We can use the standard rules of dierentiation to

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

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