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[apcalculus] Neat Taylor series application
Posted:
Sep 19, 2007 10:54 AM


I know it's way too early for anyone to be worrying about Taylor series yet, but I came across something this morning that many of you teaching BC calculus may find useful to look at later, when you get to Taylor series. If you've taught BC calculus more than a couple of times, you probably have a collection of "applications of Taylor series" in a folder (evaluating limits where L'Hopital's rule is very difficult, best quadratic and cubic and so on approximations at a point, various things you can obtain by differentiating or integrating Taylor series expansions, etc.), but what follows is one that you probably haven't seen before.
This is a problem in Clement V. Durell and Alan Robson's "Advanced Trigonometry" [Dover Publications, 1930/2003], Exercise V.f #20 on p. 102.
Using Taylor expansions, determine the "near x = 0" growth rate ordering for the following collection of 6 functions:
sin(sin x)
sin(arctan x)
tan(tan x)
tan(arcsin x)
arcsin(arcsin x)
arctan(arctan x)
Each of these functions is increasing in a sufficiently small interval about x = 0 (a nonmaximal such interval is, for example, 0.8 < x < 0.8). Moreover, the linear approximations for each of these functions at x = 0 is L(x) = x. [Technical note: It is possible for a function to be everywhere differentiable, have a linear approximation L(x) = x at x = 0, and yet not be increasing on any interval containing x = 0. An example is f(x) = x + (x^2)*sin(1/x^2).]
Therefore, linear approximations at x = 0 do not give us enough information to order these functions. However, by using Taylor expansions, we _can_ obtain enough information about their growth rates at x = 0 to put them in order.
All we need are these expansions:
sin(x) = x  (1/6)x^3 + ...
tan(x) = x + (1/3)x^3 + ...
arcsin(x) = x + (1/6)x^3 + ...
arctan(x) = x  (1/3)x^3 + ...
Note: If the trig. and arctrig. expansion symmetry catches your interest, apply "reversion of series" to y = x + Bx^3 + ... (i.e. constant and quadratic coefficients are zero, linear coefficient is 1). http://mathworld.wolfram.com/ReversionofSeries.html http://books.google.com/books?q=reversionofseries
It's now easy to get the first nonzero correction terms for each of the linear expansions of the 6 functions above. For example,
tan[arcsin x] = tan[x + (1/6)x^3 + ...]
= [x + (1/6)x^3 + ...] + (1/3)*[x + (1/6)x^3 + ...]^3
= [x + (1/6)x^3 + ...] + (1/3)*[x^3 + ...]
= x + (1/2)x^3 + ...
Do this for the other functions and then arrange them from least to greatest rate of increase at x = 0:
1. arctan(arctan x) = x  (2/3)x^3 + ...
2. sin(arctan x) = x  (1/2)x^3 + ...
3. sin(sin x) = x  (1/3)x^3 + ...
4. arcsin(arcsin x) = x + (1/3)x^3 + ...
5. tan(arcsin x) = x + (1/2)x^3 + ...
6. tan(tan x) = x + (2/3)x^3 + ...
You can see this graphically with a graphing calculator by graphing all 6 of these functions (USE RADIAN MODE) along with y = x. Use a window of a < x < a for a = 0.6, 0.7, or 0.8 (experiment some) and, if you don't have a "zoomfit" option, use b < y < b for b between 0.8 and 1.7 (depending on what you used for 'a'). Of course, the graph will look better if you use a CAS such as MAPLE or Mathematica . . .
Dave L. Renfro ==== Course related websites: http://apcentral.collegeboard.com/calculusab http://apcentral.collegeboard.com/calculusbc  apcalculus is an Electronic Discussion Group (EDG) of The College Board TO CHANGE YOUR EMAIL ADDRESS, PASSWORD OR SETTINGS, go to http://lyris.collegeboard.com/read/my_account/edit To UNSUBSCRIBE click the unsubscribe button on your Forums page: http://lyris.collegeboard.com/read/my_forums/



