text:
User:Egm6341.s11.team5.oh/HW4

= Problem 4.1 - Simpson's rule integrates the 3rd order polynomial exactly (I and I_2) = From (meeting 18 page 2)

Given

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

f(x) = P_3(x) = 1 + 3x -9x^2 +12x^3 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (1-1)
 * }
 * }

Find
 Find, I (exact) and I_2 (Simple simpson)

Solution
 We solved on our own

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

\begin{align} I &= \int_{-2}^{1} f(x) \ dx \\ &= \left[ x + \frac{3}{2} x^2 -3x^3 +3x^4\right]_{-2}^{1} \\ &= -73.5 \end{align} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (1-2)
 * }
 * }

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

 * }
 * }
 * }

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

\begin{align} I_2 &= \frac{h}{3} \left[ f(x_0) +4f(x_1) + f(x_2) \right] \\ &= \frac{1}{2} \left[ f(-2) +4f(- \frac{1}{2}) + f(1) \right] \\ &= \frac{1}{2} \left[ -137 -17 +7 \right] \\ &= -73.5 \end{align} $$
 * $$\displaystyle
 * $$\displaystyle

Where, $$\displaystyle \ \left( h = \frac{1-(-2)}{2} = \frac{3}{2}, \qquad x_0 = -2, \ x_1 = - \frac{1}{2}, \ x_2 = 1 \right)$$

$$
 * $$\displaystyle (1-3)
 * }
 * }

Author and proof-reader
[Author] Oh

[Proof-reader]

=Problem 4-2: Evaluation of $$ \alpha^{(1)}(t)$$=

Given
$$ \displaystyle\ \alpha(t)=\int_{-t}^{t} f(x(t'))dt'$$ where $$ \displaystyle\ f(x(t))=F(t)$$.

Find
Show that $$ \displaystyle\ \alpha^{(1)}(t)=F(-t)+F(t) $$.

Solution
We are asked to show that for $$ \displaystyle\ \alpha(t)=\int_{-t}^{t} f(x(t'))dt'=\int_{-t}^{t} F(t')dt',\alpha^{(1)}(t)=F(-t)+F(t)$$, where $$ \displaystyle\ f(x(t))=F(t)$$.

Using The Fundamental Theorem of Calculus (Second Part), we know that for some differentiable function $$ \displaystyle\ G(t)$$ with with derivative $$ \displaystyle\ g(t)=G'(t)$$, $$ \displaystyle\ \int_a^b g(t)dt=G(b)-G(a)$$.

Allowing $$ \displaystyle\ f(x(t))=F(t)$$ and $$ \displaystyle\ G'(t)=F(t)$$, then $$ \displaystyle\ \alpha(t)=\int_{-t}^{t} F(t')dt'=G(t)-G(-t)$$.

If the derivative of $$ \displaystyle \alpha $$ is now taken, the $$ \displaystyle\ \alpha^{(1)}(t)=\frac{d}{dt}(G(t)-G(-t))=G'(t)+G'(-t)$$.

Substituting in our definitions $$ \displaystyle\ F(t)=G'(t)$$ we see that

$$ \displaystyle\ \alpha^{(1)}(t)=F(-t)+F(t) $$.

Author
Solved and Typed by - Egm6341.s10.Team4.riherd 14:55, 18 February 2010 (UTC) .

= Problem 4-3: Evaluation of $$ G^{2}(0) $$ =

Given
Refer Lecture slide 15-2 for problem statement

From Lecture slide 15-1,

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

G^{1}(t) = e^{1}(t) - 5t^4e(1) $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (1)

 * }
 * }

where,

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

e(t) := \alpha(t) - \alpha_{2}(t) $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (2)

 * }
 * }

where $$\displaystyle \alpha(t), \alpha_{2}(t) $$ are given by,

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

\alpha(t) := \int_{-t}^{t}F(t)dt $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (3)

 * }
 * }

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

\alpha^{2}(t) := \frac{t}{3} \left[F(-t) + 4F(0)+ F(t)\right] $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (4)

 * }
 * }

and,

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

F(t) := f(x(t)) $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (5)

 * }.
 * }.

Find
Show that,

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

G^{2}(0) = 0 $$
 * $$\displaystyle

 * }.
 * }.

Solution
On differentiating Equation (3) [Please refer differentiating an integral and Problem 14 Solution below],

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

\alpha^{1}(t) = F(-t) + F(t) $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (6)

.
 * }
 * }

Again differentiating Equation (6),

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

\alpha^{2}(t) = -F(-t) + F(t) $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (7)

.
 * }
 * }

On differentiating Equation (4),
 * {| style="width:90%" border="0" align="center"

\begin{align} \Rightarrow \alpha_{2}^{1}(t) &= \frac{1}{3} \left[ F(-t) + 4F(0) + F(t) \right] + \frac{t}{3} \left[-F^{1}(-t) + \cancelto{0}{4F^{1}(0)} + F^{1}(t)\right] \\ &= \frac{1}{3} \left[ F(-t) + 4F(0) + F(t) \right] + \frac{t}{3} \left[-F^{1}(-t) + F^{1}(t)\right] \end{align} $$
 * $$\displaystyle

 * }.
 * }.

Again on differentiating the above equation,

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

\begin{align} \Rightarrow \alpha_{2}^{2}(t) &= \frac{1}{3} \left[ -F^{1}(-t) + \cancelto{0}{4F^{1}(0)} + F^{1}(t) \right] + \frac{1}{3} \left[-F^{1}(-t) + F^{1}(t)\right] + \frac{t}{3} \left[F^{2}(-t) + F^{2}(t)\right]\\ &= \frac{2}{3} \left[ -F^{1}(-t) + F^{1}(t) \right] + \frac{t}{3} \left[F^{2}(-t) + F^{2}(t)\right] \end{align} $$
 * $$\displaystyle

 * }.
 * }.

So,

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

 * $$\displaystyle

\Rightarrow \alpha_{2}^{2}(t) = -\frac{2}{3} F^{1}(-t) + \frac{2}{3} F^{1}(t) + \frac{t}{3} \left[F^{2}(-t) + F^{2}(t)\right]

$$ $$
 * $$\displaystyle \longrightarrow (8)

 * }.
 * }.

On differentiating Equation (1) we get,

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

G^{2}(t) = e^{2}(t) - 20t^3e(1) $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (9)

 * }.
 * }.

At point $$\displaystyle t = 0 $$,

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

G^{2}(0) = e^{2}(0) - \cancelto{0}{20(t)^3e(1)} \Rightarrow G^{2}(0) = e^{2}(0) $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (10)

 * }.
 * }.

$$\displaystyle e^{2}(t) $$ is given by differentiating Equation (2) twice,

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

e^{2}(t) = \alpha^{2}(t) - \alpha_{2}^{2}(t) $$ $$
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (11)

 * }.
 * }.

Substituting Equations (7) and (8) in (11),

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

\Rightarrow e^{2}(t) = -F(-t) + F(t) +\frac{2}{3} F^{1}(-t) - \frac{2}{3} F^{1}(t) - \frac{t}{3} \left[F^{2}(-t) + F^{2}(t)\right] $$
 * $$\displaystyle

 * }.
 * }.

At point $$\displaystyle t = 0 $$,

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

\begin{align} \Rightarrow e^{2}(0) &= -F(-0) + F(0) +\frac{2}{3} F^{1}(-0) - \frac{2}{3} F^{1}(0) - \cancelto{0}{\frac{t}{3} \left[F^{2}(-t) + F^{2}(t)\right]} \\ &= -\frac{1}{3} F^{1}(-0) + \frac{1}{3} F^{1}(0) \\ &= \frac{1}{3} \left[ -F^{1}(0) + F^{1}(0) \right] \end{align} $$
 * $$\displaystyle

 * }.
 * }.

So,

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

 * $$\displaystyle

\Rightarrow e^{2}(0) = 0

$$

 * }.
 * }.

which means from Equation (10),

 * {| 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 G^{2}(0) = 0

$$

 * style= |

 * }.
 * }.

Author
Solved and Typed by - Egm6341.s10.Team4.andy 21:56, 17 February 2010 (UTC) .

= Problem 4-4: Third derivative of e(t) in Lagrange Intp. Error =

Given
If we define function e(t) as the following,

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle e(t)=\int\limits_{-t}^{t} F(t)\, dt-\dfrac{t}{3}[F(-t)+4F(0)+F(t)]

 * }
 * }

Find
Show that:

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle e^{(3)}(t)=\dfrac{-t}{3}[F^{(3)}(t)-F^{(3)}(-t)]

 * }
 * }

Solution
Let e(t) be,

 * $$\displaystyle

e(t)=\alpha(t)-\alpha_2(t) $$

Basically, we have the following in the calculus for differentiating an integral:

\left({\int\limits_{g(x)}^{h(x)} f(x)\, dx}\right)'=\dfrac{d}{dx}\int\limits_{g(x)}^{h(x)} f(x)\, dx=f(h(x)).h'(x)-f(g(x)).g'(x) $$ Assuming k a point in the [-t,t] interval:

\alpha (t)=\int\limits_{-t}^{k} F(t)\, dt+\int\limits_{k}^{t} F(t)\, dt $$

\Rightarrow \alpha^{(1)}(t)=[F(k)\times0-F(-t)\times(-1)]+[F(t)\times 1-F(k)\times 0]=F(-t)+F(t) $$

\Rightarrow \alpha^{(2)}(t)=[F^{(1)}(-t)\times(-1)+F^{(1)}(t)\times 1]=F^{(1)}(t)-F^{(1)}(-t) $$

\Rightarrow \alpha^{(3)}(t)=F^{(2)}(t)+F^{(2)}(-t) $$

\alpha_2^{(1)}(t)=\dfrac{1}{3}[F(-t)+4F(0)+F(t)]-\dfrac{t}{3}F^{(1)}(-t)+0+\dfrac{t}{3}F^{(1)}(t) $$

\Rightarrow \alpha_2^{(2)}(t)=-\dfrac{1}{3}F^{(1)}(-t)+\dfrac{1}{3}F^{(1)}(t)-\dfrac{1}{3}F^{(1)}(-t)+\dfrac{t}{3}F^{(2)}(-t)+\dfrac{1}{3}F^{(1)}(t)+\dfrac{t}{3}F^{(2)}(t) $$

=-\dfrac{2}{3}F^{(1)}(-t)+\dfrac{2}{3}F^{(1)}(t)+\dfrac{t}{3}F^{(2)}(-t)+\dfrac{t}{3}F^{(2)}(t) $$

\Rightarrow \alpha_2^{(3)}(t)=\dfrac{2}{3}F^{(2)}(-t)+\dfrac{2}{3}F^{(2)}(t)+\dfrac{1}{3}F^{(2)}(-t)-\dfrac{t}{3}F^{(3)}(-t)+\dfrac{1}{3}F^{(2)}(t)+\dfrac{t}{3}F^{(3)}(t) $$

=F^{(2)}(-t)+F^{(2)}(t)+\dfrac{t}{3}[F^{(3)}(t)-F^{(3)}(-t)] $$

\Rightarrow e^{(3)}(t)=\alpha^{(3)}(t)-\alpha_2^{(3)}(t)=F^{(2)}(t)+F^{(2)}(-t)-F^{(2)}(-t)-F^{(2)}(t)+\dfrac{t}{3}[F^{(3)}(-t)-F^{(3)}(t)]= $$

=\dfrac{t}{3}[F^{(3)}(-t)-F^{(3)}(t)]=\dfrac{-t}{3}[F^{(3)}(t)-F^{(3)}(-t)] $$

 * {| style="width:60%" border="0"

$$ e^{(3)}(t)=\dfrac{-t}{3}[F^{(3)}(t)-F^{(3)}(-t)] $$
 * style="width:30%; padding:10px; border:1px solid #8888aa" |
 * style="width:30%; padding:10px; border:1px solid #8888aa" |
 * style= |
 * }

Author
Solved and Typed by - Egm6341.s10.Team4.nimaa&m 02:42, 18 February 2010 (UTC) .

= Problem 4-5: Relationship between $$ \xi$$ and $$ \zeta_4$$=

Given

 * {| style="width:70%" border="0" align="center"

 * $$ \displaystyle x=\alpha + ht$$.

