Glossary
- Page ID
- 133642
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| Words (or words that have the same definition) | The definition is case sensitive | (Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pages] | (Optional) Caption for Image | (Optional) External or Internal Link | (Optional) Source for Definition |
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| (Eg. "Genetic, Hereditary, DNA ...") | (Eg. "Relating to genes or heredity") |  |
The infamous double helix | https://bio.libretexts.org/ | CC-BY-SA; Delmar Larsen |
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Word(s) |
Definition |
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absolute Convergence |
if the series sum^∞_{n=1} a_n converges, then the series sum^∞_{n=1}a_n is said to converge absolutely |
Adapted from OpenStax Calculus for Calculus 2 | |||
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absolute error |
if B is an estimate of some quantity having an actual value of A, then the absolute error is given by |A−B| |
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| absolute extremum | if f has an absolute maximum or absolute minimum at c, we say f has an absolute extremum at c |
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| absolute maximum | if f(c)≥f(x) for all x in the domain of f, we say f has an absolute maximum at c | ||||
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absolute minimum |
if f(c)≤f(x) for all x in the domain of f, we say f has an absolute minimum at c |
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| absolute value function | f(x)=\begin{cases}−x, & \text{if } x<0\x, & \text{if } x≥0\end{cases} | ||||
| acceleration | is the rate of change of the velocity, that is, the derivative of velocity | ||||
| algebraic function | a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable x |
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| alternating series | for an alternating series of either form, if b_{n+1}≤b_n for all integers n≥1 and b_n→0, then an alternating series converges |
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| angular coordinate | θ the angle formed by a line segment connecting the origin to a point in the polar coordinate system with the positive radial (x) axis, measured counterclockwise |
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| antiderivative | a function F such that F(x)=f(x) for all x in the domain of f is an antiderivative of f |
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| arc length | the arc length of a curve can be thought of as the distance a person would travel along the path of the curve |
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| arithmetic sequence | a sequence in which the difference between every pair of consecutive terms is the same is called an arithmetic sequence |
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| average value of a function | (or f_{ave}) the average value of a function on an interval can be found by calculating the definite integral of the function and dividing that value by the length of the interval |
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| binomial series | the Maclaurin series for f(x)=(1+x)^r; it is given by (1+x)^r=\sum_{n=0}^∞(^r_n)x^n=1+rx+\dfrac{r(r−1)}{2!}x^2+⋯+\dfrac{r(r−1)⋯(r−n+1)}{n!}x^n+⋯ for |x|<1 |
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| bounded sequence | a sequence \displaystyle {a_n} is bounded if there exists a constant \displaystyle M such that \displaystyle |a_n|≤M for all positive integers \displaystyle n |
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| cardioid |
a plane curve |
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| carrying capacity | the maximum population of an organism that the environment can sustain indefinitely | ||||
| catenary | a curve in the shape of the function y=a\cdot\cosh(x/a) is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary |
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| center of mass | the point at which the total mass of the system could be concentrated without changing the moment | ||||
| centroid | the centroid of a region is the geometric center of the region; laminas are often represented by regions in the plane; if the lamina has a constant density, the center of mass of the lamina depends only on the shape of the corresponding planar region; in this case, the center of mass of the lamina corresponds to the centroid of the representative region |
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| change of variables | the substitution of a variable, such as u, for an expression in the integrand | ||||
| comparison test | If 0≤a_n≤b_n for all n≥N and \displaystyle \sum^∞_{n=1}b_n converges, then \displaystyle \sum^∞_{n=1}a_n converges; if a_n≥b_n≥0 for all n≥N and \displaystyle \sum^∞_{n=1}b_n diverges, then \displaystyle \sum^∞_{n=1}a_n diverges. | ||||
| conditional convergence | if the series \displaystyle \sum^∞_{n=1}a_n converges, but the series \displaystyle \sum^∞_{n=1}|a_n| diverges, the series \displaystyle \sum^∞_{n=1}a_n is said to converge conditionally | ||||
| conic section | a conic section is any curve formed by the intersection of a plane with a cone of two nappes |
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cycloid |
the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage | ||||
| definite integral | a primary operation of calculus; the area between the curve and the x-axis over a given interval is a definite integral | ||||
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density function |
a density function describes how mass is distributed throughout an object; it can be a linear density, expressed in terms of mass per unit length; an area density, expressed in terms of mass per unit area; or a volume density, expressed in terms of mass per unit volume; weight-density is also used to describe weight (rather than mass) per unit volume |
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disk method |
a special case of the slicing method used with solids of revolution when the slices are disks |
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divergence of a series |
a series diverges if the sequence of partial sums for that series diverges | ||||
| divergent sequence |
