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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[" {",
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    FontWeight->"Bold"],
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    FontWeight->"Bold"],
  StyleBox[" ",
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  StyleBox[",", "MR",
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  StyleBox[" ",
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      \(TraditionalForm\`}\)], "InlineFormula",
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  " \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",
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  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"],
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    FontWeight->"Bold"],
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    FontWeight->"Bold"],
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    FontSize->16,
    FontWeight->"Bold"],
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    FontSize->16,
    FontWeight->"Bold"],
  StyleBox[" ",
    FontSize->16,
    FontWeight->"Bold"],
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      \(TraditionalForm\`a\)], "InlineFormula",
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    FontWeight->"Bold"],
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    FontWeight->"Bold"],
  StyleBox[" ",
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    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  ]]
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