Glossary of calculus

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This glossary of calculus is a list of definitions about calculus, its sub-disciplines, and related fields.

A
Abel's test:

absolute convergence: a_n\right absolute maximum:

absolute minimum:

absolute value:

alternating series:

alternating series test:

annulus:

antiderivative:

arcsin:

area under a curve:

asymptote:

automatic differentiation:

average rate of change:

B
binomial coefficient: Any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient. Commonly, a binomial coefficient is indexed by a pair of integers $|x|$ and is written $\tbinom{n}{k}.$ It is the coefficient of the $|x| = x$ term in the polynomial expansion of the binomial power $|x| = −x$, and it is given by the formula


 * $\binom{n}{k} = \frac{n!}{k! (n-k)!}.$

binomial theorem (or binomial expansion): Describes the algebraic expansion of powers of a binomial. bounded function: A function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M such that bounded sequence: .

C
calculus: (From Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus) is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. Cavalieri's principle: Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: chain rule: The chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f and g are functions, then the chain rule expresses the derivative of their composition $−x$ (the function which maps x to f(g(x)) ) in terms of the derivatives of f and g and the product of functions as follows:
 * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas.
 * 3-dimensional case: Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross-sections of equal area, then the two regions have equal volumes.


 * $(f\circ g)'=(f'\circ g)\cdot g'.$

This may equivalently be expressed in terms of the variable. Let $|0| = 0$, or equivalently, $f$ for all x. Then one can also write
 * $F'(x) = f'(g(x)) g'(x).$

The chain rule may be written in Leibniz's notation in the following way. If a variable z depends on the variable y, which itself depends on the variable x, so that y and z are therefore dependent variables, then z, via the intermediate variable of y, depends on x as well. The chain rule then states,


 * $\frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx}. $

The two versions of the chain rule are related; if $z=f(y)$ and $y=g(x)$, then


 * $\frac{dz}{dx}=\frac{dz}{dy}\cdot\frac{dy}{dx} = f'(y)g'(x) = f'(g(x))g'(x).$

In integration, the counterpart to the chain rule is the substitution rule. change of variables: Is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem. cofunction:

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. constant of integration:

continuous function:

continuously differentiable:

contour integration:

convergence tests:

convergent series: S_n - \ell \right \vert \le \ \varepsilon.$

If the series is convergent, the number $\ell$ (necessarily unique) is called the sum of the series.

Any series that is not convergent is said to be divergent. convex function:

Cramer's rule: In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand-sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for an arbitrary number of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in 1748 (and possibly knew of it as early as 1729). critical point:

curve: curve sketching:

D
damped sine wave:

degree of a polynomial:

derivative:

derivative test:

differentiable function:

differential (infinitesimal):

differential calculus:

differential equation:

differential operator:

differential of a function:

differentiation rules:

direct comparison test:

Dirichlet's test: \sum^{N}_{n=1}b_n\right disc integration:

divergent series:

discontinuity:

dot product:

double integral:

E
e (mathematical constant):

elliptic integral:

essential discontinuity:

Euler method:

exponential function:

extreme value theorem:

extremum:

F
Faà di Bruno's formula:

first-degree polynomial:

first derivative test:

Fractional calculus:

frustum:

function:

function composition:

fundamental theorem of calculus: The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function. The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the antiderivatives (also called indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration. This implies the existence of antiderivatives for continuous functions. Conversely, the second part of the theorem, sometimes called the second fundamental theorem of calculus, states that the integral of a function f over some interval can be computed by using any one, say F, of its infinitely many antiderivatives. This part of the theorem has key practical applications, because explicitly finding the antiderivative of a function by symbolic integration avoids numerical integration to compute integrals. This provides generally a better numerical accuracy.

G
general Leibniz rule:

global maximum:

global minimum:

golden spiral:

gradient:

H
harmonic progression:

higher derivative:

homogeneous linear differential equation:

hyperbolic function:

I
identity function:

imaginary number:

implicit function:

improper fraction:

improper integral:

inflection point:

instantaneous rate of change:

instantaneous velocity:

integral:

integral symbol:

integrand:

integration by parts:

integration by substitution:

intermediate value theorem:

inverse trigonometric functions:

J
jump discontinuity:

L
Lebesgue integration:

L'Hôpital's rule:

limit comparison test:

limit of a function:

limits of integration:

linear combination:

linear equation:

linear system:

list of integrals:

logarithm:

logarithmic differentiation:

lower bound:

M
mean value theorem:

monotonic function:

multiple integral:

Multiplicative calculus:

multivariable calculus:

N
natural logarithm:

non-Newtonian calculus:

nonstandard calculus:

notation for differentiation:

numerical integration:

O
one-sided limit:

ordinary differential equation:

P
Pappus's centroid theorem:

parabola:

paraboloid:

partial derivative:

partial differential equation:

partial fraction decomposition:

particular solution:

piecewise-defined function:

position vector:

power rule:

product integral:

product rule:

proper fraction:

proper rational function:

Pythagorean theorem:

Pythagorean trigonometric identity:

Q
quadratic function:

quadratic polynomial:

quotient rule:

R
radian:

ratio test:

reciprocal function:

reciprocal rule:

Riemann integral:

related rates:

removable discontinuity:

Rolle's theorem:

root test:

S
scalar:

secant line:

second-degree polynomial:

second derivative:

second derivative test:

second-order differential equation:

series:

shell integration:

Simpson's rule:

sine:

sine wave:

slope field:

squeeze theorem:

sum rule in differentiation:

sum rule in integration:

summation:

supplementary angle:

surface area:

system of linear equations:

T
table of integrals:

Taylor series:

Taylor's theorem:

tangent:

third-degree polynomial:

third derivative:

toroid:

total differential:

trigonometric functions:

trigonometric identities:

trigonometric integral:

trigonometric substitution:

trigonometry:

triple integral:

U
upper bound:

V
variable:

vector:

vector calculus:

W
washer:

washer method: