Explain instantaneous velocity and discuss how it can be determined from an $x-t$ graph.

(N/A) The average velocity tells us how fast an object has been moving over a given time interval but does not tell us how fast it moves at different instants of time during that interval. For this,we define instantaneous velocity.
The velocity at an instant is defined as the limit of the average velocity as the time interval $\Delta t$ becomes infinitesimally small.
In other words,
$v = \lim_{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t}$
$v = \frac{dx}{dt} = \dot{x}$
In the language of calculus,it is the differential coefficient of $x$ with respect to $t$ and is denoted by $\frac{dx}{dt}$.
We can obtain the value of velocity at an instant either graphically or numerically.
$(1)$ Graphical Method:
Suppose the given $x-t$ graph is for non-uniform motion of a car and we want to obtain graphically the value of velocity at time $t = 4 \ s$.
As we take smaller time intervals $\Delta t_1, \Delta t_2, \Delta t_3, \dots$ around $t = 4 \ s$,the corresponding displacements are $\Delta x_1, \Delta x_2, \Delta x_3, \dots$. The slope of the secant line connecting points on the curve approaches the slope of the tangent line at $t = 4 \ s$. The instantaneous velocity at $t = 4 \ s$ is equal to the slope of the tangent to the $x-t$ graph at that point.