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9.1
Chapter Nine
The Taylor Polynomial
9.1 Introduction
Let
f
be a function and let
F
be a collection of "nice" functions.
The approximation
problem is simply to find a function
g
∈
F
that is "close" to the given function
f
.
There are
two issues immediately.
How is the collection
F
selected, and what do we mean by
"close"?
The answers depend on the problem at hand. Presumably we want to do
something to
f
that is difficult or impossible (This might be something as simple as finding
f x
( ) for some
x.
).
The collection
F
would thus consist of functions to which it is easy to
do that which we wish to do to
f
.
Our measure of how close one function is to another
would try to reflect the closeness of the results of our operations.
Now, what are we
talking about here. Suppose, for example, we wish to find
f x
( ) .
Our collection
F
of
functions should include functions that are easy to evaluate at
x
, and two function would
be "close" simply if there values are close.
We might, for instance, want to evaluate sin
x
for all
x
is some interval
I
. The collection
F
could be a collection of second degree
polynomials.
The approximation problem is then to find elements of
F
that make the
"distance" max{sin
( ):
}
x
p x
x
I

∈
as small as possible.
Similarly, we might want to find
the integral of some function
f
over an interval
I
.
Here we would want
F
to consist of
functions easily integrated and measure the distance between functions by the difference of
their integrals over
I
. In the previous chapter, we found the "best" straight line
approximation to a set of data points. In that case, the collection
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 Fall '09
 Peter

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