A function is defined by

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

Problem

A function

f

is defined by

f(x)=e^{-x^{2}}

. The

n

th–degree Maclaurin polynomial

P_{n}(x)

is used to approximate

f(0.3)

. For the purposes of this problem, assume that, for every non–negative integer

m

and for every

x

in the interval

[-0.3,0.3]

, the inequality

|f^{(m)}(x)|\le 1

holds. (a) Verify the inequality

|f^{(m)}(x)|\le 1

for

m=0

and explain why the same bound remains valid for all higher–order derivatives on

[-0.3,0.3]

. Your explanation should reference structural features of the derivatives of

e^{-x^{2}}

. (b) Using the information above, establish an inequality that bounds the magnitude of the remainder

R_{n}(0.3)=f(0.3)-P_{n}(0.3)

produced by the

n

th–degree Maclaurin polynomial. (c) Determine the least integer

n

such that

|R_{n}(0.3)|<10^{-7}

. Show all reasoning that leads to your conclusion and cite the inequality from part (b) in your justification.