CALCULUS IN A NEW ANGLE

Imagine a growing circle. What is the 'most-least' area you can add to the circle. or what is the area of the thinnest ring that can be added to the circle. It is 2*pi*r. How? Consider a rectangle. It's area is length*breadth. Suppose breadth vanishes, the area is only length. Similarly length of the thinnest ring is 2*pi*r-- the circumference. Also we can say 2pi*r is the infinitesimal change in the area of the circle.
From Dot, if you go on add big and bigger rings, a circle will be formed. If we add the areas of the all the rings, we will get area of the circle. In calculus adding is called "integrating". Hence, integrating '2pi*r' from 0 to r we get,

We got, total area by integrating small change in area. Calculus works.

Add circular sheets of diminishing area one above the other, you get a solid cone. correct? Refer figure. Here, the least decrease in volume (by the same argument) is pi *r^2. By integrating pi*r^2, we get 1/3pi*r^3. That is the volume of the solid cone. Again calculus in action.

The reverse of integration is differentiation:
By differentiating volume, we get area. By differentiating area, we get length- circumference.

Differentiation:

We know Y = x^2 gives parabola. By differentiating, we get 2x- which is the small change in Y at any point x. 2x also gives the slope at any point in the curve.
Y=x^2
dy/dx = 2x
Hence, differentiation of Y with respect to x is
1. A infinitesimal change in Y when change in x reaches the limit zero.
2. A change in Y for unit change of x, for any value of x.
3. Rate of change of Y with respect to x.
4. A gradient or slope at any point in the Y-X curve.

Integration:
1. Integration is the summation of small changes.
2. Commutative effect of gradual or continuous small changes.(increments or decrements).
3. It gives area under the curve between two limits.
4. It also gives volume enclosed by curvy sheets.

WHY CALCULUS
1. For discrete and countable things, Asthmatics and Algebra is enough.
2. When things have mathematical relationships and when things changes continuously, calculus is required.
For counting money, calculus is not required. But for finding inflation rate, calculus may be required.
------------------------------------------------------------------------------------------------

Comments

THE WORK HORSE "="

One cannot think of a mathematical step without 'is equal to ' . It balances right hand side and left hand side. It aids simplification and manipulation of a mathematical expression. example: 2(A+B) = C 2A+2B = C 2A = C-2B A = C-2B/ 2 In an electronic calculator, the pressing of ' = " sign executes an asthmatic expression or simply calculates. In computer languages, it plays very important role. A = B When a computer looks at this expression, the value stored in the location named B is just transferred to the storage named A . After execution both A and B will have the same value an...

DISORDER IS THE "ORDER OF THE DAY"

Imagine a balloon full of air. The air molecules are moving randomly inside the balloon. Let us pierce the balloon with a pin. The air rushes out. Why should not the air molecules stay inside the balloon safely and ignore the little hole? That is not the way the world works. The molecules always "want to occupy as many states as possible". Hence the air goes out in the open to occupy more volume. The things always goes into disorder (entropy) and the disorder increases with time. The above statement is what we call "second law of thermodynamics". Consider a cup of coffee on the table. Suppose the heat from entire room flows to your cup of coffee, the coffee will boil and the rest of the room will freeze. Freezing means bringing things to order and arrangement. It violates the second law. Hence it will never happen. Hence heat must flow from high ...

THE EARTH, A SUPER ORGANISM

JOIN MY COURSE: "Become a programmer in a day with python" A man called 'love lock' (what a name) proposed a theory called Gaia theory, named after Greek Goddess. It says, "Earth is a self-regulating organism like a human being. The organic life in it interacts with in-organic matter and maintains atmosphere, temperature and environment". Hence the earth is still suitable for the life to thrive. Imagine, in a particular place, there are lot of flowers. Some flowers are white and some are darkly coloured. We know, white reflects light and heat while dark absorbs the same. White flowers can thrive in hot climate. But dark flowers requires cold climate. The absorption and reflection balances and the environment reaches average, warm temperature at which both the flowers can co-exist. This is the essence of "Gaia" theory. On our earth, ...

sciencecapsules

Search This Blog