Purpose of Absolute Value and Piecewise Functions

Date: 12/09/2001 at 00:12:05
From: Kyle Hill
Subject: Purpose for Absolute Value Functions and Piecewise Functions

This is more of a word question than a mathematical question... What 
is the purpose of absolute value and of piecewise functions? I can't 
really think of a good reason to do them.

Thanks a lot.
-Kyle Hill

Date: 12/09/2001 at 23:03:09
From: Doctor Peterson
Subject: Re: Purpose for Absolute Value Functions and Piecewise 
Functions

Hi, Kyle.

A good case could be made to say that piecewise-defined functions are 
more realistic than ordinary smooth functions, so it's good to get 
used to working this way. 

There are many real-life problems in which something changes in 
different regions; for example, if you were considering the effect of 
gravity at different distances from the center of the earth, the fact 
that there is rock up to a certain distance, then air, then "empty" 
space means that you will need at least two regions with different 
formulas for gravitational acceleration. As in this example, the real 
world is not uniform, but has boundaries and sudden changes 
everywhere!

I searched our archives for applications of absolute value, to see 
what other ideas we can give. Here are two pages listing a few:

   Practical Applications of Absolute Value
   http://mathforum.org/library/drmath/view/57177.html   

   Absolute Value
   http://mathforum.org/library/drmath/view/53087.html   

This problem I helped with requires an absolute value to get it right:

   Find the Path and the Time Taken
   http://mathforum.org/library/drmath/view/53778.html   

In this case, the absolute value arises in its most basic form: the 
distance between two points.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/