Taylor polynomial question....?
Support we were to plot (about 0) the 1000th Taylor polynomial p1000 of sin(x)
(a) what do you expect would be the relationship between the graphs of p1000 and sin(x) ?
(b) Find lim (x->infinity) p1000(x). What can you say about |p1000(x) - sin(x)|for large values of x?
(c) let R>0. Use the lagrange Remainder Formula to determine an upper bound for the nth Taylor remainder rn(x) for all x in [ -R, R]
(d) What conclusion can you draw from (b) and (c)?
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(a) The graphs are similar for small |x| and become more and more different when |x| gets bigger.
(b) lim(x→∞) p₁₀₀₀(x) = -∞
|p₁₀₀₀(x) - x| gets bigger as |x| increases.
(c)
|m(x)| = |f(ηx)| |x|ⁿ⁺¹ | / (n+1)! ≤ Rⁿ⁺¹/ (n+1)!
where ±f(x) = ±sin(x) or cos(x) depending of n, and 0<η<1.
(d)
The Taylor polynomial is not the best way to calculate sin(x) for |x|>>1.