Calculus, historically known as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Ideas leading up to the notions of function, derivative, and integral were developed throughout the 17th century, but the decisive step was made by Isaac Newton and Gottfried Leibniz. Publication of Newton's main treatises took many years, whereas Leibniz published first (Nova methodus, 1684) and the whole subject was subsequently marred by a priority dispute between the two inventors of calculus.

Greek mathematicians are credited with a significant use of infinitesimals. Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea. At approximately the same time, Elea discredited infinitesimals further by his articulation of the paradoxes which they create. Antiphon and later Eudoxus are generally credited with implementing the method of exhaustion, which made it possible to compute the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes. Archimedes of Syracuse developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. (See Archimedes' Quadrature of the Parabola, The Method, Archimedes on Spheres & Cylinders.) It should not be thought that infinitesimals were put on a rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true. It was not until the time of Newton that these methods were incorporated into a general framework of integral calculus. Archimedes was the first to find the tangent to a curve, other than a circle, in a method akin to differential calculus. While...

...“The Contribution of Calculus in the Social Progress”
The history of calculus falls into several distinct time periods, most notably the ancient, medieval, and modern periods. The ancient period introduced some of the ideas of integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Egyptian Moscow papyrus (c. 1800 BC), in which an Egyptian successfully calculated the volume of a pyramidal frustum.[1][2] From the school of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus.[3] The method of exhaustion was later used in China by Liu Hui in the 3rd century AD in order to find the area of a circle. It was also used by Zu Chongzhi in the 5th century AD, who used it to find the volume of a sphere.[2]
In AD 499 the Indian mathematician Aryabhata used the notion of infinitesimals and expressed an astronomical problem in the form of a basic differential equation.[4] This equation eventually led Bhāskara II in the 12th century to develop an early derivative representing infinitesimal change, and he described an early form of "Rolle's theorem".[5] Around AD...

...
Calculus in Medicine
Calculus in Medicine
Calculus is the mathematical study of changes (Definition). Calculus is also used as a method of calculation of highly systematic methods that treat problems through specialized notations such as those used in differential and integral calculus. Calculus is used on a variety of levels such as the field of banking, data analysis, and as I will explain, in the field of medicine. Medicine is defined as the science and/or practice of the prevention, diagnosis, and treatment of physical or mental illness (Definition). The term medicine can also mean a compound or a preparation applied in treatment or control of diseases, mostly in form of a drug that is usually taken orally (Definition). Calculus has been widely used in the medical field in order to better the outcomes of both the science of medicine as well as the use of medicine as treatment. (Luchko, Mainardi & Rogosin, 2011). There has been a strong movement towards the inclusion of additional mathematical training throughout the world for future researchers in biology and medicine. It can be hard to develop new courses as well as alter major requirements, but institutions should consider the importance of a clear understanding of the function of mathematics in science. However, scientists who have not had the level of mathematical training needed to work in...

...No 1. 2. 3. 4. 5. 6. 7. 8. Code: UCCM1153 Status: Credit Hours: 3 Semester and Year Taught:
Information on Every Subject Name of Subject: Introduction to Calculus and Applications
Pre-requisite (if applicable): None Mode of Delivery: Lecture and Tutorial Valuation: Course Work Final Examination 40% 60%
9. 10.
Teaching Staff: Objective(s) of Subject: • Review the notion of function and its basic properties. • Understand the concepts of derivatives. • Understand linear approximations. • Understand the relationship between integration and differentiation and continuity. Learning Outcomes: After completing this unit, students will be able to: 1. describe the basic ideas concerning functions, their graphs, and ways of transforming and combining them; 2. use the concepts of derivatives to solve problems involving rates of change and approximation of functions; 3. apply the differential calculus to solve optimization problems; 4. relate the integral to the derivative; 5. use the integral to solve problems concerning areas.
11.
12.
Subject Synopsis: This unit covers topics on Functions and Models, Limits and Derivatives, Differentiation Rules, Applications of Differentiation and Integrals.
13.
Subject Outline and Notional Hours: Topic Learning Outcomes 1 L 4 T 1.5 P SL 6.25 TLT 11.75
Topic 1: Functions and Models
• • • • • • Functions Models and curve fitting Transformations, combinations, composition and graphs of...

