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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. Calculus More past papers & solutions for AP Calculus

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

Calculus More past papers, solutions, videos, activities & worksheets for Calculus & AP 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. Calculus More past papers & solutions for AP Calculus

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