*This preview shows
page 1. Sign up
to
view the full content.*

**Unformatted text preview: **ever, it is
actually quite easy to see that this can be done. For example, if we want p(a) = f (a) then
by noting that
p(a) = c0 + c1 (a − a) + c2 (a − a)2 = c0
we immediately see that we should let c0 = f (a).
We can use the standard rules of diﬀerentiation to show that
p (x) = c1 + 2c2 (x − a).
In order that p (a) = f (a) we only need that
f (a) = p (a) = c1 + 2c2 (a − a) = c1 .
Finally, since
p (x) = 2c2
for all x, if we let c2 = f (a )
,
2 we will get that
p (a) = 2c2 = 2( f (a)
) = f (a)
2 exactly as desired. This shows that if
p(x) = f (a) + f (a)(x − a) + f (a)
(x − a)2 ,
2 then p(x) is the unique polynomial of degree 2 or less such that
1. p(a) = f (a),
2. p (a) = f (a),
3. p (a) = f (a).
DE Math 128 315 (B. Forrest)2 4.5. Introduction to Taylor Series CHAPTER 4. Sequences and Series The polynomial p(x) is called the second degree Taylor polynomial for f (x) centered at
x = a. We denote this Taylor polynomial by T2,a (x).
Example.
Let f (x) = cos(x). Then,
f (0) = cos(0) = 1,...

View
Full
Document