Before calculus (precalculus

Precalculus

In American mathematics education, precalculus , an advanced form of secondary school algebra, is a foundational mathematical discipline. It is also called Introduction to Analysis. In many schools, precalculus is actually two separate courses: Algebra and Trigonometry...

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• Graph of a function
  Graph of a function
  In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...
  
  
• Linear function
  Linear function
  In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....
  
  
• Secant
  Secant
  Secant is a term in mathematics. It comes from the Latin secare . It can refer to:* a secant line, in geometry* the secant variety, in algebraic geometry...
  
  
• Slope
  Slope
  In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....
  
  
• Tangent
  Tangent
  In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
  
  
• Concave function
  Concave function
  In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.-Definition:...
  
  
• Finite difference
  Finite difference
  A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
  
  
• Radian
  Radian
  Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
  
  
• Factorial
  Factorial
  In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
  
  
• Binomial theorem
  Binomial theorem
  In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...
  
  
• Free variables and bound variables
  Free variables and bound variables
  In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place...
  
  

Limits

• Limit (mathematics)
  Limit (mathematics)
  In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
  
  
• Limit of a function
  Limit of a function
  In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....
  
  
  • One-sided limit
    One-sided limit
    In calculus, a one-sided limit is either of the two limits of a function f of a real variable x as x approaches a specified point either from below or from above...
    
    
• Limit of a sequence
  Limit of a sequence
  The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
  
  
• Indeterminate form
  Indeterminate form
  In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution...
  
  

• Orders of approximation
  Orders of approximation
  In science, engineering, and other quantitative disciplines, orders of approximation refer to formal or informal terms for how precise an approximation is, and to indicate progressively more refined approximations: in increasing order of precision, a zeroth order approximation, a first order...
  
  
• (ε, δ)-definition of limit

Differential calculus

Differential calculus

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

• Derivative
  Derivative
  In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
  
  
• Notations
  • Newton's notation for differentiation
    Newton's notation for differentiation
    Newton's notation for differentiation, or dot notation, uses a dot placed over a function name to denote the time derivative of that function. Newton referred to this as a fluxion.Isaac Newton's notation is mainly used in mechanics...
    
    
  • Leibniz's notation for differentiation
• Simplest rules
  • Derivative of a constant
    Derivative of a constant
    In calculus, the derivative of a constant function is zero .The rule can be justified in various ways...
    
    
  • Sum rule in differentiation
    Sum rule in differentiation
    In calculus, the sum rule in differentiation is a method of finding the derivative of a function that is the sum of two other functions for which derivatives exist. This is a part of the linearity of differentiation. The sum rule in integration follows from it...
    
    
  • Constant factor rule in differentiation
    Constant factor rule in differentiation
    In calculus, the constant factor rule in differentiation, also known as The Kutz Rule, allows you to take constants outside a derivative and concentrate on differentiating the function of x itself...
    
    
  • Linearity of differentiation
    Linearity of differentiation
    In mathematics, the linearity of differentiation is a most fundamental property of the derivative, in differential calculus. It follows from the sum rule in differentiation and the constant factor rule in differentiation...
    
    
  • Calculus with polynomials
• Derivative (examples)
• Chain rule
  Chain rule
  In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
  
  
• Product rule
  Product rule
  In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
  
  
• Quotient rule
• Inverse functions and differentiation
• Implicit differentiation
• Stationary point
  Stationary point
  In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
  
  
  • Maxima and minima
    Maxima and minima
    In mathematics, the maximum and minimum of a function, known collectively as extrema , are the largest and smallest value that the function takes at a point either within a given neighborhood or on the function domain in its entirety .More generally, the...
    
    
  • First derivative test
    First derivative test
    In calculus, the first derivative test uses the first derivative of a function to determine whether a given critical point of a function is a local maximum, a local minimum, or neither.-Intuitive explanation:...
    
    
  • Second derivative test
    Second derivative test
    In calculus, the second derivative test is a criterion often useful for determining whether a given stationary point of a function is a local maximum or a local minimum using the value of the second derivative at the point....
    
    
  • Extreme value theorem
    Extreme value theorem
    In calculus, the extreme value theorem states that if a real-valued function f is continuous in the closed and bounded interval [a,b], then f must attain its maximum and minimum value, each at least once...
    
    
• Differential equation
  Differential equation
  A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
  
  
• Differential operator
  Differential operator
  In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
  
  
• Newton's method
  Newton's method
  In numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
  
  
• Taylor's theorem
  Taylor's theorem
  In calculus, Taylor's theorem gives an approximation of a k times differentiable function around a given point by a k-th order Taylor-polynomial. For analytic functions the Taylor polynomials at a given point are finite order truncations of its Taylor's series, which completely determines the...
  
