Sample Exam Questions

AP Calculus BC Sample Exam Questions

Multiple Choice: Section I, Part A

A calculator may not be used on questions on this part of the exam.

1. A curve is defined by the parametric equations and

What is in terms of t ?

(A)

(B)

(C)

(D)

Mathematical
Practice for
Learning Objective Essential Knowledge AP Calculus

LO 2.1C: Calculate EK 2.1C7: (BC) Methods for calculating MPAC 3: Implementing

derivatives. derivatives of real-valued functions can be algebraic/
extended to vector-valued functions, parametric computational
functions, and functions in polar coordinates. processes
MPAC 2: Connecting
concepts

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 Sample Exam Questions

2.

Consider the differential equation where A is a constant.

Let be the particular solution to the differential equation with the initial condition
Euler’s method, starting at with a step size of 2, is used to approximate
Steps from this approximation are shown in the table above. What is the value of A ?

(A)

(B) 2
(C) 5

(D)

Mathematical
Practice for
Learning Objective Essential Knowledge AP Calculus

LO 2.3F: Estimate EK 2.3F2: (BC) For differential equations, Euler’s MPAC 4:

solutions to differential method provides a procedure for approximating Connecting multiple
equations. a solution or a point on a solution curve. representations
MPAC 3: Implementing
algebraic/
computational
processes
AP CALCULUS BC SAMPLE EXAM QUESTIONS

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 Sample Exam Questions

3.

(A)
(B)
(C)
(D)

Mathematical
Practice for
Learning Objective Essential Knowledge AP Calculus

LO 3.3B(a): Calculate EK 3.3B5: Techniques for finding antiderivatives MPAC 3: Implementing

antiderivatives. include algebraic manipulation such as long algebraic/
division and completing the square, substitution computational
of variables, (BC) integration by parts, and processes
nonrepeating linear partial fractions.
MPAC 5: Building
notational fluency

AP CALCULUS BC SAMPLE EXAM QUESTIONS

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 Sample Exam Questions

4. The shaded region in the figure above is bounded by the graphs of and
for Which of the following expressions gives the perimeter of the
region?

(A)

(B)

(C)

(D)
AP CALCULUS BC SAMPLE EXAM QUESTIONS

Mathematical
Practice for
Learning Objective Essential Knowledge AP Calculus

LO 3.4D: Apply EK 3.4D3: (BC) The length of a planar MPAC 2: Connecting

definite integrals to curve defined by a function or by a concepts
problems involving parametrically defined curve can be
MPAC 4: Connecting
area, volume, (BC) and calculated using a definite integral.
multiple representations
length of a curve.

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5. The number of fish in a lake is modeled by the function P that satisfies the differential equation
where t is the time in years. Which of the following could be the

graph of

(A) (B)

(C) (D)

AP CALCULUS BC SAMPLE EXAM QUESTIONS

Mathematical
Practice for
Learning Objective Essential Knowledge AP Calculus

LO 3.5B: Interpret, EK 3.5B2: (BC) The model for logistic growth that MPAC 2: Connecting
create and solve arises from the statement “The rate of change of concepts
differential equations a quantity is jointly proportional to the size of the
MPAC 4:
from problems quantity and the difference between the quantity
Connecting multiple
in context.
and the carrying capacity” is representations

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 Sample Exam Questions

6. Which of the following series is absolutely convergent?

(A)

(B)

(C)

(D)

Mathematical
Practice for
Learning Objective Essential Knowledge AP Calculus

LO 4.1A: Determine EK 4.1A4: A series may be absolutely convergent, MPAC 1: Reasoning

whether a series conditionally convergent, or divergent. with definitions
converges or diverges. and theorems
MPAC 2: Connecting
concepts
AP CALCULUS BC SAMPLE EXAM QUESTIONS

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7. Which of the following series cannot be shown to converge using the limit comparison test

with the series

(A)

(B)

(C)

(D)

Mathematical
Practice for
Learning Objective Essential Knowledge AP Calculus

LO 4.1A: Determine EK 4.1A6: In addition to examining the limit MPAC 2: Connecting

whether a series of the sequence of partial sums of the series, concepts
converges or diverges. methods for determining whether a series
MPAC 3: Implementing
of numbers converges or diverges are the
algebraic/
nth term test, the comparison test, the limit
computational
comparison test, the integral test, the ratio
processes
test, and the alternating series test.