 * }.
 * }.

where,

 * {| style="width:70%" border="0" align="center"

 * $$ \displaystyle h= \frac{(b-a)}{2}$$.

 * }.
 * }.

Find
Prove that
 * {| style="width:70%" border="0" align="center"

 * $$\displaystyle \frac{-F^4(\zeta_4)}{90}= \frac{-(b-a)^4}{1440}f^4(\xi) $$

 * }.
 * }.

and hence deduce a relationship between $$\displaystyle \xi$$ and $$\displaystyle \zeta_4$$

Solution
We have F(t)= f(x(t))= f(x). Given

 * {| style="width:70%" border="0" align="center"

 * $$ \displaystyle x=\alpha + ht$$.

 * }.
 * }.

where,

 * {| style="width:70%" border="0" align="center"

 * $$ \displaystyle h= \frac{(b-a)}{2}$$.

 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$\displaystyle F^'(t)= \frac{dF(t)}{dt}= \frac{df(x(t))}{dt}= \frac{df(x)}{dt}= \frac{df(x)}{dx}\cdot\frac{dx}{dt}= f^'(x)\cdot h$$

 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$\displaystyle F^{}(t)= \frac{d}{dt}\cdot \frac{dF(t)}{dt}= \frac{d}{dt}\cdot(f^'(x)h) = h\cdot \frac{df^'(x)}{dt}= h\cdot \frac{df^'(x)}{dx}\cdot \frac{dx}{dt}= h^2\cdot f^{}(x)$$

 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$\displaystyle F^{}(t)= \frac{d}{dt}\cdot (F^{}(t))= \frac{d}{dt}\cdot(h^2\cdot f^{}(x)) = h^2\cdot \frac{df^{}(x)}{dt}= h^2\cdot \frac{df^{}(x)}{dx}\cdot \frac{dx}{dt}= h^3\cdot f^{}(x)$$

 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$\displaystyle F^{'}(t)= \frac{d}{dt}\cdot (F^{}(t))= \frac{d}{dt}\cdot(h^3\cdot f^{}(x)) = h^3\cdot \frac{df^{}(x)}{dt}= h^3\cdot \frac{df^{}(x)}{dx}\cdot \frac{dx}{dt}= h^4\cdot f^{'}(x)$$

 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$\Rightarrow \displaystyle F^4(t)=h^4\cdot f^4(x)$$
 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$\Rightarrow \displaystyle F^4(t)=\frac{(b-a)}{16} ^4\cdot f^4(x)$$
 * }.
 * }.

Substituting $$\displaystyle \xi$$ for $$\displaystyle x$$ and $$\displaystyle \zeta_4$$ for $$\displaystyle t$$

 * {| style="width:70%" border="0" align="center"

 * $$\Rightarrow \displaystyle F^4(\zeta_4)=\frac{(b-a)}{16} ^4\cdot f^4(\xi)$$
 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

 * $$\Rightarrow \displaystyle \frac{-F^4(\zeta_4)}{90}= \frac{-(b-a)^4}{1440}f^4(\xi) $$
 * }.
 * }.

Since $$ \displaystyle x=\alpha + ht$$, We have $$ \displaystyle \xi=\alpha + h\zeta_4$$

This is the required relation between $$\displaystyle \xi$$ and $$\displaystyle \zeta_4$$.

Hence proved.

Author
Solved and Typed by - Egm6341.s10.team4.anandankala 13:43, 19 February 2010 (UTC) .

= Problem 4-6: Runge phenomenon =

Given
Refer Lecture slide 16-1 for problem statement
 * {| style="width:100%" border="0" align="left"

I = \int^{5}_{-5} \frac{1}{1+x^2} dx $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (1)

 * }.
 * }.

Find
'''(1) Find $$\displaystyle I_n$$ using Newton-Cotes for n=1, 2,. . ., 15'''

(2) Plot $$\displaystyle f, f_n $$ $$\mbox{,  }$$ for n=1, 2, 3, 8, 12

(3) Plot $$\displaystyle I_n$$ vs. $$\displaystyle n$$ and observe that $$\displaystyle I_n$$ does not converge as $$\displaystyle n \rightarrow \infty $$

(4) Observe weight $$\displaystyle W_{i,n}:=\int^{b}_{a} l_{i,n}(x),\ dx$$ are not all positive for n $$ \ge $$ 8. '''Plot $$\displaystyle l_{i,n}(x)$$ for i=1, 2, ... ,8 & n=8.'''

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

 * $$\displaystyle
 * $$\displaystyle

$$

 * }.
 * }.

Solution
'''(1) Find $$\displaystyle I_n$$ using Newton-Cotes for n=1, 2,. . ., 15'''

Newton-cotes formula $$\displaystyle P_n$$,

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

P_n = \sum_{i=0}^n l_{i,n}(x)f(x_i)$$
 * $$\displaystyle
 * $$\displaystyle

 * }.
 * }.

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

l_{i,n}(x) = \prod_{j=0,j\ne i}^n \frac{x-x_j}{x_i-x_j}$$
 * $$\displaystyle
 * $$\displaystyle

 * }.
 * }.

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

I_n = \int^{5}_{-5}P_n(x) dx = \sum_{i=0}^n \underbrace{\int^{5}_{-5}l_{i,n}(x) dx}_{W_i} f(x_i) = \sum_{i=0}^n w_i f(x_i) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (2)
 * }.
 * }.

From the lecture slide 13-1,

Although this integration can be done numerically, there ia an another approach(analyitical solution) that derive the w(i) weight values directly.

the Error of integration of Newton-Cotes formula

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

\left | E_n(f(x)) \right | = \left | I-I_n \right | \le \int^{b}_{a} \left | f(x)-P_n(x)\right |  dx \le \frac {\int^{b}_{a}\left | q_{n+1}(x) \right| dx}{(n+1)!}\Big( max f^{(n+1)}(\xi), \xi \in [a,b] \Big)$$ $$
 * $$\displaystyle
 * $$\displaystyle
 *
$$\displaystyle \longrightarrow (3)

 * }.
 * }.

From the equation (2),

We know the error is zero when the $$\displaystyle f(x) $$ is the polynomial of degree less than $$n$$

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

E_n(f(x)) = 0 $$
 * $$\displaystyle
 * $$\displaystyle

 * }.
 * }.

because, for the polynomial of degree less than $$\displaystyle n (\mathcal P_n)$$

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

if\quad f(x) \in \mathcal P_n, \qquad f^{(n+1)}(x)=0 $$
 * $$\displaystyle
 * $$\displaystyle

 * }.
 * }.

In order to find the $$\displaystyle w_i$$ in equation (2), we can use n equations and solve using matrix method like below,

for the $$\displaystyle f(x)=1,\; x, \; x^2, \; ...... \; x^n$$ (simplist and representative polynomial less than nth order), plug f(x) into equation(3)

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

E_n(f(x))= E_n(1)=E_n(x)=E_n(x^2)=......=E_n(x^n)=0 $$
 * $$\displaystyle
 * $$\displaystyle

 * }.
 * }.

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

\begin{align} &E_n(1)= \int^{5}_{-5}1 dx -[w_1(1)+w_2(1)+w_3(1)+......+w_n(1)] =0\\ &E_n(x)= \int^{5}_{-5}x dx -[w_1(x_0)+w_2(x_2)+w_3(x_3)+......+w_n(x_n)] =0\\ &E_n(x^2)= \int^{5}_{-5}x^2 dx -[w_1(x_0^2)+w_2(x_1^2)+w_3(x_3^2)+......+w_n(x_n^2)] =0\\ &E_n(x^3)= \int^{5}_{-5}x^3 dx -[w_1(x_0^3)+w_2(x_1^3)+w_3(x_3^3)+......+w_n(x_n^3)] =0\\ &.......................\\ &..............................\\ &......................................\\ &E_n(x^n)= \int^{5}_{-5}x^n dx -[w_1(x_0^n)+w_2(x_1^n)+w_3(x_3^n)+......+w_n(x_n^n)] =0 \end{align} $$
 * $$\displaystyle
 * }.
 * }.

Hence, the $$\displaystyle w_i$$ may be determined from

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

\underbrace{ \begin{bmatrix} 1 & 1 & 1 & 1 & ...... & 1\\ x_0 & x_1 & x_2 & x_3 & ...... & x_n\\ x_0^2 & x_1^2 & x_2^2 & x_3^2 & ...... & x_n^2\\ x_0^3 & x_1^3 & x_2^3 & x_3^3 & ...... & x_n^3\\ \vdots & \vdots& \vdots& \vdots& \vdots & \vdots \\ x_0^n & x_1^n & x_2^n & x_3^n & ...... & x_n^n\\ \end{bmatrix} }_{A} \underbrace{ \begin{bmatrix} w_0\\ w_1\\ w_2\\ w_3\\ \vdots\\ w_n\\ \end{bmatrix} }_{W} = \underbrace{ \begin{bmatrix} \int^{5}_{-5}1 dx\\ \int^{5}_{-5}x dx\\ \int^{5}_{-5}x^2dx\\ \int^{5}_{-5}x^3dx\\ \vdots \\ \int^{5}_{-5}x^ndx\\ \end{bmatrix} = \begin{bmatrix} (b-a)\\ \frac{(b^2-a^2)}{2}\\ \frac{(b^3-a^3)}{3}\\ \frac{(b^4-a^4)}{4}\\ \vdots \\ \frac{(b^{n+1}-a^{n+1})}{n+1} \end{bmatrix} }_{B} $$
 * $$\displaystyle
 * }.
 * }.

$$\displaystyle W_i$$ can be obtained like this,
 * {| style="width:100%" border="0" align="left"

\begin{align} &AW=B\\ &W_i=A^{-1}B \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

plug obtained $$\displaystyle W_i$$ and \displaystyle f(x)=\frac{1}{1+x^2} into equation(2), $$\displaystyle I_n$$ can be calculated,
 * {| style="width:100%" border="0" align="left"

I_n = \sum_{i=0}^n w_i f(x_i) $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

$$ \begin{array}{|c||c|c|} n & I_n \\ \hline 1&0.384615\\ 2&6.794872\\	3&2.081448\\ 4&2.374005\\ 5&2.307692\\ 6&3.870449\\	7&2.898994\\	8&1.500489\\ 9&2.398618\\ 10&4.673301\\ 11&3.244773\\ 12&-0.31294\\ 13&1.919797\\ 14&7.899545\\ 15&4.155559\\ \hline \end{array} $$

 Matlab Code:

(2) Plot $$\displaystyle f, f_n $$ $$\mbox{,  }$$ for n=1, 2, 3, 8, 12

$$\displaystyle f, f_n $$ were plotted by below matlab code.

 Matlab Code:

(3) Plot $$\displaystyle I_n$$ vs. $$\displaystyle n$$ and observe that $$\displaystyle I_n$$ does not converge as $$\displaystyle n \rightarrow \infty $$

(4) Observe weight $$\displaystyle W_{i,n}:=\int^{b}_{a} l_{i,n}(x),\ dx$$ are not all positive for n $$ \ge $$ 8. '''Plot $$\displaystyle l_{i,n}(x)$$ for i=1, 2, ... ,8 & n=8.'''

This is the table that show the $$\displaystyle W_{i,n}:=\int^{b}_{a} l_{i,n}(x),\ dx$$ for n=8

 * Plot $$\displaystyle l_{i,n}(x)$$ for i=1, 2, ... ,8 & n=8.
 * Plot $$\displaystyle l_{i,n}(x)$$ for i=1, 2, ... ,8 & n=8.

 *  Matlab Code:

.

Author
Solved - --Egm6341.s10.Team4.roni 05:15, 18 February 2010 (UTC)

Typed and reviewed by - Egm6341.s10.Team4.yunseok 02:06, 18 February 2010 (UTC) .

