Ap Calculus Essay

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 courses are among the most popular choices.

In 2016, over 300,000 of the 2.6 million students taking AP exams took the Calculus AB exam. This places it among the top 4 most popular AP exams. If you are interested in taking the AP Calculus AB 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

The AP Calculus AB course focuses on the unifying themes of calculus, including derivatives, integrals, limits, approximation, and applications and modeling, all the while providing general experience with appropriate methods and applications. Though computational competence is important, the primary emphasis of the course is on a multidimensional approach to calculus, with concepts, results, and problems expressed in numerous ways including graphically, numerically, analytically, and verbally. The course also emphasizes the importance of the connections and relationships between these 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 AB course requires a good deal of foundational knowledge in order to be successful, whether you enroll in the course or self-study. Before beginning your Calculus AB studies, you will need four years of secondary mathematics covering algebra, geometry, trigonometry, analytic geometry, and elementary functions. Your knowledge should cover linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions. You will need to be familiar with the properties of functions, the algebra of functions, the graphs of functions, and the language of functions. You will need access to a graphing calculator for the duration of the course and you should know that you cannot take both AP Calculus AB and AP Calculus BC within the same year.

The AP Calculus AB course is newly redesigned for the 2016-2017 school year. Its content remains the same, with a single addition (L’Hôpital’s Rule), but the format of the course outline has changed significantly in order 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.

About the Exam

While taking the AP Calculus AB exam, you may use a scientific calculator 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 AB exam is one of the longest AP exams, clocking in at three hours and 15 minutes. It is comprised of two sections. The first section contains 45 multiple-choice questions, spans one hour and 45 minutes, and accounts for 50% of your total score. This section is divided into two parts. Part A contains 30 questions and lasts for 60 minutes. You will not be allowed to use your calculator on this part. Part B contains 15 questions, lasts for 45 minutes, and allows the use of a calculator.

The second section is the free-response section, which lasts for one hour and 30 minutes and accounts for the remaining 50% of your score. This section is divided into six questions and, like the multiple-choice 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 AB exam seems to be one that students either grasp well or not at all. In 2016, over 55% of students taking the exam scored either at the highest end with a top score of 5 or at the lowest end with a bottom score of 1. Specifically, nearly a quarter of all test-takers scored a 5 while a third of all test-takers scored a 1. Overall, 59.5% of students who took the exam received a 3 or higher (typically considered passing).

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

Take a practice test to assess your initial knowledge of the material. This is an integral step to shaping your personalized study path. 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 AB course was recently redesigned, its content has remained almost entirely the same. For this reason, previous exams are still valuable for assessment purposes. You can find the 2012 exam, 1998 exam, and 1988 exams all available from the College Board. Alternatively, you could also use a practice or diagnostic test provided in one of the many up-to-date commercial study guides.

Once you have taken some kind of formative assessment, score it to identify the areas you already understand and those in need of improvement. From an accurate formative assessment, you will get a better idea of where to focus your studying efforts.

Step 2: Study the material

The AP Calculus AB course focuses on differential and integral calculus while relying heavily on a strong foundation in algebra, geometry, trigonometry, and elementary functions. To be successful on the exam you will need to learn the concepts and skills of limits, derivatives, definite integrals, and the Fundamental Theorem of Calculus. You should have knowledge about how to approach calculus concepts and problems when they are represented graphically, numerically, analytically, and verbally, and you should be able to use technology to help solve problems, experiment, interpret results, and support conclusions.

In order to understand the material in the AP Calculus AB course, you’ll need to understand the structure of the course outline, which is available in the course description. The outline organizes key concepts into three “big ideas” that you will need to understand completely. This comprehension should include each of 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 application. The three big ideas of the AP Calculus AB course are:

  • Limits
  • Derivatives
  • Integrals and the Fundamental Theorem of Calculus

To apply your knowledge, you will also need to master the following six mathematical practices for AP Calculus AB:

  • Reasoning with definitions and theorems
  • Connecting concepts
  • Implementing algebraic/computational processes
  • Connecting multiple representations
  • Building notational fluency
  • Communicating

To get a better idea of the course content and structure, you should review the sample course syllabi available here and here. You might also choose to review the College Board’s video overview of curriculum, assessment, and instructional approaches.

