Ultimate Guide to the AP Calculus BC Exam
In 2019, approximately 440,000 of the 2.8 million students taking AP exams took an AP Calculus exam. Most of these students took the AP Calculus AB exam, which covers pre-calculus along with differential and integral calculus, but many others took the AP Calculus BC exam, which goes into deeper detail, covering the same content plus polar coordinates, sequences and series, vectors, and an introduction to differential equations.
If you’re planning to take the AP Calculus BC exam, whether you have taken the class or self-studied, read on for a breakdown of the test and CollegeVine’s advice for how to best prepare for it.
When is the AP Calculus BC Exam?
The 2020 AP Exams have been changed to online exams due to coronavirus. Learn about these changes in our post How is Coronavirus Impacting AP Exams?
About the AP Calculus BC Exam
Much like the Calculus AB course, the AP Calculus BC course focuses on the unifying themes of calculus. These include derivatives, integrals, limits, approximation, applications and modeling, and sequences and series, as well as basic experience with appropriate methods and applications. Competence in computation is important, but the primary emphasis of the course is on a multidimensional approach to calculus. In other words, concepts, results, and problems should be expressed in numerous ways including graphically, numerically, analytically, and verbally. In this course, you will also focus on the importance of the connections and relationships between various representations of functions. The course relies heavily on technology to reinforce relationships among functions, confirm written work, implement experimentation, and assist in interpreting results.
The AP Calculus BC course is broken down into two separate components that students must master in order to demonstrate an understanding of calculus and pass the AP Calculus BC exam. The two key components are mathematical practices and big ideas.
Mathematical Practices: Mathematical practices are the four key skills a student must possess to complete the tasks asked on the AP Calculus exam. The practices of the AP Calculus BC course are:
Mathematical Practice | Description |
Percentage of Exam Score (Multiple-Choice Section) |
Percentage of Exam Score (Free-Response Section) |
Implementing Mathematical Processes | Determine expressions and values using mathematical procedures and rules | 53%-66% | 37%-59% |
Connecting Representations | Translate mathematical information from a single representation or across multiple representations | 18%-28% | 9%-16% |
Justification | Justify reasoning and solutions | 11%-18% | 37%-59% |
Communication and Notation | Use correct notation, language, and mathematical conventions to communicate results or solutions | N/A | 9%-20% |
Big Ideas: The other key component of the AP Calculus BC curriculum are big ideas. Big ideas are the foundational concepts and themes that run throughout the course vital to developing a deeper understanding of calculus. The three big ideas of the AP Calculus course are:
Big Idea | Description |
Change | The integration and differentiation as expressed in the Fundamental Theorem of Calculus—a central idea in AP Calculus |
Limits | Beginning with a discrete model and then considering the consequences of a limiting case allows us to model real-world behavior and to discover and understand important ideas, definitions, formulas, and theorems in calculus |
Analysis of Functions | Calculus allows us to analyze the behaviors of functions by relating limits to differentiation, integration, and infinite series and relating each of these concepts to the others |
During the AP Calculus BC exam, you should plan to use a scientific graphing calculator. Such use is allowed on one part of the multiple-choice section and on one part of the free-response section. Your calculator should be able to plot the graph of a function within an arbitrary viewing window, find the zeros of functions, numerically calculate the derivative of a function, and numerically calculate the value of a definite integral. More information and a list of acceptable calculator models can be found in the official Calculator Policy.
AP Calculus BC Course Content
Being an advanced course, the AP Calculus BC curriculum requires a good deal of foundational knowledge, whether you enroll in the course or self-study. You will need four years of high school level mathematics including algebra, geometry, trigonometry, analytic geometry, and elementary functions before you undertake AP Calculus BC. You should be familiar with linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions.
It’s also valuable to have some knowledge of the properties of functions, the algebra of functions, the graphs of functions, and the language of functions. In addition, because this course goes further into its subject depth than the AP Calculus AB course, you should also have basic familiarity with sequences and series, as well as some exposure to polar equations. You will need access to a graphing calculator for the duration of the course. You should also keep in mind that you cannot take both AP Calculus AB and AP Calculus BC within the same year.
