Suppose a twice-differentiable function is centered at with Maclaurin polynomials

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

Problem

Suppose a twice-differentiable function

p

is centered at

a=0

with Maclaurin polynomials

P_{n}(x)

. Consider two approximation scenarios for the same error tolerance

\varepsilon>0

: Scenario I: Approximate

p(b)

, where

|b|<R

. Scenario II: Approximate

p\!\bigl(\dfrac{b}{2}\bigr)

. Assume that an upper bound

M

for

|p^{(n+1)}(x)|

[-R,R]

is available and does not depend on

n

. (a) Explain qualitatively how the required degree

n

in Scenario II compares with that in Scenario I for achieving the same tolerance

\varepsilon

. Your answer should discuss the influence of the factor

\dfrac{b}{2}

on the magnitude of the remainder. (b) Provide a symbolic inequality that relates the two least integers

n_{1}

and

n_{2}

required in Scenarios I and II, respectively, and justify your conclusion without computing explicit numerical values.