2025 AP Calculus BC Free-Response Questions

2025 AP Calculus BC Free-Response Questions

AP Calculus BC Free-Response Questions for 2025 provide students with essential practice for the exam. This resource includes two sections: Part A, featuring questions on topics such as rates of change and polar equations, and Part B, which covers advanced calculus concepts without the use of calculators. Designed for AP Calculus BC students, this document helps reinforce understanding of calculus principles and prepares learners for the May exam. Each question is structured to challenge students' problem-solving skills and deepen their comprehension of calculus concepts.

Key Points

  • Includes free-response questions for AP Calculus BC 2025 exam preparation.
  • Covers topics such as rates of change, polar equations, and limits.
  • Structured in two parts: Part A with calculator use and Part B without.
  • Designed to enhance problem-solving skills and understanding of calculus concepts.
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2025
AP
®
Calculus BC
Free-Response Questions
© 2025 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.org.
AP Central is the official online home for the AP Program: apcentral.collegeboard.org.
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© 2025 College Board.
AP CALCULUS BC 2025
n
FREE-RESPONSE QUESTIONS
CALCULUS BC
SECTION II PART A
TIME – 30 MINUTES
Directions:
Section II, Part A has 2 free-response questions and lasts 30 minutes.
A graphing calculator is required for the questions on this part of the exam. You may use
a handheld graphing calculator or the calculator available in this application.Make sure your
calculator is in radian mode.
You may use the available paper for scratch work, but you must write your answers in the free-
response booklet. In the free-response booklet, write your solution to each part of each question in
the space provided for that part. For questions that have sub-parts, be sure to label those clearly in
your solution. Use a pencil or a pen with black or dark blue ink.
You are permitted to use your calculator to solve an equation, find the derivative of a function at
a point, or calculate the value of a definite integral. However, you must clearly indicate the setup
of your question, namely the equation, function, or integral you are using. If you use other built-in
features or programs, you must show the mathematical steps necessary to produce your results.
Show all of your work, even though a question may not explicitly remind you to do so. Clearly label
any functions, graphs, tables, or other objects that you use. Justifications require that you give
mathematical reasons and that you verify the needed conditions under which relevant theorems,
properties, definitions, or tests are applied. Your work will be scored on the correctness and
completeness of your methods as well as your answers. Answers without supporting work will
usually not receive credit.
Your work must be expressed in standard mathematical notation rather than calculator syntax. For
example,
xdx
2
1
5
y
may not be written as
,,,
fnIn
tX X15
2
_i
.
Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If you use
decimal approximations in calculations, your work will be scored on accuracy. Unless otherwise
specified, your final answers should be accurate to three places after the decimal point.
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x
for which
fx
_i
is a real number.
You can go back and forth between questions in this part until time expires. The clock will turn red
when 5 minutes remain—the proctor will not give you any time updates or warnings.
Note: This exam was originally administered digitally. It is presented here in a format optimized for
teacher and student use in the classroom.
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© 2025 College Board.
AP CALCULUS BC 2025
n
FREE-RESPONSE QUESTIONS
1. An invasive species of plant appears in a fruit grove at time
t 0
=
and begins to spread. The
function C defined by
..
arctan
Ct t
76 02
=
__ii
models the number of acres in the fruit grove
affected by the species t weeks after the species appears. It can be shown that
Ct
t
25
38
2
=
+
l
_i
.
(Note: Your calculator should be in radian mode.)
A. Find the average number of acres affected by the invasive species from time
t 0
=
to time
t 4
=
weeks. Show the setup for your calculations.
B. Find the time t when the instantaneous rate of change of C equals the average rate of change
of C over the time interval
t04##
. Show the setup for your calculations.
C. Assume that the invasive species continues to spread according to the given model for all
times
t 0>
. Write a limit expression that describes the end behavior of the rate of change in
the number of acres affected by the species. Evaluate this limit expression.
D. At time
t 4
=
weeks after the invasive species appears in the fruit grove, measures are taken
to counter the spread of the species. The function A, defined by
. lnAt Ct xdx01·
=-
,
models the number of acres affected by the species over the time interval
t43
6
##
. At what
time t, for
t43
6
##
, does A attain its maximum value? Justify your answer.
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End of Document
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Faqs of 2025 AP Calculus BC Free-Response Questions
What types of questions are included in the 2025 AP Calculus BC Free-Response section?
The 2025 AP Calculus BC Free-Response section includes a variety of questions that assess students' understanding of calculus concepts. Topics covered range from rates of change and average value to polar equations and limits. Each question requires students to show their work and justify their answers, ensuring a comprehensive understanding of the material. This format prepares students for the types of questions they will encounter on the actual exam.
How does the 2025 AP Calculus BC exam structure differ between Part A and Part B?
The 2025 AP Calculus BC exam is divided into two parts: Part A and Part B. Part A consists of two free-response questions that allow the use of a graphing calculator, focusing on practical applications of calculus concepts. In contrast, Part B includes four free-response questions where calculators are not permitted, emphasizing analytical problem-solving skills. This structure tests students' abilities to apply calculus principles in both calculator-assisted and non-calculator scenarios.
What is the significance of the average rate of change in AP Calculus BC?
The average rate of change is a fundamental concept in calculus that measures how a function changes over a specified interval. In the context of AP Calculus BC, understanding this concept is crucial for solving problems related to motion, growth, and decay. Students must be able to calculate the average rate of change using the formula (f(b) - f(a)) / (b - a) and interpret its meaning in real-world scenarios. Mastery of this concept is essential for success on the AP exam.
What are polar equations and why are they important in calculus?
Polar equations express relationships between a radius and an angle, providing a different perspective on graphing functions compared to Cartesian coordinates. In calculus, polar equations are important for understanding curves that cannot be easily represented in rectangular form. They allow students to explore topics such as area, arc length, and the behavior of functions in a circular context. Mastery of polar coordinates is essential for solving specific problems on the AP Calculus BC exam.
How can students effectively prepare for the AP Calculus BC exam using free-response questions?
Students can effectively prepare for the AP Calculus BC exam by practicing with free-response questions from previous exams and study guides. These questions help reinforce key concepts and improve problem-solving skills. It's important for students to work through each question methodically, showing all steps and justifications for their answers. Additionally, reviewing scoring guidelines and sample responses can provide insights into what graders are looking for, enhancing students' performance on the actual exam.