The AP Calculus BC course is broken down into 10 individual units. Below is a suggested structuring of the 10 units and the percentage each unit is given on the AP Calculus BC exam.
Unit | Percentage of Exam Score (Multiple-Choice Section) |
Limits and Continuity | 4%–7% |
Differentiation: Definition and Fundamental Properties | 4%–7% |
Differentiation: Composite, Implicit, and Inverse Functions | 4%–7% |
Contextual Applications of Differentiation | 6%-9% |
Analytical Applications of Differentiation | 8%–11% |
Integration and Accumulation of Change | 17%–20% |
Differential Equations | 6%–9% |
Applications of Integration | 6%–9% |
Parametric Equations, Polar Coordinates, and Vector-Valued Functions | 11%–12% |
Infinite Sequences and Series | 17%–18% |
AP Calculus BC Exam Content
The AP Calculus BC exam is one of the longest AP exams, clocking in at three hours and 15 minutes. The exam content is divided into two primary sections: one section of multiple-choice questions, and the other made up of free-response questions.
Section 1: Multiple Choice
1 hour 45 minutes | 45 questions | 50% of score
The first section takes one hour and 45 minutes, contains 45 multiple-choice questions, and accounts for 50% of your total score. This section has two separate parts. Part A consists of 30 questions, lasts for 60 minutes, and makes up 33.3% of your exam score. In this section, you will not be allowed to use your calculator. You may use your calculator on Part B, which contains 15 questions, lasts for 45 minutes, and makes up 16.7% of your exam score.
Here’s an example of a multiple-choice question that you’re not allowed to use a calculator to answer:
Source: The College Board. Answer: C
Here’s an example of a multiple-choice question that you are allowed to use a calculator to answer:
Source: The College Board. Answer: A
Section 2: Free Response
1 hour 30 minutes | 6 questions | 25% of score
The free-response section is next, and it lasts for one hour and 30 minutes, accounting for the remaining 50% of your score. Like the multiple-choice section, the free-response section is split into two parts. Part A contains two problems, which you will have 30 minutes to complete with the use of your calculator and is worth 16.7% of your exam score. Part B has four problems, which you will have 60 minutes to complete without the use of your calculator and is worth 33.3% of your exam score.
Here’s an example of a free-response question that you are allowed to use a calculator to answer:
Source: The College Board.
Here’s an example of a free-response question that you’re not allowed to use a calculator to answer:
Source: The College Board.
AP Calculus BC Score Distribution, Average Score, and Passing Rate
Exam | 5 | 4 | 3 | 2 | 1 |
AP Calculus BC | 43% | 18.5% | 19.5% | 13.9% | 5.2% |
The AP Calculus BC exam seems to be one on which students generally perform quite well. This could perhaps be due to the particularly stringent guidelines for prerequisite knowledge that ensure that most students taking the course are already well-suited for its rigor. In 2019, 43% of students taking the exam received the top score of 5, making it the highest scoring math or science AP exam. 81% of students who took the exam score a 3 or higher while just 5.2% received the bottom score of 1.
Keep in mind, credit and advanced standing based on AP scores varies widely from school to school. Though a score of 3 is typically considered passing, it is not always enough to receive credit. Regulations regarding which APs qualify for course credits or advanced placement at specific colleges and universities can be found on the College Board’s website.
Best Ways to Study for the AP Calculus BC Exam
Step 1: Assess Your Skills
You will need to begin your studying with some measure of your current knowledge. Although the AP Calculus BC course was redesigned in the 2016-2017 school year, its content has remained almost entirely the same, which is a good thing when it comes to sourcing practice exams and study materials. Previous administrations of the exam are still valuable for assessment purposes. You can find the 2012 exam and the 2008 exam available from the College Board. There are also diagnostic and practice tests provided in many of the available commercial study guides.
Step 2: Study the Material
The AP Calculus BC course focuses on differential and integral calculus along with sequences and series. It relies heavily on your strong background in algebra, geometry, trigonometry, and elementary functions. You will need to demonstrate your knowledge of the concepts and skills of limits, derivatives, definite integrals, and the Fundamental Theorem of Calculus to perform well on the exam, just as you needed to learn these for the AP Calculus AB exam.
