Once upon a time, the world of numbers

was invented by humans, by then was developed and had been widen by great

talents accompanied with the gift of wisdom. And now it is an instrument used

for such different activities that comprise survival and living by us, humans.

One of the useful discovery that math had undergone was the calculus. Calculus,

a spur and affiliate of the topic of mathematics is defined as highest standard

of accuracy when dealing with incescant matter of change. Thus we can pertain

that it can be recognized and applied in our different multiple activities. And

accordingly, it is a branch of mathematics that carries the topic of finding

and properties of derivatives and integrals of functions, using the process

which was founded from the summation of infinitesimal differences. It has two

main types namely the differential and the integral calculus. These two are of

course methods of calculations but only deals with a certain portion and each

does consist deluge defined concerns. We are living in a world that is

constantly changing, which in fact change is the only constant. We experience

such different observations around us. Just take a look around you, some things

had moved at a different pace and at different time. Some took act in a

constant motion, while some are in the state of rest. By then humans had come

up developing such several machines, equipments and tools to explain these

aesthetic phenomena. And the grand entrance of calculus prevailed as a tool

used in explicating this occurrence.

Who discovered calculus? A simple question

yet puzzled a lot of minds way back from the past until this present period of

time. A number of researchers are still paving their way and putting some great

efforts trying to uncover such a quest, a way for clearing a blurred concept

and information about an important part of history. This is an argument that

had aroused a long time, and does exist even now in the present. Is it Newton

or Leibniz or both of them? Two men, two minds

but had only been circulating in a common field which is calculus. It is a long

story which until now is a subject that doesn’t contain an ending. Products of

grand and noble universities like Oxford and Cambridge seem to conquer a great

title as scientist and mathematicians an instant example would be Isaac Newton. On the other side, Gottfried Wilhelm

Leibniz is not only known as a great

mathematician, but he was also a philosopher, scientist, logician, diplomat and

a lawyer. Now it is clearly expressed by some of the sources that the unveiling

of calculus is often attributed to these two men, who had developed each

foundation quite different and independent towards each other. Even though they

were both entitled as an instrument for this creation, they unlikely had come

up with the same fundamental concepts. While

Newton considered variables changing with time, Leibniz thought of the

variables x and y as ranging over sequences of infinitely close values. He

introduced dx and dy as differences between successive values of these

sequences. Leibniz knew that dy/dx gives the tangent but he did not use it as a

defining property. By the present, Leibniz

is well known for introducing notations that are still used in Calculus today,

such as ‘dy/dx’ and the integral symbol. On the other hand, Newton used quantities x’

and y’, which were finite velocities, to compute the tangent. Of course neither

Leibniz nor Newton thought in terms of functions, but both always thought in

terms of graphs. For Newton the calculus was geometrical while Leibniz took it

towards analysis. It is interesting to note

that Leibniz was very conscious of the importance of good notation and put a

lot of thought into the symbols he used. Newton, on the other hand, wrote more

for himself than anyone else. Consequently, he tended to use whatever notation

he thought of on that day. This turned out to be important in later

developments. Leibniz’s notation was better suited to generalizing calculus to

multiple variables and in addition it highlighted the operator aspect of the

derivative and integral. As a result, much of the notation that is used in

Calculus today is due to Leibniz. The development of Calculus can roughly be

described along a timeline which goes through three periods: Anticipation,

Development, and Rigorization. In the Anticipation stage techniques were being

used by mathematicians that involved infinite processes to find areas under

curves or maximize certain quantities. In the Development stage Newton and

Leibniz created the foundations of Calculus and brought all of these techniques

together under the umbrella of the derivative and integral. However, their

methods were not always logically sound, and it took mathematicians a long time

during the Rigorization stage to justify them and put Calculus on a sound mathematical

foundation. In their development of the calculus both Newton and Leibniz used

“infinitesimals”, quantities that are infinitely small and yet

nonzero. Of course, such infinitesimals do not really exist, but Newton and

Leibniz found it convenient to use these quantities in their computations and

their derivations of results. Although one could not argue with the success of

calculus, this concept of infinitesimals bothered mathematicians. Lord

Bishop Berkeley made serious criticisms of

the calculus referring to infinitesimals as “the ghosts of departed

quantities”. Berkeley’s criticisms were well founded and important in that

they focused the attention of mathematicians on a logical clarification of the

calculus. It was to be over 100 years, however, before Calculus was to be made

rigorous. Ultimately, Cauchy, Weierstrass,

and Riemann reformulated Calculus in terms of limits rather than

infinitesimals. Thus the need for these infinitely small (and nonexistent)

quantities was removed, and replaced by a notion of quantities being

“close” to others. The derivative and the integral were both

reformulated in terms of limits. While it may seem like a lot of work to create

rigorous justifications of computations that seemed to work fine in the first

place, this is an important development. By putting Calculus on a logical

footing, mathematicians were better able to understand and extend its results,

as well as to come to terms with some of the more subtle aspects of the theory.

