Statements of theorems
- Fundamental theorem of calculus. Let f be a real-valued In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in integrable function In mathematics a function is a relation between a given set of elements and another set of elements (the range), which associates each element in the domain with exactly one element in the range. The elements so related can be any kind of thing (words, objects, qualities) but are typically mathematical quantities, such as real numbers defined on a closed interval In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers x satisfying is an interval which contains 0 and 1, as well as all numbers between them. Other examples of intervals are the set of all real numbers , the [a, b]. If F is defined for x in [a, b] by
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- then F is continuous In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous. An intuitive though imprecise idea of continuity is given by the common on [a, b]. If f is continuous at x in [a, b], then F is differentiable Differential calculus, a field in mathematics, is the study of how functions change when their inputs change. The primary object of study in differential calculus is the derivative. A closely related notion is the differential. The derivative of a function at a chosen input value describes the behavior of the function near that input value. For a at x, and F ′(x) = f(x).
- Second fundamental theorem of calculus. Let f be a real-valued integrable function defined on a closed interval [a, b]. If F is a function such that F ′(x) = f(x) for all x in [a, b] (that is, F is an antiderivative In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i.e., F ′ = f. The process of solving for antiderivatives is antidifferentiation . Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over of f), then
- Corollary. If f is a continuous function on [a, b], then f is integrable on [a, b], and F, defined by
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- is an anti-derivative of f on [a, b]. Moreover,
<<Table of Contents Integration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral | Next>> | Show All>>
