## Monday, January 16, 2012

### Spiral

A spiral is formed with a simple equation $r=a\theta, a\neq 0$. The graph below set 'a' as positive, therefore there are two equations for you to tick. Choose whichever equation you prefer, you may choose both as well, then drag the value of 'a' according to your preference. Then, drag the value of $\theta$ to view the forming of spiral.

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### Rose

For the equation $r=k\cos (a\theta)$ or $r=k\sin(a\theta)$, a rose is formed. The number of petal depends on the value of 'a'. Try to drag on the value of 'a' and compare the results when 'a' is even and when 'a' is odd.

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## Saturday, January 14, 2012

### Lemniscate

The equations $r^2=k^2\cos a\theta$ and $r^2=k^2\sin a\theta$ form the shape of lemniscate. If you take a look at the graph (in default setting) below, you might find out that there is no graph formed for $\theta = 90^o$ till $180^o$, and from $270^o$ to $360^o$. Can you explain why? You may drag the value of 'a' to see the difference of graphs and $\theta$ for the forming of the curve. Then, choose another equation to display its graph and compare. What can you say about these two graphs?

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## Friday, January 13, 2012

### Polar

A coordinate (x,y) in xy-plane can be written in polar form $\left ( r,\theta \right )$, where r is the directed distance from (0,0) to (x, y), and $\inline \theta$ is the directed angle formed by the vector <x, y> from the positive side of x-axis:

$r=\sqrt{x^2+y^2}$

$\theta =\tan^{-1}\left ( \frac{y}{x} \right )$

There are some interesting graphs can be obtained through polar equations, such as
etc.

Click on the content alongside to look at the shapes mentioned above.

### Limaçon & Cardioid

A limaçon is a snail curve. The equations $r=a\pm b\cos\theta$ and $r=a\pm b\sin\theta$ give to the shape of it when $a\neq b$.

When $a=b$, a cardioid is formed. Try to change the values of a and b by dragging it. You may also drag the value of $\theta$ to see the forming of the curve.

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## Wednesday, January 11, 2012

### Sequence & Series

A sequence is a set of terms arranged in a particular order such as

1. $1,2,3,4,5$ (Finite sequence)
2. $2,4,6,8,10,...$ (Infinite sequence)
3. $\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},...,\frac{1}{256}$ (Finite sequence)
while a series is the summation of the terms in a sequence, e.g.

A sequence $\left \{ a_n \right \}$ is a convergence sequence if and only if there is a finite number a where $\lim_{n \to \infty}a_n = a$. Otherwise, the sequence is divergent. Likewise, a series $S_n$ is a convergence series if and only if there is a finite number S where $\lim_{n \to \infty }S_n = S$. Otherwise, the series is divergent.

For example, harmonic sequence is convergence but its series is divergent. Perhaps putting the sequence and series in graph might help you to understand better. Look at the graph below, y-coordinate of point A represents the harmonic sequence while y-coordinate of point B represents summation of the harmonic series. n is the index. Drag the value of n and observe the positions of point A & B. You'll find that point A is approaching to x-axis (y = 0) as n is getting bigger but point B keeps on raising.

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Let's take a look at another type of divergence series - Geometric series with common ratio |r| = |a| > 1 :
$a+a^2+a^3+a^4+...,\left | a \right |> 1$.

The points in purple represent sequence and points in red represent series. Tick on the series you want to look into, and drag the values of a and/or n to observe the convergence of the sequence and series.

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Now, consider 3 types of infinite convergence series:
1. Geometric series with common ratio, 0 < r < 1 : $\frac{1}{a}+\frac{1}{a^2}+\frac{1}{a^3}+\frac{1}{a^4}+..., a>1$
2. Geometric series with common ratio, -1 < r < 0 : $\frac{1}{a}+\frac{1}{a^2}+\frac{1}{a^3}+\frac{1}{a^4}+..., a<-1$
3. Alternating harmonic series : $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-...$

Graph shown below is the graph of Geometric series with 0 < r < 1. The point in purple represents the respective sequence and the red represents the series. The solid line in green gives the value of the series as n is getting bigger, whereas the dotted green line tells us the value of the current sum of the series. Try to play around with the graph below. After that, click on other choices to observe the convergence of the sequence and series given.

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