An engineer working with a micro-electromechanical system (MEMS) needs to approximate…

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

Problem

An engineer working with a micro-electromechanical system (MEMS) needs to approximate

\sin(x)

near

x=0

to six-decimal precision. Measurements ensure that the maximum input angle will not exceed

0.4

radians. The Maclaurin series for

\sin(x)

is employed. (a) Show that, for every integer

k\ge 0

and every

x

[-0.4,0.4]

, the inequality

|\sin^{(k)}(x)|\le 1

is valid. (b) Establish an inequality that bounds the absolute error when the

n

th-degree Maclaurin polynomial

S_{n}(x)

is used to approximate

\sin(x)

[-0.4,0.4]

. Explicitly state the dependence of the bound on

n

and

x

. (c) Determine the minimum degree

n

required so that

|\sin(x)-S_{n}(x)|<10^{-6}

for every

x

[-0.4,0.4]

. Provide detailed reasoning and reference the bound from part (b).