Free response with a graphing calculator (Part A)
BC: Circle and limaçon in polar form
- 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 and its sources
Use the description of the two polar curves below. (Your calculator should be in radian mode.)
Two polar curves (described in words)
The polar curve r = 4 sin θ, for 0 ≤ θ ≤ π, is a circle of radius 2 centered at the point (0, 2) in rectangular coordinates. It passes through the origin and is traced once as θ goes from 0 to π.
The polar curve r = 2 + cos θ, for 0 ≤ θ ≤ π, is the upper half of a limaçon. It starts at the point (3, 0) when θ = 0, passes through (0, 2) when θ = π/2, and ends at the point (−1, 0) when θ = π.
The two curves intersect at exactly two points in the upper half-plane, where θ = α and θ = β, with 0 < α < β < π. For α < θ < β, the circle is farther from the origin than the limaçon.
Source: Hypothetical curves
Suggested time: 15 minutes
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Part (a)
3 pointsFind α and β. Then find the area of the region that lies inside the circle r = 4 sin θ and outside the limaçon r = 2 + cos θ.
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Part (b)
2 pointsFind the area of the region that lies inside both curves.
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Part (c)
2 pointsFind the slope of the line tangent to the limaçon r = 2 + cos θ at the point where θ = π/3.
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Part (d)
2 pointsA particle moves along the limaçon r = 2 + cos θ so that at time t ≥ 0 its polar angle is θ = t²/4. At time t = 2, find dr/dt and dy/dt, where y is the particle's y-coordinate. What do these values tell you about the motion of the particle at time t = 2?
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