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 AB Exam, 2017 Edition: Proven Techniques to Help You Score a 5 provides a comprehensive and updated guide to the exam, with reviews of each unit, practice problems, and three complete 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. Many AP teachers have posted complete study guides, review sheets, and test questions. One 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

Once you have your theory down, give a try by practicing multiple-choice questions. You can find these in most study guides or through online searches. You could also try taking the multiple-choice section of another practice exam. The College Board even provides a set of sample questions with scoring explanations. Additionally, there are several study books available that consist of just multiple-choice questions with answers. The most highly rated of those is available here.

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 AB exam will test your ability to solve problems using an extended chain of reasoning. It will challenge you to demonstrate your knowledge of correct mathematical reasoning and thinking while articulating the reasoning and methods that support your answer. In most cases, 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. This means that every step you took to arrive at a solution should be clearly notated for the exam reader, who will be assessing your knowledge 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 readily apparent.

To ensure success on this section, make sure to the review the scoring rubric located on page 72 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

As you did at the very beginning of your studying, take a practice test to evaluate your progress. 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 AB 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

Kate Sundquist

Senior Blogger at CollegeVine

Kate Koch-Sundquist is a graduate of Pomona College where she studied sociology, psychology, and writing before going on to receive an M.Ed. from Lesley University. After a few forays into living abroad and afloat (sometimes at the same time), she now makes her home north of Boston where she works as a content writer and, with her husband, raises two young sons who both inspire her and challenge her on a daily basis.

Latest posts by Kate Sundquist (see all)

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One of the best ways to prepare for the AP Calculus AB exam, as well as stay on top of things in class throughout the year, is to take regular practice tests. Taking practice tests lets you estimate how well you’ll do on the AP exam, shows you the areas you need to focus your studies on, and helps you become more familiar and comfortable with the format of the AP exam.

There are a ton of AB Calc practice tests available, however; not all of them are created equally, and taking a poorly written practice test can give you a false idea of what the real AP exam will be like and cause you to study the wrong things.

You can avoid this by reading this guide to AP Calculus AB practice tests. I’ll go through every AP Calculus AB practice exam that’s available, tell you which are highest-quality, and explain how you should use practice tests when preparing for the AP exam as well as throughout the year.


Official AP Calculus AB Practice Tests

Official practice exams (those developed by the College Board) are always the best to use because you can be sure they’ll be an accurate representation of the real AP exam. There are three types of official practice resources, and each is explained below.


Complete Practice Tests

The College Board has released two complete exams from prior administrations of the AP Calculus AB exam. The tests are from 1988 and 1998. The 1988 test has an answer key included; however, for some reason, the 1998 exam does not. The College Board provided answers for the free-response questions in a separate document, but there is no official answer key available for the 1998 exam's multiple-choice section. The answer key linked below is unofficial, but no one has publicly disagreed with any of the answers, so it’s highly likely that it’s correct.

Because these exams are from a while back, they both have some format differences compared to the current AP Calculus AB exam. The AP Calculus AB exam is 3 hours and 15 minutes long and has two sections. Both of these sections are divided into two parts. For reference, here’s the current format of the exam:

Multiple-Choice Section

  • 45 questions total
  • 1 hour 45 minutes total
  • Worth 50% of your total score
  • Part A:
    • 30 questions
    • 55 minutes long
    • No calculator allowed
  • Part B:
    • 15 questions
    • 50 minutes long
    • Calculator permitted

Free-Response Section

  • Six questions total
  • 1 hour 30 minutes total
  • Worth 50% of your total score
  • Part A:
    • Two questions
    • 30 minutes long
    • Calculator permitted
  • Part B:
    • Four questions
    • 60 minutes long
    • No calculator allowed


You can only use a calculator for certain sections of the AP exam.


