AP Calculus AB (Section 2, Part A)

A spaceship moves along the x-axis in space so that its velocity at time t is given by v(t) = \frac{2t^4-5t^3+8}{t^2+2} . At time t = 0, the initial position of the spaceship is x = 15.

(a) Find the acceleration of the spaceship at time t = 3.2.
(b) Find all values of t in the interval 0\le t\le2 for which the speed of the spaceship is 2.
(c) Find the position of the spaceship at time t=5 . Is the spaceship moving toward a celestial body or away from it at time t = 5? Justify your answer.
(d) During the time interval 0<t\le5 , does the spaceship return to its initial position? Give a reason for your answer.

Maria rides a bicycle along a straight path in a 40-kilometer-long trail for 120 minutes. Maria’s velocity is modeled by v(t) = 3.5e^{-0.015t}\cos{\left(\frac{\pi}{40}t\right)} , where t is measured in minute.

(a) Find all times in the interval 0 < t < 120 at which Maria changes direction. Give a reason for your answer.
(b) Find Maria’s acceleration at time t = 80 minutes. Show the setup for your calculations, and indicate units of measure. Is Maria speeding up or slowing down at time t = 80 minutes? Give a reason for your answer.
(c) Find the distance between Maria’s position at time t = 30 minutes and her position at time t = 90 minutes. Show the setup for your calculations.
(d) Find the total distance Maria rides over the time interval 0\le t\le120 minutes. Show the setup for your calculations.