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Introduction of Calculus

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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.

History of Calculus

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The study of the history of Calculus will not make us a mathematician but makes us more familiar with the topic. It will enrich our minds, and smooth our heart. Calculus is the branch of mathematics, which includes Functions, limits, derivatives, integrals, and infinite series. The very first step in calculus was taken by Greek mathematicians. They state that number line had "holes" in it because Numbers were ratios of integers. They overcomes this difficulty using length, volumes and areas.

About 450 BC Zeno of Elea gave a number problem that motion is impossible. To understand this concept, Let's take an example, if a object moves from 'X' to 'Y' then before it reaches 'Y' it passes through the Mid-Point of 'XY', let X1. Now to move X1 it must first reach the mid-point X2 of X1Y and so on. The object move through an infinite number of distances and so it can't move.

Archimedes used the method of exhaustion to find an approximation to the Area of a Circle. This can be viewed as an early example of Integration which led to approximate values of π. Isaac Newton and Gottfried Leibniz, thought of the fundamental concepts in very different ways. Newton considered variables changing with time, while Leibniz consider that variables are sequences of infinitely close values. He gave dx and dy as differences between successive values of these sequences. Leibniz tells dy/dx gives the Slope but he did not use this in defining property. Newton used quantities, which were finite velocities, to compute the Tangent. Both did not give their thoughts in terms of Functions but give in terms of graphs.

Definition of Calculus

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Calculus is a branch of mathematics which deals with function, range, and Domain. It also deals with the rate of change, infinite sequences and orbits of planet. We can find the derivative and integral of any given function with the help of Calculus. It includes arithmetic, algebra, trigonometry, geometry, and coordinates Geometry. So, you need to know all the basic concept of Algebra and Trigonometry. Generally, calculus has two branches algebra and trigonometry. If we talk about mathematics, it is used to find Slope, area, tangents, average of some function, intercepts forms, in vector calculus and the rate of change of the graphs.