Calculus: Early Transcendentals (2nd Edition)

We draw a tangent at $(a,s(a))$ and its slope will be equal to the instantaneous velocity.
To find the instantaneous velocity at $t=a$ we measure the position $s$ at as many points as we can so that we can sketch a smooth graph of the function $s(t)$. Then we draw a line through $(a,s(a))$ an $(a,s(a+h))$. The average velocity on the time interval $[a,a+h]$ is equal to $$v=\frac{s(a+h)-s(a)}{h},$$ I.e. it will be equal to the slope of tha secant line we drew. When we shrink the interval $[a,a+h]$ (make $h$ smaller) the secant line we draw more and more starts to resemble the tangent to the graph at $(a,s(a))$ and the average velocity will become closer to the instantaneous velocity. So the goal is to successfully draw this tangent line and its slope will be equal to the instantaneous velocity.