(************** Content-type: application/mathematica **************
CreatedBy='Mathematica 4.2'
Mathematica-Compatible Notebook
This notebook can be used with any Mathematica-compatible
application, such as Mathematica, MathReader or Publicon. 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[ 20860, 701]*)
(*NotebookOutlinePosition[ 21719, 728]*)
(* CellTagsIndexPosition[ 21675, 724]*)
(*WindowFrame->Normal*)
Notebook[{
Cell[TextData[{
Cell[BoxData[{
StyleBox[\(MATH\ 257\ Calculus\ III\t\t\t\tWeek\ of\ April\ 21, \
2003\),
FontSize->14], "\n",
StyleBox[\(Lab\ 4 : \ Taylor\ Series\),
"Title"], "\n",
StyleBox[\(Name\ 1\),
"Section"], "\n",
StyleBox[\(Name\ 2\),
"Section"], "\n",
StyleBox[\(\(Section\)\(:\)\),
"Section"]}], "Input"],
"\n"
}], "Text"],
Cell[TextData[{
"I. Laboratory Objectives\n\tUse ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" to explore Taylor Series"
}], "Text"],
Cell[TextData[StyleBox["II. Formatting and Syntax Information", "Text"]], \
"Text"],
Cell[TextData[{
"\t",
StyleBox["Series[", "MR",
FontWeight->"Bold"],
StyleBox["f", "TI",
FontWeight->"Bold"],
StyleBox[",", "MR",
FontWeight->"Bold"],
StyleBox[" {",
FontWeight->"Bold"],
StyleBox["x", "TI",
FontWeight->"Bold"],
StyleBox[",", "MR",
FontWeight->"Bold"],
StyleBox[" ",
FontWeight->"Bold"],
Cell[BoxData[
\(TraditionalForm\`a\)], "InlineFormula",
FontWeight->"Bold"],
StyleBox[",", "MR",
FontWeight->"Bold"],
StyleBox[" ",
FontWeight->"Bold"],
StyleBox["n", "TI",
FontWeight->"Bold"],
Cell[BoxData[
\(TraditionalForm\`}\)], "InlineFormula",
FontWeight->"Bold"],
StyleBox["]", "MR",
FontWeight->"Bold"],
" \tThis generates a Taylor series \n\t\t\t\t\texpansion for ",
StyleBox["f", "TI"],
" about the point \n\t\t\t\t\t",
Cell[BoxData[
\(TraditionalForm\`x = a\)], "InlineFormula"],
" up to the term ",
Cell[BoxData[
\(TraditionalForm\`\((x - a)\)\^n\)], "InlineFormula"],
"\n\t",
StyleBox["Normal[expression]\t",
FontFamily->"Courier",
FontWeight->"Bold"],
"\tConverts a Taylor series to a \n\t\t\t\t\tTaylor Polynomial by \
truncating\n\t\t\t\t\tthe higher order terms"
}], "Text",
FontFamily->"Helvetica",
FontSize->14],
Cell[CellGroupData[{
Cell[TextData[StyleBox[" Taylor Series",
FontWeight->"Bold",
FontColor->RGBColor[0, 0, 1]]], "Subtitle"],
Cell[TextData[{
StyleBox["Think of the Taylor Series of a function f(x) as being a way of \
getting a different formula for f(x). \n\nFor example, we do NOT have a nice \
formula for Sin[x], do we? We always have to rely on our calculator to \
compute things like Sin[.5]. Taylor Series give us a way of trying to find a \
formula for things like Sin[x]. \n\nThe general form for a Taylor Series \
about the point x = a:", "Text",
FontFamily->"Times",
FontSize->14],
"\n\nf(x) = ",
Cell[BoxData[
FormBox[
