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**This branch of mathematics mainly deals with two fundamental and most important operations of applied mathematics, differentiation and integration, which are executed on functions.**** **

Calculus originated in the 17^{th} century by none other than I. Newton and G. W. Leibniz amid the controversies of continental proportions. Apparently, there arises the question of who did it first. It is now accepted that the two founding fathers made their discoveries independently, with some basic ideasthat first came up in Newton's head, while Leibniz was the first to publish his results and make advancements in the field. We also owe the common notations nowadays used in Calculus to Leibniz. But the story is not heartwarming at all!

**WHAT IS CALCULUS ABOUT**

The most basic answer that describes calculus isa domain of mathematics which aids us in comprehending thechanges between values that are related by a function. Calculus can help us to study constant changes in the real world. It finds patterns between equations like between the circumference of a circle = 2πr relates to a similar one area of a circle = πr^{2}. This is just one example; we can do many sorts of things with the help of calculus like:

- How does an equation (defining a real system (or a model) grow or shrink or accumulate over time?
- To use constantly changing variables (Heat, motion, energy, etc....).

In the field of pure mathematics too, calculus with algebra is a problem-solving duo.

**Let's have an example:**

Consider a circle with the circumference 2πr, and we want to find its area. So, what to do?

We can consider a filled in disc with radius r (given), now again there are two ways to draw a disc:

- Just fill the circle
- Draw a bunch of rings inside it continuously with a thick marker.

The amount of space, i.e., the area is same in each case, but how much?

Well, the largest ring has radius **r** and a circumference 2πr. As the rings shrink, the circumference shrinks, but it keeps the pattern of 2π. The final ring is more like a dot, with no circumference at all. Now, imagine unrolling the rings and lining them up. It will be like a bunch of lines making a jagged triangle(there are infinite rings in the circle). Now the top will be pointy as with 0 radius (only for imagination). The height of the triangle will exactly be r (since unrolled), and the base length will be the circumference of the largest ring, i.e., 2πr. Now the area of the triangle is given by ½(base)(height). So, the area of the jagged triangle is ½*r*2πr, i.e., πr^{2}. It is reduced to the area of the circle. This is calculus!!

**THERE ARE TWO DIFFERENT FIELDS OF CALCULUS.**

The first one is called Differential Calculus. It uses the concept of function derivatives, i.e., differentiation, to study the behaviour and rate on how different quantities change. Using differentiation, the graph of a function can be computed, analysed, and predicted.

The second one is called Integral Calculus. And just like Newton's third law of motion, every action has an equal and opposite reaction. The reverse process of differentiation is integration, concerned with the concept of the anti-derivative. These existed for centuries already. Even though these 2 subfields are generally different from each other, but the Fundamental Theorem of Calculus links these.

1.

d/dx _{a}∫^{x }f(t)dt = f(x)

This version of the theorem says is that the derivative of F is f, or F is anti-derivative of **f.**

**2.**** ** _{a}∫^{b}^{ }f(x) dx = f(b)-F(a)

Though it is complicated to use, calculus does have a lot of practical uses. One of the practical use of calculus is when plotting graphs of certain formulae or functions. Using methods of differential calculus such as the first derivative and the second derivative, a graph and its dimensions can be accurately estimated. These derivatives are utilised to predict how a graph may resemble, the direction that it is taking on a specific point, the shape of the graph at a certain point (if it isconcave or convex), to name a few and also the points of minima and maxima. And by using the methods of Integral calculus, one can easily find out the area under a curve (in 2-D) and the volume by any given number of surfaces (3-D).

Mathematical Analysis, or justAnalysis, is also the branch of pure mathematics that not onlycoversboth the subfields of calculus, but also extends the calculus to limits, infinite series, and analytic functions. The understanding of analysis develops from the pre-existing knowledge of Algebra and Geometry.

Calculus is a generalised concept of Analysis to a specific field, i.e., Analysis can explain how the Calculus is working and what is the logic behind the working. Consider the Reimann Integral it explains well enough how the integration process is working and what it is doing.

The analysis gives a perfect idea to see the functions, at what values it is defined (i.e., its Domain), and what is the nature of the function.This helps to interpret the graphs of functions, for instance, simple polynomials but also knowing rational functions, trigonometric functions, and the inter-relationship between powers & logarithms.

It also deals with the infinitesimals a basicity for calculus. And hence Analysis helped in the development of the limits to calculate the infinite integrations and differentiation. We know that differentiation of a function f(x) at a point,a is the rate of change of the function at point a. Using the concepts of limit the differentiation of a function f(x) at a point a is defined as:

** **In the end, Analysis and Calculus are like a duo, one explains the existence of others, and one shows the way to interact with the real world.

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