Let . You are to approximate using a Maclaurin polynomial of degree . A manufacturing…

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

Problem

Let

h(x)=\cos(x)

. You are to approximate

h(1)

using a Maclaurin polynomial of degree

n

. A manufacturing specification requires that the magnitude of the approximation error be less than

10^{-4}

. It is known that

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

for every integer

k\ge 0

and every real

x

. (a) Write an inequality that bounds

|h(1)-P_{n}(1)|

, where

P_{n}

denotes the

n

th-degree Maclaurin polynomial for

h

. (b) Solve the inequality from part (a) to identify the least integer

n

ensuring the required accuracy of

10^{-4}

. Show your algebraic steps clearly.