If v is much smaller than c$,$ then all terms after the first are very small when compared with the first term. If we omit them, we get

K $$ m$0$c$2$

$12$

$$

v$2$c$2$

$=$

m$0$v$22$

$.$

$($b$)$

If

x $=$

v$2$c$2$

$,$

f$($x$)=$ m$0$c$2[(1+$ x$)121],$

and M is a number such that

$|$f$\text{'}\text{'}($x$)|$ M$,$

then we can use Taylor's Inequality to write

$|$R$1($x$)|$

M$2!$

$$x$2.$

We have

f$\text{'}\text{'}($x$)=$

$34$m$0$c$21+$x $(52)$

and we are given that

$|$v$|140$ m$/$s$,$

so

$|$f$\text{'}\text{'}($x$)|=$

$3$m$0$c$24$

$1$

v$2$c$2$

$52$

$$

$3$m$0$c$24$

$1$

$1402$c$2$

$52$

$(=$ M$).$