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The acceleration function (in m/s2) and the initial velocity v(0) are given for a particle moving along a line.

a(t) = 2t + 2, v(0) = −3, 0 ≤ t ≤ 3
(a) Find the velocity at time t.
(b) Find the distance traveled during the given time interval.

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  1. if a(t) = 2t + 2
    v(t) = t^2 + 2t + c
    but v(0) = -3
    -3 = 0 + 0 + c, c = 3

    v(t) = t^2 + 2t + 3

    s(t) = (1/3)t^3 + t^2 + 3t + k
    s(0) = k
    s(3) = 9 + 9 + 9 + k =27+k

    distance travelled during the first 3 seconds = 27+k - k = 27 metres

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