The Purpose of Calculus. The History and Intent of a New Mathematics. Created nearly simultaneously by two mathematical geniuses, calculus to many is little more than a pointless numerical exercise. It’s uses, though, are vast in number.

Though some of the principles that are now known to be a part of the mathematics known as *calculus* had been established prior to this point (its most basic elements dating as far back as Archimedes), credit for the actual “invention” of this mathematics is generally given to two great men, working simultaneously in England and Germany, respectively – **Sir Isaac Newton** and **Gottfried Wilhelm Leibniz**.

While Newton was the first and most prominent to recieve credit for this achievement, and Leibnitz was at first accused of plagiarizing his work, it is now recognized that both men were responsible for the development of calculus in its modern form.

Today’s calculus borrows more heavily from Leibnitz’s version than Newton’s due to its much more simple and intuitive system of notion, though a great deal has been added and “cleaned” up since the original versions created by these two astonishing polymaths.

## The Study of Functions

At heart, today’s calculus can also be considered to be the study of *functions*. A function is similar to an equation, except instead of simply finding the “answer,” it is the search for a “behavior.” In other words, a function explores how one variable changes in terms of another.

To put it more mathematically, if one is presented with the function, *f(x) = 2x* (which reads “*f* of *x* equals two times *x*), all that is being asked is how the funciton (*f(x)*) changes as the value of x changes. When *x* = 2 (*f(2)*), then the function result is 4. These two numbers are called the *input* and output.

An input is entered into the function, and the result is the output. This is the key element to all of calculus, for it explores how something behaves under different situations. Different types of equations, such as polynomials, radicals, trigonometrics, and complex, when explored as functions, exhibit very different behaviors.

## The Exploration of the Infinite

Calculus is, in a sense, a form of mathematics which deals with infinite numbers using the idea of functions. Formally, it is the mathematics of limits, integrals, derivitives and series’, but this does little to truly explain just why it is important to both math and science.

The necessity for calculus arose from the realization that standard mathematics – geometry, arithmetic and advanced forms of algebra, were only useful when working in situations where exact quantities exist.

A question that could not be answered by these traditional mathematics would be one such as this: Given a rocket that takes off and travels *x* distance in *y* seconds, how fast was it traveling at point *x + 2*? While on the surface this does not seem like such a complicated question, it cannot be answered precisely using standard algebra, which can really only find the rocket’s average speed over a certain distance.

Even a simple answer like this, however, is plagued by the inclusion of infinities. Une could construct a curve on a standard graph which shows the velocity of the rocket throughout its journey, but this still does little to convey the exact answer to the aforementioned question. One can get a general sense of the motion of the rocket in this visual manner, but no precise conclusion.

The problem stems from the fact that the curve is filled with an infinite number of points, each of which possesses a slightly different (even if infinitesimal) change in velocity. Using algebra and geometry, a capable mathematician can discover the exact *position* of the rocket at point *x + 2*, but how can he discover its behavior at this point (of course this question does not arrive if one is considering an object traveling at a perfectly constant speed)?

Through the mathematics known as calculus, one can ask a couple of important questions which seeks to arrive at an answer to this, one of the simplest.

## The Key Elements of Calculus

First, it uses a concept known as a *limit* to discover what is happening *around* the point in question. In other words, as the path of the rocket approaches *x +* *2* from either direction, how is it changing?

Next, a derivitive can be arrived at using the limit, from which the exact rate of change can be discovered for any single point on the graph.

These are two of the most basic principles of calculus, and range from fairly simple for an easy problem like the one above, or terribly difficult for more complex systems with vast series’ of equations.

Another key element to calculus are *integrals* (which seeks to measure the area under a complex curve where Euclidean geometry would prove highly innacurate).

## Uses of Calculus

It is almost foolish to attempt to make a list of the potential uses of calculus to date. The nearly limitless potential of this powerful (yet extraordinarily difficult) mathematical tool holds incalculable value in many areas of math, science, technology, statistics, economics, etc…

In the physical sciences, particularly, such as theoretical physics, very little of what is known today could be possible without the advancements of calculus (which serves to explain how it came to be developed not by mathematicians, but by men who focused much of their studies on the sciences).

Explore any scholarly work in physics throughout the previous two centuries and one is bound to find themselves inundated with both simple and complex forms of calculus. It is how science has learned to explain the complex behavior of almost any moving object with great precision.

While the study of calculus may be (understandably) intimidating to students of mathematics, its potential value for both mathematics and actual pragmatic applications cannot possibly be overstated. Its exploration can be both excruciatingly difficult and extremely rewarding.

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