```Date: Feb 10, 2010 11:39 AM
Author: Skerbie
Subject: [ap-calculus] More on integral of 1/x and initial conditions

Greetings all,I was searching the archives looking fordiscussions regarding the antiderivative of 1/x, resulting in ln|x| vs.ln(x).  (I know Lou Talman has a nice write-up on this one ...)  At onepoint, Jon Rogawski posted the following:*****<snip from April 9, 2008>A similar issue arises in a much more common setting. Consider thestandard formulaintegral of  1/x  dx   =  ln |x|  + CThistells us that  y = ln |x| + C  is the general antiderivative of 1/x. Inother words, it's the general solution of  y' = 1/x.  But this is notcompletely correct. If we were so inclined, we could choose differentconstants for  x>0 and x<0. For example,y = ln x              for   x > 0y = ln |x| + 10     for  x < 0is a valid (but rather unnatural!) antiderivative of   1/x.*****I'mfine with all this theoretically.  My question is this ... would thisoccur in "real life"?  In other words, are there practical situationsin which practically-minded models have solutions that occur acrossboth sides of a discontinuity and hence require differing choices forthe value of the constant of integration?  I don't think a practicalsituation would have two differing initial conditions simultaneously,but could you give examples of cases from practical models in which thesame "solution" (piecewise or otherwise)  could have initial conditionsthat occur on either side of the discontinuity?  I've seen plenty ofmodeling situations in which we can have different IC's on either sideof equilibria within the same context, but that's not quite the samething.Or is it the case that in *most* practical situations, the most "typical" solution curves of interest would all be on a particular "side" of the discontinuity?  Does my question make sense?Thanks,Bill      ====Course related websites: http://apcentral.collegeboard.com/calculusabhttp://apcentral.collegeboard.com/calculusbc---To unsubscribe click here: http://lyris.collegeboard.com/read/my_forums/?forum=ap-calculusTo change your subscription address or other settings click here: http://lyris.collegeboard.com/read/my_account/edit
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