The inverse tangent function is approximated near by the third–degree Taylor…

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

Problem

The inverse tangent function

f(x)=\arctan x

is approximated near

x=1

by the third–degree Taylor polynomial

P_3

centered at

x=1

. The approximation will be used for inputs in the interval

[1,1.20]

. (a) Find

P_3(x)

for

f

centered at

x=1

. Show all derivatives used. (b) Show that for every

z

[1,1.20]

, the fourth derivative satisfies

|f^{(4)}(z)| \le 0.40

. Provide complete reasoning. (c) Employ the bound from part (b) to obtain a numerical upper bound for

|f(1.20)-P_3(1.20)|

. Give your answer to five decimal places.