text:
User:Egm6341.s10.Team4/HW7

= Problem 1: Linearization about Equilibrium Point $$\displaystyle \hat{x} = x_{max}$$ =

Given
Refer Lecture slide 38-4 for problem statement

The logistic equation for population dynamics is given by 38-3,

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$$\displaystyle \dot{x}(t) = f(x) = rx \left(1-\frac{x}{x_{max}}\right) $$
 * $$\displaystyle \longrightarrow (1) $$
 * }.
 * }.

Find
Linearize Equation (1) about the equilibrium point $$\displaystyle \hat{x} = x_{max}$$.

Solution
With a small perturbation $$\displaystyle y $$, the value of $$\displaystyle x $$ is given by,

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$$\displaystyle \begin{align} x &= \hat{x} + y \\ &= x_{max} + y \end{align} $$
 * }.
 * }.

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$$\displaystyle \begin{align} \Rightarrow \frac{dx}{dt} &= \frac{d(x_{max} + y)}{dt} \\ &= \frac{dy}{dt} \end{align} $$
 * }.
 * }.

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$$\displaystyle \begin{align} \Rightarrow \frac{dy}{dt} &= r(x_{max}+y) \left(1-\frac{(x_{max}+y)}{x_{max}}\right) \\ &= r(x_{max}+y) \left(-\frac{y}{x_{max}}\right) \\ &= -r(y+\frac{y^2}{x_{max}})

\end{align} $$
 * }.
 * }.

Since, $$\displaystyle y $$ is a small perturbation compared to $$\displaystyle x_{max} $$,

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$$\displaystyle \begin{align} \Rightarrow \frac{dy}{dt} &= -r(y+\cancelto{0}{\frac{y^2}{x_{max}}})\\ &= -ry \end{align} $$
 * }.
 * }.

And so, the solution

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$$\displaystyle \begin{align} \Rightarrow y = y_0 e^{-rt} \Rightarrow &y(t) \to 0 \ as\ t \to \infty \\ \Rightarrow &x(t) \to x_{max} \ as\ t \to \infty \end{align} $$
 * }.
 * }.

Author
Solved and typed by - Egm6341.s10.Team4.andy 06:16, 23 April 2010 (UTC)

Reviewed by - Egm6341.s10.Team4.roni 17:58, 23 April 2010 (UTC)

= Problem 2: Analytical equation for $$\displaystyle x(t) $$ in Newton-Raphson method =

Given
The relationship between population growth $$\displaystyle x $$ respect to time $$\displaystyle t $$ can be expressed as follows:

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \dot x=\frac{dx}{dt}=rx(1-\frac{x}{x_{max}})

 * }
 * }

Find
Find the analytical equation to show $$\displaystyle x $$ as a function of time $$\displaystyle t $$:

Solution

 * $$\displaystyle

\dot x=\frac{dx}{dt}=rx(1-\frac{x}{x_{max}}) $$

 * $$\displaystyle

\Rightarrow \frac{dx}{rx(1-\frac{x}{x_{max}}})=dt $$

For LHS we have:

 * $$\displaystyle

\frac{1}{x(1-\frac{1}{x_{max}}x)}=\frac{a}{x}+\frac{b}{1-\frac{1}{x_{max}}x}=\frac{a-\frac{a}{x_{max}}x+bx}{x(1-\frac{1}{x_{max}}x)} $$

 * $$\displaystyle

\Rightarrow a-\frac{a}{x_{max}}x+bx=1\Rightarrow x(b-\frac{a}{x_{max}})+a=1 $$

 * $$\displaystyle

\Rightarrow a=1 $$

 * $$\displaystyle

\Rightarrow b-\frac{a=1}{x_{max}}=0\Rightarrow b=\frac{1}{x_{max}} $$

 * $$\displaystyle

\Rightarrow \frac{1}{x(1-\frac{1}{x_{max}}x)}=\frac{1}{x}+\frac{\frac{1}{x_{max}}}{1-\frac{1}{x_{max}}x} $$

 * $$\displaystyle

\Rightarrow \frac{1}{r}\int\limits_{x_0}^{x}\frac{1}{x(1-\frac{1}{x_{max}}x)}\, dx=\frac{1}{r}\bigg[\int\limits_{x_0}^{x}\frac{1}{x}\,dx+\frac{1}{x_{max}}\int\limits_{x_0}^{x}\frac{1}{1-\frac{1}{x_{max}}x}\,dx\bigg]=\int\limits_{t_0}^{t}\,dt $$

 * $$\displaystyle

\Rightarrow \frac{1}{r}\bigg[ln(x)-ln(1-\frac{1}{x_{max}}x)]\Bigg ]^{x}_{x_0}=\frac{1}{r}\bigg[ln(\frac{x}{1-\frac{x}{x-{max}}})-ln(\frac{x_0}{1-\frac{x_0}{x_{max}}})\Bigg]=\int\limits_{t_0}^{t}\,dt $$

 * $$\displaystyle

\Rightarrow \frac{1}{r}ln(\frac{\frac{x}{1-\frac{x}{x_{max}}}}{\frac{x_0}{1-\frac{x_0}{x_{max}}}})=\frac{1}{r}ln(\frac{x(1-\frac{x_0}{x_{max}})}{x_0(1-\frac{x}{x_{max}})})=\int\limits_{t_0}^{t}\, dt=t-t_0 $$

Initial condition:

 * $$\displaystyle

t_0=0\Rightarrow x=x_0. $$

 * $$\displaystyle

\Rightarrow ln(\frac{x(1-\frac{x_0}{x_{max}})}{x_0(1-\frac{x}{x_{max}})})=r(t-t_0) $$

 * $$\displaystyle

\Rightarrow e^{rt}=\frac{x(1-\frac{x_0}{x_{max}})}{x_0(1-\frac{x}{x_{max}})}=\frac{x}{1-\frac{x}{x_{max}}}\times\frac{1-\frac{x_0}{x_{max}}}{x_0}=\frac{x_{max}.x}{x_{max}-x}\times\frac{x_{max}-x_0}{x_{max}.x_0} $$

 * $$\displaystyle

\Rightarrow \frac{x^2_{max}.x-x_{max}.x_0.x}{x^2_{max}.x_0-x_{max}.x_0.x}=e^{rt} $$

 * $$\displaystyle

\Rightarrow (x^2_{max}.x-x_{max}.x_0.x)=(x^2_{max}.x_0-x_{max}.x_0.x)e^{rt} $$

 * $$\displaystyle

\Rightarrow x(x^2_{max}-x_0.x_{max})=(x^2_{max}.x_0)e^{rt}-x_{max}.x_0.e^{rt}.x $$

 * $$\displaystyle

\Rightarrow x(x^2_{max}-x_0.x_{max}+x_{max}.x_0.e^{rt})=x^2_{max}.x_0.e^{rt} $$

 * $$\displaystyle

\Rightarrow x=\frac{x^2_{max}.x_0.e^{rt}}{x^2_{max}-x_0.x_{max}+x_{max}.x_0.e^{rt}}=\frac{x_{max}.x_0.e^{rt}}{x_{max}-x_0+x_0.e^{rt}} $$

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$$\displaystyle \Rightarrow x(t)=\frac{x_0.x_{max}.e^{rt}}{x_{max}+x_0(e^{rt}-1)} $$
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 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style= |
 * }.
 * }.

Author
Solved and typed by - Egm6341.s10.Team4.nimaa&m 15:50, 11 April 2010 (UTC) .

= Problem 3: Hermite-Simpson Rule to interpret the logistic Equation =

Given
Logistic Equation is given by $$\displaystyle \dfrac{dx}{dt}=rx(1-\dfrac{x}{x_{max}})$$

where $$\displaystyle x_{max}=10; r=1.2; x_0=2,7; t\in[0,10]$$

Find
Use Hermit Simpson rule to interpret the above equation.

Solution
This problem has been solved by writing a matlab code.

Hermit Simpson Code:

Case(1) : $$ \displaystyle x_0 = 2.0 $$ Plot: .

Case(2) : $$ \displaystyle x_0 = 7.0 $$ Plot: .

Author
Solved and typed by - Egm6341.s10.team4.anandankala 14:49, 23 April 2010 (UTC)

Proof read by -

= Problem 4: Discrete Population Dynamics - Logistic Map - Chaotic System - Period Doubling =

Given
Class notes and Pages 454-456 King et al 2003 - An Introduction to Chaotic Systems.

Given factor r -represents the growth from generation to the next [[media:Egm6341.s10.mtg39.djvu|39-3]]

Find
Run the given Code to reproduce plots 15.6 and 15.7 in King et al 2003.

Solution
original Code

Matlab Code:

Plot 1 shows the results from running the original code

Modification to the original Code

Matlab Code:

Plot 2 shows the results from running the modified code

Discussion:

From Plot 2 One can see that when r=4, a small change in the input of the initial x value would give apparently a random nature of solution and divergence after about 50 iterations. The 0 marker in blue are for x=0.1 starting point and the x markers in green are for x=0.1+1e-16. Initially (up to about 50 iterations) the x and o symbols fall one on to of the other (giving same solution) and diverging after about 50 iterations.

Author
Solved and typed by - Egm6341.s10.Team4.roni 18:49, 18 April 2010 (UTC)

Proof read by -

= Problem 5: Integration of logistic equation with $$h=2^kh $$=

Given
Refer Lecture slide 42-2 for problem statement

The logistic equation

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

\frac{d\overline{x}}{dt} = rx\left(1- \overline{x}\right) $$ where,
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

$$\displaystyle \overline{x} = \frac{x}{x_{max}} $$

Find
Integrate logistic equation already found $$\hat{h}$$ such that Hermit-simpson algorithm yields error($$ 10^{-6}$$)

1) Run Hermit-Simpson with $$\displaystyle h= 2^k \hat{h}$$

2) Develop and Run Forward Euler with $$\displaystyle h= 2^k \hat{h}$$

3) Develop and Run Backward Euler with $$\displaystyle h= 2^k \hat{h}$$

Solution
1) Run Hermit-Simpson algo $$\displaystyle h= 2^k \hat{h}$$

See problem 3

2) Develop and Run Forward Euler with $$\displaystyle h= 2^k \hat{h}$$

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\begin{align} &\frac{d\overline{x}}{dt} = r\overline{x}\left(1- \overline{x}\right)\\ &\frac{\overline{x}_{i+1}-\overline{x}_i}{h}= r\overline{x_i}\left(1- \overline{x_i}\right)\\ &\overline{x}_{i+1}= hr\overline{x}_i\left(1- \overline{x}_i\right)+\overline{x}_i\\ &\overline{x}_{i+1}= (hr+1)\overline{x}_i-hr\overline{x}^2_i\\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

MATLAB CODE:

3) Develop and Run Backward Euler with $$\displaystyle h= 2^k \hat{h}$$

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\begin{align} &\frac{d\overline{x}}{dt} = r\overline{x}\left(1- \overline{x}\right)\\ &\frac{\overline{x}_{i+1}-\overline{x}_i}{h}= r\overline{x}_{i+1}\left(1- \overline{x}_{i+1}\right)\\ &-\overline{x}_{i}= hr\overline{x}_{i+1}\left(1- \overline{x}_{i+1}\right)-\overline{x}_{i+1}\\ &=>hr\overline{x}^2_{i+1}+(1-hr)\overline{x}_{i+1}-\overline{x}_{i}= 0\\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

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

\begin{align} &\overline{x}_{i+1}=\frac{hr-1 + \sqrt{(1-hr)^2+4hrx_i}}{2hr}\\ &\overline{x}_{i+1}=\frac{hr-1 - \sqrt{(1-hr)^2+4hrx_i}}{2hr}\\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

MATLAB CODE:

Conclusion

As the step size($$ 2^kh$$) increase, the Forwaord Euler method shows the unstable properties.

on the other hands, The Hermit simpson and Backworkd Euler shows the relatively stable properties.