Your learning will also need to extend to a series of numbers, power series, and various methods to determine convergence or divergence of a series. For the AP Calculus BC exam, you will need to be familiar with Maclaurin series for common functions and general Taylor series representations. Your knowledge should include various approaches to calculus concepts and problems when they are represented graphically, numerically, analytically, and verbally. You should also be able to use the technology available to you to help solve problems, experiment, interpret results, and support conclusions.
For a more specific idea of where to focus your studying, you may consider using a commercial study guide. The Princeton Review’s Cracking the AP Calculus BC Exam, 2020 Edition: Practice Tests & Proven Techniques to Help You Score a 5 gets good reviews and includes a complete, updated guide to the exam, chapter reviews, practice problems, and three practice tests. Another solid choice is Barron’s AP Calculus, 14th Edition which covers both AB and BC calculus content in a concise and accurate way.
There are also a number of free study resources available online. Because calculus remains a popular AP choice, many teachers and students have experience with the exam already. You can capitalize on this by using the materials posted online by AP teachers and students. One study guide/calculus resource can be found on Lamar University’s website, but this is just one of the many that are out there.
Step 3: Practice Multiple-Choice Questions
After you gain a solid understanding of the basic theory required on the exam, test your knowledge by practicing multiple-choice questions. Most study guides provide practice multiple-choice questions, or you can find more through online searches. You could also try taking the multiple-choice section of another practice exam. The College Board provides a set of sample questions with scoring explanations and there are several study books available that consist entirely of multiple-choice questions with answers. One highly rated book is Multiple Choice Questions to Prepare for the AP Calculus BC Exam: 2019 Calculus BC Exam Preparation workbook 2nd Edition. Another great option is the collection of multiple-choice questions available online through Khan Academy.
The College Board course description includes many practice multiple-choice questions along with explanations of their answers. As you go through these, try to keep track of which areas are still tripping you up, and go back over this theory again. Focus on understanding what each question is asking and keep a running list of any concepts that are still unfamiliar.
Step 4: Practice Free-Response Questions
The free-response portion of the AP Calculus BC exam will test your ability to solve problems using an extended chain of reasoning. In this section, you will be expected to demonstrate a sound knowledge of mathematical reasoning and thinking. You will also articulate the reasoning and methods that support your answer. Keep in mind that an answer without supporting work will receive no credit. Some questions will ask you to justify an answer or determine whether a theorem can be applied. Each part of the free-response section is timed, and you may use a graphing calculator only for Part A. During Part B, you may not use a calculator, and though you are allowed to return to working on Part A questions during Part B, you must do so without a calculator.
When completing the free-response section, be very careful to show all of your work, even when you’re using a calculator. There should be a clear-to-follow record of every step you took to arrive at a solution. Remember that the exam reader will be evaluating your mastery through not only your final answer, but also your methods that led to it. If you use your calculator to solve an equation, compute a numerical derivative, or find a definite integral, then be sure to write the equation, derivative, or integral first. Even a correct answer will not receive full credit if your work is not clearly notated.
The College Board has the free-response questions from the AP Calculus BC exam dating back to 1998 posted on their website. In addition to the questions is scoring commentary, which is extremely useful for understanding the types of answers the College Board is looking for.
Step 5: Take Another Practice Test
Once you’ve completed the steps above, take another formative assessment. Varsity Tutors offers a handful of free diagnostic tests and practice exams. Another resource is Study.com, which offers a free 50-question practice test.
You should see a steady progression of knowledge, and it’s likely that you will see patterns identifying which areas have improved the most and which areas still need improvement. If you have time, repeat each of the steps above to incrementally increase your score.
Step 6: Exam Day Specifics
If you’re taking the AP course associated with this exam, your teacher will walk you through how to register. If you’re self-studying, check out our blog post How to Self-Register for AP Exams.
For information about what to bring to the exam, see our post What Should I Bring to My AP Exam (And What Should I Definitely Leave at Home)?
For more info about the APs, check out these other helpful CollegeVine posts:
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