User:EGM6341.s11.team1.Chiu/Mtg6

Mtg 6: Sun, 16 Jan 11

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Proof of Taylor series continued

Since

$$  \displaystyle \int_{x_0}^{x} {f}^{1}(t)dt=[ {f}^{0}(t)]_{t= {x}_{0}}^{t=x} $$     (1)
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$$  \displaystyle \Rightarrow \color{red}(e) \color{blue} p.5-3 $$

Int. by parts (1)

$$  \displaystyle \color{red}(2) \begin{cases} & \ \color{black} \int_{x_0}^{x} \color{blue}  \underset{u^{'}}{ \underbrace{1}} \underset{v}{ \underbrace{ \color{black}{{f}^{(1)}(t)} }} \color{black}dt= {[uv]}^{x}_{x_0}- \int_{}^{}{uv}^{'}   \\ & \ \color{black}=  {[tf^{(1)}{t}]}^{t=x}_{t=x_0}-  \color{blue}  \underset{ \alpha}{ \underbrace{ \color{black}{ \int_{x_0}^{x}tf^{(2)}dt} }} \\ & \  \color{black}= xf^{1}(x)-x_{0}f^{(1)}(x)0)- \color{blue} \alpha  \\ & \  \color{red}+ \color{blue} xf^{(1)}(x_0) \color{red}- \color{blue}xf^{(1)}(x_{0})\\ & \     \color{black} =\color{blue} \underset{ \int_{x_{0}}^{x}xf^{(2)}(t)dt=: \beta}{  \underbrace{\color{black}[xf^(1)(x)-xf^(1)(x_{0})] }} \color{black}+(x-x_{0}){f}^{(1)}(x_{0}) - \color{blue}   \alpha \end{cases}

$$

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Combine [ + β - α ] into a single int. use (2) p.6-1 in (2) p.5-3 :


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$$  \displaystyle f(x)=f(x_{0})+ \frac{ {(x-x_{0})}^{ \color{blue} 1}}{ \color{blue}1! } {f}^{(1)}(x_{0})+ \color{blue} \underset{ \beta-\alpha}{ \underbrace{ \color{black} \int_{x_{0}}^{x}(x-t)f^{(2)}(t)dt }} $$

(1)
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HW*2.1: 1) Do integration by parts on last term (integration) of (1) to reveal 3 more terms in Taylor series, i.e. ,

$$  \displaystyle \frac{ {x- {x}_{0}}^{2}}{2!} {f}^{(2)}( {x}_{0})+ \frac{ {x- {x}_{0}}^{3}}{3!} {f}^{(3)}( {x}_{0})+ \frac{ {x- {x}_{0}}^{4}}{4!} {f}^{(4)}( {x}_{0}) $$

plus remainder2)Use IMVT to expression remainder in terms of f(s)(ξ) for s belong[x0,x]3) Assume(3)&amp;(4)p.3-3 correct, do intergration by parts once more

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to verify(3)&amp;(4) p.3-3 for (n+1) expansionwith R(n+2)(x)<P>4) UseIMVT on(4) p.3-3to show(5)p.3-3</P>

<U>HW</U><SUP>*</SUP><U>2.2:</U>$$f(x)=sinx, x\in[0, \pi]$$ Constrast Taylor Series of f(.) around

$$  \displaystyle \begin{cases} & \ {x}_{0}= \frac{ \pi}{4} \ \color{blue} S10 \\ & \ {x}_{0}= \frac{3 \pi}{4} \ \color{blue} S11 \end{cases} $$ for n = 0,1,2,...,10

Plot these series (for each n) Find (estimate) max $$ \left|{R}_{n+1}(x=\frac{3 \pi}{4}) \right|$$

$${\color{red}(5)} {\color{blue}p.3-3} \Rightarrow \left|{R}_{n+1}(x=\frac{3 \pi}{4}) \right| \leqq \frac{{x-{x}_{0}}^{n+1}}{(n+1)!}{\color{blue}(\alpha)}$$

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$${\color{blue} \alpha}=max \left|{f}^{(n+1)}(t) \right| $$

$$\color{blue} \underset{\leqq 1 (f(x)=sinx)}{ \underbrace{ \color{black} t \in [{x}_{0},x]}}$$

<U>Note:</U>Motivation for pf of Taylor series expansion. (similar technique will be used)


 * higher order analysis of Trap. rule (not in A.)
 * Richardson extrap.
 * clenshaw-Cwetis quadrature
 * chebyshew poly (orthog.) Recent devel. using chebyshew poly to solveL2_ODE_VC (Linear 2nd order ODE with varying coefficient ) combine of symbolics + numericsReference : Trefethen's chebfun.

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$${\color{blue}\underline{Note:}} \ $$

$${\color{red} \underset{Quadrilateral}{ \underbrace{\color{blue}rature=}}{\color{black} Num. \ int.}}$$

$${\underline{Greeks}}:\ Meas. \ area$$



$${\color{red} \underset{Cube}{ \underbrace{\color{blue}ture}}}$$

$$Vol. \ = \sum \ Cubes$$

<U><STRONG>Num. Int. Using Taylor Series cont'd</STRONG></U>

$$I_{n}= \int_{0}^{1}f_{n}(x)dx=\int_{0}^{1}$$

{\color{blue} \underset{f_{n}(x)=P_{n}(x)}{ \underbrace{\color{black}dx}}}

$$ \overset{=} \sum_{j=1}^{n}\frac{1}{j!j}$$

$${\color{red}(2)} \underset{ \xi \in [0,x]}{f(x)-f_{n}(x)}=R_{n+1}(x)=\frac{(x-0)^{n+1}}{x(n+1)!}exp(\xi)$$