Questions and Worked Solutions for AP Calculus BC 2018.
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AP Calculus BC 2018 Free Response Questions - Complete Paper (pdf)
AP Calculus BC 2018 Free Response Question 1
Rate in/rate out problem. You can use a calculator. Find total entering. Use an accumulation function and the Fundamental Theorem of Calculus (FTC). Finding the absolute minimum on a closed interval.
1. People enter a line for an escalator at a rate modeled by the function r given by
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 0.7 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.
AP Calculus BC 2018 Free Response Question 2
Integrating a density function. This was a kind of new context but all the same old stuff. Also, you had a calculator to do the hard work for you. Approximating a derivative at a point. Integrating density to get a total. Using squeeze theorem (comparison test logic) to show that number is less than another number. Arc length to find distance traveled for some parametric equations.
2. Researchers on a boat are investigating plankton cells in a sea. At a depth of h meters, the density of plankton cells, in millions of cells per cubic meter, is modeled by
p(h) = 0.2h2
for 0 ≤ h ≤ 30 modeled by f(h) for h ≥ 30. The continuous function f is not explicitly given.
(a) Find p'(25). Using correct units, interpret the meaning of p'(25) in the context of the problem.
(b) Consider a vertical column of water in this sea with horizontal cross sections of constant area 3 square
meters. To the nearest million, how many plankton cells are in this column of water between h = 0 and
h = 30 meters?
(c) There is a function u such that 0 ≤ f(h) ≤ u(h) for all h > 30 and . The column of
water in part (b) is K meters deep, where K > 30. Write an expression involving one or more integrals
that gives the number of plankton cells, in millions, in the entire column. Explain why the number of
plankton cells in the column is less than or equal to 2000 million.
(d) The boat is moving on the surface of the sea. At time t ≥' 0, the position of the boat is (x(t),y(t)), where
x'(t) = 662 sin(5t) and y'(t) = 880 cos (6t). Time t is measured in hours, and x(t) and y(t) are
measured in meters. Find the total distance traveled by the boat over the time interval 0 ≤ t ≤ 1.
AP Calculus BC 2018 Free Response Question 3
Reasoning from a graph. Using the FTC and a graph geometrically. Using the FTC and a function. Finding an antiderivative. Using the first derivative to find where a function is increasing and concave up. Finding and justifying points of inflection.
3. The graph of the continuous function g, the derivative 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)?
(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.
AP Calculus BC 2018 Free Response Question 4
Reasoning from a table. Approximating a derivative at a point. Using MVT (Mean Value Theorem) to guarantee a value. Trapezoidal Rule to approximate average value. Related rates problem using a new function
4. 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 , 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 0.03 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?
AP Calculus BC 2018 Free Response Question 5
A polar question! Finding polar area between two curves. Finding the slope of the tangent line (convert to parametric and find dy/dx). Related rates--which are almost always on polar questions.
5. The graphs of the polar curves r = 4 and r = 3 + 2cosθ are shown in the figure above. The curves intersect at θ = π/3 and θ = 5π/3.
(a) Let R be the shaded region that is inside the graph of r = 4 and also outside the graph of r = 3 + 2cosθ, as shown in the figure above. Write an expression involving an integral for the area of R.
(b) Find the slope of the line tangent to the graph of r = 3 + 2cosθ at θ = π/2.
(c) A particle moves along the portion of the curve r = 3 + 2cosθ for 0 < θ < π/2. The particle moves in such a way that the distance between the particle and the origin increases at a constant rate of 3 units per second. Find the rate at which the angle θ changes with respect to time at the instant when the position of the particle corresponds θ = π/3. Indicate units of measure.
AP Calculus BC 2018 Free Response Question 6
6. The Maclaurin series for is given by .
Series question! Creating a new series from a given series (composition and multiplying by x). Interval of convergence using ratio test and testing end-points. Alternating series. Alternating series error bound. Taylor polynomial approximation.
On its interval of convergence, this series converges to ln(1 + x). Let f be the function defined by
f(x) = xln(1 + x/3)
(a) Write the first four nonzero terms and the general term of the Maclaurin series for f.
(b) Determine the interval of convergence of the Maclaurin series for f. Show the work that leads to your
(c) Let P4
be the fourth-degree Taylor polynomial for f about x = 0. Use the alternating series error
bound to find an upper bound for |P4
(2) - f(s)|.
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