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User:Egm6341.s11.team5.oh/HW2

=Problem 2 - Construct Taylor series = From [ (meeting page )]

Problem statement
$$ f(x)=sin x, \ x \in [0,\pi] $$ (i)Construct a Taylor series of $$f(x)$$ around $$x_0 = \frac{3 \pi}{8}$$ for $$\displaystyle n=0,1,2......,10.$$ (ii)Plot the series for each $$\displaystyle n $$ (iii)Estimate the maximum $$\displaystyle R(x)$$ at $$ x=\frac{3 \pi}{4}$$

Solution
 We refered to Egm6341.s10.team3 but solve by ourselves again with s11 values

Part 1. Construct
$$\displaystyle n^{th}$$ order polynomial is given by, $$\displaystyle P_n(x) = f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) + ........+ \frac{(x-x_0)^n}{n!}f^n(x_0) $$ (a)$$\displaystyle P_n(x)$$ for $$\displaystyle n=0 $$ and $$\displaystyle x_0 = \frac{3 \pi}{8}$$

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 * $$ \displaystyle P_0(x)= f(x_0) $$
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$$\displaystyle P_0(x) = sin (\frac{3 \pi}{8})$$
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(b)$$\displaystyle P_n(x)$$ for $$\displaystyle n=1 $$ and $$\displaystyle x_0 = \frac{3 \pi}{8}$$
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 * $$ \displaystyle P_1(x)= f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) $$
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$$ \displaystyle P_1(x)= sin (\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})}{1!} cos(\frac{3 \pi}{8}) $$
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(c)$$\displaystyle P_n(x)$$ for $$\displaystyle n=2 $$ and $$\displaystyle x_0 = \frac{3 \pi}{8}$$
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 * $$ \displaystyle P_2(x)= f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) + \frac{(x-x_0)^2}{2!} f^2(x_0) $$
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$$ \displaystyle P_2(x)= sin (\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})}{1!} cos(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^2}{2!} sin(\frac{3 \pi}{8})$$
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(d)$$\displaystyle P_n(x)$$ for $$\displaystyle n=3 $$ and $$\displaystyle x_0 = \frac{3 \pi}{8}$$
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 * $$ \displaystyle P_3(x)= f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) + \frac{(x-x_0)^2}{2!} f^2(x_0) + \frac{(x-x_0)^3}{3!} f^3(x_0)$$
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$$ \displaystyle P_3(x)= sin (\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})}{1!} cos(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^2}{2!} sin(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^3}{3!} cos(\frac{3 \pi}{8}) $$
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(e)$$\displaystyle P_n(x)$$ for $$\displaystyle n=4 $$ and $$\displaystyle x_0 = \frac{3 \pi}{8}$$
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 * $$ \displaystyle P_4(x)= f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) + \frac{(x-x_0)^2}{2!} f^2(x_0) + \frac{(x-x_0)^3}{3!} f^3(x_0) +  \frac{(x-x_0)^4}{4!} f^4(x_0) $$
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$$ \displaystyle P_4(x)= sin (\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})}{1!} cos(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^2}{2!} sin(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^3}{3!} cos(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^4}{4!} sin(\frac{3 \pi}{8}) $$
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(f)$$\displaystyle P_n(x)$$ for $$\displaystyle n=5 $$ and $$\displaystyle x_0 = \frac{3 \pi}{8}$$
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 * $$ \displaystyle P_5(x)= f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) + \frac{(x-x_0)^2}{2!} f^2(x_0) + \frac{(x-x_0)^3}{3!} f^3(x_0) +  \frac{(x-x_0)^4}{4!} f^4(x_0) + \frac{(x-x_0)^5}{5!} f^5(x_0)$$
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$$ \displaystyle P_5(x)= sin (\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})}{1!} cos(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^2}{2!} sin(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^3}{3!} cos(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^4}{4!} sin(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^5}{5!} cos(\frac{3 \pi}{8}) $$
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(g)$$\displaystyle P_n(x)$$ for $$\displaystyle n=6 $$ and $$\displaystyle x_0 = \frac{3 \pi}{8}$$
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 * $$ \displaystyle P_6(x)= f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) + \frac{(x-x_0)^2}{2!} f^2(x_0) + \frac{(x-x_0)^3}{3!} f^3(x_0) +  \frac{(x-x_0)^4}{4!} f^4(x_0) + \frac{(x-x_0)^5}{5!} f^5(x_0) + \frac{(x-x_0)^6}{6!} f^6(x_0)$$
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$$ \displaystyle P_6(x)= sin (\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})}{1!} cos(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^2}{2!} sin(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^3}{3!} cos(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^4}{4!} sin(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^5}{5!} cos(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^6}{6!} sin(\frac{3 \pi}{8}) $$
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(h)$$\displaystyle P_n(x)$$ for $$\displaystyle n=7 $$ and $$\displaystyle x_0 = \frac{3 \pi}{8}$$
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 * $$ \displaystyle P_7(x)= f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) + \frac{(x-x_0)^2}{2!} f^2(x_0) + \frac{(x-x_0)^3}{3!} f^3(x_0) +  \frac{(x-x_0)^4}{4!} f^4(x_0) + \frac{(x-x_0)^5}{5!} f^5(x_0) + \frac{(x-x_0)^6}{6!} f^6(x_0) $$
 * $$ \displaystyle P_7(x)= f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) + \frac{(x-x_0)^2}{2!} f^2(x_0) + \frac{(x-x_0)^3}{3!} f^3(x_0) +  \frac{(x-x_0)^4}{4!} f^4(x_0) + \frac{(x-x_0)^5}{5!} f^5(x_0) + \frac{(x-x_0)^6}{6!} f^6(x_0) $$

$$ \displaystyle + \frac{(x-x_0)^7}{7!} f^7(x_0)$$
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$$ \displaystyle P_7(x)= sin (\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})}{1!} cos(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^2}{2!} sin(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^3}{3!} cos(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^4}{4!} sin(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^5}{5!} cos(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^6}{6!} sin(\frac{3 \pi}{8})$$
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$$\displaystyle -\frac{(x-\frac{3 \pi}{8})^7}{7!} cos(\frac{3 \pi}{8}) $$
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(i)$$\displaystyle P_n(x)$$ for $$\displaystyle n=8 $$ and $$\displaystyle x_0 = \frac{3 \pi}{8}$$
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 * $$ \displaystyle P_8(x)= f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) + \frac{(x-x_0)^2}{2!} f^2(x_0) + \frac{(x-x_0)^3}{3!} f^3(x_0) +  \frac{(x-x_0)^4}{4!} f^4(x_0) + \frac{(x-x_0)^5}{5!} f^5(x_0) + \frac{(x-x_0)^6}{6!} f^6(x_0) $$
 * $$ \displaystyle P_8(x)= f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) + \frac{(x-x_0)^2}{2!} f^2(x_0) + \frac{(x-x_0)^3}{3!} f^3(x_0) +  \frac{(x-x_0)^4}{4!} f^4(x_0) + \frac{(x-x_0)^5}{5!} f^5(x_0) + \frac{(x-x_0)^6}{6!} f^6(x_0) $$

$$ \displaystyle + \frac{(x-x_0)^7}{7!} f^7(x_0) + \frac{(x-x_0)^8}{8!} f^8(x_0)$$
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$$ \displaystyle P_8(x)= sin (\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})}{1!} cos(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^2}{2!} sin(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^3}{3!} cos(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^4}{4!} sin(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^5}{5!} cos(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^6}{6!} sin(\frac{3 \pi}{8})$$
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$$\displaystyle -\frac{(x-\frac{3 \pi}{8})^7}{7!} cos(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^8}{8!} sin(\frac{3 \pi}{8}) $$
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(j)$$\displaystyle P_n(x)$$ for $$\displaystyle n=9 $$ and $$\displaystyle x_0 = \frac{3 \pi}{8}$$
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 * $$ \displaystyle P_9(x)= f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) + \frac{(x-x_0)^2}{2!} f^2(x_0) + \frac{(x-x_0)^3}{3!} f^3(x_0) +  \frac{(x-x_0)^4}{4!} f^4(x_0) + \frac{(x-x_0)^5}{5!} f^5(x_0) + \frac{(x-x_0)^6}{6!} f^6(x_0) $$
 * $$ \displaystyle P_9(x)= f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) + \frac{(x-x_0)^2}{2!} f^2(x_0) + \frac{(x-x_0)^3}{3!} f^3(x_0) +  \frac{(x-x_0)^4}{4!} f^4(x_0) + \frac{(x-x_0)^5}{5!} f^5(x_0) + \frac{(x-x_0)^6}{6!} f^6(x_0) $$

