(*********************************************************************** 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). 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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[ 385418, 9024]*) (*NotebookOutlinePosition[ 386134, 9049]*) (* CellTagsIndexPosition[ 386090, 9045]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Using ", StyleBox["Mathematica", FontSlant->"Italic"], " to Illustrate the Race Track Principle in Calculus" }], "Title", FontSize->24], Cell[TextData[{ "by Alkiviadis G. Akrita", Cell[BoxData[ \(TraditionalForm\`s\)]], " and Zamir Bavel" }], "Author", CellMargins->{{Inherited, 10}, {Inherited, Inherited}}], Cell[TextData[{ "\nAn astonishing innovation in the teaching of Calculus is the use of the \ ", StyleBox["race track principle. ", FontSlant->"Italic"], "This little-known principle is elegantly used in the Calculus and ", StyleBox["Mathematica ", FontSlant->"Italic"], "(C&M) series of books ([1], [2], [4]) to explain and prove many concepts. \ Below we present two different versions of this principle and, using ", StyleBox["Mathematica,", FontSlant->"Italic"], " we show how it is used to explain the power series expansion of a \ function and the round-off errors that appear in certain computations. " }], "Text", CellMargins->{{Inherited, 10}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman"], Cell[TextData[{ "Although implicitly used in most Calculus books (see for example [5] and \ [6]) the Race Track Principle is extensively used in the Calculus and ", StyleBox["Mathematica ", FontSlant->"Italic"], "(C&M) book series by Davis, Porta and Uhl ([1]-[4]). This series of books \ is a valuable and well thought-out method for teaching Calculus. As \ indicated by the title of this series of books, ", StyleBox["Mathematica ", FontSlant->"Italic"], "facilitates the exploration." }], "Text"], Cell[TextData[{ "Following their lead, we use ", StyleBox["Mathematica", FontSlant->"Italic"], " to present two versions of the Race Track Principle and then, in the two \ sections that follow, we show how this principle is applied." }], "Text"], Cell[CellGroupData[{ Cell["First Version of the Race Track Principle", "Subsection", CellDingbat->None], Cell[TextData[{ StyleBox["Horses: ", FontWeight->"Bold"], "If two horses start a race at the same point, then the faster horse is \ always ahead.\n", StyleBox["Functions: ", FontWeight->"Bold"], "If ", Cell[BoxData[ \(TraditionalForm\`f[a]\)]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`g[a]\)]], StyleBox[" ", FontSlant->"Italic"], "and ", Cell[BoxData[ \(TraditionalForm\`\(f'\)[x]\)]], StyleBox[" ", FontSlant->"Italic"], "\[GreaterEqual]", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(g'\)[x]\)]], StyleBox[" ", FontSlant->"Italic"], "for ", StyleBox["x ", FontSlant->"Italic"], "\[GreaterEqual]", StyleBox[" a ", FontSlant->"Italic"], "then ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], StyleBox[" \[GreaterEqual] ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`g[x]\)]], StyleBox[" ", FontSlant->"Italic"], "for ", StyleBox["x ", FontSlant->"Italic"], "\[GreaterEqual]", StyleBox[" a", FontSlant->"Italic"], ".\n\nThis version of the Race Track Principle is good for explaining why \ one function plots out above another function. 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Nb5_N`00ofmk8Fmk003oKg/QKg/00?m_Nb5_N`00ofmk8Fmk0000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.125656, -0.204741, 0.00437098, 0.0170584}}], Cell[TextData[{ "The function ", Cell[BoxData[ \(TraditionalForm\`\(f[x]\ \)\)]], StyleBox["= Sin[x]+ArcSin[x]", FontSlant->"Italic"], " (dashed line) plots above the function ", Cell[BoxData[ \(TraditionalForm\`g[x]\)]], StyleBox[" = x.", FontSlant->"Italic"], " " }], "Caption"], Cell[TextData[{ "The reason the plot in Figure 1 turned out this way can be easily \ explained using (the first version of) the Race Track Principle. Since ", Cell[BoxData[ \(TraditionalForm\`f[0]\)]], StyleBox[" ", FontSlant->"Italic"], "= ", Cell[BoxData[ \(TraditionalForm\`g[0]\)]], " = 0," }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({f[0], g[0]}\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ \({0, 