Date: Nov 18, 2011 7:42 AM
Author: James Hus
Subject: [ap-calculus] Natural Log Derivative
Good problems spur good discussions.
1973 AB 23 lim h->0 (1/h)(ln( (2+h)/2 ) provides such discussion.
We have started to call some of these problems "stealth" problems because some of the parts are missing or don't appear in function form i.e. the ln (2/2) in this case.
We got past the preliminary questions of "What does this limit represent...etc"
The part of this functions slope that the students found interesting is the way to look at the transformations that can occur with or without using log properties. The slope of f(x) = ln (x/2) has the same slopes as g(x) = ln (x) or h(x) = ln (ax) (a=0). When using log properties they saw the family of functions that were shifted vertically. With no log properties they discussed the relationship to the curves with stretches/compression in the x direction.
I would love to hear about other favorite DQ problems (definition of a derivative) as well.
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