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AP Calculus Question of the Month: August
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by Ben Klein
Davidson College
Davidson, North Carolina
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The Particle from 2004's AB3
Problem 3 on the 2004 AB examination asked questions about the motion of a particle on the y-axis given the initial position and velocity of the particle. Students did not do very well on this problem. In this question of the month, we will provide some additional explorations in the context of AB3 that we hope will help students deal with problems of this type on future AP Examinations. Even though you can download the original problem for yourself, we will reproduce it here. We will also provide an unofficial solution on the Answers Page.
AB3
A particle moves along the y-axis so that its velocity v at time t ≥ 0 is given by v(t) = 1 - tan-1(et). At time t = 0, the particle is at y = -1. (Note: tan-1x = arctan x.)
(A) Find the acceleration of the particle at time t = 2.
(B) Is the speed of the particle increasing or decreasing at time t = 2? Give a reason for your answer.
(C) Find the time t ≥ 0 at which the particle reaches its highest point. Justify your answer.
(D) Find the position of the particle at time t = 2. Is the particle moving toward the origin or away from the origin at time t = 2? Justify your answer.
The original AB3 ended here. The first of the new parts below is directly related to part (C); it did not appear on the examination in part because it is so similar to the first part of (D). The other new parts take the problem in new directions. In these new parts, we let y(t) denote the position of the particle at time t ≥ 0 .
(E) Find the y-coordinate of the highest point reached by the particle. Use your answer to confirm your answer to the question in (D) about whether the particle is moving toward or away from the origin at time t = 2.
(F) Show that
. [HINT: Do not try to do this by direct integration. Instead, show that there is a constant v1 and a positive value t1 such that v(t) ≤ v1 < 0 for all t ≥ t1.]
(G) Suppose that the initial position of the particle did not have to be -1. If the origin was the highest point reached by the particle, what would the value of y(0) have to be?
(H) Suppose that the velocity of the particle at time t≥0 was given by v(t) = a - tan1 (et), where a is a parameter. If y(0) = y(2), what is the value of the parameter a?
(I) Suppose that the velocity of the particle at time t ≥ 0 was given by v(t) = π/2 - tan-1(et). Then v is positive for all t ≥ 0 . Show that the particle travels a finite distance as t goes from 0 to infinity. [HINT: Again, do not try to use direct integration here; instead note that:
for all t ≥0.]
Click here to view the answers and commentary!
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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