Introduction Of Calculus

Calculus is the study of the Functions at a particular value or at a nearest time. To learn Calculus following topics should be studied in detail:

·Limits

·Limits & Continuity

·Differentiation

·Integration

To start up, we first talk about Limits. Let us consider a function:

f(x) =( x^2 -1 ) / (x - 1)

Here, if we place the value of x=1, then

f(1)= (1^2 -1) / (1-1)

 = 0 / 0 , which is not meaningful value.

We know that  ( x^2-1 ) = (x - 1) (x+1)

So , f(x) =(x^2-1) / (x - 1)

        = (x+1) (x-1) /(x-1)

Cancelling (x-1) from numerator and denominator, we get

 f(x) = (x+1), only if x ≠1.

Further, let us imagine that we are giving a value to x a little more than x=1, then we can observe that the value of f(x) will be a little more than 2. Slowly if we go on sliding the value of x nearer to 1, but not exactly 1, then the value of function f(x) will go on sliding towards 2. Let’s see how this change occurs:

If we take x=1.1, then the value of          f(x) =2.1,

If we take x= 1.01, then the value of       f(x)= 2.01,

If we take x= 1.001, then the value of     f(x)= 2.001,

Proceeding in the same way,

If we take x=1.0000001, then the value of f(x)= 2.0000001

We conclude that as the value of x approaches 1, the value of f(x) approaches to 2. We write it as:

x→1, then f(x) →2

Now let us try to move towards another direction and observe the change. Let us take the value of x a little lesser than 1, we find that the value of f(x) is a little lesser than 2. So,

If we take x=0.9, then the value of          f(x) =1.9,

If we take x= 0.09, then the value of       f(x)= 1.09,

If we take x= 0.009, then the value of       f(x)= 1.009,

Proceeding in the same way,

If we take x=0.0000009, then the value of f(x)= 1.0000009

We conclude that as the value of x approaches 1, the value of f(x) approaches to 2. We write it as:

x→1, then f(x) →2

  lim f(x)=m, means if x →a, f(x) →m. 

  ^x→a

While finding the limits of a given function, certain rules are to be remembered. They are as follows:

· We simply put the value x=a in a given function and check if f(x)  is a  definite value, then simply

               Lim f(x) = f(a)
               ^x→a

· In case, we find f(x) as a rational number, then we simply factorize the numerator and the denominator, then cancel the common factors and finally place the value of x=a.

· In case, we find the given function contains a surd, then we simplify the function by multiplying the numerator and the denominator of the given function with the conjugate of the given surd. Then we simplify it and finally put x=a in it to get the solution.

· In case, a function contains a series, which can be expanded, then expand it, simplify it, cancel the common factors of the numerator and denominator and finally put the value x=a to get the solution.