Let and denote the second- and fourth-degree Taylor polynomials for centered at

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

Problem

Let

P_{2}

and

P_{4}

denote the second- and fourth-degree Taylor polynomials for

\ln x

centered at

x=1

. A student uses these polynomials to approximate

\ln(1.1)

. Which statement is most accurate? (a)

P_{2}

and

P_{4}

give identical results at

x=1.1

because the third- and fourth-degree coefficients of the series are zero. (b)

P_{2}

is more accurate than

P_{4}

since the alternating higher-order terms in

P_{4}

make the approximation overshoot when

x>1

. (c)

P_{4}

is more accurate than

P_{2}

because the additional higher-degree terms reduce the truncation error near

x=1

. (d) Both

P_{2}

and

P_{4}

are equally inaccurate because the point

x=1.1

lies outside the radius of convergence of the series centered at

x=1