0}\)], "Output", CellLabel->"Out[10]="] }, Open ]], Cell[TextData[{ "the two functions start their race at ", StyleBox["x = 0 ", FontSlant->"Italic"], "together. Now, ", Cell[BoxData[ \(TraditionalForm\`\(f'\)[x]\)]], StyleBox[" ", FontSlant->"Italic"], "and", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(g'\)[x]\)]], StyleBox[" ", FontSlant->"Italic"], "for ", StyleBox["x ", FontSlant->"Italic"], "\[GreaterEqual]", StyleBox[" 0 ", FontSlant->"Italic"], "come into play:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({\(f'\)[x], \(g'\)[x]}\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \({1\/\@\(1 - x\^2\) + Cos[x], 1}\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell[TextData[{ "For 0 \[LessEqual]", StyleBox[" x ", FontSlant->"Italic"], "\[LessEqual]", StyleBox[" 1", FontSlant->"Italic"], " we have ", Cell[BoxData[ \(TraditionalForm\`\(f'\)[x]\)]], StyleBox[" \[GreaterEqual] 2 ", FontSlant->"Italic"], ">", StyleBox[" 1 = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(g'\)[x]\)]], ";", StyleBox[" ", FontSlant->"Italic"], "that is to say,", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`f[x]\)]], StyleBox[" = Sin[x]+ArcSin[x]", FontSlant->"Italic"], " grows faster than ", Cell[BoxData[ \(TraditionalForm\`g[x]\)]], StyleBox[" = x ", FontSlant->"Italic"], "for 0 \[LessEqual]", StyleBox[" x ", FontSlant->"Italic"], "\[LessEqual]", StyleBox[" 1. ", FontSlant->"Italic"], "By the Race Track Principle,", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`f[x]\ \[GreaterEqual] \ g[x]\)]], StyleBox[" ", FontSlant->"Italic"], "for 0 \[LessEqual]", StyleBox[" x ", FontSlant->"Italic"], "\[LessEqual]", StyleBox[" 1 ", FontSlant->"Italic"], "and this explains the plot." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Second Version of the Race Track Principle", "Subsection", CellDingbat->None], Cell[TextData[{ StyleBox["Horses: ", FontWeight->"Bold"], "If two horses are tied at one point, and they run at the same speed at \ this point, then they run close together near this point.\n", StyleBox["Functions: ", FontWeight->"Bold"], "If ", Cell[BoxData[ \(TraditionalForm\`f[a]\)]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`g[a]\)]], StyleBox[" ", FontSlant->"Italic"], "and ", Cell[BoxData[ \(TraditionalForm\`\(f'\)[a]\)]], StyleBox[" ", FontSlant->"Italic"], "=", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(g'\)[a]\)]], ", then the two functions plot out nearly the same as ", StyleBox["x ", FontSlant->"Italic"], "advances from a little bit to the left of ", StyleBox["x = a", FontSlant->"Italic"], " to a little bit to the right of ", StyleBox["x = a.", FontSlant->"Italic"], "\n\n(The usage of the expressions \"nearly\", \"from a little bit to the \ left of x\" and \"to a little bit to the right of ", StyleBox["x", FontSlant->"Italic"], "\"", StyleBox[" ", FontSlant->"Italic"], "is based on intuition. With the help of ", StyleBox["Mathematica", FontSlant->"Italic"], " students can \"zoom in\" and get a clear estimate of the numerical values \ involved. We do not present the zooming process in this article.)\nThis \ version of the Race Track Principle is good for explaining why at the point ", StyleBox["x = a ", FontSlant->"Italic"], "we have a smooth transition from ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], StyleBox[" ", FontSlant->"Italic"], "to ", Cell[BoxData[ \(TraditionalForm\`g[x]\)]], StyleBox[" ", FontSlant->"Italic"], "(or vice-versa) as ", StyleBox["x ", FontSlant->"Italic"], "advances accross ", StyleBox["x = a. ", FontSlant->"Italic"], "For example, consider the functions ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`Cosh[x]\)], FontSlant->"Italic"], StyleBox["+Cos[x]", FontSlant->"Italic"], " and ", Cell[BoxData[ \(TraditionalForm\`g[x]\)]], StyleBox[" = 2 ", FontSlant->"Italic"], "and look at their values and the values of their first few derivatives at \ ", StyleBox["x = 0", FontSlant->"Italic"], ": " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Clear[f, g, x]\), \(f[x_] = Cosh[x] + Cos[x]; \ng[x_] = 2\ ; \n\n{f[0], g[0]}\)}], "Input", CellLabel->"In[67]:="], Cell[BoxData[ \({2, 2}\)], "Output", CellLabel->"Out[68]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({\(f'\)[0], \(g'\)[0]}\)], "Input", CellLabel->"In[57]:="], Cell[BoxData[ \({0, 0}\)], "Output", CellLabel->"Out[57]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({\(\(f'\)'\)[0], \(\(g'\)'\)[0]}\)], "Input", CellLabel->"In[61]:="], Cell[BoxData[ \({0, 0}\)], "Output", CellLabel->"Out[61]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({\(\(\(f'\)'\)'\)[0], \(\(\(g'\)'\)'\)[0]}\)], "Input", CellLabel->"In[62]:="], Cell[BoxData[ \({0, 0}\)], "Output", CellLabel->"Out[62]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({\(\(\(\(f'\)'\)'\)'\)[0], \(\(\(\(g'\)'\)'\)'\)[0]}\)], "Input", CellLabel->"In[63]:="], Cell[BoxData[ \({2, 0}\)], "Output", CellLabel->"Out[63]="] }, Open ]], Cell[TextData[{ "The first three derivatives are the same at ", StyleBox["x = 0", FontSlant->"Italic"], ". Both functions start their race at ", StyleBox["x = 0 ", FontSlant->"Italic"], "together and their growth rates are the same at ", StyleBox["x = 0", FontSlant->"Italic"], ". According to (the second version of) the Race Track Principle the two \ functions plot out nearly the same as ", StyleBox["x ", FontSlant->"Italic"], "advances from a little bit to the left of ", StyleBox["x = 0", FontSlant->"Italic"], " to a little bit to the right of ", StyleBox["x = 0. 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{-2.86837, -2.20315, 0.0211079, 0.0213458}}], Cell[TextData[{ "The functions ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`Cos[x]\)]], " (dashed line) and ", Cell[BoxData[ \(TraditionalForm\`g[x]\)]], StyleBox[" = 1 - ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^2\/2\)]], StyleBox[" ", FontSlant->"Italic"], "have strikingly similar plots for ", StyleBox["-1 ", FontSlant->"Italic"], "\[LessEqual]", StyleBox[" x ", FontSlant->"Italic"], "\[LessEqual]", StyleBox[" 1", FontSlant->"Italic"], " " }], "Caption"], Cell[TextData[{ "The two functions are nearly identical near ", StyleBox["x = 0. ", FontSlant->"Italic"], " As ", StyleBox["x", FontSlant->"Italic"], " advances from the left of ", StyleBox["0", FontSlant->"Italic"], " to the right of ", StyleBox["0", FontSlant->"Italic"], ", we can smoothly transfer from one curve to the other." }], "Text"], Cell[TextData[{ "This phenomenon has to do with derivatives. Consider the two functions at \ ", StyleBox["x = 0 ", FontSlant->"Italic"], " and their first four derivatives there:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({f[0], g[0]}\)], "Input", CellLabel->"In[26]:="], Cell[BoxData[ \({1, 1}\)], "Output", CellLabel->"Out[26]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({\(f'\)[0], \(g'\)[0]}\)], "Input", CellLabel->"In[27]:="], Cell[BoxData[ \({0, 0}\)], "Output", CellLabel->"Out[27]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({\(\(f'\)'\)[0], \(\(g'\)'\)[0]}\)], "Input", CellLabel->"In[28]:="], Cell[BoxData[ \({\(-1\), \(-1\)}\)], "Output", CellLabel->"Out[28]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({\(\(\(f'\)'\)'\)[0], \(\(\(g'\)'\)'\)[0]}\)], "Input", CellLabel->"In[29]:="], Cell[BoxData[ \({0, 0}\)], "Output", CellLabel->"Out[29]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({\(\(\(\(f'\)'\)'\)'\)[0], \(\(\(\(g'\)'\)'\)'\)[0]}\)], "Input", CellLabel->"In[30]:="], Cell[BoxData[ \({1, 0}\)], "Output", CellLabel->"Out[30]="] }, Open ]], Cell[TextData[{ "Both functions go through ", Cell[BoxData[ \({0, 1}\)]], " and they have the same first, second, and third derivatives at ", Cell[BoxData[ \(x\ = \ 0\)]], ". They differ only when we get to the fourth derivatives." }], "Text"], Cell[TextData[{ "We are now ready for the following definition: We say that ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], StyleBox[" ", FontSlant->"Italic"], "and ", Cell[BoxData[ \(TraditionalForm\`g[x]\)]], StyleBox[" ", FontSlant->"Italic"], "have ", StyleBox["order of contact", FontSlant->"Italic"], " ", StyleBox["m", FontSlant->"Italic"], " at ", StyleBox["x = a", FontSlant->"Italic"], " if :\n\t", Cell[BoxData[ \(TraditionalForm\`f[a]\ = \ g[a]\)]], StyleBox[",\n\t", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(f'\)[a]\)]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(g'\)[a]\)]], StyleBox[",\n\t", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\(f'\)'\)[a]\)]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\(g'\)'\)[a]\)]], StyleBox[",\n\t...