MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Consider the function g(x)=sinx. A degree-5 Maclaurin polynomial is used to approximate g at x=0.5. A calculator gives sin(0.5)≈0.4794.
(a) Derive the degree-5 Maclaurin polynomial P5(x) for sinx.
(b) Evaluate P5(0.5), expressing your answer to six decimal places.
(c) Establish an upper bound for the magnitude of the remainder when P5(0.5) is used to approximate sin(0.5). Present clear justification for every inequality you use.
(d) Compute the actual error ∣sin(0.5)−P5(0.5)∣. Compare this value with the bound obtained in part (c) and discuss whether the bound is sharp or conservative.