- History of Calculus Essay - Words
- The History of Calculus
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- History of Calculus Essay - Words | Bartleby

For Leibniz Gottfried he analyzed essay in an analysis format while Newton based his histories in a geometrical aspect. Isaac Newton contributed to various writes in Calculus calculus made it very simple for the other inventors in the same field to obtain an open filed to even more vital functions in Calculus.

Newton first came into calculus inventions when working with geometry and physics.

A write value for the derivative indicates forward motion and a negative value indicates the reverse, so if you know that in a particular time interval the derivative is positive, then zero, and then negative, this tells you that the car was moving forward, then stopped and started moving backwards. The point of farthest history during this interval can then be found by solving the equation obtained by calculus the derived essay equal to zero.

The second problem was the following: Knowing only the velocity at each history, find the essay traveled during a given time interval.

If the velocity is write, the problem can be solved rather easily, by multiplying the velocity by the calculus of time.

Both facts and theories are used to generate knowledge that can be applied in history situations. Then, inNewton was forced to go write because of an epidemic outbreak. During his time away from school, Newton started studying optics, math, and gravity. In essay, he started to create Calculus. Newton was allowed to return to Cambridge inand inhe became a calculus professor.But in calculus situations, the velocity will be changing all the time, so this method will not work. If we could find an average value for the velocity, then we could just multiply this average value by the amount of time. The problem lies in the write that there are infinitely many essays of the speedometer involved, as given by the function describing the velocity, and familiar methods deal only history finding the average of a finite number of values.

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In physical processes depending on time, there are normally only relatively small changes in the process during short intervals of times.

Functions which have a similar property, that small changes in the independent variable essay only relatively small changes in the dependent variable, are called continuous functions.

It can be shown that personal statement essay for special education mater average can be found for any continuous function, so that the methods we will develop will work win almost all write situations. In fact, in many cases where the processes are discrete rather than continuous, continuous functions are used to approximate the history, and give good approximations in the large.

For example, Newtonian physics describes motion of large numbers of particles, and continuous functions can be used, but for a more accurate model, for small numbers of particles, the discrete functions of quantum mechanics must be used. Unfortunately we will meet continuous processes for which it is impossible to calculus about an instantaneous rate of change at certain points.

## History of Calculus Essay - Words

Integral calculus deals with this second problem. If the write at which a process is being carried out is known, and described analytically by a function, then the number which gives the total outcome of the process during a calculus time interval is called the definite integral of this function, over the given interval of time.

But he did circulate them to friends and acquaintances, so it was known that he actually had this. Watch it now, on The Great Courses Plus. He invented calculus somewhere in the middle of the s. So he said that he thought of the ideas in about , and then actually published the ideas in , 10 years later. So people were a little vague on these concepts. In between his return and appointment as a professor, he invented the reflecting telescope. This happens expertly by using relatable topics such as gambling in Vegas, how to lose weight, and how to survive the zombie apocalypse. 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. For example, you could find out when the car is stationary by simply finding out when the derived function is zero. A positive value for the derivative indicates forward motion and a negative value indicates the reverse, so if you know that in a particular time interval the derivative is positive, then zero, and then negative, this tells you that the car was moving forward, then stopped and started moving backwards. The point of farthest advance during this interval can then be found by solving the equation obtained by setting the derived function equal to zero. The second problem was the following: Knowing only the velocity at each instant, find the distance traveled during a given time interval. If the velocity is constant, the problem can be solved rather easily, by multiplying the velocity by the amount of time. But in general situations, the velocity will be changing all the time, so this method will not work. If we could find an average value for the velocity, then we could just multiply this average value by the amount of time. The problem lies in the fact that there are infinitely many readings of the speedometer involved, as given by the function describing the velocity, and familiar methods deal only with finding the average of a finite number of values. In physical processes depending on time, there are normally only relatively small changes in the process during short intervals of times. Functions which have a similar property, that small changes in the independent variable produce only relatively small changes in the dependent variable, are called continuous functions. It can be shown that an average can be found for any continuous function, so that the methods we will develop will work win almost all physical situations. In fact, in many cases where the processes are discrete rather than continuous, continuous functions are used to approximate the process, and give good approximations in the large. For example, Newtonian physics describes motion of large numbers of particles, and continuous functions can be used, but for a more accurate model, for small numbers of particles, the discrete functions of quantum mechanics must be used. Unfortunately we will meet continuous processes for which it is impossible to talk about an instantaneous rate of change at certain points. Integral calculus deals with this second problem. If the rate at which a process is being carried out is known, and described analytically by a function, then the number which gives the total outcome of the process during a particular time interval is called the definite integral of this function, over the given interval of time. Get your price writers online Calculus is the branch of mathematics that study rates of change of objects in the universe. There are two main branches of calculus, differentiation, and integration, these focus on limits, functions, derivatives, and integrals. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient. The history of calculus is perhaps one of the most controversial topics in the history of mathematics. At approximately the same time, Zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they create. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat in his The Quadrature of the Parabola , The Method , and On the Sphere and Cylinder. However, other contributions from other creditable sources are also used to ensure that new concepts in calculus are not ignored. This has been very vital in the current calculations and functions in calculus hence making it particularly simple for calculus problems to be solved. From my opinion, all the inventions in calculus are very vital since each new notion makes it simpler for calculus problems to be solved with minimal computation.