=Problem 4.7: Determining the Upper Bound of the Error in the Composite Simpson's Rule =

''' This problem was solved without referring to S10 homework. '''

Given: The Error as the Difference between Actual and Numeric
The error $$ \displaystyle E_n $$ in the Composite Simpson's rule

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

$$ \displaystyle \begin{align} {E_n} &= I-I_n \\ &= \int\limits_{a}^{b} f(x)dx - \dfrac{h}{3}[f(x_0)+4f(x_1)+2f(x_2)+...+2f(x_{n-2})+4f(x_{n-1})+f(x_n)]\\ \end{align}$$ (7.0)
 * style="width:98%" |
 * style="width:98%" |
 *

 * }

Show: The Error for the the Composite Simpson's Rule is Bounded Above
That the error for the Composite Simpson's Rule is given by,
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$$ \displaystyle \left| {E}_{n}^{2} \right| \leq \frac{(b-a)^{5}}{2880 n^{4}} M_{4} = \frac{(b-a)h^{4}}{2880} M_{4} \quad where \quad M_{4} := max \left| f^{4}(\xi) \right| \quad s.t. \quad \xi \in [a,b] $$     (7.1)
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Solution: Relating the Error in the Simple Simpson's Rule to the Partitioning of Intervals in the Composite Simpson's Rule
Consider the error $$ \displaystyle E_n $$ which is defined as the difference between the actual value of the integral, $$ \displaystyle I $$ and the value obtained from numerically integrating using the Composite Simpson's Rule, $$ \displaystyle I_n $$ where $$ \displaystyle n \geq 2 $$ is an even integer,

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$$ \displaystyle \begin{align} {E_n} &= I-I_n \\ &= \int\limits_{a}^{b} f(x)dx - \dfrac{h}{3}[f(x_0)+4f(x_1)+2f(x_2)+...+2f(x_{n-2})+4f(x_{n-1})+f(x_n)]\\ \end{align}$$ (7.2) Observe that the following pattern arises from the terms $$ \displaystyle I_n $$ in (7.2)
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$$ \begin{align}\displaystyle & f(x_0)+4f(x_1)+{\color{red}2f(x_2)}+...+{\color{green}2f(x_{n-2})}+4f(x_{n-1})+f(x_n) = \\ & \left[ f(x_0)+4f(x_1) + {\color{red}f(x_2)} \right] + \left[ {\color{red} f(x_2)}+4f(x_3) + f(x_4) \right] + \cdots \\ & \left[ f(x_{n-4})+4f(x_{n-3}) + {\color{green}f(x_{n-2})} \right] + \left[ {\color{green}f(x_{n-2})}+4f(x_{n-1}) + f(x_{n}) \right] \end{align} $$ The above relation allows us to write the terms inside of the summation as what is shown in (7.3). We must also take precautions to ensure that the partitioning of not only the summation, but also the partitioning of the integral contain the same subintervals namely, $$ \displaystyle [x_{2i-2},x_{2i}] $$. Both are accounted for defining $$ \displaystyle h:=\frac{b-a}{n} $$ and letting $$ \displaystyle x_i = a + ih $$ where $$ \displaystyle i = 0,1,2,...,n $$.
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$$ \displaystyle \begin{align} {E_n} &= \sum_{i=1}^{n/2}\left[ \int\limits_{x_{2i-2}}^{x_{2i}} f(x)dx \right] - \dfrac{h}{3}\sum_{i=1}^{n/2} \left[ f(x_{2i-2})+4f(x_{2i-1})+f(x_{2i}) \right]\\ &= \sum_{i=1}^{n/2} \left[ \int\limits_{x_{2i-2}}^{x_{2i}} f(x)dx-\dfrac{h}{3}[f(x_{2i-2})+4f(x_{2i-1})+f(x_{2i})]\right]\\ where \quad h&=\frac{b-a}{n} = \frac{x_i - x_{i-1}}{n} \end{align}$$ (7.3)
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We know that for the Simple Simson's Rule the error associated with the Lagrange Interpolation in, $$ \displaystyle E_2 $$ is,
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$$ \displaystyle {E_2} = \frac{x_{i+1}-x_{i-1}}{2}f^{(4)}(\xi) \quad s.t. \quad \xi \in [x_{i-1},x_{i+1}] $$     (7.4)
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Which implies for the Composite Simpson's Rule
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$$ \displaystyle \begin{align} \left| {E_n} \right| &\leq  \sum_{i=1}^{n/2} max \left| \frac{(x_{i+1}-x_{i-1})^5}{90\cdot 2^5}f^{(4)}(\xi) \right| \quad s.t. \quad \xi \in [x_{i-1},x_{i+1}]\\ &= \frac{(x_{i+1}-x_{i-1})^5}{90} \cdot \sum_{i=1}^{n/2} {\color{red}\underbrace{max \left| f^{(4)}(\xi)\right|}_{M_4}} \quad s.t. \quad \xi \in [x_{i-1},x_{i+1}] \end{align} $$ (7.5)
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Where $$ M_{4} $$ is defined on $$ \displaystyle [a,b] $$ as
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$$ \displaystyle M_{4} := max \left| f^{(4)}(\xi)\right| \quad s.t. \quad \xi \in [a,b] $$     (7.6) Next we will consider $$ \displaystyle \bar{M}_{4} \leq n \cdot M_{4} $$ allowing us to write $$ \displaystyle E_n $$ as,
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$$ \begin{align} \displaystyle \left| {E_n} \right| & \leq \left| \frac{(b-a)^{5}}{2880n^{5}}nM_{4} \right| \\ &= \left| \left(\frac{(b-a)}{2880}\right) {\color{red} \left( \frac{(b-a)^4} {n^4}\right)} M_{4}\right| \\ &= \left| \frac{(b-a){\color{red}h^{4}}}{2880}M_{4} \right| \end{align} $$ (7.7)
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Therefore showing that the error associated with the Composite Simpson's Rule is

=Problem 4.9 proof of SSET in different cases= Refer to lecture slide [[media:nm1.s11.mtg22.djvu|mtg-22]] for the problem statement.

Objectives
A). Redo the proof for two cases:

1).
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$$ \displaystyle
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G(t):=e(t)-{{t}^{4}}e(1)

$$ 2). :{| style="width:100%" border="0" $$ \displaystyle
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G(t):=e(t)-{{t}^{6}}e(1)

$$ Point out where proof breaks down.
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B). for G(t) as in (1) p.19-1(w/t5), find G(3)(0) and follow same steps in proof to see what happen.

Solutions
A). Redo the proof for two cases: 1). $$\displaystyle G(t):=e(t)-{{t}^{4}}e(1)$$
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$$ \displaystyle
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e(t)=\int\limits_{-t}^{+t}{F(t)dt}-\frac{t}{3}[F(-t)+4F(0)+F(t)]

$$ Where $$\displaystyle F(t)=f(x(t))$$
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$$ \displaystyle
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\text{G}\left( 0 \right)=e(0)-{{0}^{4}}e(1)=\int\limits_{-0}^{+0}{F(t)dt}-\frac{0}{3}[F(-0)+4F(0)+F(0)]=0

$$
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$$ \displaystyle
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\text{G}\left( 1 \right)=e(1)-{{1}^{4}}e(1)=0

$$ Apply Rolle ‘ s theorem, it yields:
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$$ \displaystyle
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\exists {{\xi }_{1}}\in \left( 0,1 \right)\begin{matrix} {} & st. \\ \end{matrix}{{G}^{\left( 1 \right)}}({{\xi }_{1}})=0

$$
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$$ \displaystyle
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\begin{align} & {{\text{G}}^{(1)}}\left( t \right)={{e}^{(1)}}(t)-4{{t}^{3}}e(1)={{A}^{(1)}}(t)-A{{_{2}^{L}}^{(1)}}(t)-4{{t}^{3}}e(1) \\ & \begin{matrix} \begin{matrix} {} & {} \\ \end{matrix} & {}  \\ \end{matrix}=F(-t)+F(t)-\{\frac{1}{3}[F(-t)+4F(0)+F(t)]+\frac{t}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)]\}-4{{t}^{3}}e(1) \\ \end{align}

$$ So,
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$$ \displaystyle
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\begin{align} & {{\text{G}}^{(1)}}\left( 0 \right)=F(-0)+F(0)-\{\frac{1}{3}[F(-0)+4F(0)+F(0)]+\frac{0}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)]\}-4\centerdot {{0}^{3}}e(1) \\ & \begin{matrix} {} & {} & {} \\ \end{matrix}=0 \\ \end{align}

$$ Apply Rolle ‘ s theorem, it yields:
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$$ \displaystyle
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\exists {{\xi }_{2}}\in \left( 0,{{\xi }_{1}} \right)\begin{matrix} {} & st. \\ \end{matrix}{{G}^{\left( 2 \right)}}({{\xi }_{2}})=0

$$
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$$ \displaystyle
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\begin{align} & {{\text{G}}^{(2)}}\left( t \right)={{e}^{(2)}}(t)-12{{t}^{2}}e(1)={{A}^{(2)}}(t)-A{{_{2}^{L}}^{(2)}}(t)-12{{t}^{2}}e(1) \\ & \begin{matrix} \begin{matrix} {} & {} \\ \end{matrix} & {}  \\ \end{matrix}=-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)-\{\frac{1}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)] \\ & \begin{matrix} {} & {} & {} & {} \\ \end{matrix}+\frac{1}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)]+\frac{t}{3}[{{F}^{(2)}}(-t)+{{F}^{(2)}}(t)]\}-12{{t}^{2}}e(1) \\ & \begin{matrix} \begin{matrix} {} & {} \\ \end{matrix} & {}  \\ \end{matrix}=\frac{1}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)]-\frac{t}{3}[{{F}^{(2)}}(-t)+{{F}^{(2)}}(t)]\}-12{{t}^{2}}e(1) \\ \end{align}

$$ So,
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$$ \displaystyle
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\begin{align} & {{\text{G}}^{(2)}}\left( 0 \right)=\frac{1}{3}[-{{F}^{(1)}}(-0)+{{F}^{(1)}}(0)]-\frac{0}{3}[{{F}^{(2)}}(-0)+{{F}^{(2)}}(0)]\}-12\centerdot {{0}^{2}}e(1) \\ & \begin{matrix} {} & {} & = \\ \end{matrix}0 \\ \end{align}

$$ Apply Rolle ‘ s theorem, it yields:
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$$ \displaystyle
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\exists {{\xi }_{3}}\in \left( 0,{{\xi }_{2}} \right)\begin{matrix} {} & st. \\ \end{matrix}{{G}^{\left( 3 \right)}}({{\xi }_{3}})=0

$$
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$$ \displaystyle
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\begin{align} & {{\text{G}}^{(3)}}\left( t \right)=\frac{1}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)]-\frac{t}{3}[{{F}^{(2)}}(-t)+{{F}^{(2)}}(t)]-24te(1) \\ & \begin{matrix} {} & {} & = \\ \end{matrix}\frac{1}{3}[{{F}^{(2)}}(-t)+{{F}^{(2)}}(t)]-\frac{1}{3}[{{F}^{(2)}}(-t)+{{F}^{(2)}}(t)]-\frac{t}{3}[-{{F}^{(3)}}(-t)+{{F}^{(3)}}(t)]-24te(1) \\ & \begin{matrix} {} & {} & = \\ \end{matrix}-\frac{t}{3}[-{{F}^{(3)}}(-t)+{{F}^{(3)}}(t)]-24te(1) \\ \end{align}

$$ So, $$\displaystyle {{\text{G}}^{(3)}}\left( {{\xi }_{3}} \right)=\frac{3}[{{F}^{(3)}}(-{{\xi }_{3}})-{{F}^{(3)}}({{\xi }_{3}})]-24\left( {{\xi }_{3}} \right)e(1)=0$$, then using DMVT:
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$$ \displaystyle
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e(1)=\frac{\frac{3}[{{F}^{(3)}}(-{{\xi }_{3}})-{{F}^{(3)}}({{\xi }_{3}})]}{24\left( {{\xi }_{3}} \right)}=\frac{\frac{3}[-2{{\xi }_{3}}{{F}^{(4)}}({{\xi }_{4}})]}{24\left( {{\xi }_{3}} \right)}=\frac{-{{\xi }_{3}}{{F}^{(4)}}({{\xi }_{4}})}{36}

