s=3t^2, (s'=velocity=6t). To work out the average velocity algebraically, we calculate s for t and t+h. s=3t^2 and 3(t+h)^2. The difference is 3(t+h-t)(t+h+t)=3h(2t+h)=6ht neglecting h^2 when h is very small. So the difference in distance is 6ht+3h^2 and the average velocity is (distance)/(time)=6t+3h.
a i) Velocity at t=5 is 6*5=30, so 3h=0.3 making the average velocity=30.3.
a ii) The average velocity is 30+3*0.01=30.03.
a iii) The average velocity is 30.003.
b. The instantaneous velocity at t=5 is 30 because h approaches zero.
c. The limit definition is (6ht+3h^2)/h=6t+3h which becomes 6t as h approaches the limit of zero. So the instantaneous velocity at t=5 is 30.