334 Chapter 6 • Differentiable FunctionsDefinition 6.5.9 Suppose f and its first n derivatives f', f", · · · , f(n) exist
in an open interval I containing a. Then, Vx E I , we define the nth Taylor
remainder for f about a to beRn(x) = f (x) -Tn(x).
Thus,. Vx E I,
f (x) = Tn(x) + Rn(x).
We can get some preliminary insight by looking at T 0 (x), the "zeroth»
Taylor polynomial for f about a in light of the mean value theorem, studied in
Section 6.4. Suppose f is differentiable in an interval I containing a. Let x E I,
x -/:-a. By the mean value theorem applied to f on the closed interval between
x and a, 3 c in the open interval between x and a such that
Equivalently,f(x) - f(a) = J'(c).
x-af(x) = f(a) + f'(c)(x - a)
f(x) - f(a) = f'(c)(x - a)
f(x) - Ta(x) = f'(c)(x - a)
Ro(x) = f'(c)(x - a).
Thus, the mean value theorem could be rephrased as a statement about
the remainder Ro ( x).Lemma 6.5.10 (Mean Value Theorem Rephrased) Suppose f is differentiable
in an interval I containing a. Then, for all x -/:-a in I , 3 c between x and a
such that Ro(x) = f'(c)(x - a).We are now at the point where we can state and prove Taylor's theorem.
It is a statement about the remainder Rn ( x), the difference between f ( x) and
Tn(x). It can be considered as a generalized form of the mean value theorem; in
fact, its proof will remind you of the proof of that theorem. We will use Rolle's
theorem just as we used it to prove the mean value theorem (6.4.3).
Theorem 6.5.11 (Taylor's Theorem) Suppose f is n times differentiable
on an open interval containing a and x, where x-/:-a, and f(n+l)(t) exists for
all t in the open interval I between a and x. If Tn(x) and Rn(x) are as defined
above, then 3 c E I 3f(n+l)( )
R (x ) = c (x - a)n+^1
n (n+l)! · (5)[Formula (5) is called the "Lagrange form"^13 of the remainder.]- There are other formulas for Rn(x ) that look quite different from this one. For example,
see Theorem 7.6.16.