Mean value theorem, the Glossary

Table of Contents

1. 57 relations: Applied Probability Trust, Augustin-Louis Cauchy, Bhāskara II, Cauchy–Schwarz inequality, Chord (geometry), Constant function, Continuous function, Cube root, Curve, Cusp (singularity), Derivative, Differentiable function, Dot product, Extreme value theorem, Fundamental theorem of calculus, Govindasvāmi, Gradient, Henstock–Kurzweil integral, Holomorphic function, India, Interior (topology), Intermediate value theorem, Interval (mathematics), Jacobian matrix and determinant, Jean Dieudonné, Joseph-Louis Lagrange, Kerala school of astronomy and mathematics, Khan Academy, L'Hôpital's rule, Limit of a function, Lipschitz continuity, MacTutor History of Mathematics Archive, Mathematics, Mean value theorem (divided differences), Measurable function, Michel Rolle, Monotonic function, Newmark-beta method, Parallel (geometry), Parameshvara Nambudiri, Probability density function, Racetrack principle, Random variable, Real analysis, Riemann integral, Rolle's theorem, Secant line, Semi-differentiability, Serge Lang, Slope, ... Expand index (7 more) »

2. Augustin-Louis Cauchy
3. Theorems in calculus
4. Theorems in real analysis

Applied Probability Trust

The Applied Probability Trust is a UK-based non-profit foundation for study and research in the mathematical sciences, founded in 1964 and based in the School of Mathematics and Statistics at the University of Sheffield, which it has been affiliated with since 1964.

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Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy (France:, ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist.

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Bhāskara II

Bhāskara II (1114–1185), also known as Bhāskarāchārya, was an Indian polymath, mathematician, astronomer and engineer.

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Cauchy–Schwarz inequality

The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. Mean value theorem and Cauchy–Schwarz inequality are Augustin-Louis Cauchy.

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Chord (geometry)

A chord (from the Latin chorda, meaning "bowstring") of a circle is a straight line segment whose endpoints both lie on a circular arc.

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Constant function

In mathematics, a constant function is a function whose (output) value is the same for every input value.

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Continuous function

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function.

See Mean value theorem and Continuous function

Cube root

In mathematics, a cube root of a number is a number such that.

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Curve

In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.

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Cusp (singularity)

In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction.

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Derivative

The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function's output with respect to its input.

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Differentiable function

In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

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Dot product

In mathematics, the dot product or scalar productThe term scalar product means literally "product with a scalar as a result".

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Extreme value theorem

In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed and bounded interval, then f must attain a maximum and a minimum, each at least once. Mean value theorem and extreme value theorem are theorems in calculus and theorems in real analysis.

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Fundamental theorem of calculus

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each point in time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). Mean value theorem and fundamental theorem of calculus are theorems in calculus and theorems in real analysis.

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Govindasvāmi

Govindasvāmi (or Govindasvāmin, Govindaswami) (c. 800 – c. 860) was an Indian mathematical astronomer most famous for his Bhashya, a commentary on the Mahābhāskarīya of Bhāskara I, written around 830.

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Gradient

In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase.

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Henstock–Kurzweil integral

In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced), Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral – is one of a number of inequivalent definitions of the integral of a function.

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Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space.

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India

India, officially the Republic of India (ISO), is a country in South Asia.

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Interior (topology)

In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in.

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In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval, then it takes on any given value between f(a) and f(b) at some point within the interval. Mean value theorem and intermediate value theorem are theorems in calculus and theorems in real analysis.

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Interval (mathematics)

In mathematics, a (real) interval is the set of all real numbers lying between two fixed endpoints with no "gaps".

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Jacobian matrix and determinant

In vector calculus, the Jacobian matrix of a vector-valued function of several variables is the matrix of all its first-order partial derivatives.

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Jean Dieudonné

Jean Alexandre Eugène Dieudonné (1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of mathematics, particularly in the fields of functional analysis and algebraic topology.

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Joseph-Louis Lagrange

Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia, Encyclopædia Britannica or Giuseppe Ludovico De la Grange Tournier; 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia, was an Italian mathematician, physicist and astronomer, later naturalized French.

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Kerala school of astronomy and mathematics

The Kerala school of astronomy and mathematics or the Kerala school was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Tirur, Malappuram, Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar.

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Khan Academy

Khan Academy is an American non-profit educational organization created in 2006 by Sal Khan.

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L'Hôpital's rule

L'Hôpital's rule or L'Hospital's rule, also known as Bernoulli's rule, is a mathematical theorem that allows evaluating limits of indeterminate forms using derivatives. Mean value theorem and L'Hôpital's rule are theorems in calculus and theorems in real analysis.

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Limit of a function

Although the function is not defined at zero, as becomes closer and closer to zero, becomes arbitrarily close to 1.

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Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions.

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MacTutor History of Mathematics Archive

The MacTutor History of Mathematics Archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland.

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Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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Mean value theorem (divided differences)

In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives.

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Measurable function

In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable.

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Michel Rolle

Michel Rolle (21 April 1652 – 8 November 1719) was a French mathematician.

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Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.

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Newmark-beta method

The Newmark-beta method is a method of numerical integration used to solve certain differential equations.

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Parallel (geometry)

In geometry, parallel lines are coplanar infinite straight lines that do not intersect at any point.

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Parameshvara Nambudiri

Vatasseri Parameshvara Nambudiri (1380–1460) was a major Indian mathematician and astronomer of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama.

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Probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would be equal to that sample.

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Racetrack principle

In calculus, the racetrack principle describes the movement and growth of two functions in terms of their derivatives.

