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Calculus AB Practice FRQ

AP Calculus AB Practice FRQ - 6 Practice Questions from past exam

Course

Calculus 1 (MA201)

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Arcadia University

Academic year: 2020/2021

Listed booksCalculus: Single and Multivariable Junqueira's Basic Histology

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2021

AP

Calculus AB

Free-Response Questions

© 2021 College Board. College Board, Advanced Placement, AP, AP Central, and the acorn logo are registered trademarks of College Board. Visit College Board on the web: collegeboard. AP Central is the official online home for the AP Program: apcentral.collegeboard.

CALCULUS AB

SECTION II, Part A

**Time—30 minutes ** **2 Questions **

**A GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS. **

GO ON TO THE NEXT PAGE.

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2. A particle, P, is moving along the xaxis. The velocity of particle Pat time tis given by vtP ()=sin () t1.

for 0 ££t p . At time t = 0 , particle Pis at position x= 5 . A second particle, Q, also moves along the xaxis. The velocity of particle Qat time tis given by

vtQ ()=( − t 1)◊1 0 ££t p . At time t = 0 , particle Qis at position x= 10.

(a) Find the positions of particles Pand Qat time t= 1 .

(b) Are particles Pand Qmoving toward each other or away from each other at time t= 1 ? Explain your reasoning. (c) Find the acceleration of particle Qat time t= 1 . Is the speed of particle Qincreasing or decreasing at time t= 1 ? Explain your reasoning.

(d) Find the total distance traveled by particle Pover the time interval 0 ££t p.

**Write your responses to this question only on the designated pages in the separate Free Response ** **booklet. Write your solution to each part in the space provided for that part. **

GO ON TO THE NEXT PAGE.

END OF PART A

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3. A company designs spinning toys using the family of functions y = cx 4 − x 2 , where c is a positive

constant. The figure above shows the region in the first quadrant bounded by the xaxis and the graph of y = cx 4 − x 2 , for some c. Each spinning toy is in the shape of the solid generated when such a region is revolved about the xaxis. Both x and y are measured in inches.

(a) Find the area of the region in the first quadrant bounded by the xaxis and the graph of y = cx 4 − x 2 for c = 6 .

(b) It is known that, for y = cx 4 − x 2 , c( 4 − 2 x )

2 dy = dx 4 − x 2 . For a particular spinning toy, the radius of the largest crosssectional circular slice is 1 inches. What is the value of c for this spinning toy?

**Write your responses to this question only on the designated pages in the separate Free Response ** **booklet. Write your solution to each part in the space provided for that part. **

GO ON TO THE NEXT PAGE.

__________________________________________________________

4. Let f be a continuous function defined on the closed interval − 4 £x £ 6 . The graph of f, consisting of four

line segments, is shown above. Let Gbe the function defined by Gx()= ∫ 0 x f t()dt.

(a) On what open intervals is the graph of Gconcave up? Give a reason for your answer.

(b) Let Pbe the function defined by Px()= Gx()◊f x(). Find P¢( 3 ).

.

(d) Find the average rate of change of Gon the interval [−4, 2]. Does the Mean Value Theorem guarantee a value c, − 4 <c < 2 , for which G¢(c)is equal to this average rate of change? Justify your answer.

**Write your responses to this question only on the designated pages in the separate Free Response ** **booklet. Write your solution to each part in the space provided for that part. **

GO ON TO THE NEXT PAGE.

6. A medication is administered to a patient. The amount, in milligrams, of the medication in the patient at

time t hours is modeled by a function y = A(t )that satisfies the differential equation dy dt = 12 3 −y. At time t = 0 hours, there are 0 milligrams of the medication in the patient.

(a) A portion of the slope field for the differential equation dy dt = 12 3 −yis given below. Sketch the solution curve through the point (0, 0).

(b) Using correct units, interpret the statement tlim →∞ At()= 12 in the context of this problem.

with initial condition A() 0 = 0 .

GO ON TO THE NEXT PAGE.

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(d) A different procedure is used to administer the medication to a second patient. The amount, in milligrams,

of the medication in the second patient at time t hours is modeled by a function y = B(t) that satisfies the differential equation dy dt = 3 − t +y 2 . At time t = 1 hour, there are 2 milligrams of the medication in the second patient. Is the rate of change of the amount of medication in the second patient increasing or decreasing at time t = 1 ? Give a reason for your answer.

**Write your responses to this question only on the designated pages in the separate Free Response ** **booklet. Write your solution to each part in the space provided for that part. **

GO ON TO THE NEXT PAGE.

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