Calculus
Calculus is defined as “the branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences. The two main types are differential calculus and integral calculus.”
The first thing to note in the definition is calculus concerns infinitesimal differences, which are defined using the notion of limits and continuity, and will be explained later. The other is differential and integral calculus. The summation of infinitesimal differences is called integration, and by differentiation we examine the rate of change of a given function.
Sir Isaac Newton and Gottfried Wilhelm Leibniz would be the most notable mathematicians that contributed to the invention and development of calculus. While in their time there was a controversy, each claiming the other stole his own work, it turned out in fact their work coincidentally revolved around similar ideas while working independently in the late 17th century. Leibniz’s legacy primarily lies in establishing useful concepts and developing consistent notations, and Newton in application of integral calculus to physics.
The domain where calculus is utilized is almost universal in science. To name a few, they are physical sciences, actuarial science, computer science, statistics, engineering, economics, business, medicine, and any fields finding an optimal solution with mathematical models are applicable. Most of the times we are interested in finding, or converting between the total change and the rate of change.
In physics, specifically in classical mechanics, the first derivative of a displacement function is a velocity function, and the second derivative is an acceleration function. In probability theory of statistics, the probability of a continuous random variable is obtained as the area under the curve of the probability density function (PDF), by integrating the PDF at a given interval. The notion of PDF is also used in chemistry as well. The Valence Shell Electron Pair Repulsion (VSEPR) Theory is a model to predict the location(s) of electron(s) with probability found in the areas called orbital.