$$ Where, $$\displaystyle {{\xi }_{4}}\in \left( -{{\xi }_{3}},{{\xi }_{3}} \right)$$
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$$ \displaystyle
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{{\text{E}}_{2}}=h\centerdot e(1)=h\frac{-{{\xi }_{3}}{{F}^{(4)}}({{\xi }_{4}})}{36}

$$     (Eq ) The proof will break down here because the magnitude of error is related to $$\displaystyle {{\xi }_{3}}$$in the above equation.
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2)

$$\displaystyle G(t):=e(t)-{{t}^{6}}e(1)$$
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$$ \displaystyle
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e(t)=\int\limits_{-t}^{+t}{F(t)dt}-\frac{t}{3}[F(-t)+4F(0)+F(t)]

$$ Where $$\displaystyle F(t)=f(x(t))$$
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$$ \displaystyle
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\text{G}\left( 0 \right)=e(0)-{{0}^{6}}e(1)=\int\limits_{-0}^{+0}{F(t)dt}-\frac{0}{3}[F(-0)+4F(0)+F(0)]=0

$$
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$$ \displaystyle
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\text{G}\left( 1 \right)=e(1)-{{1}^{6}}e(1)=0

$$ Apply Rolle ‘ s theorem, it yields:
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$$ \displaystyle
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\exists {{\xi }_{1}}\in \left( 0,1 \right)\begin{matrix} {} & st. \\ \end{matrix}{{G}^{\left( 1 \right)}}({{\xi }_{1}})=0

$$
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$$ \displaystyle
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\begin{align} & {{\text{G}}^{(1)}}\left( t \right)={{e}^{(1)}}(t)-6{{t}^{5}}e(1)={{A}^{(1)}}(t)-A{{_{2}^{L}}^{(1)}}(t)-6{{t}^{5}}e(1) \\ & \begin{matrix} \begin{matrix} {} & {} \\ \end{matrix} & {}  \\ \end{matrix}=F(-t)+F(t)-\{\frac{1}{3}[F(-t)+4F(0)+F(t)]+\frac{t}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)]\}-6{{t}^{5}}e(1) \\ \end{align}

$$ So,
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$$ \displaystyle
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\begin{align} & {{\text{G}}^{(1)}}\left( 0 \right)=F(-0)+F(0)-\{\frac{1}{3}[F(-0)+4F(0)+F(0)]+\frac{0}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)]\}-6\centerdot {{0}^{5}}e(1) \\ & \begin{matrix} {} & {} & {} \\ \end{matrix}=0 \\ \end{align}

$$ Apply Rolle ‘ s theorem, it yields:
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$$ \displaystyle
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\exists {{\xi }_{2}}\in \left( 0,{{\xi }_{1}} \right)\begin{matrix} {} & st. \\ \end{matrix}{{G}^{\left( 2 \right)}}({{\xi }_{2}})=0

$$
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$$ \displaystyle
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\begin{align} & {{\text{G}}^{(2)}}\left( t \right)={{e}^{(2)}}(t)-30{{t}^{4}}e(1)={{A}^{(2)}}(t)-A{{_{2}^{L}}^{(2)}}(t)-30{{t}^{4}}e(1) \\ & \begin{matrix} \begin{matrix} {} & {} \\ \end{matrix} & {}  \\ \end{matrix}=-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)-\{\frac{1}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)] \\ & \begin{matrix} {} & {} & {} & {} \\ \end{matrix}+\frac{1}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)]+\frac{t}{3}[{{F}^{(2)}}(-t)+{{F}^{(2)}}(t)]\}-30{{t}^{4}}e(1) \\ & \begin{matrix} \begin{matrix} {} & {} \\ \end{matrix} & {}  \\ \end{matrix}=\frac{1}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)]-\frac{t}{3}[{{F}^{(2)}}(-t)+{{F}^{(2)}}(t)]\}-30{{t}^{4}}e(1) \\ \end{align}

$$ So,
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$$ \displaystyle
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\begin{align} & {{\text{G}}^{(2)}}\left( 0 \right)=\frac{1}{3}[-{{F}^{(1)}}(-0)+{{F}^{(1)}}(0)]-\frac{0}{3}[{{F}^{(2)}}(-0)+{{F}^{(2)}}(0)]\}-30\centerdot {{0}^{4}}e(1) \\ & \begin{matrix} {} & {} & = \\ \end{matrix}0 \\ \end{align}

$$ Apply Rolle ‘ s theorem, it yields:
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$$ \displaystyle
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\exists {{\xi }_{3}}\in \left( 0,{{\xi }_{2}} \right)\begin{matrix} {} & st. \\ \end{matrix}{{G}^{\left( 3 \right)}}({{\xi }_{3}})=0

$$
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$$ \displaystyle
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\begin{align} & {{\text{G}}^{(3)}}\left( t \right)=\frac{1}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)]-\frac{t}{3}[{{F}^{(2)}}(-t)+{{F}^{(2)}}(t)]-120{{t}^{3}}e(1) \\ & \begin{matrix} {} & {} & = \\ \end{matrix}\frac{1}{3}[{{F}^{(2)}}(-t)+{{F}^{(2)}}(t)]-\frac{1}{3}[{{F}^{(2)}}(-t)+{{F}^{(2)}}(t)]-\frac{t}{3}[-{{F}^{(3)}}(-t)+{{F}^{(3)}}(t)]-120{{t}^{3}}e(1) \\ & \begin{matrix} {} & {} & = \\ \end{matrix}-\frac{t}{3}[-{{F}^{(3)}}(-t)+{{F}^{(3)}}(t)]-120{{t}^{3}}e(1) \\ \end{align}

$$ So, $$\displaystyle {{\text{G}}^{(3)}}\left( {{\xi }_{3}} \right)=\frac{3}[{{F}^{(3)}}(-{{\xi }_{3}})-{{F}^{(3)}}({{\xi }_{3}})]-120{{\left( {{\xi }_{3}} \right)}^{3}}e(1)=0$$, then using DMVT:
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$$ \displaystyle
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e(1)=\frac{\frac{3}[{{F}^{(3)}}(-{{\xi }_{3}})-{{F}^{(3)}}({{\xi }_{3}})]}{120{{\left( {{\xi }_{3}} \right)}^{3}}}=\frac{\frac{3}[-2{{\xi }_{3}}{{F}^{(4)}}({{\xi }_{4}})]}{120{{\left( {{\xi }_{3}} \right)}^{3}}}=\frac{-{{F}^{(4)}}({{\xi }_{4}})}{180{{\xi }_{3}}}

$$ Where, $$\displaystyle {{\xi }_{4}}\in \left( -{{\xi }_{3}},{{\xi }_{3}} \right)$$
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$$ \displaystyle
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{{\text{E}}_{2}}=h\centerdot e(1)=h\frac{-{{F}^{(4)}}({{\xi }_{4}})}{180{{\xi }_{3}}}

$$
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B)
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$$ \displaystyle
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G(t):=e(t)-{{t}^{5}}e(1)

$$
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$$ \displaystyle
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e(t)=\int\limits_{-t}^{+t}{F(t)dt}-\frac{t}{3}[F(-t)+4F(0)+F(t)]

$$ Where $$\displaystyle F(t)=f(x(t))$$
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$$ \displaystyle
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\text{G}\left( 0 \right)=e(0)-{{0}^{5}}e(1)=\int\limits_{-0}^{+0}{F(t)dt}-\frac{0}{3}[F(-0)+4F(0)+F(0)]=0

$$
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$$ \displaystyle
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\text{G}\left( 1 \right)=e(1)-{{1}^{5}}e(1)=0

$$ Apply Rolle ‘ s theorem, it yields:
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$$ \displaystyle
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\exists {{\xi }_{1}}\in \left( 0,1 \right)\begin{matrix} {} & st. \\ \end{matrix}{{G}^{\left( 1 \right)}}({{\xi }_{1}})=0

$$
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$$ \displaystyle
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\begin{align} & {{\text{G}}^{(1)}}\left( t \right)={{e}^{(1)}}(t)-5{{t}^{4}}e(1)={{A}^{(1)}}(t)-A{{_{2}^{L}}^{(1)}}(t)-5{{t}^{4}}e(1) \\ & \begin{matrix} \begin{matrix} {} & {} \\ \end{matrix} & {}  \\ \end{matrix}=F(-t)+F(t)-\{\frac{1}{3}[F(-t)+4F(0)+F(t)]+\frac{t}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)]\}-5{{t}^{4}}e(1) \\ \end{align}

$$ So,
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$$ \displaystyle
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\begin{align} & {{\text{G}}^{(1)}}\left( 0 \right)=F(-0)+F(0)-\{\frac{1}{3}[F(-0)+4F(0)+F(0)]+\frac{0}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)]\}-5\centerdot {{0}^{4}}e(1) \\ & \begin{matrix} {} & {} & {} \\ \end{matrix}=0 \\ \end{align}

$$ Apply Rolle ‘ s theorem, it yields:
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$$ \displaystyle
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\exists {{\xi }_{2}}\in \left( 0,{{\xi }_{1}} \right)\begin{matrix} {} & st. \\ \end{matrix}{{G}^{\left( 2 \right)}}({{\xi }_{2}})=0

$$
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$$ \displaystyle
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\begin{align} & {{\text{G}}^{(2)}}\left( t \right)={{e}^{(2)}}(t)-20{{t}^{3}}e(1)={{A}^{(2)}}(t)-A{{_{2}^{L}}^{(2)}}(t)-20{{t}^{3}}e(1) \\ & \begin{matrix} \begin{matrix} {} & {} \\ \end{matrix} & {}  \\ \end{matrix}=-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)-\{\frac{1}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)] \\ & \begin{matrix} {} & {} & {} & {} \\ \end{matrix}+\frac{1}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)]+\frac{t}{3}[{{F}^{(2)}}(-t)+{{F}^{(2)}}(t)]\}-20{{t}^{3}}e(1) \\ & \begin{matrix} \begin{matrix} {} & {} \\ \end{matrix} & {}  \\ \end{matrix}=\frac{1}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)]-\frac{t}{3}[{{F}^{(2)}}(-t)+{{F}^{(2)}}(t)]\}-20{{t}^{3}}e(1) \\ \end{align}

$$ So,
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$$ \displaystyle
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\begin{align} & {{\text{G}}^{(2)}}\left( 0 \right)=\frac{1}{3}[-{{F}^{(1)}}(-0)+{{F}^{(1)}}(0)]-\frac{0}{3}[{{F}^{(2)}}(-0)+{{F}^{(2)}}(0)]\}-20\centerdot {{0}^{3}}e(1) \\ & \begin{matrix} {} & {} & = \\ \end{matrix}0 \\ \end{align}

$$ Apply Rolle ‘ s theorem, it yields:
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$$ \displaystyle
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\exists {{\xi }_{3}}\in \left( 0,{{\xi }_{2}} \right)\begin{matrix} {} & st. \\ \end{matrix}{{G}^{\left( 3 \right)}}({{\xi }_{3}})=0

$$
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$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} & {{\text{G}}^{(3)}}\left( t \right)=\frac{1}{3}[-{{F}^{(1)}}(-t)+{{F}^{(1)}}(t)]-\frac{t}{3}[{{F}^{(2)}}(-t)+{{F}^{(2)}}(t)]-60{{t}^{2}}e(1) \\ & \begin{matrix} {} & {} & = \\ \end{matrix}\frac{1}{3}[{{F}^{(2)}}(-t)+{{F}^{(2)}}(t)]-\frac{1}{3}[{{F}^{(2)}}(-t)+{{F}^{(2)}}(t)]-\frac{t}{3}[-{{F}^{(3)}}(-t)+{{F}^{(3)}}(t)]-60{{t}^{2}}e(1) \\ & \begin{matrix} {} & {} & = \\ \end{matrix}-\frac{t}{3}[-{{F}^{(3)}}(-t)+{{F}^{(3)}}(t)]-60{{t}^{2}}e(1) \\ \end{align}

$$ So,
 *

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

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

{{\text{G}}^{(3)}}\left( 0 \right)=-\frac{0}{3}[-{{F}^{(3)}}(-t)+{{F}^{(3)}}(t)]-60\centerdot {{0}^{2}}e(1)=0

$$ Apply Rolle ‘ s theorem, it yields:
 *

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

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

\exists {{\xi }_{4}}\in \left( 0,{{\xi }_{3}} \right)\begin{matrix} {} & st. \\ \end{matrix}{{G}^{\left( 4 \right)}}({{\xi }_{4}})=0

$$
 *

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

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

{{\text{G}}^{(4)}}\left( t \right)=\frac{1}{3}[{{F}^{(3)}}(-t)-{{F}^{(3)}}(t)]-\frac{t}{3}[{{F}^{(4)}}(-t)+{{F}^{(4)}}(t)]-120te(1)

$$ So,
 *

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

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

{{\text{G}}^{(4)}}\left( {{\xi }_{4}} \right)=\frac{1}{3}[{{F}^{(3)}}(-{{\xi }_{4}})-{{F}^{(3)}}({{\xi }_{4}})]-\frac{3}[{{F}^{(4)}}(-{{\xi }_{4}})+{{F}^{(4)}}({{\xi }_{4}})]-120{{\xi }_{4}}e(1)=0

$$ Using DMVT:
 *

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

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

e(1)=\frac{\frac{1}{3}[-2{{\xi }_{4}}{{F}^{(4)}}({{\xi }_{5}})]-\frac{3}[{{F}^{(4)}}(-{{\xi }_{4}})+{{F}^{(4)}}({{\xi }_{4}})]}{120{{\xi }_{4}}}=\frac{-[2{{F}^{(4)}}({{\xi }_{5}})+{{F}^{(4)}}(-{{\xi }_{4}})+{{F}^{(4)}}({{\xi }_{4}})]}{360}

$$ Where, $$\displaystyle {{\xi }_{5}}\in \left( -{{\xi }_{4}},{{\xi }_{4}} \right)$$
 *

