Calculus, the Latin word for "pebble", refers to several mathematical disciplines. Most commonly, it refers to the detailed analysis of the 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.
The first, differential calculus, is concerned with finding the instantaneous rate of change (or derivative) of a function's value with respect to changes in its argument (roughly speaking, how much the value of a function changes with a small change in its argument). This derivative can also be interpreted as the slope of the function's graph at a specific point.
Initially, the derivative is defined via a process involving taking the limit of secant slopes as the two points defining the secant converge and the secant turns into a tangent line. This formula is called the difference quotient or Newton quotient, after Sir Issac Newton, who discovered it. The Newton quotient is:
f'(x)= lim (f(x+h) - f(x)) h->0 h
where h is the distance between the two secant points.
Since immediately substituting 0 for h would lead to 0/0, which cannot be computed, the numerator must first be simplified until h can be factored out and then canceled against the h of the denominator. The resulting equation is the derivative of the function, or the compilation of the instantaneous slopes at each point x.
These messy limit calculations can be avoided however because of powerful differentiation rules which allow us to find derivatives easily using simple algebraic manipulations. See derivative for the details.
At a maximal or minimal point, a function's derivative must be zero, and this yields a very useful optimization method.
The second branch of calculus, integral calculus, studies methods for finding the integral of a function. An integral may be defined as the limit of a sum of terms which correspond to areas under the graph of a function. Considered as such, integration allows us to calculate the area under a curve and the surface area and volume of solids such as spheres and cones.
The Fundamental Theorem of Calculus states that derivatives and intergrals are inverse operations. This allows us to recover the total change in a function over some interval from its instantaneous rate of change, by integrating the latter. The fundamental theorem also provides a method to compute many integrals algebraically, without actually performing the limit process, by finding antiderivatives. It also allows us to solve some differential equations, equations that relate an unknown function to its derivative. Differential equations are ubiquitous in the sciences.
The conceptual foundations of calculus include the function, limit, infinite sequences, infinite series and continuity. Its tools include the Symbol Manipulation techniques associated with School Algebra, and mathematical induction.
Gottfried Wilhelm Leibniz and Sir Isaac Newton independently invented differential and integral calculus in the late 1600's. Newton (who represented derivatives as f', f'', etc.) provided a host of applications in physics, but Leibniz' superior notation (df/dx, d^2f/(dx)^2, etc.) was eventually adopted.
Other disciplines called "calculus" include
- Lambda calculus (a formulation of the theory of computation)
- Predicate calculus (the rules governing the logic of predicates, see symbolic logic)