$$ \displaystyle + \frac{(x-x_0)^7}{7!} f^7(x_0) + \frac{(x-x_0)^8}{8!} f^8(x_0) + \frac{(x-x_0)^9}{9!} f^9(x_0)$$
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$$ \displaystyle P_9(x)= sin (\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})}{1!} cos(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^2}{2!} sin(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^3}{3!} cos(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^4}{4!} sin(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^5}{5!} cos(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^6}{6!} sin(\frac{3 \pi}{8})$$
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$$\displaystyle -\frac{(x-\frac{3 \pi}{8})^7}{7!} cos(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^8}{8!} sin(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^9}{9!} cos(\frac{3 \pi}{8}) $$
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(k)$$\displaystyle P_n(x)$$ for $$\displaystyle n=10 $$ and $$\displaystyle x_0 = \frac{3 \pi}{8}$$
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 * $$ \displaystyle P_{10}(x)= f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) + \frac{(x-x_0)^2}{2!} f^2(x_0) + \frac{(x-x_0)^3}{3!} f^3(x_0) +  \frac{(x-x_0)^4}{4!} f^4(x_0) + \frac{(x-x_0)^5}{5!} f^5(x_0) + \frac{(x-x_0)^6}{6!} f^6(x_0) $$
 * $$ \displaystyle P_{10}(x)= f(x_0) + \frac{(x-x_0)}{1!} f^1(x_0) + \frac{(x-x_0)^2}{2!} f^2(x_0) + \frac{(x-x_0)^3}{3!} f^3(x_0) +  \frac{(x-x_0)^4}{4!} f^4(x_0) + \frac{(x-x_0)^5}{5!} f^5(x_0) + \frac{(x-x_0)^6}{6!} f^6(x_0) $$

$$ \displaystyle + \frac{(x-x_0)^7}{7!} f^7(x_0) + \frac{(x-x_0)^8}{8!} f^8(x_0) + \frac{(x-x_0)^9}{9!} f^9(x_0) + \frac{(x-x_0)^{10}}{10!} f^{10}(x_0)$$
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$$ \displaystyle P_{10}(x)= sin (\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})}{1!} cos(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^2}{2!} sin(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^3}{3!} cos(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^4}{4!} sin(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^5}{5!} cos(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^6}{6!} sin(\frac{3 \pi}{8})$$
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$$\displaystyle -\frac{(x-\frac{3 \pi}{8})^7}{7!} cos(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^8}{8!} sin(\frac{3 \pi}{8}) + \frac{(x-\frac{3 \pi}{8})^9}{9!} cos(\frac{3 \pi}{8}) - \frac{(x-\frac{3 \pi}{8})^{10}}{10!} sin(\frac{3 \pi}{8}) $$
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Part 3. Error Estimation
$$\displaystyle R_{n+1}(x) = \frac{(x-x_0)^{n+1}}{(n+1)!} f^{n+1}(\xi)$$ $$\displaystyle max R_{n+1}(x) = \frac{(x-x_0)^{n+1}}{(n+1)!} max \bigg| f^{n+1}(\xi)\bigg|$$

Since $$\displaystyle f(x) = sin (x)$$  $$\displaystyle max \bigg| f^{n+1}(\xi)\bigg| = 1  $$ for all $$\displaystyle n $$

(a)$$\displaystyle R_{n+1}(x)$$ for $$\displaystyle n=0 $$ ,$$\displaystyle x_0 = \frac{3 \pi}{8} $$ and $$\displaystyle x = \frac{3 \pi}{4} $$

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 * $$ \displaystyle R_{n+1}(x)_{max}= \frac{x-x_0}{1!} $$
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$$ \displaystyle = \frac{\frac{3 \pi}{4}-\frac{3 \pi}{8}}{1!} $$
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$$\displaystyle R_1(x)_{max} = 1.1781 $$ ( WA result )
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(b)$$\displaystyle R_{n+1}(x)_{max}$$ for $$\displaystyle n=1 $$ ,$$\displaystyle x_0 = \frac{3 \pi}{8} $$ and $$\displaystyle x = \frac{3 \pi}{4} $$

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 * $$ \displaystyle R_{2}(x)_{max}= \frac{(x-x_0)^2}{2!}  $$
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$$ \displaystyle = \frac{(\frac{3 \pi}{4}-\frac{3 \pi}{8})^2}{2!} $$
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$$\displaystyle R_2(x)_{max} = 0.69396 $$ ( WA result )
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(c)$$\displaystyle R_{n+1}(x)_{max}$$ for $$\displaystyle n=2 $$ ,$$\displaystyle x_0 = \frac{3 \pi}{8} $$ and $$\displaystyle x = \frac{3 \pi}{4} $$

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 * $$ \displaystyle R_{3}(x)_{max}= \frac{(x-x_0)^3}{3!}  $$
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$$ \displaystyle = \frac{(\frac{3 \pi}{4}-\frac{3 \pi}{8})^3}{3!} $$
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$$\displaystyle R_3(x)_{max} = 0.27252 $$ ( WA result )
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(d)$$\displaystyle R_{n+1}(x)_{max}$$ for $$\displaystyle n=3 $$ ,$$\displaystyle x_0 = \frac{3 \pi}{8} $$ and $$\displaystyle x = \frac{3 \pi}{4} $$

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 * $$ \displaystyle R_{4}(x)_{max}= \frac{(x-x_0)^4}{4!}  $$
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$$ \displaystyle = \frac{(\frac{3 \pi}{4}-\frac{3 \pi}{8})^4}{4!} $$
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$$\displaystyle R_4(x)_{max} = 0.08026 $$ ( WA result )
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(e)$$\displaystyle R_{n+1}(x)_{max}$$ for $$\displaystyle n=4 $$ ,$$\displaystyle x_0 = \frac{3 \pi}{8} $$ and $$\displaystyle x = \frac{3 \pi}{4} $$

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 * $$ \displaystyle R_{5}(x)_{max}= \frac{(x-x_0)^5}{5!}  $$
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$$ \displaystyle = \frac{(\frac{3 \pi}{4}-\frac{3 \pi}{8})^5}{5!} $$
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$$\displaystyle R_5(x)_{max} = 0.01891 $$ ( WA result )
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(f)$$\displaystyle R_{n+1}(x)_{max}$$ for $$\displaystyle n=5 $$ ,$$\displaystyle x_0 = \frac{3 \pi}{8} $$ and $$\displaystyle x = \frac{3 \pi}{4} $$

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 * $$ \displaystyle R_{6}(x)_{max}= \frac{(x-x_0)^6}{6!}  $$
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$$ \displaystyle = \frac{(\frac{3 \pi}{4}-\frac{3 \pi}{8})^6}{6!} $$
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$$\displaystyle R_6(x)_{max} = 0.00371 $$ ( WA result )
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(g)$$\displaystyle R_{n+1}(x)_{max}$$ for $$\displaystyle n=6 $$ ,$$\displaystyle x_0 = \frac{3 \pi}{8} $$ and $$\displaystyle x = \frac{3 \pi}{4} $$

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 * $$ \displaystyle R_{7}(x)_{max}= \frac{(x-x_0)^7}{7!}  $$
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$$ \displaystyle = \frac{(\frac{3 \pi}{4}-\frac{3 \pi}{8})^7}{7!} $$
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$$\displaystyle R_7(x)_{max} = 6.24939 * 10^{-4} $$ ( WA result )
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(h)$$\displaystyle R_{n+1}(x)_{max}$$ for $$\displaystyle n=7 $$ ,$$\displaystyle x_0 = \frac{3 \pi}{8} $$ and $$\displaystyle x = \frac{3 \pi}{4} $$

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 * $$ \displaystyle R_{8}(x)_{max}= \frac{(x-x_0)^8}{8!}  $$
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$$ \displaystyle = \frac{(\frac{3 \pi}{4}-\frac{3 \pi}{8})^8}{8!} $$
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$$\displaystyle R_8(x)_{max} = 9.20298 * 10^{-5} $$ ( WA result )
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(i)$$\displaystyle R_{n+1}(x)_{max}$$ for $$\displaystyle n=8 $$ ,$$\displaystyle x_0 = \frac{3 \pi}{8} $$ and $$\displaystyle x = \frac{3 \pi}{4} $$

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 * $$ \displaystyle R_{9}(x)_{max}= \frac{(x-x_0)^9}{9!}  $$
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 * }

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

$$ \displaystyle = \frac{(\frac{3 \pi}{4}-\frac{3 \pi}{8})^9}{9!} $$
 * }
 * }

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

$$\displaystyle R_9(x)_{max} = 1.20467 * 10^{-5}$$ ( WA result )
 * style="width:35%; padding:10px; border:2px solid #8888aa" |
 * style="width:35%; padding:10px; border:2px solid #8888aa" |
 * style= |
 * }
 * }

