MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Consider the function f(x)=ln(1+x), which is undefined at x=−1. Let T2(x) and T4(x) denote the Maclaurin polynomials of degrees 2 and 4, respectively, for f.
(a) Derive explicit expressions for T2(x) and T4(x).
(b) Compute f(0.50), T2(0.50), and T4(0.50), each to six decimal places. Then determine which polynomial is closer to the exact value and calculate its absolute error.
(c) Repeat the calculations in part (b) for x=1.0. Be sure to comment on any difference in the relative accuracy of the two polynomials compared with the case x=0.50. Support your answer quantitatively.
(d) Explain, using properties of Taylor series, why the accuracy of a fixed-degree polynomial changes as x moves farther from 0.