Free response with a graphing calculator (Part A)
BC: Polar curve area
- 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
The curve C is given by the polar equation r = 2 + sin(2θ) for 0 ≤ θ ≤ π/2. The curve starts at the point (2, 0) on the positive x-axis when θ = 0 and ends at the point (0, 2) on the positive y-axis when θ = π/2. Let R be the region in the first quadrant bounded by C, the positive x-axis and the positive y-axis. (Your calculator should be in radian mode.)
Suggested time: 15 minutes
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Part (a)
2 pointsFind the area of R.
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Part (b)
3 pointsThe circle r = 5/2 intersects C at two points in the first quadrant. Find the area of the region that lies inside C and outside the circle r = 5/2.
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Part (c)
2 pointsFind the value of dr/dθ at θ = π/3. Explain what this value tells you about the distance between the origin and a point moving along C as θ increases through π/3.
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Part (d)
2 pointsFind the greatest y-coordinate of any point on C. Justify your answer.
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