(j)$$\displaystyle R_{n+1}(x)_{max}$$ for $$\displaystyle n=9 $$ ,$$\displaystyle x_0 = \frac{3 \pi}{8} $$ and $$\displaystyle x = \frac{3 \pi}{4} $$

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

 * $$ \displaystyle R_{10}(x)_{max}= \frac{(x-x_0)^{10}}{10!}  $$
 * }
 * }
 * }

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

$$ \displaystyle = \frac{(\frac{3 \pi}{4}-\frac{3 \pi}{8})^{10}}{10!} $$
 * }
 * }

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

$$\displaystyle R_{10}(x)_{max} = 1.41922 * 10^{-6} $$ ( WA result )
 * style="width:35%; padding:10px; border:2px solid #8888aa" |
 * style="width:35%; padding:10px; border:2px solid #8888aa" |
 * style= |
 * }
 * }

(k)$$\displaystyle R_{n+1}(x)_{max}$$ for $$\displaystyle n=10 $$ ,$$\displaystyle x_0 = \frac{3 \pi}{8} $$ and $$\displaystyle x = \frac{3 \pi}{4} $$

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

 * $$ \displaystyle R_{11}(x)_{max}= \frac{(x-x_0)^{11}}{11!}  $$
 * }
 * }
 * }

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

$$ \displaystyle = \frac{(\frac{3 \pi}{4}-\frac{3 \pi}{8})^{11}}{11!} $$
 * }
 * }

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

$$\displaystyle R_{11}(x)_{max} = 1.519976 * 10^{-7} $$ ( WA result )
 * style="width:35%; padding:10px; border:2px solid #8888aa" |
 * style="width:35%; padding:10px; border:2px solid #8888aa" |
 * style= |
 * }
 * }

Author and proof-reader
[Author] Oh

[Proof-reader]

=Problem 6 - Verification genernal form of Legendre poly.= From [ (meeting page )]

Given

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

P_0(x) = 1 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (6-1)
 * }
 * }

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

P_1(x) = x $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (6-2)
 * }
 * }

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

P_2(x) = \frac{1}{2} \left(3 x ^ 2 - 1\right) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (6-3)
 * }
 * }

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

P_3(x) = \frac{1}{2} \left(5 x ^ 3 - 3 x\right) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (6-4)
 * }
 * }

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

P_4(x) = \frac{35}{8} x ^ 4 - \frac{15}{4} x ^ 2 + \frac{3}{8} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (6-5)
 * }
 * }

Find
Verify that (6-1) - (6-5) can be written as

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

P_n(x) = \sum_{i=0}^{[n/2]} \frac{(-1)^i (2 n - 2 i)! x^{n - 2i}}{2^n i! (n-i)! (n-2i)!} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (6-6)
 * }
 * }

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

 * }
 * }
 * }

$$\displaystyle \ [\frac{n}{2}]$$ = interger part of  $$ \displaystyle \ \frac{n}{2} $$

Solution
''' We refered to [http://en.wikiversity.org/w/index.php?title=User_talk:Egm6321.f10.team5.oh/hw6#Problem_8_-_Verification_genernal_form_of_Legendre_poly. Egm6321.f10.team5.oh] and verify the solution with WA'''

Background knowledge :  Factorial , Floor and ceiling functions

1. In case of $$\displaystyle n = 0 $$

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

P_0(x) = \sum_{i=0}^{[0/2]} \frac{(-1)^i (2 (0) - 2 i)! x^{0 - 2i}}{2^0 i! (0-i)! (0-2i)!} \ = \sum_{i=0}^{0} \frac{(-1)^i ( - 2 i)! x^{- 2i}}{i! (-i)! (-2i)!} \ = \frac{(-1)^0 \cdot 0! \cdot x^0}{0! \cdot 0! \cdot 0!}$$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }
 * }

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

$$\displaystyle P_0(x) = 1 $$ ( WA result )
 * style="width:25%; padding:10px; border:2px solid #8888aa" |
 * style="width:25%; padding:10px; border:2px solid #8888aa" |
 * }
 * }
 * }

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

 * }
 * }
 * }

2. In case of $$\displaystyle n = 1 $$

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

P_1(x) = \sum_{i=0}^{[1/2]} \frac{(-1)^i (2 (1) - 2 i)! x^{1 - 2i}}{2^1 i! (1-i)! (1-2i)!}= \sum_{i=0}^{0} \frac{(-1)^i \ (2 - 2 i)! \ x^{1 - 2i}}{2 \ i! \ (1-i)! \ (1-2i)!}= \frac{(-1)^0 \cdot 2! \cdot x^1}{2 \cdot 0! \cdot 1! \cdot 1!} $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }
 * }

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

$$\displaystyle P_1(x) = x $$ ( WA result )
 * style="width:25%; padding:10px; border:2px solid #8888aa" |
 * style="width:25%; padding:10px; border:2px solid #8888aa" |
 * }
 * }
 * }

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

 * }
 * }
 * }

3. In case of $$\displaystyle n = 2 $$

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

P_2(x) = \sum_{i=0}^{[2/2]} \frac{(-1)^i (2 (2) - 2 i)! x^{2 - 2i}}{2^2 i! (2-i)! (2-2i)!}= \sum_{i=0}^{1} \frac{(-1)^i \ (4 - 2 i)! \ x^{2 - 2i}}{4 \ i! \ (2-i)! \ (2-2i)!}= \underbrace{\frac{(-1)^0 \cdot 4! \cdot x^2}{4 \cdot 0! \cdot 2! \cdot 2!}}_{\frac{3}{2}x^2} \ + \ \underbrace{\frac{(-1)^1 \cdot 2! \cdot x^0}{4 \cdot 1! \cdot 1! \cdot 1!}}_{-\frac{1}{2}} $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }
 * }

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

$$\displaystyle P_2(x) = \frac{1}{2}(3x^2-1) $$ ( WA result )
 * style="width:45%; padding:10px; border:2px solid #8888aa" |
 * style="width:45%; padding:10px; border:2px solid #8888aa" |
 * }
 * }
 * }

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

 * }
 * }
 * }

4. In case of $$\displaystyle n = 3 $$
 * {| style="width:100%" border="0" align="left"

P_3(x) = \sum_{i=0}^{[3/2]} \frac{(-1)^i (2 (3) - 2 i)! x^{3 - 2i}}{2^3 i! (3-i)! (3-2i)!}= \sum_{i=0}^{1} \frac{(-1)^i (6 - 2 i)! x^{3 - 2i}}{8 \ i! \ (3-i)! \ (3-2i)!} = \underbrace{\frac{(-1)^0 \cdot 6! \cdot x^{3}}{8 \cdot 0! \cdot 3! \cdot 3!}}_{\frac{5}{2}x^3} + \underbrace{\frac{(-1)^1 \cdot 4! \cdot x^1}{8 \cdot 1! \cdot 2! \cdot 1!}}_{-\frac{3}{2}x}$$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }
 * }

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

$$\displaystyle P_3(x) = \frac{1}{2}(5x^3-3x) $$ ( WA result )
 * style="width:45%; padding:10px; border:2px solid #8888aa" |
 * style="width:45%; padding:10px; border:2px solid #8888aa" |
 * }
 * }
 * }

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

 * }
 * }
 * }

5. In case of $$\displaystyle n = 5 $$
 * {| style="width:100%" border="0" align="left"

P_4(x) = \sum_{i=0}^{[4/2]} \frac{(-1)^i (2 (4) - 2 i)! x^{4 - 2i}}{2^4 i! (4-i)! (4-2i)!}= \sum_{i=0}^{2} \frac{(-1)^i \ (8 - 2 i)! \ x^{4 - 2i}}{16 \ i! \ (4-i)! \ (4-2i)!}$$
 * $$\displaystyle
 * $$\displaystyle

$$\displaystyle = \underbrace{\frac{(-1)^0 \cdot 8! \cdot x^4}{16 \cdot 0! \cdot 4! \cdot 4!}}_{\frac{35}{8}x^4} \ + \ \underbrace{\frac{(-1)^1 \cdot 6! \cdot x^{2}}{16 \cdot 1! \cdot 3! \cdot 2!}}_{- \frac{15}{4}x^2} \ + \ \underbrace{\frac{(-1)^2 \cdot 4! \cdot x^0}{16 \cdot 2! \cdot 2! \cdot 0!}}_{\frac{3}{8}}$$
 * }
 * }
 * }

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

$$\displaystyle P_4(x) = \frac{35}{8} x ^ 4 - \frac{15}{4} x ^ 2 + \frac{3}{8} $$ ( WA result )
 * style="width:45%; padding:10px; border:2px solid #8888aa" |
 * style="width:45%; padding:10px; border:2px solid #8888aa" |
 * }
 * }
 * }

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

 * }
 * }
 * }

Author and proof-reader
[Author] Oh

[Proof-reader]

=Problem 8 - Trapezoidal rule (simple) = From [ (meeting page )]

Given

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

I = I_1(f) = \int\limits_{a}^{b} f_1(x) \, dx = (\int\limits_{a}^{b} l_0(x) \ dx)f(x_0) + (\int\limits_{a}^{b} l_1(x) \ dx)f(x_1)] $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (8-1)
 * }
 * }

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

\Rightarrow I_1 = \dfrac{b-a}{2}[f(a)+f(b)] $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (8-2)
 * }
 * }

Find
Show $$ (8-1) ==> (8.2) $$

==> Trap. rule(simple)

Solution
We refered to Egm6341.s10.Team4 but we solved again by ourselves and discussed about background knowledge and tried to make more clearly.

