The natural logarithm function satisfies

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

Problem

\ln(0.95)\approx-0.051293

. Centered at

x=1

, the first-degree Taylor polynomial for

\ln x

P_{1}(x)=x-1

and the second-degree polynomial is

P_{2}(x)=x-1-\dfrac{(x-1)^{2}}{2}

. Which of the following correctly lists the two polynomial approximations of

\ln(0.95)

and identifies which has the smaller absolute error? (a)

P_{1}\approx -0.0513

and

P_{2}\approx -0.0500

;

P_{1}

has the smaller error. (b)

P_{1}\approx -0.0500

and

P_{2}\approx -0.05125

;

P_{2}

has the smaller error. (c)

P_{1}\approx -0.0500

and

P_{2}\approx -0.04875

;

P_{2}

has the smaller error. (d)

P_{1}\approx -0.0500

and

P_{2}\approx -0.05125

;

P_{1}

has the smaller error.