%% This is the LaTeX source code for "Approximating Polynomials"
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%% History
%% 2009/03/16  First version
%% 2009/03/18  Posted
%% 2009/04/08  Testing tex4ht fixes
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\documentclass[12pt]{article}
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\title{Approximating Polynomials}
\author{Robert A.\ Beezer\\University of Puget Sound}
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\begin{document}
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\maketitle
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\section*{Introduction}
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This is a short introduction to the notion of using polynomials to approximate more complicated functions.  It is entirely informal, with the intent of motivating a careful study of infinite series prior to learning about Taylor polynomials and Taylor series.
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\section{A Very Basic Approximating Polynomial}
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Consider the following algebra centering on polynomial multiplication,
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\begin{align*}
(1-x)(1+x+x^2+x^3+\cdots+x^n)
&=1+x+x^2+x^3+\cdots+x^n\\
&\quad\quad-(x+x^2+x^3+\cdots+x^n+x^{n+1})\\
&=1+(x-x)+(x^2-x^2)+\cdots+(x^n-x^n)-x^{n+1}\\
&=1-x^{n+1}\\
&\approx 1
\end{align*}
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The approximation in the last step is valid if $x^{n+1}$ is small, which will be the case if $-1<x<1$ and $n$ is large.  Keep those conditions in mind as we continue.\par
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If we assume $x\neq 1$ and divide both sides of the above by $1-x$ we obtain
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\begin{align}
\frac{1}{1-x}&\approx 1+x+x^2+x^3+\cdots+x^n\label{geometric}
\end{align}
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This will be the basis of all but one of our approximations.  In the demonstration below notice the following:
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\begin{itemize}
\item The approximation gets better as the degree, $n$, increases.
\item No matter how large the degree is, the approximation appears limited to $-1<x<1$.
\item For even versus odd degrees, the left end of the approximating polynomial approaches $\pm\infty$.
\item The degree 1 approximation is just the tangent line at the point $(0,1)$.
\end{itemize}
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% Work around
\par
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\begin{sageverbatim}
%hide
%auto
a=-1.25
b= 0.95
original_color='blue'
approx_color='red'
@interact
def _( n = slider(0, 20, 1, 2 , label = "Degree") ):
    var('x')
    f(x)=1/(1-x)
    approx(x)=0
    for i in srange(n+1):
        approx(x)=approx(x)+x^i
    original_plot = plot( f(x), a, b ,color=original_color)
    approx_plot = plot( approx(x),  a, b, color=approx_color)
    html("Function: <font color='%s'>$%s$</font>$" % (original_color, latex(f(x))) )
    html("Approximation: <font color='%s'>$%s$</font>$" % (approx_color, latex(approx(x))) )
    show(original_plot+approx_plot, xmin=a, xmax=b, ymin=0, ymax=10)
\end{sageverbatim}
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\section{Approximating a Rational Function}
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We can use the approximation above to create an approximation of a rational function (a fraction of two polynomials).  With a systematic use of partial fractions, this method can be extended to more complicated examples.  Consider the following:
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\begin{align*}
\frac{1+x^2}{1-x^2}
&=\frac{1-x^2}{1-x^2}+\frac{2x^2}{1-x^2}\\
&=1+2x^2\frac{1}{1-x^2}\\
\end{align*}
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Employ equation~(\ref{geometric}) where we replace $x$ by $x^2$,
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\begin{align*}
&\approx 1+2x^2\left(1+x^2+(x^2)^2+(x^2)^3+\cdots+(x^2)^n\right)\\
&=1+2x^2\left(1+x^2+x^4+x^6+\cdots+x^{2n}\right)\\
&=1+2x^2+2x^4+2x^6+2x^8+\cdots+2x^{2n+2}
\end{align*}
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Notice that our approximation should again be best when $n$ is large and now we would require $-1<x^2<1$ which simply translates back to $-1<x<1$.  In the demonstration below we only plot the function for $-1<x<1$.  The full graph would have vertical asymptotes at $x=-1$ and $x=1$ and has two branches below the $x$-axis --- one for $x<-1$ and another for $x>1$.
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\begin{sageverbatim}
%hide
%auto
a=-0.9999
b= 0.9999
original_color='blue'
approx_color='red'
@interact
def _( n = slider(0, 20, 2, 2 , label = "Degree") ):
    var('x')
    f(x)=(1+x^2)/(1-x^2)
    approx(x)=1
    for i in srange(2,n+1,2):
        approx(x)=approx(x)+2*x^i
    original_plot = plot( f(x), a, b ,color=original_color)
    approx_plot = plot( approx(x),  a, b, color=approx_color)
    html("Function: <font color='%s'>$%s$</font>$" % (original_color, latex(f(x))) )
    html("Approximation: <font color='%s'>$%s$</font>$" % (approx_color, latex(approx(x))) )
    show(original_plot+approx_plot, xmin=a, xmax=b, ymin=0, ymax=10)
\end{sageverbatim}
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\section{Approximating a Transcendental Function}
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We begin with equation~(\ref{geometric}), replacing $x$ by $-t^2$, then form a definite integral that equals the inverse tangent.  Again, we would expect larger values of $n$, with $-1<x<1$, to yield better approximations.  First,
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\begin{align*}
\frac{1}{1+t^2}
&=\frac{1}{1-(-t^2)}\\
&\approx 1+(-t^2)+(-t^2)^2+(-t^2)^3+\cdots+(-t^2)^n\\
&=1-t^2+t^4-t^6+\cdots+(-1)^{n}t^{2n}
\end{align*}
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We will now use a definite integral and the derivative of the inverse tangent in a novel way,
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\begin{align*}
\arctan(x)
&=\arctan(x)-\arctan(0)\\
&=\int_0^x\frac{1}{1+t^2}dt\\
&\approx\int_0^x\,1-t^2+t^4-t^6+\cdots+(-1)^{n}t^{2n}\,dt\\
&=\left.t-\frac{t^3}{3}+\frac{t^5}{5}-\frac{t^7}{7}+\cdots+\frac{(-1)^{n}t^{2n+1}}{2n+1}\,\right\vert_0^x\\
&=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots+\frac{(-1)^{n}x^{2n+1}}{2n+1}
\end{align*}
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In the demonstration below we have drawn attention to the value of the function and the approximating polynomial at $x=1$.  It is of course debatable if the approximation is even valid at $x=1$, but we will examine this question carefully later.  We know that in a 45-45-90 right triangle, the two non-hypotenuse sides are equal.  Expressing angles in radians we formulate this fact as $\tan\left(\frac{\pi}{4}\right)=1$, or equivalently, $\arctan(1)=\frac{\pi}{4}$.  So if we evaluate our approximating polynomial at $x=1$ we should get a reasonable approximation of $\frac{\pi}{4}$, and by extension, multiplying by 4 we could obtain an estimate of $\pi$.  Consider that $\pi$ is defined as the ratio of a circle's circumfrence to its diameter.  Could we derive the necessary trigonometric facts, limits, derivatives and integrals used above without ever computing an actual value for $\pi$?  I think so.  Hmmmmmmmmm.  The ``real'' value of $\frac{\pi}{4}$, and the approximation, are given below, in addition to being pinpointed by specific points on the plot.
