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History, Differential Calculus, Integral Calculus, Indefinite Integral, Definite Integral

Calculus is the branch of mathematics that deals with rates of change and motion. It grew out of a desire to understand various physical phenomena, such as the orbits of planets, and the effects of gravity. The immediate success of calculus in formulating physical laws and predicting their consequences led to development of a new division in mathematics called analysis, of which calculus remains a large part. Today, calculus is the essential language of science and engineering, providing the means by which physical laws are expressed in mathematical terms. As a scientific tool it is invaluable in the further analysis of physical laws, in predicting the behavior of electrical and mechanical systems governed by those laws, and in discovering new laws.

Calculus divides naturally into two parts, differential calculus and integral calculus. Differential calculus is concerned with finding the instantaneous rate at which one quantity changes with respect to another, called the derivative of the first quantity with respect to the second. For example, determining the speed of a falling body at a particular instant of time, say that of a skydiver or bungi jumper, is equivalent to calculating the instantaneous rate of change in his or her position with respect to time. In general, evaluating the derivative of a function, f(x), involves finding another function, f'(x), such that f'(x) is equal to the slope of the tangent to the graph of f(x) at each x. This is accomplished, for each 2, by determining the slope of an approximating line segment in the limit that its length approaches zero.

Integral calculus deals with the inverse of the derivative, namely, finding a function when its rate of change is known. For example, if a skydiver's velocity is a known function of time, then we may ask what is his or her position at any given time after jumping. Finding the original function, given its derivative, is called integration, and the function is called the indefinite integral. Evaluating the indefinite integral of any function between specific limits to definition of the definite integral, which is equal to the area under the graph of the function between the specified limits. The latter is developed as a natural consequence of approximating an area by summing the areas of a number of inscribed rectangles. The approximation becomes exact in the limit that the number of rectangles approaches infinity. Thus, both differential and integral calculus are based on the theory of limits.

The usefulness of calculus is indicated by its widespread application. For example, it is used in the design of navigation systems, particle accelerators, and synchrotron light sources. It is used to predict rocket trajectories, and the orbits of communications satellites. Calculus is the mathematical tool used to test theories about the origins of the universe, the development of tornadoes and hurricanes, and salt fingering in the oceans. It has even found extensive application in business, where it is used, among other things, to optimize production.

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