Background knowledge :  Trapezoidal_rule , Lagrange

According to the lecture mtg9 (9-2) and (9-3)

 * $$\displaystyle

f_1(x) = P_1(x) = \sum_{i=0}^{n=1} l_i(x)f(x_i) $$
 * $$\displaystyle

f_1(x) = \underbrace{l_0(x)}_{\frac{x-x_1}{x_0-x_1}}f(x_0) + \underbrace{l_1(x)}_{\frac{x-x_1}{x_1-x_0}}f(x_1) $$

Simply define

 * $$\displaystyle

x_0=a;   x_1=b $$

 * $$\displaystyle

(8-1) \Rightarrow\ I= \int\limits_{a}^{b}[\dfrac{x-b}{a-b}f(a)+\dfrac{x-a}{b-a}f(b)]\, dx $$

\dfrac{1}{b-a} \int\limits_{a}^{b}[(b-x)f(a)+(x-a)f(b)]\, dx $$

=\dfrac{1}{b-a} [\int\limits_{a}^{b}(b-x)f(a)\, dx]+[\int\limits_{a}^{b}(x-a)f(b)\, dx] $$

Now, integration can be easily solved in its interval:

=\dfrac{1}{b-a}[f(a)(bx-\dfrac{x^2}{2})+f(b)(\dfrac{x^2}{2}-ax)]_{a}^{b}=\dfrac{1}{b-a}[f(a)(\dfrac{b^2}{2}-ab+\dfrac{a^2}{2})+f(b)(\dfrac{b^2}{2}-ab+\dfrac{a^2}{2})] $$

=\dfrac{1}{2(b-a)}[f(a)(b-a)^2+f(b)(b-a)^2]=\dfrac{b-a}{2}[f(a)+f(b)] $$

So,

 * {| style="width:20%" border="0"

$$\displaystyle \Rightarrow I = \dfrac{b-a}{2}[f(a)+f(b)] $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

 * style= |

 * }.
 * }.

Author and proof-reader
[Author]

[Proof-reader]

=Problem 10 - Lagrange Basis Approximation= From [ (meeting page )]

Given

 * {| style="width:90%" border="0" align="center"

p_2(x) = c_2 x^2 + c_1 x + c_0 $$ $$
 * $$\displaystyle
 * $$\displaystyle (10-1)

 * }
 * }

 * {| style="width:90%" border="0" align="center"

p_2(x) = f(x_i), \ i=0,1,2 $$ $$
 * $$\displaystyle
 * $$\displaystyle (10-2)

 * }
 * }

Find
We need to approximate Equation (1) using,

 * {| style="width:90%" border="0" align="center"

p_2(x) = \sum_{i=0}^{2} l_i(x_j)f(x_i) $$ $$
 * $$\displaystyle
 * $$\displaystyle (10-3)

 * }.
 * }.

where,

 * {| style="width:90%" border="0" align="center"

l_i(x) = \prod_{j=0,i \ne j}^{2} \frac{x-x_j}{x_i-x_j} $$ $$
 * $$\displaystyle
 * $$\displaystyle (10-4)

 * }.
 * }.

and find $$\displaystyle c_0, c_1 \ and\ c_2 $$ in terms of the given points,

 * {| style="width:90%" border="0" align="center"

(x_i,f(x_i)), \ i = 0,1,2 $$
 * $$\displaystyle

 * }.
 * }.

Solution
 We refered to Egm6341.s10.Team4 but we solved again by ourselves and discussed about background knowledge

Background knowledge :  Lagrange_polynomial ,

Equation (10-3) is given by the following 3 Equations,

 * {| style="width:90%" border="0" align="center"

l_0(x) = \frac{(x-x_1)(x-x_2)}{(x_0-x_1)(x_0-x_2)} $$ $$
 * $$\displaystyle
 * $$\displaystyle (10-5)

 * }
 * }

 * {| style="width:90%" border="0" align="center"

l_1(x) = \frac{(x-x_0)(x-x_2)}{(x_1-x_0)(x_1-x_2)} $$ $$
 * $$\displaystyle
 * $$\displaystyle (10-6)

 * }
 * }

 * {| style="width:90%" border="0" align="center"

l_2(x) = \frac{(x-x_0)(x-x_1)}{(x_2-x_0)(x_2-x_1)} $$ $$
 * $$\displaystyle
 * $$\displaystyle (10-7)

 * }
 * }

Substituting Equations (10-5), (10-6) and (10-7) in (10-3),

 * {| style="width:90%" border="0" align="center"

p_2(x) = \frac{(x^2-(x_1+x_2)x+x_1x_2)}{(x_0-x_1)(x_0-x_2)} f(x_0) + \frac{(x^2-(x_0+x_2)x+x_0x_2)}{(x_1-x_0)(x_1-x_2)} f(x_1) + \frac{(x^2-(x_0+x_1)x+x_0x_1)}{(x_2-x_0)(x_2-x_1)} f(x_2) $$ $$
 * $$\displaystyle
 * $$\displaystyle (10-8)

 * }
 * }

Rearranging,

 * {| style="width:90%" border="0" align="center"

\begin{align} \Rightarrow p_2(x) =  & x^2 \underbrace{\left [\frac{f(x_0)}{(x_0-x_1)(x_0-x_2)}  + \frac{f(x_1)}{(x_1-x_0)(x_1-x_2)}  + \frac{f(x_2)}{(x_2-x_0)(x_2-x_1)} \right]}_{c_2} +  \\ & x \underbrace{\left [\frac{-f(x_0)(x_1+x_2)}{(x_0-x_1)(x_0-x_2)}  + \frac{-f(x_1)(x_0+x_2)}{(x_1-x_0)(x_1-x_2)}  + \frac{-f(x_2)(x_0+x_1)}{(x_2-x_0)(x_2-x_1)} \right]}_{c_1}+ \\ & \underbrace{\left [\frac{f(x_0)(x_1x_2)}{(x_0-x_1)(x_0-x_2)} + \frac{f(x_1)(x_0x_2)}{(x_1-x_0)(x_1-x_2)}  + \frac{f(x_2)(x_0x_1)}{(x_2-x_0)(x_2-x_1)} \right]}_{c_0}
 * $$\displaystyle

\end{align} $$

 * }
 * }

So,

 * {| style="width:90%" border="0" align="center"

$$\displaystyle
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

\Rightarrow c_0 = \frac{f(x_0)(x_1x_2)}{(x_0-x_1)(x_0-x_2)}  + \frac{f(x_1)(x_0x_2)}{(x_1-x_0)(x_1-x_2)}  + \frac{f(x_2)(x_0x_1)}{(x_2-x_0)(x_2-x_1)}

$$

 * style= |

 * }
 * }

 * {| style="width:90%" border="0" align="center"

$$\displaystyle
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

\Rightarrow c_1 = \frac{-f(x_0)(x_1+x_2)}{(x_0-x_1)(x_0-x_2)} + \frac{-f(x_1)(x_0+x_2)}{(x_1-x_0)(x_1-x_2)}  + \frac{-f(x_2)(x_0+x_1)}{(x_2-x_0)(x_2-x_1)}

$$
 * style= |
 * }
 * }

 * {| style="width:90%" border="0" align="center"

$$\displaystyle \Rightarrow c_2 = \frac{f(x_0)}{(x_0-x_1)(x_0-x_2)} + \frac{f(x_1)}{(x_1-x_0)(x_1-x_2)}  + \frac{f(x_2)}{(x_2-x_0)(x_2-x_1)} $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style= |
 * }
 * }