Author
Solved and typed by - Egm6341.s10.Team4.yunseok 19:42, 23 April 2010 (UTC) .

= Problem 6 - Hermite-Simpson Integration of Coupled ODE's =

Given
This is a flight control problem. The dynamics of flight are well documented and will not be described in detail here.

General Method:

By using collocation to force our Hermite-Simpson interpolation to exactly solve the given ODE's at the mid point $$ \displaystyle z_{i+1/2}$$, we find that our time stepping method takes the form

$$ \displaystyle \mathbf{z_{i+1}}-\mathbf{z_{i}}=\frac{\tfrac{h}{2}}{3}(\mathbf{f_i}+4\mathbf{f_{i+1/2}+\mathbf{f_{i_1}}})$$

Additionally, using our interpolation $$ \displaystyle \mathbf{f_{i+1/2}}=\tfrac{1}{2}(\mathbf{z_i}+\mathbf{z_{i=1}}+\tfrac{h}{8}(\mathbf{f_i}-\mathbf{f_{i+1}}))$$.

However, no exact solution exists in order to determine $$ \displaystyle \mathbf{z_{i+1}}$$. Given $$ \displaystyle \mathbf{z_{i}}$$, the Newton-Rhapson method can be employed in order to find $$ \displaystyle \mathbf{z_{i+1}}$$. By converting $$ \displaystyle \mathbf{z_{i+1}}-\mathbf{z_{i}}=\frac{h}{6}(\mathbf{f_i}+4\mathbf{f_{i+1/2}+\mathbf{f_{i_1}}})$$ to $$ \displaystyle \mathbf{F(z_i,z_{i+1})}=\mathbf{0} $$, the Newton-Rhapson method takes the form

$$ \displaystyle \mathbf{z_{i+1}^{k+1}}=\mathbf{z_{i+1}^{k}}- \begin{bmatrix}\mathbf{\frac{dF(z_i,z_{i+1}^{k})}{dz_{i+1}}}\end{bmatrix}^{-1}\mathbf{F(z_i,z_{i+1}^{k})} $$

where

$$ \displaystyle \mathbf{F(z_i,z_{i+1})}= \mathbf{z_{i+1}}-\mathbf{z_{i}}- \frac{h}{6}(\mathbf{f_i}+4\mathbf{f_{i+1/2}+\mathbf{f_{i_1}}})=0 $$

Equations of Motion:

Let

$$ \displaystyle \begin{bmatrix} z \end{bmatrix} = \begin{bmatrix} x \\ y \\ v \\ \gamma\end{bmatrix}$$

The equations of motion that we are concerned with are

$$ \displaystyle \begin{bmatrix} \dot{z} \end{bmatrix} = \begin{bmatrix} \dot{x} \\ \dot{y} \\ \dot{v} \\ \dot{\gamma} \end{bmatrix} = \begin{bmatrix} v cos \gamma \\ v sin \gamma \\ \frac{1}{m}((T-D) cos \alpha - L sin \alpha - g sin \gamma)\\ \frac{T-D}{m}sin \alpha+\frac{L}{mv}cos \alpha -\frac{g}{v}cos \gamma \end{bmatrix} $$

The physical parameters for which we will solve using are

$$ \displaystyle D=\tfrac{1}{2}C_d\rho v^2 S_{ref} $$

$$ \displaystyle L=\tfrac{1}{2}C_l\rho v^2 S_{ref} $$

where

$$ \displaystyle C_d=a_1\alpha^2+a_2\alpha+a_3 $$

$$ \displaystyle C_l=b_1\alpha+b_2 $$

$$ \displaystyle \rho=c_1y^2+c_2y+c_3 $$

$$ \displaystyle a_1=-1.9431 \tfrac{1}{rad^2}$$

$$ \displaystyle a_2=-0.1499 \tfrac{1}{rad}$$

$$ \displaystyle a_3=0.2359 $$

$$ \displaystyle b_1=21.9 \tfrac{1}{rad}$$

$$ \displaystyle b_2=0.0 $$

$$ \displaystyle c_1=3.312E-9 \tfrac{kg}{m}$$

$$ \displaystyle c_2=-1.142E-4 \tfrac{kg}{m^2}$$

$$ \displaystyle c_3=1.224 \tfrac{kg}{m^3}$$

The initial conditions are

$$ \displaystyle x(t=0)=0.0 m$$

$$ \displaystyle y(t=0)=30.0 m $$

$$ \displaystyle v(t=0)=272.0 \tfrac{m}{s}$$

$$ \displaystyle \gamma(t=0)=0.0 rad $$

The flight is controlled using the angle of attack and the thrust, which are shown below

Find
We are asked to determine x,y,v, and \gamma using the Hermite-Simpson/Newton-Rhapson Method and then verify this with MATLAB's ode solver ode45.

Solution
In performing this analysis, we see that our Jacobian, $$ \displaystyle |\mathbf{\tfrac{dF}{dz_{i+1}}}|$$ can be described as

$$ \displaystyle \mathbf{\frac{dF(z_{i},z_{i+1})}{dz_{i+1}}}= \mathbf{\frac{dz_{i+1}}{dz_{i+1}}}-\mathbf{\frac{dz_{i}}{dz_{i+1}}}- \frac{h}{6}(\mathbf{\frac{df_{i}}{dz_{i+1}}}+4\mathbf{\frac{df_{i+1/2}}{dz_{i+1}}}+ \mathbf{\frac{df_{i+1}}{dz_{i+1}}})$$

Using the interpolation for z_{i+1/2} as a function of z_{i} and z_{i+1} described above, this becomes

$$ \displaystyle \mathbf{\frac{dF(z_{i},z_{i+1})}{dz_{i+1}}}= \mathbf{\frac{dz_{i+1}}{dz_{i+1}}}-\mathbf{\frac{dz_{i}}{dz_{i+1}}}- \frac{h}{6}(\mathbf{\frac{df_{i}}{dz_{i+1}}}+4\mathbf{\frac{df_{i+1/2}}{dz_{i+1/2}}\frac{dz_{i+1/2/2}}{dz_{i+1}}}+ \mathbf{\frac{df_{i+1}}{dz_{i+1}}})$$

In evaluating this equation, our Jacobian can be described as

$$ \displaystyle \mathbf{\frac{dF(z_{i},z_{i+1})}{dz_{i+1}}}= \mathbf{I}- \frac{h}{6}(3\mathbf{I}-\frac{h}{2}\mathbf{\frac{df_{i+1}}{dz_{i+1}}}) \mathbf{\frac{df_{i+1}}{dz_{i+1}}}$$

Thus, as long as $$ \displaystyle \tfrac{df}{dz} $$ can be calculated, the Jacobian can be found.

The obvious next step is to calculate $$ \displaystyle \tfrac{df}{dz} $$. The full expansion of $$ \displaystyle \tfrac{df}{dz} $$ can be seen to be

$$ \displaystyle

\frac{\mathbf{df}}{\mathbf{dz}}= \begin{bmatrix} 0 & 0 & cos(\gamma) & -v sin(\gamma) \\ 0 & 0 & sin(\gamma) & v cos(\gamma) \\ 0 & \frac{1}{m}(-\frac{dD}{dy}cos\alpha -\frac{dL}{dy}cos\alpha)& \frac{1}{m}(-\frac{dD}{dv}cos\alpha -\frac{dv}{dy}cos\alpha & \frac{1}{m}(-gcos\gamma)\\ 0 & \frac{1}{mv}(\frac{dD}{dy}sin\alpha+\frac{dL}{dy}cos\alpha)& -\frac{T}{mv^2}sin\alpha -\frac{1}{m}\frac{d}{dv}(\frac{D}{v})cos\alpha+\frac{1}{m}\frac{d}{dv}(\frac{L}{v})cos\alpha+\frac{g}{v^2}cos\gamma & \frac{g}{v}sin\gamma\\ \end{bmatrix}

$$

In performing the evaluations we can see that the two forms of numerical integration produce a very similar output.

Source Codes

Main Program

Calculate the Jacobian

Calculate f

Function for ode 45 to use

Author
Solved and typed by Egm6341.s10.Team4.riherd 15:24, 23 April 2010 (UTC)

= Problem 7: Logistic Equation: Inconsistent (Trapezoidal) Simpson's rule & Newton-Raphson =

Given
Refer Lecture slide 41-2 for problem statement

The logistic equation for population dynamics is given by 38-3,

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

$$\displaystyle \dot{x}(t) = f(x) = rx \left(1-\frac{x}{x_{max}}\right) $$
 * $$\displaystyle \longrightarrow (1) $$
 * }.
 * }.

Here, $$\displaystyle x_{max}=10;\ r = 1.2,\ $$ and $$\displaystyle t \in [0,10] $$

The analytical solution is given by,

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

$$\displaystyle x(t) = \frac{x_0x_{max}e^{rt}}{x_{max} + x_0(e^{rt}-1)} $$ $$
 * $$\displaystyle \longrightarrow (2)
 * }.
 * }.

Find
To solve logistic equation defined by Equation (1) for two initial conditions, $$\displaystyle x_0=2\ and\ x_0 = 7,\ $$, using Inconsistent(Trapezoidal) Simpson's rule given by,

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$$\displaystyle z_{i+1} = z_{i} + \frac{h/2}{3} \left[f_i + 4f_{(i+1/2)} +f_{i+1}\right] $$
 * $$\displaystyle \longrightarrow (3) $$
 * }.
 * }.

The Inconsistent rule (Trapezoidal) is given by,

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$$\displaystyle z_{i+1/2} = \frac{1}{2} [z_{i} + z_{i+1}] $$
 * }.
 * }.

Solution
Consider the Equation (1) in which $$\displaystyle f_{(i+1/2)}$$ is a function of $$\displaystyle z_{i}$$ and $$\displaystyle z_{i+1}$$, where $$\displaystyle z_i, z_{i+1} \rightarrow$$ values of $$\displaystyle x$$ at $$\displaystyle t = t_i$$ and $$\displaystyle t = t_{i+1}$$.

