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:

finding the acceleration and velocity of a freefalling body at a particular moment

finding the optimal number of units a company should produce to maximize its profit
Examples of integral calculus problems include:

finding areas and volumes

finding the amount of water pumped by a pump with a set power input but varying conditions of pumping losses and pressure

finding the amount of parking lot plowed by a snowplow of given power with varying rates of snowfall
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.
Additional information:
For a much better overview and history of calculus, go to the
MathWorld
This site claims to be the web's most extensive mathematics resource, and a mathematician we know says it's a pretty good site. It doesn't deal with arithmetic, but the more advanced aspects of mathematics, like statistics and calculus.
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