RowBox[{\(\[Sum]\+\(n = 0\)\%\[Infinity]\),
RowBox[{
StyleBox[\(\(\(f\^\((n)\)\)(a)\)\/\(n!\)\),
FontSize->18], \(\((x - a)\)\^n\)}]}], TraditionalForm]]],
"\n\n",
StyleBox["This is just a Power Series whose coefficiants are given by ",
"Text",
FontFamily->"Times",
FontSize->14],
StyleBox[Cell[BoxData[
\(TraditionalForm\`c\_n\)], "Text",
FontFamily->"Times",
FontSize->14], "Text"],
StyleBox[" = ", "Text",
FontFamily->"Times",
FontSize->14],
StyleBox[Cell[BoxData[
FormBox[
StyleBox[\(\(\(f\^\((n)\)\)(a)\)\/\(n!\)\),
FontSize->18], TraditionalForm]], "Text",
FontFamily->"Times",
FontSize->14], "Text"],
StyleBox[".\nLuckily, ", "Text",
FontFamily->"Times",
FontSize->14],
StyleBox["Mathematica", "Text",
FontFamily->"Times",
FontSize->14,
FontSlant->"Italic"],
StyleBox[" will compute this formula for us, so we don't have to! (But you \
wil need to know this for your regular class tests and such...)\n", "Text",
FontFamily->"Times",
FontSize->14]
}], "Section"],
Cell[CellGroupData[{
Cell[TextData[{
StyleBox["A Taylor Polynomial of degree k is the truncated Taylor Series of \
that degree. A Taylor Polynomial is a finite polynomial. We use Taylor \
polynomials because it's usually too messy to deal with the complete Taylor \
Series.\n\nThus the Taylor Polynomial of degree k is \t", "Text",
FontFamily->"Times",
FontSize->14],
"\t\t\t\t",
Cell[BoxData[
FormBox[
RowBox[{\(\[Sum]\+\(n = 0\)\%K\),
RowBox[{
StyleBox[\(\(\(f\^\((n)\)\)(a)\)\/\(n!\)\),
FontSize->18], \(\((x - a)\)\^n\)}]}], TraditionalForm]]],
"\n"
}], "Section"],
Cell[TextData[{
"We will use the command ",
StyleBox["Series",
FontSize->14,
FontWeight->"Bold",
FontSlant->"Italic"],
" to create a Taylor Series expansion of f(x) about x = a. \n",
StyleBox["Series[", "MR",
FontSize->16,
FontWeight->"Bold"],
StyleBox["f", "TI",
FontSize->16,
FontWeight->"Bold"],
StyleBox[",", "MR",
FontSize->16,
FontWeight->"Bold"],
StyleBox[" {",
FontSize->16,
FontWeight->"Bold"],
StyleBox["x", "TI",
FontSize->16,
FontWeight->"Bold"],
StyleBox[",", "MR",
FontSize->16,
FontWeight->"Bold"],
StyleBox[" ",
FontSize->16,
FontWeight->"Bold"],
Cell[BoxData[
\(TraditionalForm\`a\)], "InlineFormula",
FontSize->16,
FontWeight->"Bold"],
StyleBox[",", "MR",
FontSize->16,
FontWeight->"Bold"],
StyleBox[" ",
FontSize->16,
FontWeight->"Bold"],
StyleBox["n", "TI",
FontSize->16,
FontWeight->"Bold"],
Cell[BoxData[
\(TraditionalForm\`}\)], "InlineFormula",
FontSize->16,
FontWeight->"Bold"],
StyleBox["]", "MR",
FontSize->16,
FontWeight->"Bold"],
" generates a power series expansion for ",
StyleBox["f", "TI"],
" about the point ",
Cell[BoxData[
\(TraditionalForm\`x = a\)], "InlineFormula"],
" to order ",
Cell[BoxData[
\(TraditionalForm\`\((x - a)\)\^n\)], "InlineFormula"],
". In other words, it gives is the Taylor polynomial of degree n."
