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In my 2nd year studying Maths at Uni and revising for a Numerical Analysis final exam. We're given 1 past paper but no solutions, and I can't answer this question:

Use the error term of a Taylor polynomial to estimate the degree of the Taylor polynomial which approximates cox for |x|<_PIE/4, with an error of no greater than 10^-5.

Any help would be much appreciated!

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  • $\begingroup$ Do you have any expression for the error term of Taylor polynomials at hand? (They come in different flavours) $\endgroup$ May 8, 2013 at 11:24
  • $\begingroup$ I didn't know this, and I don't. Do I have to go through lecture notes rather than using books then? $\endgroup$
    – glottore
    May 8, 2013 at 11:26

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Since $$\cos x = 1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\frac{x^8}{40320}-\dots$$ and $\pi/4$ is roughly $1$, it's obvious that the answer is $6$ ... although for best results we should think of this $6$th degree polynomial as $T_7$ rather than $T_6$. Using the $T_6$ error bound is for dummies. The popular Lagrange form of the remainder for $T_7$ bounds the error by $$\max_{[-\pi/4,\pi/4]}|f^{(8)}| \frac{(\pi/4)^8}{40320}<0.0000036 $$

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