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Notebook[{
Cell[CellGroupData[{
Cell["\<\
The Symbolic Microscope
for
Functions of 1 Variable\
\>", "Title",
Evaluatable->False,
AspectRatioFixed->True],
Cell["\<\
by
K. D. Stroyan
University of Iowa\
\>", "Subsubtitle",
Evaluatable->False,
AspectRatioFixed->True],
Cell["\<\
copyright 1997 by Academic Press, Inc. - All rights reserved.\
\>", "Text",
Evaluatable->False,
AspectRatioFixed->True,
FontFamily->"Times"],
Cell[CellGroupData[{
Cell[TextData[{
"Special ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" Functions used in this NoteBook"
}], "Subsubsection",
Evaluatable->False,
AspectRatioFixed->True],
Cell["Needs[\"Graphics`Colors`\"];", "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell[" Notebook Overview", "Section",
Evaluatable->False,
AspectRatioFixed->True],
Cell["\<\
This NoteBook is a Mathematica animation of the microscope interpretation the \
linear approximation given by the first derivative in one variable. The \
effect is similar to the NoteBooks Zoom and SecantGapZ of Chapter 3. This \
calculation is completely \"hands on\" to emphasize the prediction YOU can \
make with symbolic calculus.\
\>", "Text",
Evaluatable->False,
AspectRatioFixed->True,
FontFamily->"Times"],
Cell[TextData[
"Algebraically the \"microscopic\" error or \"gap\" is \[Epsilon] in the \
formula: "], "Text",
Evaluatable->False,
AspectRatioFixed->True,
FontFamily->"Times"],
Cell[BoxData[
RowBox[{\(f[x + \[Delta]x] - f[x]\), "=",
RowBox[{
RowBox[{
RowBox[{
SuperscriptBox["f", "\[Prime]",
MultilineFunction->None], "[", "x", "]"}], " ", "\[Delta]x"}],
"+", \(\[Epsilon]\[CenterDot]\[Delta]x\)}]}]], "Text",
Evaluatable->False,
AspectRatioFixed->True,
FontFamily->"Times"],
Cell[TextData[
"Geometrically,the error \[Epsilon] is what you see as the difference \
between the graph and its tangent approximation when you look at the graph \
under a microscope of magnification 1/\[Delta]x.The function is smooth if \
\[Epsilon] is small,\[Epsilon]\[TildeTilde]0,when \[Delta]x is small,\
\[Delta]x\[TildeTilde]0. Under an ideal \"infinitesimal microscope\" of \
infinite power 1/\[Delta]x the error term \[Epsilon] would be \
infinitesimal.This means that you would not see the error \[Epsilon],so that \
the graph would appear to be linear-in the change (or difference) \
\[Delta]x."], "Text",
Evaluatable->False,
AspectRatioFixed->True,
FontFamily->"Times"],
Cell["\<\
A linear function in local variables (dx,dy) centered at a fixed (x,y) \
with y = f[x] has the form\
\>", "Text"],
Cell[BoxData[
RowBox[{"dy", "=",
RowBox[{
RowBox[{
SuperscriptBox["f", "\[Prime]",
MultilineFunction->None], "[", "x", "]"}], "\[CenterDot]",
"dx"}]}]], "Text"],
Cell["\<\
with slope m=f'[x],the derivative at the fixed x,in the basic formula dy=m \
dx.\
\>", "Text",
Evaluatable->False],
Cell[TextData[
"The linear function is the first term of the formula above for the change in \
f[x]. At a small value of dx = \[Delta]x \[TildeTilde] 0, the second term \
is small even compared with the linear term, so the change is nearly linear. \
In other words, the graph of a smooth function is \"locally linear.\" This \
is the main approximation of differential calculus."], "Text",
Evaluatable->False],
Cell["\<\
The rules of calculus let us \"see\" in the microscope without first seeing \
the graph. In other words, if the rules of differentiation apply, yielding a \
valid formula, then we know that sufficiently powerful magnification will \
result in a straight line of slope given by the formula obtained from the \
rules. This NoteBook and Problem 6.9 combine the ideas of Chapters 4, 5 and \
6.\
\>", "Text",
Evaluatable->False],
Cell["A 3-D Microscope is in the Chapter 18 folder.", "Text",
Evaluatable->False],
