A Car is travelling on a straight path and its position is given by [math] S(t)=t3+2t2−4t [/math] When is the particle at rest? The particle is at rest when the velocity is equal to zero. We can find the velocity by taking the derivative of [math]S(t)[/math] [math]V(t)=S′(t)=3t2+4t−4[/math] [math]0=3t2+4t−4[/math] [math]0=(t+2)(3t−2)[/math] [math]t=−2,t=2/3[/math] We cannot have a negative time, so we reject[math] t=−2[/math], leaving us with the car being at rest at[math] t=2/3[/math] When is the car speeding up and slowing down? The car is speeding up when the Acceleration is posisitve, we can find the acceleration by taking the derivative of the velocity. [math]A(t)=V′(t)=6t+4[/math] [math]0=6t+4[/math] [math]t=−4/6[/math] We cannot have a negative time, so we reject [math]t=−4/6[/math], telling us that the car is constantly accelerating on this interval.