A differentiable function satisfies , ,

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

Problem

A differentiable function

f

satisfies

f(0)=1

f'(0)=2

f''(0)=4

, and

f'''(0)=8

. A cubic polynomial

Q_3

is constructed so that

Q_3

and

f

have the same value and first three derivatives at

x=0

. The calculator value of

f(0.2)

1.491825

(rounded to 6 decimal places). (a) Write an explicit expression for

Q_3(x)

. Express all coefficients as exact numbers—not decimals. (b) Use

Q_3

to approximate

f(0.2)

. Give your result to 6 decimal places. (c) Compute the absolute error

|f(0.2)-Q_3(0.2)|

, correct to 4 decimal places.