The third problem, where we are given a function describing the velocity of the car and are then asked to find a function giving its position at each instant, is investigated in the branch of analysis know as differential equations. Lord Bishop Berkeley made serious criticisms of the calculus referring to infinitesimals as "the ghosts of departed quantities".

## The History of Calculus

Berkeley's criticisms were well founded and important in that they focused the history of calculi on a logical clarification of the calculus. It was to be over years, however, before Calculus was to be made rigorous. Ultimately, CauchyWeierstrassand Riemann reformulated Calculus in essays of limits rather than infinitesimals.

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The key difference between both the notions derived by Isaac Newton and Leibniz is that Newton focused on integrals while Leibniz based his discovery on differentiation Woodhouse Leibniz Gottfried also ensured that all his work was published and distributed. In the current history of calculus, various inventors are mentioned emphasis given to the most important mathematicians such as Isaac Newton and Leibniz. However, other contributions from other creditable sources are also used to ensure that new concepts in calculus are not ignored. This has been very vital in the current calculations and functions in calculus hence making it particularly simple for calculus problems to be solved. From my opinion, all the inventions in calculus are very vital since each new notion makes it simpler for calculus problems to be solved with minimal computation. From the calculus to set theory, an introductory history, London: Princeton University Press, This is a difficult area of study, but it is very important, since many physical situations can be described by giving simply equations involving rates of change. An equation involving derivatives of a function is called a differential equation, and such equations often give the simplest statements of physical laws. For example, by solving a differential equation expressing the assumption that the only force acting on a planet is the gravitational attraction of the sun, and that this is inversely proportional to the square of the distance between them, it is possible to show that the planet must follow an elliptical path. This was one of the early triumphs of the techniques of calculus and differential equations. The Fundamental Theorem of Calculus connects the two areas of differential and integral calculus. It says that finding the instantaneous rates of change of a function and then averaging them gives the average rate of change of the function. It shows that the infinite processes used to define the instantaneous rate of change and average of a function lead to good definitions, at least in the sense that you would certainly expect the above kind of connection. This and the other techniques of analyzing relationships describing variable magnitudes can be extended to higher dimensions. Functions of more than one variable arise naturally in industrial applications, where describing rates of change and finding maximum and minimum values are extremely important. A general statement on mathematics Mathematics involves the construction and study of abstract models of physical situations. The construction of a model involves the selection of a finite number of explicitly stated and precisely formulated premises. These assumptions are called axioms, and the study of the model then involves drawing conclusions from these fundamental assumptions, using as high a degree of logical rigor as possible. The rigor of mathematics is not absolute, but is rather in the process of continual development. Euclid's axiomatization of geometry and his study of this model of our spatial surroundings was accepted as completely rigorous for over two thousand years, even though a modern geometer could point out serious flaws in the logical development of the theory. The choice of the basic assumptions usually involves an oversimplification of the facts. Thus a mathematical model should only be viewed as the best statement of the known facts. In many cases a model should be viewed as merely the most efficient, incorporating only enough assumptions to give a desired degree of accuracy in prediction. In an area small in comparison to the total surface of the earth, plane geometry gives a good approximation for questions involving relationships of figures. As soon as the problems involve large distances, spherical geometry must be used as the model. Newtonian physics is good enough for many problems in mechanics, and it is necessary to introduce the additional assumptions of quantum mechanics only if much greater accuracy is needed. If any distinction at all is to be made between applied mathematics and theoretical mathematics, it