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

$$ \displaystyle
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

{{\text{E}}_{2}}=h\centerdot e(1)=h\frac{-[2{{F}^{(4)}}({{\xi }_{5}})+{{F}^{(4)}}(-{{\xi }_{4}})+{{F}^{(4)}}({{\xi }_{4}})]}{360}

$$ Here, it is observed that the magnitude of error is related to the value of function at these two points$$\displaystyle {{\xi }_{4}},{{\xi }_{5}}$$in the above equation.
 *

 * }

title: User:Egm6341.s11.team5.oh/HW4


text:
OCAD

OCAD is software for drawing orienteering maps.

Rarely does creation of an orienteering map need to start from scratch with field work. One or more existing maps (background maps) can be used as sources. Possum Trot Orienteering Club (of Kansas City) has an outline of the steps involved and an online tutorial.

In OCAD 10, creating a new orienteering map can be greatly simplified if you start with a georeferenced background map, and you know the current magnetic declination of the location.

Using background maps
OCAD supports the use of multiple background maps. Each background map is treated as a separate layer. If georeferenced background maps are imported, OCAD will align them to each other automatically. Else, you will need to align each new background map to the previous map(s) by hand.

Background maps are not copied into the OCAD map file (file type .ocd). This means that if the map file is to be given to someone else to work on it, the background map files need to be given too.

Declination adjustment
All orienteering maps are oriented to magnetic north. However, many existing maps that are used as sources are oriented to true north or grid north. Thus, some adjustment may be necessary to adjust source map data so that it is oriented to magnetic north. You can ignore the difference between true north and grid north.

Because the earth's magnetic poles wander, if your source is an older map you may want to not rely on the declination printed on the map. Instead, look up the current declination adjustment for the location. A free online tool is provided by the NOAA Geophysical Data Center. Even if the source map is oriented to magnetic north, you may need to look up the current declination and make an adjustment if magnetic north has changed more than 1 or 2 degrees since the map was made.

Declination adjustment can be done in OCAD.

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

title: OCAD


text:
User:Podlington

Student

title: User:Podlington


text:
Microkernel/Project planning

Research
This is the text of the first meeting which was on the Puredarwin IRC (freenode). It is also an example of wiki deconstruction, a converstion about the correct pathway for open and free OS development. This will be pulled apart and turned into a flow chart of development options: XNU (or OSX Darwin microk seems like the choice of choice, but has barriers. GNU Mach also looks good.  The developed base microkernel (MK) will be the development platform for purer MKs especially L4, which boasts a billion copies out there.  Others that may be applicable but have languished are LibOS and ExOS.  Beyond that are paradigms to differentiate the project from the convoluted mess we deal with now, such as data centricity over disconnected applications.

Puredarwin.org IRC
I was ty'n whoever can get xnu to boot on x86, so I can get started w/ gcc

dumb question, is I/O kit compile in gcc c++?

sorry, does I/O kit..

 I believe that G++ is used to compile XNU in its entirety.

good to know, thanks

your honest opinion, what is the easiest way to get xnu running as an independent desktop?

 No problem. It's been a long time since I've attempted to build it, though.

I am attempting a microkernel project, and it would be good to have something to show

 Bootstrapping it from a Mac OS X system using the PD scripts is probably the easiest method, at the moment.

what would be the earliest model Mac to do this? (ebay)

 Adapting one of the ISOs or VMDKs might work, though - but you're sure to hit limitations.

 It might be possible to cross-compile from a PowerPC-based Mac, but it's a case of trial-and-error.

actually x86 might be better

 An early MacBook or x86-based Mac Mini running 10.5.x should do the trick, though.

ty

 If it works under 10.4, then you might be able to obtain something even cheaper.

I was fine with x and Toms wm on less than 300mz and 1/4 gig mem

gcc too, I suppose

compiles can always swap while I visit my "reality" of psychopathology

3 weeks to go

 If you're willing to strip out the optimisations for S/SSE/2/3 and workaround the CPU checks, you could probably run it on a fairly modest system.

 *work around

its a project-course to introduce microkernel, and I still believe in supporting the 3rd world!

 (The Voodoo fork of XNU may hold some clues with regards to that). :)

what about these guys in Malaysia with the pirate logos?

 If all else fails, there's always Minix...

does minix swap yet?

 Minix 3 does, apparently.

been, like only 22 yrs, ugh

 Or at least there's talk of swap partition support.

during my last IT incarnation, I was looking at ExOS

 Bah. http://groups.google.com/group/minix3/browse_thread/thread/2ff644ae70ce3782?pli=1 seems to suggest rudimentary paging support, but not "swap partition" support.

rudimentary does not inspire one

LibOS too, and I was interested in the Perl VM as it uses registers instead of stacks

and a compiler writer told me that a register-only system would be factors more efficient

 Until they decided to close access to the codebase again, QNX Neutrino might have been a nice fit.

yeah, that angered me, QNX is a nice OS

All things considered L4 seems a good path, as it is marginally related to XNU and Mac

 Lots of microkernel-based OSes seem to implement support for demand-paging (or some other paging technique), but not swapping to disk.

<vmlemon_> :(

<vmlemon_> For what it's worth, we were contemplating adding swapping/disk-bound demand paging to Symbian OS; but the consensus was that it'd be a bad idea, since NAND Flash chips would quickly burn out under the workload.

<vmlemon_> (Not what you'd want to happen in an expensive smartphone, of course!) :)

<vmlemon_> Still, at least a demand paging mechanism made its way in, prior to the Open Source release.

heh, as it happens I have an early Mako that is the nicest piece of hardware I own, beautiful construction

who is "we?" are you in the symbian sphere?

<vmlemon_> "We" was the Symbian Foundation and the community that formed around it, prior to its recent closure.

yet another "bites the dust"

 * duvjones has quit (Quit: Leaving)

 * machdev (~machdev@ip68-4-241-106.oc.oc.cox.net) has joined #puredarwin

my hardware scope is a little larger that smart phones, possibly recycled laptop equipment, as I know my audience has that lying around

<vmlemon_> (The SF ran an "Ideas" site where folks both internally and externally could throw around thoughts on improving the Symbian Platform - although it eventually devolved into a bunch of spamming idiots, and whiners who wanted us to make all but an iPhone OS clone; and both the staff and the visitor community abandoned it quickly after the signal-to-noise ratio plummeted).

<vmlemon_> A lot of high-quality ideas and debate was generated when it was in private beta, though.

<vmlemon_> The moral of the story is that letting in the unwashed masses quickly ruins things...

sounds like the Wikiversity, except the non-whiners have sufficent critical mass

<vmlemon_> Heh

the fact that it is a virtual mental hospital is benefiting me now that I am studying to be a counselor

<vmlemon_> Interesting stuff

my better half and I want to open a group home, and knowing most kids are gaming junkies only becuase they are frustrated OS hacks

I will make IT a component of it

<vmlemon_> Aah

 * vmlemon_ never really got into gaming

the other component will be working shelter animals

<vmlemon_> Nice

me neither, though I liked Sonic the Hedgehog

being in canada, my fantasy is to hitch the dogs up to sleds, and go from there!

<vmlemon_> The last games console we bought was a Wii - which is now seldom used, and the last one I played with any regularity was the PlayStation 2.

<== we are talking sega dreamcast

<vmlemon_> I sometimes play the odd puzzle game on my phone, though. (I've got Snakes and a Dutch Tetris clone installed on it, at the moment).

I actually text now that I have a keyboard on a phone, though I swore I never would

and I tried snowboarding

looks like the gen II is winning me over

so the least I can do is give them a decent OS to get acquainted with

<vmlemon_> Make sense

hate it when that happens
 * vmlemon_ had a GameBoy Advance emulator installed on the Flash card in his old phone, until he lost it somewhere

<vmlemon_> What about Haiku? It seems to have a fairly minimalistic kernel, and a UNIX-like userland.

I actually lost a pair of steel edge XC skis

OK, noted

<vmlemon_> Bah

If you don't mind me jumping in, what are you looking for?

<vmlemon_> Internally, it seems to be implemented in a very different manner to "classical UNIX clones", though. :)

student project that can work on old x86 hardware that seems to have a future

L4 seems to have the most "future" potential

<vmlemon_> (Like the Symbian OS, it's mostly written in C++, and has its own set of APIs that are preferred over the use of POSIX ones, where possible).

I use to teach CS classes, my advice is to either use linux (which runs decently on old hardware) or to turn those x86 machines into dumb terminals and connect to a TS such as WTS (Windows), LTSP (Linux), ACTS (Mac)

did I forget to mention microkernel?

also interested in I/O kit

lol

Then in that case there are 2 options still:

<vmlemon_> An L4-derived microkernel and Genode might be worth a look, FWIW.

1. Run GNU Mach (bootable now)

<vmlemon_> (I managed to get an L4Ka::Pistachio-based system built fairly quickly using it; and it supports running Linux applications when using OKL4 instead - but that's a PITA to build).

GNU has switched strategies so many times...

2. ACTS which can be used to develeop Mac apps/kernel stuff (using virtualbox and xcode)

True but the GNU Mach system is 80% the same as the XNU's Mach System (both based on Mach 3.0)

this is a constructivist project, which means starting with the most basic components and then building up to something usable

just my opinion of course

usable meaning something a kid can hold and say "this does something useful"

If thats the case then I would avoid the Darwin and related (XNU) platforms

The darwin platform (no offense) is not complete enough for usefulness and Apple won't really let you build a XNU platform (OSX)

on the surface, darwin and XNU seem the best route

and/or

OK, dumb(er) question: If XNU is open source, why do they think they can prevent an x86 boot?

if I phrased the correctly

that thats not it, Apple's "opensource" projects are in complete and even if they were would still not yield a very useful platfrom since all the really good pieces are not opensoured

actually, apple being a corp, why does "it" think that?

<vmlemon_> They're not actively preventing it, but they're not focussing on it as much as they used to, since they've already got a Darwin-based x86-platform that works for them internally.

<vmlemon_> (Mac OS X, of course).

<vmlemon_> Darwin was probably considered to be a proof-of-concept/Open Source experiment, in my opinion.

what do you mean by "really good pieces?"

CoreGraphics, CoreAudio, CoreAnimation, CoreVideo, CoreImage, QuartzCore, etc...

For example

<vmlemon_> A case in point being that there isn't even a working audio subsystem in Darwin, unlike Linux.

