Let and consider its fourth–degree Maclaurin polynomial

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

Problem

Let

f(x)=e^{x}

and consider its fourth–degree Maclaurin polynomial

P_4(x)=1+x+\dfrac{x^{2}}{2!}+\dfrac{x^{3}}{3!}+\dfrac{x^{4}}{4!}

. An approximation to

e^{0.3}

is obtained by evaluating

P_4(0.3)

. (a) Calculate

P_4(0.3)

, rounding your answer to six decimal places. (b) Derive an explicit upper bound for the absolute error

|e^{0.3}-P_4(0.3)|

that depends on a single value of a derivative of

f

on the interval

[0,0.3]

. Show all reasoning that leads to your bound and present the bound as a numerical value rounded to seven decimal places. (c) Determine whether the approximation in part (a) is guaranteed to be within

10^{-5}

of the exact value of

e^{0.3}

. Justify your conclusion in a single, coherent argument.

Related theory

Power Series
Maclaurin Series

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