Natural logarithms are used in modeling relative growth
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Natural logarithms are used in modeling relative growth. In this problem you will construct a Taylor polynomial centered at x=1 for the natural log function.
(a) Find the first three non-zero derivatives of g(x)=lnx at x=1.
(b) Write the third-degree Taylor polynomial T3(x) for g centered at x=1.
(c) Use T3 to approximate ln(1.2). Express your answer as a decimal rounded to five places.
(d) Without performing an error bound calculation, give a reasoned argument based on the sign of an appropriate derivative to indicate whether your approximation in part (c) is greater than or less than the exact value of ln(1.2).