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Calculus-From Wikipedia, the free encyclopedia
For other uses, see Calculus (disambiguation). Topics in calculus Fundamental theorem Limits of functions Continuity Vector calculus Tensor calculus Mean value theorem
Differentiation Product rule Quotient rule Chain rule Implicit differentiation Taylor's theorem Related rates Table of derivatives
Integration Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution
Calculus [from Latin, literally "pebble" (used in reckoning)] is a major area in mathematics. The methods of calculus are used to solve problems which cannot be solved using algebra, problems that in algebra lead to paradoxes involving division by zero or sums of infinitely many numbers. These paradoxes arise, for example, in the study of velocity and of area. The ancient Greek philosopher Zeno gave examples of such paradoxes; calculus provides tools, especially the limit, the derivative, the integral, and the infinite series, which resolve the paradoxes and lead to the solution of such problems.
Calculus builds on analytic geometry. The two major branches of calculus, differential calculus and integral calculus, are related by the Fundamental Theorem of Calculus.
Applications of differential calculus include computations involving position, velocity, and acceleration, the slope of a curve, related rates, and optimization. Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and force. More advanced applications include power series and Fourier series. Calculus can be used to compute the trajectory of a shuttle docking at a space station or the amount of snow in a driveway.
While some of the ideas of calculus were developed earlier, in ancient Greece and in India, the modern use of calculus began in Europe, during the 17th century, when Isaac Newton and Gottfried Wilhelm Leibniz built on the work of earlier mathematicians to introduce the basic principles of calculus. This work had a strong impact on the development of physics.
Contents 1 Differentiation 2 Integration 3 Fundamental theorem 4 Foundations 5 History 6 Applications 7 See also 7.1 Lists 7.2 Related topics 8 References 9 Further reading 10 External links 10.1 Online books 10.2 Web pages
Differentiation Main article: Derivative Differential calculus defines a linear operator called the derivative, and describes its properties and applications.
Tangent line at (x, f(x)). The derivative f ' (x) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point.The concept of the derivative is fundamentally more advanced than the concepts encountered in algebra. In algebra, students learn about functions which input a number and output another number. For example, if the doubling function inputs 3, then it outputs 6, while if the squaring function inputs 3, it outputs 9. But the derivative inputs a function and outputs another function. For example, if the derivative inputs the squaring function, then it outputs the doubling function, because the doubling functions gives the slope of the squaring function at any given point.
To understand the derivative, students must master mathematical notation. In mathematical notation, one common symbol for the derivative of a function is an apostrophe-like mark called "prime". Thus the derivative of f is f ' (f prime). The last sentence of the preceding paragraph, in mathematical notation, would be written:
. The derivative of a function measures the slope of the function. If the input of the function is time, then the derivative measures the rate at which the function changes.
If a function is linear (that is, if the graph of the function is a straight line), then the function can be written y = m x + b, where:
. This gives an exact value for the slope of a straight line. If the function is not a straight line, however then the change in y divided by the change in x varies, and we need calculus to find an exact value at a given point. (Note that y and f(x) are two different notations for the same thing: the output of the function.) A line through two points on a curve is called a secant line. The slope, or rise over run, of a secant line can be expressed as:
where the coordinates of the first point are (x, f(x)) and h is the horizontal distance between the two points.
To determine the instantaneous rate of change we use the limit:
Working out one particular case, we find the slope of the squaring function at the point where the input is 3 and the output is 9. (i.e. f(x) = x2)
The slope of the squaring function at the point (3, 9) is 6, that is to say, it is going up six times as fast as it is going to the right.
For more technical information on the limit and the derivative, see their respective articles.
Integration Main article: Integral Integration is applied in two (related) ways.
The indefinite integral is the antiderivative, the inverse operation to the derivative. F is an antiderivative of f when f is a derivative of F. (This use of upper- and lower-case letters is common in calculus.)
The definite integral is the limit of a sum, called a Riemann sum. A motivating example is the distances traveled in a given time.
If the speed is constant, only multiplication is needed, but if the speed changes, then we need a more powerful method of finding the distance. One such method is to approximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum (Riemann sum) of the approximate distance traveled in each interval. The basic idea is that if only a short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact distance traveled.
Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b) by subdividing the area into ever-smaller slices and then adding them all up.If f(x) in the diagram on the left represents speed as it varies over time, the distance traveled between the times represented by a and b is the area of the shaded region s.
To approximate that area, an intuitive method would be to divide up the distance between a and b in to a number of equal segments, the length of each segment represented by the symbol Δx. For each small segment, we can choose one value of the function f(x). Call that value h. Then the area of the rectangle with base Δx and height h gives the distance (time Δx multiplied by speed h) traveled in that segment. Associated with each segment is the average value of the function above it, f(x)=h. The sum of all such rectangles gives an approximation of the area between the axis and the curve, which is an approximation of the total distance traveled. A smaller value for Δx will give more rectangles and in most cases a better approximation, but for an exact answer we need to take a limit as Δx goes to zero.
The symbol of integration is , an elongated S (which stands for "sum"). The definite integral is written as:
and is read "the integral from a to b of f-of-x with respect to x."
The indefinite integral, or antiderivative, is written:
. Since the derivative of the function y = x2 + C is y ' = 2x (where C is any constant)
. For a more technical discussion, see the main article, integral.
Fundamental theorem Main article: Fundamental theorem of calculus The fundamental theorem of calculus states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the Fundamental Theorem of Calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration.
The Fundamental Theorem of Calculus states: If a function f is continuous on the interval [a, b] and if F is a function whose derivative is f on the interval (a, b), then
Furthermore, for every x in the interval (a, b),
This realization, made by both Newton and Leibniz, who based their results on earlier work by Isaac Barrow, was key to the massive proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.
Foundations In mathematics, foundations refers to the rigorous development of a subject from precise axioms and definitions. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz and is still to some extent an active area of research today.
There is more than one rigorous approach to the foundation of calculus. The usual one is via the concept of limits defined on the continuum of real numbers. An alternative is nonstandard analysis, in which the real number system is augmented with infinitesimal and infinite numbers. The foundations of calculus are included in the field of real analysis, which contains full definitions and proofs of the theorems of calculus as well as generalisations such as measure theory and distribution theory.
History Main article: History of calculus The history of calculus falls into several distinct periods, most notably the Greek, Indian, and European periods.
The Greek period introduced some of the ideas of integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Eudoxus (circa 408 BCE - circa 355 BCE) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes. Archimedes (circa 287 BCE - 212 BCE) developed this idea further, inventing heuristics which resemble integral calculus.[1]
Indian mathematics, largely unknown in the west until the 20th century, produced a number of works with some ideas of calculus. The mathematician-astronomer Aryabhata in 499 CE used a notion of infinitesimals and expressed an astronomical problem in the form of a basic differential equation.[2] Manjula, in the 10th century, elaborated on this differential equation in a commentary. This equation eventually led Bhaskara in the 12th century to develop a proto-derivative representing infinitesimal change, and describe an early form of "Rolle's theorem".[3] In the 14th century, Madhava, along with other mathematician-astronomers of the Kerala School, described special cases of Taylor series[4] which are treated in the text Yuktibhasa.[5][6][7]
Sir Isaac Newton Gottfried Wilhelm LeibnizThe second half of the 17th century was a time of major innovation in Europe. Calculus provided a new opportunity in mathematical physics to solve long-standing problems. Several mathematicians contributed to these breakthroughs, notably John Wallis and Isaac Barrow. James Gregory proved a special case of the second fundamental theorem of calculus in 1668.
Leibniz and Newton pulled these ideas together into a coherent whole and they are usually credited with the independent and nearly simultaneous invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today; he often spent days determining appropriate symbols for concepts. The fundamental insight that both Newton and Leibniz had was the fundamental theorem of calculus.
When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first, but Leibniz published first. Newton claimed Leibniz stole ideas from his unpublished notes. This controversy divided English-speaking mathematicians from continental mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. Today, both Newton and Leibniz are given credit for developing calculus independently. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus the "the science of fluxions".
Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Cauchy, Riemann, and Weierstrass. It was also during this period that the ideas of calculus were generalized to Euclidean space and the complex plane. Lebesgue further generalized the notion of the integral.
