Consider the Maclaurin polynomial of degree for

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

Problem

Consider the Maclaurin polynomial

T_n(x)

of degree

n

for

e^{x}

. You wish to approximate

e^{x}

on the interval

[-1,1]

so that the absolute error does not exceed

10^{-5}

on the entire interval. (a) Derive an upper bound for the remainder

R_n(x)=e^{x}-T_n(x)

[-1,1]

in terms of

n

. Clearly state any theorem or inequality you use in your reasoning. (b) Determine the least integer

n

that guarantees the bound

|R_n(x)|<10^{-5}

for all

x

[-1,1]

. Show all steps that lead to your conclusion. (c) Using the degree

n

found in part (b), write the explicit form of

T_n(x)

and use it to approximate

e^{0.8}

. Present the approximation to six decimal places.