In its modern sense, calculus includes several mathematical disciplines. Most commonly, it means the detailed analysis of rates of change in functions and is of utmost importance in all sciences. It is usually divided into two (closely related) branches: differential calculus and integral calculus.
Calculus is concerned with concepts such as the rate of change of one variable quantity with respect to another, the slope of a curve at a prescribed point, the computation of the maximum and minimum values of functions, and the calculation of the area bounded by curves. Evolved from algebra, arithmetic, and geometry, it is the basis of that part of mathematics called analysis. Analytical topics depend on the concepts of calculus, such as differential equations, vector analysis, real analysis, and complex analysis.
Differential calculus involves derivatives and deals with such topics as maxima and minima and rates of change of functions. Calculus can be applied to many problems involving the notion of extreme amounts, such as the fastest, the most, the slowest, or the least. These maximum or minimum amounts may be described as values for which a certain rate of change (increase or decrease) is zero.
By using calculus it is possible to determine how high a projectile will go by finding the point at which its change of altitude with respect to time, that is, its velocity, is equal to zero.
Many general principles governing the behavior of physical processes are formulated almost invariably in terms of rates of change.
The definite integral, studied in integral calculus, can be used to find areas and volumes of irregular figures, to find lengths of curves, and to determine convergence or divergence of infinite series of numbers.
Calculus is widely employed in the physical, biological, and social sciences. It is used in physics to study the speed of a falling body, the rates of change in a chemical reaction, or the rate of decay of a radioactive material. In biology a problem such as the rate of growth of a colony of bacteria as a function of time is easily solved using calculus. In the social sciences calculus is widely used in the study of statistics and probability.
The fundamental concept of calculus, which distinguishes it from other branches of mathematics and is the source from which all its theory and applications are developed, is the theory of limits of functions of variables.
