cell[BoxData[ \(\[Integral]\_0\%6\( E\^\(-Sin[4 x]\)\) \[DifferentialD]x\)]] }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "No fair! You want a function ", Cell[BoxData[ \(f[x]\)]], " with \n ", Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "=", \(E\^\(-Sin[4 x]\)\)}]]], "\nBut no function ", Cell[BoxData[ \(f[x]\)]], " with\n ", Cell[BoxData[ \(\(\(\(f'\)[x]\)\(\ \)\)\)]], " anything like ", Cell[BoxData[ \(E\^\(-Sin[4 x]\)\)]], " comes to mind. \n See whether ", StyleBox["Mathematica", FontSlant->"Italic"], " can handle it." }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_0\%6\( E\^\(-Sin[4\ x]\)\) \[DifferentialD]x\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\[Integral]\_0\%6\( \[ExponentialE]\^\(-Sin[ 4\ x]\)\) \[DifferentialD]x\)], "Output"] }, Open ]], Cell["When you get tired of waiting, abort the calculation.", "Special2"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can't do it, either. The trouble is not with us, and the trouble is not \ with ", StyleBox["Mathematica", FontSlant->"Italic"], " or with mathematics. The real trouble is that ", Cell[BoxData[ \(f[x]\)]], " is not neatly expressible in terms of the everyday functions everyone has \ heard about. \nThe fundamental formula is of no help here.\nTo calculate this \ integral, you are forced to forgo an exact answer in favor of an approximate \ estimate:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(NIntegrate[E\^\(-Sin[4\ x]\), {x, 0, 6}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(7.467353218590222`\)], "Output"] }, Open ]], Cell["\<\ This number is an approximation of the measurement of the shaded \ area in the following plot:\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(\(FilledPlot[E\^\(-Sin[4\ x]\), {x, 0, 6}, \n\t AspectRatio \[Rule] 1\/GoldenRatio];\)\)], "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 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Exponential growth is kicking in. Today you owe this many dollars:\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Debt[10]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(242112.60468920428`\)], "Output"] }, Open ]], Cell[TextData[{ "That's a lot of bread - almost a quarter of a million smackers.\nIf you \ keep going on this way, here's how many dollars you'll owe ", Cell[BoxData[ \(10\)]], " years from now:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Debt[20]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1.2823691282125113`*^6\)], "Output"] }, Open ]], Cell["Out of sight.", "SmallText"] }, Open ]], Cell["T.3.a.ii)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Get the formula for ", Cell[BoxData[ \(Debt[t]\)]], " in part a.i) as the solution of a differential equation." }], "Text"], Cell["Answer:", "Special1"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[Debt, t, x, Derivative, r];\)\), "\n", \(Simplify[ DSolve[{\(Debt'\)[t]\ == \ 10000\ E\^\(r\ t\), \n\ \ \ \ \ \ \ \ Debt[0]\ == \ 10000}, Debt[t], t] /. r -> 0.15]\n\)}], "Input"], Cell[BoxData[ \({{Debt[t] \[Rule] \(-56666.66666666667`\) + 66666.66666666667`\ \[ExponentialE]\^\(0.15`\ t\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "This is the same formula you got in part i).\nThis is no accident because \ the fundamental formula says\n ", Cell[BoxData[ \(Debt[t] - 10000\)]], " \n ", Cell[BoxData[ \(\(\(=\)\(Debt[t] - Debt[0]\)\)\)]], " \n ", Cell[BoxData[ \(\(\(=\)\(\[Integral]\_o\%t\( Debt'\)[x] \[DifferentialD]x\)\)\)]], " \n ", Cell[BoxData[ \(\(\(=\)\(\[Integral]\_o\%t 10000 \( E\^\(0.15 x\)\) \[DifferentialD]x\)\)\)]], ".