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Adventures in the Amusement Park: AB2/BC2 from the 2002 AP Calculus Exams
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by David Bressoud
AP Calculus
Macalester College
St. Paul, Minnesota
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2002 AB2/BC2
The rate at which people enter an amusement park on a given day is modeled by the function E defined by
The rate at which people leave the same amusement park is modeled by the function L defined by
Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for 
, the hours during which the park is open. At time t = 9, there are no people in the park.
- How many people have entered the park by 5 p.m. (t=17)? Round answer to the nearest whole number.
- The price of admission to the park is $15 until 5:00 p.m. (t=17). After 5:00 p.m., the price of admission to the park is $11. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number.
- Let
The value of H(17) to the nearest whole number is 3725. Find the value of H'(17) and explain the meaning of H(17) and H'(17) in the context of the park.
- At what time t, for
, does the model predict that the number of people in the park is a maximum?
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I have followed this problem since its birth. I was a member of the College Board's AP Calculus Development Committee that spent over a year developing and refining it. I served as the Question Leader for it at the 2002 Reading at Colorado State University in Fort Collins, working with others to finalize the standards for scoring student work and explaining those scoring standards to the Readers. And then I took over as chair of the Calculus Development Committee, thus needing to internalize the lessons learned from this problem that can help guide the development of future exams.
I believe that others, especially AP high school teachers, may be interested in the life of this problem. I have written the story of 2002: AB2/BC2 from its beginnings through its final version, describing how the scoring standards were set, talking about where students had the most difficulty, and detailing how they performed. I discuss the lessons we have learned -- both for ourselves (about writing and scoring questions) and for the students (what easy steps can help improve their grades). I end with ideas for using this problem as a jumping off point for classroom projects.
You can find all of this in the PDF file AB2/BC2.pdf that can be downloaded in "More," below.
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