Since, this is an initial value problem, $$\displaystyle z_i$$ is known. So, Equation (3) as a whole becomes a function of $$\displaystyle z_{i+1}$$ given by,

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

$$\displaystyle F(z_{i+1}) = 0 $$
 * $$\displaystyle \longrightarrow (4) $$
 * }.
 * }.

We find the root $$\displaystyle z_{i+1}$$ of Equation (4) using Newton-Raphson method 40-3,

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

$$\displaystyle z_{i+1}^{(k+1)} = z_{i+1}^{(k)} - \left[\frac{dF(z_{i+1}^{(k)})}{dz}\right]^{-1} F(z_{i+1}^{(k)}) $$ $$
 * $$\displaystyle \longrightarrow (5)
 * }.
 * }.

The Newton-Raphson iteration, starting with an initial guess as $$\displaystyle z_{i+1}^{0} = z_{i}$$, is stopped once the absolute tolerance

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

$$\displaystyle \begin{align} AbsTol &= ||z_{i+1}^{(k+1)} - z_{i+1}^{(k)}|| \\ &\le 10^{-6} \end{align} $$
 * }.
 * }.

is satisfied.

In the below Matlab code, for each of the h values $$\displaystyle x_{Actual} $$ is the vector calculated from Equation (3) in order to calculate the error max (for reference), $$\displaystyle e_{Max} = || x_{Actual} - x ||_{\infty} $$ where $$\displaystyle x $$ is calculated from Inconsistent Simpson's rule - Newton Raphson Method described above, for the given interval $$\displaystyle t \in [0,10] $$.

Matlab Code:

$$\displaystyle x_0 = 2 $$ :- A comparison on $$\displaystyle h $$ values:

The plot for $$\displaystyle h \approxeq 0.001 $$:

$$\displaystyle x_0 = 7 $$ :- A comparison on $$\displaystyle h $$ values:

The plot for $$\displaystyle h \approxeq 0.001 $$:

Author
Solved and typed by - Egm6341.s10.Team4.andy 05:49, 23 April 2010 (UTC) Reviewed by - Egm6341.s10.Team4.roni 18:00, 23 April 2010 (UTC)

= Problem 8: Rate of momentum change for optimal control problem =

Given
Envisage the below figure as free body diagram of an aircraft:

Find
Show that $$\displaystyle \bar{CD}=\bar{AB} $$ + higher order terms of $$\displaystyle d\gamma $$.

Solution
According to the above figure, we can survey these two cases at $$\displaystyle t $$ and $$\displaystyle t+dt $$, the velocity of the aircraft after $$\displaystyle dt $$ will reach to $$\displaystyle v+dv $$ and the angle between the aircraft and horizontal axis will reach to the $$\displaystyle \gamma+d\gamma $$. Thus, for two generated triangles $$\displaystyle \triangle OAB $$ and $$\displaystyle \triangle ODC $$ following relationships can be written:

 * $$\displaystyle

tan(d\gamma)=\frac{\bar {AB}}{\bar {OA}}=\frac{\bar {AB}}{v} $$

Likewise we have below relationship in $$\displaystyle \triangle ODC $$:

 * $$\displaystyle

sin(d\gamma)=\frac{\bar {CD}}{\bar {OD}}=\frac{\bar {CD}}{v+dv} $$

Hence, by applying Taylor series for these two cases:

 * $$\displaystyle

tan(d\gamma)=d\gamma+\frac{d\gamma^3}{3}+\frac{2d\gamma^5}{15}+...=\frac{\bar{AB}}{v} $$

 * $$\displaystyle

\Rightarrow \bar{AB}=v(d\gamma+\frac{d\gamma^3}{3}+...)=v.d\gamma+v\frac{d\gamma^3}{3}+v\frac{2d\gamma^5}{15}+... $$

 * $$\displaystyle

sin(d\gamma)=d\gamma-\frac{d\gamma^3}{3!}+\frac{d\gamma^5}{5!}+...=\frac{\bar{CD}}{v+dv} $$

 * $$\displaystyle

\Rightarrow \bar{CD}=(v+dv)(d\gamma-\frac{d\gamma^3}{3!}+\frac{d\gamma^5}{5!}...)=v.d\gamma-v\frac{d\gamma^3}{3!}+v\frac{d\gamma^5}{5!}+... $$

 * $$\displaystyle

\Rightarrow \bar{CD}-\bar{AB}=-\frac{1}{2}vd\gamma^3-\frac{1}{8}vd\gamma^5... $$

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

$$\displaystyle \Rightarrow \bar{CD}=\bar{AB}+v.f(d\gamma^3)+v.f(d\gamma^5)+... $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style= |
 * }.
 * }.

Author
Solved and typed by - Egm6341.s10.Team4.nimaa&m 22:19, 22 April 2010 (UTC) .

= Problem 9: Parameterization of ellipse=

Given
$$\displaystyle x=a cost; y=b sint.$$

Prove That
$$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2}=1$$

Solution
We have $$\displaystyle x=acost; y=bsint.$$

$$\displaystyle x=a cost \Rightarrow x^2=a^2(cost)^2$$

$$\displaystyle y=b sint \Rightarrow y^2=b^2(sint)^2$$

$$\displaystyle \Rightarrow \frac{x^2}{a^2}=(cost)^2 and \frac{y^2}{b^2}=(sint)^2 $$

$$\displaystyle \Rightarrow \frac{x^2}{a^2}+ \frac{y^2}{b^2}=(cost)^2+(sint)^2 $$

$$\displaystyle \Rightarrow \frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1 $$

Author
Solved and typed by - Egm6341.s10.team4.anandankala 14:50, 23 April 2010 (UTC) .

= Problem 10: Parameterization of an Ellipse =

Given
Result from HW problem 7.9 [[media:Egm6341.s10.mtg42.djvu|42-1]], which is:

$$\displaystyle Eq. (1)  \qquad    \frac{x^2}{a^2}+ \frac{y^2}{b^2}=1 $$

Show
That the arc length dl comes to:

HW 7.10 on slide [[media:Egm6341.s10.mtg42.djvu|42-2]]

$$\displaystyle

Eq. (2) \qquad  dl= \sqrt{dx^2+dy^2} $$

Solution
Geometrically:

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

 * [[Image:RP_HW_7_10c.png]]
 * }.
 * }.
 * }.

Eq. (3) \qquad  dl^2= {dx^2+dy^2} $$

Eq. (4) \qquad  dl=  \sqrt{dx^2+dy^2} $$

and also:

Eq. (5) \qquad  dl =   \left( \sqrt{1+  \left( \frac{dy}{dx}\right)^2} \right) dx

$$

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

$$\displaystyle dl= \sqrt{dx^2+dy^2} $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

 * style= |
 * }
 * }

Author
Solved and typed by - Egm6341.s10.Team4.roni 18:50, 18 April 2010 (UTC) Reviewed by -

= Problem 11: Circumference of Ellipse =

Given
Lecture notes [[media:Egm6341.s10.mtg42.djvu|p.42-2]]

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

dl = [dx+dy]^{1/2}\,dt$$
 * $$\displaystyle
 * $$\displaystyle

 * $$ \displaystyle x = a cost $$

 * $$ \displaystyle y = b sint $$

 * }
 * }

Find

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

$$\displaystyle C = \int dl = a \int_{0}^{2\pi} [1-e^2cos^2t]^{1/2}\,dt$$ where, Eccentricity, $$ \displaystyle e = \Bigg[1-\frac{b^2}{a^2}\Bigg]^{1/2} $$
 * Show that,
 * Show that,
 * }
 * }

Solution

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

$$\displaystyle \frac {dy}{dt} = b cos t $$  and
 * $$\displaystyle \frac {dx}{dt} = -a sin t $$
 * $$\displaystyle \frac {dx}{dt} = -a sin t $$
 * }
 * }

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

\begin{align} dl &= [dx^2 + dy^2 ]^{1/2}\\ &= \Bigg[\bigg(\frac{dx}{dt}\bigg)^2 + \bigg(\frac{dy}{dt}\bigg)^2 \Bigg]^{1/2} dt\\ &=\Bigg[(-a sin t)^2 + (b cos t)^2\Bigg]^{1/2} dt\\ &= \Bigg[(a^2 sin^2 t) + (b^2 cos^2 t)\Bigg]^{1/2} dt\\ &= a \Bigg[sin^2 t + \left(\frac{b^2}{a^2} cos^2 t\right)\Bigg]^{1/2} dt\\ &= a \Bigg[sin^2 t + cos^2 t + \frac{b^2}{a^2} cos^2 t- cos^2 t \Bigg]^{1/2} dt\\ &= a \Bigg[1 -\left(1-\frac{b^2}{a^2}\right) cos^2 t \Bigg]^{1/2} dt\\ &= a \Bigg[1 - e^2 cos^2 t \Bigg]^{1/2} dt\\ \end{align} $$ So,
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.
 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle C = \int_{0}^{2\pi} dl = a \int_{0}^{2\pi} [1-e^2cos^2t]^{1/2}\,dt$$
 * }
 * }

Author
Solved and typed by - Egm6341.s10.Team4.yunseok 19:42, 23 April 2010 (UTC) .

= Problem 12 - Manipulation of Elliptical Integral =

Given
$$ \displaystyle C=a\int_{t=0}^{2\pi}\sqrt{1+e^2cos^2t} dt $$

Find
We are asked to show that $$ \displaystyle C=a\int_{t=0}^{2\pi}\sqrt{1+e^2cos^2t} dt =4 a\int_{t=0}^{\tfrac{\pi}{2}}\sqrt{1+e^2sin^2\alpha} d\alpha $$

Solution
Let $$ \displaystyle t=cos^{-1}(sin\alpha)$$ and $$ \displaystyle dt=cos\alpha\frac{1}{\sqrt{1-sin^2\alpha}}d\alpha=cos\alpha\frac{1}{cos\alpha}d\alpha=d\alpha $$.

Putting this into our equation,

$$ \displaystyle C = a \int_{t=0}^{2\pi} \sqrt{1+e^2 sin^2 \alpha} d\alpha $$

The function $$ \displaystyle \sqrt{1+e^2 sin^2 \alpha}$$ is periodic over a period of $$ \displaystyle \pi$$ and in this period the function is even around the point $$ \displaystyle \alpha=\tfrac{\pi}{2}$$. Thus, by symmetry, this integral can be broken up into 4 parts of equal magnitude such that

$$ \displaystyle C= 4 a\int_{t=0}^{\tfrac{\pi}{2}}\sqrt{1+e^2sin^2\alpha} d\alpha $$

Author
Solved and typed by Egm6341.s10.Team4.riherd 15:50, 23 April 2010 (UTC)

= Problem 13: Chebychev - Change of Variable =

Given
Refer Lecture slide 42-2 for problem statement

The integral for a function $$\displaystyle f(x)$$ is,

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

$$\displaystyle I = \int_{-1}^{+1}f(x)dx $$
 * $$\displaystyle \longrightarrow (1)$$
 * }.
 * }.

between the interval $$\displaystyle x \in [-1,1]$$

Find
With a change of variable $$\displaystyle x = cos\theta$$, show that the integral in Equation (1) is,

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

$$\displaystyle I = \int_{0}^{\pi} f(cos\theta)sin\theta d\theta $$
 * }.
 * }.

