2011-2012 Particle Motion Definition and Calculus The position of a particle (in inches) moving along the x -axis after t seconds have elapsed is given by the following equation: s = f (t) = t4 – 2 t3 – 6 t2 + 9 t (a) Calculate the velocity of the particle at time t. The position of particle Pis given by x P(t) = ln(t22t+ 10) when 0 t8, so its velocity is given by x0 P(t) = 2t 2 t22t+10 = 2(t 1) (t 1)2+9: (21) The denominator of this fraction is never negative, being 9larger than a square. Therefore, the sign of the derivative (which is velocity) is determined by the sign of its numerator.
Questions And Worked Solutions For AP Calculus AB 2019
AP Calculus AB 2019 Free Response Question 1
Rate in, rate out problem. Integral of a rate; average value of a function
- Fish enter a lake at a rate modeled by the function E given by E(t) = 20 + 15 sin(πt/6). Fish leave the lake at a rate modeled by the function L given by L(t) = 4 + 20.1t2. Both E(t) and L(t) are measured in fish per hour, and t is measured in hours since midnight (t = 0).
(a) How many fish enter the lake over the 5-hour period from midnight (t = 0) to 5 A.M. (t = 5)? Give youranswer to the nearest whole number.
(b) What is the average number of fish that leave the lake per hour over the 5-hour period frommidnight (t = 0) to 5 A.M. (t = 5)?
(c) At what time t, for 0 ≤ t ≤ 8, is the greatest number of fish in the lake? Justify your answer.
(d) Is the rate of change in the number of fish in the lake increasing or decreasing at 5 A.M. (t = 5)? Explainyour reasoning.
AP Calculus AB 2019 Free Response Question 2
2. The velocity of a particle, P, moving along the x-axis is given by the differentiable function Vp, where Vp(t) is measured in meters per hour and t is measured in hours. Selected values of Vp(t) are shown in the tableabove. Particle P is at the origin at time t = 0.
(a) Justify why there must be at least one time t, for 0.3 ≤ t ≤ 2.8, at which VP'(t), the acceleration ofparticle P, equals 0 meters per hour per hour.
(b) Use a trapezoidal sum with the three subintervals [0, 0.3], [0.3, 1.7], and [1.7, 2.8] to approximate thevalue of .
(c) A second particle, Q, also moves along the x-axis so that its velocity for 0 ≤ t ≤ 4 is given bymeters per hour. Find the time interval during which the velocity of particle Q is at least 60 meters per hour. Find the distance traveled by particle Q during the interval when the velocity of particle Q is at least 60 meters per hour.
(d) At time t = 0, particle Q is at position x = −90. Using the result from part (b) and the function VQ frompart (c), approximate the distance between particles P and Q at time t = 2.8.
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AP Calculus AB 2019 Free Response Question 3
3. The continuous function f is defined on the closed interval −6 ≤ x ≤ 5. The figure above shows a portion ofthe graph of f, consisting of two line segments and a quarter of a circle centered at the point (5, 3). It isknown that the point (3, 3 −√5 ) is on the graph of f.
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AP Calculus AB 2019 Free Response Question 4
4. A cylindrical barrel with a diameter of 2 feet contains collected rainwater, as shown in the figure above. Thewater drains out through a valve (not shown) at the bottom of the barrel. The rate of change of the height h ofthe water in the barrel with respect to time t is modeled bydh/dt = -1/10 √, where h is measured in feet and t is measured in seconds. (The volume V of a cylinder with radius r and height h is V = πr2«/sup>h.)
(a) Find the rate of change of the volume of water in the barrel with respect to time when the height of thewater is 4 feet. Indicate units of measure.
(b) When the height of the water is 3 feet, is the rate of change of the height of the water with respect to timeincreasing or decreasing? Explain your reasoning.
(c) At time t = 0 seconds, the height of the water is 5 feet. Use separation of variables to find an expressionfor h in terms of t.
AP Calculus AB 2019 Free Response Question 5
5. Let R be the region enclosed by the graphs of g(x)) = −2 + 3 + cos(π/2 x) and h(x) = 6 - 2(x - 1)2, they-axis, and the vertical line x = 2, as shown in the figure above.
(a) Find the area of R.
(b) Region R is the base of a solid. For the solid, at each x the cross section perpendicular to the x-axis hasarea A(x) = 1/(x + 3). Find the volume of the solid.
(c) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R isrotated about the horizontal line y = 6.
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AP Calculus AB 2019 Free Response Question 6
6. Functions f, g, and h are twice-differentiable functions with g(2) = h(2) = 4. The line y = 4 + 2/3 (x - 2) is tangent to both the graph of g at x = 2 and the graph of h at x = 2.
(a) Find h'(2).
(b) Let a be the function given by a(x) = 3x3h(x). Write an expression for a'(x). Find a'(2).
(d) It is known that g(x) ≤ h(x) for 1 < x < 3. Let k be a function satisfying g(x) ≤ k(x) ≤ h(x) for1 < x < 3. Is k continuous at x = 2? Justify your answer.
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Here are the resources you will need for the AP Calculus AB free response questions. Be sure to try out the prior year questions and check out the video solutions. Working through these problems is an important part of your FRQ test prep.
2017 & 2018 Free Response
At the College Board website you can access the prior year free response questions. You should definitely review these as part of your FRQ test prep.
2018 FRQ Solutions
In this series of videos, Tom Cochran solves the 2018 AP Calculus AB free response questions. Clear solutions that are easy to understand.
Mario Particle Motion Ap Calculus 2018 Problems
2017 FRQ Solutions
More great videos from Mr. Cochran. In these he solves the 2017 AP Calc free response questions. Be sure to try the questions first, then watch the videos.
Sample Questions
The sample question packet from the College Board includes some samples of the updated AP Calculus AB free response questions. Good for test prep and review.
Course Description
The official course description includes some free response practice questions as well. The AB FRQ questions start on page 67 of this packet.
Mario Particle Motion Ap Calculus Problems
Mario Particle Motion Ap Calculus Notes
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