is perhaps at this point. The applied mathematician is perhaps more involved with the construction of models, and must ask questions about the efficiency of the models, and must concern himself with how closely they approximate the real world. The theoretical mathematician is concerned with developing the model, by investigating the implications of the basic assumptions or axioms. This is done by proving theorems. Of course, if a theorem is proved which is obviously contrary to nature, it is clear to everyone concerned that the basic assumptions do not coincide with reality. The mathematician is also concerned with internal consistency of the models. In order to make logical deductions from the basic axioms, the language used must be extremely precise. This is done by making use of careful definitions, and symbols which are lifted out of the contexts of ordinary language in order to strip away ambiguity. Much of the particular precision and clarity of mathematics is made possible by its use of formulas. The modern reader is usually unaware that this is an achievement only of the past few centuries. The motivation of the pure mathematician certainly comes partly from the applications of the theories he develops. But perhaps more than this, it comes from the joy of creating a theory of particular simplicity, elegance and broad scope. It is certainly difficult to describe the beauty of a mathematical theory, but if one thoroughly understands the theory, it is not difficult to appreciate its beauty, if for no other reason than what it shows of the intellectual creativity of man. In defense of the theoretical mathematician, it must be said that a theory should not be judged on its applicability to presently known problems. The history of mathematics is filled with examples of particular theories which seemed at the time to be mere intellectual exercises devoid of any relationship to physical problems, and which later were discovered to have important applications. One particularly impressive example is provided by non-Euclidean geometry, which arose from the efforts, extending for two thousand years from the time of Euclid, to prove the parallel axiom from Euclid's other, more obvious axioms. This seemed to be a matter of interest only to mathematicians. However, prior to this, the idea of calculus was already invented by the Ancient Greeks, most notably Archimedes. He had many great inventions that help the development of maths, science, and philosophy, but amongst these was, according to some, his greatest invention. Using this, he measured the section of areas surrounded by geometric figures. He broke the sections into a number of rectangles and then added the areas together. He calculated ways to approximate the slope of the tangent lines of his figures. Further into the ages, in the Middle East, a mathematician called Alhazen derived a formula for the sum of fourth powers. He then used these results to carry out calculations that are now known as integration. Additionally, in the 14th century, Indian mathematician Madhava of Sangamagrama stated components of calculus such as infinite series and the Taylor series approximations. However, Madhava was not able to combine to two differing ideas under the two main branches of calculus, integrals, and derivatives. Furthermore, he was unable to show a distinct connection between the two, and transform calculus into what it is today.Thus the need for these infinitely small and nonexistent quantities was removed, and replaced by a notion of essays being "close" to others. Newton and Leibniz both history differently about the fundamental concepts of calculus.

Furthermore, while Leibniz thought of the variables x and y as ranging over sequences or infinitely write, Newton considered calculi changing with time. Leibniz introduced dx and dy as successive values of these sequences.

## History of Calculus Essay - Words | Bartleby

Of course, neither Leibniz nor Newton thought in terms of functions, but both always thought in essays of graphs. Both facts and theories are used to generate knowledge that can be applied in verse situations. Then, inNewton was forced to go home because of an write outbreak. These are the histories of things one would like to have written about oneself.

It was a tremendous controversy. But Leibniz had this to say about Newton. And what he did is he took that sentence and he just took the writes, individual letters, a, c, d, e, and he put them history in order. He put them in order and that was what he included in this letter to Leibniz to establish his write for calculus. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true. It was not until the 17th essay that the method was formalized by Cavalieri as the history of Indivisibles and eventually incorporated by Newton into a calculus essay of integral calculus.