University of Florida/Egm6341/s11.team1.Gong/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)

$\displaystyle \Rightarrow \color {red}(e)\color {blue}p.5-3$

Int. by parts (1)

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

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

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

(1)

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 remainder

2)Use IMVT to expression remainder in terms of f^(s)(ξ) for s belong[x₀,x]

3) Assume(3)&(4)p.3-3 correct , do intergration by parts once more

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to verify(3)&(4) p.3-3 for (n+1) expansionwith R_(n+2)(x)

4) UseIMVT on(4) p.3-3to show(5)p.3-3

IMVT:$\int _{a}^{b}w(x)g(x)dx{\overset {\color {red}(1)}{=}}g(\xi )\int _{a}^{b}w(x)dx$ $\xi \in [a,b]$ Use "g(x)" instead of "f(x)" to avoid confusion with "f(x)" is a Taylor Series.

HW^*2.2:$f(x)=sinx,x\in [0,\pi ]$ 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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${\color {blue}\alpha }=max\left|{f}^{(n+1)}(t)\right|$

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

Note: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:}}}\$

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

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

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

$Vol.\ =\sum \ Cubes$

Num. Int. Using Taylor Series cont'd

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

{\color{blue} \underset{f_{n}(x)=P_{n}(x)}{ \underbrace{{\color{black} \sum_{j=0}^{\n}\frac{x^{j-1}}{j!}}}{\color{black}dx}}}

${\overset {\color {red}(1)}{=}}\sum _{j=1}^{n}{\frac {1}{j!j}}$

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