#### Answer

1

#### Work Step by Step

$R(\theta)=\sqrt{4(\theta)+1} , [0,2]$
$$\frac{\Delta{y}}{\Delta{x}}=\frac{R(\theta_{2})-R(\theta_{1})}{\theta_{2}-\theta_{1}}$$
$R(\theta_{2})=R(2)=\sqrt{4(2)+1}$
$R(2)=\sqrt{9}$
$R(2)=3$
$R(\theta_{1})=R(0)=\sqrt{4(0)+1}$
$R(0)=\sqrt{1}$
$R(0)=1$
$$\frac{\Delta{y}}{\Delta{x}}=\frac{R(2)-R(0)}{\theta_{2}-\theta_{1}}$$
$$\frac{\Delta{y}}{\Delta{x}}=\frac{3-1}{2-0}$$
$$\frac{\Delta{y}}{\Delta{x}}=\frac{2}{2}=1$$