\n\t", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(f\^\((m - 1)\)\), "TraditionalForm"], "[", "a", "]"}], TraditionalForm]]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(g\^\((m - 1)\)\), "TraditionalForm"], "[", "a", "]"}], TraditionalForm]]], ", and\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(f\^\((m)\)\), "TraditionalForm"], "[", "a", "]"}], " "}], TraditionalForm]]], StyleBox["= ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(g\^\((m)\)\), "TraditionalForm"], "[", "a", "]"}], TraditionalForm]]], StyleBox[",", FontSlant->"Italic"], "\nso that the functions and their first ", Cell[BoxData[ \(m\)]], " derivatives agree at ", StyleBox["x = a.", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ "Now, according to (the second version of) the Race Track Principle, if ", Cell[BoxData[ \(TraditionalForm\`f[a]\)]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`g[a]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(f'\)[a]\)]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(g'\)[a]\)]], ", then the two functions plot out nearly the same as ", StyleBox["x", FontSlant->"Italic"], " advances from a little bit to the left of ", StyleBox["x = a", FontSlant->"Italic"], " to a little bit to the right of ", StyleBox["x = a", FontSlant->"Italic"], ". This explains why, when we have order of contact ", StyleBox["1", FontSlant->"Italic"], " at ", StyleBox["x = a", FontSlant->"Italic"], ", then we have a smooth transition as ", StyleBox["x", FontSlant->"Italic"], " advances from the left of ", StyleBox["x = a", FontSlant->"Italic"], " to the right of ", StyleBox["x = a", FontSlant->"Italic"], ". (See also the example in the second version of the Race Track Principle \ in the Introduction.)\nTo see why it is that, when we have order of contact ", StyleBox["2", FontSlant->"Italic"], " at ", StyleBox["x = a", FontSlant->"Italic"], ", then we can expect an even smoother transition as ", StyleBox["x", FontSlant->"Italic"], " advances from the left of ", StyleBox["x = a", FontSlant->"Italic"], " to the right of ", StyleBox["x = a", FontSlant->"Italic"], ", recall that order of contact ", StyleBox["2", FontSlant->"Italic"], " at ", StyleBox["x = a", FontSlant->"Italic"], " means\n\t", Cell[BoxData[ \(TraditionalForm\`\(f[a]\ \)\)]], StyleBox["= ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(g[a], \)\)]], StyleBox["\n\t", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(f'\)[a]\)]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(g'\)[a]\)]], StyleBox[", ", FontSlant->"Italic"], "and\n\t", Cell[BoxData[ \(TraditionalForm\`\(\(f'\)'\)[a]\)]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\(g'\)'\)[a]\)]], StyleBox[".\n", FontSlant->"Italic"], "The fact that ", Cell[BoxData[ \(TraditionalForm\`\(\(f'\)'\)[a]\)]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\(g'\)'\)[a]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(f'\)[a]\)]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(g'\)[a]\)]], " ensures that ", Cell[BoxData[ \(TraditionalForm\`\(f'\)[x]\)]], " is very close to ", Cell[BoxData[ \(TraditionalForm\`\(g'\)[x]\)]], " as ", StyleBox["x", FontSlant->"Italic"], " advances accross ", StyleBox["x = a", FontSlant->"Italic"], ". This, in turn, means that ", Cell[BoxData[ \(TraditionalForm\`\(f[x]\ \ \)\)]], "is very, very close to ", Cell[BoxData[ \(TraditionalForm\`g[x]\)]], " as ", StyleBox["x", FontSlant->"Italic"], " advances from the left of ", StyleBox["x = a", FontSlant->"Italic"], " to the right of ", StyleBox["x = a", FontSlant->"Italic"], ". (It is even more so for order of contact 3 at ", StyleBox["x = a ", FontSlant->"Italic"], " as was the case for the above example.)" }], "Text"], Cell[TextData[{ "It is now a simple matter to compute the power series expansion of a \ function. Given a function ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], ", the expansion of ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], " in powers of ", StyleBox["x ", FontSlant->"Italic"], "is the expression \n\t", Cell[BoxData[ \(TraditionalForm\`a[0]\)]], StyleBox[" + ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`a[1]\)]], StyleBox[" x + ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`a[2]\)]], StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`a[3]\)]], StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^3\)]], " + \[CenterEllipsis]+", Cell[BoxData[ \(TraditionalForm\`\(\ a[k]\)\)]], StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^k\)]], " + ", Cell[BoxData[ \(TraditionalForm\`a[k + 1]\)]], StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^\(k + 1\)\)]], " + \[CenterEllipsis]\nwhere the numbers \n\t", Cell[BoxData[ \(TraditionalForm\`a[0]\)]], StyleBox[", ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`a[1]\)]], StyleBox[", ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`a[2]\)]], StyleBox[",", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`a[3]\)]], StyleBox[",", FontSlant->"Italic"], " ...,", Cell[BoxData[ \(TraditionalForm\`\(\ a[k]\)\)]], StyleBox[",", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`a[k + 1]\)]], StyleBox[", ...\n", FontSlant->"Italic"], "are chosen so that for every positive integer ", StyleBox["m", FontSlant->"Italic"], ", the function ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], " and the polynomial\n\t", Cell[BoxData[ \(TraditionalForm\`a[0]\)]], StyleBox[" + ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`a[1]\)]], StyleBox[" x + ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`a[2]\)]], StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`a[3]\)]], StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^3\)]], " + \[CenterEllipsis]+ ", Cell[BoxData[ \(TraditionalForm\`a[m - 1]\)]], StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^\(m - 1\)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`a[m]\)]], StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^m\)]], "\nhave order of contact", StyleBox[" m ", FontSlant->"Italic"], "at", StyleBox[" x = 0.\n\n", FontSlant->"Italic"], "Let us expand the function ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`Cos[x]\)]], " in powers of ", StyleBox["x", FontSlant->"Italic"], ",", StyleBox[" ", FontSlant->"Italic"], "up to degree 4. Notice that ", StyleBox["Mathematica", FontSlant->"Italic"], " has the special function ", StyleBox["Series", FontFamily->"Courier"], " for obtaining such expansions up to any degree. Below we expand Cos[x] \ up to degree 6:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[Cos[x], {x, 0, 6}] // Normal\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[ \(1 - x\^2\/2 + x\^4\/24 - x\^6\/720\)], "Output", CellLabel->"Out[5]="] }, Open ]], Cell[TextData[{ "and note that 24 = 4! and 720 = 6!. \n\nTo get an idea how this algorithm \ works, we start by entering ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], StyleBox[" ", FontSlant->"Italic"], "and the fourth degree polynomial of the form ", Cell[BoxData[ \(TraditionalForm\`a[0]\)]], StyleBox[" + ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`a[1]\)]], StyleBox[" x + ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`a[2]\)]], StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`a[3]\)]], StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^3\)]], " + ", Cell[BoxData[ \(TraditionalForm\`a[4]\)]], StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^4\)]], ": " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Clear[f, g, x, a, k]\), \(f[x_] = Cos[x]; \ng[x_] = \[Sum]\+\(k = 0\)\%4 a[k]\ x\^k\)}], "Input", CellLabel->"In[31]:="], Cell[BoxData[ \(a[0] + x\ a[1] + x\^2\ a[2] + x\^3\ a[3] + x\^4\ a[4]\)], "Output", CellLabel->"Out[32]="] }, Open ]], Cell["\<\ According to the preceding discussion we need the following equations: \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eq1 = f[0] == g[0]\)], "Input", CellLabel->"In[33]:="], Cell[BoxData[ \(1 == a[0]\)], "Output", CellLabel->"Out[33]="] }, Open ]], Cell["and ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eq2 = \(f'\)[0] == \(g'\)[0]\)], "Input", CellLabel->"In[34]:="], Cell[BoxData[ \(0 == a[1]\)], "Output", CellLabel->"Out[34]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eq3 = \(\(f'\)'\)[0] == \(\(g'\)'\)[0]\)], "Input", CellLabel->"In[35]:="], Cell[BoxData[ \(\(-1\) == 2\ a[2]\)], "Output", CellLabel->"Out[35]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eq4 = \(\(\(f'\)'\)'\)[0] == \(\(\(g'\)'\)'\)[0]\)], "Input", CellLabel->"In[36]:="], Cell[BoxData[ \(0 == 6\ a[3]\)], "Output", CellLabel->"Out[36]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eq5 = \(\(\(\(f'\)'\)'\)'\)[0] == \(\(\(\(g'\)'\)'\)'\)[0]\)], "Input", CellLabel->"In[37]:="], Cell[BoxData[ \(1 == 24\ a[4]\)], "Output", CellLabel->"Out[37]="] }, Open ]], Cell[TextData[{ "Now we have five equations with five unknowns ", Cell[BoxData[ \(TraditionalForm\`a[0]\)]], StyleBox[", ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`a[1]\)]], StyleBox[", ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`a[2]\)]], StyleBox[",", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`a[3]\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`a[4]\)]], ". We next solve the equations, to obtain: " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(coefficients = Solve[{eq1, eq2, eq3, eq4, eq5}]\)], "Input", CellLabel->"In[41]:="], Cell[BoxData[ \({{a[0] \[Rule] 1, a[1] \[Rule] 0, a[2] \[Rule] \(-\(1\/2\)\), a[3] \[Rule] 0, a[4] \[Rule] 1\/24}}\)], "Output", CellLabel->"Out[41]="] }, Open ]], Cell["The fourth degree polynomial we seek is then ", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Clear[polynomial]\), \(polynomial[x_] = g[x] /. coefficients[\([1]\)]\)}], "Input", CellLabel->"In[42]:="], Cell[BoxData[ \(1 - x\^2\/2 + x\^4\/24\)], "Output", CellLabel->"Out[43]="] }, Open ]], Cell[TextData[{ "Indeed, it checks out that ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], StyleBox[" ", FontSlant->"Italic"], "and ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["polynomial", FontSlant->"Italic"], "[", "x", "]"}], TraditionalForm]]], StyleBox[" ", FontSlant->"Italic"], "have order of contact 4: " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{D[f[x], {x, k}], D[polynomial[x], {x, k}]} /. x \[Rule] 0, {k, 0, 4}]\)], "Input", CellLabel->"In[44]:="], Cell[BoxData[ \({{1, 1}, {0, 0}, {\(-1\), \(-1\)}, {0, 0}, {1, 1}}\)], "Output", CellLabel->"Out[44]="] }, Open ]], Cell[TextData[{ "In Figure 5 below we see the plots of ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], StyleBox[" ", FontSlant->"Italic"], "and ", StyleBox["polynomial", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\([x]\)\)]], " in the interval ", StyleBox["-2 ", FontSlant->"Italic"], "\[LessEqual] ", StyleBox["x ", FontSlant->"Italic"], "\[LessEqual] ", StyleBox["2", FontSlant->"Italic"], ": " }], "Text"], Cell[BoxData[{ \(Clear[f, polynomial, x]\), \(f[x_] = Cos[x]; \npolynomial[x_] = 1 - x\^2\/2 + x\^4\/24; \n\n Plot[{f[x], polynomial[x]}, {x, \(-2.5\), 2.5}, PlotStyle \[Rule] {{RGBColor[1, 0, 0], \ Thickness[0.01], Dashing[{0.05, 0.05}]}, Thickness[0.01]}, AspectRatio \[Rule] 1/GoldenRatio, AxesLabel \[Rule] {"\", "\< \>"}]; \)}], "Input", 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StyleBox["polynomial", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\([x]\)\)]], StyleBox[" = 1 - ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^2\/2 + x\^4\/24\)]], " we computed matches the first three (lower) terms of the expansion of ", StyleBox["Cos", FontSlant->"Italic"], "[", StyleBox["x", FontSlant->"Italic"], "] = ", StyleBox["1 - ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(x\^2\/2 + x\^4\/24\), "TraditionalForm"], "-", \(x\^6\/720\)}], TraditionalForm]]], " we computed earlier. In general, working as above we can compute an \ arbitrary number of terms. " }], "Text", CellDingbat->"\[LightBulb]"], Cell[TextData[{ "As we use more and more terms from the expansion in powers of ", StyleBox["x", FontSlant->"Italic"], ", we increase the order of contact between ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], " and the corresponding polynomial at ", StyleBox["x = 0", FontSlant->"Italic"], "." }], "Text", CellDingbat->"\[LightBulb]"], Cell[TextData[{ "As we increase the order of contact at ", StyleBox["x = 0", FontSlant->"Italic"], ", we increase the quality of the transition from one curve to the other at \ ", StyleBox["x = 0.", FontSlant->"Italic"] }], "Text", CellDingbat->"\[LightBulb]"] }, Open ]], Cell[CellGroupData[{ Cell["The Race Track Principle and Estimation of Roundoff Errors", "Section"], Cell[TextData[{ "Roundoff errors and calculators