> 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.