A particle moves along a straight line. For 0 <= t <= 5, the position of the particle is given by s(t) = 3t + 4t^2 - t^3 What is the average velocity of the particle over the interval 0 <= t <= 4?
Find a general expression for the velocity of the particle as a function of time, v(t), and determine the value of v at t=2. Is the particle traveling to the left or the right at this point? Justify your answer.
Find all times t during the interval 0 <= t<= 5 that the particle changes direction. Justify your answer.
At t = 4, is the speed of the particle increasing or decreasing? Justify your answer

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The answer is    (d) 7/2======we have    we also  have an integral      notice how function f splits up at x = 0. there's a left piece and a right pieceso we'll need to do the same thing with our integral: split it at x = 0we use a property of integrals that allows us to split up and sum:     notice that for the piecewise function f, the function has the equationf(x) = x when x  ≤ 0. so for the integral  , f(x) = x so it changes into      for the piecewise function f, the function has the equationf(x) = x+1  when x  > 0. for the integral  , we have    therefore, back to our splitted integral: