Aim for FiveCalculus AB

Free response without a calculator (Part B)

Two particles on a line

  • Units 4, 6 and 8
  • 9 points
  • About 15 minutes

A multi-part problem you solve by hand, often from a graph, a table, an equation or a differential equation. You show your work, use exact values, and justify answers with calculus reasons such as a sign change in a derivative or the conditions of a theorem. On the exam: 4 questions in Part B of the free-response section (60 minutes, no calculator). The whole free-response section is 6 questions in 90 minutes and counts for 50% of the score. AB and BC use the same format; on BC, Part B usually includes a series question and a question on BC-only topics.

The question

Particle P moves along the x-axis so that its position at time t ≥ 0 is p(t) = t³ − 6t² + 9t + 1. Particle Q also moves along the x-axis. Its velocity at time t is q′(t) = π cos(πt/3), and its position at time t = 0 is q(0) = 0.

Suggested time: 15 minutes

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Part (a)

1 point

For 0 < t < 5, on what open interval or intervals is particle P moving to the left? Give a reason for your answer.

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Part (b)

2 points

Is the speed of particle P increasing or decreasing at time t = 2.5? Give a reason for your answer.

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Part (c)

3 points

Find the total distance traveled by particle P over the time interval 0 ≤ t ≤ 5.

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Part (d)

3 points

Find the position of particle Q at time t = 2. At time t = 2, is the distance between particles P and Q increasing or decreasing? Give a reason for your answer.

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