Mathematics

Problem

Let

T_4(x)=1+x+\dfrac{x^{2}}{2}+\dfrac{x^{3}}{6}+\dfrac{x^{4}}{24}

be the fourth-degree Maclaurin polynomial for

e^{x}

. Determine the largest number

R>0

such that

|e^{x}-T_4(x)|\le 1.0\times10^{-4}\quad\text{for all }|x|\le R.

Provide

R

accurate to two decimal places.