The instantaneous velocity at time t is defined as the
average velocity in an
infinitessimally small time interval close to t:
What we have written down is simply the first derivative of the
function x(t) so in the second line of the equation I have used the notation employed in math for this quantity.
Geometrically the instantaneous velocity at time t is simply the slope of the tangential line to the curve at the time t.
I can now proceed to sketch the instantaneous velocity versus time by estimating the slope of tangents to the x(t) curve as a function of time. (draw v(t) curve under x(t) curve). We see that the instantaneous velocity is increasing linearly with time.
Amusingly I can actually also set up these timers to measure the time of passage for the cart. Knowing the length of the cart I can then derive the velocity as the cart passed that point along the track. This gives me four new independently determined points on the velocity curve which we see agree well with what we derived by the graphical technique.