do not mix very well. For example, note \ what happens when we feed ", StyleBox["8", FontSlant->"Italic"], " decimals of ", StyleBox["10 e", FontSlant->"Italic"], " into a calculator and then compute ", Cell[BoxData[ \(TraditionalForm\`\((10\ e)\)\^2\)]], " on that basis:" }], "Text"], Cell[CellGroupData[{ Cell["N[10 E,10]", "Input", CellLabel->"In[133]:="], Cell[BoxData[ StyleBox["27.1828182845904509`", StyleBoxAutoDelete->True, PrintPrecision->10]], "Output", CellLabel->"Out[133]="] }, Open ]], Cell[TextData[{ "Compute the square of the above number and compare the result with ", Cell[BoxData[ \(TraditionalForm\`e\^2\)]], " to 7 decimals. (Computing ", Cell[BoxData[ \(TraditionalForm\`e\^2\)]], " and then rounding is a more accurate procedure.)" }], "Caption"], Cell[CellGroupData[{ Cell["{N[(27.18281828)^2,10],N[(10 E)^2,10]}", "Input", CellLabel->"In[134]:="], Cell[BoxData[ RowBox[{"{", RowBox[{ StyleBox["738.905609643502181`", StyleBoxAutoDelete->True, PrintPrecision->10], ",", StyleBox["738.905609893064951`", StyleBoxAutoDelete->True, PrintPrecision->10]}], "}"}]], "Output", CellLabel->"Out[134]="] }, Open ]], Cell["Only the first 6 decimals match.", "Caption"], Cell[TextData[{ "This behavior can be explained with (the first version of) the Race Track \ Principle. More precisely, we can use (the first version of) the Race Track \ Principle to predict the accuracy of ", StyleBox["a", FontSlant->"Italic"], " needed to maintain a desired accuracy of ", Cell[BoxData[ \(TraditionalForm\`f[a]\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], StyleBox[" ", FontSlant->"Italic"], "can be any function. 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", FontSlant->"Italic"], Cell[BoxData[ FormBox[ StyleBox[\(10\^2\), FontSlant->"Italic"], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm \`\(\[VerticalSeparator]\( x\ - \ a \[VerticalSeparator] \)\)\)]], StyleBox["\n", FontSlant->"Italic"], "Thus, for ", StyleBox["a = 10 e", FontSlant->"Italic"], ", the discrepancy between ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], StyleBox[" ", FontSlant->"Italic"], "and ", Cell[BoxData[ \(TraditionalForm\`f[a]\)]], StyleBox[" ", FontSlant->"Italic"], "is smaller than ", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ StyleBox[\(10\^2\), FontSlant->"Italic"], TraditionalForm]]], "\[VerticalSeparator]x - a\[VerticalSeparator]. " }], "Text"], Cell[TextData[{ "Now we use this information to determine the number of decimals of ", StyleBox["10", FontSlant->"Italic"], " ", StyleBox["e", FontSlant->"Italic"], " we need in order to guarantee accuracy to 4", StyleBox[" ", FontSlant->"Italic"], "decimals of", Cell[BoxData[ \(TraditionalForm\`\(\ f[10\ e]\)\)]], StyleBox[".", FontSlant->"Italic"], " The discrepancy between ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], StyleBox[" ", FontSlant->"Italic"], "and ", Cell[BoxData[ \(TraditionalForm\`f[a]\)]], StyleBox[" ", FontSlant->"Italic"], "is no more than ", Cell[BoxData[ FormBox[ StyleBox[\(10\^2\), FontSlant->"Italic"], TraditionalForm]]], "\[VerticalSeparator]x - a\[VerticalSeparator]. Therefore, if ", StyleBox["x", FontSlant->"Italic"], " is accurate to 6 decimals, that is \[VerticalSeparator]x - a\ \[VerticalSeparator]", StyleBox[" ", FontSlant->"Italic"], "\[LessEqual] ", Cell[BoxData[ \(TraditionalForm\`10\^\(-6\)\)], FontSlant->"Italic"], ", then the discrepancy betwee", Cell[BoxData[ \(TraditionalForm\`n\ \(f[x\)\)]], "]", StyleBox[" ", FontSlant->"Italic"], "and ", Cell[BoxData[ \(TraditionalForm\`f[a]\)]], " is less than", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(10\^2\), "TraditionalForm"], "\[Star]", \(10\^\(-6\)\)}], TraditionalForm]], FontSlant->"Italic"], StyleBox[" = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`10\^\(-4\)\)], FontSlant->"Italic"], "." }], "Text"], Cell[TextData[{ "Consequently, if ", StyleBox["x", FontSlant->"Italic"], " approximates ", StyleBox["a", FontSlant->"Italic"], " to ", Cell[BoxData[ \(6\)]], " decimals, then ", Cell[BoxData[ \(TraditionalForm\`f[x]\)]], " approximates ", Cell[BoxData[ \(TraditionalForm\`f[a]\)]], " to ", Cell[BoxData[ \(4\)]], " decimals." }], "Text"], Cell["In the actual computation:", "Text"], Cell[CellGroupData[{ Cell["a", "Input", CellLabel->"In[135]:="], Cell[BoxData[ \(10\ E\)], "Output", CellLabel->"Out[135]="] }, Open ]], Cell[CellGroupData[{ Cell["N[a,8]", "Input", CellLabel->"In[136]:="], Cell[BoxData[ StyleBox["27.1828182845904509`", StyleBoxAutoDelete->True, PrintPrecision->8]], "Output", CellLabel->"Out[136]="] }, Open ]], Cell[TextData[{ Cell[BoxData[ StyleBox["27.1828182845904509`", StyleBoxAutoDelete->True, PrintPrecision->8, FontSlant->"Italic"]]], " is ", Cell[BoxData[ StyleBox[\(a = 10\ e\), FontSlant->"Italic"]]], " to ", Cell[BoxData[ \(6\)]], " decimals; compare:" }], "Text"], Cell[CellGroupData[{ Cell["{N[f[a],8],N[f[27.182818],8]}", "Input", CellLabel->"In[138]:="], Cell[BoxData[ RowBox[{"{", RowBox[{ StyleBox["738.905609893064951`", StyleBoxAutoDelete->True, PrintPrecision->8], ",", StyleBox["738.905594421124067`", StyleBoxAutoDelete->True, PrintPrecision->8]}], "}"}]], "Output", CellLabel->"Out[138]="] }, Open ]], Cell[TextData[{ "If we round these to ", Cell[BoxData[ \(4\)]], " decimals, the results are the same, just as predicted." }], "Text"], Cell["\<\ In the same manner, the (first version of the) Race Track Principle may be \ used to determine the accuracy needed for a desired accuracy in computing any \ function evaluated at any point. (However, care should be taken not to \ introduce other rounding errors when calculating by machine.) \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Conclusion", "Section", CellMargins->{{Inherited, 10}, {Inherited, Inherited}}, ShowCellLabel->False], Cell["\<\ We have presented only two versions and examples of the use of the Race Track \ Principle. Additional versions and examples may be found in [1]-[4].\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section", CellMargins->{{Inherited, 10}, {Inherited, Inherited}}, ShowCellLabel->False], Cell[TextData[{ StyleBox["1", FontWeight->"Bold"], ". Davis, B., Porta, H. and J. Uhl: Calculus&", StyleBox["Mathematica", FontSlant->"Italic"], " / Derivatives: Measuring Growth. Addison-Wesley, Reading, Massachusetts, \ 1994." }], "Reference"], Cell[TextData[{ StyleBox["2", FontWeight->"Bold"], ". Davis, B., Porta, H. and J. Uhl: Calculus&", StyleBox["Mathematica", FontSlant->"Italic"], " / Integrals: Measuring Growth Accumulation. Addison-Wesley, Reading, \ Massachusetts, 1994." }], "Reference"], Cell[TextData[{ StyleBox["3", FontWeight->"Bold"], ". Davis, B., Porta, H. and J. Uhl: Calculus&", StyleBox["Mathematica", FontSlant->"Italic"], " / Vector Calculus: Measuring in Two and Three Dimensions. \ Addison-Wesley, Reading, Massachusetts, 1994." }], "Reference"], Cell[TextData[{ StyleBox["4", FontWeight->"Bold"], ". Davis, B., Porta, H. and J. Uhl: Calculus&", StyleBox["Mathematica", FontSlant->"Italic"], " / Approximation: Measuring Nearness. Addison-Wesley, Reading, \ Massachusetts, 1994." }], "Reference"], Cell[TextData[{ StyleBox["5", FontWeight->"Bold"], ". Hughes-Hallett, D., Gleason, A. M., et al: Calculus. J. Wiley \ International Edition, New York, NY, 1994." }], "Reference"], Cell[TextData[{ StyleBox["6", FontWeight->"Bold"], ". Thomas, G. B. and R. L. Finney: Calculus and Analytic Geometry. \ Addison-Wesley, Reading, Massachusetts, 1986.", StyleBox[" ", FontSlant->"Italic"] }], "Reference"] }, Open ]], Cell["About the Authors", "Section"], Cell[CellGroupData[{ Cell[TextData[{ "by Alkiviadis G. Akrita", Cell[BoxData[ \(TraditionalForm\`s\)]], " and Zamir Bavel" }], "Author", CellMargins->{{Inherited, 10}, {Inherited, Inherited}}], Cell["\<\ University of Kansas / EECS-IPS 415 Snow Hall Lawrence, Kansas 66045, USA\ \>", "Address", CellMargins->{{Inherited, 10}, {Inherited, Inherited}}], Cell["\<\ Akritas is on leave at the University of Thessaly, Department of Theoretical \ and Applied Sciences, GR-38221 Volos, Greece\ \>", "Address", CellMargins->{{Inherited, 10}, {Inherited, Inherited}}, FontSize->10] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowSize->{689, 625}, WindowMargins->{{41, Automatic}, {Automatic, 5}}, ShowCellLabel->False, StyleDefinitions -> "ARTICLEMODERN.NB" ] (*********************************************************************** 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. 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