Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



Re: True/False?
Posted:
Sep 6, 1999 3:19 PM


neenag@cableol.co.uk writes: > True or False? : > > There exists a polynomial P such that: > >  P(t)  cos(t)  <= 10^6.
Presumably missing "for all t"?
> I said false. Because P(t) could be very large.
Indeed. When you go to very large t (large positive or negative t), only the term with the largest power of t will matter. This mean that P will go off towards plus or minus infinity for sufficiently large t. The only exception to this are constant polynomials like P(t)=1, which obviously won't work.
So, there is no such P.
> cos(t) = 1  x^2/2! + x^4/4!  x^6/6! + ... > > and this is valid for all values of t, so perhaps there could be a > polynomial (I'm not sure what that would be).
The Taylor series is infinite, and has to be so to work. (My argument above won't work, because there is no largestpower term)
Polynomials are defined to have only a finite number of terms.
If you have some easy way of plotting graphs you might want to make a graph of the sum up the fifth term or so of the Taylor series. It will show a graph that follows cos(t) well near t=0 but breaks off when t gets larger.
On the other hand, one often isn't interested in _all_ t, but only in t in an interval like e.g. pi < t <= pi, or even 0 <= t <= pi/2. (think about it)
Then there _are_ good solutions for P(t). The first terms of the Taylor series is one possibility, but there are others, slightly changing the terms to make the difference a bit smaller. (I don't know how many terms are needed to get below 10^6)
Stig Hemmer, Jack of a Few Trades.



