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Topic: [ap-calculus] Approximation using local linearization
Replies: 1   Last Post: Dec 6, 2007 7:22 AM

 Slomer, David Posts: 3,048 Registered: 12/6/04
RE: [ap-calculus] Approximation using local linearization
Posted: Dec 6, 2007 7:22 AM

> why it's useful to
> be able to approximate a value using local linearization

Given that x^3 + x y + y^3 = 3, find an approximate value for y near
1 when x = 1.02.
Linearization is also the basis for many other applications of
calculus. In particular, Newton's Method and Euler's Method both
rely on it. In multivariable calculus, the condition that there be a
linearization becomes the *definition* of differentiability.
--Lou Talman

Linearization works well enough to really un-nasty some problems,
including real-life, yes? Henry Pollak once showed at NCTM[?][20 years
ago? pre-handheld technology, anyway] why tractor trailers get stuck
under overpasses that the rig is in fact shorter than--the posted height
not taking into consideration slope of roadway [assuming it levels off
just before the overpass] and length of trailer. I suppose with today's
technology, who cares to merely approximate trig functions with lines.
But it was a cool demo of linearization. Maybe that was all it was and
that with mere linear approximations trucks would still get stuck...
but, as I recall, his argument included that the interval of convergence
for his "linear Taylor series" would indeed make linearization give
reliable enough results.

Dave Slomer

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Course related websites:
http://apcentral.collegeboard.com/calculusab
http://apcentral.collegeboard.com/calculusbc