Exactly

<vmlemon_> (Same deal with Bluetooth and 802.11, too).

but that OSX, and I would go with Xwin

oh 802.11, thats a biggie

and sound

doesn't matter, the User Land daemons and kernel extensions are far from complete and some components are out right missing

is I/O kit portable?

<vmlemon_> Heck, some of the most critical components are essentially "reimplementations for Open Source".

<vmlemon_> (CoreFoundation comes to mind, there).

For example the IOAudioFamily kernel extension is there but it can't hook in to the kernel properly since a dependency is missing between the IOAudioFamily and the XNU kernel

IOKit is found in XNU/Darwin, FreeBSD (latest), GNU Mach

<vmlemon_> Some folks from NICTA managed to weld it onto an experimental hybrid XNU/L4-based system (Darbat), if I remember correctly.

I heard OpenBSD was also porting IOKit but not sure

and BSD is now a debian release

FreeBSD is NOT a microkernel though

but a step along the way

but it does have an version of IOKit now

FreeBSD won't ever become a microkernel (FYI)

hey vmlemon_, who is running the things for puredarwin anyways?

it would have to become mach, which I think* is where mach came from (??)

<vmlemon_> machdev: probono's mostly running things, but he's busy nowadays.

ahh

<vmlemon_> Heh, yes. XNU could probably be considered a "family reunion" of the two. ;)

I jumped on here a day or 2 ago to see what "blockers" are being worked on so I could start on ones that are not being worked on

I got an ok from my work to do some OSS work on the side and since my work is 90% having to deal with low-level Apple components I though this would be a good place to start

thanks, I have to read about girls who "cut" who have been abused by religious figures (not plaster ones)

I will look into your suggests, I think IOkit is a key component

good luck johnbessa

xnu/darwin seemed a perfect world

I would agree if the project was complete enough

seeing how many would benefit from it, possibly there is a pathway to create open versions of all missing or disconnected pieces

thats the point of puredarwin (I think)

there is no reason that it has to comply with the OSX model

vmlemon_: has anyone thought about porting GNUStep for the AppKit related pieces?

bing!

or bingo

<vmlemon_> Good question. I tried to build it under Darwin 8, and didn't get very far.

<vmlemon_> I haven't reattempted it, since.

vmlemon_: currently you are unaware of anyone working on any current blockers, correct?

<vmlemon_> Most of the work done so far as been ad-hoc stuff on getting booting under VirtualBox working.

<vmlemon_> Theoretically, there's nothing to prevent you from extracting the I/OKit components from XNU, and implementing a compatibility layer on top of another kernel/OS, in order to use said components...

<vmlemon_> machdev: Of course, anyone's free to work on stuff that they find interesting.

<btQuark> greetings guys - have you, by chance had any chance of trying the new vbox 4.x additions for osx server?

vmlemon_: I'm just trying to see where my time would be best spent, all of the work in the blockers are not interesting to me but it has to be done. (I spend all day doing OSX Low Level dev work)

I think that a criteria for developing strategies is showing that the ideas have potential for long-term value

in other words, are popular

<vmlemon_> That said, we've had a successful boot on real non-Apple hardware.

btQuark: I have not tried it, however early vbox 4.0 seemed to be very flaky for me

<vmlemon_> (Of Xmas, which is one of the most complex PureDarwin-based systems so far).

thats great

would access to a remote OSX box help?

you mean, via the Internet?

<vmlemon_> Pertinent question, since we've been trying to get a build machine of any sort running, for ages.

SSH?

<vmlemon_> (probono came close, but he had to take said machine down, and couldn't find a new location to host it in).

I probably could rig up a Mac Terminal Server (Graphical) / SSH server

<vmlemon_> Bart- has also investigated various things, but hasn't quite brought anything to fruition.

<btQuark> machdev the 4.0 seems quite ok

<btQuark> the final one it may help with our vm we could provide anyhow, it could be run tool-less if needed too

<btQuark> anyhow it needs some testing

I could put an extra MacBook Pro (Core2Duo - 4gb RAM - OSX 10.6 Server + ACTS) I have, probably on Tuesday or Wed (after I get it from my lab at work).

<vmlemon_> I'm sure that probono would appreciate any solution that you can come up with. k if you are not running apache, then using a DSL connection is hardly a bother

or cable modem

The box will have RDP and SSH access

<vmlemon_> Great! :) It is literally sitting in the corner collecting dust, I would have to host it at either my home or a friends (via cable modem) since I won't be allowed to host it at the office

How would I get the information to you guys when I have it set?

well, be thankful you have an IT job (I assume it is, I wound up driving big rigs for $$)

I haven't seen probono

lol

My job is nice but the pay sucks

<vmlemon_> Just a case of setting up Dynamic DNS; and maybe adding a CNAME record to the PureDarwin.org DNS server, I guess.

<vmlemon_> (Although I'm not responsible for that).

lol, no I mean how do I get the login credentials to you guys (I don't want the box used for downloading porn, etc)

my psych fork may actually be a business model, esp if we can cure some rich kid with grateful parents

so I don't want to "publish" it

<vmlemon_> Giving them to probono via e-mail is probably the best way.

k

<vmlemon_> He can then redistribute them to genuinely needy folks.

<vmlemon_> :)

lol

The box won't be that powerful so it could handle only like 5 to 10 remote RDP sessions at 1 time

<vmlemon_> (I trust that most folks here won't abuse the box).

RDP = Remote Desktop Protocol (btw)

Each RDP session would have its own desktop so each can run xcode, etc

Ahhh, speaking of abuse... I really have to research cutting, inserting, injecting...

lol, johnbessa, that just sounds bad

<vmlemon_> Creation of FrankenKids? ;)

better someone else!

 * vmlemon_ hides

lol

you should see the xrays

whole tool boxes inside humans

<vmlemon_> Ugh

I would rather not

but they survive!

humans are really resiliant, and so is DNA

<vmlemon_> And I thought that the story about the guy with the knife lodged in his skull for years was grotesque...

probably a story

omg you saw that! It gave me the creeps

<vmlemon_> http://www.spokesman.com/blogs/hbo/2011/feb/18/man-discovers-knife-lodged-skull/

I kept scratching my head for hours

<vmlemon_> It made one of the national newspapers here.

<vmlemon_> (In the UK, that is).

<vmlemon_> *at least one

self-inserted?

<vmlemon_> No clue.

no idea

still, probably needs therapy

seems traumatizing

Canada hands therapy out for free

not here (USA)

makes a big difference

yup, I am just 5 miles past the end of Interstate I-95

and it makes a big difference

like the difference between Detroit and nearby Windsor/London

lol

I'm near mexico

 * vmlemon_ notes that Nokia Remote Device Access is a nice idea, but the implementation could do with some improvement

in the USA still

haven't seen it vmlemon_

<vmlemon_> Aah. I'm working on a Qt-based application, but I don't have a handset capable of running Qt. :(

I hear Mexico is suffering from drug saturation worse than the US

Mexico is in a state of civil war (just no one wants to admit it)

well, anyway, thanks for the material, this provides a good start towards creating a project path, I will edit it down to wiki bytecode

<vmlemon_> Good luck :)

good luck

Yeah Qt doesn't work well in an ACTS session

so I couldn't help you there

title: Microkernel/Project planning


text:
User talk:Topeka

 Hello and Welcome to Wikiversity Topeka! You can contact us with questions at the colloquium or me personally when you need help. Please remember to sign and date your finished comments when participating in discussions. The signature icon above the edit window makes it simple. All users are expected to abide by our Privacy, Civility, and the Terms of Use policies while at Wikiversity.

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Concerns were raised that your user page may have been a violation of Privacy policy. I've deleted the page according to this policy. You are welcome to contribute to Wikiversity, but please do not reveal personal identifying information about anyone under 18 years of age. Thanks! -- Dave Braunschweig (discuss • contribs) 15:16, 13 November 2014 (UTC)

title: User talk:Topeka


text: User:Jamie.Alban

title: User:Jamie.Alban


text:
User:Brylie/ESL

This page outlines an open ended journey towards communication and language comprehension.

Start Small

 * 1) Take an inventory of phonemes in the learner(s) native language(s).
 * 2) * Pick 1 language to start, e.g. Japanese.
 * 3) Determine the second language for learning purposes.
 * 4) Identify overlapping phonemes in the languages.
 * 5) * Is this what you're thinking?
 * 6) Practice vocabulary that consists primarily of phonemes from the overlapping set.
 * 7) Practice new phonemes to attain accurate articulation.

Patterns and Rhythms
Each class can explore a piece of poetry that is accessible to the learners' current proficiency while still being challenging and enjoyable.
 * Dave's ESL Cafe poetry links

Encourage learners to speak the language they have learned so far and to create spoken word poetry. There are no incorrect words or utterances in these poems, this is a generative and freeform excercise.

title: User:Brylie/ESL


text:
User:Wouter~enwikiversity/BlogGSC

Inleiding
Team GSC houdt op deze pagina een blog bij die wekelijks geupdate zal worden. Hierdoor kan u alle ontwikkelingen binnen ons team en de vooruitgang van ons project op de voet volgen.

Blog
15 februari 2011

Vandaag vond het eerste seminarie plaats en werden de team verdeeld. De teamleden konden kennismaken met elkaar en met de coach. In eerste instantie werd gezocht naar ieders goede kanten zodat de taken optimaal verdeeld konden worden. Pieterjan De Feyter werd gekozen als teamleider en zal de leiding over het team op zich nemen. De benodigde documenten voor de eerste deadline werden besproken en de taken werden verdeeld. Een eerste versie van de WBS werd in team opgesteld en ook over de Gant Chart werd nagedacht. Tine Deckers zal de WBS verder verfijnen en het samenwerkinscontract opstellen, Frederik Claes staat in voor de Gant Chart en Wouter Aerts kreeg de taak van notulist en is verantwoordelijk voor PVA.

18 februari 2011

Eerste deadline is vandaag. Team GSC kwam in de namiddag samen om de laatste hand te leggen aan de verschillende documenten. De WBS werd afgewerkt en de taken van de Gant Chart werden verdeeld over de teamleden. Al de documenten werden beschikbaar gesteld op de Wikiversity pagina van ons team. Wiki Pagina Team GSC

22 februari 2011 Start van de eerste seminaries over race strategie, DC-motor en overbrenging. De mensen van het Umicore solar team kwamen praten over hun ervaring met de zonnewagen en er werd informatie verschaft over het gebruik van het fablab. Het zonnepaneel en DC-motor werd uitgedeeld aan de teams. Vervolgens werden de eerste metingen uitgevoerd met het zonnepaneel om also de U/I-grafiek te bepalen. Hieruit kunnen de nodige parameters gehaald worden om te starten aan case SSV 1. Verder werd er in groep een eerste brainstorm sessie gehouden over de verschillende aspecten van onze SSV. Er werden ideeen bedacht voor de aandrijving, het frame, wielen, plaatsing en koeling zonnepaneel, ... De documeten van de vorige deadline werden besproken met de coach en de nodige aanpassingen werden aangebracht. De verbeterde documenten werden op onze wiki pagina gepost.

25 februari 2011

De verantwoordelijke voor case 1 kwamen samem om een eerste maal de nodige berekeningen onder de loep te nemen. Niet zo simpel. Volgende week wordt hier aan verder gewerkt. 1 maart 2011

Vandaag werden de aanpassingen van de documenten met de coach besproken en goed bevonden. De brainstorm sessie van vorige week werd verder gezet en de teamleden brachten nieuwe opmerkingen en ideeen aan. Er moet een keuze gemaakt worden op vele domeinen: Hoeveel wielen gebruiken we? Welke soort aandrijving: riem of tandwieloverbrenging? Voor of achteraandrijving? Vorm en materiaal van het frame? Hoe plaatsen we het zonnepanneel? In groep werden de voor- en nadelen van de verschillende keuzes afgewogen en uiteindelijk bedachten we een voorlopig en primitief eerste model. Na overleg met de coach is er besloten om vooral de nadruk te leggen op een goede overbrenging met zo weinig mogelijk wrijving en verliezen. Frederik en Wouter gingen verder met de nodige berekeningen terwijl de overige teamleden het eerste model verder uitwerkte. Het is belangrijk dat de teamleden informatie vergaren bij iedereen die iets kan bijbrengen tot dit project en deze informatie delen met het team op de volgende vergadering.

