Free response with a graphing calculator (Part A)
BC: Particle on a parametric path
- Unit 9
- 9 points
- About 15 minutes
- BC only
You can use a calculator on this question, just like on exam day.
A multi-part problem, usually set in a real-world context, where you need a graphing calculator for things like definite integrals, derivatives at a point and solving equations. You show your setup, give decimal answers to three places, and explain or justify your conclusions. On the exam: 2 questions in Part A of the free-response section (30 minutes, graphing calculator required). 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, one of these is usually a parametric, polar or vector question.
The question
A particle moves along a curve in the xy-plane. Its position at time t, for 0 ≤ t ≤ 4, is (x(t), y(t)), where x(t) = t² − 4 ln(1 + t) and y(t) = 3 sin t + t. (Your calculator should be in radian mode.)
Suggested time: 15 minutes
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Part (a)
2 pointsFind dy/dx at time t = 2. Write an equation for the line tangent to the curve at the point where t = 2.
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Part (b)
2 pointsFind the value of d²y/dx² at time t = 2. Is the curve concave up or concave down at the point where t = 2? Give a reason for your answer.
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Part (c)
2 pointsFind the total distance traveled by the particle over the time interval 0 ≤ t ≤ 4.
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Part (d)
3 pointsAt what time t, for 0 ≤ t ≤ 4, is the particle farthest to the left? Find the coordinates of the particle at that time. Justify your answer.
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