Integral Applications
1. A squirrel starts a building A at time t = 0 and travels along a straight, horizontal wire connected to building B. For 0 ≤ t ≤ 18, the squirrel’s velocity is modeled by the piecewise-linear function defined by the graph above.
a. At what times in the interval 0 ≤ t ≤ 18, if any, does the squirrel change direction? Give a reason for your answer.
b. At what time in the interval 0 ≤ t ≤ 18 is the squirrel farthest from building A? How far from building A is the squirrel at that time?
c. Find the total distance the squirrel travels during the time interval 0 ≤ t ≤ 18.
d. Write expressions for the squirrel’s acceleration a(t), velocity v(t), and distance x(t) from building A that are valid for the time interval 7 < t < 10.
2. (2001 AB3)
A car is traveling on a straight road with velocity 55ft/sec at time t=0. For seconds, the car’s acceleration a(t), in , is the piecewise function defined by the graph above.
a) Is the velocity of the car increasing at t=2 seconds? Why or why not?
b) At what time in the interval , other than t=0, is the velocity of the car 55ft/sec? Why?
c) On the time interval , what is the car’s absolute maximum velocity and at what time does it occur? Justify your answer.
d) At what times in the interval , if any is the car’s velocity equal to zero? Justify your answer.
3. The graph below is of the velocity in m/sec of a particle moving along the x-axis. The particle starts at x = 2 at t = 0.
a. Find the position of the particle at t = 5.
b. Find the position of the particle at t = 10.
c. Find the total distance traveled by the particle.
d. Find the acceleration of the particle at t = 2.
e. Find the acceleration of the particle at t = 9.
4. A particle moves along the x-axis so that its velocity at time t is given by
.
At time t = 0, the particle is at position x = 1.
a. Find the acceleration of the particle at time t = 2. Is the speed of the particle increasing at t = 2? Why or why not?
b. Find all the times t in the open interval 0 < t < 3 when the particle changes direction. Justify your answer.
c. Find the total distance traveled by the particle from t = 0 until time t = 3.
d. During the time interval 0 ≤ t ≤ 3, what is the greatest distance between the particle and the origin? Show the work that leads to your answer.