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Topic: True/False?
Replies: 4   Last Post: Sep 7, 1999 12:28 PM

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Stig Hemmer

Posts: 21
Registered: 12/12/04
Re: True/False?
Posted: Sep 6, 1999 3:19 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply 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 largest-power 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.

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