Solution
With a change of variable $$\displaystyle x = cos\theta \Rightarrow dx = -sin\theta d\theta ; cos(\pi) = x = -1 \Rightarrow \theta = \pi; cos(0) = x = 1 \Rightarrow \theta = 0$$ Equation (1) is,

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

$$\displaystyle \begin{align}

I &= \int_{-1}^{1} f(x)dx \\ &= \int_{\pi}^{0} f(cos\theta) \left(-sin\theta d\theta \right) \\ &= - \int_{\pi}^{0}f(cos\theta) sin\theta d\theta \\ &= \int_{0}^{\pi}f(cos\theta) sin\theta d\theta \\

\end{align} $$
 * }.
 * }.

Author
Solved and typed by - Egm6341.s10.Team4.andy 06:32, 23 April 2010 (UTC)

= Problem 14: Obtaining $$\displaystyle a_k $$ in cosine fourier transform =

Given
The Cosine series as Fourier transform for function $$\displaystyle f(x) $$ is expressed as follows:

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle f(cos\theta)=\frac{a_0}{2}+\sum_{k=1} a_k.cos(k\theta)

 * }
 * }

Find
Demonstrate the following expression for $$\displaystyle a_k $$:

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle a_k=\frac{2}{\pi}\int\limits_{0}^{\pi} f(cos\theta).cos(k\theta)\, d\theta

 * }
 * }

Solution
Considering $$\displaystyle x=cos(\theta) $$, cosine Fourier series for $$\displaystyle f(x) $$ is written as:

 * $$\displaystyle

f(x=cos(\theta))=\frac{a_0}{2}+\sum_{k=1} a_k.cos(k\theta) $$

 * $$\displaystyle

\xrightarrow{\times cos(k\theta)}f(x).cos(k\theta)=\frac{a_0}{2}.cos(k\theta)+\sum_{k=1} a_k.cos^2(k\theta) $$

 * $$\displaystyle

\xrightarrow{\int\limits_{0}^{\pi}}\int\limits_{0}^{\pi}f(x).cos(k\theta)\, d\theta=\frac{a_0}{2}\int\limits_{0}^{\pi}cos(k\theta)\, d\theta+\sum_{k=1} a_k\int\limits_{0}^{\pi}cos^2(k\theta)\, d\theta $$

 * $$\displaystyle

\frac{a_0}{2}\times\underbrace{\frac{1}{k}[sin(k\theta)]_{0}^{\pi}}_{=0}+\sum_{k=1}a_k\int\limits_{0}^{\pi}cos^2(k\theta)\, d\theta=\sum_{k=1}a_k\int\limits_{0}^{\pi}cos^2(k\theta)\, d\theta $$

 * $$\displaystyle

cos(2\theta)=cos^2(\theta)-sin^2(\theta)=2cos^2(\theta)-1\Rightarrow cos^2(\theta)=\frac{cos(2\theta)+1}{2} $$

 * $$\displaystyle

\Rightarrow \int\limits_{0}^{\pi}f(x).cos(k\theta)\, d\theta=\sum_{k=1} \lbrack a_k\times\frac{1}{2}\int\limits_{0}^{\pi}(cos(2\theta)+1)\, d\theta\rbrack=\frac{1}{2}\sum_{k=1}a_k\lbrack\int\limits_{0}^{\pi}cos(2\theta)\, d\theta+\int\limits_{0}^{\pi}\, d\theta]=\frac{1}{2}\sum_{k=1}a_k\lbrack[\underbrace{\frac{1}{2}sin(2\theta)]_{0}^{\pi}}_{=0}+\underbrace{[\theta]_{0}^{\pi}}_{\pi}\rbrack=\frac{\pi}{2}\sum_{k=1}a_k $$

 * $$\displaystyle

\Rightarrow a_k=\frac{2}{\pi}\int\limits_{0}^{\pi}f(x).cos(k\theta)\, d\theta $$

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

$$\displaystyle a_k=\frac{2}{\pi}\int\limits_{0}^{\pi}f(x).cos(k\theta)\, d\theta $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style= |
 * }.
 * }.

Author
Solved and typed by - Egm6341.s10.Team4.nimaa&m 19:15, 18 April 2010 (UTC)

Reviewed by - Egm6341.s10.Team4.roni 18:02, 23 April 2010 (UTC)

= Problem 15: Integral Evaluation=

Given
$$\displaystyle f(cos\theta)= \frac{a_0}{2} + \sum_{k=1}^{\infty} a_kcos(k\theta)$$

Prove that
$$\displaystyle I= a_0 + \sum_{j=1}^{\infty} \frac{2a_{2j}}{1-(2j)^2}$$

Solution
$$\displaystyle I= \int_{-1}^{1}f(x)dx= \int_{0}^{\pi}f(cos\theta)sin\theta d\theta$$

$$\displaystyle \Rightarrow I= \int_{0}^{\pi}(\frac{a_0}{2}+ \sum_{k=1}^{\infty}a_kcos(k\theta))sin\theta d\theta$$

$$\displaystyle \Rightarrow I= \int_{0}^{\pi} \frac{a_0}{2} sin\theta d\theta + \sum_{k=1}^{\infty}\int_{0}^{\pi} a_kcos(k\theta))sin\theta d\theta $$

$$\displaystyle \Rightarrow I= \dfrac{a_0}{2} [cos\theta]_{\pi}^{0}+ \sum_{k=1}^{\infty}\int_{0}^{\pi} \dfrac{a_k}{2} [sin((k+1)\theta) -sin((k-1)\theta)]d\theta $$

$$\displaystyle \Rightarrow I= a_0+ \sum_{k=1}^{\infty}\dfrac{a_k}{2} [\int_{0}^{\pi} sin((k+1)\theta)d\theta-\int_{0}^{\pi} sin((k-1)\theta)d\theta] $$

$$\displaystyle \Rightarrow I= a_0+ \sum_{k=1}^{\infty} \dfrac{a_k}{2} [[\dfrac{cos((k+1)\theta)}{(k+1)}]_{\pi}^{0} -[\dfrac{cos((k-1)\theta)}{(k-1)}]_{\pi}^{0}]$$

Since all the cosine terms for odd values of k become zero. The above expression is modified as below.

$$\displaystyle \Rightarrow I= a_0+ \sum_{j=1}^{\infty} \dfrac{a_{2j}}{2} [[\dfrac{cos((2j+1)\theta)}{(2j+1)}]_{\pi}^{0} -[\dfrac{cos((2j-1)\theta)}{(2k-1)}]_{\pi}^{0}]$$

$$\displaystyle \Rightarrow I= a_0+ \sum_{j=1}^{\infty} \dfrac{a_{2j}}{2} \dfrac{2}{(2j+1)}]-[\dfrac{2}{(2k-1)}$$

$$\displaystyle \Rightarrow I= a_0+ \sum_{j=1}^{\infty} a_{2j}\dfrac{1}{(2j+1)}]-[\dfrac{1}{(2j-1)}$$

$$\displaystyle \Rightarrow I= a_0+ \sum_{j=1}^{\infty} \dfrac{2a_{2j}}{1-(2j)^2}.$$

Hence the proof.

Author
Solved and typed by - Egm6341.s10.team4.anandankala 14:51, 23 April 2010 (UTC) .

= Contributing Team Members =

Egm6341.s10.Team4.andy 06:35, 23 April 2010 (UTC) - Author of Problems:1, 7, 13 ; Proof Read: 2, 3, 14, 15

Egm6341.s10.Team4.nimaa&m 14:30, 23 April 2010 (UTC) - Author of Problems:2, 8, 14 ; Proof Read:15

Egm6341.s10.Team4.riherd 15:25, 23 April 2010 (UTC) - Author of Problem 6, 12. Proof read 10, 11.

Egm6341.s10.Team4.roni 18:04, 23 April 2010 (UTC) - Author of Problem 4, 10; Proof Read: 1,7,4.

Egm6341.s10.team4.anandankala 14:49, 23 April 2010 (UTC)- Author of Problem 3, 9, 15; Proof Read:2, 13, 14.

Egm6341.s10.Team4.yunseok 19:40, 23 April 2010 (UTC) - Author of Problem 5, 11, Proof Read:3

title: User:Egm6341.s10.Team4/HW7


text:
User talk:Kense~enwikiversity

Greeting! My name is Nuttapon Chansrikesorn or you can call me 'Knot'. I am in Environmental science major.

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Kense. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Kense~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:35, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:13, 19 April 2015 (UTC)

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

title: User talk:Kense~enwikiversity


text:
User:Egm6341.s10.team4.anandankala/HW7-3

=Hermit-Simpson Rule to interpret the logistic Equation=

Given
Logistic Equation is given by $$\displaystyle \dfrac{dx}{dt}=rx(1-\dfrac{x}{x_{max}})$$

where $$\displaystyle x_{max}=10; r=1.2; x_0=2,7; t\in[0,10]$$

Find
Use Hermit Simpson rule to interpret the above equation.

Solution
This problem has been solved by writing a matlab code.

Hermit Simpson Code:

Case(1) : $$ \displaystyle x_0 = 2.0 $$ Plot:

.

Case(2) : $$ \displaystyle x_0 = 7.0 $$ Plot:

.


text:
User talk:Smita tendulkar

 Hello Smita tendulkar, 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. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 07:17, 23 April 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 07:17, 23 April 2010 (UTC)

title: User talk:Smita tendulkar


text:
User talk:Thanida09

 Hello Thanida09, 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. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 07:17, 23 April 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 07:17, 23 April 2010 (UTC)

Which is the best smart phone operating system and why?
There are many kinds of smart phone operating system; however, I think the IPhone 3G which is based on version of OSX works best.

The reason is because it can zoom in and zoom out the screen while opening the Internet browser by using HTC touch Dual. Zooming in and out was equally easy using the two-fingered pinch motion. Since Safari uses the entire screen and automatically rotates to landscape mode when the phone is turned 90 degrees, many Web pages were readable without the need to zoom at all.Browsing, using the iPhone's version of Apple's Safari, is very similar to browsing on a desktop, a claim none of the other smart phones in this group can make. Pages rendered accurately and quickly

title: User talk:Thanida09


text:
User talk:Jitiwan518

Hello! I'm Jitiwan Nitiyothin I'm a 2nd year student. My ID number is 5180458. my birthday is 27 February 1990.

Homework: Cloud Computing 1. How is cloud computing different than using a normal computer? ANS: The differences between the Cloud computing and a normal computer is that, the cloud computing allows us to use applications and share files on the internet. Also, anyone can access to those at anytime on any computers. In contrast, a normal computer, the files we make and save will be privately storedd on the computer we use, and they can't be accessed by people that use other computers.