  
• L'Hôpital's rule
  L'Hôpital's rule
  In calculus, l'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms. Application of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit...
  
  
• Leibniz's rule
  Leibniz's rule
  Leibniz's rule is either of the following:* The General Leibniz rule, a form of the product rule in differential calculus* The Leibniz integral rule...
  
  
• Mean value theorem
  Mean value theorem
  In calculus, the mean value theorem states, roughly, that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the...
  
  
• Logarithmic derivative
  Logarithmic derivative
  In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formulawhere f ′ is the derivative of f....
  
  
• Differential (calculus)
  Differential (calculus)
  In calculus, a differential is traditionally an infinitesimally small change in a variable. For example, if x is a variable, then a change in the value of x is often denoted Δx . The differential dx represents such a change, but is infinitely small...
  
  
• Related rates
  Related rates
  In differential calculus, related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time. Because science and engineering often relate quantities to each...
  
  
• Regiomontanus' angle maximization problem
  Regiomontanus' angle maximization problem
  In mathematics, the Regiomontanus' angle maximization problem, is a famous optimization problem posed by the 15th-century German mathematician Johannes Müller...
  
  

Integral calculus

• Antiderivative
  Antiderivative
  In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...
  
  , Indefinite integral
• Simplest rules
  • Sum rule in integration
    Sum rule in integration
    In calculus, the sum rule in integration states that the integral of a sum of two functions is equal to the sum of their integrals. It is of particular use for the integration of sums, and is one part of the linearity of integration....
    
    
  • Constant factor rule in integration
  • Linearity of integration
    Linearity of integration
    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. Linearity of integration is related to the linearity of summation, since integrals are thought of as infinite sums.Let ƒ and g be functions...
    
    
• Arbitrary constant of integration
  Arbitrary constant of integration
  In calculus, the indefinite integral of a given function is only defined up to an additive constant, the constant of integration. This constant expresses an ambiguity inherent in the construction of antiderivatives...
  
  
• Fundamental theorem of calculus
  Fundamental theorem of calculus
  The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
  
  
• Integration by parts
  Integration by parts
  In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
  
  
• Inverse chain rule method
• Integration by substitution
  Integration by substitution
  In calculus, integration by substitution is a method for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool for mathematicians...
  
  
  • Weierstrass substitution
    Weierstrass substitution
    In integral calculus, the Weierstrass substitution, named after Karl Weierstrass, is used for finding antiderivatives, and hence definite integrals, of rational functions of trigonometric functions. No generality is lost by taking these to be rational functions of the sine and cosine. The...
    
    
• Differentiation under the integral sign
• Trigonometric substitution
  Trigonometric substitution
  In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities to simplify certain integrals containing radical expressions:...
  
  
• Partial fractions in integration
  Partial fractions in integration
  In integral calculus, partial fraction expansions provide an approach to integrating a general rational function. Any rational function of a real variable can be written as the sum of a polynomial function and a finite number of algebraic fractions...
  
  
  • Quadratic integral
• Proof that 22/7 exceeds π
• Trapezium rule
• Integral of the secant function
  Integral of the secant function
  The integral of the secant function of trigonometry was the subject of one of the "outstanding open problems of the mid-seventeenth century", solved in 1668 by James Gregory. In 1599, Edward Wright evaluated the integral by numerical methods – what today we would call Riemann sums...
  
  
• Integral of secant cubed
• Arclength

Special functions and numbers

• Natural logarithm
  Natural logarithm
  The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
  
  
• e (mathematical constant)
  E (mathematical constant)
  The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
  
  
• Exponential function
  Exponential function
  In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
  
  
• Stirling's approximation
  Stirling's approximation
  In mathematics, Stirling's approximation is an approximation for large factorials. It is named after James Stirling.The formula as typically used in applications is\ln n! = n\ln n - n +O\...
  
  
• Bernoulli number
  Bernoulli number
  In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....
  
  s

Numerical integration

Numerical integration

In numerical analysis, numerical integration constitutes 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 the numerical solution of differential equations. This article focuses on calculation of...

• Rectangle method
  Rectangle method
  In mathematics, specifically in integral calculus, the rectangle method computes an approximation to a definite integral, made by finding the area of a collection of rectangles whose heights are determined by the values of the function.Specifically, the interval over which the function is to be...
  