}], "Text",
FontSize->14],
Cell["To begin, let f(x) = ln(x).", "Text",
FontSize->14],
Cell[BoxData[
\(Clear[f]\)], "Input"],
Cell[BoxData[
\(f[x_] := Log[x]\)], "Input"],
Cell[TextData[{
"Let's now create the 10th degree Taylor polynomial of f(x)=ln(x) about x = \
1. First, create a Taylor Series called \"taylseries\". The syntax ",
StyleBox["Series[f[x], {x, ", "Input"],
StyleBox[Cell[BoxData[
\(TraditionalForm\`1\)], "Input"], "Input"],
StyleBox[", 10", "Input"],
StyleBox[Cell[BoxData[
\(TraditionalForm\`}\)], "Input"], "Input"],
StyleBox["]", "Input"],
StyleBox[" ", "MR"],
"will give us the first 10 powers of the infinite polynomial plus the \
term",
StyleBox[" O", "MR"],
Cell[BoxData[
\(TraditionalForm\`\([x - x\_0]\)\^11\)]],
"which indicated the complete Taylor Series continues on past the 10th \
degree Taylor polynomial. "
}], "Text",
FontSize->14],
Cell[BoxData[
\(tayseries = Series[f[x], {x, 1, 10}]\)], "Input"],
Cell[TextData[{
"Normal[series] truncates the power series and converts it to a normal \
expression for the Taylor Polynomial. Notice the ",
Cell[BoxData[
RowBox[{"+",
InterpretationBox[\(O[x - 1]\^11\),
SeriesData[ x, 1, {}, 1, 11, 1]]}]]],
" part. This is meant to symbollically represent all the higher ordered \
terms in the Taylor Series in ",
StyleBox["Mathematica",
FontSlant->"Italic"],
". ",
StyleBox["Note:",
FontWeight->"Bold"],
" this is a complicated, different data structure in ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" than a regular function. In order to turn it into an everyday function \
(so we can graph it and such), we need to use the ",
StyleBox["Normal[]", "Input"],
" command:"
}], "Text",
FontSize->14],
Cell[BoxData[
\(taypoly10[x_] = Normal[taylseries]\)], "Input"],
Cell[CellGroupData[{
Cell[TextData[{
"From this work we should be able to determine the general form of the \
Taylor Series for ln(x) about the point x = 1. The series is alternating, it \
has powers of (x-1) in every term, and the denomenator of coefficient of ",
Cell[BoxData[
\(TraditionalForm\`\((x - 1)\)\^n\)]],
" is just n. So we get the infinite Taylor Series:\n\n\t\t\t",
Cell[BoxData[
\(ln \((x)\) = \ \[Sum]\+\(n = 1\)\%\[Infinity]\(\(\((\(-1\))\)\^\(n - \
1\)\) \((x - 1)\)\^n\)\/n\)]]
}], "Subsection"],
Cell[TextData[{
"We need to find the radius of convergence of this series. We could do \
this by hand, but it's pretty swank to get ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" to do it for us! \nIn preparation for using the Ratio Test, let's define \
the nth term of the series ",
Cell[BoxData[
\(TraditionalForm\`a\_n\)]],
"\n(This should look familiar from the last lab.)"
}], "Text",
FontSize->14],
Cell[BoxData[
\(a[n_, x_] = Abs[x - 1]\^n\/n\)], "Input"],
Cell[TextData[{
"Now we can form the ratio of the absolute value of ",
Cell[BoxData[
\(TraditionalForm\`a\_\(n + 1\)\/a\_\(\(n\)\(\ \)\)\)]],
"giving it the name \"ratio.\""
}], "Text",
FontSize->14],
Cell[BoxData[
\(ratio = a[n + 1, x]\/a[n, x]\)], "Input"],
Cell["\<\
Next take the limit of this ratio as n\[Rule] \[Infinity].\
\>", \
"Text",
FontSize->14],
Cell[BoxData[
\(Limit[ratio, n \[Rule] \[Infinity]]\)], "Input"],
Cell["\<\
The series will converge when the answer is less than 1. To solve \
the inequality
|x - 1| < 1, we must load the package \"Inequality Solve\" in the Algebra \
package. This is done below.\
\>", "Text",
FontSize->14],
Cell[BoxData[
\(<< Algebra`InequalitySolve`\)], "Input"],
Cell["To solve the inequality we use the syntax below:", "Text",
FontSize->14],
Cell[BoxData[
\(InequalitySolve[Abs[\(-1\) + x] < 1, x]\)], "Input"],
Cell[TextData[{
"So now we know that ",
Cell[BoxData[
\(ln \((x)\) = \ \[Sum]\+\(n = 1\)\%\[Infinity]\(\(\((\(-1\))\)\^\(n - \
1\)\) \((x - 1)\)\^n\)\/n\)]],
" is a convergent Taylor Series expansion on the interval 0 < x < 2. We \
DON'T know what is happening at the end points, at x = 0 and at x = 2. You'd \
need to check these endpoints separately."
