- Integrals and the Fundamental Theorem of Calculus
- Reasoning with definitions and theorems
- Connecting concepts
- Implementing algebraic/computational processes
- Connecting multiple representations
- Building notational fluency
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Ultimate Guide to the AP Calculus BC Exam
As you probably already know by this point in your high school career, Advanced Placement (AP) courses and exams are administered each year under the oversight of the College Board. AP Calculus coursework remains a popular choice, though AP Calculus BC holds more prestige than its counterpart, AP Calculus AB.
In 2016, over 450,000 of the 2.6 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 are interested in taking the AP Calculus BC exam, whether you have taken the class or are planning to self-study, read on for a breakdown of the test and CollegeVine’s advice for how you can prepare for it.
About the Course
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, and also includes 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.
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. You should also 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 it 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.
Both AP Calculus courses were newly redesigned for the 2016-2017 school year. The course content remains the same, with a single addition (L’Hôpital’s Rule), but the format of the course outlines has changed to make essential mathematical practices explicit and to directly align course content with demonstrable learning objectives. For a complete description of the revisions, see AP Calculus Updates: Key Changes.
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.
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. The first section take 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 and lasts for 60 minutes. On this section, you will not be allowed to use your calculator. You may use your calculator on Part B, which contains 15 questions and lasts for 45 minutes.
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. Part B has four problems, which you will have 60 minutes to complete without the use of your calculator.
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 2016, 48.5% of students taking the exam received the top score of 5, making it the highest scoring math or science AP exam. 85.1% of students who took the exam score a 3 or higher while just 13.2% 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 here.
A full course description that can help to guide your studying and understanding of the knowledge required for the exam can be found in the College Board course description.
Read on for tips for preparing for the exam.
Step 1: Assess Your Skills
You will need to begin your studying with some measure of your current knowlege. To learn more about the importance of formative assessments and how you can use one to get your studying off on the right foot, check out the CollegeVine article What Is a Formative Assessment and Why Should I Use One to Study?
Although the AP Calculus BC course was recently redesigned, 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.
One great way to begin your studying is to review the structure of the course outline, which is available in the course description. This outline organizes core content into “big ideas” that you will need to grasp thoroughly. Your knowledge for each should also include the “enduring understandings” (falling under each big idea) and examples of essential knowledge to support them. The big ideas encompass core mathematical principles and theories, along with their applications. The four big ideas of the AP Calculus BC course are:
You will demonstrate your knowledge of the big ideas by applying the six mathematical practices for AP Calculus BC:
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, 2017 Edition: 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, 13th 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 can be found here and another here, but these are just a few of the many that are out there.
Another new, fun way to study is to use one of the recently developed apps for AP exams. These range in price from $0.99 to $4.99, but they provide a fun and easy way to quiz yourself. Make sure you read reviews before choosing one – their quality varies widely.
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. The most highly rated of those is available here. 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 vocabulary that is 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. On 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.
To ensure success on this section, make sure to the review the scoring rubric for the sample questions located on page 91 of the course description. You can also get a better understanding of the scoring by reading scoring commentary from the Development Team and authentic examples of student responses and their scoring explanations from previous exam administrations.
Step 5: Take another practice test
Once you’ve completed the steps above, take another formative assessment. 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
In 2017, the AP Calculus BC Exam will be administered on Tuesday, May 9 at 8 AM.
For complete registration instructions, check out CollegeVine’s How to Register for AP Exams (Even If You Didn’t Take the Class).
For information about what to bring to the exam, see CollegeVine’s What Should I Bring to My AP Exam (And What Should I Definitely Leave at Home)?
If you feel like you still need more help or you are not sure that you can do it on your own, look no further. For personalized AP tutoring, check out the CollegeVine Academic Tutoring Program, where students who are intimately familiar with the exam can help you ace it too, just like they did.
For more about APs, check out these CollegeVine posts