\nThis is the same as saying\n ", Cell[BoxData[ \(\(Debt'\)[t] = 10000 E\^\(0.15 t\)\)]], " and ", Cell[BoxData[ \(Debt[0] = 10000\)]], ". " }], "SmallText"] }, Closed]], Cell["T.3.b.i)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Ten years ago (", Cell[BoxData[ \(t = 0\)]], ") you owed ", Cell[BoxData[ \($10, 000\)]], " and you continued to borrow at a rate of \n ", Cell[BoxData[ \(10000 E\^\(0.25 t\)\)]], "\nadditional dollars per year for ", Cell[BoxData[ \(0 \[LessEqual] t \[LessEqual] 10\)]], ".\nFortunately your dowager auntie loved you enough to give you a trust \ fund that pays you at the rate of ", Cell[BoxData[ \($50, 000\)]], " per year for life.\nTen years ago you decided to use this trust money to \ help pay off your debt.\nPlot your debt as a function of ", Cell[BoxData[ \(t\)]], " for ", Cell[BoxData[ \(0 \[LessEqual] t \[LessEqual] 10\)]], ". \nDo you ever get out of debt? \nHow much do you owe today?\nWhat are \ your prospects ", Cell[BoxData[ \(10\)]], " years from now?" }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Measure ", Cell[BoxData[ \(t\)]], " in years with ", 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If you keep going on this way, here's how many dollars \ you'll owe ten years from now:\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Debt[20]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(4.906526364103064`*^6\)], "Output"] }, Open ]], Cell["\<\ Almost five million dollars. Better find another rich aunt soon.\ \>", "SmallText"] }, Closed]], Cell["T.3.b.ii)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Get the formula for ", Cell[BoxData[ \(Debt[t]\)]], " in part b.i) as the solution of a differential equation." }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Ten years ago (", Cell[BoxData[ \(t = 0\)]], ") you owed ", Cell[BoxData[ \($10, 000\)]], " and you continued to borrow at a rate of \n ", Cell[BoxData[ \(10000 E\^\(0.25 t\)\)]], " additional dollars per year for ", Cell[BoxData[ \(0 \[LessEqual] t \[LessEqual] 10\)]], ".\nThe trust fund that pays you at the rate of ", Cell[BoxData[ \($50, 000\)]], " per year for life." }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[Debt, t, x, r, Derivative];\)\), "\n", \(Simplify[ DSolve[{\(Debt'\)[t]\ == \ 10000\ E\^\(r\ t\)\ - \ 50000, \n\ \ \ \ \ \ \ \ Debt[ 0]\ == \ 10000}, Debt[t], t] /. r -> 0.25]\)}], "Input"], Cell[BoxData[ \({{Debt[t] \[Rule] \(-30000.`\) + 40000.`\ \[ExponentialE]\^\(0.25`\ t\) - 50000.`\ t}}\)], "Output"] }, Open ]], Cell[TextData[{ "This is the same formula you got in part i).\nThis is no accident because \ the fundamental formula says\n ", Cell[BoxData[ \(Debt[t] - 10000\)]], " \n ", Cell[BoxData[ \(\(\(=\)\(Debt[t] - Debt[0]\)\)\)]], " \n ", Cell[BoxData[ \(\(\(=\)\(\[Integral]\_o\%t\( Debt'\)[x] \[DifferentialD]x\)\)\)]], " \n ", Cell[BoxData[ \(\(\(=\)\(\[Integral]\_0\%t 10000 E\^\(0.25 x\) - 50000 \[DifferentialD]x\)\)\)]], ".\nThis is the same as saying\n ", Cell[BoxData[ \(\(Debt'\)[t] = 10000 E\^\(0.25 t\) - 50000\)]], " and ", Cell[BoxData[ \(Debt[0] = 10000\)]], ". " }], "SmallText"] }, Closed]] }, Closed]], Cell[" Give It a Try Problems", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "G.2) How does ", StyleBox["Mathematica", FontSlant->"Italic"], " calculate an integral?\n How do you calculate an integral by \ hand?*" }], "Subsection", CellTags->"2.02.G2"], Cell[TextData[{ "The fundmental formula\n ", Cell[BoxData[ \(\[Integral]\_a\%b\( f'\)[x] \[DifferentialD]x = f[b] - f[a]\)]], "\nis so important that it is programmed into ", StyleBox["Mathematica", FontSlant->"Italic"], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x, a, b, Derivative];\)\), "\n", \(\[Integral]\_a\%b\( f'\)[x] \[DifferentialD]x\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-f[a]\) + f[b]\)], "Output"] }, Open ]], Cell[TextData[{ "In fact, ", StyleBox["Mathematica ", FontSlant->"Italic"], "uses the fundamental formula to calculate integrals. Here's how it goes:\n\ When you feed a formula for a function ", Cell[BoxData[ \(g[x]\)]], " into ", StyleBox["Mathematica", FontSlant->"Italic"], " and ask ", StyleBox["Mathematica", FontSlant->"Italic"], " to calculate ", Cell[BoxData[ \(\[Integral]\_a\%t g[x] \[DifferentialD]x\)]], ", \n", StyleBox["Mathematica", FontSlant->"Italic"], " looks for a function ", Cell[BoxData[ \(f[t]\)]], " with \n ", Cell[BoxData[ \(\(f'\)[t] = g[t]\)]], " \nand then it reports \n ", Cell[BoxData[ \(\[Integral]\_a\%t g[x] \[DifferentialD]x = f[t] - f[a]\)]], ".