2. How could you turn a classroom into a Cloud computing classroom? ANS:The first step for turning a classroom into a cloud computing classroom is that everyone must create their own accounts where we can upload and share files on the internet. The account will allow us to share and access other people’s works, such as friends’ document or teachers’ lectures, at anytime and from any computers. By having a cloud computing classroom, it will be convenient for students to share ideas and learn things from their classmates and teachers. For example, if any student post a question or explaination, other students also can get benefit from the post.

Copyright problem with File:Jitiwan.jpg
Homework What is the best smartphone marketer? ans In my opinion, I think blackberry is the best smartphone marketer. There are 4 reasons to support my opinion. Firstly, the E-mail service of Blackberry is the best. This is because it has intelligent push email service which will help users to immediately download the message and has LED indicates when a new message is received. Secondly, every version of blackberry have a full QWERTY keyboard which will help users to type faster and more accurate same as using keyboard on notebook. Thirdly, a camera of blackberry is quite good because it has many functions such as auto-focus, white balance, flash, image stabilization, and variable zoom. Lastly, using Blackberry is more suitable for teenagers who like to chat because there are many chatting programs available such as BBM, MSN, IM, and Skype. Moreover, facebook and twitter can be downloaded for using in Blackberry which help users to widely and quickly contact with others at the same time.

title: User talk:Jitiwan518


text:
User talk:Thanchanok518

Greeting! I'm Thanchanok Jintanatanyachat (Noey) ID 5180319

Questions,, 1. How is cloud computing different from using a normal computer?  Ans. The difference is that using a cloud computing allows users to access information or data from the same database from everywhere in every time. However, using a normal computer, user cannot access or share information to others wherever and whenever they want. This is because user must use the same computer to share or access it. Therefore, using cloud computing is more comfortable and convenience than using the normal computer. For example, user can use Google Document, E-mail to access or share information and data.

2. How could you turn a classroom into a cloud computing classroom? Ans. To turn a classroom into a cloud computing classroom, we then using internet as medium. We have to create a classroom account on a particular website, for example, Google Document, to share information to other classmate to access.

title: User talk:Thanchanok518


text:
User:MrMcDermott

Greetings.

I'm here for the benefit of knowledge and am thankful for those of you who share it willingly.

Over the period of the next year I will be using Wikiversity, Wikibooks, and Wikipedia to learn material pertaining to the subject

of Computer Science. I'm going to be looking for work instead of attending school. I plan to pick up a net book in the near future

do to the fact that the city I live in down town is wired with hot spots.

Mr McDermott

title: User:MrMcDermott


text:
User talk:MrMcDermott

 Hello MrMcDermott, 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. 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.

 * 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.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. Jtneill - Talk - c 08:00, 23 April 2010 (UTC)

 Hello , you can call me Jacob Frank. I'd like to be your mascot and welcome you to Wikiversity. Even though I was a plain and untutored man back in 18th-century Europe, that still didn't stop me from contributing my ideas to the world. Neither should it stop you. As a religious leader who developed the doctrine of "purification through transgression," my advice to you is to be bold and to ignore all rules—yes, these are two of Wikiversity's official policies! Happy editing.

title: User talk:MrMcDermott


text:
User:Escalad3

HELLOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

IM PETE ID: 5180738 I love Computer and technology :) but sadly i use a Mac

Answers for social networking

Q1: What site you selected?

A: DIGG

Q2: What is the site's purpose and why is it considered a web 2.0 site?

A: Digg is the site that people can share or post anything. For instance, we can post any news, picture, video, links and informations. It's also the place that every people can view the posted topic and they can share their thoughts and opinion. In conclusion, this site helps people in communicating with each other in a particular topic.

Q3: What is your favorite website and why?

A: Personally, I like the web that I've done which is digg.com. What I like about this site is that any people can share things that are interesting or things that they are interested in. In addition, people can view and discuss about that topic and we are able to know other people by discussing the topic.

The Best Pc in The World

Alienware Area-51 ALX

Motherboard                     >Asus P6T WS Pro

Processor                           >Intel Core i7 960

Memory                              >6 GB DDR3

Case                                   >Alienware Area 51 ALX

Hard drive                         >1.2 TB

CD/DVD/? >24x DVD+/-RW

Keyboard                           >Razer Tarantula

Mouse                                >Razer Mamba

Monitor                              >LG W2486L

Sound card                        > Integrated 7.1

Video card                         >Dual ATI Radeon HD 5870     With 2 GB Video Memory

title: User:Escalad3


text:
User talk:U5280331

Hi, My name is Nataporn Suwanpanont or you can call me Ormmii. I'm 1st year student. ID 5180331 My account of wikiversity is u5280331

Answer the questions 26/04/10
1.) How is cloud computing defferent than using a normal computer?     Normal Computer        - off-line working        - update by user        - individual access information      Cloud Computing        - on-line working, through internet        - software is updated automatically        - storage and share information on internet        - can access information by anyone and anywhere

2.) How cloud you turn a classroom into a cloud computing classroom?     In a classroom. Suppose each person is a cloud computing. When teacher assign works to students, they can update their works or information to other students by presenting their work in front of class. That suppose like automatic update information to public user in cloud computing. Morover, other students can give comments which can compare with sharing information that any can comment in cloud computing.

My Image 26/04/10

Hello!!!! Ormmii is my name**

Ps. sorry that I use this pic, but i have only one pic in my thumb drive :p

title: User talk:U5280331


text:
User:Phithiwat

Best Spec computer for me Graphic card NVIDIA GeForce GTX 480 1536 MB GDDR5 the best for playing game, it is HDMI support for super graphic 4 GB of RAM Windows XP because every game work on it but some don't work for Window 7 Intel i7 core to make OS work fast DVD RW ROM is enough (if you want to watch full HD movie then use blu-ray ROM and the latest version of Power DVD) CRT 22 inch monitor is cool for game 2TB HDD speed up to 10000 RPM and Port SATA This is best for game and Full HD movie --- BEST SMART PHONE OPERATING SYSTEM

iOS Free update of the OS version Take less than half GB of storage Keep the program running while making a phone call, (program doesn't interfere with the phone calling) easily manage with the program iTunes that can be install in every computer operating system Easy used interface with graphical program list Home screen where all applications are kept is easily access with one hardware button Application can be rearrange according to our want, user can also delete icon or program easily by holding the screen and touch X at the corner of the icon Having best multi motion touch screen and easy to use User can zoom in out using two finger or drag drop change pages easily While listing to music and phone call come the song will fade out and fade in when end call Screen will tilt with the phone Graphic of many application are similar to Mac PC Can connect to Wi-Fi and access internet without opening GPRS service ( if that area have Wi-Fi connection) Internet connection, phone call can be easily deactivate by the airplane mode It have virtual keyboard where user can set the type of keyboard by themself there a lot of good graphic and application available in iAppStore (there are ways to get those application free)

-- Social Networking due May 7 1.What is the site you selected? Meetup.com - Users: Phithiwat 2.What is the site's purpose and why is it considered a web 2.0 site? Purpose of the website is to arrane meeting for people of same interest within area that users are willing to travel, for example Bangkok Coffee and Movie Meetup, Bangkok Spanish Club, Bangkok Social Networking Meetup Group, etc. And it is a web 2.0 site because it is a web base community, it allowed users to meet with each other as content to the website. 3.What is your favorite website and why? My favorite website is www.2bbit.com, because there are lots of torrent files for me to download, there is a chat room for me to meet new friends and they are helpful, and there is forum for people to share infromation and knowledge with each other.

--

Home Work due on May 1, 2010 Introduction of myself MY PICTURE

Name: Phithiwat Phongwattanavichit Nick: Pop Age: 18 Status: single (I'm not a gay) Birth day: 11/9/1991 (11 Sep 1991) Nationality: Thai Region: Southern-Thailand Religion: Bhuddhist Place of birth: Bangkok Hobbies: car racing, watch cartoon, play consol, computer, and board game, and extream sports Car in use: All new Honda City 08 silver Thing I like: computer, game, all that stated in hobbies Pet Peeve: lizard Interest: computer, graphic, photo graph

---

Class work on April 26,2010 assignment on cloud computing

Ans 1: To share or work on the data on normal computer we have to many add on hard ware and software, in some case we need to have legal software also, and it uses physical transportaion to transfer data to another place or computer.Sometime we might have forgot where we put the data. But in cloud computing we use internet as a medium to transfer, share, and work on the data, so all the data can be found on the internet. Cloud computing is a lot more comfortable and cheaper kind of computing. There are sofware proovide on the website, so it save money from buying legle sofware by sacrifying much lower price of money to rent a space on the website or server. And data can be access anytime and anywhere if there is an internet connection.

Ans 2: We can turn a classroom into a cloud computing classroom by using server or website who offer software and storage online to let student access class material of the class anywhere and anytime. Student don't have to bother of coming late to class or missing class material, it all there on the website or server. We can conduct a normal classroof for a practical work that need real time instruction from teacher. Each student should have an account for the subject to avoid unwanted users to access and edit the class meterial.

http://en.wikiversity.org/wiki/User:Icns141

title: User:Phithiwat


text:
User talk:U5180385

left text center text right text Block quote center text Hi My name is Phupisit Smittinet or you can call me Warm. I'm 2nd year student. ID 5180385 My account of wikiversity is u5180385

Home Work Cloud computing is a new way to do your work on the computer. It involves working, storing and sharing information online. There are many differences in working with cloud computing when compared to normal computing. To begin with, cloud computing gives u access to information anytime, anywhere, and at any computer with just one condition, access to the internet. One great thing about cloud computing is the constantly updated software online. This save time and give the consumer the latest security, working and storing software. However because everything you do is online, information can never be safe. Even with the latest window defender or kaspersky you information can be hack. Furthermore, information that you store online can be physically damage or lost if the actually tangible server is damage (which happens before). Beside this, cloud computing is extremely slow if your computer and internet is slow, so only stable infrastructure are recommended. All in all, cloud computing can be a cheap way to share, work, and store information if you daily use it.

This website is one good start to cloud computing. It is access to teacher which mean it could be access to student. Because of information can be share, student could have a mutual benefits of class note, information and interest on the website. Cloud computing isn't a hard thing to do, but we just need many people to do it. What can a black berry messenger be with just a phone? it is share server that matter. So join now?

