Let and denote the third– and fifth–degree Maclaurin polynomials for

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

Problem

Let

P_{3}(x)

and

P_{5}(x)

denote the third– and fifth–degree Maclaurin polynomials for

\cos x

, respectively. When these polynomials are used to approximate

\cos(0.2)

, which statement is most accurate? (a)

P_{3}(0.2)

and

P_{5}(0.2)

give exactly the same value because the Maclaurin series of

\cos x

contains no odd–power terms. (b)

P_{3}(0.2)

is more accurate since stopping after fewer terms reduces both truncation and rounding error. (c)

P_{5}(0.2)

is more accurate because adding the

x^{4}

term decreases the truncation error for values near

x=0

. (d) Neither polynomial is appropriate because the Maclaurin series for

\cos x

converges only when

|x|<1