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Topic: Back to slope fields again, sorry
Replies: 9   Last Post: Aug 27, 1997 7:42 PM

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Patrick Cully

Posts: 31
Registered: 12/6/04
Back to slope fields again, sorry
Posted: Aug 21, 1997 5:23 PM
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Why we are putting emphasis on slope fields? Unless they are particulaly
simple it is sometimes hard to figure out by inspection what function
they represent (ie, they can be frustrating to use). Mathematically,
they do not seem to convey much in comparison to the phase plane
(graphing a function against its derivative). The phase plane for the
pendulum, for example, is an ellipse, and gets particularly interesting
when you look at the effect of damping or resonance.

Another reason to question the emphasis on slope fields is that we are
already teaching students how to reconstruct a family of functions using
the graph of a derivative. It seems that this could be sufficient for
understanding what the derivative is graphically.

Finally, it will be important later on for students to become familiar
with vector fields. Won't slope fields just add to their confusion when
they study vector fields? Maybe we should teach vector fields instead of
slope fields, since they are a nice calculus application of parametrics
as well as vectors, both of which are currently receiving emphasis on the
AP. This could be done at a level suitable for HS students.

I've looked at a number of books trying to see if I'm just missing
something and almost wonder if our interest in slope fields stems from
the fact that they are easy to graph on the calculator (tail wagging
dog). Any thoughts on this would be appreciated.

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