AP CALCULUS BC SAMPLE EXAM QUESTIONS

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8. The third-degree Taylor polynomial for the function f about is

Which of the following tables gives the values of f and its first
three derivatives at

(a) x

0 3 6

(b) x

0 3 2

(c) x

0 3 4

(d) x

0 3 4

Mathematical
Practice for
Learning Objective Essential Knowledge AP Calculus

LO 4.2A: Construct and EK 4.2A1: The coefficient of the nth-degree MPAC 1: Reasoning
use Taylor polynomials. term in a Taylor polynomial centered at with definitions
and theorems
for the function f is .
MPAC 4:
Connecting multiple
AP CALCULUS BC SAMPLE EXAM QUESTIONS

representations

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9. What is the interval of convergence for the power series

(A)
(B)
(C)
(D)

Mathematical
Practice for
Learning Objective Essential Knowledge AP Calculus

LO 4.2C: Determine EK 4.2C2: The ratio test can be used to determine MPAC 3: Implementing
the radius and interval the radius of convergence of a power series. algebraic/
of convergence of computational
a power series. processes
MPAC 1: Reasoning
LO 4.1A: Determine EK 4.1A6: In addition to examining the limit with definitions
whether a series of the sequence of partial sums of the series, and theorems
converges or diverges. methods for determining whether a series
of numbers converges or diverges are the
nth term test, the comparison test, the limit
comparison test, the integral test, the ratio
test, and the alternating series test.

AP CALCULUS BC SAMPLE EXAM QUESTIONS

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 Sample Exam Questions

Multiple Choice: Section I, Part B

A graphing calculator is required for some questions on this part of the exam.

10. For time seconds, the position of an object traveling along a curve in the xy-plane is
given by the parametric equations and where and
At what time t is the speed of the object 10 units per second?
(A)
(B)
(C)
(D)

Mathematical
Practice for
Learning Objective Essential Knowledge AP Calculus

LO 2.3C: Solve EK 2.3C4: (BC) Derivatives can be used to MPAC 2: Connecting

problems involving determine velocity, speed, and acceleration concepts
related rates, for a particle moving along curves given by
MPAC 3: Implementing
optimization, rectilinear parametric or vector-valued functions.
algebraic/
motion, (BC) and
computational
planar motion.
processes
AP CALCULUS BC SAMPLE EXAM QUESTIONS

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11. A particle moving in the xy-plane has velocity vector given by for time
What is the magnitude of the displacement of the particle between time and

(A)
(B)
(C)
(D)

Mathematical
Practice for
Learning Objective Essential Knowledge AP Calculus

LO 3.4C: Apply definite EK 3.4C2: (BC) The definite integral can be MPAC 1: Reasoning
integrals to problems used to determine displacement, distance, and with definitions
involving motion. position of a particle moving along a curve and theorems
given by parametric or vector-valued functions.
MPAC 3: Implementing
algebraic/
computational
processes

AP CALCULUS BC SAMPLE EXAM QUESTIONS

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12. Consider the series where for all n. Which of the following conditions

guarantees that the series converges?

(A)

(B)

(C) for all n

(D) converges, where for all n

Mathematical
Practice for
Learning Objective Essential Knowledge AP Calculus

LO 4.1A: Determine EK 4.1A6: In addition to examining the limit MPAC 1: Reasoning

whether a series of the sequence of partial sums of the series, with definitions
converges or diverges. methods for determining whether a series and theorems
of numbers converges or diverges are the
MPAC 5: Building
nth term test, the comparison test, the limit
notational fluency
comparison test, the integral test, the ratio
test, and the alternating series test.

LO 4.1A: Determine EK 4.1A5: If a series converges

whether a series absolutely, then it converges.
converges or diverges.
AP CALCULUS BC SAMPLE EXAM QUESTIONS

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Free Response: Section II, Part A

A graphing calculator is required for problems on this part of the exam.

1. Let r be the function given by for The graph of r in polar

coordinates consists of two loops, as shown in the figure above. Point P is on the graph of r and
the y-axis.
(A) Find the rate of change of the x-coordinate with respect to at the point P.
(B) Find the area of the region between the inner and outer loops of the graph.

(C) The function r satisfies . For find the value of that

gives the point on the graph that is farthest from the origin. Justify your answer.

AP CALCULUS BC SAMPLE EXAM QUESTIONS

Mathematical
Practice for
Learning Objective Essential Knowledge AP Calculus

LO 2.2A: Use EK 2.2A4: (BC) For a curve given by a polar MPAC 1: Reasoning
derivatives to analyze equation derivatives of r, x, and with definitions
properties of a function. y with respect to and first and second and theorems
derivatives of y with respect to x can MPAC 2: Connecting
provide information about the curve. concepts
MPAC 3: Implementing
LO 2.3C: Solve EK 2.3C3: The derivative can be used algebraic/
problems involving to solve optimization problems, that is, computational
related rates, finding a maximum or minimum value processes
optimization, rectilinear of a function over a given interval.
motion, (BC) and MPAC 4:
planar motion. Connecting multiple
representations
LO 3.4D: Apply EK 3.4D1: Areas of certain regions in the plane MPAC 5: Building
definite integrals to can be calculated with definite integrals. notational fluency
problems involving (BC) Areas bounded by polar curves can
area, volume, (BC) and be calculated with definite integrals. MPAC 6:
length of a curve. Communicating

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Free Response: Section II, Part B

No calculator is allowed for problems on this part of the exam.

