3 What is Calculus?From Latin, calculus, a small stone used for countingA branch of mathematics including limits, derivatives, integrals, and infinite sumsUsed in science, economics, and engineeringBuilds on algebra, geometry, and trig with two major branches differential calculus and integral calculusCalculus comes from the Latin word calculus which were stones used for counting.Includes limits, derivatives, integrals and infinite sums. You will study the first three topics in detail next year. We will study limits in detail and derivatives a bit.Used in many science fields and economics for optimization and cost analysis.Builds on algebra, geometry and trig skills. That is why pre-calc focused so much on improving those skills. All of your previous math will come in use as you study calculus.
4 Ancient HistoryIn the earliest years, integral calculus was being used as an idea, but was not yet formalized into a system.Calculating volumes and areas can be traced to the Egyptian Moscow papyrus (1820 BC).The idea of calculating areas and volumes by using easier, known formulas has been used since ancient times. These ideas later became formal integration.The Egyptian Moscow papyrus is one of the oldest examples of calculating the surface area of a curved surface.
5 Ancient GreeksGreek mathematician Eudoxus ( BC) used the method of exhaustion, a precursor to limits, to calculate area and volumeArchimedes ( BC) continued Eudoxus’ idea and invented heuristics, similar to integration, to calculate area.Eudoxus formalized a method to find the area of a circle using inscribed polygons. Increase the number of the sides to reach the area of the circle.Archimedes used this method later to calculate pi, find other areas and volumes.
6 Medieval HistoryIn about 1000 AD, Islamic mathematician, Ibn al-Haytham (Alhacen) derived a formula for the sum of the fourth powers of an arithmetic progression, later used to perform integration.In the 12th century, Indian mathematician Bhaskara II developed an early derivative. He described an early form of what will later be “Rolle’s Theorem”Also in the 12th century, Persian mathematician Saraf al-Din al-Tusi discovered the derivative of a cubic polynomialThe studies in the medieval period were more focused on the derivative side of calculus rather than the integral side. This is a shift to more formal study of abstract mathematics. Notice, most of the study was done in the far east as opposed to Europe.
7 Modern HistoryBonaventure Cavalieri argued that volumes be computed by the sums of the volumes of cross sections. (This was similar to Archimedes’s).However, Cavalieri’s work was not well respected, so his infinitesimal quantities were not accepted at first.Cavalieri corresponded with Galileo and considered himself a disciple of Galileo. His work on infinitesimal quantities was correct but lacked mathematical rigor. Thus it was criticized and not fully accepted in his time.
8 Modern HistoryFormal study combined Cavalieri’s infinitesimal quantities with finite differences in Europe. This was done by John Wallis, Isaac Barrow, and James GregoryBarrow and Gregory would later prove the 2nd Fundamental Theorem of Calculus in 1675.Wallis used arithmetic to show his area under a curve as an integral (the first time in print). Wallis also introduced the symbol for infinity in his text.Barrow improved on Fermat’s work for a method of finding tangents (later derivatives). He and Gregory would prove the 2nd Fundamental Th. Of calculus. Although Gregory understood the ideas of calculus, he could not express them well and therefore credit for the discovery was not given to him.
9 Enter Newton…Isaac Newton (English) is credited with many of the beginnings of calculus. He introduced product rule, chain rule and higher derivatives to solve physics problems.He replaced the calculus of infinitesimals with geometric representations.He used calculus to explain many physics problems in his book Principia Mathematica, however he had developed many other calculus explanations that he did not formally publish.Newton attended Barrow’s lectures and later was recommended for his chair when Barrow retired from Trinity College. His study with astronomy and physics led to his formalizing of calculus as a mathematical study.
10 …and LeibnizGottfried Wilhelm Leibniz (German) systemized the ideas of calculus of infinitesimals. Unlike Newton, Leibniz provided a clear set of rules to manipulate infinitesimals.Leibniz spent time determining appropriate symbols and paid more attention to formality.His work leads to formulas for product and chain rule as well as rules for derivatives and integrals.Leibniz work gives us much of the common notation used in calculus today. His work was published one year before Newton’s book.
11 Newton vs. LeibnizThere was much controversy over who (and thus which country) should be credited with calculus since both worked at the same time.Newton derived his results first, but Leibniz published first.The debate continued past the deaths of both men.
12 Newton vs. LeibnizNewton claimed Leibniz stole ideas from unpublished notes written to the Royal Society.This divided English-speaking math and continental math for many years.
13 Newton vs. LeibnizToday it is known that Newton began his work with derivatives and Leibniz began with integrals. Both arrived at the same conclusions independently.The name of the study was given by Leibniz, Newton called it “the science of fluxions”.
14 Since then…There have been many contributions to build upon Newton and Leibniz.Calculus was put on a more rigorous footing by mathematicians such as Cauchy, Riemann, and WeierstrassCalculus has also been generalized for the Euclidean and complex space.
15 In conclusion…We will stand on the shoulders of those that came before us and study their findings to possibly apply to our modern world!