Answer
$0.75$
Work Step by Step
We are given the function:
$y(t)=\sqrt{3t+1}$
The average rate of change on $[t_0,t_1]$ is:
$\dfrac{\Delta y}{\Delta t}=\dfrac{y(t_1)-y(t_0)}{t_1-t_0}$
The instantaneous rate of change is the limit of the average rate of change.
In order to estimate the instantaneous rate of of change at $t=1$, consider intervals $[t_1,t_0],[t_0,t_1]$ for $t_1$ close to $t_0$, where $t_0=1$:
$[0.9,1]$: $\dfrac{\Delta y}{\Delta x}=\dfrac{y(0.9)-y(1)}{0.9-1}=\dfrac{\sqrt{3(1.9)+1}-\sqrt{3(1)+1}}{-0.1}\approx 0.76461594$
See table
From the table we find that the instantaneous rate of of change at $t=1$ is approximately $075$.