}], "Text",
FontSize->14],
Cell["\<\
To see what all this means visually, we will plot both the natural \
log function and the Taylor Polynomial of degree 10 on the interval .01 < x < \
3. You should be able to see that close to the center (x = 1) the two graphs \
are nearly identical and that they continue to be similar throughout the \
interval of convergence. Outside of the interval of convergence the graphs \
look dissimilar. The PlotStyle command sets helps to differentiate between \
the curves: the natural log function is plotted thicher than normal using \
\"Thickness\" and the Taylor Polynomial is plotted using a dashed line using \
\"Dashing.\"\
\>", "Text",
FontSize->14],
Cell[BoxData[
\(Plot[{f[x], taypoly10[x]}, {x, 0.01, 3},
PlotStyle \[Rule] {{Thickness[ .01]}, {Dashing[{ .02, .02}]}}]\)], \
"Input"]
}, Open ]]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[StyleBox["IV. Problems to Solve",
FontFamily->"Helvetica",
FontSize->18,
FontColor->RGBColor[0, 0, 1]]], "Subtitle",
FontFamily->"Times",
FontSize->14],
Cell[TextData[{
StyleBox["Question 1",
FontWeight->"Bold"],
" Now analyze some data points along the two curves represented by \n\t",
StyleBox["Log[x]", "Input"],
" and ",
StyleBox["taypoly10[x] ", "Input"],
"(You HAVE to have evaluated taypoly10 above to use it here!)"
}], "Text",
FontSize->14,
FontColor->RGBColor[1, 0, 0]],
Cell[TextData[{
"By evaluating data points in these functions you are testing out the \
Taylor Polynomial of degree 10 to see how accurate it is against the function \
ln(x). \n",
StyleBox["a. ",
FontWeight->"Bold"],
" Evaluate both ",
StyleBox["Log[x]", "Input"],
" and ",
StyleBox["taypoly10[x]", "Input"],
" for the following values of x:\n\tx = 1/4\n\tx = 1/2\n\tx = 3/4\n\tx = 1\n\
\tx = 5/4\n\tx = 3/2\n\tx = 2\nTip: You can enter a list of values into a \
function, like ",
StyleBox["Log[{1/4, 1/2, 3/4}]",
FontFamily->"Courier",
FontWeight->"Bold"],
" and so on. You can also use the ",
StyleBox["TableForm[ ]", "Input"],
" command to make your work look neat."
}], "Text",
FontSize->14,
FontColor->RGBColor[1, 0, 0]],
Cell[BoxData[
\(\[IndentingNewLine]\)], "Input"],
Cell[TextData[{
StyleBox["b.",
FontWeight->"Bold"],
" Where is the Taylor Polynomial closest to the value of the natural log \
function?"
}], "Text",
FontSize->14,
FontColor->RGBColor[1, 0, 0]],
Cell[BoxData[
\(\[IndentingNewLine]\)], "Input"],
Cell[TextData[{
StyleBox["Question 2",
FontWeight->"Bold"],
" ",
StyleBox["a.",
FontWeight->"Bold"],
" Now it's ",
StyleBox["your turn",
FontWeight->"Bold"],
" to try creating a Taylor Polynomial. Find the 8th degree Taylor \
Polynomial for ",
StyleBox["Sin[x]", "Input"],
" about ",
StyleBox["a =",
FontSlant->"Italic"],
Cell[BoxData[
\(TraditionalForm\`\[Pi]\/4\)]],
". Follow the steps above to first create the Taylor Series and then \
truncate it using the ",
StyleBox["Normal", "Input"],
" command."