  
• Trapezium rule
• Simpson's rule
  Simpson's rule
  In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation:...
  
  
• Newton–Cotes formulas
• Gaussian quadrature
  Gaussian 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

• Table of common limits
• Table of derivatives
• Table of integrals
• Table of mathematical symbols
  Table of mathematical symbols
  This is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in , as an image.-Symbols:-Variations:...
  
  
• List of integrals
• List of integrals of rational functions
• List of integrals of irrational functions
• List of integrals of trigonometric functions
• List of integrals of inverse trigonometric functions
• List of integrals of hyperbolic functions
• List of integrals of exponential functions
• List of integrals of logarithmic functions
• List of integrals of area functions

Multivariable

• Partial derivative
  Partial derivative
  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...
  
  
• Disk integration
  Disk integration
  Disk integration, , is a means of calculating the volume of a solid of revolution of a solid-state material, when integrating along the axis of revolution. This method models the generated 3 dimensional shape as a "stack" of an infinite number of disks of infinitesimal thickness...
  
  
• Shell integration
  Shell integration
  Shell integration is a means of calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution.It makes use of the so-called "representative cylinder"...
  
  
• Gabriel's horn
  Gabriel's Horn
  Gabriel's Horn is a geometric figure which has infinite surface area but encloses a finite volume. The name refers to the tradition identifying the Archangel Gabriel as the angel who blows the horn to announce Judgment Day, associating the divine, or infinite, with the finite...
  
  
• Jacobian matrix
• Hessian matrix
  Hessian matrix
  In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...
  
  
• Curvature
  Curvature
  In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
  
  
• Green's theorem
  Green's theorem
  In mathematics, Green's 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...
  
  
• Divergence theorem
  Divergence theorem
  In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...
  
  
• Stokes' theorem
  Stokes' theorem
  In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
  
  

Series

• Infinite series
• Maclaurin series, Taylor series
  Taylor series
  In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
  
  
• Fourier series
  Fourier series
  In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
  
  
• Euler–Maclaurin formula

History

• Adequality
  Adequality
  In the history of infinitesimal calculus, adequality is a technique developed by Pierre de Fermat. Fermat said he borrowed the term from Diophantus. Adequality was a technique first used to find maxima for functions and then adapted to find tangent lines to curves...
  
  
• Infinitesimal
  Infinitesimal
  Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
  
  
  • Archimedes' use of infinitesimals
    Archimedes' use of infinitesimals
    The Method of Mechanical Theorems is a work by Archimedes which contains the first attested explicit use of infinitesimals. The work was originally thought to be lost, but was rediscovered in the celebrated Archimedes Palimpsest...
    
    
• Gottfried Leibniz
  Gottfried Leibniz
  Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
  
  
• Isaac Newton
  Isaac Newton
  Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
  
  
• Method of Fluxions
  Method of Fluxions
  Method of Fluxions is a book by Isaac Newton. The book was completed in 1671, and published in 1736. Fluxions is Newton's term for differential calculus...
  
  
• Infinitesimal calculus
  Infinitesimal calculus
  Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...
  
  
• Brook Taylor
  Brook Taylor
  Brook Taylor FRS was an English mathematician who is best known for Taylor's theorem and the Taylor series.- Life and work :...
  
  
• Colin Maclaurin
  Colin Maclaurin
  Colin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a special case of the Taylor series, are named after him....
  
  
• Leonhard Euler
  Leonhard Euler
  Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
  
  
• Law of continuity
  Law of Continuity
  The Law of Continuity is a heuristic principle introduced by Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler. It is the principle that "whatever succeeds for the finite, also succeeds for the infinite"...
  
  
• History of calculus
  History of calculus
  Calculus, historically known as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Ideas leading up to the notions of function, derivative, and integral were developed throughout the 17th century, but the decisive step was...
  
  
• Generality of algebra
  Generality of algebra
  In the history of mathematics, the generality of algebra is phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler and Joseph Lagrange...
  
  

Nonstandard calculus

• Elementary Calculus: An Infinitesimal Approach
  Elementary Calculus: An Infinitesimal Approach
  Elementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of Abraham Robinson's non-standard analysis...
  
  
• Nonstandard calculus
• Infinitesimal
  Infinitesimal
  Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
  
  
• Archimedes' use of infinitesimals
  Archimedes' use of infinitesimals
  The Method of Mechanical Theorems is a work by Archimedes which contains the first attested explicit use of infinitesimals. The work was originally thought to be lost, but was rediscovered in the celebrated Archimedes Palimpsest...