Calc2 - MATH 104 study guide PDF

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MATH 104 study guide...

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Limit (or, Convergence) the value that a function or sequence "approaches" as the input or index approaches some value. these are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals. can be used to resolve Zeno's paradoxes Sequence an ordered collection of objects in which repetitions are allowed. it contains members (also called elements, or terms). the number of elements (possibly infinite) is called its length. order matters, and exactly the same elements can appear multiple times at different positions in this collection. Series the sum of the terms of a sequence. Isaac Newton an English physicist and mathematician (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time and as a key figure in the scientific revolution. considered to be one of two people who developed modern calculus in the 17th century. he developed the use of calculus in his laws of motion and gravitation. The product rule and chain rule, the notion of higher derivatives, Taylor series, and analytical functions were introduced by him in an idiosyncratic notation which he used to solve problems of mathematical physics. In his works, he rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, and many other problems discussed in his Principia Mathematica (1687). In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable

Gottfried Leibniz a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy. considered to be one of two people who developed modern calculus in the 17th century. he was the first to publish his results on the development of calculus. He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. He paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. he developed much of the notation used in calculus today. Science a systematic enterprise that creates, builds and organizes knowledge in the form of testable explanations and predictions about the universe Engineering the application of mathematics, empirical evidence and scientific, economic, social, and practical knowledge in order to invent, innovate, design, build, maintain, research, and improve structures, machines, tools, systems, components, materials, and processes Economics the social science that describes the factors that determine the production, distribution and consumption of goods and services...