8 maart 2011

De sessie startte met een seminarie over matlab/simulink. Hierin werd uitgelegd hoe dit gebruikt moest worden doorheen ons verdere project. Nadien hadden we een afspraak met onze coach die zei dat we goed op weg waren en onze berekeningen goedkeurde. Daarna konden we zelf nog wat verder werken, een heleboel taken werden verdeel en er werd ook beslist om volgende week een kijkje te gaan nemen in het fablab om toch al iets of wat te weten in welk materiaal we het uiteindelijke wagentje zullen laten spuiten. Tegen volgende meeting zou iedereen al goed opgeschoten moeten zijn met het werk en staan we er zeer goed voor!

15 maart 2011

Vandaag werd de voortuitgang van ons project besproken. De berekeningen werden afgerond en het Sankey diagram werd in groep besproken en voorgelegd aan de onze coach. Voor het sankey diagram zouden nog enkele schattingen en berekeningen nuitgevoerd moeten worden zodat dit tegen volgende vergadering afgewerkt kan worden. Quentin en Pieterjan brachten het team op de hoogte over de vooruitgang in Simulink. Tine verzorgt het procesverlag.

22 maart 2011

Op de vergadering werd de laatste hand gelegd aan de verschillende cases en het procesverslag. Voor het procesverslag werden de moeilijkheden en de uitdagingen van dit project besproken. Tine en Michael werken dit verder af. De eerste versie van het sankey diagram werd in groep overlopen en besproken met de coach. De nodige aanpassingen zullen doorgevoerd worden zodat het diagram tijdig af is. Bram bracht het team op de hoogte over de vooruitgang van het frame. Een eerste versie van het frame werd ontwikkeld in solid edge. Het simulink gedeelte wordt zo spoedig mogelijk afgerond.

 10 mei 2011

Op de vergadering werd duidelijk dat er nog een hele boel werk was. Bram en Wouter overliepen de technische tekeningen en Bram zal de laatste aanpassingen aanbrengen zodat deze klaar zijn voor het eindverslag. Michael en Tine werkte het procesverslag af en Michael maakte zijn deel over de overbrening inorde. Het sankey diagram van de Umicar, die door Frederik werden gemaakt, werden in groep besproken. Case 2: Umicar werd hiermee afgesloten. Wouter werkte als laatste het marktonderzoek af.

 12 mei 2011

Alle hens aan dek voor het eindverslag! Nog 24u te gaan en nog heel wat werk. Alle documenten werden samengevoegd en da layout van het eindverslag werd opgemaakt. De teamleden werkte tot 3u 's nachts door om het verslag in orde te maken.

13 mei 2011

Het eindrapport werd nog vlug gecontroleerd door de teamleden: de laatste grammatica en spelfouten werden eruit gehaald en problemen met de layout werden opgelost. Het verslag werd op tijd doorgemaild naar de coach en op de wiki gepost. Ook werd er een papiere versie overhandigd aan de coach. Dit was dan het laatste onderdeel (op de test na) van EE4.

title: User:Wouter~enwikiversity/BlogGSC


text:
Talk:Psycholinguistics/Aphasia

Peer Review
Your page seems to be very well constructed and I like that the introduction is very clear and is relevant to what the reader will continue to read. The topics seem to be in a great logical order so that the reader can easily follow along. I really like the table you used to compare the various aphasias as it makes it clear to the reader what the differences are between different aphasias. Although I think you should probably cite the table. I feel like it may also be of benefit to you to include citations throughout your page so that persons reading your page will be able to find or refer to the material without looking through all of the references to find the applicable knowledge. It says in the rubric that the citations need to be linked to the references at the bottom of the page. You will need to take out ...., and anything in bold fill in with the correct info. This is the format I used to do mine (I only put .... there so wiki would not convert it) <....ref name="Author, Year">put the APA reference here and any citations after that you would use <....ref name="Author, Year"> at the bottom of your page under your reference section put <....references/> once and Wikiversity will automatically put all your references there and link them to your citation that you placed throughout your page. This is an example of one of my citations <....ref name="Pang, 2003">Pang, E., Muaka, A., Bernhardt, E., Kamil, M. (2003). Teaching Reading. Brussels, The International Academy of Education. and to reference it again I used <....ref name="Pang, 2003"> There are a few other ways people have done this so you can look at their pages too or use the help button when editing your page to figure it out. I did a bit of editing on your page and there are some terms that you may want to explain to the reader a bit more or link to a Wikipedia page so the reader can easily access the terms they do not understand. Also, I believe we are not supposed to cite the textbook but rather use the references that the textbook used which you can find in the back of the textbook or deem it common knowledge and not cite it at all. It was good idea to link the treatment to another persons page. I think it would be beneficial to the reader to have examples of what each type of aphasia would look like. There is a place where you can add pictures to your page as well, which may also enhance the appearance of your page. Maybe a picture of where the Broca and Wernicke’s area are would be helpful to the reader. Here is a link to a picture of both areas if you wanted to add it to your page http://commons.wikimedia.org/wiki/File:Brain_Surface_Gyri.SVG You may also want to find a few more sources or something of interest to the reader that was not in the textbook as well. Other than those few things your page is well organized and looks similar to other pages on Wiki so that is great so have fun with the different tools on Wiki!

Example References
<....references/> Stepheny 18:49, 20 February 2011 (UTC)

title: Talk:Psycholinguistics/Aphasia


title: File:Funkadoodle101.png


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Observational astronomy/Meteor observing

A Meteor, or Shooting Star, is a streak of light seen in the sky. It is caused by a small particle of interplanetary material (a Meteoroid) hitting the Earth's atmosphere and burning up.

Occasionally, an exceptionally large meteoroid will not burn up completely, and part will hit the ground. This is called a Meteorite.

Meteors are of two types:

 * Sporadic meteors: These can occur at any time of year and in any part of the sky. They tend to be more frequent just before dawn rather than earlier in the night, because at that time the observer is facing towards the Earth's direction of motion around the Sun and so meteoroids strike head on.

 * Meteor showers: These are due to clouds of meteoroids in orbit round the Sun. Such clouds are believed to be expelled from comets, and most of them can be identified with a particular comet.  The Earth will pass through a cloud once a year, and observers will see an enhanced rate of meteors.  Due to perspective, the paths of meteors from a shower will appear to diverge from a point in the sky known as the shower radiant.  Many showers are named after the constellation in which the radiant is located.  For example, the Leonids (pictured) have their radiant in Leo.  (Their parent comet is Tempel-Tuttle.)

Guidelines on Measuring Meteor Activity

It is very valuable to have a record of the level of meteor activity, hour by hour and day by day, particularly at the time of major showers so that their activity can be monitored. For example, showers are often more active when their associated comet is near perihelion (the point in the orbit of a planet or comet where it is nearest to the sun). Monitoring meteors can be done easily by amateurs with minimal equipment, but it does require commitment, as observing sessions need to last for several hours.

Observe the sky from a place as far as possible from lights, as these can drown out faint meteors. Wrap yourself with warm clothing, especially in winter, though you can feel chilled even apparently warm summer nights and it may get colder after midnight. Sit in a deck chair or lie flat on the ground so that you can see a large part of the sky easily. It is essential to be comfortable as you will be observing for a long time. It is a good plan to have two or three observers in the same spot, so that the whole sky is covered between them. With enough people, you can have observers to work on rotation basis to share the effort.

Before observing, try to decide how bright are the faintest visible stars. This will help those analyzing the observations to assess whether an apparently high or low meteor rate is distorted by particularly good or bad observing conditions. Ideally, re-assess this visibility limit at intervals through-out the night.

Make a note of each meteor you see, and the approximate time so you can deduce hourly rates. If there is a shower, try to see if the meteor is headed away from the position of the radiant; if not, it is a sporadic meteor and should be recorded separately so as not to inflate estimates of shower activity. Note if it is particularly bright or otherwise unusual. For example, some meteors have a flare due to the meteoroid exploding rather than burning up steadily. Notes may be taken on paper or a tape recorder. It would be ideal to have a person do this separately from the observers.

To make these observations of value, they must be reported regularly to some group that can collate and publish them. Some of the leading groups have their web sites listed below.

Zenithal Hourly Rate
If the radiant of a shower is at a low altitude, the meteors will tend to be further away and seen through a greater thickness of atmosphere, hence the rate of meteors will appear to be lower. The less the altitude, the fewer meteors will be seen. To complicate matters further, the altitude of the radiant will vary during the night. Shower rates are often expressed as the Zenithal Hourly Rate or ZHR, the rate to be expected if the radiant were at the zenith. If you are inexperienced, there is no need to worry; the collating group will be able to do the adjustment. To get the ZHR, multiply the observed rate by 1.25 if the altitude of the radiant is 52º, 1.67 (altitude 35º), 2 (altitude 27º), 5 (altitude 8.6º) and 10 (altitude 2.6º).

Observing activities
To find out when the next meteor shower will occur see the Weekly Meteor Activity Outlook.


text:
User:Jef753

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title: User:Jef753


text:
A Translation of the Bible/Apocrypha

Apocrypha (Greek Apokruphos, "hidden") are books regarded as part of the Old Testament by many Christians but not regarded as Biblical by Jews or Protestants. The term was invented by St. Jerome ("Hierome") to refer to books that were regarded as part of the Old Testament by Roman Catholics but which he discovered were not accepted by the Jews. The rule of the Church of England is given in the sixth of the 39 Articles:

 * And the other Books (as Hierome saith) the Church doth read for example of life and instruction of manners; but yet doth it not apply them to establish any doctrine; such are these following:

 * The Third Book of Esdras,
 * The Fourth Book of Esdras,
 * The Book of Tobias,
 * The Book of Judith,
 * The rest of the Book of Esther,
 * The Book of Wisdom,
 * Jesus the Son of Sirach,
 * Baruch the Prophet,
 * The Song of the Three Children,
 * The Story of Susannah,
 * Of Bel and the Dragon,
 * The Prayer of Manasses,
 * The First Book of Maccabees,
 * The Second Book of Maccabees.

They are included in many editions of the King James Bible between the Old and New Testaments, but not in most Protestant Bibles. Except for the first two, they are included in Roman Catholic bibles as integral parts of the Old Testament. The third and fourth books of Esdras are now normally called the first and second books; what were the first two books are the books of Ezra and Nehemiah.

The Wikisource Bible wrongly includes other books (3 Maccabees, 4 Maccabees, Psalm 151) in the Apocrypha, although these books have never been recognised by the Church of England as part of the Apocrypha or included in the King James Version or any Roman Catholic Bibles. This error appears to have arisen because some printings of the Revised Standard Version and the New Revised Standard Version have an "expanded" collection of "The Apocrypha/Deuterocanonical Books". However, it is clear from The Oxford Companion to the Bible, edited by Bruce Metzger, the chairman of the NRSV translators, that these books are not part of the standard Apocrypha.

title: A Translation of the Bible/Apocrypha


text:
User:Mattisse

 * Technical Writing Making Wiki Tables
 * Learning the basics of French/Simple Grammar Rules
 * Web design

title: User:Mattisse


text:
User talk:Nobelium

 Hello Nobelium, 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

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.
 * Explore Wikiversity with the links to your left.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

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) 19:01, 20 February 2011 (UTC)

title: User talk:Nobelium


text:
Elementary Math 3/1

Numbers (to ten)
A number is a symbol that represents how many objects there are in a set.

A number can be written as a symbol (1, 2, 3, 4, 5, 6, 7 ...) or a word (one, two, three, four, five, six, seven ...)