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Random variable

A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events.

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Real analysis

In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions.

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Riemann integral

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.

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Rolle's theorem

In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one point, somewhere between them, at which the slope of the tangent line is zero. Mean value theorem and Rolle's theorem are theorems in calculus and theorems in real analysis.

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Secant line

In geometry, a secant is a line that intersects a curve at a minimum of two distinct points.

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Semi-differentiability

In calculus, the notions of one-sided differentiability and semi-differentiability of a real-valued function f of a real variable are weaker than differentiability.

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Serge Lang

Serge Lang (May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career.

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Slope

In mathematics, the slope or gradient of a line is a number that describes the direction and steepness of the line.

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Stationary point

In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero.

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Stochastic ordering

In probability theory and statistics, a stochastic order quantifies the concept of one random variable being "bigger" than another.

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Stolarsky mean

In mathematics, the Stolarsky mean is a generalization of the logarithmic mean.

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Tangent

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point.

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Taylor's theorem

In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the k-th-order Taylor polynomial. Mean value theorem and Taylor's theorem are theorems in calculus and theorems in real analysis.

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Theorem

In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven.

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Uniform continuity

In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want.

See Mean value theorem and Uniform continuity

See also

Augustin-Louis Cauchy

• Augustin-Louis Cauchy
• Cauchy (crater)
• Cauchy condensation test
• Cauchy distribution
• Cauchy formula for repeated integration
• Cauchy principal value
• Cauchy product
• Cauchy sequence
• Cauchy theorem
• Cauchy's equation
• Cauchy's integral formula
• Cauchy's integral theorem
• Cauchy's theorem (geometry)
• Cauchy's theorem (group theory)
• Cauchy–Binet formula
• Cauchy–Hadamard theorem
• Cauchy–Kovalevskaya theorem
• Cauchy–Riemann equations
• Cauchy–Schwarz inequality
• Integral test for convergence
• Mean value theorem
• Peano existence theorem
• Picard–Lindelöf theorem
• Restricted sumset
• Root test

Theorems in calculus

• Bioche's rules
• Cantor's intersection theorem
• Chain rule
• Darboux's theorem (analysis)
• Differentiation of integrals
• Differentiation rules
• Divergence theorem
• Extreme value theorem
• Fermat's theorem (stationary points)
• Fubini's theorem
• Fundamental theorem of calculus
• General Leibniz rule
• Generalized Stokes theorem
• Gradient theorem
• Green's theorem
• Helmholtz decomposition
• Implicit function theorem
• Increment theorem
• Integral of inverse functions
• Integration by parts
• Integration using Euler's formula
• Intermediate value theorem
• Inverse function rule
• Inverse function theorem
• L'Hôpital's rule
• Linearity of differentiation
• Mean value theorem
• Monotone convergence theorem
• Pappus's centroid theorem
• Power rule
• Product rule
• Quotient rule
• Reciprocal rule
• Rolle's theorem
• Stokes' theorem
• Taylor's theorem
• Triple product rule
• Uniqueness theorem for Poisson's equation

Theorems in real analysis

• Abel's theorem
• Anderson's theorem
• Arzelà–Ascoli theorem
• Bernstein's theorem on monotone functions
• Blumberg theorem
• Bohr–Favard inequality
• Caristi fixed-point theorem
• Darboux's theorem (analysis)
• Dini's theorem
• Discontinuities of monotone functions
• Dominated convergence theorem
• Extreme value theorem
• Fatou–Lebesgue theorem
• Fermat's theorem (stationary points)
• Fubini's theorem on differentiation
• Fundamental theorem of calculus
• Glaeser's composition theorem
• Hardy's inequality
• Heine–Borel theorem
• Identity theorem
• Implicit function theorem
• Intermediate value theorem
• Inverse function theorem
• Kirszbraun theorem
• Kolmogorov–Arnold representation theorem
• L'Hôpital's rule
• Lagrange inversion theorem
• Lebesgue differentiation theorem
• Lusin's theorem
• Mean value theorem
• Monotone convergence theorem
• Nested intervals
• Riemann series theorem
• Riesz–Fischer theorem
• Rolle's theorem
• Routh–Hurwitz theorem
• Steinhaus theorem
• Sturm's theorem
• Taylor's theorem
• Titchmarsh convolution theorem
• Uniform limit theorem
• Vitali–Carathéodory theorem
• Watson's lemma
• Zahorski theorem
• Śleszyński–Pringsheim theorem

References

[1] https://en.wikipedia.org/wiki/Mean_value_theorem

Also known as Cauchy MVT, Cauchy mean theorem, Cauchy mean value theorem, Cauchy's Mean Value Theorem, Cauchy's mean theorem, Cauchy's mean-value theorem, Cauchys mean value theorem, Cauchys mean-value theorem, Extended mean value theorem, Extended mean-value theorem, First mean value theorem, First mean value theorem for definite integrals, First mean value theorem for integrals, First mean value theorem for integration, Lagrange's mean value theorem, Law of the Mean, Mean value inequality, Mean value theorem for definite integrals, Mean value theorem for integrals, Mean value theorem for integration, Mean value theorems for definite integrals, Mean value theorems for integrals, Mean value theorems for integration, Mean value thm, Mean-Value Theorem, Second mean value theorem, Second mean value theorem for definite integrals, Second mean value theorem for integrals, Second mean value theorem for integration.

, Stationary point, Stochastic ordering, Stolarsky mean, Tangent, Taylor's theorem, Theorem, Uniform continuity.