When we first study Calculus we often learn its concepts in an order that is

somewhat backwards to its development. We wish to take advantage of the

hundreds of years of thought that have gone into it. As a result, we often

begin by learning about limits. Afterward we define the derivative and integral

developed by Newton and Leibniz. But unlike Newton and Leibniz we define them

in the modern way — in terms of limits. Afterward we see how the derivative

and integral can be used to solve many of the problems that precipitated the

development of Calculus.

The fact that we may use

calculus in a lot of ways, there are some real life contexts that embodied the

application of this field which most of the students had been questioning all

this time when encountering such great difficulties in the problems in math

given unto them. This mathematical model of

change and has amazing prediction powers, is extremely useful in our everyday

life. You would need some practice to know how to use it well in everyday life,

but once you mastered it, it helps greatly to weed out irrationality, clarify

your life choices, predict your future, or simply beat your friends in various

computer games. Some of us today, regarding about the stage we’re going through

basically have problems about love life. A relationship toward an opposite sex

is somehow the prevailing issue one could encounter when at this stage of

looking for intimacy rather than isolation. Thus relating this to the calculus let

us say you and your partner is in the state of having a frequent situation of

times being in bad conditions and after that you two would be in a good terms

and then bad and then good and then bad and so on. So luckily with the help of

calculus you resolve it. Let us say X is the amount of your love for your

partner, and Y is the amount of love your partner has for you his. Both X and Y

are functions of time t. We are all normal individuals so with your partner. If

she/he felt loved, then the love that will return would increase and if not

then it will decrease. So to put this on equation we have Y’=X. Sometimes there’s an instance of taking things

for granted, which affects the love itself. So if your love for a person would

actually decrease if this other person loved you too much and on the other

hand, you like things that are hard to get. So to sum up, we have X’= -Y. Now,

X’=Y and Y’=X form a very classical system of differential equations. Now you

can easily see the periodical nature of your relationship. Your relationship is

periodic because of your attitude towards love. In real life, some reality X is

unsatisfactory, and we cannot change it. But by changing attitude and changing

habits, you can change the derivative of X, and eventually change X over time.

Another sample would be playing a mobile game in your pc, laptop or cellular

phones. One of my favorite game is cybersphere, in which you had a chosen

character and then your ultimate objective is to kill all the attacker of your

camp. When you shoot a moving enemy, it is very duly to miss. So you need to

predict where they might be in the next moment, and then shoot there. That’s

how you get them. How would you predict? You predict by checking out the

direction and speed of their movement, example the derivative of their

movement. Then when you predict their eventual location, you are

unintentionally doing a mental integration. That’s how you “predict” their

movement and get your shot. It was a fun mobile game but unfortunately I think

no one else would play it and sadly I had deleted it due to storage problem.

Lastly, one of the activities that would apply calculus is studying. Not only

relating to the fact that you need to study the lessons in calculus but it also

deals on how you should study given in a fixed speed. For an example I myself

would firstly gather the information and materials needed to start up the

session and start reading then let myself find ease and interest about that

certain lesson, for if I will not I would end up being bored and expected to

halt. After that I would gain motivations to pursue what I have started. So the

knowledge there increases as the stage also does. Let us say G’ is constant and

we are increasing G at a constant speed. So G grows like x where G stands for

knowledge and the process goes by until it will reach its expected limitations.

Gradually, I would infer that it is a good technique in studying. I did not

invent this method for I only gotten it from one of my studious peers. These examples would really satisfy that calculus

is extremely useful in our everyday life. Like a graph of a mathematical

function, life can have several discontinuities at times. But like a removable

discontinuity, we can always rely on our selves for being determinate to

redefine our life function when our path is clearly rough. We would only need

some practice to know how to use it well, but once one had mastered it, it

helps greatly to weed out great strategies dealing with lifetime struggles.