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What Is Calculus?

According to our Accounting Assignment Help experts, it is a branch of mathematics that deals with the study of the increases in variables, slopes of curves, maximum and minimum values of functions, and the determination of lengths, areas, and volumes. Its use is very extensive, especially in sciences such as economics and engineering, as long as there are quantities that vary continuously.

Historical Evolution Of Calculus

The calculation is derived from ancient Greek geometry. Democritus calculated the volume of pyramids and cones, it is believed that considering them formed by an infinite number of sections of an infinitesimal thickness (infinitely small), and Eudoxus and Archimedes used the "exhaustion method" to find the area of a circle with the required accuracy through the use of inscribed polygons. However, the difficulties of working with irrational numbers and the paradoxes of Zeno of Elea prevented the formulation of a systematic theory of calculus.

In the 17th century, Cavalieri and Torricelli expanded the use of infinitesimals, and Descartes and Fermat used algebra to find areas and tangents (integration and differentiation in modern terms). Fermat and Barrow were sure that both calculations were related but it was Newton (in 1660) and Leibniz (in 1670) who proved that they are inverses, which is known as the fundamental theorem of calculus. Newton's discovery, based on his theory of gravity, predated Leibniz's, but the delay in its publication still provokes disputes about who was the first. However, Leibniz's notation was eventually adopted.

In the 18th century the number of applications of calculus increased considerably, but the imprecise use of infinite and infinitesimal quantities, as well as geometric intuition, still caused confusion and controversy over its foundations. One of his most notable critics was the philosopher Berkeley. In the nineteenth century, mathematical analysts replaced this vagueness with solid foundations based on finite quantities: Bolzano and Cauchy precisely defined limits and derivatives, Cauchy and Riemann did the same with integrals and Dedekind and Weierstrass with real numbers. For example, it was learned that differentiable functions are continuous and that continuous functions are integral, although the reciprocals are false. In the 20th century, unconventional analysis legitimized the use of infinitesimals.

What Are Derivatives And Integrals?

The infinitesimal calculus is divided into two main branches: differential calculus and integral calculus.

The differential, using the derivative as its fundamental tool, is dedicated to the study of instantaneous exchange rates or, geometrically, to determining the slope of the tangent line to a curve at a given point.

On the other hand, the integral calculus, using integrals as its fundamental unit, focuses on the study of the area under a curve.

The most surprising thing is that, although the two geometric problems that calculus solves at first glance seem disconnected, they are amalgamated perfectly through the fundamental theorem of calculus, which relates the derivatives with the integrals.

What Are Its Applications?

Calculus has several applications in many fields such as physics, engineering, economics, electronics, statistics, biology, medicine, and even sociology, among many others.

Its applications range from the simple optimal design of a can to the description of the movement of the planets in the solar system.

It provides simple techniques for the study of topics of fundamental importance within the different branches of mathematics, physics, Calculus, economics, etc.

For Example:

• In analytical geometry; allows you to determine the equations of the tangent line and normal to a curve at a point.
• In physics; define instantaneous velocity and acceleration.
• In Calculus; define the reaction rate.
• In economics; define the variation rate.
• Allows the calculation of indeterminate limits.
• Allows studying functions through which we can obtain their graphs.
• Allows calculating errors.

Integral calculus allows solving the problem of determining a function from information about how fast it changes, calculating the area of the figure enclosed by a curve, determining the work done by a variable force, finding areas, volumes, etc.

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