Both released exams have the same total number of multiple-choice and free-response questions as the current exam. However, the 1998 test does not have separate parts for the free-response section, and students were allowed to use a calculator to answer all six questions.

Neither the multiple-choice nor the free-response sections of the 1988 exam were separated into different parts, and students were allowed to use their calculator for the entire exam. The multiple-choice section was also only 90 minutes long, instead of 105 minutes.

When you take these exams for practice, it’s not worth the time and effort needed to try and figure out which questions you wouldn’t be allowed to solve with a calculator today. Instead, take the tests with the calculator and timing rules that were in place when the tests were administered.

These variations between current and past exams do mean that these two complete released exams don’t give quite as accurate a representation of the current AP exam as the complete released exams for other AP subjects do.

However, they are still very useful because they cover the same content and are worded the same way as the current exam. Towards the end of this guide I’ll explain exactly how to use these resources and others.


AP Calculus AB Multiple-Choice Sample Questions

The College Board often reuses multiple-choice questions for multiple exams, so there are typically few official multiple-choice problems available for any AP exam, AP Calculus AB included.

Besides the complete practice tests discussed above, there are no full official multiple-choice sections available, but you can check out these official sample questions for Calculus AB. (The questions start on page 5, and there are Calculus BC questions listed after the AB questions; be sure you’re not accidentally looking at those.) This document contains 16 multiple-choice problems, along with answers and the major skills each question tests. There are also two free-response questions.


AP Calculus AB Free-Response Sample Questions

Fortunately, there are more official free-response questions available and, since they are recent, they provide you with a very accurate idea of what to expect on the real exam.

The College Board has released free-response questions from 2002-2016, along with scoring guidelines for each set of questions. These are a great resource, and you should definitely make use of them during your review.


Khan Academy Resources

Khan Academy has recently partnered with the College Board to provide study resources for the PSAT, SAT, and some AP exams. This includes study resources for AB Calc.

On Khan Academy’s website, there are explanation videos for several dozen previously administered questions, both multiple choice and free response. These videos can be particularly helpful if you’ve gotten stuck on one of the official practice problems or if you just want to learn step-by-step how to solve a particular problem.



Unofficial AP Calculus AB Practice Tests and Quizzes

While not developed by the College Board, unofficial practice resources can still be very useful for your studying, particularly because there are so many resources available. For each resource listed below, I explain what is offered as well as how you should make use of the resource. They are roughly listed from highest quality to lowest quality.



Barron’s has a complete and free practice test that you can take in timed or untimed mode. The multiple-choice section will be automatically scored, and there are answer explanations for the free-response section so you can self-score. This is a high-quality practice exam with questions that have a similar format and cover the same topics as the real AP exam. You’ll definitely want to use it in your review (more on how to do that in the next section).



Shmoop is the only resource listed in this guide that requires a fee to access any of its resources. Paying its fee of $24.68 a month gets you access to a diagnostic exam, as well as eight complete practice tests and additional practice questions. It also gets to access to Shmoop’s study materials for other AP exams, as well as the SAT and ACT.


Varsity Tutors

Varsity Tutors has a collection of three diagnostic tests and 139 short practice quizzes, organized by topic, such as the chain rule and finding the second derivative of a function. Difficulty levels are also given for each of the quizzes. The diagnostic tests are 40-45 questions long (all multiple-choice). They pretty closely represent what questions from the actual AP exam are like, and, as a bonus, the score results show you how well you did in each topic area so you can focus your future studying on the areas you need the most work in. However, these diagnostic tests don’t have calculator and no calculator sections.



This site organizes quizzes into the three Big Ideas of Calculus AB, as well as more specific tags you can select (you don’t need to worry about the Series quizzes, that’s just for BC Calc). After creating a free account you can access their hundreds of practice questions (some of the more challenging questions require a paid account). Questions are ranked as easy, moderate, or difficult, they are not timed, and you see the correct answer (plus a detailed explanation) after you answer each question.