Hi

Social network website HW
What is the site you selected? Â DIGG What is the site's purpose and why is it considered a web 2.0 site? Like any wedsite, Digg is a social website to fit the interest of the people inorder to bring their share interest together. After the first 5 minute of looking through, i can see that this website is quite user friendly with main topic that fits most people interest. what i mean by this is that the topic they focus on are topic that many people shares interest. For example game and technology. After that i went to look at "about digg" on the site and they said that their main objective is to bring the people blog to public to make people have fun. What is your favorite website and why? I will let the group decide, because for me who doesn't like sharing much blog or main stream, mi blog wont be famous. =)

title: User talk:U5180385


text:
Career Exploration Using Second Life

Exploring Culture and Career Using 'Second Life'

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 * bgcolor="#b4b4b4" style="border: 0px #cccccc solid;" |Subject: Career Exploration
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 * bgcolor="#b4b4b4" style="border: 0px #cccccc solid;" |Sub-Subject: AVID, EOPS Counseling, Sociology, Cultural Anthropology, etc. (sub-subjects will requires lesson-plan editing)
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 * bgcolor="#b4b4b4" style="border: 0px #cccccc solid;" |Length/Duration: roughly 1 week pre-discussion and planning, added to 42 hours on Second Life (minimum)
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 * bgcolor="#b4b4b4" style="border: 0px #cccccc solid;" |Technologies Used: Second Life, Internet, Photoshop Elements (optional)
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Introduction
Career exploration is a necessity for both high school-aged and college-aged students. At times it is difficult for a person to know exactly what a career is like or just if s/he has an interest in a topic unless s/he gets to experience it first. That experience is unfortunately not something that comes by easily in real life due to -a lack of a program that offers such opportunity to students, - community unemployment levels, -community industry variety, -or available student guidance and/or low self-esteem, so why not do it virtually? The following lesson is an extensive 'career shadowing' process for students. In conjunction to this lesson of career exploration is also cultural exploration. Seeing what career you wish to follow is a wonderful experience in itself, but is that experience going to be different based on your physical appearance? In finding themselves, their values, their personality, skills, and interests, students are encouraged to ask themselves as they search, "does the online experience differ from the real-life experience when it comes to dealing/working/socializing with others and if so, how?" and to individually compare what they see in life to what they will experience virtually in completing this lesson.

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Goals and Purpose
The purpose of this lesson is to use college/career ready standards in addition to or instead of career shadowing. The desire is to offer goal setting skills, life exploration, and career search.

It's been researched that the online setting lacks "triggers of differences", that a person's race, culture, language, gender, etc. is ignored thereby implying that there are such cultural triggers in real life. This lesson will have students explore careers using a specified avatar and to record any experiences they had based on the looks of the avatar. This lesson may result in a real-life versus online-life differences (pros/cons) when looking for a job or choosing a career. {| cellpadding="5" cellspacing="5" style="width: 90%; background-color: inherit; margin-left: auto; margin-right: auto"
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Definition(s)
'Triggers of difference' according to Merryfield are described as "When they [teachers] hear a Chinese accent, see a Jordanian woman's head cover, observe a Brazilian's body language, smell curry on an Indian's breath, some Americans automatically register a consciousness of difference that may trigger discomfort, stereotypes, xenophobia, or recognition of their own ignorance of other cultures. The triggers of visual and aural differences often subconsciously make people uncomfortable or otherwise constrain people's abilities to listen, interact, and learn from others" (Merryfield, 2003, p. 160).

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Lesson Steps
Quote: "Closed online environments provide a secure place for people to take risks, share personal experiences, admit to the realities of prejudice and discrimination..." (Merryfield, 2003, p.158). CONTEXT After having picked a variety of career categories (variety depends on class number and personality test results), the teacher will break the class into either groups of 3 or 5, where each person will represent a particular race or cultural group. Note that this lesson plan may be used both as a project in career and life success as well as a case study. PREFACE Pre-step: Students will be offered a pretest, a questionnaire of sorts in order to gauge their "self-concepts". Students are then set up and signed up to Second Life before starting the Second Life project. With the avatar of a specific ethnicity, students will walk/fly/teleport to different settings and locations in Second Life for a least 1-2 hours a day on three separate occasions. During this time, students will visit specific locations (given by teacher) as well as miscellaneous destinations of their choosing. Students are to keep a journal where they are expected to catalog their experiences. The goals are: (1) to shadow a potential career, (2) to be exposed to career options, (3) to observe the type of treatment each avatar receives from its wanderings and (4) whether or not the student's perception of him/herself changes depending on his/her avatar.

Teacher study: Further experiences in Second Life during the project will continue to be recorded by each student. The teacher may also choose to keep a record of student behavior throughout the project using a "behavior checklist." Doing this will allow the teacher to observe whether or not the students' virtual experience of using "superior" versus "inferior" avatars affects their self-image since according to Merryfield's text "Like A Veil: Cross-cultural experiential learning online", the virtual experience lacks "triggers of differences" while PBS's 2008 Frontline report "A Digital Nation" mentions how positive self-concepts attained from one would argue, superior, avatars in the virtual world, carries to real life. After the project, students will repeat the "self-concepts" test followed by a self reflection. PROJECT Step 1. Students complete the Self Concepts Questionnaire.

Step 2. Complete a personality test (either Copes, Cops, Caps [for Interest, Values, and Skills], ASVAB, or Myers Briggs) in order to get an idea of careers that would match the individual.

Step 3. Select with teacher encouragement, at least three job categories and locate areas/businesses in Second Life, that fit the careers options chosen.

Step 4. Complete an entire length of training/internship/course or at least 42 hours online. This will be recorded by offering the teacher a series of pictures (be sure that date/time are included in image) of exploration completed at the approved training posts, along with login hours for proof.

Step 5. Complete outside research. For each job selected, find the pay rate, level of education necessary, duties etc. In doing the research, students are required to interview a professional in the careers they are exploring and virtually experiencing. Suggested interview questions will be provided by the teacher and the professional being interviewed may be the career 'mentor' (person training student(s) on career) on Second Life or a person within the community.

Step 6. Have each student present/report back to the class their findings, experience, level of difficulty, and career goals set based on online experiences.

Step 7. Students retake Self-Concepts Questionnaire.

Step 8. Only now do students discover that the teacher recorded behavioral and/or self-concepts changes (if any) of students while continuing the project. Teacher presents students with the data recorded and opens the floor up for discussion in regards to the presence or lack of 'triggers of difference'.

TO DISCUSS: (1) Were students of differing avatars treated differently from one another during their Second Life career exploration? (2) Does the 'real world' versus Second Life provide people with more, less, or the same amount of prejudice in the workplace? (3) Should 'triggers of differences' deter me from persuing a particular career option? (4) What advocacy groups, programs, and/or businesses offer support for persons of varying ethnicities in persuance of 'nontraditional' career options?

For the questions students are unable to answer, the teacher will provide them with prompts, suggestions, and contact information for that community.

Step 9. Reflect back to values, interests etc. fulfilled by the job exploration as well as the overall experience and what was learned from personal exploration and what other students with different avatars went through.

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Activity Details
AVATAR ETHNICITIES The class will break into groups of either 3 or 5 where each student within the group will possess an avatar of a different ethnicity. This may also be done with a focus on height, clothing, hair, or any other physical trait or look that can constitute a trigger of difference.' ABOUT SECOND LIFE and REGISTERING Click Here to learn more about this process. SELF-CONCEPTS QUESTIONNAIRE A teacher created questionnaire may be offered to students that ask such as questions as: Rating from 1 to 4 where 1 is Not True, 2 is Moderately True, 3 is True, and 4 is Very True, answer the following... (1) On the whole, I am satisfied with myself. (2) I feel I have much to be proud of. (3) I'm always apologizing for things. (4) If I can't do something well, there is no point in doing it at all. Otherwise, students can be guided to take the test online on such sites as: a) How is Your Self Esteem? at Queendom Length: 79 Questions > > Click Here b)  How is Your Self Esteem? at Psychology Today Length: 79 Questions > > Click Here c) Self Esteem Quz at Self Esteem Awareness Length: 15 Questions > > Click Here PLACES TO VISIT: EXPLORING BEFORE THE PROJECT Before getting started with the project, it is important to get a feel for one's surroundings.  For this reason, students will be encouraged once they sign up to Second Life and create their Avatar, to explore a series of at least 8 locations during 3 separate 1-2 hour long sessions on their own.  The initial exploration will take place as a group guided tour with the teacher.  >> Night Club >> Fraternity/sorority house or meeting place >> University or college setting >> Church or religious setting >> Store or shop (2 different types such as a grocery store, a department store etc.) >> Office environment >> cafe (or area of socialization thta is similar to cafe)  JOURNAL ENTRY EXAMPLE Date:  ... Time (start and end times):  ... Second Life image(s) of session (print screen image for example)  ... I went to (LOCATION NAME) at (COORDINATES). - Location description and rating: ... - What I saw:  ... - What I heard:  ... - What I learned:  ... - How the experience affected me and why: ... (ex) "The online discussions are like a veil that protects me and Yang; I feel safe..." (Merryfield, 2003, p.146). CAREER INTERESTS AND VALUES A. Myers-Briggs Online The Myers Briggs test is a known personality test whose "...customized report will assist you with career planning, goal setting, making decisions, time management, understanding your social interactions and much more!" The link above is one that offers the test through "Your Life's Path", a career counseling and consulting firm. Please note that the test is available at several online locations including myersbriggs.org B. ASVAB Career Exploration As described on the site, "THE ASVAB CAREER EXPLORATION PROGRAM is a career planning and exploration program that combines a multiple-aptitude test with an interest self-assessment and a wide range of career exploration tools. And it's FREE to participating schools!" CAREER LOCATION EXAMPLE - Angola University (Animation, DJ'ing, Realtor, Achitecture). Opportunites are further described in a separate section of the lesson titled "Angola University" SUGGESTED INTERVEW QUESTIONS Some good lists can be found online on the Kent School District site, at Aspire Oregon's PDF list, and the American University in Cairo webpage. Examples each site offers for questions are: 1. What is the typical entry-level position(s) for this job? 2. How important is it to have the "right" degree or courses to have a chance to develop a career like yours today? 3. What are other qualifications that you as an employer/employee consider important? 4. Does your organization offer summer or part-time employment (or volunteer opportunities) for students who are looking for related experience? 5. Can you suggest other work/volunteer experiences that would be advantageous to have? 6. What is a typical day like for you? 7. What do you see as the pros and cons of this type of work? 8. What are the qualifications and educational requirements for this type of work? REPORT REQUIREMENTS (PRESENTED BY STUDENT(S)) In the presentation to the class before writing the reflection piece, students are to present the following information: I. Experiences on Second Life II. Ultimate career goals III. How to obtain goals IV. Education necessary for career option(s) V. Income possibilties VI. Pros/Cons of career observed DISCUSSION QUESTIONS Refer back to step 8 'to discuss' under the LESSON STEPS category of the lesson plan for teacher-driven discussion questions followed by the following questions below--of which the first is from one posed by Devers in his chapter 5 dissertation discussion "Students perspectives and learning from a cross-cultural online course: the influence of a critial pedagogical approach", and a few others to take the discussion a step further... (1)  What did I (the student) learn in a cross-cultural online environment such as Second Life. (2) What did I (the student) learn about my chances of career success/opportunity in an online environment? (3) Did my career choices or interests vary when training or exposed to options in the online environment? Did it help be to think 'outside the box' in respect to career possibilities? (4) Did the exercise widden my understanding of the career? OTHER The teacher may offer a evaluation for students to fill out about the process. The evaluation can either be teacher created or use Survey Monkey.
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Angola University
What's available career exploration-wise... DJ'ing Animation Architecture Real Estate Agent ...FROM THE SECOND LIFE NOTECARD ON ANGOLA UNIVERSITY... Classes at the Angola Virtual Learning Center are free all classes on in-world subjects will continue to be free, and language classes are free first 4 weeks. AVLC is here for those in sl who are new, and want to learn to navigate sl, learn building basics, and language learning. Also, starting test preparation courses soon.

