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This is a list of calculus topics. For other uses of the term calculus see calculus (disambiguation) Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ...
Note: the ordering of topics in sections is a suggestion to students.
From the point of view of mathematical education, pre-calculus is a foundational mathematical discipline. ...
In mathematics, the graph of a function f(x1, x2, ..., xn) is the collection of all tuples (x1, x2, ..., xn, f(x1, ..., xn)). In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc. ...
A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ...
Secant can refer to: a secant line secant, a trigonometric function, equivalent to sec(x) = 1/cos(x) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
In mathematics, the slope (or gradient, especially where three or more dimensions are discussed) of a straight line (within a Cartesian coordinate system) is a measure for the steepness of said line. ...
In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ...
In geometry, concavity is a property of certain geometric figures, and in calculus, a property of certain graphs of functions. ...
There are two subfields of mathematics that concern themselves with finite differences. ...
In mathematics and physics, the radian is a unit of angle measure. ...
In mathematics, the factorial of a natural number n is the product of all positive integers less than and equal to n. ...
In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation for a place or places in an expression, into which some definite substitution may take place, or with respect to which some operation (summation or quantification, to give two...
Limits In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ...
Limit of a sequence is one of the oldest concepts in mathematical analysis. ...
In mathematical analysis, and in particular in elementary calculus, certain expressions are indeterminate forms and must be treated as symbolic only, until more careful discussion has taken place. ...
Often in science, engineering, or other quantitative disciplines, it is necessary to make approximations with various degrees of precision. ...
In mathematics, a trigonometric rational function is a rational function in the functions sin θ and cos θ. ...
Differential calculus The derivative in mathematics (specifically, differential calculus) is a quantity that measures, on continuous functions, the limit of a rate of change, , as approaches 0. ...
It has been suggested that this article or section be merged with Dot notation for differentiation. ...
See Leibniz notation and separation of variables for, among other things, an account of certain advantages of this notation over others. ...
It has been suggested that this article or section be merged with Newtons notation for differentiation. ...
In calculus, the derivative of a constant function is zero. ...
The sum rule in differentiation is possibly the most useful rule in differentiation. ...
In calculus, the constant factor rule in differentiation allows you to take constants outside a derivative and concentrate on differentiating the function of x itself. ...
In mathematics, the linearity of differentiation is a most fundamental property of the derivative, in differential calculus. ...
In mathematics, polynomials are perhaps the simplest functions with which to do calculus. ...
// Example 1 Consider f(x) = 5: The derivative of a constant is zero. ...
In calculus, the chain rule is a formula for the derivative of the composition of two functions. ...
In mathematics, the product rule of calculus, which is also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...
In calculus, the quotient rule is a method of finding the derivative of a function which is the quotient of two other functions for which derivatives exist. ...
In mathematics, the inverse of a function is a function that, in some fashion, undoes the effect of (see inverse function for a formal and detailed definition). ...
In mathematics, to give an implicit function f is to give the graph of a function, as a relation. ...
In mathematics, particularly in calculus, a stationary point is a point on the graph of a function where the tangent to the graph is parallel to the x-axis or, equivalently, where the derivative of the function equals zero (known as a critical number). ...
A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some ε > 0 such that f(x*) ≥ f(x) for all x with |x-x*| < ε. ...
In calculus, a branch of mathematics, the first derivative test determines whether a given critical point of a function is a maximum, a minimum, or neither. ...
In calculus, a branch of mathematics, the second derivative test determines whether a given stationary point of a function (where its first derivative is zero) is a maximum, a minimum, or neither. ...
In calculus, the extreme value theorem states that if a function f(x) is continuous in the closed interval [a,b] then f(x) must attain its maximum and minimum value, each at least once. ...
In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...
In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...
In numerical analysis, Newtons method (or the Newton-Raphson method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ...
In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ...
In calculus, lHôpitals rule uses derivatives to help compute limits with indeterminate forms. ...
At least two results in calculus are called Leibnizs rule or the Leibniz rule, in honor of Gottfried Leibniz. ...
For any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c. ...
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula f′/f where f′ is the derivative of f. ...
In mathematics, the word differential has various meanings: In calculus, a differential is an infinitesimal change in the value of a function. ...
In differential calculus, related rates problems involve ratios of derivatives of two or more related variables that are changing with respect to time. ...
Integral calculus In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...
In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...
In calculus the sum rule in integration states that It is of constant use in going from the left hand to the right-hand side, to integrate sums. ...
The constant factor rule in integration is a dual of the constant factor rule in differentiation, and is a consequence of the linearity of integration Start by noticing that, from the definition of integration as the inverse process of differentiation: Now multiply both sides by a constant k. ...
In calculus, linearity is a fundamental property of the integral that follows from the sum rule in integration and the constant factor rule in integration. ...
In calculus, the indefinite integral of a given function (i. ...
The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other. ...
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...
In calculus, the inverse chain rule is a method of integrating a function which relies on guessing the integral of that function, and then differentiating back using the chain rule. ...
In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ...
In mathematics, there is a formula for differentiation under the integral sign in calculus. ...
In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. ...
In integral calculus, the use of partial fractions is required to integrate the general rational function. ...
In mathematics, a quadratic integral of the form may be computed by completing the square in the denominator. ...