McGraw-Hill offers a 25-question quiz (although questions are randomly selected from a larger bank, so if you take the quiz multiple times you may get more than 25 questions out of it). The quiz is automatically graded and has answer explanations (although some explanations can be pretty brief and may not help you figure out what you did wrong). Additionally, this test is only available in untimed mode, although you can always independently time yourself (give yourself about an hour to answer the questions). Overall, the questions themselves are high-quality, but this quiz has significant differences from the actual AP exam in terms of format, namely the fact that there are only 25 questions, the quiz is untimed, and there’s no indication of when you should and shouldn’t use a calculator.



GetaFive offers 182 Calculus AB practice questions, along with 54 review lessons. This is more of an online review course rather than just practice questions, so the questions themselves are spread across the lessons, but if you’re looking for more in-depth explanations of topics, you may find it useful to watch the videos and then answer the accompanying questions.




This site has a 50-question multiple-choice test. The questions typically easier and more basic than those you’d find on the actual AP exam, but if you’re just starting your review or want to brush up on the basics, this can be a good resource to use.


Free Test Online

This site has four short quizzes, each 5-8 questions long, along with answer explanations. Two quizzes are multiple-choice, and two are free-response. The free-response questions are much shorter than what you’d encounter on the real AP exam, but you can treat them like slightly more involved multiple-choice questions. The quizzes aren’t long enough for an in-depth practice session, but, unlike many of the other practice materials linked here, they also separate the quizzes on whether or not you’re allowed a calculator.


Analyze Math

This is a 20-question multiple-choice quiz. The questions are a bit overly simplistic, and it’s not automatically graded, but if you’re just looking for a quick study session, this fits the bill.



This is a short quiz, and, unfortunately, it’s not very high-quality. The questions are pretty basic and not nearly as complex or as in-depth as the ones you’ll find on the AP exam. Additionally, the format of this quiz is very poor, and it can be difficult to read. I wouldn’t recommend using this quiz unless you’re really desperate for review questions or you need a very basic quiz to get you started with your review.


How to Use These AP Calculus AB Practice Tests

Knowing how to use each of these practice exams and quizzes will make your studying much more effective, as well as prepare you for what the real AP Calculus AB exam will involve. Below is a guide for when and how to use the resources, organized by semester.


First Semester

During your first semester of Calc AB, you don’t know enough material for it to be useful to take a complete practice exam. Therefore, you should spend this semester answering quizzes and free-response questions on topics you’ve already covered. You’ll probably want to begin answering practice questions about halfway through the semester.


Free-Response Practice

For free-response questions, use the official released free-response questions in the Official Resources section. Look through them to find questions you can answer based on what you’ve already learned. It’s best if you can take a group of them (up to six) together at a time in order to get the most realistic preparation for the real AP exam.

It also helps to time yourself when answering these questions, particularly as it gets later in the year. On the real AP exam, you’ll have about 15 minutes to answer each free-response question, so try to answer practice questions under those same time restrictions.


Multiple-Choice Practice

For multiple-choice practice, take unofficial quizzes that let you choose the subject(s) you want to be tested on. This will allow you to review content you’ve already learned and not have to answer questions on material you haven’t covered yet. The best resources for this are Albert and Varsity Tutors because their quizzes are clearly broken up by specific subject.


Sometimes the numbers can get overwhelming. Don't forget to take a break every now and then.


Second Semester

Second semester is when you can begin to take complete practice exams and continuing to review content you’ve learned throughout the year.


Step 1: Take and Score Your First Complete Practice Exam

Early on in this semester, when you have covered a majority of the content you need to know for the AP exam, take your first complete practice exam. This test should be taken in one sitting and with official timing rules (see how the AP test is formatted above).

For this first practice test, I recommend using the Barron’s exam and saving the official practice exams for down the line. After you take this practice test, correct the exam and see what score you earned on the test.

This is a good time to set a score goal if you haven’t already. The minimum score you should be aiming for is a 3, since this is the lowest passing score. However, if you scored a 3 or higher on this first practice exam, it’s a good idea to set your goal score even higher, to a 4 or 5. Getting a higher score on the AP Calculus AB exam looks more impressive to colleges, and it can sometimes get you more college credit.


