1. Add two numbers
2+2 = 4
2. Add the number repeatedly
4+4 +4 = 12 3*4 = 12
That is multiplication
3. Now multiply repeatedly
12*12*12 = 1728 12ˆ3 = 1728
That is taking power
Let us go backwards.
1. Take cube root of 1728, we get 12
(dividing repeatedly)
2. Divide 12 by 3, we get 4
(subtracting repeatedly)
3. Subtract 2 from 4, we get 2
Addition is the basic operation, in moving forward. But subtraction is the basic operation when moving backward.
Using addition and subtraction, we can move to and fro in mathematical process and get the desired answers.
Calculus designed in similar fashion.
As an example consider a freely falling body. We know the acceleration is 9.8 m /sˆ2
1. Acceleration=g= 9.8 m/sˆ2
if we integrate acceleration with respect to time,
2. we get gt
Velocity of freely falling body.
If we further integrate velocity with respect to time.
3. we get (1/2)g t^2
Distance traveled = (1/2)gtˆ2
We get distance traveled by falling body.
Now, Differentiate 'distance traveled' with time.
1. differentiating (1/2)gtˆ2
we get
2. velocity =gt
Now, further differentiate velocity
we get
3. acceleration =g
Hence, using integration, we moved forward. With differentiation we moved backward.
Integration is a kind of repeated addition. differentiation is a kind of repeated subtraction.
Why calculus? Calculus consists mainly above two mathematical operations applied to continuously changing quantities like temperature, time, current velocity etc. Arithmetic and algebra is applied to countable quantities like books, money, pens etc.
Hence calculus provides easy and elegant way to get answers.
FOOT NOTE: Let us say, we have a curve[graph] for a mathematical function. The area under the curve is integration. The slope at any point in the curve is differentiation. y = x^2 gives a parabolic curve.
------------------------------------------------------------------------------------------------