Free response with a graphing calculator (Part A)
Region described with functions of y
- Units 5 and 8
- 9 points
- About 15 minutes
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
Let R be the region bounded by the graphs of x = 4y − y² and x = e^(y/2) − 1. The two graphs intersect at the origin and at one other point, where y = B with B > 0. For 0 < y < B, the graph of x = 4y − y² lies to the right of the graph of x = e^(y/2) − 1.
Suggested time: 15 minutes
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Part (a)
2 pointsFind the area of R.
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Part (b)
3 pointsFind the volume of the solid generated when R is revolved about the vertical line x = −1.
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Part (c)
2 pointsRegion R is the base of a solid. For this solid, each cross section perpendicular to the y-axis is a semicircle whose diameter lies in R. Find the volume of the solid.
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Part (d)
2 pointsFor 0 ≤ y ≤ B, let w(y) be the horizontal width of R at height y. Find the value of y at which w(y) is greatest, and find the greatest width.
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