a sequence that is not convergent is divergent |
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| divergence test | if \displaystyle \lim_{n→∞}a_n≠0, then the series \displaystyle \sum^∞_{n=1}a_n diverges | ||||
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eccentricity |
the eccentricity is defined as the distance from any point on the conic section to its focus divided by the perpendicular distance from that point to the nearest directrix |
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explicit formula |
a sequencemay be defined by an explicit formula such that \displaystyle a_n=f(n) |
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focus |
a focus (plural: foci) is a point used to construct and define a conic section; a parabola has one focus; an ellipse and a hyperbola have two |
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fundamental theorem of calculus |
(also, evaluation theorem) we can evaluate a definite integral by evaluating the antiderivative of the integrand at the endpoints of the interval and subtracting uses a definite integral to define an antiderivative of a function the theorem, central to the entire development of calculus, that establishes the relationship between differentiationand integration |
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geometric sequence |
a sequence \displaystyle {a_n} in which the ratio \displaystyle a_{n+1}/a_n is the same for all positive integers \displaystyle n is called a geometric sequence |
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| geometric series | a geometric series is a series that can be written in the form \displaystyle \sum_{n=1}^∞ar^{n−1}=a+ar+ar^2+ar^3+⋯ |
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harmonic series |
the harmonic series takes the form \displaystyle \sum_{n=1}^∞\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+⋯ |
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improper integral |
an integral over an infinite interval or an integral of a function containing an infinite discontinuity on the interval; an improper integral is defined in terms of a limit. The improper integral converges if this limit is a finite real number; otherwise, the improper integral diverges |
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indeterminate forms |
When evaluating a limit, the forms \dfrac{0}{0},∞/∞, 0⋅∞, ∞−∞, 0^0, ∞^0, and 1^∞ are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is. |
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infinite series |
an infinite series is an expression of the form \displaystyle a_1+a_2+a_3+⋯=\sum_{n=1}^∞a_n |
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integration by parts |
a technique of integration that allows the exchange of one integral for another using the formula \displaystyle ∫u\,dv=uv−∫v\,du | ||||
| integration by substitution | a technique for integration that allows integration of functions that are the result of a chain-rule derivative | ||||
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interval of convergence |
the set of real numbers x for which a power series converges | ||||
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lamina |
a thin sheet of material; laminas are thin enough that, for mathematical purposes, they can be treated as if they are two-dimensional | ||||
| limaçon | the graph of the equation r=a+b\sin θ or r=a+b\cos θ. If a=b then the graph is a cardioid | ||||
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limit comparison test |
Suppose a_n,b_n≥0 for all n≥1. If \displaystyle \lim_{n→∞}a_n/b_n→L≠0, then \displaystyle \sum^∞_{n=1}a_n and \displaystyle \sum^∞_{n=1}b_n both converge or both diverge; if \displaystyle \lim_{n→∞}a_n/b_n→0 and \displaystyle \sum^∞_{n=1}b_n converges, then \displaystyle \sum^∞_{n=1}a_n converges. If \displaystyle \lim_{n→∞}a_n/b_n→∞, and \displaystyle \sum^∞_{n=1}b_n diverges, then \displaystyle \sum^∞_{n=1}a_n diverges. | ||||
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L’Hôpital’s rule |
If f and g are differentiable functions over an interval a, except possibly at a, and \displaystyle \lim_{x→a}f(x)=0=\lim_{x→a}g(x) or \displaystyle \lim_{x→a}f(x) and \displaystyle \lim_{x→a}g(x) are infinite, then \displaystyle \lim_{x→a}\dfrac{f(x)}{g(x)}=\lim_{x→a}\dfrac{f′(x)}{g′(x)}, assuming the limiton the right exists or is ∞ or −∞. |
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Maclaurin polynomial |
a Taylor polynomial centered at 0; the n^{\text{th}}-degree Taylor polynomial for f at 0 is the n^{\text{th}}-degree Maclaurin polynomial for f |
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Maclaurin series |
a Taylor series for a function f at x=0 is known as a Maclaurin series for f |
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major axis |
The major axis of a conic section passes through the vertex in the case of a parabola or through the two vertices in the case of an ellipse or hyperbola; it is also an axis of symmetry of the conic; also called transverse axis |
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minor axis |
the minor axis is perpendicular to the major axis and intersects the major axis at the center of the conic, or at the vertex in the case of the parabola; also called the conjugate axis |
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moment |
if n masses are arranged on a number line, the moment of the system with respect to the origin is given by \displaystyle M=\sum^n_{i=1}m_ix_i; if, instead, we consider a region in the plane, bounded above by a function f(x) over an interval [a,b], then the moments of the region with respect to the x- and y-axes are given by \displaystyle M_x=ρ∫^b_a\dfrac{[f(x)]^2}{2}\,dx and \displaystyle M_y=ρ∫^b_axf(x)\,dx, respectively |
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monotone sequence |
an increasing or decreasing sequence | ||||
| p-series | a series of the form \displaystyle \sum^∞_{n=1}1/n^p | ||||
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parameter |
an independent variable thatboth x and y depend on in a parametric curve; usually represented by the variable t |
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parameterization of a curve |
rewriting the equation of a curve defined by a function y=f(x) as parametric equations | ||||
