3 4 5 x2 x4 2 24 t40 x 1 an important observation

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Unformatted text preview: 0. Hence T3,0 (x) = 1 + 0(x − 0) + −1 0 (x − 0)2 + (x − 0)3 2! 3! x2 2 = T2,0 (x) = 1− We also have that T4,0 (x) = 1 + 0(x − 0) + = 1− DE Math 128 −1 0 1 (x − 0)2 + (x − 0)3 + (x − 0)4 2! 3! 4! x2 x4 + 2 24 319 (B. Forrest)2 4.5. Introduction to Taylor Series CHAPTER 4. Sequences and Series and T5,0 (x) = 1 + 0(x − 0) + 0 1 0 −1 (x − 0)2 + (x − 0)3 + (x − 0)4 + (x − 0)5 2! 3! 4! 5! x2 x4 + 2 24 = T4,0 (x) = 1− An important observation to make is that not all of these polynomials are distinct. In fact, T0,0 (x) = T1,0 (x), T2,0 (x) = T3,0 (x) and T4,0 (x) = T5,0 (x). In general, this equality of different order Taylor polynomials happens when one of the derivatives is 0 at x = a. (In this example at x = 0.) This can be easily seen by observing that for any n Tn+1,a (x) = Tn,a (x) + f (n+1) (a) (x − a)n+1 (n + 1)! so if f (n+1) (a) = 0, we get Tn+1,a (x) = Tn,a (x). The diagram below shows cos(x) and its Taylor polynomials up to degree 5. You will notice that there are only four distinct graphs. In the next example, we wi...
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This document was uploaded on 03/23/2013.

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