Unformatted text preview: s of the form
p(x) = c0 + c1 (x − a),
it is the only one with both properties 1 and 2 and as such, the only one that encodes both
the value of the function at x = a and its derivative.
We know that for x near a that
f (x) ∼ La (x).
=
This means that we can use the simple function La (x) to estimate the value of what could
be a rather complicated function f (x) at points near x = a. However, any time we use a
process to approximate a given value it is best that we understand as much as possible about
DE Math 128 313 (B. Forrest)2 4.5. Introduction to Taylor Series CHAPTER 4. Sequences and Series what the error in our procedure might be. In this case, the error in linear approximation
is
Error(x) = f (x) − La (x) 
and we know that at x = a our estimate is exact since La (a) = f (a).
We also know that there are two basic factors that aﬀect the potential size of our error in
using linear approximation. These are
1. How far x is away from a? That is, how large is  x − a ?
2. How curved the graph is near x = a?
Since the larger  f (x)  is, the more curved the graph will be, the second factor can
be expressed in t...
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 Spring '09
 Math, Derivative, Power Series, Taylor Series, Sequences And Series, Taylor's theorem, B. Forrest

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