\nExamples:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[g, x, t, a];\)\), "\n", \(\(g[x_] = Cos[x];\)\), "\n", \(\[Integral]\_a\%t g[x] \[DifferentialD]x\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-Sin[a]\) + Sin[t]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[g, x, t, a];\)\), "\n", \(\(g[x_] = x;\)\), "\n", \(\[Integral]\_a\%t g[x] \[DifferentialD]x\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-\(a\^2\/2\)\) + t\^2\/2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[g, x, t, a];\)\), "\n", \(\(g[x_] = E\^x;\)\), "\n", \(\[Integral]\_a\%t g[x] \[DifferentialD]x\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-\[ExponentialE]\^a\) + \[ExponentialE]\^t\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[g, x, t, a];\)\), "\n", \(\(g[x_] = \(2\ E\^\(-x\^2\)\)\/\@\[Pi];\)\), "\n", \(\[Integral]\_a\%t g[x] \[DifferentialD]x\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(2\ \((\(-\(1\/2\)\)\ \@\[Pi]\ Erf[a] + 1\/2\ \@\[Pi]\ Erf[t])\)\)\/\@\ \[Pi]\)], "Output"] }, Open ]], Cell[TextData[{ "Once ", Cell[BoxData[ \(f[t]\)]], " with ", Cell[BoxData[ \(\(f'\)[t] = g[t]\)]], " is found, then the reported answer is guaranteed to be correct because \ the fundamental formula says that\n ", Cell[BoxData[ \(\[Integral]\_a\%t g[ x] \[DifferentialD]x = \(\[Integral]\_a\%t\( f'\)[ x] \[DifferentialD]x = f[t] - f[a]\)\)]], ".\nWhen you calculate integrals, you do the same thing. The only \ difference is that ", StyleBox["Mathematica", FontSlant->"Italic"], " has a great advantage over you because it is faster than you and has a \ bigger collection of functions to choose from than you have.\nNevertheless, \ for simple cases, you should be able to beat ", StyleBox["Mathematica", FontSlant->"Italic"], " by doing integrals in your head faster than you can type them into ", StyleBox["Mathematica", FontSlant->"Italic"], ".\nScientists who cannot calculate an integral like \n ", Cell[BoxData[ \(\[Integral]\_0\%1\( x\^2\) \[DifferentialD]x\)]], " \nin their heads quickly without machine help are labeled illiterate just \ as people who can't spell \"cat\" are labeled illiterate.\nFind the exact \ values of the following integrals in your head; check yourself with ", StyleBox["Mathematica", FontSlant->"Italic"], ". 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MultilineFunction->None], "[", "t", "]"}], "=", \(g[t]\)}]]], " and ", Cell[BoxData[ \(y[0] = 0\)]], "\ntogether with the plot of ", Cell[BoxData[ \(f[t]\)]], " from above" }], "Text"], Cell["This code is from C&M Lesson 1.07", "Special5"], Cell[CellGroupData[{ Cell[BoxData[{ \(\n\(a\ = \ 0;\)\), "\n", \(\(b\ = \ 5;\)\), "\n", \(\(starter\ = \ 0;\)\n\), "\n", \(\(y[a]\ = \ starter;\)\), "\n", \(\(beginner\ = \ {a, y[a]};\)\n\), "\n", \(\(Clear[t, y, next, jump, iterations, point, k];\)\n\), "\n", \(\(jump[iterations_]\ := \ \(b - a\)\/iterations;\)\n\), "\n", \(\(next[{t_, y_}, iterations_]\ := \ \n\ \ \ \ \ \ \ {t, y}\ + \ jump[iterations] {1, \ g[t]};\)\n\ \ \ \ \ \ \ \), "\n", \(\(point[0, iterations_]\ = \ beginner;\)\), "\n", \(\(point[k_, iterations_]\ := \ \(point[k, iterations]\ = \n\ \ \ \ \ \ \ \ N[ next[point[k - 1, iterations], iterations]]\);\)\n\ \ \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \), \ "\n", \(\(Clear[Euler];\)\), "\n", \(\(Euler[iterations_]\ := \ \n 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