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8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES
and thus
c 2 =
f ′′ (a)
2
.
Therefore, the quadratic approximation P 2 (x) to f centered at x = 0 is
P 2 (x) = f (a) + f
′ (a)(x − a) +
f ′′ (a)
2!
(x − a)
2 .
This approach extends naturally to polynomials of higher degree. In this situation, we
define polynomials
P 3 (x) = P 2 (x) + c 3 (x − a)
3
,
P 4 (x) = P 3 (x) + c 4 (x − a)
4
,
P 5 (x) = P 4 (x) + c 5 (x − a)
5
,
and so on, with the general one being
P n (x) = P n−1 (x) + c n (x − a)
n .
The defining property of these polynomials is that for each n, P n (x) must have its value
and all its first n derivatives agree with those of f at x = a. In other words we require that
P
(k)
n (a) = f
(k) (a)
for all k from 0 to n.
To see the conditions under which this happens, suppose
P n (x) = c 0 + c 1 (x − a) + c 2 (x − a)
2 + · · · + c n (x − a)
n .
Then
P
(0)
n (a) = c 0
P
(1)
n (a) = c 1
P
(2)
n (a) = 2c 2
P
(3)
n (a) = (2)(3)c 3
P
(4)
n (a) = (2)(3)(4)c 4
P
(5)
n (a) = (2)(3)(4)(5)c 5
and, in general,
P
(k)
n (a) = (2)(3)(4) · · · (k − 1)(k)c k = k!c k .