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Figure:
Graphical Taylor analysis, the approximation includes only the linear term

Often times you are working with awful functions as part
of some analysis. They are impossible to integrate or differentiate or to
find the roots of. As with many things,
the trick is to find a reasonable approximation that is simple enough
to work with. The Taylor Series allows you to do just that. With it
you can decompose any complex function,
, into an infinite series. Often, a truncated series will be used for
for the analysis. For example, you might end up using only the first
two terms of the series (the linear portion of the approximation). The difference between
the value obtained from the infinite series and the truncated series can
be called "error" or noise.
The Taylor Series, or Taylor Expansion of any function, is
defined as:
The first two terms are the most important
because this is the linear approximation. It says that for any function,
f(x), you can create an approximation of it around any point, , by
looking at the slope of the function at and multiplying it
by the distance between and another point
you want to know something about. It is a straight line approximation, the
value of the function at the point, added to the derivative
.
Approximate Exponentialno_title
To demonstrate the how the Taylor series can generate an approximation
function for , we will define as an exponential
function and create an approximate function around zero. That is, we
will let
and . Since ,
, the linear portion of the Taylor expansion is:
If we want
to know
at , when
then
Compare this approximate answer to the correct answer to 6 decimal places, 0.818731.
Note, if
is positive, then the series will
diverge. However,
when the exponent is negative, the series converges. That is, the signs
of each term in the approximation will alternate.
Thus, for some analyses, we can replace
with
and continue
the analysis.
Subsections
Next: Reverse Engineering
Up: How to create a
Previous: Introduction
Index
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Frank Starmer
20040519