# What is the average speed of an object that is still at #t=0# and accelerates at a rate of #a(t) = t^3-2t+2# from #t in [2, 3]#?

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The average speed of an object that accelerates at a rate of ( a(t) = t^3 - 2t + 2 ) from ( t ) in [2, 3] can be calculated using the formula for average speed:

[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} ]

To find the total distance traveled, integrate the velocity function over the given interval [2, 3]. Then divide the total distance by the total time, which is the difference between the final and initial times.

First, integrate the acceleration function ( a(t) = t^3 - 2t + 2 ) to find the velocity function ( v(t) ). Then integrate the velocity function over the interval [2, 3] to find the total distance traveled.

Next, calculate the total time, which is ( t_f - t_i = 3 - 2 = 1 ).

Finally, divide the total distance by the total time to find the average speed.

This process will provide the average speed of the object over the given time interval.

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