Author and proof-reader
[Author]

[Proof-reader]

=References=

title: User:Egm6341.s11.team5.oh/HW2


text:
User:Egm6341.spring-11.5.D

=  Class Lecture Plan page = http://en.wikiversity.org/wiki/User:Egm6341.s11/Lecture_plan

=  Team page = Team #5

= Team 5 Members  = Junsu Shin

Sangmin Oh

Cavalcanti,Eunice H

Aditya Raghunathan

Reiss, Phillip J

=  Homework page = Homework #1

Homework #2

Homework #3

Homework #4

Homework #5

Homework #6

Homework #7

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Egm6341.spring-11.5.D


title: File:Nm1.s11.team2.HW1.fig4-1.png


text:
User:Egm6341.spring-11.5.D/HW1

Find

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

\int _a^bf(x) w(x)dx=f(\zeta ) \int_a^b w(x) \, dx $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Solution
'''Solved via combination of Wikipedia http://en.wikipedia.org/wiki/Mean_value_theorem#Mean_value_theorems_for_integration and Egm6341.s10.team3.sa/HW1 http://en.wikiversity.org/w/index.php?title=User:Egm6341.s10.team3.sa/HW1&oldid=524038#Prove_IMVT'''

First, since w(x) is arbitrary and non negative we will set it equal to I

$$ I=\int_a^b w(x) \, dx$$

$$\displaystyle

\int_a^b w(x) f(x) \, dx=f(\zeta ) I $$

Next we will set constants at the extreme values of $$ f(x)\in [a,b]$$

$$\displaystyle M=\max (f(x)),m=\min (f(x)) $$ Therefore, from the Extreme Value Theorem

http://en.wikipedia.org/wiki/Extreme_value_theorem

$$\displaystyle

m I \leq \int_a^b w(x) f(x) \, dx\leq M  I $$ Evaluating and subsituting in from above

$$\displaystyle m \leq \frac{1}{ I } \int_a^b w(x) f(x) \, dx\leq M $$

$$Z=\frac{1}{ I } \int_a^b w(x) f(x) \, dx$$ $$\displaystyle \min (f(x))\leq f(\zeta )\leq \max (f(x)) $$

Since f(x)is continous from M to m there exists $$ \zeta \in [a,b] at   f(\zeta)=Z $$

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

$$\therefore f(\zeta)=\frac{1}{\int_a^b w(x) \, dx} \int_a^b w(x) f(x) \, dx$$
 * style="width:50%; padding:10px; border:2px solid #8888aa" align="left" |
 * style="width:50%; padding:10px; border:2px solid #8888aa" align="left" |

 * }
 * }
 * }

Egm6341.spring-11.5.D 19:30, 24 January 2011 (UTC)

If w(x) is strictly negative

$$ f(\zeta)=\frac{1}{(-1) \cdot \int_a^b w(x) \, dx} \cdot (-1) \cdot \int_a^b w(x) f(x) \, dx$$

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

 * style="width:50%; padding:10px; border:2px solid #8888aa" align="left" |
 * style="width:50%; padding:10px; border:2px solid #8888aa" align="left" |

$$\therefore f(\zeta)=\frac{1}{\int_a^b w(x) \, dx} \int_a^b w(x) f(x) \, dx$$

 * }
 * }
 * }

lecture hyperlink
http://clesm.mae.ufl.edu/~vql/videos/nm1.s11.mtg29.wmv

http://clesm.ippd.ufl.edu/~vql/videos/nm1.s11.mtg29.wmv

http://clesm.mae.ufl.edu/~vql/videos/nm1.s11.mtg30.wmv

http://clesm.ippd.ufl.edu/~vql/videos/nm1.s11.mtg30.wmv

title: User:Egm6341.spring-11.5.D/HW1


title: File:Nm1.s11.team2.HW1.fig4-1a.png


text: User:Eml5526.s11.team7.flater/test.subpage


title: File:Nm1.s11.team2.HW1.fig4-2.png


title: File:Nm1.s11.team2.HW1.fig4-3.png


text:
Solution Architecture/Envisioning

Determining the Business Opportunity

 * 1) Project Portfolio Management
 * 2) The Business Case

Related Readings

 * 1) http://en.wikipedia.org/wiki/IBM_Rational_Unified_Process
 * 2) http://en.wikipedia.org/wiki/Zachman_Framework
 * 3) http://en.wikipedia.org/wiki/Agile_software_development

title: Solution Architecture/Envisioning


text:
Solution Architecture/Envisioning/BusinessCase

The Business Case
Template A Business Case template can be found in google docs; Sample
 * https://docs.google.com/document/d/1ukYsbXlGeaLv-jbuFHTZyYT5wLaIRLOHHmiJHMLIux8/edit?hl=en

Related Readings

 * 1) http://en.wikiversity.org/wiki/Business_plan
 * 2) http://en.wikipedia.org/wiki/Business_case
 * 3) http://www.ucalgary.ca/pmoancillary/Templates/Business%20Case.doc


text:
User talk:Chase me ladies, I'm the Cavalry

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text:
User talk:Egm6341.s11.team5.raghunathan

 Hello Egm6341.s11.team5.raghunathan, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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title: User talk:Egm6341.s11.team5.raghunathan


text:
User:EGM6341.s11.TEAM1.Yoon.HW2

1st term
Derive the first term down below from

Taking integration by parts from the integrating part of the LHS,

Back to the main equation including the factorial term,

2nd term
Derive the first term down below from

Note: Integrating by parts

Taking integration by parts from the integrating part of the LHS,

Back to the main equation including the factorial term,

3rd term
Derive the first term down below from

Taking integration by parts from the integrating part of the LHS,

Back to the main equation including the factorial term,

In conclusion,

Part.2)
Consider the remainder as a function that we derived in HW1, prob3

Applying the remainder function to below equation,

Integrating the integration part of the RHS of the equation yields,

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

$$\displaystyle \begin{align} \therefore \int_{x_0}^{x} (x-t)^4 \cdot f^{(5)}(t) dx = f^{(5)}(\xi) \frac{(x-x0)^5}{5} \quad _{for} \quad w(x) \geqq 0 \end{align} $$
 * style="width:10%; padding:10px; border:2px solid #8888aa" |
 * style="width:10%; padding:10px; border:2px solid #8888aa" |
 * style = | (1.17)
 * }
 * }

Part.3)
Derive the first term down below from

With same method as used in part a), use integrating by parts again,

Back to the main equation including the factorial term,

Part.4)
Consider the remainder as a function that we derived in HW1, prob3

Applying below IMVT to equation (1.22)

Integrating the integration part of the RHS of the equation yields,

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

$$\displaystyle \begin{align} \therefore \frac{1}{(n+1)!}\int_{x_0}^{x} (x-t)^{n+1} \cdot f^{(n+2)}(t) dx = \frac{(x-x_0)^{n+2}}{(n+2)!}f^{(n+2)}(\xi) \quad _{for} \quad w(x) \geqq 0 \end{align} $$
 * style="width:10%; padding:10px; border:2px solid #8888aa" |
 * style="width:10%; padding:10px; border:2px solid #8888aa" |
 * style = | (1.25)
 * }
 * }

Solution
When n = 0:

When n = 1:

When n = 2:

When n = 3:

When n = 4:

title: User:EGM6341.s11.TEAM1.Yoon.HW2


text:
Solution Architecture/Envisioning/ProjectPortfolioManagement

Innovations, Demand & Ideas
This are where the new project initiatives will come from. All innovations, demands and ideas should be honored and recorded. New projects can come from many sources;
 * Either internal or external innovations are identified from many sources. it can be a hallway conversation, while having coffee, a structured idea generating meeting, another project meeting or from a vendor.
 * A customer demand is something requested by a customer or by identifying a customer need through there activity. This demand could become a new project.
 * Ideas come from everywhere, they should not be lost.

Executive & Governance
Final decision making will come from the executive level with guidance from strong governance practices. The main idea being that with inputs from new innovations, demand, ideas, retiring applications, business cases and learned lessons that the projects most likely of success with a good ROI should become projects.

