Posts Tagged ‘functions’
Tuesday, October 6th, 2009
The Application of Bessel Functions
Description
A detailed tutorial on the application of Bessel functions. Step by step tutorial including several examples of the application of Bessel functions for reference.
Overview
Bessel functions are the solution to Bessel’s equation – although they were created by the same person, the question and the answer are seperated into two different things. This is because Bessel functions are also used to solve other things, such as Laplace’s equation and the Helmholtz equation. It can also be used with spherical and cylindrical coordinates. There are many different definitions of Bessel functions, but the most common one involves a Taylor series expansion around x = 0. Mathematically this is expressed as:
Tags: Bessel, cylindrical, differential equations, equation, expansion, function, functions, helmholtz equation, Laplace's Equation, Math, series, spherical, Taylor
Posted in Differential Equations | No Comments »
Tuesday, October 6th, 2009
Plotting Points in the Polar Coordinate System
Description
A detailed tutorial on plotting points in the polar coordinate system. Step by step tutorial including several examples of how to plot points on the polar coordinate system for reference.
Overview
By this point, everyone should know how to plot points on a normal graph. But what about a circular graph? This circular graph is called the polar coordinate system or the polar plane. Instead of using the points (x, y), the polar coordinate system uses the points (r, theta). Theta is a greek letter that looks like a zero with a horizontal line drawn through the center. Most of the points you will be finding for the polar coordinate system will be used with trigonometric functions – sine, cosine, and tangent. Graphing occurs in about the same way as it would on a normal graph – just match up the points, even if they are on a circle.
Tags: Calculus, circle, coordinate, cosine, function, functions, graph, Math, points, polar, r, sine, system, tangent, theta, trig, trigonometric, x, y
Posted in Calculus | No Comments »
Thursday, September 24th, 2009
Introduction to the Fourier Series
Description
A detailed tutorial on the solving of Fourier series. Step by step tutorial including several examples of how to solve the Fourier series for reference.
Overview
The Fourier series is very common in differential equations and partial differential equations. What a Fourier series does is decompose a periodic function into the sum of simple oscillating functions, like sine and cosine. Fourier series are part of Fourier analysis and were first introduced by Joseph Fourier to solve the heat equation. So another name for the Fourier series is the heat equation, although they are considered different things. Both the heat equation and the Fourier series are partial differential equations.
Tags: cosine, decompose, differential equations, Fourier analysis, Fourier series, function, functions, heat equation, Joseph Fourier, Math, oscillating, partial differential equations, periodic function, simple, sine, sum
Posted in Differential Equations | 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
Squeeze Theorem Explained
Description
A detailed tutorial on the solving of the Squeeze Theorem. Step by step tutorial including several examples of how to solve the Sqeeze Theorem for reference. Knowledge of the Squueze Theorem is required for calculus.
Overview
The squeeze theorem is a theorem regarding the limit of a function. It is very important in calculus proofs. The Squeeze Theorem states: Let l be an interval containing point a. Let f, g, and h be functions defined on l, except possibly at a itself. Suppose that for every x in l not equal to a, we have:
And also suppose that:
Then, 
Tags: Calculus, functions, Math, pinching theorem, sandwich rule, sandwich theorem, squeeze lemma, squeeze theorem
Posted in Calculus | No Comments »
Friday, September 11th, 2009
How to Find the Inverse of a Function
Description
A detailed tutorial on how to find the inverse of a function. Step by step tutorial including several examples of inverse functions for reference.
Overview
Most people are familiar with inverse operations – subtraction is the inverse of addition, etc. But what about inverse functions? Inverse functions are functions that are an exact opposite of another function. If you have two functions, set them equal to each other – if they cancel out, then they are inverse functions. What if you are given one function, and told to find the inverse? There is a mathematical trick to finding the inverse of a function. Set f(x) equal to y, and then switch x and y. Then solve for y. Your value for y is your inverse function. This works because f(y) is the inverse function of f(x), and you are simply solving backwards for it.
Tags: algebra, f(x), f(y), functions, inverse, inverse of a function, Math, x, y
Posted in Algebra | No Comments »
Friday, September 11th, 2009
How to Use the Vertical Line Test
Description
A detailed tutorial on how to use the Vertical Line Test for grphing functions. Step by step tutorial including several examples of how to use the Vertical Line Test for reference.
Overview
The Vertical Line Test is a test used to find out if a graph is a function or not. If the vertical line only passes through the line or curve once, then the graph is a function. If the vertical line passes through the line or curve more than once, or not at all, then the graph is not a function. The vertical line only hitting the graph once must be true for every single place the vertical line could be drawn or the graph is not a function.
Tags: algebra, functions, graphing, graphs, line, Math, point, vertical, vertical line test
Posted in Algebra | No Comments »
Thursday, September 10th, 2009
How to Perform Operations of Functions
Description
Step-by-step video tutorial on how to solve various operations of functions. Several example problems are provided. Knowledge of functions and operations of functions are required for grade-school algebra.
Overview
I’m sure you are familiar with the normal form of a function – f(x) = (equation or number). If you have a second one, it will be expressed as g(x) = (equation or number). But what happens if you are told to combine the functions through an operation? You follow these basic patterns:
(f + g)(x) = f(x) + g(x)
(f – g)(x) = f(x) – g(x)
(f * g)(x) = f(x) * g(x)
(f / g)(x) = f(x) / g(x)
These are all very basic, and very easy to solve. There is one other pattern, which can seem rather confusing. This pattern is (f o g)(x) = f[g(x)]. The circle means “of”, so you would read that to be “f of g of x”. This means that for every x you see in the f function, you replace it with the entire g function.
Tags: addition, arithmetic, division, functions, Math, multiplication, operations, operations of functions, subtraction
Posted in Algebra, Arithmetic | No Comments »
