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Cell[CellGroupData[{
Cell["Taylor Polynomials", "Title",
  CellFrame->True,
  TextAlignment->Center,
  Background->RGBColor[0, 1, 0]],

Cell["\<\
This notebook is by Steven Amgott.  Please send any questions or comments to \
samgott1@swarthmore.edu.  Feel free to use and distribute this notebook, but \
keep this author information in any copy you use or distribute.\
\>", "SmallText"],

Cell[TextData[{
  "In the sections below, anything in ",
  StyleBox["magenta",
    FontColor->RGBColor[1, 0, 1]],
  " is something you can, and possibly should, change.  The two ",
  StyleBox["Off",
    FontColor->RGBColor[1, 0, 0]],
  " commands prevent a few annoying warning messages."
}], "Text"],

Cell[BoxData[{
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Cell[CellGroupData[{

Cell["Finding the Linear and Quadratic polynomials", "Section"],

Cell["\<\
The cell below finds the first and second order partial derivatives of f(x,y) \
at the point (a,b), and then constructs the linear (l[x,y]) and quadratic \
(q[x,y]) Taylor polynomials for the function centered at (a,b).\
\>", "Text"],

Cell[BoxData[{\(Clear[f, x, y, a, b, fx, fy, fxx, fyy, fxy, l, q]\), "\n", 
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        StyleBox["0",
          FontColor->RGBColor[1, 0, 1]]}], 
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          y -> b}\), "\n", \(fy = \[PartialD]\_y\ f[x, y] /. {x -> a, 
          y -> b}\), "\n", \(fxx = \[PartialD]\_{x, 2}\ f[x, y] /. {x -> a, 
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          y -> b}\), "\n", \(l[x_, y_] = 
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        fy \((y - b)\) + \(fxx\/2\) \((x - a)\)\^2 + 
        fxy \((x - a)\) \((y - b)\) + \(fyy\/2\) \((y - b)\)\^2\)}], "Input"]
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Cell["3D plots", "Section"],

Cell[TextData[{
  "The cell below produces plots of the function, the linear approximation, \
and the quadratic approximation on a rectangle whose center is (a,b).  Make \
sure you have evaluated the cell in the first section defining the linear and \
quadratic approximations before evaluating this one.  You can \"zoom-in\" by \
changing the value of ",
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  " (and/or ",
  StyleBox["k",
    FontColor->RGBColor[1, 0, 1]],
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}], "Text"],

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        PlotLabel \[Rule] "\<linear approximation\>"];\), "\n", \(Plot3D[
        q[x, y], {x, a - h, a + h}, {y, b - k, b + k}, 
        PlotLabel \[Rule] "\<quadratic approximation\>"];\)}], "Input"],

Cell[TextData[{
  "These plots may not be the best to see how decreasing ",
  StyleBox["h",
    FontColor->RGBColor[1, 0, 1]],
  " improves the approximations.  It might be better to look at the contour \
plots, which is done in the next section."
}], "Text"]
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Cell[CellGroupData[{

Cell["Contour plots", "Section"],

Cell[TextData[{
  "The next cell produces contour plots of the function, the ",
  StyleBox["linear approximation",
    FontColor->RGBColor[1, 0, 0]],
  ", and the ",
  StyleBox["quadratic approximation",
    FontColor->RGBColor[0, 1, 0]],
  " on a rectangle whose center is (a,b).  Make sure you have evaluated the \
cell in the first section defining the linear and quadratic approximations \
before evaluating this one."
}], "Text"],

Cell[BoxData[{\(Clear[h, k, plotf, plotl, plotq]\), "\n", 
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        StyleBox["h",
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        ContourPlot[f[x, y], {x, a - h, a + h}, {y, b - k, b + k}, 
          ContourShading \[Rule] False];\), "\n", \(plotl = 
        ContourPlot[l[x, y], {x, a - h, a + h}, {y, b - k, b + k}, 
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          ContourStyle \[Rule] RGBColor[1, 0, 0]];\), "\n", \(plotq = 
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          ContourStyle \[Rule] RGBColor[0, 1, 0]];\)}], "Input"],

Cell[TextData[{
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approximation in the first plot, the function and the quadratic approximation \
in the second plot, and all three together in the third plot.  To zoom in, \
change ",
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  " (and/or ",
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}], "Text"],

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}, Closed]],

Cell[CellGroupData[{

Cell["Tables of values and errors", "Section"],

Cell["\<\
The cell below produces a table of values of the function and the \
approximations, and computes the errors in the approximations as we get \
closer to the point (a,b).   Make sure you have evaluated the cell in the \
first section defining the linear and quadratic approximations before \
evaluating this one.  What are the factors by which the errors in the linear \
and quadratic approximations decrease as the distance we are from the center \
point decreases by a factor of 10?\
\>", "Text"],

Cell[BoxData[
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      N[Table[{10\^\(-n\), f[a + 10\^\(-n\), b + 10\^\(-n\)], 
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            l[a + 10\^\(-n\), b + 10\^\(-n\)] - 
              f[a + 10\^\(-n\), b + 10\^\(-n\)], 
            q[a + 10\^\(-n\), b + 10\^\(-n\)], 
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              f[a + 10\^\(-n\), b + 10\^\(-n\)]}, {n, 0, 6}]], 
      TableHeadings \[Rule] {{}, {"\<h\>", "\<f(a+h,b+h)\>", \
"\<l(a+h,b+h)\>", "\<linear error\>", "\<q(a+h,b+h)\>", "\<quadratic \
error\>"}}]\)], "Input"]
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