NeXT Education Software Sampler 1992 Fall/NeXT Education Software Sampler 1992 Fall.iso/Mathematics/Notebooks/BookSamples/CalcUsingMathematica/Microscope.ma

Text File | 1992-08-12 | 3.1 MB | 164,924 lines

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Copyright ª 1992 by Academic Press, Inc.

Text Examples & Exercises 16.3, 16.4, ..., 16.40
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Infinitesimal Microscopes
in
Two and Three Dimensions
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by
K. D. Stroyan
University of Iowa
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Copyright 1992 by Academic Press, Inc.  All rights reserved.
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Overview of the NoteBook
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    This NoteBook is a Mathematica animation of the microscope interpretation of Landau's small oh form of the linear approximation given by the first derivative in one or two independent variables.  It quickly runs thru a sequence of pre-drawn plots like a movie.  Algebraically the error is  e  in the formula

                f(x + dx) = f(x) + f '(x) dx + e dx
or
        f(x + dx , y+ dy) = f(x,y) + fx(x,y) dx + fy(x,y) dy + e |dX|   ,      dX = (dx,dy)

                       = f(x,y) + fx(x,y) dx + fy(x,y) dy + e Sqrt(dx^2 + dy^2)

Geometrically, the error  e  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/dx  (or in the vector case, 1/ |dX|).
    In either case, the function is smooth if  e  is small when  dx  (or  |dX|)  is small.  Under an ideal "infinitesimal microscope" of infinite power  1/dx  the error term  e  would be infinitesimal.  This means that you would not see the error  e, so that the graph would appear to be linear.  In other words, the graph of a smooth function is "locally linear."  This is the main approximation of differential calculus.

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2 D Microscopic view of  the explicit graph  y = f(x)
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    Following is a sequence of magnifications of the graph of the function  y=f[x] centered over the point  x=a  on its graph.  Highlight the closed cell next to the graph and then press the <Command>-key together with the <Y>-key.
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SetOptions[Plot, AxesOrigin -> {a,f[a]},
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Do[{delta = 5/(1.5^n),
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Clear[x,y]                                                            ;
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a = +1.0                                                                ;
b = -1.0                                                                ;
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Animation
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    Following is a sequence of magnifications of the explicit graph of the function  z=f[x,y], centered over the point  (x,y)=(a,b).  Highlight the closed cell next to the graph below and then press the <Command>-key together with the <Y>-key.
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SetOptions[Plot3D, Boxed -> False, Axes -> None];
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0.904 0.79 0.725 setrgbcolor
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0.826 0.74 0.77 setrgbcolor
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0.928 0.798 0.696 setrgbcolor
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0.976 0.879 0.671 setrgbcolor
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0.975 0.918 0.605 setrgbcolor
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0.944 0.915 0.54 setrgbcolor
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0.983 0.904 0.644 setrgbcolor
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0.975 0.879 0.674 setrgbcolor
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0.958 0.85 0.693 setrgbcolor
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0.938 0.823 0.705 setrgbcolor
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0.904 0.784 0.718 setrgbcolor
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0.879 0.763 0.729 setrgbcolor
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0.894 0.774 0.722 setrgbcolor
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0.892 0.773 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.724 setrgbcolor
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0.891 0.772 0.724 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.724 setrgbcolor
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0.891 0.772 0.724 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.724 setrgbcolor
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0.891 0.772 0.724 setrgbcolor
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0.891 0.772 0.724 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.723 setrgbcolor
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0.891 0.772 0.724 setrgbcolor
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0.891 0.772 0.724 setrgbcolor
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0.891 0.772 0.724 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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0.892 0.772 0.723 setrgbcolor
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