...discoveries, calculus was the culmination of centuries of work rather than an instant epiphany. Mathematicians all over the world contributed to its development, but the two most recognized discoverers of calculus are Isaac Newton and Gottfried Wilhelm Leibniz. Although the credit is currently given to both men, there was a time when the debate over which of them truly deserved the recognition was both heated and widespread. Evidence also shows that Newton was the first to establish the general method called the "theory of fluxions" was the first to state the fundamental theorem of calculus and was also the first to explore applications of both integration and differentiation in a single work (Struik, 1948). However, since Leibniz was the first to publish a dissertation on calculus, he was given the total credit for the discovery for a number of years. This later led, of course, to accusations of plagiarism being hurled relentlessly in the direction of Leibniz. It is also known that Leibniz and Newton corresponded by letter quite regularly, and they most often discussed the subject of mathematics (Boyer, 1968). In fact, Newton first described his methods, formulas and concepts of calculus, including his binomial theorem, fluxions and tangents, in letters he wrote to Leibniz (Ball, 1908). However an examination of Leibniz' unpublished manuscripts provided evidence that despite his correspondence...

...THE HISTORY OF CALCULUS
The discovery of calculus is often attributed to two men, Isaac Newton and Gottfried Leibniz, who independently developed its foundations. Although they both were instrumental in its creation, they thought of the fundamental concepts in very different ways. 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. 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...

...mathematician. With Leibniz’s ideas and theories, he became a prominent figure in mathematics and was a major mathematician that greatly contributed to calculus.
Leibniz’s contribution to the invention of infinitesimal calculus was monumental and is widely recognized as modern mathematics starting point. Leibniz’s Nova Methodus pro Maximis et Minimis, itemque Tangentibus… in Acta Eruditorum, in 1684, published Leibniz’s details of his ideas of differentialcalculus. The paper contained the d notation, the derivatives of powers rule, the quotient rule, and product rule. But the journal did not contain any proof of the ideas. In Acta Eruditorum, a paper was published by Leibniz which dealt with calculus integrals and had the first appearance of the integral notation. The next year Newton’s Principia appeared and contained Newton’s ‘method of fluxions’ but Newton failed to get it published in 1671 which resulted in a major dispute between Newton and Leibniz. Newton claimed Leibniz’s ideas were plagiarized from his work and ideas but there was no proof of the claim. Newton’s approach to it was geometrical, while Leibniz’s approach was algebraical. Leibniz’s language was appropriate for the ideas presented. Leibniz’s crystallization of infinitesimal calculus is considered as his designing of a new problem solving process. His discovery of that calculus extended the mathematical treatment to...

...History of Calculus
The history of calculus falls into several distinct time periods, most notably the ancient, medieval, and modern periods. The ancient period introduced some of the ideas of integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Egyptian Moscow papyrus (c. 1800 BC), in which an Egyptian successfully calculated the volume of a pyramidal frustum.[1][2] From the school of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus.[3] The method of exhaustion was later used in China by Liu Hui in the 3rd century AD in order to find the area of a circle. It was also used by Zu Chongzhi in the 5th century AD, who used it to find the volume of a sphere.[2]
In AD 499 the Indian mathematician Aryabhata used the notion of infinitesimals and expressed an astronomical problem in the form of a basic differential equation.[4] This equation eventually led Bhāskara II in the 12th century to develop an early derivative representing infinitesimal change, and he described an early form of "Rolle's theorem".[5] Around AD 1000, the Islamic...

...History of Differential Calculus
Universidad Iberoamericana
September 20, 2013
Ever since men felt the need to count, the history of calculus begins, which together with Mathematics is one of the oldest and most useful science. Since men felt that need for counting objects, this need led to the creation of systems that allowed them to maintain control of their properties. They initially did it with the use of fingers, legs, or stones. But as humans continued developing intellectually, they achieved to implement systems or more advanced forms that allowed them to solve problems.
The Egyptians were the first civilization to develop mathematical knowledge. They devised numeral systems through hieroglyphs, representing the numbers 1, 10 and 100 through sticks and human figures. This system evolved into what we know today as the Roman system. Other important civilizations in history, such as the Babylonians, created other numeral systems, where the solution to the problem of counting the objects was solved with the implementation of a sexagesimal method. Civilizations as ancient China and ancient India used a hieroglyph decimal system, with the characteristic that these implemented the number zero.
The progress achieved ever since each culture implemented their numeral system are still used today. The algebraic advance of the Egyptians resulted in the resolution to significant equations. The correct...