}], "Text",
FontSize->14,
FontColor->RGBColor[1, 0, 0]],
Cell[BoxData[
\(\[IndentingNewLine]\)], "Input"],
Cell[TextData[{
"b. In preparation for using the Ratio Test, define the nth term of the \
series ",
Cell[BoxData[
\(TraditionalForm\`a\_n\)]],
" using the model of the previous problem. "
}], "Text",
FontSize->14,
FontColor->RGBColor[1, 0, 0]],
Cell[BoxData[
\(\(\(a[n_, x_]\)\(=\)\(\[IndentingNewLine]\)\)\)], "Input"],
Cell[TextData[{
"c. Now form the ratio of the absolute value of ",
Cell[BoxData[
\(TraditionalForm\`a\_\(n + 1\)\/a\_\(\(n\)\(\ \)\)\)]],
"giving it the name \"ratio.\""
}], "Text",
FontSize->14,
FontColor->RGBColor[1, 0, 0]],
Cell[BoxData[
\(\[IndentingNewLine]\)], "Input"],
Cell[TextData[{
"d. Finally you're ready to take the limit of the ratio. You'll find that \
the ratio still includes a factorial term in the numerator and denominator. \
In order to take the limit you'll need a simplified version of \"ratio\" \
which will cancel the factorials. ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" does this in a roundabout way, so I'll give you the syntax: "
}], "Text",
FontSize->14,
FontColor->RGBColor[1, 0, 0]],
Cell[BoxData[
\(\(\(\[IndentingNewLine]\)\(Limit[FullSimplify[ratio], \
n \[Rule] \[Infinity]]\)\)\)], "Input"],
Cell["\<\
e. Explain what this answer to Ratio Test means in a complete \
sentence. Be sure write what the interval of convergence is for this Taylor \
Series.\
\>", "Text",
FontSize->14,
FontColor->RGBColor[1, 0, 0]],
Cell[BoxData[
\(\[IndentingNewLine]\)], "Input"],
Cell["\<\
f. Finally, plot the two functions, f[x] = Sin[x] and your 8th \
degree Taylor Polynomial using a thick solid curve for Sin[x] and a thin \
dashing curve for the Taylor Polynomial. Plot them on the x interval \
[-2\[Pi], 2\[Pi]]. Does this plot confirm what you think the interval of \
convergence is? Explain.\
\>", "Text",
FontSize->14,
FontColor->RGBColor[1, 0, 0]],
Cell[BoxData[
\(\[IndentingNewLine]\)], "Input"],
Cell[TextData[{
StyleBox["Question 3 ",
FontWeight->"Bold"],
"Recall that the function y=",
Cell[BoxData[
\(TraditionalForm\`\[ExponentialE]\^\(-x\^2\)\)]],
" has no closed-form antiderivative. ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" has ways of dealing with antiderivatives of this function. To see this, \
execute the following:"
}], "Text",
FontSize->14,
FontColor->RGBColor[1, 0, 0]],
Cell[BoxData[
\(\(\(\[IndentingNewLine]\)\(\(2\/\@\[Pi]\) \(\[Integral]\_0\%x\( \
\[ExponentialE]\^\(-t\^2\)\) \[DifferentialD]t\)\)\)\)], "Input",
FontSize->14],
Cell[TextData[{
"You should get the function ",
StyleBox["Erf[x]", "Input"],
" back. This is a special function that symbolically represents the \
antiderivative of ",
Cell[BoxData[
\(TraditionalForm\`\[ExponentialE]\^\(-x\^2\)\)]],
". Even though ",
StyleBox["Erf[x]", "Input"],
" is a tricky function, it is easy to approximate it with Taylor \
