User:EGM6341.S11.team5.cavalcanti/Mtg3

 Mtg 3: Mon, 10 Jan 11 [[media: Nm1.s11.mtg3.djvu | Page 3-1]]

Taylor Series (cont'd)}
 * Students who just joined
 * Example of computation for insight: Nature 2007 paper by Bottke et al. on origin of Chicxulub crater → extinction of dinosaurs

Note: HW * was assigned in S10.

Boxed comment: Either – you did HW* on your own, w/o the help of S10 HW reports. Or – if you did look at S10 HW reports, link to that S10 HW rep., and indicate improvements you brought to their soln. Avoid plagiarism.  "End Note" 

[[media: Nm1.s11.mtg3.djvu | Page 3-2]]

Taylor series

Theorem: On Taylor series (A., p.4, Thm 1.4)

Any $$f\left(.\right)$$st $${f}^{\left(n+1\right)}$$exists and continuous. $$\color{blue}{{f}^{\left(n+1\right)}\left(x\right)\mathrm{\colon }=\frac{{d}^{\left(n+1\right)}}{{\mathit{dx}}^{\left(n+1\right)}}f\left(x\right)}$$

$$f\left(x\right)={p}_{n}\left(x\right)+{R}_{\left(n+1\right)}\left(x\right)$$ → lowercase “p”, mnemonic for “polynomial” of order n $${p}_{n}\in {P}_{n}$$ = set of poly. of deg. ≤ n

[[media: Nm1.s11.mtg3.djvu | Page 3-3]]

Here, $${f}_{n}\left(x\right)\equiv {p}_{n}\left(x\right)$$. For Taylor series of $$f\left(.\right)$$at $${x}_{0}$$:

$$=\frac{{\left(x-{x}_{0}\right)}^{n+1}}{\left(n+1\right)!}{f}^{\left(n+1\right)}(\xi )$$for $$\xi \in \left\lbrack {x}_{0},x\right\rbrack $$

[[media: Nm1.s11.mtg3.djvu | Page 3-4]]