Chapter 11.7, Problem 51E

(a)

To determine

To show: That the error bound is $\left|f\left(x\right)-T_3\left(x\right)\right|\le \frac{x^4}{24}$ for all $x\ge 0$ for the function $f\left(x\right)=e^{-x}\text{ and }{T}_3\left(x\right)=1-x+\frac{x^2}{2}-\frac{x^3}{6}$ .

(b)

To determine

To sketch: The inequality $y=f\left(x\right)-T_3\left(x\right)\text{ and }y=\frac{x^4}{24}$ together over $\left[0,1\right]$ .