The third‐degree Maclaurin polynomial is used to approximate

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

The third‐degree Maclaurin polynomial P3(x)=x−x36P_3(x)=x-\dfrac{x^{3}}{6} is used to approximate sin⁡(0.5)\sin(0.5). (i) On 0≤x≤0.50\le x\le0.5, assume ∣sin⁡(4)(x)∣=∣sin⁡x∣≤0.5|\sin^{(4)}(x)|=|\sin x|\le0.5. (ii) The Maclaurin series for sin⁡x\sin x alternates in sign for x>0x>0. Which choice lists, in order, the Lagrange error bound and the alternating–series error bound at x=0.5x=0.5, and identifies which bound is tighter? (a) 2.6×10−4;  1.3×10−3;2.6\times10^{-4};\;1.3\times10^{-3}; the alternating–series bound is tighter (b) 1.3×10−3;  2.6×10−4;1.3\times10^{-3};\;2.6\times10^{-4}; the Lagrange bound is tighter (c) 1.3×10−3;  2.6×10−4;1.3\times10^{-3};\;2.6\times10^{-4}; the alternating–series bound is tighter (d) 2.6×10−3;  2.6×10−4;2.6\times10^{-3};\;2.6\times10^{-4}; the alternating–series bound is tighter

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