AP Calculus AB - mrbittmanap.files.wordpress.com · 2019 AP® CALCULUS AB FREERESPONSE QUESTIONS CALCULUS AB SECTION II, Part B Time—1 hour Number of questions—4 NO CALCULATOR

2019

AP®

Calculus ABFree-Response Questions

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2019 AP® CALCULUS AB FREERESPONSE QUESTIONS

CALCULUS AB

SECTION II, Part A

Time—30 minutes

Number of questions—2

A GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS.

1. Fish enter a lake at a rate modeled by the function E given by p t

E t( ) = 20 + 15 sin( 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 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.

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2. The velocity of a particle, P, moving along the x-axis is given by the differentiable function Pv , where P v t( )is measured in meters per hour and t is measured in hours. Selected values of v t( )P 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 v t dt2.8

∫0 P( ) .

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

Q ( )t cos 0.063t2 v t( ) = 45 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 vQ from part (c), approximate the distance between particles P and Q at time t = 2.8.

END OF PART A OF SECTION II



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CALCULUS AB

SECTION II, Part B

Time—1 hour

Number of questions—4

NO CALCULATOR IS ALLOWED FOR THESE QUESTIONS.

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.

(a) If 5

f x d = 7( ) x ∫−6 , f ind the value of

−2 f x d( )∫ x

−6 . Show the work that leads to your answer.

(b) Evaluate 5

∫ (2f x 4 x¢( ) + ) d3

.

(c) The function g is given by −

g x( ) = f t d( ) t∫x

2 . Find the absolute maximum value of g on the interval

2 x − £ 5£ . Justify your answer.

(d) Find ( ) ¢x 10 − 3f x

lim x→1 f x( ) − arctan x

.



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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 1

= − hdt 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 = pr 2h.)

(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.



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5. Let R be the region enclosed by the graphs of g x( ) = −2 3+ cos(p x

2 ) and 2h x( ) = 6 2− ( −x 1) , 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 1

A x( ) = 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.



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6. Functions f, g, and h are twice-differentiable functions with ( ) 2 g 2 = h( ) = 4. The line 2

y = + ( −x 4 2 3

) 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 h x( ) 3 ( ). Write an expression for a (¢ x). Find ¢(a 2).

(c) The function h satisfies 2 x − 4

h x( ) = 1 f x − ( ( ))3 for x π 2. It is known that lim h x(

→2 x ) can be evaluated using

L’Hospital’s Rule. Use lim h x(x→2

) to find f (2 ) and f ¢(2 ). Show the work that leads to your answers.

(d) It is known that g x h x(( ) £ ) for 1 < 3<x . Let k be a function satisfying ( ) £ k x £ (g x ( ) h x) for 1 x < 3< . Is k continuous at = 2 x ? Justify your answer.

STOP

END OF EXAM

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AP Calculus AB - mrbittmanap.files.wordpress.com · 2019 AP® CALCULUS AB FREERESPONSE QUESTIONS CALCULUS AB SECTION II, Part B Time—1 hour Number of questions—4 NO CALCULATOR