polynomials because we can easily take its derivatives!\n\n(a) Why is it \
easier to find ",
StyleBox["derivatives",
FontSlant->"Italic"],
" of ",
StyleBox["Erf[x]", "Input"],
" than to find ",
StyleBox["values",
FontSlant->"Italic"],
" of ",
StyleBox["Erf[x]", "Input"],
"? If you don't know, then try asking ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" to calculate a few derivatives of ",
StyleBox["Erf[x]", "Input"],
"; it might jog your memory.\n"
}], "Text",
FontSize->14,
FontColor->RGBColor[1, 0, 0]],
Cell["\<\
\
\>", "Text"],
Cell[TextData[{
"\n(b) Compute some Taylor polynomials of ",
StyleBox["Erf[x]", "Input"],
" centered at x=0. Make sure you do a few big ones, like 10 or 20 terms. \
What is the power series for ",
StyleBox["Erf[x]", "Input"],
" written in summation notation? (Hint: Factor out the ",
Cell[BoxData[
\(TraditionalForm\`2\/\@\[Pi]\)]],
" and write it as ",
Cell[BoxData[
\(TraditionalForm\`\(2\/\@\[Pi]\) \(\[Sum]\+\(n = 0\)\%\[Infinity]
STUFF\)\)]],
". Make sure to think about factorials.)"
}], "Text",
FontSize->14,
FontColor->RGBColor[1, 0, 0]],
Cell[BoxData[
\(\[IndentingNewLine]\)], "Input"],
Cell[TextData[{
"(c) Make a NICE plot of ",
StyleBox["Erf[x]", "Input"],
" and some (at least 3) of your Taylor polynomials on the same axis. To \
make it pretty, I suggest using the command \n",
StyleBox["Plot[{Erf[x], f1, f2}, {x,-3,3}, PlotStyle-> {RGBColor[0,0,0], \
RGBColor[1,0,0], RGBColor[0,1,0]}]", "Input"],
". This will make",
StyleBox[" Erf[x]", "Input"],
" be black and the other functions different colors. Preeeettty. \n"
}], "Text",
FontSize->14,
FontColor->RGBColor[1, 0, 0]],
Cell[BoxData[
\(\[IndentingNewLine]\)], "Input"],
Cell[TextData[{
"(d) Based on your graph, what do you think the interval of convergence of \
the ",
StyleBox["Erf[x]", "Input"],
" Taylor series is?\n"
}], "Text",
FontSize->14,
FontColor->RGBColor[1, 0, 0]],
Cell[BoxData[
\(\[IndentingNewLine]\)], "Input"],
Cell[TextData[{
StyleBox["Now that you're done, ",
FontSize->18],
"\n\t(1) \t",
StyleBox["Clean up your work",
FontWeight->"Bold"],
" by deleting everything that's not needed. \n\t\tThere should only be the \
title, your names, section number, and the questions and your answers\n\t\t\
(with any needed explanations, graphs, etc.).\n\t(2) \tSave this to your \
disk.\n\t(3)\tGo back to Blackboard and upload your final lab report to the \
",
StyleBox["Digital Drop Box",
FontWeight->"Bold"],
", located\n\t\tin the ",
StyleBox["User Tools",
FontWeight->"Bold"],
" area. Remember that you have to first ",
StyleBox["ADD",
FontWeight->"Bold"],
" your file to your digital drop box and \n\t\tthen ",
StyleBox["SEND it to me!",
FontWeight->"Bold"]
}], "Text",
CellFrame->{{0, 0}, {0, 2}},
FontSize->12]
}, Open ]]
},
FrontEndVersion->"4.2 for Macintosh",
ScreenRectangle->{{0, 800}, {0, 580}},
WindowToolbars->"EditBar",
