$[(0)/(16)$ Points$]$

SCALCET$9$

$11\mathrm{.}11\mathrm{.}019.$MI$.$SA$.$

Consider the following function.

f$($x$)=$e$^(2$x$^(2)),$a$=0,$nx

Exercise$($a$)$

Approximate f by a Taylor polynomial with degree n at the number a$.$

Step $1$

The Taylor polynomial with degree n$=3$ is

T$\_(3)($x$)=$f$($a$)+$f$^(\text{'})($a$)($x$-$a$)+($f$^(\text{'}\text{'})($a$))/(2!)($x$-$a$)^(2)+($f$^(\text{'}\text{'}\text{'})($a$))/(31)($x$-$a$)^(3).$

The function f$($x$)=$e$^(2$x$^(2))$ has derivatives

f$^(\text{'})($x$)=($x$)$e$^(2$x$^(2))$

f$^(\text{'}\text{'})($x$)=($x$)$e$^(2$x$^(2)),$ and

f$^(\text{'}\text{'}\text{'})($x$)=($x$)$e$^(2$x$^(2))$

SKIP $($YOU CANNOT COME BACK$)$

Exercise $($b$)$

Use Taylor's Inequality to estimate the accuracy of the approximation f~~$\backslash$tau $\_($n$)($x$)$ when x lies in the given interval.

Exercise $($c$)$

Check your result in part $($b$)$ by graphing $|$R$\_($n$)($x$)|.$

Windows $$