User:Egm6341.s10.team3.heejun/Meeting 2 transcript

MTG 2: Tue. 5 Jan 10

[[media: Egm6341.s10.mtg2.djvu | Page 2-1]] 

2 ways to construct $$ \displaystyle fn $$:

1) Taylor series

2) Interpolating polynomial

$$ \displaystyle fn $$ as Taylor series of $$ \displaystyle f $$:

Example:
 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle I = \int_{0}^{1}\frac{e^{x}-1}{x}dx $$
 * }
 * }
 * }


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle f(x)=\frac{e^{x}-1}{x} $$
 * }.
 * }.
 * }.

The above function $ \displaystyle \color{blue}{f(x)} $ has a removable singularity at $ \displaystyle \color{blue}{x=0} $

It means that $ \displaystyle lim f(x) $ is finite when x is going to zero, $ \displaystyle x\rightarrow 0. $


 * {| style="width:100%" border="0" align="left"

HW: $$ \displaystyle Find\,\, lim f(x)\,\, as\,\, x\rightarrow 0.\,\, Plot\,\, f(x),\,\, x \in [0,1]\,\,\,\, $$
 * style="width:100%; padding:10px; border:2px solid #8888aa" |
 * style="width:100%; padding:10px; border:2px solid #8888aa" |
 * style = |
 * }.
 * }.

Review of Taylor series:
 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle e^{x} = \sum_{j=0}^{\infty}\frac{x^{j}}{j!} = \underbrace{1}_{\color{blue}{\frac{x^{0}}{0!}=\frac{1}{1} (j=0)}} + \sum_{j=1}^{\infty}\frac{x^{j}}{j!} $$
 * }
 * }
 * }


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle e^{x}-1 = \sum_{j=1}^{\infty}\frac{x^{j}}{j!} $$
 * }.
 * }.
 * }.

[[media: Egm6341.s10.mtg2.djvu | Page 2-2]] 


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle f(x)=\frac{e^{x}-1}{x} = \sum_{j=1}^{\infty}\frac{x^{j-1}}{j!} $$
 *  (1)
 * }.
 * }.
 * }.

Thm. on Taylor series (A. P.4, Thm 1.4)
 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle f(\cdot)\,\, such\,\, that\,\, f^{(n+1)}\,\, exists\,\, and\,\, continuous $$
 * }
 * }
 * }


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle \color{blue}{f^{(n+1)}(x):= \frac{d^{n+1}}{dx^{n+1}}f(x)} $$
 *  (2)
 * }
 * }
 * }


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle f(x) = \underbrace{P_{n}(x)}_{\color{blue}{poly.\, of\, order\, n}} + R_{n+1}(x) $$
 *  (3)
 * }
 * }
 * }


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle {\color{blue} P_{n}(x) \equiv f_{n}(x)\,\,\, (\equiv\,\, \leftarrow ident.\,\, or\,\, equiv.\,\, to)} $$
 * }
 * }
 * }


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle \color{blue}{The\,\, remainder\,\, R_{n+1}(x)\,\, can\,\, be\,\, thought\,\, as\,\, error.\,\,\,\, It\,\, will\,\, be\,\, used\,\, for\,\, error\,\, analysis.} $$
 * }
 * }
 * }


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle P_{n}(x) := f(x_{0}) + \frac{(x-x_{0})}{1!}F^{(1)}(x_{0}) + ... + \frac{(x-x_{0})^{n}}{n!}F^{(n)}(x_{0})\,\, \rightarrow\,\, This\,\, is\,\, the\,\, expansion\,\, of\,\, Taylor\,\, series\,\, at\,\, x=0 $$
 *  (4)
 * }
 * }
 * }


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle R_{n+1}(x) := \frac{1}{n!}\int_{x_{0}}^{x}\underbrace{(x-t)^{n}}_{\color{blue}{w(t)}}\underbrace{f^{(n+1)}(t)}_{\color{blue}{g(t)}} dt $$
 *  (5)
 * }.
 * }.
 * }.

[[media: Egm6341.s10.mtg2.djvu | Page 2-3]] 


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle R_{n+1}(x)=\frac{(x-x_{0})^{n+1}}{(n+1)!}f^{n+1}(\xi)\,\, for\,\, \xi \in [x_{0}, x] $$
 *  (1)
 * }.
 * }.
 * }.

Note: (5)   p.2-2 =   (1)   p.2-3   by int. mean value thm $ \color{blue}{\equiv} $ IMVT

Int. Mean Value Thm (IMVT)


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle \int_{a}^{b}w(x)f(x)dx = f(\xi)\int_{a}^{b}w(x)dx\,\, with\,\, \underbrace{w(x)\geq 0\,\, All\,\, x \in [a,b]}_{\color{blue}{w(\cdot)\,\, is\,\, a\,\, weighting\,\, function\,\, and\,\, non-negative}} $$
 * }.
 * }.
 * }.

Special case: $$ \displaystyle \color{blue}{w(x)=1\,\, All\,\, x \in [a,b]}. $$


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle \int_{a}^{b}f(x)dx = f(\xi)\underbrace{(b-a)}_{\color{blue}{\int_{a}^{b}1dx}} $$
 * }.
 * }.
 * }.

[[media: Egm6341.s10.mtg2.djvu | Page 2-4]] 




 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle The\,\, area\,\, under\,\, f(x)\,\, on\,\, left\,\, Fig.\,\, is\,\, the\,\, same\,\, with\,\, the\,\, area\,\, under\,\, f(\xi)\,\, on\,\, right\,\, Fig.\,\, between\,\, a\,\, and\,\, b\,\, $$
 * }
 * }
 * }


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle \int_{a}^{b}f(x)dx\,\, =\,\, f(\xi)(b-a) $$
 * }
 * }
 * }


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle f(\cdot)\,\, cont.\,\, on\,\, [a,b] $$
 * }
 * }
 * }


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle m \leq f(x) \leq M\,\, All\,\, x \in [a,b]\,\,\,\, where\,\, m:=min_{x}f(x)\,\, and\,\, M:=max_{x}f(x) $$
 * }
 * }
 * }


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle \Rightarrow \underbrace{\int_{a}^{b}mdx}_{\color{blue}{m(b-a)}} \leq \int_{a}^{b}f(x)dx \leq \underbrace{\int_{a}^{b}Mdx}_{\color{blue}{M(b-a)}} $$
 * }
 * }
 * }


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle \underbrace{\exists}_{\color{blue}{'there\,\, exists'}} \xi \in [a,b]\,\, such\,\, that\,\, m(b-a) \leq f(\xi)(b-a) \leq M(b-a) $$
 * }.
 * }.
 * }.