If you are interested in taking classes @ AVLC, Please leave an IM or notecard with Nzingha Dreddmor.

AVLC is very newbie friendly!!

Coordinates: Pooch Deezul, Gream (210, 163, 1501) For other awesome lesson plan ideas, click here! {| cellpadding="5" cellspacing="5" style="width: 90%; background-color: inherit; margin-left: auto; margin-right: auto"
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Reflection
(I) Explain what you believe was the purpose of the project and answer in detail: How have you and your classmates' personal views about the future, about life, your values, interests, goals, and about success (in any environment) changed throughout the course of the Virtual Project? (II) How does what you learned in school connect with your career choice? {| cellpadding="5" cellspacing="5" style="width: 90%; background-color: inherit; margin-left: auto; margin-right: auto"
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Alternative(s)
(1) Teacher may pick a set of careers for the entire class and supposing the class picks five careers, divide the class into five groups but students are still expected to complete the research and training independently.

(2) School may chose to purchase land and create its own settings rather than to use what is already there, and then use the space to post student findings, experiences, and reports in the Second Life land location for public availability.

(3) Accumulate written student presentations to post online on a secure site such as wikiversity as a record over the years of career choice information which may be used for a school organized career fair or virtual fair.

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Other
ARTICLES AND/OR SITES THAT SUPPORTING JOB SHADOWING: (1) "Research Companies and Careers Through Job Shadowing". It's an online article discussing the benefits and steps of job shadowing. (2) Project Shadow by Grossmont. It describes steps to consider when job shadowing. (3) Job Shadow, a free online resource. "Job Shadow is an academically motivating activity designed to give kids the unique opportunity of an up-close look at the world of work and provide the answer to the commonly asked question, “Why do I have to learn this?”."

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Other Lesson Plans

title: Career Exploration Using Second Life


text:
University of Florida/Egm6341/s10.team3.aks/HW7

Ref Lecture p.40.2

Problem Statement
Integrate the logistic equation and use the code of question 3 for HS and run the code for $$ h = 2^k h $$ and also develop Forward Eular and Back Eular methods for the same step sizes.

Solution
Matlab code with change in step size is given below.

MATLAB CODE:

For k = 1

For k = 2

For k=3

We can see that as we increase the value of k due to decrease in step size our curve shifted to the left side.

Below is the code for Forward Eular method

MATLAB CODE:

Forward Eular breaks down at higher values of step size h .This shows that it is unstable in nature. Below is the code for Backword Eular Method

MATLAB CODE:

For k = 1

For k=2

(12)Convert circumference of ellipse formula from t to α variable
Ref Lecture 42.2

Problem Statement
Prove that $$ a \int\limits_{t=0}^{2\pi}[1-e^2 cos^2 t]^{1/2}dt = 4a \int\limits_{\alpha=0}^{\pi/2} [1-e^2 sin^2 \alpha]^{1/2}d\alpha   $$

Solution
$$ a \int\limits_{t=0}^{2\pi}[1-e^2 cos^2 t]^{1/2}dt $$

$$ = 4a \int\limits_{t=0}^{\pi/2}[1-e^2 cos^2 t]^{1/2}dt $$

Let $$ cos t = Sin \alpha $$

$$ - sin t dt = cos \alpha d\alpha $$

Using above values we obtain

$$ = 4a (-) \int\limits_{\alpha=0}^{\pi/2} [1-e^2 sin^2 \alpha]^{1/2} (-) \frac {cos \alpha}{sin t} d\alpha  $$

$$ = 4a (-) \int\limits_{\alpha=0}^{\pi/2} [1-e^2 sin^2 \alpha]^{1/2} (-) \frac {cos \alpha}{sin t} d\alpha   $$

$$ = 4a \int\limits_{\alpha=0}^{\pi/2} [1-e^2 sin^2 \alpha]^{1/2} \frac {cos \alpha}{(1- sin^2 \alpha)^(1/2)} d\alpha         $$

$$ = 4a \int\limits_{\alpha=0}^{\pi/2} [1-e^2 sin^2 \alpha]^{1/2} \frac {cos \alpha}{cos \alpha} d\alpha        $$

$$ = 4a \int\limits_{\alpha=0}^{\pi/2} [1-e^2 sin^2 \alpha]^{1/2}  d\alpha        $$

Hence Proved


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User talk:Nicola4869

Hi everyone,

I'm Nick from Thailand, 2nd year Biology Science Student at International College Mahidol University.

Hi, Thanaporn. --Icns141 06:40, 24 April 2010 (UTC)

Welcome
 Hello Nicola4869, 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. 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.

 * 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.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Ottava Rima (talk) 13:47, 23 April 2010 (UTC)

Classwork on April 26
1. The different between cloud computing and a normal computer - The cloud computing is internet-based computing. Not like the software in a normal computer, the cloud computing is faster, more memory, and only pay for the software and information to computers and other devices in amount of what we really use, like a public utility. Moreover,we can access and shared the information everywhere, basically around the world. There are like an attached e-mail that could run application on the share data center. The example of cloud computing are yahoo, g-mail, zoho, and salesforce.

2. How could you turn a classroom into cloud computing? - We can turn the classroom into cloud computing by create a site for the student in the class to access the information provided by the teacher. For example, classroom online, wikiuniversity. Even though they did not sit in the class. The students could access the information anytime such as their class schedule, homework assignment, and reading list. Additionally, they could easily turn in their works in the cloud computing system as well.

Homework Due Jun 18 by 11:59 PM
What is the best smart phone operating system and why?

- The answer depends on what you most need your smart phone for. I think the best smart phone OS is BlackBerry OS. According to NPD Group, unit sales for BlackBerry OS smartphones ranked first among all smartphone OS handsets sold in the U.S. in the first quarter of 2010. The operating system provides multitasking and supports specialized input devices that have been adopted by RIM for use in its handhelds, particularly the trackwheel, trackball, and most recently, the trackpad and touchscreen.

Best PC in the world
- A custom made computer is the best way to go because even the good brands like Dell can be rather cheaply made. No company makes all parts of the computer, so they have to buy some of the part from other companies. In addition, they cannot make all parts the best. I think we should build PC ourselves and not a store as they are not the best and basically just made to sell. Some examples of the best brand: motherboard is Asus, processor is Pentium, hard drive is Seagate and etc.

title: User talk:Nicola4869


text: File:BE k=1.png

title: File:BE k=1.png


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text:
User talk:J601130342000

Hi everyone!!! This is Ting-Yun Wang or you can just call me Wawa. I'm a Taiwanese and I have been in Thailand for about 7 years. I went to international school in Thailand for high school. I have been in pre-college for 2 quarters before I get into college.

Thanks, Ting-Yun. --Icns141 06:36, 24 April 2010 (UTC)

Welcome
 Hello J601130342000, 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. 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.

 * 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.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Ottava Rima (talk) 13:47, 23 April 2010 (UTC)

Me
My name is Ting-Yun Wang or you can just call me Wawa. I'm a Taiwanese and I have been in Thailand for about 7 years. I went to international school in Thailand for high school. I have been in pre-college for 2 quarters before I get into college.

title: User talk:J601130342000


text:
User talk:Akarapong

Hi! I'm Akarapong or you can call me Pui.

I'm studying at Mahidol University IC, TIM major.

I take Comp Ess class in this term...

Love it!

see you in class, Mr. Brian.

Homework#3

What is the best smartphone operating system and why?

The best smartphone operating system that I choose is iPhone OS, which belong to iPhone 3GS. This iPhone Os is one of the smartphone operating system because all the power and sophistication of an advanced operating system that gives you access to true desktop-class applications and software, including rich HTML email, applications such as widgets, Safari, calendar, text messaging, Notes, and Address Book etc. In addition, it can do multi-tasking, which mean you can read a web page while downloading your email in the background.

Homework#2

Here's my picture...

Homework#1

How is clound computing different than using a normal computer?

Cloud computing differs from a normal computer because information in the cloud computing can be access anywhere and by anybody through internet. This means that information is shared. Moreover, you don't even have to bring your computer with you to get access to the information or file. Unliked cloud computing, the normal computer is more private.

How could you turn a classroom into clound computing classroom?

Turning a classroom into a cloud computing classroom can be done in many ways. Instead of having a physical classroom, students can have class on the internet. For example,  wikiversity is one way for students to share information, homework, etc. Furthermore, the instructor can give assignments and class materials on this website, and he can also keep track of his students.

Welcome
 Hello Akarapong, 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. 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.

 * 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.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Ottava Rima (talk) 13:47, 23 April 2010 (UTC)

title: User talk:Akarapong


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User talk:Llnichall

Specification CPU : Intel Core2 Extreme QX9770 3.20 Ghz X     Mainborad : Asus Striker II Extreme Memory : Corsair 2048 Twin PC8600 X2              Harddisk : Western WD10EACS 1TB SATA II X2 (Raid 0) VGA : NVdia Quadro FX 5600 1.5 GB  X2            Rom :  LG 20X  [GAS-E60L] Sound : Creative Gigaworks THX 7.1 PSU : Coolermaster RP M1000 M/B : GIGABYTE P55-UD3 RAM : DDR3 2G/1333Kingston 1750 HDD : SATAII 500/7200 Western/KS. 1450 Case: ATX READER CARD : all in one APCER 200 UPS : Power 530 W URANO 1N0X 1450 Cache L2: 3MB L3 cache Chipset : Intel Calpella HM57 Optical: Blu-Ray ROM DVD+/- RW SuperMulti DL LightScribe Drive Communication: 56K V.92 Modem, LAN 10/100/1000Mb, Intel 802.11a/b/g/n I2 Card ,Bluetooth Graphic: ATI Mobility Raedon HD 4350 with 512MB DDR3-800MHz Operating Systems: DOs

Symbian is the best type of smartphone operating system because symbian features pre-emptive multitasking andmemory protection. In addition, symbian operating system was created with the three systems design principles, the first one is that it's dominant in the integrity and security of user data. Another is that user time must not be waste and the last is that all resource are limited. Symbian used a microkernel which has a request and callback approach to services, and maintains seperation between user and engine which these things made the symbian best to follow all the three principles.

Nicha Santadkarn ID 5180530 Study in Social Science major

1. Cloud computing is a system that base on users’ preferences which they can assign to meet their requirements through the software system of cloud computing. After that the software system will automatically arrange resources to match the requirements and this makes it more convenient for the users because they do not have to do things by themselves. In addition, they can store and share all their ideas or information on the internet which makes it easy for them to back up and retrieve information because all information are stored in the central server on the internet. Moreover, people can update data through the internet which allow other people to immediately see and also allow them to give comments. In contrast, people usually save their data on hard disk which they have to do everything by themselves. In addition, all the information will become a private property which other people will not see your data and there will be no interaction between you and other people. Moreover, people have to use Flash drives to save and transfer their information when they want to carry and use that data in other places.

