< 3π, and 3π < t < 4±. When 0 < t < π, the velocity is positive, so the particle is moving to the
right. When π < t < 3π, the velocity is negative, so the particle is moving to the left. When 3π <
t < 4π, the velocity is positive, so the particle is moving to the right. Therefore, the particle is
changing direction at t = π and t = 3π.
- The distance is 69.
In order to find the distance that the particle travels, we need to look at the position of the
particle at and at t = 2 and at t = 5. We also need to see if the particle changes direction
anywhere on the interval between the two times. If so, we will need to look at the particle’s
position at those “turning points” as well. The way to find out if the particle is changing
direction is to look at the velocity of the particle, which we find by taking the derivative of the
position function. We get = v(t) = 6t + 2. If we set the velocity equal to zero, we get t = − ,
which is not in the time interval. This means that the velocity doesn’t change signs, and thus the
particle does not change direction. Now we look at the position of the particle on the interval.
At t = 2, the particle’s position is: x = 3(2)^2 + 2(2) + 4 = 20. At t = 5, the particle’s position is:
x = 3(5)^2 + 2(5) + 4 = 89. Therefore, the particle travels a distance of 69.
- The distance is 48.
In order to find the distance that the particle travels, we need to look at the position of the
particle at t = 0 and at t = 4. We also need to see if the particle changes direction anywhere on
the interval between the two times. If so, we will need to look at the particle’s position at those
“turning points” as well. The way to find out if the particle is changing direction is to look at
the velocity of the particle, which we find by taking the derivative of the position function. We
get = v(t) = 2t + 8. If we set the velocity equal to zero, we get t = −4, which is not in the
time interval. This means that the velocity doesn’t change signs, and thus the particle does not
change direction. Now we look at the position of the particle on the interval. At t = 0, the
particle’s position is x = (0)^2 + 8(0) = 0. At t = 4, the particle’s position is: x = (4)^2 + 8(4) =
