Consider the function

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

f(x)=\ln(1+x)

, which is analytic on the interval

[0,0.30]

. A second–degree Taylor polynomial

P_2

is centered at

x=0

and used to approximate

f

on that interval. (a) Compute

P_2(x)

. (b) Derive an explicit expression for

f'''(x)

and determine a real number

M

such that

|f'''(z)|\le M

for every

z

[0,0.30]

. Justify the choice of

M

. (Algebraic or numeric justification earns full credit.) (c) Using your value of

M

, obtain a numerical upper bound for

|f(0.30)-P_2(0.30)|

. Give your answer to at least four decimal places.