There were isolated instances mathematicians in other cultures who made independent discoveries in calculus. For example, in Japan Seki Kowa (circa 1637 - 1708) expanded further upon Eudoxus's method of exhaustion.
Today, calculus is studied around the world, and mathematicians worldwide contribute to new advances in calculus.
Applications Calculus is used in every branch of the physical sciences, in computer science, statistics, engineering, economics, business, medicine, and in other fields wherever a problem can be mathematically modeled and an optimal solution is desired.
Physics makes particular use of calculus; all concepts in classical mechanics are interrelated through calculus. Newton's second law of motion expressly uses the term "rate of change" which is the derivative: The rate of change of momentum of a body is equal to the resultant force acting on the body and is in the same direction. Even the common expression of Newton's second law as: Force = Mass × Acceleration, involves differential calculus because acceleration is the derivative of velocity. Maxwell's theory of electromagnetism and Einstein's theory of general relativity are also expressed in the language of differential calculus.
In analytic geometry, the study of graphs of functions, calculus is used to find high points and low points (maxes and mins), slope, concavity and inflection points.
Calculus is used to find approximate solutions to equations, in methods such as Newton's method and fixed point iteration.
See also
Lists List of basic calculus topics List of basic calculus equations and formulas List of calculus topics Publications in calculus
Related topics Calculus with polynomials Differential geometry Mathematics Multivariable calculus Non-standard analysis Precalculus (mathematical education)
References ^ Archimedes, Method, in The Works of Archimedes ISBN 978-0-521-66160-7 ^ Aryabhata the Elder ^ Ian G. Pearce. Bhaskaracharya II. ^ Madhava. Biography of Madhava. School of Mathematics and Statistics University of St Andrews, Scotland. Retrieved on 2006 September 13. ^ An overview of Indian mathematics. Indian Maths. School of Mathematics and Statistics University of St Andrews, Scotland. Retrieved on 2006 July 7. ^ Science and technology in free India. Government of Kerala — Kerala Call, September 2004. Prof.C.G.Ramachandran Nair. Retrieved on 2006 July 9. ^ Charles Whish (1835). Transactions of the Royal Asiatic Society of Great Britain and Ireland. Donald A. McQuarrie (2003). Mathematical Methods for Scientists and Engineers, University Science Books. ISBN 9781891389245 James Stewart (2002). Calculus: Early Transcendentals, 5th ed., Brooks Cole. ISBN 9780534393212
Further reading Robert A. Adams. (1999) ISBN 978-0-201-39607-2 Calculus: A complete course. Albers, Donald J.; Richard D. Anderson and Don O. Loftsgaarden, ed. (1986) Undergraduate Programs in the Mathematics and Computer Sciences: The 1985-1986 Survey, Mathematical Association of America No. 7, John L. Bell: A Primer of Infinitesimal Analysis, Cambridge University Press, 1998. ISBN 978-0-521-62401-5. Uses synthetic differential geometry and nilpotent infinitesimals Florian Cajori, "The History of Notations of the Calculus." Annals of Mathematics, 2nd Ser., Vol. 25, No. 1 (Sep., 1923), pp. 1-46. Leonid P. Lebedev and Michael J. Cloud: "Approximating Perfection: a Mathematician's Journey into the World of Mechanics, Ch. 1: The Tools of Calculus", Princeton Univ. Press, 2004 Cliff Pickover. (2003) ISBN 978-0-471-26987-8 Calculus and Pizza: A Math Cookbook for the Hungry Mind. Michael Spivak. (Sept 1994) ISBN 978-0-914098-89-8 Calculus. Publish or Perish publishing. Silvanus P. Thompson and Martin Gardner. (1998) ISBN 978-0-312-18548-0 Calculus Made Easy. Mathematical Association of America. (1988) Calculus for a New Century; A Pump, Not a Filter, The Association, Stony Brook, NY. ED 300 252. Thomas/Finney. (1996) ISBN 978-0-201-53174-9 Calculus and Analytic geometry 9th, Addison Wesley. Weisstein, Eric W. "Second Fundamental Theorem of Calculus." From MathWorld--A Wolfram Web Resource. ..
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