2. We can turn classroom into a cloud computing classroom by specifying our need to the software system of cloud computing. After that we have to create accounts for each student after the system has set up the main webpage, homepage, for us which everyone can visit on the internet. In addition, we can post information or ideas on that webpage and also allow other people to give us feedbacks.

My name is nicha santadkarn I study in social science My ID 5180530 My freetime I like to listen music Like red and green color --Llnichall 14:19, 23 April 2010 (UTC)

Thanks, Nicha --Icns141 06:27, 24 April 2010 (UTC)

title: User talk:Llnichall


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User:EGM6341.s10.Team1.Kumanchik/HW7

=Problem 5: Determine the limit on the step size in the Hermite-Simpson method, Forward Euler, Backward Euler=

Statement
Hermite-Simpson algorithm was used to find $$\hat{h}=0.125$$ to give an error of order $$10^{-6}$$ from the true value where $$\hat{h}$$ is the step size between nodes in the numerical solution. The function to be integrated was the population logistic equation,

 * $$\dot{x}=rx\left ( 1-\frac{x}{x_{max}} \right )$$

The numerical integration using the Hermite-Simpson algorithm was compared to the true value for x,

 * $$x\left ( t \right )=\frac{x_0x_{max}e^{rt}}{x_{max}+x_0\left ( e^{rt}-1 \right )}$$

For this problem, the step size is increased at a rate of $$h_{new}=2^k\hat{h}$$ where $$k=1,2,3,...$$

Solution
The differential equation is first integrated using ODE45 (at MATLAB command).

Next the Hermite-Simpson algorithm is run using increasing values of step size. The max step size is limited by the size of the interval $$t\in [0,10]$$. The step size maxes out at 8. A step size of 16 is not allowed. Therefore this algorithm is stable at integrating this function.

Next the Forward Euler integration is used. The step size maxes at 2 at which point it wildly oscillates about $$x_{max}=10$$

Finally, backward Euler integration is used. The step size is allowed to go to 8 so it is stable like the Hermite-Simpson algorithm. Notice, though, that the answer is less accurate for the same number of steps compared to the Hermite-Simpson method

=Problem 6: Interceptor Simulation using Hermite-Simpson algorithm=

Statement
Use the Hermite-Simpson algorithm to determine the trajectory of an interceptor given the following inputs.

 * $$T(t)=\left\{\begin{matrix}

6000 N, & 0\leq t< 27, & 33\leq t\leq 40\\ 1000 N, & 27\leq t< 33 & \end{matrix}\right.$$

 * $$\alpha(t)=\left\{\begin{matrix}

0.03 rad & 0\leq t< 21\\ \frac{0.13-0.09}{27-21}(t-21)+0.09 rad & 21\leq t< 27\\ \frac{-0.2+0.13}{33-27}(t-27)-0.13 rad & 27\leq t< 33\\ \frac{-0.13+0.2}{40-33}(t-33)-0.2 rad & 33\leq t\leq 40 \end{matrix}\right.$$

 * $$\mathbf{f}=

\begin{bmatrix} \dot{\gamma}\\ \dot{v}\\ \dot{x}\\ \dot{y} \end{bmatrix}=\begin{bmatrix} \frac{T-D}{mv}sin(\alpha)+\frac{L}{mv}cos(\alpha)-\frac{g}{v}cos(\gamma)\\ \frac{T-D}{m}cos(\alpha)+\frac{L}{m}sin(\alpha)-gsin(\gamma)\\ vcos(\gamma)\\ vsin(\gamma) \end{bmatrix}$$

 * $$\mathbf{z}=\begin{bmatrix}

\gamma\\ v\\ x\\ y \end{bmatrix}$$

 * $$DL=\left\{\begin{matrix}

D(\alpha,v,y)=0.5C_d\rho v^2S_{ref}\\ L(\alpha,v,y)=0.5C_c\rho v^2S_{ref} \end{matrix}\right. $$

 * $$\left\{\begin{matrix}

C_d(\alpha)=A_1\alpha^2+A_2\alpha+A_3\\ C_l(\alpha)=B_1\alpha+B_2\\ \rho(y)=C_1y^2+C_2y+C_3 \end{matrix}\right.$$

 * $$\left\{\begin{matrix}

m=1005 kg\\ g=9.81 \frac{m}{s^2}\\ S_{ref}=0.3376 m^2\\ A_1=-1.9431\\ A_2=-0.1499\\ A_3=0.2359\\ B_1=21.9\\ B_2=0\\ C_1=3.312e^{-9} \frac{kg}{m^5}\\ C_2=-1.142e^{-4} \frac{kg}{m^4}\\ C_3=1.224 \frac{kg}{m^3} \end{matrix}\right.$$

 * $$t\in [0,40]$$ seconds
 * $$h=1$$ seconds

Solution
The control inputs are the thrust, T, and the angle of attack, $$\alpha$$. They are plotted below.

The Hermite-Simpson algorithm takes the solution to a differential equation and subdivides it into nodes. Each node is related to the next by,

 * $$z_{i+1}=z_{i}+\frac{h}{6}\left [ f_i+4f_{i+\frac{1}{2}}+f_{i+1} \right ]$$
 * $$z_{i+\frac{1}{2}}=\frac{1}{2}\left [ z_i+z_{i+1} \right ]+\frac{h}{8}\left [ f_i-f_{i+1} \right ]$$

However, this equation is dependent on the $$z_{i+1} \,$$ node which appears in the $$f_{i+1} \,$$ and $$f_{i+\frac{1}{2}}$$ terms. Therefore, the equation is solved for by the Newton Raphson method,

 * $$F=-z_{i+1}+z_{i}+\frac{h}{6}\left [ f_i+4f_{i+\frac{1}{2}}+f_{i+1} \right ]=0$$

Intelligently choosing values for $$z_{i+1} \,$$ allows $$F \,$$ to approach 0. The intelligent algorithm is,

 * $$z^{(k+1)}=z^{(k)}-\frac{dF(z^{(k)})}{dz}^{-1}F(z^{(k)})$$

where $$k $$ is the previous guess and $$k+1$$ is the next guess. This algorithm can search for $$z_{i+1}$$ by starting with $$z^{k}=z_i$$. Using $$F$$ as a metric, the algorithm can stop when $$F\approx0$$ or for this problem we choose $$F$$ of $$O(10^{-9})$$.

This solution can be extended to this interceptor problem which solves a system of differential equations. Placing the differential equations in matrix form, the solution remains identical. Keep in mind that the differential of a matrix is the Jacobian. The result of the interceptor problem is plotted below. The time step is 1 second between 0 and 40 seconds.

The source code of the interceptor algorithm

Thrust and angle functions (control variables)

=Problem 7: Integrate the logistic equation using an inconsistent trapezoidal-Simpson algorithm=

Statement
Integrate the logistic equation,
 * $$\dot{x}=rx\left ( 1-\frac{x}{x_{max}} \right )$$

Using the Hermite-Simpson algorithm and the inconsistent Trapezoidal-Simpson algorithm, compare their errors for the same step size, $$h=0.125$$. Both methods require the Newton-Raphson method so use a error for each node of $$O(10^{-9})$$. Compare the integrations to the true integral of the logistic equation,
 * $$x\left ( t \right )=\frac{x_0x_{max}e^{rt}}{x_{max}+x_0\left ( e^{rt}-1 \right )}$$

Solution
The Hermite-Simpson algorithm was adjusted to create the inconsistent trapezoidal-Simpson algorithm. Both solutions were subtracted from the true solution. The error in the Hermite-Simpson algorithm was designed from problem 5 as less than $$O(10^{-6})$$. Using the same parameters on the inconsistent trapezoidal-Simpson algorithm yields worse error by about 3 orders of magnitude. The errors from the true value are plotted below. The line on the bottom is the Hermite-Simpson error. It is much smaller than the inconsistent trapezoidal-Simpson error.

title: User:EGM6341.s10.Team1.Kumanchik/HW7


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User talk:Phawat5280484

center text Me Myself My name is Phawat Sekekul. My nickname is Meng. I am 19 years old. I do not have any brothers or sisters. I am studying at Mahidol University International College. My student ID is 5280484. My major is International Business. I used to study in pre-college program for 9 months. I graduated from English Program at Yothinburana School. I like to wash cars and listen to music in my leisure time. My favorite movie is The Italian Job. My favorite food is papaya salad. I like to go to the beach during my holiday. I would like to be the prime minister of Thailand in order to solve the conflicts inside the country.

Nice to meet you Phawat. --Icns141 06:21, 24 April 2010 (UTC)

How is cloud computing different than using a normal computer? Ans Cloud computing is different from normal computer in terms of privacy. Cloud computing is based on internet, so everyone can access the information which is published through cloud computing. Moreover, people can upload and download data from cloud computing easily. cloud computing can be an online storage devices because the information which is put on cloud computing will never loose except the owner delete it. On the other hand, a normal computer is individual. Other people cannot access the information.

How could you turn a classroom into a cloud computing classroom? Ans Firstly, we have to find a channel or website that we can use to be a classroom. Most of these websites need to be gateways for people to communicate and discuss their ideas. Some examples of these websites are wikiversity and turnitin. After we got a gateway, we have to register for an account in order to know who r we. Finally, you can exchange your ideas with people in your classroom or members of that website

Hi! I am Meng.

Nice to meet you all :-)

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User:Aleena C.

Assignment
1. How is cloud computing different than using a normal computer?

The differences between the Cloud Computing and a normal computer is that, the cloud computing allows us to use applications and share files on the internet. Also, anyone can access to those files at anytime on any computers. A normal computer, in contrast, the files we make and save will be stored on that computer we use, and they can’t be accessed by people that use other computers.

2. How could you turn a classroom into a cloud computing classroom?

The first step for turning a classroom into a cloud computing classroom is that everyone must create their own accounts where we can upload and share files on the internet. The account will allow us to share and access other people’s works, such as friends’ document or teachers’ lectures, at anytime and from any computers. By having a cloud computing classroom, it will be convenient for students to share ideas and learn things from their classmates and teachers

Assignment
http://www.meetup.com/

1.) MeetUp.com

2.) This site helps to arrange a meeting. Also, it shares many information about certain places good for meetings.

3.) Youtube, because it is globally cover everything as well as knowledge and entertainments.

Assignment
The Best PC On Earth (In My Opinion...)

MainBoard          : Intel Sandusky 945PSNLK

CPU                : Intel P-Dual 3GHz

Memory             : 1GB DDR2 Geil Dual Kit.

Storage            : two WD 250GB HDDs SATA II

GFX card           : pcx7900GTX DDR3 512MB 256Bit card with imp clock speed (don't remember what speed exactly).

Sound Card         : Creative X-Fi Fatality full bundle

CDROM              : Dual Layer DVDRW

Case               : Asus Vento 3600

WaterCooling system : ThermalTake Aquarius III (already had that on my priveous machine).

Mouse              : mx518 Logitech

Monitor            : Toshiba 32' plazma monitor.

title: User:Aleena C.


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