Let , defined for . The third-degree Taylor polynomial of about is used to…

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

Problem

Let

g(x)=\dfrac{1}{x}

, defined for

x>0

. The third-degree Taylor polynomial of

g

about

x=1

is used to approximate

g(x)

. For a numerical routine, the absolute error of this approximation must be less than

5\times10^{-5}

. (a) Starting with the general form of the remainder for a Taylor polynomial centered at

x=1

, write an inequality that must hold to ensure the desired accuracy. (b) Show that for

x

(1-r,\,1+r)

with

0<r<1

, the fourth derivative of

g

satisfies

|g^{(4)}(x)|\le 24\,(1-r)^{-5}

. Explain each step of your estimation carefully. (c) Use the result from part (b) to obtain an implicit inequality for

r

that guarantees the error requirement. Then solve the inequality to find the largest possible value of

r

. (d) State the corresponding open interval centered at

x=1

on which the third-degree Taylor polynomial approximates

\dfrac{1}{x}

within the specified error tolerance.