Free response without a calculator (Part B)
BC: Motion along a parametric path, no calculator
- Unit 9
- 9 points
- About 15 minutes
- BC only
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
A particle moves in the xy-plane. For t ≥ 0, its position vector is ⟨x(t), y(t)⟩ = ⟨t³ − 3t, 3t²⟩.
Suggested time: 15 minutes
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Part (a)
2 pointsFind the acceleration vector of the particle at time t = 1. Find the position of the particle at the time t > 0 when the line tangent to its path is vertical. Justify your answer.
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Part (b)
2 pointsWrite an equation for the line tangent to the path of the particle at time t = 2.
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Part (c)
2 pointsFind d²y/dx² in terms of t. Is the path of the particle concave up or concave down at time t = 2? Give a reason for your answer.
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Part (d)
3 pointsFind the total distance traveled by the particle from t = 0 to t = 2.
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