Mathematics

Mathematics

Problem

For f(x)=sin⁡xf(x)=\sin x centered at a=0a=0, approximate f(0.5)f(0.5) with a degree-nn Maclaurin polynomial. Knowing that ∣f(n+1)(c)∣≤1|f^{(n+1)}(c)|\le1 for all real cc, determine the smallest integer nn that ensures the absolute error does not exceed 1.0×10−41.0\times10^{-4}.

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