Taylor Series

# We can use the standard rules of dierentiation to

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

This is the end of the preview. Sign up to access the rest of the document.

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

## This document was uploaded on 03/23/2013.

Ask a homework question - tutors are online