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parametric curve |
the graph of the parametric equations x(t) and y(t) over an interval a≤t≤b combined with the equations |
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parametric equations |
the equations x=x(t) and y=y(t) that define a parametric curve | ||||
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partial fraction decomposition |
a technique used to break down a rational function into the sum of simple rational functions | ||||
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partial sum |
the kth partial sum of the infinite series \displaystyle \sum^∞_{n=1}a_n is the finite sum \displaystyle S_k=\sum_{n=1}^ka_n=a_1+a_2+a_3+⋯+a_k |
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polar coordinate system |
| a system for locating points in the plane. The coordinates are r, the radial coordinate, and θ, the angular coordinate | ||||
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polar equation |
an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system |
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power series |
a series of the form \sum_{n=0}^∞c_nx^n is a power series centered at x=0; a series of the form \sum_{n=0}^∞c_n(x−a)^n is a power series centered at x=a |
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radius of convergence |
if there exists a real number R>0 such that a power series centered at x=a converges for |x−a|<R and diverges for |x−a|>R, then R is the radius of convergence; if the power series only converges at x=a, the radius of convergence is R=0; if the power series converges for all real numbers x, the radius of convergence is R=∞ |
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ratio test |
for a series \displaystyle \sum^∞_{n=1}a_n with nonzero terms, let \displaystyle ρ=\lim_{n→∞}|a_{n+1}/a_n|; if 0≤ρ<1, the series converges absolutely; if ρ>1, the series diverges; if ρ=1, the test is inconclusive | ||||
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recurrence relation |
a recurrence relation is a relationship in which a term a_n in a sequence is defined in terms of earlier terms in the sequence |
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sequence |
an ordered list of numbers of the form \displaystyle a_1,a_2,a_3,… is a sequence | ||||
| Simpson's Rule | a method of numerical integration that involves partitioning a curve using the area under a piecewise quadratic function | ||||
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slicing method |
a method of calculating the volume of a solid that involves cutting the solid into pieces, estimating the volume of each piece, then adding these estimates to arrive at an estimate of the total volume; as the number of slices goes to infinity, this estimate becomes an integral that gives the exact value of the volume | ||||
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solid of revolution |
a solid generated by revolving a region in a plane around a line in that plane |
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Taylor polynomials
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the n^{\text{th}}-degree Taylor polynomial for f at x=a is p_n(x)=f(a)+f′(a)(x−a)+\dfrac{f''(a)}{2!}(x−a)^2+⋯+\dfrac{f^{(n)}(a)}{n!}(x−a)^n | ||||
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Taylor series |
a power series at a that converges to a function f on some open interval containing a. |
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Taylor’s theorem with remainder | |
for a functionf and the n^{\text{th}}-degree Taylor polynomial for f at x=a, the remainder R_n(x)=f(x)−p_n(x) satisfies R_n(x)=\dfrac{f^{(n+1)}(c)}{(n+1)!}(x−a)^{n+1} for somec between x and a; if there exists an interval I containing a and a real number M such that ∣f^{(n+1)}(x)∣≤M for all x in I, then |R_n(x)|≤\dfrac{M}{(n+1)!}|x−a|^{n+1} |
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telescoping series |
a telescoping series is one in which most of the terms cancel in each of the partial sums |
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term |
the number \displaystyle a_n in the sequence \displaystyle {a_n} is called the \displaystyle nth term of the sequence |
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term-by-term differentiation of a power series |
a technique for evaluating the derivative of a power series \displaystyle \sum_{n=0}^∞c_n(x−a)^n by evaluating the derivative of each term separately to create the new power series \displaystyle \sum_{n=1}^∞nc_n(x−a)^{n−1} |
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term-by-term integration of a power series |
a technique for integrating a power series \displaystyle \sum_{n=0}^∞c_n(x−a)^n by integrating each term separately to create the new power series \displaystyle C+\sum_{n=0}^∞c_n\dfrac{(x−a)^{n+1}}{n+1} |
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trapezoidal rule |
a rule that approximates \displaystyle ∫^b_af(x)\,dx using the area of trapezoids. The approximation T_n to \displaystyle ∫^b_af(x)\,dx is given by T_n=\frac{Δx}{2}\big(f(x_0)+2\, f(x_1)+2\, f(x_2)+⋯+2\, f(x_{n−1})+f(x_n)\big). \nonumbe | ||||
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trigonometric substitution |
an integration technique that converts an algebraic integral containing expressions of the form \sqrt{a^2−x^2}, \sqrt{a^2+x^2}, or \sqrt{x^2−a^2} into a trigonometric integral | ||||
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unbounded sequence |
a sequence that is not bounded is called unbounded. There does not exist an integer M such that the terms of the sequence are between -M and M |
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vertex |
a vertex is an extreme point on a conic section; a parabola has one vertex at its turning point. An ellipse has two vertices, one at each end of the major axis; a hyperbola has two vertices, one at the turning point of each branch |
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washer method |
a special case of the slicing method used with solids of revolution when the slices are washers | ||||
| work | the amount of energy it takes to move an object; in physics, when a force is constant, work is expressed as the product of force and distance |