9887

Detachment and Tendency of a Single Variable Function

Depicted are the pointwise operators: "detachment from left or right" and "tendency" applied to a given function. In the suggested graphs, the value of the one-sided detachments is colored in green, and the value of the tendency is colored in red.
The definition of the detachment is intuitive: the detachment from the left or right is the limit (from the left or right) of the sign of . The definition of the tendency is based on the one-sided detachments: if they are equal, then the tendency is 0; otherwise, the tendency equals the right detachment. Thus, a function's tendency in many cases simulates the sign of the derivative—without calculating the derivative first (this calculation saving has advantages in theory and applications).

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Ever since the early 1980s, computer scientists have been using discrete versions of Green's and Stokes's theorems, also known as the "summed area tables" algorithms. These theorems were shown to provide a tremendous computational gain, since they fit precisely to the needs of discrete geometry researchers due to their discrete nature. It was suggested that these theorems are actually derived from a differently defined calculus, namely the "calculus of detachment". The main operator of this theory is defined by a mixture of the discrete and continuous to form a semi-discrete version of the familiar derivative. This approach to analyze functions is hence more suitable for computers (in order to save computation time), and the simplicity of the definition allows further research in other areas of classical analysis. A basic and not very comprehensive video introduction to this theory is available, and a preprint is also available.


RELATED LINKS

    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+