Rethinking Calculus
Mathematics can sometimes seem scary for me, and I am sure that a lot of other high school students feel the same way. Maybe, it’s because we often see math as merely a series of problems to be solved and rules to master and apply. Calculus is one of the branches of math that some students like me find intimidating to learn.
This paper aims to establish an appreciation and better understanding of calculus by reviewing its historical groundings and giving the practical application of the subject.
The foundation of calculus did not just appear in history, in fact, mathematicians had encountered numerous difficulties and problems that had led to their desire to find ways in which to offer solutions. It is the case that although Isaac Newton and Gottfried Leibniz were the ones to formulate the theorems of Calculus we know today, a fair share of mathematicians began utilizing concepts of calculus as early as the greek period. Calculus was developed from ancient Greek geometry.
It was mainly use to Democritus calculated the volumes of pyramids and cones, probably by regarding them as consisting of infinitely many cross-sections of infinitesimal (infinitely small) thickness, and Eudoxus and Archimedes used the “method of exhaustion”, finding the area of a circle by approximating it arbitrarily closely with inscribed polygons. In fact it was Archimedes who was the first person to find an approximation of the area of the circle using the “method of exhaustion”; it was the first samples of integration and led to the approximated values of ?
(pi). In line with the developments in the field of theoretical mathematics, it can be said that mathematicians encountered their own difficulties with math problems before they were able to actually find the answers through calculus. It was not until the 16th century when mathematicians found the need to further develop the methods that could be used to calculate areas bounded by curves and spheres.
Johannes Kepler for example had to find the area of the sectors of the ellipse in order for him to proceed with his work in planetary motion. He was lucky enough to find the answer in two tries despite the then crude methods of calculus. Imagine if he was unable to compute the area of ellipses during that time, chances are there would have been a delay in the development of astronomical science. It was through Kepler’s exploration of integration that laid groundwork for the further study of Cavalieri, Roberval, and Fermat.
The latter especially contributed a great deal to calculus by generalizing the parabola and hyperbola as y/a = (x/b)2 to (y/a)n = (x/b)m and y/a = b/x to (y/a)n = (b/x)m respectively. It is the case that some mathematicians (like Joseph Louis Langrange) consider Fermat to be the father of calculus, especially with his formulation of the method used in acquiring the maxima and minima by calculating when the derivative of the function was 0; this method is not far from that which we use today in solving such equations.
The formulas we use today to determine motion at variable speeds use calculus. Toricelli and Barrow were the first mathematicians to explore the problem of motion by implicitly applying the inverse of differentiation, integral and derivative as inverses of each other in asserting that the derivative of distance is velocity and vice versa. Newton and Leibniz are considered to be the inventors of calculus because of their discovery of the fundamental theorems of calculus.
However though both shares credit for the latter, Newton was able to apply it further showing its use both in his works in physics and planetary motion which are considered the most significant of all his contributions. The three laws of motion echoed if not are born out of the notion that since the world changes and derivatives are the rates of changes, and then the latter becomes pivotal to any scientific endeavor that attempts to understand the world. Newton was able to use calculus in determine a lot of things during his time.
We must remember though, that in voicing Newton it is good to reminisce his advice that abstractions and concepts don’t stand alone, they’re pieced together with other ideas to find a solution, an answer. This goes with his Newtonian laws, which if we are to really understand we must see how it relates with his law of gravitational force. Calculus bridges the gaps between theoretical math and the applied sciences/mathematics; if we are to look at it exclusively then we would miss the entire point of why we use it as such fail to realize its true value.
Calculus plays a role in the natural, physical as well as the social sciences; it is being employed in solving numerous problems that wishes to determine the maximum and minimum rates of change. It is capable of describing the physical processes that occur around us. It has even been used to solve paradoxes created during the time of Zeno in ancient Greece. It is impossible to imagine how we can be able to understand the world today without the calculus as one of our tools in acquiring knowledge. We may perhaps still be slaves to mystical forces that were claimed to be the cause of change in this world.
Mathematics would remain to us mere abstractions if calculus was not introduced to become the mediator of thought and practice. The development of other disciplines would have not followed without first establishing the existence of the fundamental concepts of calculus. Things which in history were thought to be inconceivable were able to have a figure that man can understand and therefore have the capacity to manipulate though not complete control. Students like me get frustrated when trying to solve a mathematical problem and failing once or twice.
Reading on the history of calculus made me realize that mathematicians would not have come up with the theorems and methods we use today if they too decided to simply get frustrated. In as much as Calculus teaches you at what rate things change and how the infinite can be understood, one could also learn the value of knowing something even if exclusively it seems unimportant. In order for us to appreciate the subject we must look at it as part of the greater system of knowledge, without it all things would not be coherent.