The title given to this article is incorrect due to technical limitations. ...
The function f(x) (in blue) is approximated by a linear function (in red). ...
Special functions and numbers The natural logarithm is the logarithm to the base e, where e is approximately equal to 2. ...
Eulers number (or Napiers constant) is the base of the natural logarithm function. ...
The exponential function is one of the most important functions in mathematics. ...
In mathematics, Stirlings approximation (or Stirlings formula) is an approximation for large factorials. ...
In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ...
See also list of numerical analysis topics In numerical analysis, the term numerical integration is used to describe a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe numerical algorithms for solving differential equations. ...
This is a list of numerical analysis topics, by Wikipedia page. ...
In mathematics, the rectangle method of integral calculus uses an approximation to a definite integral, made by finding the area of a series of rectangles. ...
The function f(x) (in blue) is approximated by a linear function (in red). ...
In numerical analysis, Simpsons rule (named after Thomas Simpson) is a way to get an approximation of an integral: Basics Simpsons rule works by approximating by the quadratic polynomial which takes the same values as at a, b, and the midpoint m=(a+b)/2. ...
In numerical analysis, the Newton-Cotes formulas, also called the Newton-Cotes rules, are a group of formulas for numerical integration (also called quadrature). ...
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. ...
Lists and tables The primary operation in differential calculus is finding a derivative. ...
Integration is one of the two basic operations in calculus and since it, unlike differentiation, is non-trivial, tables of known integrals are often useful. ...
In mathematics, a set of symbols is frequently used in mathematical expressions. ...
See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of hyperbolic functions List of integrals of exponential functions List of integrals of logarithmic functions List of integrals of arc functions...
The following is a list of integrals (antiderivative functions) of rational functions. ...
The following is a list of integrals (antiderivative functions) of irrational functions. ...
The following is a list of integrals (antiderivative functions) of trigonometric functions. ...
The following is a list of integrals (antiderivative functions) of hyperbolic functions. ...
The following is a list of integrals (antiderivative functions) of exponential functions. ...
The following is a list of integrals (antiderivative functions) of logarithmic functions. ...
The following is a list of integrals (antiderivative functions) of arc functions. ...
The following is a list of integrals (antiderivative functions) of area functions. ...
Multivariable In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant. ...
In mathematics, in particular integral calculus, disk integration (the disk method) is a means of calculating the volume of a solid of revolution. ...
Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution. ...
Gabriels Horn (also called Torricellis trumpet) is a figure invented by Evangelista Torricelli which has infinite surface area, but finite volume. ...
In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...
In mathematics, the Hessian matrix is the square matrix of second partial derivatives of a scalar-valued function. ...
Curvature is the amount by which a geometric object deviates from being flat. ...
In physics and mathematics, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Greens Theorem was named after British scientist George Green and is a special case of the more...
In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Ostrogradsky-Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. ...
The Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...
Series In mathematics, a series is a sum of a sequence of terms. ...
As the degree of the taylor series rises, it approaches the correct function. ...
As the degree of the Taylor series rises, it approaches the correct function. ...
The Fourier series, named in honor of Joseph Fourier (1768-1830), is an extremely useful mathematical tool. ...
In mathematics, the Euler-Maclaurin formula provides a powerful connection between integrals (see calculus) and sums. ...
History In mathematics, an infinitesimal, or arbitrarily small number, is a number that is greater in absolute value than zero yet smaller than any positive real number. ...
The ancient Greek mathematician, physicist, and engineer Archimedes of Syracuse was the first mathematician to make explicit use of infinitesimals. ...
Gottfried Leibniz Gottfried Wilhelm von Leibniz (also Leibnitz) (Leipzig July 1 (June 21 O.S.), 1646 – November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ...
Sir Isaac Newton in Godfrey Knellers 1689 portrait Sir Isaac Newton (25 December 1642 – 20 March 1727 by the Julian calendar in use in England at the time; or 4 January 1643 – 31 March 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemist who...
Method of Fluxions was a book by Isaac Newton. ...
Infinitesimal calculus is an area of mathematics pioneered by Gottfried Leibniz based on the concept of infinitesimals, as opposed to the calculus of Isaac Newton, which is based upon the concept of the limit. ...
Brook Taylor (August 18, 1685 – December 29, 1731) was an English mathematician. ...
Colin Maclaurin Colin Maclaurin (February, 1698 - June 14, 1746) was a Scottish mathematician. ...
Leonhard Euler by Emanuel Handmann Leonhard Euler [oilər] (April 15, 1707 - September 18, 1783) was a Swiss mathematician and physicist. ...
Nonstandard calculus For further developments: see list of real analysis topics, list of complex analysis topics, list of multivariable calculus topics. In mathematics non-standard calculus is the application of non-standard analysis techniques to differential and integral calculus. ...
In mathematics, an infinitesimal, or arbitrarily small number, is a number that is greater in absolute value than zero yet smaller than any positive real number. ...
The ancient Greek mathematician, physicist, and engineer Archimedes of Syracuse was the first mathematician to make explicit use of infinitesimals. ...
This is a list of real analysis topics by Wikipedia page NB The topics are in a deliberately chosen order, for the use of students. ...
This is a list of complex analysis topics, by Wikipedia page. ...
This is a list of multivariable calculus topics, by Wikipedia page. ...
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