Cell[CellGroupData[{
Cell["References to the Text", "Subsection",
Evaluatable->False],
Cell["This NoteBook goes with Chapters 5, 6 & 7 of the text.", "Text",
Evaluatable->False]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["Step-by-step Computations ", "Section",
Evaluatable->False],
Cell[CellGroupData[{
Cell["The graph of f[x] near a fixed x:", "Subsubsection",
Evaluatable->False],
Cell["\<\
Clear[f,x,dx];
f[x_] := x Cos[x^2-x]/2;
Print[\"f[x] = \",f[x]];
x = 1.5;
Print[\" near x = \",x];
SetOptions[Plot, AxesOrigin -> {0,0},AxesLabel -> {\"dx\",\"dy\"},
\tAspectRatio -> Automatic ];
Plot[f[x+dx]-f[x], {dx,-5,5}, PlotRange -> {-5,5},PlotStyle->{Blue}];\
\>", "Input",
InitializationCell->True]
}, Closed]],
Cell[CellGroupData[{
Cell["Animation", "Subsubsection",
Evaluatable->False],
Cell["\<\
Inside this cell there is a sequence of magnifications of the graph of the \
function y=f[x] defined above, centered over the point x on its graph. \
(The value of x is also given above.) Notice that this does not use the \
derivative function f'[x]. \
\>", "Text",
Evaluatable->False],
Cell["\<\
You can see an animation by highlighting the closed cell next to the graphs \
and then pressing the -key together with the -key. (Or select \
\"animate selected graphics\" from the \"Graph\" menu.)\
\>", "Text",
Evaluatable->False],
Cell["\<\
SetOptions[Plot,Ticks -> None];
Print[\"Magnifying near x = \",x];
Do[delta = 5/(1.5^n);
\tPlot[f[x+dx]-f[x],{dx,-delta,delta},
\t\tPlotRange -> {-delta,delta},PlotStyle->{Thickness[0.01],Blue}];
\t,{n,0,16}]\
\>", "Input",
InitializationCell->True]
}, Closed]],
Cell[CellGroupData[{
Cell["Differential", "Subsubsection",
Evaluatable->False],
Cell["\<\
When we can compute the differential of y = f[x] by rules and both f[x] \
and f'[x] are defined near x, the magnified graph of y = f[x] near this \
value looks the same as the local linear graph dy = f'[x] dx.\
\>", "Text",
Evaluatable->False],
Cell["\<\
Compare the following \"prediction\" with the last frame of the previous \
animation. \
\>", "Text",
Evaluatable->False],
Cell["\<\
Clear[df,dx];
df[dx_] := f'[x] dx;
Print[\"dy = \",df[dx]];
Plot[ df[dx], {dx, -5, 5},PlotRange -> {-5,5},PlotStyle->{Red}] ; \
\>", "Input",
InitializationCell->True]
}, Closed]],
Cell[CellGroupData[{
Cell["The Symbolic Differential", "Subsubsection",
Evaluatable->False],
Cell["\<\
Clear[x];
df[dx]\
\>", "Input",
InitializationCell->True]
}, Closed]],
Cell[CellGroupData[{
Cell["Summary", "Subsubsection"],
Cell["\<\
If the formulas for f[x] and f'[x] are valid in an interval around the \
x-value of interest, we know the magnified graph looks linear. In this case, \
evaluate the specific number m = f'[x] and graph your predicted line in \
local coordinates:\
\>", "Text"],
Cell["\tdy = m dx", "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["Combined Computation", "Section",
Evaluatable->False],
Cell[CellGroupData[{
Cell["Theorem 5.4 ", "Subsection",
Evaluatable->False],
Cell["\<\
If f[x] is a smooth real function on the real line, then under an \
sufficiently powerful microscope focused over a fixed x, the graph y=f[x] \
appears to be the same as the linear graph given in the microscope \
(dx,dy)-coordinates by dy=m dx , where m=f'[x] and the center of the \
microscope is focused at the (x,y)-point (x,f[x]). \
\>", "Text",
Evaluatable->False]
}, Closed]],
Cell[CellGroupData[{
Cell["Procedure", "Subsection"],
Cell["To predict the view in an ideal microscope we:", "Text",
Evaluatable->False],
Cell["\<\
1) Compute f'[x] from y = f[x] by the symbolic rules of Chapter 6.\
\>", "Text",
Evaluatable->False],
Cell["\<\
2) Verify that f[x] and f'[x] are valid formulas in an interval about \
the focus value of x.\
\>", "Text",
Evaluatable->False],
Cell["\<\