IT PROJECTS become IT ASSETS
It's like managing an investment portfolio. Keeping an eye to what to retire, what to re-invest in and what should stay the same. Once a project goes live it becomes an asset and needs to be well managed to stay healthy.


text: File:Eml5526.s11.team7.hw1.1 3.png

title: File:Eml5526.s11.team7.hw1.1 3.png


title: File:Nm1.s11.team4.HW1.fig9.jpg


title: File:Nm1s11.team4.HW1.fig10.jpg


text:
User talk:333prime.curios

 Hello 333prime.curios, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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title: User talk:333prime.curios


text:
Paideia High School/English Bill of Rights

English Bill of Rights on Wikisources

Bill of Rights 1689 on Wikipedia

''The Bill of Rights 1689 is an Act of the Parliament of England (1 Will. & Mar. sess. 2 c. 2) with the long title An Act Declaring the Rights and Liberties of the Subject and Settling the Succession of the Crown and also known by its short title, the Bill of Rights and English Bill of Rights. It is one of the basic documents of English constitutional law, alongside Magna Carta, the  Act of Settlement 1701 and the Parliament Acts of 1911 and 1949. It also forms part of the law of some other Commonwealth nations, such as New Zealand. A separate but similar document applies in Scotland: the Claim of Right.''

An Act Declaring the Rights and Liberties of the Subject and Settling the Succession of the Crown.

Whereas the Lords Spiritual and Temporal and Commons assembled at Westminster, lawfully, fully and freely representing all the estates of the people of this realm, did upon the thirteenth day of February in the year of our Lord one thousand six hundred eighty-eight present unto their Majesties, then called and known by the names and style of William and Mary, prince and princess of Orange, being present in their proper persons, a certain declaration in writing made by the said Lords and Commons in the words following, viz.:

 * Whereas the late King James the Second, by the assistance of divers evil counsellors, judges and ministers employed by him, did endeavour to subvert and extirpate the Protestant religion and the laws and liberties of this kingdom;
 * By assuming and exercising a power of dispensing with and suspending of laws and the execution of laws without consent of Parliament;
 * By committing and prosecuting divers worthy prelates for humbly petitioning to be excused from concurring to the said assumed power;
 * By issuing and causing to be executed a commission under the great seal for erecting a court called the Court of Commissioners for Ecclesiastical Causes;
 * By levying money for and to the use of the Crown by pretence of prerogative for other time and in other manner than the same was granted by Parliament;
 * By raising and keeping a standing army within this kingdom in time of peace without consent of Parliament, and quartering soldiers contrary to law;
 * By causing several good subjects being Protestants to be disarmed at the same time when papists were both armed and employed contrary to law;
 * By violating the freedom of election of members to serve in Parliament;
 * By prosecutions in the Court of King's Bench for matters and causes cognizable only in Parliament, and by divers other arbitrary and illegal courses;
 * And whereas of late years partial corrupt and unqualified persons have been returned and served on juries in trials, and particularly divers jurors in trials for high treason which were not freeholders;
 * And excessive bail hath been required of persons committed in criminal cases to elude the benefit of the laws made for the liberty of the subjects;
 * And excessive fines have been imposed;
 * And illegal and cruel punishments inflicted;
 * And several grants and promises made of fines and forfeitures before any conviction or judgment against the persons upon whom the same were to be levied;
 * All which are utterly and directly contrary to the known laws and statutes and freedom of this realm;

And whereas the said late King James the Second having abdicated the government and the throne being thereby vacant, his Highness the prince of Orange (whom it hath pleased Almighty God to make the glorious instrument of delivering this kingdom from popery and arbitrary power) did (by the advice of the Lords Spiritual and Temporal and divers principal persons of the Commons) cause letters to be written to the Lords Spiritual and Temporal being Protestants, and other letters to the several counties, cities, universities, boroughs and cinque ports, for the choosing of such persons to represent them as were of right to be sent to Parliament, to meet and sit at Westminster upon the two and twentieth day of January in this year one thousand six hundred eighty and eight [old style date], in order to such an establishment as that their religion, laws and liberties might not again be in danger of being subverted, upon which letters elections having been accordingly made;

And thereupon the said Lords Spiritual and Temporal and Commons, pursuant to their respective letters and elections, being now assembled in a full and free representative of this nation, taking into their most serious consideration the best means for attaining the ends aforesaid, do in the first place (as their ancestors in like case have usually done) for the vindicating and asserting their ancient rights and liberties declare:

 * That the pretended power of suspending the laws or the execution of laws by regal authority without consent of Parliament is illegal;
 * That the pretended power of dispensing with laws or the execution of laws by regal authority, as it hath been assumed and exercised of late, is illegal;
 * That the commission for erecting the late Court of Commissioners for Ecclesiastical Causes, and all other commissions and courts of like nature, are illegal and pernicious;
 * That levying money for or to the use of the Crown by pretence of prerogative, without grant of Parliament, for longer time, or in other manner than the same is or shall be granted, is illegal;
 * That it is the right of the subjects to petition the king, and all commitments and prosecutions for such petitioning are illegal;
 * That the raising or keeping a standing army within the kingdom in time of peace, unless it be with consent of Parliament, is against law;
 * That the subjects which are Protestants may have arms for their defence suitable to their conditions and as allowed by law;
 * That election of members of Parliament ought to be free;
 * That the freedom of speech and debates or proceedings in Parliament ought not to be impeached or questioned in any court or place out of Parliament;
 * That excessive bail ought not to be required, nor excessive fines imposed, nor cruel and unusual punishments inflicted;
 * That jurors ought to be duly impanelled and returned, and jurors which pass upon men in trials for high treason ought to be freeholders;
 * That all grants and promises of fines and forfeitures of particular persons before conviction are illegal and void;
 * And that for redress of all grievances, and for the amending, strengthening and preserving of the laws, Parliaments ought to be held frequently.

And they do claim, demand and insist upon all and singular the premises as their undoubted rights and liberties, and that no declarations, judgments, doings or proceedings to the prejudice of the people in any of the said premises ought in any wise to be drawn hereafter into consequence or example; to which demand of their rights they are particularly encouraged by the declaration of his Highness the prince of Orange as being the only means for obtaining a full redress and remedy therein.

Having therefore an entire confidence that his said Highness the prince of Orange will perfect the deliverance so far advanced by him, and will still preserve them from the violation of their rights which they have here asserted, and from all other attempts upon their religion, rights and liberties, the said Lords Spiritual and Temporal and Commons assembled at Westminster do resolve that William and Mary, prince and princess of Orange, be and be declared king and queen of England, France and Ireland and the dominions thereunto belonging, to hold the crown and royal dignity of the said kingdoms and dominions to them, the said prince and princess, during their lives and the life of the survivor to them, and that the sole and full exercise of the regal power be only in and executed by the said prince of Orange in the names of the said prince and princess during their joint lives, and after their deceases the said crown and royal dignity of the same kingdoms and dominions to be to the heirs of the body of the said princess, and for default of such issue to the Princess Anne of Denmark and the heirs of her body, and for default of such issue to the heirs of the body of the said prince of Orange. And the Lords Spiritual and Temporal and Commons do pray the said prince and princess to accept the same accordingly.

And that the oaths hereafter mentioned be taken by all persons of whom the oaths have allegiance and supremacy might be required by law, instead of them; and that the said oaths of allegiance and supremacy be abrogated.

 * "I, A.B., do sincerely promise and swear that I will be faithful and bear true allegiance to their Majesties King William and Queen Mary. So help me God."

 * "I, A.B., do swear that I do from my heart abhor, detest and abjure as impious and heretical this damnable doctrine and position, that princes excommunicated or deprived by the Pope or any authority of the see of Rome may be deposed or murdered by their subjects or any other whatsoever. And I do declare that no foreign prince, person, prelate, state or potentate hath or ought to have any jurisdiction, power, superiority, pre-eminence or authority, ecclesiastical or spiritual, within this realm. So help me God."

Upon which their said Majesties did accept the crown and royal dignity of the kingdoms of England, France and Ireland, and the dominions thereunto belonging, according to the resolution and desire of the said Lords and Commons contained in the said declaration.

And thereupon their Majesties were pleased that the said Lords Spiritual and Temporal and Commons, being the two Houses of Parliament, should continue to sit, and with their Majesties' royal concurrence make effectual provision for the settlement of the religion, laws and liberties of this kingdom, so that the same for the future might not be in danger again of being subverted, to which the said Lords Spiritual and Temporal and Commons did agree, and proceed to act accordingly.

Now in pursuance of the premises the said Lords Spiritual and Temporal and Commons in Parliament assembled, for the ratifying, confirming and establishing the said declaration and the articles, clauses, matters and things therein contained by the force of law made in due form by authority of Parliament, do pray that it may be declared and enacted that all and singular the rights and liberties asserted and claimed in the said declaration are the true, ancient and indubitable rights and liberties of the people of this kingdom, and so shall be esteemed, allowed, adjudged, deemed and taken to be; and that all and every the particulars aforesaid shall be firmly and strictly holden and observed as they are expressed in the said declaration, and all officers and ministers whatsoever shall serve their Majesties and their successors according to the same in all time to come.

