A differentiable function is analytic at

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

Problem

A differentiable function

q

is analytic at

x=2

. Its fourth–degree Taylor polynomial about

x=2

P_4(x)=7+3(x-2)-5(x-2)^2+8(x-2)^3+\lambda(x-2)^4,

where

\lambda

is a constant. (a) Given that

q^{(4)}(2)=48

, determine the value of

\lambda

. (b) Use

P_4

to approximate

q(2.1)

. Show all substitution steps clearly. (c) Explain why your approximation in part (b) is likely to be accurate to at least three decimal places when

|x-2|\le 0.1

. Base your reasoning on the degree of the polynomial and the analyticity of

q

. No explicit remainder bound computation is required.