User:EGM6341.s11.TEAM1.Yoon.HW1

Part (A)
Define m,M as below,

Consider,

Integrating each side of the non-equality, where, m and M are constants by definition of (3.1)

Deviding (3.3) by $$\displaystyle \frac{1}{\int^b_a w(x)dx}$$ yields

Applying Intermediate Value Theorem,

Thus,


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$$\displaystyle \begin{align} \therefore \int_a^b w(x) \cdot f(x)dx = f(\xi ) \int_a^b w(x)dx \quad _{for} \quad w(x) \geqq 0 \end{align} $$
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 * style = | $\displaystyle \color{red}{\begin{align} (3.7) \end{align}}$|undefined
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Part (B)
Define m,M as below,

Consider,

Integrating each side of the non-equality, where, m and M are constants by definition of (3.8)

Deviding (3.10) by $$\displaystyle \frac{1}{\int^b_a |w(x)|dx}$$ yields

Since $$\displaystyle w(x)<0 $$, $$\displaystyle |w(x)| $$ becomes $$\displaystyle - w(x) $$,

Applying Intermediate Value Theorem,

Thus,


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$$\displaystyle \begin{align} \therefore \int_a^b w(x) \cdot f(x)dx = f(\xi ) \int_a^b w(x)dx \quad _{for} \quad w(x) < 0 \end{align} $$
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 * style = | $\displaystyle \color{red}{\begin{align} (3.15) \end{align}}$|undefined
 * }
 * }