Most places I've seen give average velocity as: $$v_a=\frac{\Delta x}{\Delta t}=\frac{x_f-x_i}{t_f-t_i}$$
Which is great and I understand perfectly. I've seen average velocity given by this though and I'm not certain I know what it is saying.
Average velocity of the object over the time interval t to t + $\Delta t$ $$\frac{x(t + \Delta t) - x(t)}{\Delta t} = \frac{change\ in\ position}{change\ in\ time}$$
Isn't $t + \Delta t = t_i + t_f - t_i = t_f$ and $x(t + \Delta t) = x(t_f)$ So couldn't it be written: $$\frac{x(t_f) - x(t_i)}{t_f - t_i}$$
And save people a lot of confusion, is there some reason to write it the way they chose? If so why over complicate it so much what is it trying to say?
After a lot of searching I found an answer in another textbook.
The quantity that tells us how fast an object is moving anywhere along its path is the instantaneous velocity, usually called simply velocity. It is the average velocity between two points on the path in the limit that the time (and therefore the displacement) between the two points approaches zero. To illustrate this idea mathematically, we need to express position x as a continuous function of t denoted by x(t). The expression for the average velocity between two points using this notation is $$v=\frac{x(t_2) - x(t_1)}{t_2 - t_1}$$
To find the instantaneous velocity at any position, we let t1 = t and t2 = t + Δt. After inserting these expressions into the equation for the average velocity and taking the limit as Δt → 0, we find the expression for the instantaneous velocity:
$$v(t)=\lim\limits_{\Delta t \to 0} \frac{x(t+Δt)−x(t)}{Δt}=\frac{dx(t)}{dt}$$
This format matches the definition of a derivative, if you replace t with x and $\Delta t$ with h.
$$f'(x)=\lim\limits_{x \to 0} \frac{f(x+h)−f(x)}{h}$$ Source here.