$($a$)$ Let $f(x)=e^x.$ Find a bound on the magnitude of the error when $f(x)$ is approximated using $P_3(x),$ its Taylor approximation of degree $3$ around $0$ over the interval $-2,2.$

$($b$)$ What is the actual maximum error in approximating $f(x)$ by $P_3(x)$ over the interval $-2,2?$