What exactly does instantaneous velocity mean?

The velocity at any given point.

 

The average velocity is for the entire period of motion; i.e. when I drove from Sarasota to Miami in 2.3 hours, my average velocity was 206/2.3 = 89.56mph; but this doesn't reflect that my velocity was 80mph when I was picking something up on my radar detector and going 100 when I wasn't.

 

Instantaneous velocity is the value of the derivative of the position vector at any given moment, independent of :delta:t

 

 

Say the equation describing my position was 2t^2. My average velocity between 0 and 10 is 400/10 = 40. My velocity at 5 seconds, d/dt = 4t, is 20.

ok quick question:

 

in your example you gave the equation 2t^2.

 

do you mean 2(t^2) or (2t)^2?

 

if you meant the first equation [which i'm assuming you did] wouldn't your average velocity be 200/10 or 20...

 

but what would the derivative of (2t)^2 be?

 

EDIT: eh, disregard that. (2t)^2 => 4t^2 which would be 8t