$!$

Required information

Problem $04\mathrm{.}009-$ Taylor Series Fundamentals

The following infinite series can be used to approximate $e^x$ :

$e^x$~$=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+$cdots$+\frac{x^n}{n!}$

Problem $04\mathrm{.}009.$b $-$ Calculate the relative error associated with different order Taylor estimates.

Use the Taylor series to estimate $f(x)=e^{-x}$ at $x_{i+1}=1$ for $x_i=0\mathrm{.}25.$ Employ zero$-,$ first$-,$ second$-,$ and third$-$order versions and

compute $|{}_t|$ for each case.

For zero$-$order, $|{}_t|=$

For first$-$order, $|{}_t|=$

$\%.$

For second$-$order, $|{}_t|=$

$\%.$

For third$-$order, $|{}_l|=$

$\%.$