And the said Lords Spiritual and Temporal and Commons, seriously considering how it hath pleased Almighty God in his marvellous providence and merciful goodness to this nation to provide and preserve their said Majesties' royal persons most happily to reign over us upon the throne of their ancestors, for which they render unto him from the bottom of their hearts their humblest thanks and praises, do truly, firmly, assuredly and in the sincerity of their hearts think, and do hereby recognize, acknowledge and declare, that King James the Second having abdicated the government, and their Majesties having accepted the crown and royal dignity as aforesaid, their said Majesties did become, were, are and of right ought to be by the laws of this realm our sovereign liege lord and lady, king and queen of England, France and Ireland and the dominions thereunto belonging, in and to whose princely persons the royal state, crown and dignity of the said realms with all honours, styles, titles, regalities, prerogatives, powers, jurisdictions and authorities to the same belonging and appertaining are most fully, rightfully and entirely invested and incorporated, united and annexed. And for preventing all questions and divisions in this realm by reason of any pretended titles to the crown, and for preserving a certainty in the succession thereof, in and upon which the unity, peace, tranquility and safety of this nation doth under God wholly consist and depend, the said Lords Spiritual and Temporal and Commons do beseech their Majesties that it may be enacted, established and declared, that the crown and regal government of the said kingdoms and dominions, with all and singular the premises thereunto belonging and appertaining, shall be and continue to their said Majesties and the survivor of them during their lives and the life of the survivor of them, and that the entire, perfect and full exercise of the regal power and government be only in and executed by his Majesty in the names of both their Majesties during their joint lives; and after their deceases the said crown and premises shall be and remain to the heirs of the body of her Majesty, and for default of such issue to her Royal Highness the Princess Anne of Denmark and the heirs of the body of his said Majesty; and thereunto the said Lords Spiritual and Temporal and Commons do in the name of all the people aforesaid most humbly and faithfully submit themselves, their heirs and posterities for ever, and do faithfully promise that they will stand to, maintain and defend their said Majesties, and also the limitation and succession of the crown herein specified and contained, to the utmost of their powers with their lives and estates against all persons whatsoever that shall attempt anything to the contrary.

And whereas it hath been found by experience that it is inconsistent with the safety and welfare of this Protestant kingdom to be governed by a popish prince, or by any king or queen marrying a papist, the said Lords Spiritual and Temporal and Commons do further pray that it may be enacted, that all and every person and persons that is, are or shall be reconciled to or shall hold communion with the see or Church of Rome, or shall profess the popish religion, or shall marry a papist, shall be excluded and be for ever incapable to inherit, possess or enjoy the crown and government of this realm and Ireland and the dominions thereunto belonging or any part of the same, or to have, use or exercise any regal power, authority or jurisdiction within the same; and in all and every such case or cases the people of these realms shall be and are hereby absolved of their allegiance; and the said crown and government shall from time to time descend to and be enjoyed by such person or persons being Protestants as should have inherited and enjoyed the same in case the said person or persons so reconciled, holding communion or professing or marrying as aforesaid were naturally dead; and that every king and queen of this realm who at any time hereafter shall come to and succeed in the imperial crown of this kingdom shall on the first day of the meeting of the first Parliament next after his or her coming to the crown, sitting in his or her throne in the House of Peers in the presence of the Lords and Commons therein assembled, or at his or her coronation before such person or persons who shall administer the coronation oath to him or her at the time of his or her taking the said oath (which shall first happen), make, subscribe and audibly repeat the declaration mentioned in the statute made in the thirtieth year of the reign of King Charles the Second entitled, An Act for the more effectual preserving the king's person and government by disabling papists from sitting in either House of Parliament. But if it shall happen that such king or queen upon his or her succession to the crown of this realm shall be under the age of twelve years, then every such king or queen shall make, subscribe and audibly repeat the same declaration at his or her coronation or the first day of the meeting of the first Parliament as aforesaid which shall first happen after such king or queen shall have attained the said age of twelve years.

All which their Majesties are contented and pleased shall be declared, enacted and established by authority of this present Parliament, and shall stand, remain and be the law of this realm for ever; and the same are by their said Majesties, by and with the advice and consent of the Lords Spiritual and Temporal and Commons in Parliament assembled and by the authority of the same, declared, enacted and established accordingly.

II. And be it further declared and enacted by the authority aforesaid, that from and after this present session of Parliament no dispensation by non obstante of or to any statute or any part thereof shall be allowed, but that the same shall be held void and of no effect, except a dispensation be allowed of in such statute, and except in such cases as shall be specially provided for by one or more bill or bills to be passed during this present session of Parliament.

III. Provided that no charter or grant or pardon granted before the three and twentieth day of October in the year of our Lord one thousand six hundred eighty-nine shall be any ways impeached or invalidated by this Act, but that the same shall be and remain of the same force and effect in law and no other than as if this Act had never been made.


text:
User talk:Egm5526.s11.team-2.langpm

hello

title: User talk:Egm5526.s11.team-2.langpm


text:
User talk:Eml5526.s11.team2.brenner

 Hello Eml5526.s11.team2.brenner, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

To get started, you may

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You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Ottava Rima (talk) 02:21, 26 January 2011 (UTC)

title: User talk:Eml5526.s11.team2.brenner


text:
User:Eml5526.s11.team2.brenner

Test
This is a test ... Hello wiki! Eml5526.s11.team2.brenner 03:15, 26 January 2011 (UTC)

Useful Links
Team2 page

Egm6341 Course Wiki

Eml5526 Course Wiki

Egm6321 Team03 Example Page

Homework
Report table

Spring 2010 Version

Homework #1

Homework #2

Homework #3

Homework #4

Homework #5

Homework #6

title: User:Eml5526.s11.team2.brenner


title: File:Nm1.s11.team5.HW1.fig1a.png


title: File:Nm1.s11.team5.HW1.fig1b.png


text:
Paideia High School/Cosmic View

A Paideia Unit Plan based on Kees Boeke's Cosmic View (1956).

Exordium
The Exordium is the teacher's introduction to the work that is the subject of a Paideia Unit Plan. This introduction consists of both an oral and written presentation of the text. For longer works, the teacher may limit the oral presentation to key passages. The teacher should read the text live distinctly, accurately, and intelligently. In addition, the teacher should provide high quality recordings of the text if possible. Students should have a consumable written copy of the text both electronically and in hard-copy if possible.

Oral Presentation
There is currently no audio recording of Cosmic View available. Since the work is still under copyright protection, it may not be legal to post a recording. The text of the work is relatively short and, thus, easily read live. Because Cosmic View is as much a work of visual artistry as it is a written work, oral presentations must be accompanied by the pictures if at all possible.

Written Presentation

 * Cosmic View (An online version of the classic book)
 * Available in Gateway to the Great Books
 * The original book is out of print but may be available through used book vendors.

Definitions of Key Terms
December 21, 1951, 12:00 Noon: Every image in Cosmic View is developed with this precise date in mind.

General Vocabulary
Cosmic View: Glossary of Terms

Biographical References

 * Information about the author, Kees Boeke.
 * The biographical sketch in Gateway to the Great Books is worth reading if you can find a copy.

Film References
Cosmic View inspired two films: Cosmic Zoom and Powers of Ten. Both are freely available on the Internet.

Geographical References (with Maps)

 * See the position of The Netherlands in both Europe and the world to the right.
 * See map of the provinces and municipalities of The Netherlands to the right.
 * Bilthoven, the home of Werkplaats Children's Community and Kees and Betty Boeke.

Historical References

 * Kees and Betty Boeke hosted the founding of three international peace groups in their Bilthoven home during the aftermath of World War I: the International Fellowship of Reconciliation in 1919, Service Civil International in 1920, and War Resisters' International in 1921.

 * The experimental school De Werkplaats, founded by Kees Boeke, has been based in Bilthoven since 1926. Cosmic View was published in 1956.

 * The Dutch Queen Beatrix went to school in Bilthoven. She attended the Werkplaats, the same school as the little girl in Cosmic View.

Literary References

 * An article on Cosmic View.

 * "Although these views are as true to reality as they can be made withour present knowledge, they portray a wonderland as full of marvels as that which Alice saw in her dreams" (emphasis added; see Alice in Wonderland).

Great Ideas Addressed in this Work
The term "great ideas" refers to the list of 102 ideas prominent in Great Books of the Western World. and the 4000+ ideas listed in the "Index of Terms" in the same work

Great Ideas Addressed in Cosmic View

For each entry below, see Great Books of the Western World's Syntopicon at the volume and page numbers indicated. This information is quoted from Gateway to the Great Books.

