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Unformatted text preview: linear approximation. Taylor’s Theorem shows that for some c, | Rn,a (x) |= f 2(c) (x − a)2 .
This shows explicitly how the error in linear approximation depends on the potential size
of f (x) and on | x − a |, the distance from x to a.
The second observation involves the case when n = 0. In this case, the theorem requires
that f (x) be diﬀerentiable on I and its conclusion states that for any x ∈ I there exists a
point c between x and a such that
f (x) − T0,a (x) = f (a)(x − a).
But T0,a (x) = f (a), so we have
f (x) − f (a) = f (a)(x − a).
DE Math 128 325 (B. Forrest)2 4.5. Introduction to Taylor Series CHAPTER 4. Sequences and Series Dividing by x − a shows that there is a point c between x and a such that
f (x) − f (a)
= f (c).
This is exactly the statement of the Mean Value Theorem. Therefore, Taylor’s Theorem is
really a higher-order version of the MVT. Finally Taylor’s Theorem does not tell us how to ﬁnd the point c, but rather that such a
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