User:EGM6341.s11.team1.Chiu/Mtg4

Mtg 4: Wed, 12 Jan 11

[[media: Nm1.s11.mtg4.djvu| page4-1]]

Intermediate Mean Value Theorem : ​


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$$  \displaystyle
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\int_{a}^{b}w \left( x \right)f \left( x \right)dx = f \left( \xi \right) \int_{a}^{b} w \left( x \right)dx

$$     (1)
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with $$ \underbrace{w \left( x \right) \geq 0, \forall x \in \left[ a,b \right]}_{ \color{blue} w(.)\ is \ non-negative \ function} \ $$

Special case: $$ \color{blue} w \left( x \right) = 1, \forall x \in \left[ a,b \right] \ $$


 * nm1.s11.Mtg4.pg1.fig1.svg​

Consider $$ f(.) \ $$ continues on $$ [a,b] \ $$, $$ m:= \underset{x}{min} f \left( x \right),  \forall x \in \left[ a,b \right]  $$

​

[[media: Nm1.s11.mtg4.djvu| page4-2]]​

"$ M:= max f \left( x \right), x \in \left[ a,b \right] \ $"​

"$ m \leq f \left( x \right) \leq M, \forall x \in \left[ a,b \right] \ $"​

Integration over $$ \left[ a,b \right] \ $$:


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$$  \displaystyle \underbrace{\int_{a}^{b} \underset{\color{red} \underset{constant}{\uparrow}} m dx} _{ \color{blue} {m(b-a)}} \leq \underbrace{ \int_{a}^{b} f \left( x \right) dx}_{ \underset{ \color{red} \underset{ {(2)} \ \color{blue} {p.4-1  IMVT. MVT}}{\uparrow}} = { \color{blue} f \left( \xi \right) \left( b-a \right)}} \leq \underbrace{ \int_{a}^{b} \underset{ \color{red} \underset{constant}{\uparrow}} M dx}_{ \color{blue} {M \left( b-a \right)}}
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for \ \xi \in \left[ a,b \right]

$$


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Goal: Prove this equality i.e. IMVT


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$$  \displaystyle
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\underset{ \color{blue} f \left( . \right) \ constant \ by \ IVT}{\Rightarrow} m \leq \underbrace{ \frac {1}{(b-a)} \int_{a}^{b} f \left( x \right) dx}_{ \color{blue} =: \xi (zeta)} \leq M

$$
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$$  \displaystyle
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\exists \xi \in \left[ a,b \right] \ at \ f \left( \xi \right) =  \xi = \frac{1}{(b-a)} \int_{a}^{b} f \left( x \right) dx \color{red} \Rightarrow IMVT

$$


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Intermediate Value Theorem (Recall)

​

[[media:Nm1.s11.mtg4.djvu| page4-3]]​

Given $$ f(.) \ $$ continuous and $$ \xi \ $$ at $$ m \leq \xi \leq M $$, $$  \exists \xi \in  \left[ a,b \right]  \ $$ at $$ f \left( \xi \right) =  \xi  $$.


 * nm1.s11.Mtg4.pg3.fig1.svg​

End Intermediate Mean Value Theorem ​


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HW* 1.3:

a) Prove IMVT [[media: Nm1.s11.mtg4.djvu| page4-1]] for non-negative function $$ w(.) \ $$ i.e. $$ w \left( x \right) \geq 0,  \forall x \in \left[ a,b \right] \ $$

b) Another version of IMVT $$ w \left( x \right)  \neq 0,  \forall x \in \left[ a,b \right] \qquad $$   (1)

\ Either $$ w > 0 \qquad $$  (done in (a) with $$ w \geq 0 \ $$)   or $$ w < 0 \ $$


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​

[[media: Nm1.s11.mtg4.djvu| page4-4]]​

Question: Is the following function possible?


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does not satisfy [[media: Nm1.s11.mtg4.djvu| (1) page4-3]]

$$ \color{red} \rightarrow \ $$not acceptable

only 2 possibilities


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End Question

More on norms:(see [[media: Nm1.s11.mtg2.djvu| page2-1]])

Scalar (inner) product

Vectors: $$ \mathbf{u}, \mathbf{v} \in \mathbb{R}^{n} \ $$, $$ \left\{ \mathbf{e}_{i} \right\} \ $$ orthonormal basis

"​$ \mathbf{u}= \sum_{i}^{}{u}_{i} { \mathbf{e}}_{i} \color{blue} {\rightarrow} \color{black} ^{ \color{red} {T}} \ $|undefined"​

"​$ \mathbf{v} = \sum_{j}^{}{v}_{j} { \mathbf{e}}_{j} \color{blue} {\rightarrow}  \color{black} ^{ \color{red} {T}} \ $|undefined"​

​

[[media: Nm1.s11.mtg4.djvu| page4-5]]​

"​$ \underbrace{\mathbf{u} \cdot  \mathbf{v}}_{ \color{blue} {<\mathbf{u}, \mathbf{v}>}} =  \sum_{i}^{} {u}_{i} {v}_{j} \ $|undefined"​

functions: $$ f, g: \left[ a,b \right] \rightarrow \mathbb{R} \ $$

"​$ = \int_{a}^{b} f \left( x \right) g \left( x \right) dx \ $"​


 * nm1.s11.Mtg4.pg5.fig1.svg

"​$ { \color{blue} {\left \lfloor f \left( {x}_{0} \right), f \left( {x}_{1} \right) f \left(  {x}_{2} \right), \ldots, f \left( {x}_{n} \right) \right \rfloor}}^{ \color{red} {T}} \ $|undefined"​

"​$ { \color{black} {\left \lfloor g \left( {x}_{0} \right), g \left( {x}_{1} \right) g \left(  {x}_{2} \right), \ldots, g \left( {x}_{n} \right) \right \rfloor}}^{ \color{red} {T}} \ $​|undefined"​

"​$ \sum_{i=0}^{n} f \left( {x}_{i} \right) g \left( {x}_{i} \right) \ $ may go to $ \color{blue} \infty \ $ as $ \color{blue} h \rightarrow 0 \qquad $    (or $ \color{blue} n \rightarrow \infty \ $ )"​

Example: $$ \color{blue} f \left( x \right)=1, \forall x \ $$

"$ \color{blue} \qquad g \left( x \right)=1, \forall x \ $"

"​$ \Rightarrow \sum_{i=0}^{n} f \left( {x}_{i} \right) g \left( {x}_{i} \right)= \ $ $ \color{blue} n+1 \rightarrow \infty \ $ as $ \ \color{blue} n \rightarrow \infty \ $"​

​ End Example 

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