Posts Tagged ‘continuous’
Thursday, October 22nd, 2009
Inductive Sets in Set Theory
Description
A detailed tutorial on inductive sets in set theory. Step by step tutorial including several examples of inductive sets in set theory for reference.
Overview
An inductive set is a continuous set of natural numbers that follows a basic pattern of n + 1. This means that for all numbers in the set, that number plus the number one must also be included in the set.The set does not need to include all natural numbers – that is, the set may start at any natural number provided it is greater than or equal to one. However, the set must continue to infinity or it cannot be considered an inductive set.
Tags: -1, addition, complete, continuous, discrete math, element, equal, greater, induction, inductive, infinity, mathematical, natural, numbers, one, pattern, principle, set, subset, theory
Posted in Discrete Math | No Comments »
Tuesday, October 6th, 2009
How to Solve a Dirichlet Problem
Description
A detailed tutorial of solving Dirichlet problems. Step by step tutorial including several examples of how to solve Dirichlet problems for reference.
Overview
A Dirichlet problem is a problem of finding a function which solves a specified partial differential equation in the interior of a given region that takes prescribed values on the boundary of the region. It was originally supposed to be used for Laplace’s equation, although other equations can use it as well. The Dirichlet problem can be stated as: given a function f that has values everywhere on the boundary of a region in R^n, is there a unique continuous function u twice continuously differentiable in the interior and continuous on the boundary, such that u is harmonic in the interior and u = f on the boundary? A mathematical solution can be expressed as:
Tags: bounded, continuous, differential equations, Dirichlet, equation, harmonic, interior, Laplace, Math, partial differential equation, problem, region, solution, value
Posted in Differential Equations | No Comments »
Thursday, October 1st, 2009
Boundedness Theorem Explained
Description
A detailed tutorial of the boundedness theorem. Step by step tutorial including an explanation of the boundedness theorem for reference. Knowledge of the boundedness theorem is required in calculus.
Overview
The boundedness theorem is a theorem that is very closely linked to the extreme value theorem. The boundedness theorem states that a continuous function f in the closed interval [a, b] is bounded on that interval. In mathematical terms, this means that there exist real numbers m and M such that ![m \le f(x) \le M\quad\text{for all }x \in [a,b].\,](http://s.wordpress.com/latex.php?latex=m%20%5Cle%20f%28x%29%20%5Cle%20M%5Cquad%5Ctext%7Bfor%20all%20%7Dx%20%5Cin%20%5Ba%2Cb%5D.%5C%2C&bg=ffffff&fg=000000&s=0 "m \le f(x) \le M\quad\text{for all }x \in [a,b].\,")
This translates to mean “m is less than or equal to f(x) which is less than or equal to M for all x belonging to [a, b]“.
Tags: a, b, bounded, boundedness theorem, c, Calculus, closed, continuous, d, EVT, extreme value theorem, f(c), f(d), f(x), function, graph, local, m, Math, maxima, maximum, minima, minumum, value, x
Posted in Calculus | No Comments »
Thursday, October 1st, 2009
Extreme Value Theorem Explained
Description
A detailed tutorial of the extreme value theorem. Step by step tutorial including an explanation of the extreme value theorem for reference. Knowledge of the extreme value theorem is required in calculus.
Overview
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 at least once. In mathematical terms, this means that there exist numbers c and d in [a, b] such that ![f(c) \ge f(x) \ge f(d)\quad\text{for all }x\in [a,b].\,](http://s.wordpress.com/latex.php?latex=f%28c%29%20%5Cge%20f%28x%29%20%5Cge%20f%28d%29%5Cquad%5Ctext%7Bfor%20all%20%7Dx%5Cin%20%5Ba%2Cb%5D.%5C%2C&bg=ffffff&fg=000000&s=0 "f(c) \ge f(x) \ge f(d)\quad\text{for all }x\in [a,b].\,")
The translation of that formula is “f(c) is greater than or equal to f(x) which is greater than or equal to f(d), for all x belonging to [a, b]“. In order for something to belong to an interval, it must be found in the interval.
Tags: a, b, bounded, c, Calculus, closed, continuous, d, EVT, extreme value theorem, f(c), f(d), f(x), function, graph, local, Math, maxima, maximum, minima, minumum, value, x
Posted in Calculus | No Comments »
Thursday, September 24th, 2009
An Overview of Rolle’s Theorem
Description
A detailed tutorial on how to solve problems using Rolle’s Theorem. Step by step tutorial including examples of how to solve problems using Rolle’s Theorem for reference.
Overview
Rolle’s Theorem is a special instance of the Mean Value Theorem, and can be used to prove the Mean Value Theorem. Rolle’s Theorem states that a differentiable and continuous function, which attains equal values at two points, must have a point somewhere between them where the slope of the tangent line to the graph of the function is zero. Mathematically this can be expressed as if a real-valued function f is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists a c in the open interval (a, b) such that f ‘(c) = 0.
Tags: Calculus, closed, continuous, differentiable, function, graph, interval, Math, mean value theorem, open, real-valued function, rolle's theorem, slope, tangent line, zero
Posted in Calculus | No Comments »
Thursday, September 24th, 2009
Intermediate Value Theorem Explained
Description
A detailed tutorial of the intermediate value theorem. Step by step tutorial including an explanation of the intermediate value theorem for reference. Knowledge of the intermediate value theorem is required in calculus.
Overview
The intermediate value theorem states that for each value between the upper bound and the greatest lower bound of the graph of a continuous function that there is a corresponding value in its domain. In mathematical terms, the intermediate value theorem states that if f is a continuous function on the closed interval [a, b] and M is a number between f(a) and f(b), then there exists at least one number c that f(c) = M. When writing proofs in calculus, you can say that something has been proven by the IVT if you used the intermediate value theorem to reach your conclusion.
Tags: a, b, c, Calculus, continuous, corresponding, domain, f(a), f(b), f(c), function, graph, greatest lower bound, intermediate value theorem, IVT, m, Math, upper bound, value
Posted in Calculus | 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
Definition of the Mean Value Theorem
Description
A detailed tutorial on the solving of the Mean Value Theorem. Step by step tutorial including several examples of how to solve the Mean Value Theorem for reference.
Overview
You can easily figure out what the Mean Value Theorem is by looking at the word mean – a mean is an average. The Mean Value Theorem states that there is at least one point on the graph of a function where the derivative is equal to the average slope of the entire section of the graph you are looking at. The requirements are that the graph is both continuous and differentiable on the interval [a, b], where a < b. Then there exists some c in (a, b) such that:
f ‘(c) = [f(b) - f(a)] / [b - a]
The Mean Value Theorem is very similar to Rolle’s Theorem, which is a more specific theorem stating the same thing.
Tags: a, average, b, c, Calculus, continuous, derivative, differentiable, interval, Math, mean, mean value theorem, rolle's theorem, slope, theorem, value
Posted in Calculus | No Comments »
