Abstract
The paper outlines the general properties and considerations involved in the functions of bounded and generalized bounded variation. The paper then discusses the Fourier series and provides examples where some of the Fourier properties can be best seen.

Outline:
Introduction
Functions of Bounded and Generalized Bounded Variation - Considerations
General Properties of Functions of Generalized Bounded Variation
Fourier series

From the Paper
"As such, a function of bounded variation should be the first notion before touching on functions of generalized bounded variation and, subsequently, their connection to the Fourier series. The function of bounded variation, as defined on a set E if sup F (bi) - F (ai) < , where "the supremum is taken over all sequences {[ai, bi]} of nonoverlapping intervals with endpoints in E" . In less mathematical terms, a function is of bounded variation if its total variation is finite over a certain interval for its variables.
"Moving this over an interval from 0 to 2 , described as [0, 2 ] and characteristic of Fourier series, then a function f of bounded p-variation on [0, 2 ], where p<=1, and to belong to the class Vp if Vp (f) = sup { f (Ii)) p }1/p < , where sup is the supremum over all nonoverlapping subintervals of [0, 2 ].
"With L.C. Young, the concept was generalized with the introduction of a continuous function defined on the interval [0; ) and strictly increasing from 0 to . Young proposed that "the -variation of a function f on [0,2] is the supremum V (f) of the sums f (Ii))."

Tags:subset, integers, sine, cosine

Abstract
The paper defines the unit circle as a key instrument in learning about trigonometric functions, values and concepts. The paper lists the steps to making a unit circle and provides detailed examples and graphs.

Outline:
What is the Unit Circle?
How Do I Make a Unit Circle?
How To Find Coordinates
How To Find a Reference Angle
Negative Values
In Conclusion

From the Paper
"Well, to first understand the Unit Circle, you must first understand basic graphing, because the Unit Circle is based off the circular graph x2 + y2 = 1. The Unit Circle is a circle whose values are counted counterclockwise starting from the point (1,0). Then the values- in degree and radian measure (don't worry all of this will be further explained later, so don't worry if your lost)- are used to solve trigonometry problems and equations. The values on the Unit Circle are used to find sine, cosine and tangent values as well as to find compliment and supplement angles. Overall, the Unit Circle is one of the most helpful things to know when doing the ever so complicated trigonometry. An easy was to think of the Unit Circle is that the Unit Circle is a box of primary colors, it's your red, blue and yellow. With this Unit Circle/primary color box you are able to make and understand all sorts of other colors and concepts."

Tags:pre, calculous, angles, equations, sine, cosine, tangent, values