WindowSize->{685, 524},
WindowMargins->{{1, Automatic}, {Automatic, 1}},
MacintoshSystemPageSetup->"\<\
01L0001804P000000^l2@?okonh34`9B;@85:0?l0@00009H0UP00P0067@B`001
0@00I0010000000030000BL?0040000000000000000006L001000`00000@0?oH
ofXIX1=F8040800000400000000000l1\>"
]
(*******************************************************************
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->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[1754, 51, 420, 14, 143, "Text"],
Cell[2177, 67, 149, 5, 48, "Text"],
Cell[2329, 74, 84, 1, 30, "Text"],
Cell[2416, 77, 1269, 46, 125, "Text"],
Cell[CellGroupData[{
Cell[3710, 127, 108, 2, 66, "Subtitle"],
Cell[3821, 131, 1655, 45, 340, "Section"],
Cell[CellGroupData[{
Cell[5501, 180, 616, 15, 190, "Section"],
Cell[6120, 197, 1473, 58, 72, "Text"],
Cell[7596, 257, 59, 1, 32, "Text"],
Cell[7658, 260, 41, 1, 27, "Input"],
Cell[7702, 263, 48, 1, 27, "Input"],
Cell[7753, 266, 743, 19, 86, "Text"],
Cell[8499, 287, 69, 1, 27, "Input"],
Cell[8571, 290, 806, 22, 106, "Text"],
Cell[9380, 314, 67, 1, 27, "Input"],
Cell[CellGroupData[{
Cell[9472, 319, 515, 10, 126, "Subsection"],
Cell[9990, 331, 432, 11, 87, "Text"],
Cell[10425, 344, 61, 1, 43, "Input"],
Cell[10489, 347, 213, 6, 36, "Text"],
Cell[10705, 355, 61, 1, 44, "Input"],
Cell[10769, 358, 100, 4, 32, "Text"],
Cell[10872, 364, 68, 1, 27, "Input"],
Cell[10943, 367, 230, 6, 50, "Text"],
Cell[11176, 375, 60, 1, 27, "Input"],
Cell[11239, 378, 80, 1, 32, "Text"],
Cell[11322, 381, 72, 1, 27, "Input"],
Cell[11397, 384, 395, 9, 76, "Text"],
Cell[11795, 395, 667, 11, 122, "Text"],
Cell[12465, 408, 146, 3, 43, "Input"]
}, Open ]]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell[12672, 418, 176, 5, 55, "Subtitle"],
Cell[12851, 425, 348, 10, 52, "Text"],
Cell[13202, 437, 770, 21, 232, "Text"],
Cell[13975, 460, 52, 1, 43, "Input"],
Cell[14030, 463, 206, 7, 34, "Text"],
Cell[14239, 472, 52, 1, 43, "Input"],
Cell[14294, 475, 617, 23, 73, "Text"],
Cell[14914, 500, 52, 1, 43, "Input"],
Cell[14969, 503, 262, 8, 50, "Text"],
Cell[15234, 513, 78, 1, 43, "Input"],
Cell[15315, 516, 242, 7, 36, "Text"],
Cell[15560, 525, 52, 1, 43, "Input"],
Cell[15615, 528, 465, 10, 69, "Text"],
Cell[16083, 540, 122, 2, 43, "Input"],
Cell[16208, 544, 223, 6, 50, "Text"],
Cell[16434, 552, 52, 1, 43, "Input"],
Cell[16489, 555, 386, 8, 68, "Text"],
Cell[16878, 565, 52, 1, 43, "Input"],
Cell[16933, 568, 430, 13, 53, "Text"],
Cell[17366, 583, 167, 3, 69, "Input"],
Cell[17536, 588, 925, 29, 145, "Text"],
Cell[18464, 619, 25, 3, 46, "Text"],
Cell[18492, 624, 596, 16, 108, "Text"],
Cell[19091, 642, 52, 1, 43, "Input"],
Cell[19146, 645, 515, 12, 122, "Text"],
Cell[19664, 659, 52, 1, 43, "Input"],
Cell[19719, 662, 217, 7, 50, "Text"],
Cell[19939, 671, 52, 1, 43, "Input"],
Cell[19994, 674, 850, 24, 166, "Text"]
}, Open ]]
}
]
*)
(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)