Once be recognized and applied in our

         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. 

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      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. 

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