If the “curve” happens to be a straight line, the value of will be the same at all points along it. In other words its slope is constant.If a curve is one that turns more upwards as it goes along to the right, the values of will become greater and greater with the increasing steepness.
Calculus Made Easy is a book on infinitesimal calculus originally published in 1910 by Silvanus P. Thompson. The original text continues to be available as of 2008 from Macmillan and Co., but a 1998 update by Martin Gardner is available from St. Martin's Press which provides an introduction; three preliminary chapters explaining functions, limits, and derivatives; an appendix of recreational calculus problems; and notes for modern readers. Gardner changes "fifth form boys" to the more American sounding (and gender neutral) "high school students," updates many now obsolescent mathematical notations or terms, and uses American decimal dollars and cents in currency examples.
Calculus Made Easy ignores the use of limits with its epsilon-delta definition, replacing it with a method of approximating (to arbitrary precision) directly to the correct answer in the infinitesimal spirit of Leibniz, now formally justified in modern nonstandard analysis and smooth infinitesimal analysis.
The original text is now in the public domain under US copyright law (although Macmillan's copyright under UK law is reproduced in the 2008 edition from St. Martin's Press). It can be freely accessed on Project Gutenberg.
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation.
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Nonstandard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number is said to be finite if, and only if, for some integer . is said to be infinitesimal if, and only if, for all integers . The term "hyper-real" was introduced by Edwin Hewitt in 1948.
In mathematics, an infinitesimal number is a quantity that is closer to 0 than what any standard non-zero real number is, but is not 0. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinity-th" item in a sequence.
The Analyst is a book by George Berkeley. It was first published in 1734, first by J. Tonson (London), then by S. Fuller (Dublin). The "infidel mathematician" is believed to have been Edmond Halley, though others have speculated Sir Isaac Newton was intended.
In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively.
In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic.
Calculus, originally called infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Many elements of calculus appeared in ancient Greece, then in China and the Middle East, and still later again in medieval Europe and in India. Infinitesimal calculus was developed in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz independently of each other. An argument over priority led to the Leibniz–Newton calculus controversy which continued until the death of Leibniz in 1716. The development of calculus and its uses within the sciences have continued to the present.
Silvanus Phillips Thompson was an English professor of physics at the City and Guilds Technical College in Finsbury, England. He was elected to the Royal Society in 1891 and was known for his work as an electrical engineer and as an author. Thompson's most enduring publication is his 1910 text Calculus Made Easy, which teaches the fundamentals of infinitesimal calculus, and is still in print. Thompson also wrote a popular physics text, Elementary Lessons in Electricity and Magnetism, as well as biographies of Lord Kelvin and Michael Faraday.
Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete entities. As a theory, it is a subset of synthetic differential geometry. Terence Tao has referred to this concept under the name "cheap nonstandard analysis."
William Fogg Osgood was an American mathematician.
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of contemporary mathematics education. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.
In physics, the differential coefficient of a function f(x) is what is now called its derivative df(x)/dx, the (not necessarily constant) multiplicative factor or coefficient of the differential dx in the differential df(x).
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal , the unique real infinitely close to it, i.e. is infinitesimal. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat, as well as Leibniz's Transcendental law of homogeneity.
Nonstandard analysis and its offshoot, nonstandard calculus, have been criticized by several authors, notably Errett Bishop, Paul Halmos, and Alain Connes. These criticisms are analyzed below.
Infinity is something which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Elementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of the hyperreal number system of Abraham Robinson and is sometimes given as An approach using infinitesimals. The book is available freely online and is currently published by Dover.
In mathematical analysis, the staircase paradox is a pathological example showing that limits of curves do not necessarily preserve their length. It consists of a sequence of "staircase" polygonal chains in a unit square, formed from horizontal and vertical line segments of decreasing length, so that these staircases converge uniformly to the diagonal of the square. However, each staircase has length two, while the length of the diagonal is the square root of 2, so the sequence of staircase lengths does not converge to the length of the diagonal. Martin Gardner calls this "an ancient geometrical paradox". It shows that, for curves under uniform convergence, the length of a curve is not a continuous function of the curve.
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