Step 2: Analyze Your Score Results

After you’ve figured out your score, look over each problem you answered incorrectly and try to figure out why you got the question wrong.

As you’re doing this, look for patterns in your results. Are you finding that you got a lot of questions on antiderivatives wrong? Did you do well on multiple choice but struggled with free response? Did you get slowed down by questions you couldn’t use a calculator to answer?

Figuring out which problems you got wrong and why is the best way to stop repeating your mistakes and begin to make significant improvements. Don’t be tempted to skip this step!


Step 3: Focus Your Studying on Weak Areas

You should now have a good idea of what subject areas or skills you need to work on in order to raise your score. If there are specific content areas you need to work on, review them by going over your notes, reading a review book, and answering multiple-choice and free-response questions that focus specifically on those topics.

If you’re struggling with your test-taking techniques, for example, running out of time on the exam or misreading questions, the best way to combat these issues is to answer a lot of practice questions under realistic testing conditions.

Take timed quizzes or time yourself for quizzes that aren’t automatically timed. (On the real exam, you’ll get about two minutes for multiple-choice questions you can’t use a calculator to solve, a little more than three minutes for multiple-choice questions where you can use a calculator, and 15 minutes per free-response question.) Taking multiple practice quizzes and tests will help you become more familiar with the pacing needed for the AP exam.


Step 4: Take and Score Another Practice Exam

After you’ve identified your weak areas and spent time to improve them, it’s time to see how all your hard work paid off.

Take and score another complete practice exam, timed and taken in one sitting. I’d recommend using either an official released practice exam or, if you want more recently-created questions, creating your own practice test by combining a set of unofficial multiple-choice questions (such as the Varsity Tutors or 4Tests exam) with a set of official free-response questions. If you choose the second option, you should have a total of 45 multiple-choice questions for the first part of the exam. As with the first test, this should be taken timed and in one sitting.

When you take this second practice exam, remember that it won’t be formatted exactly the same way as the real AP test, where the multiple-choice and free-response sections will both be broken into two parts, only one of which you can use a calculator on.


Step 5: Review Your Results to Determine Your Future Study Plan

Now you’re able to see how much you’ve improved, and in which areas, since you took your first complete practice exam. If you’ve made improvements and have reached or are close to your target score, you may only need to do some light studying from now until the AP exam.

However, if you haven’t made much improvement, or you’re still far from your score goal, you’ll need to analyze the way you’ve been reviewing and think of ways to improve. The most common reason for not improving is not actively studying, but only passively leafing through your notes or reviewing missed questions. Even though it may seem to take a while, in the long run, carefully analyzing why you made the mistakes you did and devising ways to improve is really the only significant way to raise your score.

As you’re studying, be sure to really understand exactly where you made a mistake for every practice question you answer incorrectly. Also, when you’re reviewing notes, pause every few minutes and mentally go over what you just learned to make sure you’re really retaining the information.

You can repeat these steps as many times as you need to in order to make improvements and reach your target score.


Studying With AP Calculus AB Practice Exams: Key Tips

It would be difficult to score well on the AP Calculus AB exam without completing any practice exams. Official resources are the best to use, but there are plenty of high-quality unofficial quizzes and tests out there as well.

During your first semester, you should focus on answering free-response and multiple-choice questions on topics you’ve already covered in class.

During your second semester, follow these steps:

    • Take and score your first complete practice exam
    • Analyze your score results
    • Focus your studying on weak areas
    • Take and score another complete practice exam
    • Review your results to determine your future study plan


What's Next?

Now that you have your practice tests, do you want to know more about the AP Calculus AB Exam? Our guide explains the complete format of the test, the question types you'll see, and how to best prepare for the exam. (coming soon)

How many AP classes should you take? Get your answer based on your interests and your college goals.

Wondering how challenging other AP classes will be? Learn what the easiest AP classes are and what the hardest AP classes are so that you're prepared!


Want to improve your SAT score by 160 points or your ACT score by 4 points? We've written a guide for each test about the top 5 strategies you must be using to have a shot at improving your score. Download it for free now:


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