definition
(noun)
a formalization of the notion of the limit of functions
Examples of definition in the following topics:
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Precise Definition of a Limit
- The $(\varepsilon,\delta)$-definition of limit (the "epsilon-delta definition") is a formalization of the notion of limit.
- The $(\varepsilon,\delta)$-definition of limit ("epsilon-delta definition of limit") is a formalization of the notion of limit.
- The definitive modern statement was ultimately provided by Karl Weierstrass.
- The $(\varepsilon,\delta)$-definition of limit is a formalization of the notion of limit.
- This definition also works for functions with more than one input value.
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Approximate Integration
- The trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral $\int_{a}^{b} f(x)\, dx$.
- Numerical integration, in some instances also known as numerical quadrature, asks for the value of a definite integral.
- The trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral $\int_{a}^{b} f(x)\,dx$.
- Use the trapezoidal rule to approximate the value of a definite integral
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The Definite Integral
- A definite integral is the area of the region in the $xy$-plane bound by the graph of $f$, the $x$-axis, and the vertical lines $x=a$ and $x=b$.
- Integrals such as these are termed definite integrals.
- Definite integrals appear in many practical situations.
- A definite integral of a function can be represented as the signed area of the region bounded by its graph.
- Compute the definite integral of a function over a set interval
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Improper Integrals
- An Improper integral is the limit of a definite integral as an endpoint of the integral interval approaches either a real number or $\infty$ or $-\infty$.
- Such an integral is often written symbolically just like a standard definite integral, perhaps with infinity as a limit of integration.
- The original definition of the Riemann integral does not apply to a function such as $\frac{1}{x^2}$ on the interval $[1, \infty]$, because in this case the domain of integration is unbounded.
- The narrow definition of the Riemann integral also does not cover the function $\frac{1}{\sqrt{x}}$ on the interval $[0, 1]$.
- The problem here is that the integrand is unbounded in the domain of integration (the definition requires that both the domain of integration and the integrand be bounded).
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Numerical Integration
- Numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral.
- 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 definite integrals.
- The basic problem considered by numerical integration is to compute an approximate solution to a definite integral:
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Conic Sections in Polar Coordinates
- There are a number of other geometric definitions possible.
- One of the most useful definitions, in that it involves only the plane, is that a conic consists of those points whose distances to some point—called a focus—and some line—called a directrix—are in a fixed ratio, called the eccentricity.
- In the focus-directrix definition of a conic, the circle is a limiting case with eccentricity 0.
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Numerical Integration
- Numerical integration is a method of approximating the value of a definite integral.
- These integrals are termed "definite integrals."
- Numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral.
- A definite integral of a function can be represented as the signed area of the region bounded by its graph.
- Solve for the definite integral of a continuous function over a closed interval
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Basic Integration Principles
- Given a function $f$ of a real variable $x$, and an interval $[a, b]$ of the real line, the definite integral $\int_a^b \!
- More rigorously, once an anti-derivative $F$ of $f$ is known for a continuous real-valued function $f$ defined on a closed interval $[a, b]$, the definite integral of $f$ over that interval is given by
- If we are going to use integration by substitution to calculate a definite integral, we must change the upper and lower bounds of integration accordingly.
- A definite integral of a function can be represented as the signed area of the region bounded by its graph.
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Series
- By definition, the series $\sum_{n=0}^{\infty} a_n$ converges to a limit $L$ if and only if the associated sequence of partial sums $\{S_k\}$ converges to $L$.
- This definition is usually written as follows:
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Infinite Limits
- Limits involving infinity can be formally defined using a slight variation of the $(\varepsilon, \delta)$-definition.
- Limits involving infinity can be formally defined using a slight variation of the $(\varepsilon, \delta)$-definition.