Question

suppose that a particle moves along the x-axis so that at time t, the particle is located xt=2t^3-4t^2 meters to the right of the origin. Find the average velocity of the particle between t=4 and t=5.













Find the average velocity of the particle between t=4 and t=4.01.

Find the instantaneous velocity of the particle at t=4.

Answer 1

The average value of a function A is given by:
\[\frac{1}{b-a}\int\limits_a^b A dt \]
The velocity is given by:
\[\frac{d}{dt}(x(t))=v(t)\]
if x(t) is the position.
The instantaneous velocity at a time t_0 is
\[\frac{d}{dt}x(t) |_{t=t_0}=v(t_0) \]

Answer 2

they give you the displacement vector:
r=2t^3-4t^2
from this you need to find velocities: so remember
r'=v(t)
r'=d/dt (2t^3-4t^2)
r'=v(t)=6t^2-8t

now that we have velocity, plug in for the needed times and average where needed and such.