Let and be the degree and degree Taylor polynomials for centered at

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

Problem

Let

P_3(x)

and

P_5(x)

be the degree

3

and degree

5

Taylor polynomials for

e^{x}

centered at

x=0

. A student uses these polynomials to approximate

e^{1.2}

. Which of the following statements is true about the two approximations? (a) Both polynomials overestimate

e^{1.2}

, and the degree

5

polynomial does so by the larger amount. (b) Because the Maclaurin series for

e^{x}

converges only for

|x|\le1

, neither

P_{3}(1.2)

nor

P_{5}(1.2)

provides a reliable approximation to

e^{1.2}

. (c)

P_{5}(1.2)

is closer to

e^{1.2}

than

P_{3}(1.2)

by roughly

0.1

. (d) Both polynomials underestimate

e^{1.2}

, but

P_{3}(1.2)

is the better approximation.