 1,  "one"    -

 2, "two"  -

 3, "three"  -

 4, "four"  -

 5, "five"  -

 6, "six"  -

 7, "seven"  -

 8, "eight"  -

 9, "nine"  -

 10, "ten"  -

Math Symbols
Plus sign (+)  +  The addition symbol tells you to combine the two numbers on either side. 2 + 2 is the same as saying "two plus two" Minus sign (-)  -  The subtraction symbol tells you to take away the number on the right from the number on the left. 2 - 1 is the same as saying "two minus one"

title: Elementary Math 3/1


text:
User talk:Jamie.Alban

 Hello Jamie.Alban, 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

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.
 * Explore Wikiversity with the links to your left.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

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) 19:33, 20 February 2011 (UTC)

title: User talk:Jamie.Alban


text:
User:Mf2011

A student in the Higher Education and Leadership Department with a concentration in Media Literacy at Appalachian State University.

title: User:Mf2011


text:
User:Thefamousmoe

Personal background: I am from a small town in North Carolina. I grew up in a log cabin and spent most of my youth interacting with media in some way. I didn't have any neighbors my age, so to entertain myself I would watch tv or play video games. I suppose that is why I am so interested in how the media shapes (and in turn is shaped by) its audience.

Professional/Academic background: I received a BS from Appalachian State University in Electronic Media / Broadcasting. I am currently pursuing my masters in Media Literacy.

Information/Technology background: I do not have any professional experience in this field, but I do have a significant amount of theoretical experience from the classes I have taken.

Funny story or interesting item about yourself to remember you by: I directed, produced, and edited an 8-minute documentary about zombies.

Anything else you'd like to share: I am in the process of revising my paper, "Middle Eastern Representation in Post 9/11 Video Games", so that it may be submitted for publication.

title: User:Thefamousmoe


text:
Talk:Sociolinguistics/Introduction

Textbook Review: Language and Society

Research The author carefully selected sources that pertain to the subject matter. Although the sources are few in number, they provide a lot of scientific evidence to support each section of the chapter. The author selected both articles and books, providing well rounded evidence to support the chapter. When revising the chapter, I suggest that the author focuses on finding some current articles to include in the chapter. Many of the sources in this chapter are out of date (1996, 1989, 1990 etc). Because society is constantly changing, I think it is important to report the most up to date information. I conducted a brief literature review on the topic and did note that there is not a lot of published research in this area. An interesting area of research that you may want to investigate when looking for current literature is “sociolinguistics”. See link:

Organization The headings and subheadings that were selected divide the chapter in a logical manner. This makes the development of the chapter and each subtopic very clear to the reader from the outset. The subheadings clearly indicate the content of that section of the chapter and provide a well-rounded description of the topic. I also liked the inclusion of a table. This helps to break up the text and provides a good summary of the results of Well’s study. If possible, I would suggest that the table is enlarged. Even when opened in another screen I found that the text was too small to read. Answers Questions The chapter does an excellent job of using research to support and explain concepts. The topics of slang, taboo and sexual language are well-presented. To improve on the content of the chapter, I would recommend adding some information about the role of language in society from a more general perspective, before you focus in on the topics of slang, taboo, etc. history of taboo words. In the section on the history of taboo words, I think that you should include a few more examples of taboo. The explanation of history skips from biblical times to 1966; I think it would be useful to explain the progression in a bit more detail.

Writing Style This chapter is well written. The writing style is sophisticated, yet very logical and easy to follow. I liked the use of numbered lists within the text. These make the text easier to understand. I did not note any grammatical or spelling mistakes. The only area of the text that I would suggest further revision to is the first few sentences. I found the sentence structure of these paragraphs to be a bit confusing. I would also verify the word count of the chapter (min 2500 words).

Structure I would suggest that you re-work the introduction and/or body of the chapter to match the outline that you present in the introduction. The introduction says that the chapter will discuss how “psychological, social and affective meaning are related to one’s cultural lexicon”. However, this is not touched on in the chapter. I would suggest that you add a paragraph with more general information re: language and society if you keep the introduction as it is current written. If do you not add another section, I would suggest rewriting the last section of the introduction so that it is clear that your chapter will focus on slang, sexual language and taboo. I am unsure if this would be appropriate for the wiki format, but I think that it would be useful to include a conclusion or summary paragraph to wrap up the chapter. Arichter 21:35, 20 February 2011 (UTC)

title: Talk:Sociolinguistics/Introduction


text:
User:Nhaas

27 years old Married, mother of two Teacher Graduate Student

title: User:Nhaas


text:
User:Dawid Deutschland

 * Meta.
 * In georgischer Wikipedia.
 * In georgischem Wikiwörterbuch.
 * In deutscher Wikipedia.

title: User:Dawid Deutschland


text:
Talk:Mean

Further info
For further information about any school-level math, just contact me. Special thanks to my math teacher,--;Hiba;1 18:18, 23 February 2009 (UTC)

title: Talk:Mean


text:
Digital Age

The Digital Age course was designed primarily for students studying INFT 1018: Exploring the Digital Age at the University of South Australia, but the materials are being developed here under a free content license and thus are open to anyone. The course is an overview of a number of loosely related technology issues, ranging from subjects such as ePublishing through to Artificial Intelligence and definitions of knowledge. Thus it represents a diverse and ever-changing subject area.

Instructor
The primary instructor here is Adam Jenkins, or Bilby as he is known here, from the University of South Australia's School of Computer and Information Science.

Materials
Materials generally consist of provided text and recommended readings, with some research questions to be considered at the end of each part. It is hoped that the text will be developed over time by those involved in both teaching and learning the material.

Answering the research question, whether openly through the discussion pages, or privately, should help further your understanding of the material. Enrolled students will also have assessment questions based on the subject area.

Course plan

 * Moore's Law: Discusses Moore's Law and some of the issues such a rate of technological development may raise.
 * Technology Acceptance Model: Introduces the Technology Acceptance Model (or TAM) as a possible framework to investigate the potential uptake and impact of new technologies.
 * Web 2.0: is it nothing more than a marketing term, or is there something significant in the concept?
 * Massively Multiplayer Game: We have to do games at some point, just for the fun if nothing else, and here we look at the history of the multiplayer variety.
 * Knowledge Management and Wikis: What is Knowledge Management, and how do Wikis fit into our understanding of knowledge?

title: Digital Age


text:
Digital Age/Technology Acceptance Model

Predicting when – and understanding why – users accept a new technology or system is a major issue for Information Systems (IS) researchers. Web sites, as one example, have as a key feature their interactivity. But for them to work users must choose to interact with the site. And thus web developers should always keep in mind the factors which would encourage such choice being made. Various models have been proposed which purport to predict or explain the level of acceptance that users may have for a new system, and thus, in the case of web sites, the likelihood that they will engage with and use the site. One of the more influential models is the Technology Acceptance Model, or TAM as it is more commonly known.

TAM Version 1
According to TAM, whether or not an individual is inclined to use a given product is dependent on two factors – the perceived usefulness of the system, and the perceived ease of use. Davis defines perceived usefulness as:

 * "[T]he degree to which a person believes that using a particular system would enhance his or her job performance." (Davis, 1989, pp 320)

It is interesting that Davis chooses to tie "usefulness" to work, but this doesn't appear to be a necessary aspect of the definition. The other key factor, perceived ease of use, is defined by Davis as:

 * "[T]he degree to which a person believes that using a particular system would be free of effort" (Davis, 1989, pp 320)

This is a more straight-forward definition, and doesn’t tend to raise significant concerns. However, it is worth noting the "believes" part of the two definitions. What is significant here is not the actuality – whether the system really is easy to use or is genuinely useful – but how the user perceives these aspects of the system.

These two factors – perceived ease of use and perceived usefulness – combine to form an "intention to use". Based on the presence of such an intention, the user will have a particular attitude towards using the system. And this attitude will affect their actual system use. This is modeled in Figure 1. Figure 1 also includes a connection between perceived usefulness and perceived ease of use – TAM argues that the degree of usefulness is, in part, dependent upon how easy the system is to use. A system which is difficult to use is simply not going to be very useful, as it will take too much effort to reap the rewards. Figure 1 also shows two other interesting aspects: first, it is clear that perceived usefulness not only directly influences a person’s intent to use the system (BI) but that it also directly affects someone’s attitude to the system; and second, both the perception of usefulness and the perception of ease of use will be influenced by external factors (an issue which is explored in more detail in TAM2).

In practice, TAM tells us that we need to make sure that any given system finds the correct level of each of these two factors. A system that is extremely useful may, for example, be accepted by users, even if it is difficult to use. Equally, a system that is not particularly useful may still be employed if it is very easy to use. But a system which receives high marks for both perceived ease of use and perceived usefulness is one that will tend to be the most readily accepted. Real world examples abound: personally, I’ve never found eBay to be particularly easy to use, yet it is so useful that I’m willing to put the effort into working with the system, in spite of the difficulties that it entails. Google is both very easy to use and extremely useful: a combination which led it to become the most widely used search engine online. And, until the recent redesign, the Channel Nine website always seemed to me to be neither particularly useful (in terms of what I wanted it to do) nor easy to use, and thus I chose to go elsewhere when looking for information.

TAM 2
The "External Variables" in Figure 1 are interesting. As pointed out by Moon & Kim, Davis "argued that future technology acceptance research needs to address how other variables affect usefulness, ease of use, and user acceptance" (Moon & Kim, 2001, pp 217). These are the external variables in TAM, and the exploration of these was a task that many researchers were happy to undertake. Moon & Kim (2001) went on to explore the issue of "playfulness", while Lin & Lu (2000), in applying TAM to web sites, look at "IS quality" as a factor for both perceived usefulness and perceived ease of use, focusing on three aspects: information quality, response time and system accessibility.

In 2000, Venkatesh and Davis offered TAM2: an extension of the original TAM which adds additional factors which may influence the perceived usefulness of a system. This extended model is presented in Figure 2.

Figure 2 will go here

Venkatesh and Davis define "Subjective Norm" as, effectively, "peer pressure' – if other people believe that someone should use (or not use) a given system, that person’s willingness to use it will be affected.

 * Voluntariness is the degree to which a person may choose to use (or not use) a given system. For example, enrolling online at a university is not something which is particularly voluntary – if you wish to attend the university you may have to enroll online, whether or not you wish to. But using a discussion forum may be voluntary. (Unless, of course, a mark is attached to your use of the forum, in which case it becomes less voluntary).
 * Image refers to how a person is seen within a group – if using a system is good for his or her "image", the person may perceive the system as being "more useful" than otherwise.
 * Experience refers to ongoing use of a given system – the more experience someone has, the more likely they are to continue to use it.
 * Job relevance returns to the earlier definition of "perceived usefulness" in TAM – if something is clearly relevant to a person’s job, they will view it as "more useful" than something which isn’t.
 * How well the system performs tasks is measured as "output quality" by Venkatesh and Davis (2000, pp 191). A system that performs well is likely to be seen as more useful that a system which does not, all else being equal.
 * Finally, result demonstrability refers to the degree by which the benefits of using a system can be seen.

Research questions

 * Can you think of some popular technologies and fit them into this equation. Are they easy to use? Are they useful?
 * Can something be both difficult to use and not very useful, yet still accepted? If so, why do you think they are employed?
 * Let's consider some specific technologies - how do they fit into this scheme?
 * SMS on mobile phones
 * Email
 * Skype
 * Driving a car
 * Wikipedia/Wikiversity (for reading)
 * Wikipedia/Wikiversity (for writing/editing)
 * eBooks

 * In each case, is there a possible distinction between "perceived" usefulness and ease of use, as opposed to "actual" usefulness and ease of use?
 * Sometimes TAM (especially version 1) seems remarkably simple. Is it possible that the model is too simple to be useful? IN which case, could TAM fail TAM?
 * Both models seem to be missing any notion of fun - perhaps some technologies are successful because they are enjoyable. Should fun be included in the models, or is it already encompassed within them?
 * Perceptions are either builds up collecting tangibles and intangibles of technologies. Driving factors for people to shape the perception activly (both positive and negative) tend to be inflenced by Personal belief, Organisational Norm, Degree of personal involvement towards the generic and specific areas of technonlogies (assuming technology as Tangible and Intangible). IF we borrowing notion of perception, the model should understand dynamics of PECEPTION build, change and some being fixated. Well, there can be strong attitude built, related to the desired perception for the given tech, there is a job to be done how to inflence those attitudes in conjuction with full adoption by one individual and goups of such individuals. How can we add this complexity into the model?

title: Digital Age/Technology Acceptance Model