Unformatted text preview: and f (0) = − sin(0) = 0 and
f (0) = − cos(0) = −1.
It follows that
L0 (x) = f (0) + f (0)(x − 0) = 1 + 0(x − 0) = 1
for all x while
T2,0 (x) = f (0) + f (0)(x − 0) +
= 1 + 0(x − 0) +
= 1− f (0)
(x − 0)2
2 −1
(x − 0)2
2 x2
.
2 The following diagram shows cos(x) with its linear approximation and its second degree
Taylor polynomial centered at x = 0. It is very easy to see that the second degree Taylor polynomial does a much better job
approximating cos(x) over the interval [−2, 2] than does the linear approximation.
We might guess that if f (x) has a third derivative at x = a, then by encoding the value
f (a) along with f (a), f (a) and f (a), we may do an even better job of approximating
f (x) near x = a than we did with either La (x) or with T2,a (x). As such we would be looking
for a polynomial
p(x) = c0 + c1 (x − a) + c2 (x − a)2 + c3 (x − a)3
such that
DE Math 128 316 (B. Forrest)2 CHAPTER 4. Sequences and Series 4.5. Introduction to Taylor Series 1. p(a) = f (a),
2. p (a) = f (a),
3. p (a) = f (a),
4. p (a) = f (a).
To ﬁnd such a p(x), we follow the steps tha...
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 Spring '09
 Math, Derivative, Power Series, Taylor Series, Sequences And Series, Taylor's theorem, B. Forrest

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