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AP Calculus Question of the Month: January
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by Ben Klein
Davidson College
Davidson, North Carolina
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The Two Runners: 2000 AB2/BC2 Revisited
AB2/BC2, reproduced below, from the 2000 AP Calculus Exam was a popular problem with the Exam Readers, if not with the students who took the exam. It is a good example of problems that use multiple representations. We are going to propose some additional parts that follow naturally from the parts that actually appeared in the problem. In fact, it's arguable that the additional parts are more natural than the part (c) that was actually used.
Two runners, A and B, run on a straight racetrack for
seconds. The graph above, which consists of
two line segments, shows the velocity, in meters per second, of Runner A. The velocity, in meters per second, of
Runner B is given by the function v defined by
(a) Find the velocity of Runner A and the velocity of Runner B at time t = 2 seconds. Indicate units of measure.
(b) Find the acceleration of Runner A and the acceleration of Runner B at time t = 2 seconds. Indicate units
of measure.
(c) Find the total distance run by Runner A and the total distance run by Runner B over the time interval
seconds. Indicate units of measure.
You should first work the three original parts of the problem and then work the three following additional parts. We will provide solutions for the first three parts as part of our answer, but remember that you can download solutions, scoring guidelines, and sample responses for these questions right here on AP Central.
 The Calculus AB Exam
The Calculus BC Exam
Here, with some commentary, are the three "new" parts to AB2/BC2. In doing these parts, assume that the velocity of Runner A is 10 meters per second for all
Before you tackle these new parts, you might want to find explicit expressions for
, the positions of Runners A and B, respectively, at time
(d) Which runner would win a 100-meter dash?
Comment: This problem is much more natural than the part (c) that actually appeared on the examination. However, a student whose calculator has a Computer Algebra System (CAS) capability can solve the problem without finding
explicitly, providing a so-called CAS-advantage that the Development Committee tries to avoid.
(e) Which runner would win a marathon, which we think of in this context as a 42,195-meter dash?
Comment: You can do this problem in your head.
(f) What is the longest race the two runners could run that would end in a tie?
Complete the question before viewing the answers and explanation!
Ben Klein is currently the Beverly F. Dolan Professor of Mathematics at Davidson College in Davidson, North Carolina, where he has taught since 1971. Ben's relationship with AP Calculus began in 1990 when he served as a Reader at Clemson University. He has attended every Reading since then and has served as a Table Leader in recent years. He just completed a four-year term on the AP Calculus Development Committee.
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