LIMITLESS LEADS TO LIMIT - The birth of calculus

If you go on increase the number of sides of a polygon. the circle will be the limit.

1/2^1 + 1/2^2 + 1/2^3 + 1/2^4+........ =1
or
Sum of all the terms of 1/2^n = 1

When n goes from 1 to infinity, the sum of the above series approaches 'one'; but never touches 'one' or crosses 'one', however bigger the 'n' may be. Hence we can say, the limit of the sum of the series is 1.

(1+1/1)^1+ (1+1/2)^2 +(1+1/3)^3 + ..... =2.71... = e
The limit of the above series is the famous Euler constant e.

The concept of limit is the corner stone of calculus. Let us see an example.
Let y= x^2
Let dx be the smallest change in x , let dy be the corresponding small change in y.
I want to find dy/dx
Let x be 3, let the small change be 0.1 (dx) The new x is 3.1
corresponding change in y is

y=x^2
old x: 3 9
new x: 3.1 9.61

hence dy = 9.61 - 9 = .61
dy/dx = 0.61/0.1 = 6.1
Now let us give still more small change in x, that is 0.01
    y=x^2
old x: 3 9
new x: 3.01 9.0601

Now dy = 9.0601 - 9 = 0.0601
again, dy/dx = 0.0601/0.01 = 6.01

If we go on give more and more smaller change in 'x' (like .01,.0033,.0001), the dy/dx will approach 6(the limit). Since x = 3; 6 = 2x. Hence dy/dx =2x
That is small change in y divided by small change in x yields 2x whatever may be the value of x.
This is the operation 'division' for continuous functions like growth of a plant, fall of temperature, increase of electric current. We call this operation as "differentiation". That is calculus.
The ordinary 'division' is applicable to only countable things like books, pens, chocolates etc.

Note: 1. if y=sinx
dy/dx = cosx
2. If y = x^3
dy/dx = 3x^2
3. If y= 5x
dy/dx =5   and on and on
The above are proved results from calculus.

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LISSAJOUS FIGURES

Definition: "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures". We know sine wave motion and circular motion is basically same. Hence we draw two circles A and B perpendicular to each other. The circle B rotates twice faster than circle A. That is, frequency of circle B is two times than that of A. A particle at the intersection of two circles is subjected to two sine wave motion A and B at 90 degree simultaneously. The particle will describe figures depending on the frequency and phase of A and B . In our case, the ratio of frequency is 1:2 and the two waves are in phase. To draw lissajous figures : A moving point in both the circles are chosen. Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two. Hence the points are marked on the circles according to their speed. Then straight lines

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CASINO'S GAME

Let us find out how the casino survives with mathematics. Say, your friend invite you for a game of dice. You must bet (wager) 2 dollars. If you roll 'six' you will get back 8 dollars. The game will go on for 30 rounds. All sounds good. The probability of rolling 'six' is 1/6. Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5. That is, you are expected to win 5 rounds out of 30. Hence your gain will be 5 * 8 =40 dollars. ok. This also implies that you will loose 25 rounds. Hence your loss will be 25*2 =50 dollars. Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars. There will be a loss of -1/3 or 0.33 dollars per round. It is not a fair game. Let us make a simple formula to calculate 'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect

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