Consider the Maclaurin polynomial of degree for

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider the Maclaurin polynomial of degree

5

for

\sin x

. A designer of a motion-tracking device wishes the polynomial to approximate

\sin x

with an absolute error of at most

10^{-7}

. (a) Write an inequality involving

|R_5(x)|

that must be satisfied for the error requirement to hold. (b) Determine the largest real number

r>0

such that the error requirement is met for every

x

satisfying

|x|<r

. Show all steps of your reasoning, beginning with an appropriate bound for

|\sin^{(6)}(x)|

. (c) State the resulting symmetric interval about

0

on which the degree-

5

polynomial meets the design specification.