MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Let f(x)=xln(1+3x) for x=0 and f(0)=x→0limxln(1+3x).
(1) Determine the first three non-zero terms of the Maclaurin series for f(x) and write them in ascending powers of x. Show all reasoning.
(2) Use your series from part (a) to evaluate f(0) and verify that the limit defining f(0) exists.
(3) Using the third-degree Maclaurin polynomial, approximate f(0.05). Give your result to five decimal places.
(4) State whether the approximation in part (c) is an overestimate or an underestimate of the exact value of f(0.05). Justify your answer analytically.