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Questions and Worked Solutions for AP Calculus AB 2019.

Rate in, rate out problem. Integral of a rate; average value of a function

1. 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 + 2^{0.1t2}. 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 your answer to the nearest whole number.

(b) What is the average number of fish that leave the lake per hour over the 5-hour period from midnight (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)? Explain your 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 V_{p}, where V_{p}(t) is measured in meters per hour and t is measured in hours. Selected values of V_{p}(t) are shown in the table
above. 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 V_{P}'(t), the acceleration of
particle 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 the value of .

(c) A second particle, Q, also moves along the x-axis so that its velocity for 0 ≤ t ≤ 4 is given by meters 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 V_{Q} from
part (c), approximate the distance between particles P and Q at time t = 2.8.

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 of the graph of f, consisting of two line segments and a quarter of a circle centered at the point (5, 3). It is known that the point (3, 3 −√5 ) is on the graph of f. 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. The water drains out through a valve (not shown) at the bottom of the barrel. The rate of change of the height h of the water in the barrel with respect to time t is modeled by dh/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 = πr^{2<}h.)

(a) Find the rate of change of the volume of water in the barrel with respect to time when the height of the water 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 time increasing 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 expression for 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}, the
y-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 has area 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 is rotated about the horizontal line y = 6.

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) = 3x^{3}h(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) for 1 < x < 3. Is k continuous at x = 2? Justify your answer.

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More videos, activities and worksheets that are suitable for Calculus

Questions and Worked Solutions for AP Calculus AB 2019.

AP Calculus AB 2019 Free Response Questions - Complete Paper (pdf)

AP Calculus AB 2019 Free Response Question 1Rate in, rate out problem. Integral of a rate; average value of a function

1. 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 + 2

(a) How many fish enter the lake over the 5-hour period from midnight (t = 0) to 5 A.M. (t = 5)? Give your answer to the nearest whole number.

(b) What is the average number of fish that leave the lake per hour over the 5-hour period from midnight (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)? Explain your 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 V

(a) Justify why there must be at least one time t, for 0.3 ≤ t ≤ 2.8, at which V

(b) Use a trapezoidal sum with the three subintervals [0, 0.3], [0.3, 1.7], and [1.7, 2.8] to approximate the value of .

(c) A second particle, Q, also moves along the x-axis so that its velocity for 0 ≤ t ≤ 4 is given by meters 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 V

3. The continuous function f is defined on the closed interval −6 ≤ x ≤ 5. The figure above shows a portion of the graph of f, consisting of two line segments and a quarter of a circle centered at the point (5, 3). It is known that the point (3, 3 −√5 ) is on the graph of f. 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. The water drains out through a valve (not shown) at the bottom of the barrel. The rate of change of the height h of the water in the barrel with respect to time t is modeled by dh/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 = πr

(a) Find the rate of change of the volume of water in the barrel with respect to time when the height of the water 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 time increasing 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 expression for 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)

(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 has area 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 is rotated about the horizontal line y = 6.

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) = 3x

(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) for 1 < x < 3. Is k continuous at x = 2? Justify your answer.

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