Posts Tagged ‘limit’
Thursday, November 5th, 2009
Overview of the Monotone Convergence Theorem
Description
A detailed tutorial on the monotone convergence theorem. Step by step tutorial including several examples of the monotone convergence theorem for reference.
Overview
There are several different theorems that the term “monotone convergence” can apply to. However, the most important one, and the one most common called the monotone convergence theorem, is the Lebesgue Monotone Convergence Theorem. This particular monotone convergence theorem deals with calculus, and with integrals and limits specifically. It is a more general form of the other two monotone convergence theorems, which is why it is considered to be the most important.
Tags: Calculus, converge, convergence, form, general, integral, Lebesgue, limit, monotone, number, real, sequence, series, theorem
Posted in Calculus | No Comments »
Tuesday, October 27th, 2009
Introduction to Hilbert Space
Description
A detailed tutorial on on the application of Hilbert space. Step by step tutorial including several examples of Hilbert space for reference.
Overview
A Hilbert space is commonly used in vector algebra and calculus to generalize the notion of Euclidean space. It is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Hilbert spaces are also complete, which is a property that allows enough limits in the space for calculus to be used accurately.
Tags: abstract, algebra, angle, Calculus, complete, Hilbert, inner, length, limit, product, space, structure, vector
Posted in Algebra | No Comments »
Thursday, October 22nd, 2009
How to Find Nonlinear Asymptotes
Description
A detailed tutorial on finding nonlinear asymptotes. Step by step tutorial including several examples of how to find nonlinear asymptotes for reference.
Overview
An asymptote is used to describe the behavior of a curve as it heads away from the origin and towards infinity. Typically it is meant to describe two curves that are doing this, and these curves are said to be asymptotic. In most cases, the asymptote is linear – which means the curves have the same behavior. Whenever someone is talking about an asymptote, they are talking about a linear asymptote unless they specify a different type of asymptote. In rare cases, asymptotes are nonlinear. Both curves are still heading towards infinity, but they do not have the same behavior. This can be determined by the limit of either the subtraction or the division of these curves.
Tags: algebra, asymptote, asymptotic, behavior, curve, division, function, horizontal, infinity, limit, linear, nonlinear, oblique, origin, subtraction, vertical
Posted in Algebra | No Comments »
Tuesday, October 6th, 2009
How to Test for Convergence Using the Alternating Series Test
Description
A detailed tutorial on testing for convergence using the alternating series test. Step by step tutorial including several examples of testing for convergence using the alternating series test for reference.
Overview
The alternating series test, like all convergence and divergence tests, is fairly easy. The hardest part is figuring out if you should use the AST, or a different test. An easy way to tell is, is the equation negative? What would happen if you pulled a negative one out? Or maybe, there is already a negative one outside of the equation. If you see any fraction, function, or any equation at all with a -1 to an odd power at the front (or at the front of the numerator, in a fraction) then you should use the alternating series test for it. If the series is decreasing over time, and the limit is approaching zero, then the series is convergent. The alternating series test is normally used in conjunction with another test for convergence.
Tags: -1, alternating, AST, Calculus, converge, convergence, decreasing, diverge, divergence, fraction, function, limit, Math, negative, one, series, test, zero
Posted in Calculus | No Comments »
Tuesday, October 6th, 2009
How to Find Oblique Asymptotes
Description
A detailed tutorial on how to find oblique asymptotes. Step by step tutorial including several examples of how to find oblique asymptotes for reference.
Overview
There are several different types of asymptotes. In this tutorial, we will be discussing oblique asymptotes. In order to find the oblique asymptotes of a function, you must first determine if the asymptote slants. If the numerator of a rational function has exactly one degree greater than the denominator, then the function slants and therefore has an oblique asymptote. When you divide the numerator and the denominator, the term or polynomial you get is the oblique asymptote.
