|
Calculus is a central branch of mathematics. The word stems from the nascent development of mathematics: the early Greeks used pebbles arranged in patterns to learn arithmetic and geometry, and the Latin word for "pebble" is "calculus." Calculus is built on two major complementary ideas, both of which rely critically on the concept of limits. The first is differential calculus, which is concerned with the instantaneous rate of change of quantities with respect to other quantities, or more precisely, the local behavior of functions. This can be illustrated by the slope of a function's graph. The second is integral calculus, which studies the accumulation of quantities, such as areas under a curve, linear distance traveled, or volume displaced. These two processes act inversely to each other, as shown by the fundamental theorem of calculus. Examples of typical differential calculus problems include: Examples of integral calculus problems include: Today, calculus is used in every branch of the physical sciences, in computer science, in statistics, and in engineering; in economics, business, and medicine; and as a general method whenever the goal is an optimal solution to a problem that can be given in mathematical form.
Differential calculus The derivative measures the sensitivity of one variable to small changes in another variable. Consider the formula: for an object moving at constant speed. The speed of a car, as measured by the speedometer, is the derivative of the car's distance traveled, as measured by the odometer, as a function of time. Calculus is a mathematical tool for dealing with this complex but natural and familiar situation. Differential calculus can be used to determine the instantaneous speed at any given instant, while the formula "speed = distance divided by time" only gives the average speed, and cannot be applied to an instant in time because it then gives an undefined quotient zero divided by zero. Calculus avoids division by zero by using the concept of the limit which, roughly speaking, is a method of controlling an otherwise uncontrollable output, such as division by zero or multiplication by infinity. More formally, differential calculus defines the instantaneous rate of change (the derivative) of a mathematical function's value, with respect to changes of the variable. The derivative is defined as a limit of a difference quotient. The derivative of a function, if it exists, gives information about its graph. It is useful for finding optimum solutions to problems, called maxima and minima of a function. It is proved mathematically that these optimum solutions exist either where the tangent of the graph is flat, so that the slope is zero; or where the graph has a sharp turn (cusp) where the derivative does not exist; or at the endpoints of the graph. Another application of differential calculus is Newton's method, a powerful equation solving algorithm. Differential calculus has been applied to many questions that were first formulated in other areas, such as business or medicine. The derivative lies at the heart of the physical sciences. 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. (See Differential equation.) Maxwell's theory of electromagnetism and Einstein's theory of general relativity are also expressed in the language of differential calculus, as is the basic theory of electrical circuits and much of engineering. It is also applied to problems in biology, economics, and many other areas. The derivative of a function y = f(x) with respect to x is usually expressed as either y ′ (read "y-prime"), f ' (x) (read "f-prime of x") or as which is commonly shortened to: Integral calculus There are two types of integral in calculus, the indefinite and the definite. The indefinite integral is simply the antiderivative. That is, F is an antiderivative of f when f is a derivative of F. (This use of capital letters and lower case letters is common in calculus. The lower case letter represents the derivative of the capital letter.) The definite integral evaluates the cumulative effect of many small changes in a quantity. The simplest instance is the formula for calculating the distance a car moves during a period of time when it is traveling at constant speed. The distance moved is the cumulative effect of the small distances moved in each instant. Calculus is also able to deal with the natural situation in which the car moves with changing speed. Integral calculus determines the exact distance traveled during an interval of time by creating a series of better and better approximations, called Riemann sums, that approach the exact distance as a limit. More formally, we say that the definite integral of a function on an interval is a limit of Riemann sum approximations. Applications of integral calculus arise whenever the problem is to compute a number that is in principle (approximately) equal to the sum of a large number of small quantities. The classic geometric application is to area computations. In principle, the area of a region can be approximated by chopping it up into many pieces (typically rectangles, or, in polar coordinates, circular sectors), and then adding the areas of those pieces. The length of an arc, the area of a surface, and the volume of a solid can also be expressed as definite integrals. Probability, the basis for statistics, provides another important application of integral calculus. The symbol of integration is ∫, a stretched s (which stands for "sum"). The precise meanings of expressions involving integrals can be found in the main article Integral. The definite integral, written as: is read "the integral from a to b of f-of-x dx". Foundations 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 tools of calculus include techniques associated with elementary algebra, and mathematical induction. The foundations of calculus are included in the field of real analysis, which contains all full definitions and proofs of the theorems of calculus as well as generalisations such as measure theory and distribution theory. Fundamental theorem of calculus The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. More precisely, if one defines one function as the integral of another continuous function, then differentiating the newly defined function returns the function you started with. Furthermore, if you want to find the value of a definite integral, you usually do so by evaluating an antiderivative. Here is the mathematical formulation of the Fundamental Theorem of Calculus: 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 Also, for every x in the interval ''a'', ''b'', This realization, made by both Newton and Leibniz, 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. Applications The development and use of calculus has had wide reaching effects on all areas of modern living. It underlies nearly all of the sciences, especially physics. Virtually all modern developments such as building techniques, aviation, and other technologies make fundamental use of calculus. Many algebraic formulas now used for ballistics, heating and cooling, and other practical sciences were worked out through the use of calculus. In a handbook, an algebraic formula based on calculus methods may be applied without knowing its origins. The success of calculus has been extended over time to differential equations, vector calculus, calculus of variations, complex analysis, and differential topology. History
See also Further reading Online books Web pages | ||||||||||
|
| |||||||||||
 |
|
| |