3) Fix x in the differential dy = f'[x] dx (that is, compute the specific \
number m = f'[x]) and plot the line dy = m \[CenterDot] dx in \
(dx,dy)-coordinates.\
\>", "Text",
Evaluatable->False],
Cell[TextData[{
"This line is what the graph looks like at \"sufficiently large\" \
magnification. This can be done with ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" as follows:"
}], "Text",
Evaluatable->False],
Cell[BoxData[{\(Clear[f, x, dx, df]; \),
\(f[x_] := 1\/2\ x\ Cos[x\^2 - x]; \),
RowBox[{
RowBox[{\(df[dx_]\), ":=",
RowBox[{
RowBox[{
SuperscriptBox["f", "\[Prime]",
MultilineFunction->None], "[", "x", "]"}], " ", "dx"}]}],
";"}], \(Print["\", f[x]]; \), \(x = 1.0; \),
\(Print["\", x]; \),
\(Print["\", df[dx]]; \),
\(Plot[df[dx], {dx, \(-1\), 1}, Ticks \[Rule] None,
AxesLabel \[Rule] {"\", "\"}, AspectRatio \[Rule] Automatic,
PlotRange \[Rule] {\(-1\), 1}, PlotStyle \[Rule] {Red}]; \)}], "Input"],
Cell["You could compare as follows to be sure:", "Text"],
Cell[BoxData[
\(scale\ = \ 0.25; \n
Plot[{\((f[x + dx] - f[x])\), df[dx]}, {dx, \(-scale\), scale},
PlotRange -> {\(-scale\), scale},
PlotStyle -> {{Thickness[0.01], Blue}, {Red}}]; \)], "Input"],
Cell[BoxData[
\(scale\ = \ 0.01; \n
Plot[{\((f[x + dx] - f[x])\), df[dx]}, {dx, \(-scale\), scale},
PlotRange -> {\(-scale\), scale},
PlotStyle -> {{Thickness[0.01], Blue}, {Red}}]; \)], "Input"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["Exercises", "Section",
Evaluatable->False],
Cell[CellGroupData[{
Cell["Problem 5.1 ", "Subsection",
Evaluatable->False],
Cell["\<\
Predict the view in a powerful microscope focused over x = 2 on the graphs \
below. Explain how your computations use your computations from Problem 5.1 \
(or rules of calculus) to \"see\" in the microscope without actually \
magnifying.\
\>", "Text",
Evaluatable->False],
Cell[TextData[{
"\t",
Cell[BoxData[
\(y = \(f[x] = x\^n\)\)]],
"\n\t",
Cell[BoxData[
\(y = \(f[x] = 1\/x\^2\)\)]],
"\n\t",
Cell[BoxData[
\(y = \(f[x] = \@x\%3\)\)]]
}], "Text",
Evaluatable->False]
}, Closed]],
Cell[CellGroupData[{
Cell["Problem 5.2", "Subsection",
Evaluatable->False],
Cell[TextData[{
"1) Calculus cannot predict the view in a powerful microscope focused over \
x = -1 on the graph y = ",
Cell[BoxData[
\(\@\(x\^2 + 2\ x + 1\)\)]],
". Explain why not using rules computed by hand. Check your work with the \
commands (put in an input cell) \nClear[x,f];\nf[x_] := ",
Cell[BoxData[
\(\@\(x\^2 + 2\ x + 1\)\)]],
"; \nf[x]\nf'[x]\nf'[-1]\n(HINT: What is the line dy = m dx when x = \
-1?)"
}], "Text",
Evaluatable->False],
Cell[TextData[{
"2) Repeate the step-by-step computations above for the function f[x_] := \
",
Cell[BoxData[
\(\@\(x\^2 + 2\ x + 1\)\)]],
" , magnifying in at the point x = -1, verifying that f[x] is not \
approximately linear near x = -1."
}], "Text",
Evaluatable->False]
}, Closed]],
Cell[CellGroupData[{
Cell["Problem 6.12 ", "Subsection",
Evaluatable->False],
Cell[TextData[{
"1) Repeate the step-by-step computations above for the function f[x_] := \
",
Cell[BoxData[
\(x\^2\ Cos[\[Pi]\/x]\)]],
" , zooming in at the point x = 1."
}], "Text",
Evaluatable->False],
Cell["\<\
2) Compare the machine computation to your hand calculations from the text \
Problem.\
\>", "Text"]
}, Closed]]
}, Closed]]
}, Open ]]
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StyleDefinitions -> "CalcTLCStyle.nb"
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(***********************************************************************
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.
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(***********************************************************************
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***********************************************************************)