User:EGM6341.S11.team5.cavalcanti/L3

Mtg 3: Mon, 10 Jan 11 3-1

Taylor Series (cont'd)}
 * Students who just joined
 * Ex of comp. for insight: Nature 2007 paper by Bottke et al. on origin of Chicxulub crater → extinction of dinosaurs
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HW* 1.1: Find $$\lim f\left(x\right),x\rightarrow 0$$.

Plot f(x), $$x\in \left\lbrack \mathrm{0,1}\right\rbrack $$.
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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.

3-2

Taylor series

$${e}^{x}=\sum _{j=0}^{\infty }\frac{{x}^{j}}{j!}=1+\sum _{j=0}^{\infty }\frac{{x}^{j}}{j!}$$ (1)

$${e}^{x}-1=\sum _{j=1}^{\infty }\frac{{x}^{j}}{j!}$$ (2)

$$f\left(x\right)=\frac{{e}^{x}-1}{x}=\sum _{j=1}^{\infty }\frac{{x}^{j-1}}{j!}$$ (3)

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

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

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

3-3

$${p}_{n}\left(x\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\mathrm{...}+{a}_{1}{x}^{1}+{a}_{0}{x}^{0}$$ (1) $$=\sum _{j=0}^{n}{a}_{j}{x}^{j}$$ (2)

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

$${p}_{n}\left(x\right)=f\left({x}_{0}\right)+\frac{\left(x-{x}_{0}\right)}{1!}{f}^{\left(1\right)}\left({x}_{0}\right)+\mathrm{...}+\frac{{\left(x-{x}_{0}\right)}^{n}}{n!}{f}^{\left(n\right)}\left({x}_{0}\right)$$ (3)

$${R}_{\left(n+1\right)}\left(x\right)\mathrm{\colon }=\frac{1}{n!}{\int }_{{x}_{0}}^{x}{\left(x-t\right)}^{n}{f}^{\left(n+1\right)}\left(t\right)\mathit{dt}$$ (4)

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

3-4
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HW 1.2: Find $${p}_{n}\left(x\right)$$and $${R}_{n+1}\left(x\right)$$of $${e}^{x}$$(S10)
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$$f\left(x\right)=\frac{{e}^{x}-1}{x}$$S11 at $${x}_{0}=0$$


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