Detailed concept of calculus

Isaac Newton and Gottfried Wilhelm Leibniz independently created infinitesimal calculus in the late 17th century. Later work, such as codifying the concept of limits, established a more solid conceptual foundation for these advancements.  The word calculus comes from the Latin word calculus, which means “little pebble.” The word evolved to denote a method of calculating since such stones were used for counting distances, tallying votes, and doing abacus arithmetic. It was used in English in this sense as early as 1672, several years before Leibniz and Newton published their works.The term is also used to refer to specific methods of computation and related theories, such as propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus, in addition to differential and integral calculus.

Differential Calculus:

Differential Calculus is concerned with the issues of determining a function’s rate of change in relation to other variables. Derivatives are used to identify the maxima and minima values of a function in order to find the best solution. The study of a quotient’s limit gives rise to differential calculus. It deals with variables like x and y, as well as functions like f(x) and the changes in the variables x and y. Differentials are represented by the symbols dy and dx.. The function is defined as the derivative of y with respect to the variable x.Let’s have a look at some of the key concepts covered in fundamental differential calculus.

• Limits:

The degree to which a value or a word is approaching.

xcf(x)=A

The limit formula is usually stated as “the limit of f of x as x approaches c equals A.”

• Derivatives:

The rate of change of one quantity in proportion to another.A function’s derivative is represented as:

xcf(x+h)-f(x)h=A

• Continuity:

If the following three conditions are met, a function f(x) is said to be continuous at the point x = a.

•  f(a) is defined

• xaf(x) exists

• xa–f(x) =xa+f(x)=f(a)

Integral Calculus:

The study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral, is known as integral calculus. The process of determining the value of an integral is known as integration.Integral calculus is a branch of mathematics that investigates two related linear operators.The antiderivative, or indefinite integral, is the inverse operation of the derivative. When f is a derivative of F, F is an indefinite integral of f.

The definite integral takes a function as input and returns a number that represents the algebraic sum of areas between the input graph and the x-axis.

The distance traveled in a particular amount of time is an inspiring example. If the speed remains constant, all that is required is multiplication:

Distance = speed x time

Integration is symbolized by an elongated S,(the S stands for “sum”). The definite 

integral is denoted by the symbol:

“The integral from a to b of f-of-x with regard to x,” it says.

Fundamental theorem:

Differentiation and integration are inverse operations, according to the fundamental theorem of calculus. It connects the values of antiderivatives to definite integrals more explicitly. The fundamental theorem of calculus provides a practical technique of computing definite integrals since it is frequently quicker to compute an antiderivative than to apply the definition of a definite integral. It can also be read as a clear assertion that differentiation is the polar opposite of integration.

If a function f is continuous on the interval [a, b] and F is a function whose derivative is f on the interval (a, b), then:

Furthermore, for every x in the interval (a, b),

Application:

Calculus is particularly useful in physics; it connects all concepts in classical mechanics and electromagnetic. Calculus may be used to determine the mass of an object of known density, the moment of inertia of objects, and the potential energy due to gravitational and electromagnetic forces. Newton’s second law of motion, which states that the derivative of an object’s momentum with respect to time equals the net force acting on it, is an example of calculus in mechanics. The net force is equal to the object’s mass times its acceleration, which is the time derivative of velocity and hence the second time derivative of spatial location, according to Newton’s second law.

Calculus is utilised in actuarial science, computer science, statistics, engineering, economics, business, medicine, demography, and other professions where a problem may be mathematically modelled and an optimal solution is needed. It allows one to get from (non-constant) rates of change to total change or vice versa, and we often know one and are looking for the other while analysing a subject. Calculus is a mathematical field that can be utilised in conjunction with others.

Conclusion:

Despite its early origins, until recently, the term “calculus” was employed in mathematics without a precise universal meaning. With the rise of mathematical logic, there was a requirement for a broad theory of calculus, as well as a refinement of the idea of “calculus” itself, which was formalised more thoroughly. However, in most circumstances, the following conception of a calculus (originating with D. Hilbert) suffices. Consider a specific alphabet (usually speaking, infinite, though it might alternatively be represented by a finite number of symbols), whose elements, known as letters, are used to construct calculus formulae (also known as words or expressions) using explicitly established formation rules.