Thursday, May 24, 2012

Natural Logarithm

Some book authors define the natural logarithm function, ln (x) as

.

If we take a look at the graphs below, the red curve represents the graph of , and the dark green curve represents the graph of . The purple region falls below the red curve is bounded between x = h and 1, while the blue region is bounded between x = 1 and k.

Now scroll the value of k, do you notice that point K moves according to the k value? The y-coordinate of point K is the area of the blue region. Same concept applies to the value of h and point H. Here comes the question, do you know how to get negative y-coordinate for point H?


Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Betty, Created with GeoGebra

Monday, January 16, 2012

Spiral

A spiral is formed with a simple equation . 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 to view the forming of spiral.


Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)



Betty, Created with GeoGebra

Rose

For the equation or , 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.


Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)


Betty, Created with GeoGebra

Saturday, January 14, 2012

Lemniscate

The equations and 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 till , and from to . Can you explain why? You may drag the value of 'a' to see the difference of graphs and for the forming of the curve. Then, choose another equation to display its graph and compare. What can you say about these two graphs?



Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)


Betty, Created with GeoGebra

Friday, January 13, 2012

Polar

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






There are some interesting graphs can be obtained through polar equations, such as
  1. Limaçon : or , where
  2. Cardioid : or
  3. Rose : or
  4. Lemniscate : or
  5. Spiral :
etc.

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

Limaçon & Cardioid

A limaçon is a snail curve. The equations and give to the shape of it when .

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



Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)



Betty, Created with GeoGebra

Wednesday, January 11, 2012

Sequence & Series

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

  1. (Finite sequence)
  2. (Infinite sequence)
  3. (Finite sequence)
while a series is the summation of the terms in a sequence, e.g.
  1. (Finite series)
  2. (Infinite series)
  3. (Finite series)

A sequence is a convergence sequence if and only if there is a finite number a where . Otherwise, the sequence is divergent. Likewise, a series is a convergence series if and only if there is a finite number S where . 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.


Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)


Let's take a look at another type of divergence series - Geometric series with common ratio |r| = |a| > 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.


Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)



Now, consider 3 types of infinite convergence series:
  1. Geometric series with common ratio, 0 < r < 1 :
  2. Geometric series with common ratio, -1 < r < 0 :
  3. Alternating harmonic series :

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.


Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)


Betty, Created with GeoGebra