Calculus

Calculus uses limits to study quantities that vary continuously. A derivative describes how quickly a function changes at a particular input; geometrically, it is the slope of the tangent to the function's graph at that point. For example, if position is given as a function of time, its derivative gives instantaneous velocity. An integral, by contrast, can represent an accumulated total, such as displacement obtained from a changing velocity or the area beneath a curve. Definite integrals can be understood through limits of sums of many small contributions.

The fundamental theorem of calculus establishes a deep connection between differentiation and integration. In one part, it relates the derivative of an accumulated integral to the function being accumulated; in another, it allows a definite integral to be evaluated using an antiderivative, when the theorem's conditions are met. This connection turns some problems about totals into problems of finding functions whose derivatives are known. The theorem does not make every integral easy to calculate, but it provides a general framework linking the two main branches of calculus.

Ideas that later contributed to calculus appeared over many centuries. Greek mathematicians developed methods for reasoning about areas and infinite processes; Archimedes used a method of exhaustion to find areas and volumes. In the seventeenth century, Newton and Leibniz independently developed systematic forms of calculus, building on earlier work by mathematicians including Fermat, Cavalieri and Barrow. Newton's approach emphasized quantities changing through time, while Leibniz developed notation such as d and the integral sign that remains widely used. Later mathematicians worked to make calculus foundations more rigorous, notably through nineteenth-century analysis.

Calculus helps model motion, changing populations, rates of chemical reactions, optimization, and the accumulation of quantities such as area or total distance. It is used in fields including physics, engineering, biology and economics. Calculus supplies mathematical tools, not a guarantee that a model describes the world accurately: conclusions depend on the assumptions, data and domain in which a model is applied. More advanced work extends introductory ideas to several variables, vector fields and differential equations.