A particle moves along the x axis so that its velocity at time t is given by v(t)= −(t + 1) sin(t2/2). At time t = 0, the particle is at position x = 1. Find the acceleration of the particle at time t = 4. Is the speed of the particle increasing at t = 4? Why or why not?
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a = 0.7267 ,  acceleration is positive therefore the speed is increasing  

Explanation:

The definition of acceleration is

         a = dv / dt

they give us the function of speed

         v = - (t-1) sin (t² / 2)

         a = - sin (t²/2) -  (t-1) cos (t²/2)  2t / 2

         a = - sin (t²/2) - t (t-1)  cos (t²/2)

the acceleration for t = 4 s

          a = - sin (4²/2) - 4 (4-1) cos (4²/2)

          a = -sin 8 - 12 cos 8

remember that the angles are in radians

          a = 0.7267

the problem does not indicate the units, but to be correct they must be m/s²

We see that the acceleration is positive therefore the speed is increasing

e/c, where e is its total energy