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\begin{sageverbatim}
%hide
%auto
a=-1.75
b= 1.75
original_color='blue'
approx_color='red'
pi_true = point((1, float(pi/4)), rgbcolor='black', pointsize=20)
@interact
def _( n = slider(1, 21, 2, 1 , label = "Degree") ):
    var('x')
    f(x)=arctan(x)
    approx(x)=0
    sign=1
    for i in srange(1,n+1,2):
        approx(x)=approx(x)+sign*x^i/i
        sign=-1*sign
    pi_approx = point( (1, float(approx(1))), rgbcolor='green', pointsize=20)
    original_plot = plot( f(x), a, b ,color=original_color)
    approx_plot = plot( approx(x),  a, b, color=approx_color)
    html("Function: <font color='%s'>$%s$</font>$" % (original_color, latex(f(x))) )
    html("Approximation: <font color='%s'>$%s$</font>$" % (approx_color, latex(approx(x))) )
    print
    html("<font color='%s'>$\\frac{\\pi}{4}=\\arctan(1)=%s$</font>" % (original_color, latex(float(pi/4))) ) 
    html("<font color='%s'>$P_{%s}(1)=%s$</font>" % (approx_color, latex(n), latex(float(approx(1)))) ) 
    show(original_plot+approx_plot+pi_true+pi_approx, xmin=a, xmax=b, ymin=-1.5, ymax=1.5)
\end{sageverbatim}
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\section{An Approximation Valid Everywhere}
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Our previous approximating polynomials were each valid, at best, on the interval $-1<x<1$.  We will change our approach for this final example by simply producing a very interesting polynomial and examining its properties.  Recall that ``$n$-factorial'' is defined by $n!=n(n-1)(n-2)\cdots3\cdot2\cdot1$, and by convention $0!=1$.  Consider the polynomials, indexed by their degree $n$,
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\begin{align*}
P_n(x)&=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots+\frac{x^n}{n!}
\end{align*}
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Each of these polynomials has a derivative, which we will compute.  (Notice how fractions with factorials simplify nicely.)
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\begin{align*}
P_n'(x)
&=\frac{d}{dx}\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots+\frac{x^n}{n!}\right)\\
&=0+1+\frac{2x}{2!}+\frac{3x^2}{3!}+\frac{4x^3}{4!}+\cdots+\frac{nx^{n-1}}{n!}\\
&=1+\frac{2x}{2(1!)}+\frac{3x^2}{3(2!)}+\frac{4x^3}{4(3!)}+\cdots+\frac{nx^{n-1}}{n((n-1)!)}\\
&=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots+\frac{x^{n-1}}{(n-1)!}
\end{align*}
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So $P_n(x)$ and its derivative, $P_n'(x)$, are very similar, differing only in the term $\displaystyle\frac{x^n}{n!}$.  
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\begin{align*}
P_n(x)-P_n'(x)&=\frac{x^n}{n!}
\end{align*}
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Even for fixed values of $x$ greater than 1, if we let $n$ get large enough, the denominator of this fraction will overwhelm the numerator, and this difference will be small and tend to zero.  So $P_n(x)$ is very nearly equal to its derivative.  Do we know any functions like this?  Ah, yes, $e^x$!  The demonstration below examines the possibility that these polynomials might be good approximations to the exponential function.
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\begin{sageverbatim}
%hide
%auto
a=-2
b= 5
original_color='blue'
approx_color='red'
@interact
def _( n = slider(0, 10, 1, 2 , label = "Degree") ):
    var('x')
    f(x)=e^x
    approx(x)=0
    for i in srange(n+1):
        approx(x)=approx(x)+x^i/factorial(i)
    original_plot = plot( f(x), a, b ,color=original_color)
    approx_plot = plot( approx(x),  a, b, color=approx_color)
    html("Function: <font color='%s'>$%s$</font>$" % (original_color, latex(f(x))) )
    html("Approximation: <font color='%s'>$%s$</font>$" % (approx_color, latex(approx(x))) )
    show(original_plot+approx_plot, xmin=a, xmax=b, ymin=0, ymax=100)
\end{sageverbatim}
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\end{document}