2. Consider the function f given by for all

(A) Find
(B) Find the maximum value of f for Justify your answer.

(C) Evaluate or show that the integral diverges.

Mathematical
Practice for
Learning Objective Essential Knowledge AP Calculus

LO 1.1D: Deduce and EK 1.1D2: Relative magnitudes of MPAC 1: Reasoning

interpret behavior of functions and their rates of change with definitions
functions using limits. can be compared using limits. and theorems
MPAC 2: Connecting
LO 2.2A: Use EK 2.2A1: First and second derivatives of a concepts
derivatives to analyze function can provide information about the
properties of a function. function and its graph including intervals of MPAC 3: Implementing
increase or decrease, local (relative) and global algebraic/
(absolute) extrema, intervals of upward or computational
downward concavity, and points of inflection. processes
MPAC 4: Building
LO 3.2D: (BC) Evaluate EK 3.2D2: (BC) Improper integrals can be notational fluency
an improper integral or determined using limits of definite integrals.
MPAC 6:
show that an improper
Communicating
integral diverges.

LO 3.3B(b): Evaluate EK 3.3B5: Techniques for finding antiderivatives

definite integrals. include algebraic manipulation such as long
division and completing the square, substitution
of variables, (BC) integration by parts, and
AP CALCULUS BC SAMPLE EXAM QUESTIONS

nonrepeating linear partial fractions.

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3. The function f is defined by the power series

for all real numbers x for which the series converges.

(A) Determine the interval of convergence of the power series for f. Show the work that leads
to your answer.
(B) Find the value of
(C) Use the first three nonzero terms of the power series for f to approximate Use the
alternating series error bound to show that this approximation differs from by less
than

Mathematical
Practice for
Learning Objective Essential Knowledge AP Calculus

LO 4.1A: Determine EK 4.1A3: Common series of numbers MPAC 1: Reasoning

whether a series include geometric series, the with definitions
converges or diverges. harmonic series, and p-series. and theorems
MPAC 2: Connecting
LO 4.1A: Determine EK 4.1A4: A series may be absolutely convergent, concepts
whether a series conditionally convergent, or divergent.
converges or diverges. MPAC 3: Implementing
algebraic/computational
LO 4.1B: Determine EK 4.1B2: If an alternating series converges by the processes
or estimate the alternating series test, then the alternating series MPAC 5: Building
sum of a series. error bound can be used to estimate how close a notational fluency
partial sum is to the value of the infinite series.
MPAC 6: Communicating

AP CALCULUS BC SAMPLE EXAM QUESTIONS

LO 4.2A: Construct and EK 4.2A1: The coefficient of the nth-degree
use Taylor polynomials. term in a Taylor polynomial centered at

for the function f is .

LO 4.2C: Determine EK 4.2C1: If a power series converges,

the radius and interval it either converges at a single point or
of convergence of has an interval of convergence.
a power series.

LO 4.2C: Determine EK 4.2C2: The ratio test can be used to determine

the radius and interval the radius of convergence of a power series.
of convergence of
a power series.

LO 4.2C: Determine EK 4.2C3: If a power series has a positive

the radius and interval radius of convergence, then the power
of convergence of series is the Taylor series of the function to
a power series. which it converges over the open interval.

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 Sample Exam Questions

Answers and Rubrics (BC)

Answers to Multiple-Choice Questions

1 A

2 B

3 A

4 C

5 A

6 D

7 D

8 C

9 C

10 B

11 B

12 B
AP CALCULUS BC SAMPLE EXAM QUESTIONS

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Rubrics for Free-Response Questions

Question 1

Solutions Point Allocation

(A) 

At point P,

(B)

(C) 

0 0
2.028758 5.459117
4.913180
0

The value gives the point on the graph that is farthest from
the origin.

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Question 2

Solutions Point Allocation

(A)

(B)

exists for all

Because for and

for the maximum value of for is

(C)

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Question 3

Solutions Point Allocation

(A)

The series converges when

When the series is

1 1 1 …
1    .
2 3 4
This is the alternating harmonic series, which
converges conditionally.

1 1 1 …
When the series is 1     .
2 3 4
This is the harmonic series, which diverges.

Therefore, the interval of convergence is

 e power series given is the Taylor series for f

(B) Th
about Thus,

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 Sample Exam Questions

(C)

The power series for f evaluated at is

an alternating series whose terms decrease in
absolute value to 0. The alternating series error
bound is the absolute value of the fourth term of
the series.

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