2018 ap ab frq solutions – 2018 AP Calculus AB Free-Response Questions © 2018 The College Board. – Studocu
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2018
AP Calculus AB
Free-Response Questions
© 2018 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo
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2018 AP® CALCULUS AB FREE-RESPONSE QUESTIONS
CALCULUS AB
SECTION II, Part A
Time—30 minutes
Number of questions—
A GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS.
- People enter a line for an escalator at a rate modeled by the function r given by
t 3 t 7
44 1 for 0 £ t£
r t( ) = ( 100 ) ( − 300 ) 300
0 for t > 300,
where r t( ) is measured in people per second and t is measured in seconds. As people get on the escalator,
they exit the line at a constant rate of 0 person per second. There are 20 people in line at time t = 0.
(a) How many people enter the line for the escalator during the time interval 0 £ t £ 300?
(b) During the time interval 0 £ t £ 300 , there are always people in line for the escalator. How many people
are in line at time t = 300?
(c) For t > 300 , what is the first time t that there are no people in line for the escalator?
(d) For 0 £ t £ 300 , at what time t is the number of people in line a minimum? To the nearest whole
number, find the number of people in line at this time. Justify your answer.
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-2-
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2018 AP® CALCULUS AB FREE-RESPONSE QUESTIONS
CALCULUS AB
SECTION II, Part B
Time—1 hour
Number of questions—
NO CALCULATOR IS ALLOWED FOR THESE QUESTIONS.
- The graph of the continuous function g, the derivative of the function f, is shown above. The function g is
piecewise linear for − 5 £ x < 3 , and g x( ) = 2 ( x − 4 ) 2 for 3 £x £ 6.
(a) If f ( ) 1 = 3 , what is the value of f (− 5 )?
(b) Evaluate ∫ g x( ) dx
1
6
.
(c) For − 5 < x< 6 , on what open intervals, if any, is the graph of f both increasing and concave up? Give a
reason for your answer.
(d) Find the x-coordinate of each point of inflection of the graph of f. Give a reason for your answer.
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—
© 2018 The College Board.
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2018 AP® CALCULUS AB FREE-RESPONSE QUESTIONS
- The height of a tree at time t is given by a twice-differentiable function H, where H t( ) is measured in meters
and t is measured in years. Selected values of H t( ) are given in the table above.
(a) Use the data in the table to estimate H ¢ ( ) 6. Using correct units, interpret the meaning of H ¢( 6 ) in the
context of the problem.
(b) Explain why there must be at least one time t, for 2 < t< 10 , such that H ¢( )t = 2.
(c) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate the
average height of the tree over the time interval 2 £ t£ 10.
(d) The height of the tree, in meters, can also be modeled by the function G, given by G x( ) = 100 x
1 + x
, where
x is the diameter of the base of the tree, in meters. When the tree is 50 meters tall, the diameter of the
base of the tree is increasing at a rate of 0 meter per year. According to this model, what is the rate of
change of the height of the tree with respect to time, in meters per year, at the time when the tree is
50 meters tall?
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© 2018 The College Board.
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2018 AP® CALCULUS AB FREE-RESPONSE QUESTIONS
- Consider the differential equation )
dy 1
dx = 3 x y ( − 2 .
(a) A slope field for the given differential equation is shown below. Sketch the solution curve that passes
through the point (0, 2), and sketch the solution curve that passes through the point (1, 0).
(b) Let y = f x( ) be the particular solution to the given differential equation with initial condition f ( ) 1 = 0.
Write an equation for the line tangent to the graph of y = f ( )x at x = 1. Use your equation to
approximate f (0).
(c) Find the particular solution y = f x( ) to the given differential equation with initial condition f ( ) 1 = 0.
STOP
END OF EXAM
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