The natural logarithm function satisfies for

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

Problem

The natural logarithm function satisfies

\ln(1+x)=x-\tfrac{x^{2}}{2}+\tfrac{x^{3}}{3}-\tfrac{x^{4}}{4}+\dots

for

|x|<1

. An engineer wishes to use the quadratic approximation

Q_2(x)=x-\tfrac{x^{2}}{2}

to estimate

\ln(1+x)

for small positive

x

with an absolute error not exceeding

0.001

. (a) Derive an inequality involving the remainder

|R_2(x)|

that must be satisfied to meet the engineer’s accuracy requirement. (b) Use an appropriate bound for the third derivative of

\ln(1+x)

to determine the largest positive value

b

such that

|x|<b

guarantees

|R_2(x)|<0.001

. (c) Taking into account the domain restriction

x>-1

, state the final interval on which the quadratic approximation meets the accuracy goal.