A particle moves along a straight line and its position at time t is given by s(t)=2t^3−24t^2+90t t>=0 where s is measured in feet and t in seconds. (B) Use interval notation to indicate the time interval(s) when the particle is speeding up and slowing down. (A) Use interval notation to indicate the time interval or union of time intervals when the particle is moving forward and backward

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A particle moves along a straight line and its position at time t is given by s(t)=2t^3−24t^2+90t t>=0 where s is measured in feet and t in seconds. (B) Use interval notation to indicate the time interval(s) when the particle is speeding up and slowing down. (A) Use interval notation to indicate the time interval or union of time intervals when the particle is moving forward and backward

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(A) find the derivative of s(t), which will represent the velocity of the particle (call it v(t)).. make a sign test for the new function v(t).. the positive intervals is the intervals at which the particle is moving forward, and the negative intervals are the intervals at which the particle is slowing down.
\[s'(t)=v(t)=6t^2-48t+90=6(t-5)(t-3)\] the function is negative at t=[3,5], and positive elsewhere.. so the particle is moving backward in the interval \[t=[-3,5]\] and moving forward else where, that's \[t=[0,3]\cup[5,\infty)\]
you can solve (B) in a similar way, after finding the 2nd derivative, which will represent the acceleration a.

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