Tags: algebra, asymptote, asymptotes, closer, curves, degree, denominator, distance, farther, function, horizontal, infinity, limit, linear, lines, Math, negative, nonlinear, numerator, oblique, origin, polynomial, positive, slant, straight, vertical, zero
Posted in Algebra | No Comments »
Tuesday, September 29th, 2009
How to Find Horizontal Asymptotes
Description
A detailed tutorial on how to find horizontal asymptotes. Step by step tutorial including several examples of how to find horizontal asymptotes for reference.
Overview
There are several different types of asymptotes. In this tutorial, we will be discussing horizontal asymptotes. In order to find the horizontal asymptotes of a function, take the limit of the function to infinity. Every function has a horizontal asymptote if it has a limit to infinity. The limit is your horizontal asymptote.
Tags: algebra, asymptotes, closer, curves, distance, farther, horizontal, infinity, limit, linear, lines, Math, negative, nonlinear, oblique, origin, postive, straight, vertical, zero
Posted in Algebra | No Comments »
Tuesday, September 29th, 2009
How to Find Vertical Asymptotes
Description
A detailed tutorial on how to find vertical asymptotes. Step by step tutorial including several examples of how to find vertical asymptotes for reference.
Overview
There are several different types of asymptotes. In this tutorial, we will be discussing vertical asymptotes. In order to find the vertical asymptotes of a function, we must first determine if there is a vertical asymptote. There is only a vertical asymptote if the limit of the function is equal to positive or negative infinity. If that is true, then the limit will reveal the vertical asymptote.
Tags: algebra, asymptotes, closer, curves, distance, farther, horizontal, infinity, limit, linear, lines, Math, negative, nonlinear, oblique, origin, postive, straight, vertical, zero
Posted in Algebra | No Comments »
Thursday, September 24th, 2009
An Overview of Uniform Convergence
Description
A detailed tutorial of uniform convergence. Step by step tutorial including several example problems of uniform convergence for reference.
Overview
Uniform convergence is a very strong type of convergence, even stronger than pointwise convergence. A sequence {fn} of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x. This concept is important because several properties of these functions are transferred to the limit f if the convergence is uniform.
Tags: converge, convergence, differential equations, functions, limit, Math, pointwise convergence, sequence, speed, uniform convergence
Posted in Differential Equations | No Comments »
Thursday, September 17th, 2009
Definition of Expected Value
Description
A detailed tutorial on the solving of Expected Value. Step by step tutorial including several examples of how to solve Expected Value for reference.
Overview
The expected value of a variable is the integral of the variable with respect to its probability measure. It amounts to either the probability-weighted sum or the probability-weighted integral of all possible values of the variable, depending on whether you are using it for discrete random variables or continuous random variables. The expected value does not exist for all variables, but it is always the limit of a sample mean, or average, of the possible solutions for the variable.
Tags: average, continuous, discrete math, expected, expected value, limit, Math, mean, probability, random, sample, solutions, statistics, value, variable
Posted in Statistics | No Comments »
Thursday, September 17th, 2009
How to Solve the Equation of a Tangent Line
Description
A detailed tutorial on the solving of the equation of a tangent line. Step by step tutorial including several examples of how to solve the equation of a tangent line for reference.
Overview
A tangent line is the straight line to a curve at any given point that just touches the curve at that point. In a mathematical sense, at that point the tangent line is going in the same direction as the curve. To solve the equation of a tangent line, say that the curve is the graph of the function y = f(x). The point at which the tangent line intersects the curve is p = (a, f(a)). Now, take another point on the curve that is close to the line, which can be expressed as q = (a + h, f(a + h)). The secant line passes through both of these points, and the slope of the secant line is equal to the difference quotient. The difference quotient is expressed as:
Those who have already studied limits will recognize the difference quotient to be the definition of a limit function.
Tags: Calculus, curve, equation, equation of a tangent line, function, graph, limit, line, Math, secant, slope, slope of secant line, tangent
Posted in Calculus | No Comments »
