Calculus (by S.P.Nirala)
Calculus is that branch of mathematics which mainly deals with the study of change in the value of a function as the points in the domain change. Here , mainly we discuss about derivative, integration and differential equation in brief.
Derivatives
By knowing the position of a body at various time intervals it is possible to find the rate at which the position of the body is changing. It is of very general interest to know a certain parameter at various instants of time and try to find the rate at which it is changing. There are several real life situations where such a process needs to be carried out. For instance, people maintaining a reservoir need to know when will a reservoir overflow knowing the depth of the water at several instances of time. Rocket scientists need to compute the precise velocity with which the satellite needs to be shot out from the rocket knowing the height of the rocket at various times. Financial institutions need to predict the changes in the value of a particular stock knowing its present value. In these, and many such cases it is desirable to know how a particular parameter is changing with respect to some other parameter. The heart of the matter is derivative of a function at a given point in its domain of definition.
Definition: suppose f is a real valued function and a is a point in its domain of definition. The derivative of f at a is defined by
Provided this limit exists. Derivative of f(x) at x=a is denoted by f’(a).
Historical note
In the history of mathematics two names are prominent to share the credit for inventing calculus, Issac Newton(1642-1727) and G.W.Leibnitz(1646-1717). Both of them independently invented calculus around the seventeenth century. After the advent of calculus many mathematicians contributed for further development of calculus. The rigorous concept is mainly attributed to the great mathematicians, A.L.Cauchy, J.I.Lagrange andKarl Weierstress. Cauchy gave the foundation of calculus as we have now generally accepted in our text book
Before 1900, it was thought that calculus is quite difficult to teach. So calculus became beyond the reach of youngsters. But just in 1900, John Perry and others in England started propagating the view that essential ideas and methods of calculus were simple and could be taught even in schools. F.L. Griffin pioneered the teaching of calculus to first year students. This was regarded as one of the most daring act in those days.
Today not only the mathematics but many other subjects such as Physics, Chem istry, Economics and Biological sciences are enjoying the fruit of calculus.
Integration
Integral calculus is motivated by the problem of defining and calculating the area of the region bounded by the graph of the functions.
If a function f is differentiable in an interval I then a natural question arises that given f’ at each point of I, can we determine the function. The functions that could possibly have given function as a derivative are called anti derivatives or primitive of the function. Further , the formula that gives all these anti derivatives is called the indefinite integral of the function and such a process of finding antiderivative is called integration. For instance, if we know the instantaneous velocity of an object at any instant, then there arises a natural question i.e, can we determine the position of the object at any instant ?
The development of integral calculus arises out of the efforts of solving the problems of the following types:
i) The problem of finding a function whenever its derivative is given
ii) The problem of finding the area bounded by the graph of a function under certain conditions
These two problems lead to the two forms of the integrals indefinite and definite integrals, which together constitute the Integral calculus.
Differential equation
One of the principal languages of science is that of differential equation. The date of birth of differential equation is taken to be November11,1675, whenLeibnitz first put in black and white identity , thereby introducing the symbols. Leibnitz was actually interested in the problem of finding a curve whose tangents were prescribed. This led him to discover the method of separation of variables 1691. A year latter he formulated the ‘ method of solving homogeneous diff. equation of first order’. He went further in a very short time to the discovery ‘the method of solving linear differential equation of the first order’. How surprising is it that all these methods came from a single man and that too within 25 years of the birth of differential equations.
References: text book of mathematics for 11 and 12 NCERT
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