Helen Yang 1/19/15
AP Calculus BC: Extra Credit Pd 2&3
The History of Calculus
Calculus is the study of how things change. This branch of mathematics mainly focuses on limits, functions, derivatives, integrals and infinite series. When discussing the history of calculus, we often give Newton all the credit for the discovery, however it was actually developed independently by two different men in the seventeenth century, Gottfried Wilhelm Leibniz, a self-taught German mathematician, and Isaac Newton, an English scientist. They established the basic principles of calculus, and, with the help of other mathematicians, it was refined using the concept of the limit. Although both men were fundamental in the creation of calculus, they had different views. Newton considered variables changing with time while Leibniz thought of the variables x and y as ranging over sequences of infinitely close values. Leibniz developed dx and dy as the difference between values of these sequence to find the tangent. Newtown used x’ and y’ as finite velocities to compute the tangent. Neither Leibniz nor Newtown thought in terms of functions but instead in terms of graphs. Newton had a geometrical viewpoint of calculus, relating it to the physical world while Leibniz had an analytical viewpoint. In their development of calculus, both used “infinitesimals”, which are quantities that are infinitely small but nonzero. Although it was helpful with their computations and derivations, mathematicians disapprove of this concept of “infinitesimals” because they do not really exist. Lord Bishop Berkeley referred to them as “the ghost of departed quantities”. Berkeley’s criticisms played an important role in the development of calculus because later on, Cauchy, Weierstrass, and Riemann replaced infinitesimals with limits, the notion of quantities being “close” to others. The development of calculus goes through a timeline of three periods: Anticipation, Development, and Rigorization. During the Anticipation stage mathematicians used techniques that involved infinite processes to find areas under curves. In the Development stage Newton and Leibniz created the foundations of calculus and tied all the techniques together under derivative and integral. The ideas were sloppy and not always logically correct. It took many years during the Rigorization stage to refine and clean up those ideas and finally finalize the mathematical foundation of calculus. Newton built his theory of calculus on earlier works by René Descartes, Pierre de Fermat, and other Continental mathematicians. In 1664 he laid the foundations of the differential calculus, which he described as the "method of fluxions. ”The initial problem Newton dealt with was calculation the exact slope
The History of the Calculus The Calculus, being a difficult subject requires much more than the intuition and genius of one man. It took the work and ideas of many great men to establish the advanced concepts now known as calculus. The history of the Calculus can be traced back to c. 1820 BC to the Egyptian Moscow papyrus, in which an Egyptian successfully calculated the volume of a pyramidal frustum. [1][2] Calculating volumes and areas, the basic function of integral calculus, can…
The History of Calculus According to the Miriam-Webster dictionary calculus is a method of computation or calculation in a special notation (as of logic or symbolic logic). It is the field of mathematics that analyzes aspects of change in processes or systems that can be modeled by functions. Through its two primary tools—the derivative and the integral —it allows precise calculation of rates of change and of the total amount of change in such a system. The derivative and the integral grew out of…
Newton vs. Leibniz; The Calculus Controversy Like most 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. As the renowned…
to the healthy controlled group showed higher DMFT and DMFS ranges. The examiners proclaimed “significant build of plaque, gingivitis, and calculus compared with the controlled group”. The study proclaimed the high calculus and plaque is due to an increase in the levels of calcium and phosphorous in submaxillary and parotid saliva. The high prevalence of calculus tends to lead to increased gingival conditions, such as inflammation. It is safe to make a link between gingival conditions such as inflammation…
Daasebre N. Asante Mr. Nazarro Honors World History 3/21/13 Isaac Newton Isaac Newton had a huge impact on Europe through his discoveries. His discovery of formulas that could be used to find the areas of curved objects led to many of the mathematical breakthroughs in Europe at the time; as well as discovering the laws of motion and the laws gravity, in the field of science. All of these discoveries by Isaac Newton lead to great new ways of European thinking in Science, Math, and even Theology…
Pre-Nursing Advising Academic Area Courses Required Area A – Essential Skills ENGL 1101 and ENGL 1102 (English Comp I and II) MATH 1111 (College Algebra) or MATH 1113 (Pre-Calculus) or MATH 2200 (Calculus) Area B – Institutional Option ITEC 1001-Introduction to Computing Area C – Humanities and Arts MUSC 1100 (Music Appreciation) or ARTS 1100 (Art Appreciation) FILM 1005 (Intro to Film) or ENGL 2110 (World Literature) or ENGL 2100 (Trans-Atlantic English Literature) RELN…
Augustin- Louis Cauchy Augustin-Louis Cauchy, born in 1789, was a French mathematician who published 789 papers and seven books over the course of his lifetime and sculpted elementary calculus limit concepts such as continuity, the derivative and the integral. He is known as a “foremost French mathematician of the nineteenth century.” Augustin-Louis Cauchy was born into a lower-class family one month prior to the French Revolution. His father worked for the government but lost his job after…
Den 102 Final Test 1. Personal History, health history, dental history, oral inspection, gingival evaluation, probing chart/bleeding points, restorative exam/dental chart, patient classifications, disease risk factors, CAMBRA, conversation with the patient, PHP scores, radiographs 2. Five Principles of Instruction a. Present small amounts of info at one time b. Supervise the pt. practice/get them involved (VIP) anterior/posteriors c. Let the pt. set his/her own pace d. Use positive reinforcement…
devised the “Hedonic Calculus” this considered seven principles, each of which could be given a numerical score. The key to utilitarianism is to consider the consequences of a particular action rather than just saying that an action is intrinsically wrong in and of itself. Actions are morally neutral only their consequences have any moral value, they are not intrinsically good or bad. Bentham argued that these consequences could be calculated by the hedonic calculus. The Hedonic Calculus consisted of seven…
III Pre-Calculus Pre-Calculus Pre-AP AP Calculus AP Statistics Math 116-WKU AP Computer Science Foundations in Personal Finance Diploma Choices Merit Diploma 22 credits minimum required Comprehensive Diploma 24 credits minimum required Merit Diploma English English I, II, III, IV 4 credits Science Any three science classes 3 credits to include: life science, physical science, and earth/space science. Math Alg. I, Geometry, Alg. II Social Studies World Civ., U. S. History, Political…