ELEMENT, Vol. 2, pp. 400-412, especially
 * Topic 3a: Element and atom: qualitative and quantitative indivisibility

INFINITY, Vol 2, pp. 816-834, especially
 * Topic 3: The infinite in quantity
 * Topic 4b: The infinite divisibility of matter: the issue concerning atoms

MATHEMATICS, Vol. 3, pp. 42-62, especially
 * Topic 5a: The art of measurement

QUANTITY, Vol. 3, pp 527-545

WORLD, Vol. 3, pp. 1118-1140, especially
 * Topic 2: The universe and man: macrocosm and microcosm
 * Topic 5: The number of worlds: the uniqueness of this world; the possibility or actuality of other worlds
 * Topic 7: The space of the world: astronomical theories concerning the size or extent of the universe

Potential Seminar Questions
The following table serves to guide teachers in understanding the types of questions that could guide a good seminar on Boeke's Cosmic View. In addition, the table may serve to trigger additional question ideas and offer a guide to writing good questions. Good seminars follow the general structure given by Mortimer Adler and Charles Van Doren in their classic How to Read a Book published by Simon & Schuster in 1972. Column one gives the four main questions that a demanding reader should ask of any book. These questions guide the types of questions and the purpose of each type used in good seminars. The last column serves to illustrate questions that might be asked of Boeke's Cosmic View.

Again, the questions in the last common are examples. A good seminar will not use all of these questions. A quality seminar plan should be built around one or two questions from the first row, one question from the last row, and one to three questions from the middle two rows.

A Sample High School Seminar
Introduction to the Seminar

Seminars are “conversations, conducted in an orderly manner by the teacher who acts as leader or moderator of the discussion”. The purpose of seminars is for participants to deepen their understanding of the great ideas that shape us as human persons and citizens. Through discussion we learn, share, and deepen our grasp of concepts. Rather than simply seeking to remember information which we are likely to forget, we form our understanding in ways that will last the rest of our lives when we engage each other in dialogue.

Participating well in a seminar requires effort and skill. The effort is a choice, and skills develop with practice. In this seminar, your two leaders will press you to express yourself clearly at three levels: words, sentences, and arguments. If we press you to make clear what you mean by a particular word, take it in stride. Do not feel pressured or "put on the spot." We're merely trying to help you practice an important skill, namely using words well to express your thoughts. We may also ask you to repeat yourself or to say the same thing in a different way. We may stick with you, press you, insist on greater clarity of expression. Again, we're your teachers, and we're trying to help you develop excellent communication skills. Smile, and work with us. Finally, we will ask that you make your points and draw conclusions logically. We'll ask that you think about your points and the premises they rest on. Do your points have reasons? Are your assumptions plausible? We ask that you work with us in a conversational spirit while understanding that there is method to our madness :-). We want to help you learn.

Introduction to the Work

Welcome to today's seminar on Cosmic View by Kees Boeke. By discussing this classic 1957 work, we seek to enlarge our understanding of a number of great ideas. Within the scope of one seminar, however, we must limit ourselves because time will not permit us to delve deeply into all of the ideas in this important work.

Boeke's Cosmic View is indeed classic. It has inspired people for three generations. The creators of two short films, Cosmic Zoom and Powers of Ten, credit Boeke for their inspiration. Not only does Boeke obviously discuss great ideas like ELEMENT, INFINITY, MATHEMATICS, QUANTITY, and WORLD, but also, more subtly, he explores EDUCATION, LIFE AND DEATH, WAR AND PEACE--timeless ideas that will speak to three more generations and beyond! Cosmic View is a great and important work, relevant to our war-torn and violent world today, relevant to the struggle for democracy we see developing in Egypt, relevant to education both there and in our own country. It is worthy of our study and discussion.

Introductory Questions

 * 1) I mentioned two short films inspired by Cosmic View. The title of the first film by Eva Szasz, produced by the National Film Board of Canada in 1968 is Cosmic Zoom. The title of the second one by Charles and Ray Eames, sponsored by IBM, is Powers of Ten, originally produced in 1968 and rereleased in 1977. We'll watch this latter one now. As we do so, I want you think about the three titles. Write them down now: Cosmic View, Cosmic Zoom, and the one we'll view, Powers of Ten. Think about how these titles are the same and how they're different. Think about the ideas these titles summarize. Would one of the other titles better express the ideas in Cosmic View? At this point, simply answer "yes" or "no"; we'll go into more depth and reasons later.
 * Now I would like like you to consider the full definition of the word Cosmos from which Boeke gets the word Cosmic in his title.
 * cos·mos
 * Pronunciation: käzms; in senses 1 and 2 also -ms or -mäs
 * Function: noun
 * Etymology: German kosmos, from Greek, order, ornament, universe
 * 1 -es a : the universe conceived as an orderly and harmonious system -- contrasted with chaos b : ORDER, HARMONY
 * 2 -es : a self-inclusive system characterized by order and harmony amid complexity of detail
 * 3 capitalized [New Latin, from Greek kosmos] : a genus of tropical American herbs (family Compositae) having opposite leaves, flowers solitary in loose corymbose panicles, and flower heads with prominent rays most cultivated varieties of which are derived from a Mexican species (C. bipinnatus) and are popular fall-blooming annuals
 * 4 plural cosmos \-ms, -mz\ also cosmoses : any plant or flower of the genus Cosmos
 * Now lets go into more depth on the title. Would you change your mind now? What would be the best title for Boeke's original Cosmic View? Would one of the other titles have been better? Why or why not?
 * What ideas is Boeke including that the film we viewed is not? Support your answers from the text.

 * 1) [figure out how to get this to start with "2"] Who would like to read the second paragraph of the introduction aloud?
 * "At school we are introduced to many different spheres of existence, but they are often not connected with each other, so that we are in danger of collecting a large number of images without realizing that they all join together in one great whole. It is therefore important in our education to find the means of developing a wider and more connected view of our world and a truly cosmic view of the universe and our place in it."
 * Boeke uses the title of his work near the end of this passage? How does this use shed light on his purpose for writing Cosmic View? In addition to considering this text, you should support your answers about Boeke's purpose by citing additional lines or passages.

Boeke uses the word cosmic several times and, yet again, he repeats the title of the work in the last paragraph. Why? What does Boeke want to accomplish? What changes does he hope to produce in his students? His readers? What are his main ideas and his overall purpose?
 * 1) Who would like to read the last five paragraphs of the work? The conclusion?
 * "And so our journey ends at the nucleus of the atom, that mysterious, utterly small, and incredibly powerful center of energy which only recently has unveiled some of its mighty secrets to mankind. Whereas at the end of our first journey we stood in awe before the imposing greatness of the dimensions of the universe, and felt as nothing in comparison to their immensity, the conditions are now completely different. True, we feel as much awe and reverence when we attempt to think of the miracles of dynamic power that are hidden in these domains of the smallest existing entities, but our own dimensions are now indescribably colossal compared with what we see.
 * Thus on the scale of the last drawing, the height of the little girl would be about 15,000 million kilometers, that is, more than the diameter of the solar system! If we add the thought that man is beginning to control and use these limitless nuclear powers, it is clear that unthinkable possibilities are within his reach. When we thus think in cosmic terms, we realize that man, if he is to become really human, must combine in his being the greatest humility with the most careful and considerate use of the cosmic powers that are at his disposal.
 * The problem, however, is that primitive man at first tends to use the power put in his hands for himself, instead of spending his energy and life for the good of the whole growing human family, which has to live together in the limited space of our planet. It therefore is a matter of life and death for the whole of mankind that we learn to live together, caring for each other regardless of birth or upbringing. No difference of nationality, of race, creed, or conviction, age or sex may weaken our effort as human beings to live and work for the good of all.
 * It is therefore an urgent need that we all, children and grown ups alike, be educated in this spirit and toward this goal. Learning to live together in mutual respect and with the definite aim to further the happiness of all, without privilege for any, is a clear duty for mankind, and it is imperative that education shall be brought onto this plane.
 * In this education the development of a cosmic view is an important and necessary element; and to develop such a wide, all-embracing view, the expedition we have made in these "forty jumps through the universe" may help just a little. If so, let us hope that many will make it!"

 * 1) Boeke uses the ideas of scale, mathematics, and measurement extensively. How? Cite examples? Why? Does this add to or detract from his purpose in writing the work?

 * 1) In what ways is Boeke's work like poetry? In what ways is it not? Does Boeke have a creative purpose in his use of math, science, and measurement?

 * 1) Does Boekes work contain truth? Falsity? How? Why? Cite examples.

 * 1) Does Cosmic View make a difference? Does it make the difference Boeke hoped for? Does it make a difference for you?

title: Paideia High School/Cosmic View