A student uses the third-degree Maclaurin polynomial to approximate at two points

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

Problem

A student uses the third-degree Maclaurin polynomial

P_{3}(x)

to approximate

\sin x

at two points,

x=0.2

and

x=2

. Which of the following best explains why the approximation is far more accurate at

x=0.2

than at

x=2

? (a) Because the next neglected term in the series is

\dfrac{x^{5}}{5!}

, which is tiny at

x=0.2

but large at

x=2

. (b) Because the Maclaurin series for

\sin x

has radius of convergence

|x|<1

, so

x=0.2

is inside the interval of convergence while

x=2

is outside. (c) Because the next neglected term,

\dfrac{x^{4}}{4!}

, is much smaller in magnitude when

x=0.2

than when

x=2

. (d) Because higher–order derivatives of

\sin x

are much smaller at

x=0.2

than at

x=2

, leading to a smaller remainder there.