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AB Calculus Manual (Revised 12/2019)

In addition to the Essentials section of the manual that provide complete coverage of all topics for the AB exam, there is a Non-Essentials section as well that givers topics that are relevant to AB calculus but not on the actual AP ® exam. There is also a Review section that goes over the most common algebraic concepts that give students trouble.

This manual can easily replace an expensive textbook. Teachers teach right from it and students write in it.

The Solution Manual is exactly the same as the student manual except that the solutions with all important steps are shown. There is a one-to-one relationship between the pages of the student manual and the solution manual. So, for example, page 73 will have a series of problems and blank space for the students to write in the solutions. The solution manual's page 73 will have the same problems but with the solutions shown.

In addition, a set of answer pages (no shown work, just the answer) comes with the solution manual for the homework problems.  It is also available in download form as a stand-alone product.

New!  A new student section has been written called Calculus from 30,000 feet . Borrowed from Steven Strogatz's book Infinite Powers , this addendum is meant to be used after the AP exam and gives students a greater insight into how the Infinity Guideline breaks down comples probems into simper ones and then reassembles them. No homework or problems - just an attempy to see the Calculus forest from the trees.

School administrations are increasingly concerned with standards. This material in this manual is aligned to the AP ® standards as defined in the their 2019 document: AP ® Calculus AB and BC Course and Exam Description available on the website: www.apcentral.collegeboard.org. Although this is included with purchase of the AB Solutions in any format, you can download information about the standards at: AB Standards :

LET ME HELP YOU TEACH YOUR CLASS!

Because of the devastating impact to our way of life because of COVID-19, schools are in a particularly difficult position. It is still up in the air how safe they can be for the students but, no matter what,  classes have to be taught be adults. And those who teach courses like AP calculus are more likely to have years of experience, may be older and simply have to take fewer risks. For a number of schools and communities, the virtual classroom is the safest for everyone.

With that in mind, I have put together YouTube Videos that will teach your class the basic concepts of AP calculus. What is most exciting is that these will follow the classwork section of the AP Calculus manual. Students can download (for free) and print the AB manual student pages. They can download the video for each section and have thorough explanations of just about all the problems in the classwork section.  They can stop the video and write in solutions in their printed student manual. They can replay any section to better understand the steps. If they have questions, they can write them down and when you "meet" with them, you can help clear up their confusion.

They are narrated by me and the style is the same as if I were teaching a class. I show the thought process involved in solving calculus problems. The solutions are the same that appear in the solution manual, but these are explained audibly as well.  Unlike other math videos, you will not see me taking time to write solutions. They appear on the screen instantly and are legible and professional.   Videos will range from 10 minutes to 45 minutes depending on their complexity. Some of the more involved concepts are split into two videos, similar to what you would do when teaching - take several days to cover a topic.  When you "meet" with the students, you can then concentrate on the homework problems.

This is a huge but very exciting undertaking. These videos took time to construct, usually one every two days.  All the videos in the AB Essentials and Non-Essentials sections are complete, as well as the Precalculus Review . There is close to 24 hours of instruction.

I suggest that you first view the introduction video to understand how the YouTube videos coordinate with the manual. Then view several of the videos to get a feel for them. Feel free to direct students to this page so that they can download them directly. Have them feel free to make comments (keeping them clean!) for each video. And of course if there is a mistake or I didn't make something clear enough or if they have a question, they can mention that in the comment or, ask you or me directly.

Here is my YouTube channel:

https://www.youtube.com/channel/UC-y4_ldufRdVd5r_CuCzDCQ

AB Calc Cover Pages
Pages 1-26 - Revised 11/30/20 01. Introduction 02. Tangent Lines 03. Average and Instanteous Rate of Change 04. Limits graphical 05. Limits Algebraic ABStudentsp001-026.pdf  4.4MB
Pages 27-46 - Revised 12/01/20 06. Derivative definition 07. Derivatives w/ technology 08. Differentiation techniques 09. Chain Rule
Pages 47-69 - Revised 12/01/20 10. Derivatives of trig functions 11. Implicit differentiation 12. Derivatives transcendentals   13. Derivatives of inverses 14. Linear approximations
Pages 70-93 - Revised 12/02/20 15. Continuity/differentiability 16. Related rates
Pages 94-121 - Revised 12/02/20 17. Straight-line motion   18. 3 important theorems 19. Function analysis
Pages 122-151 - Revised 12/04/20 20. Curve sketching 21. Absolute extrema 22. Optimization 23. Economic optimization 24. L'Hospital's rule
Pages 152-174 - Revised 12/05/20 25. Indefinite integration 26. -substitution 27. Definite Integral and Area 28. The Accumulation Function MB
Pages 175-198 - Revised 12/06/20 29. Riemann Sums   30. Fundamental Theorem 31. Definite integrals -sub 32. Integrating transcendentals 33. Motion revisited
Pages 199-212 - Revised 12/06/20 34. Average value, 2nd FTC 35. Inverse Trig 36. Area between curves
Pages 213-240 - Revised 12/06/20 37. Volumes   38. Integral Applications 39. Slope fields 40. DEQ's 41. Exponential growth 42. Other growth models
Pages 241-267 - Revised 1/010/21 1a. Introduction 12a. Logarithmic differentiation 14a. Differentials 26a. L'Hospital advanced 24b. Newton's Method 29a. Exact area w/limits 37a. Volume shells
Pages 268-278 Complex fractions Inverses Exponentials/logs Graphical solutions Sigma notation
Limits and the Infinity Guideline: Addendum 1                 Derivatives as a Measure of Change: Addendum 2         Infinitesimals and Differentials:                          Integrals - an Accumulation of Change: Addendum 3
Calc Review page:

• AB Multiple Choice Exam Prep
• AB Free Response Exam Prep
• BC Multiple Choice Exam Prep
• BC Free Response Exam Prep
• Diving In to AP ® Calculus
• AB Step by Step
• BC Step by Step
• AB Calculus Exams & Quizzes
• BC Calculus Exams & Quizzes
• Calc AB Manual
• Calc BC Manual
• Combined AB/BC Calc Manual
• AB Calculus Powerpoints
• BC Calculus Powerpoints
• Calculus OnTheWall Intro Video
• Calculus Jeopardy
• Calculus Across & Down
• Word Search Poster
• Test Your Strength: AB Version
• Test Your Strength: BC Version
• Test Your Strength: Precalc
• Math Wars AB
• Math Wars BC

AP Calculus AB

Learn all about the course and exam. Already enrolled? Join your class in My AP.

Not a Student?

Go to AP Central for resources for teachers, administrators, and coordinators.

About the Exam

The AP Calculus AB Exam will test your understanding of the mathematical concepts covered in the course units, as well as your ability to determine the proper formulas and procedures to use to solve problems and communicate your work with the correct notations. A graphing calculator is permitted for parts of the exam. Note: You may not take both AP Calculus AB and Calculus BC Exams within the same year.

Mon, May 12, 2025

AP Calculus AB Exam

This is the regularly scheduled date for the AP Calculus AB Exam.

Exam Components

Section i: multiple choice.

45 questions 50% of Score

• Part A: Graphing calculator not permitted (33.3% of score)
• Part B: Graphing calculator required for some questions (16.7% of score) 
• Questions include algebraic, exponential, logarithmic, trigonometric, and general types of functions.
• Questions include analytical, graphical, tabular, and verbal types of representations.

Section II: Free Response

6 questions 50% of Score

• Part A: 2 problems | Graphing calculator required (16.7% of score) 
• Part B: 4 problems | Graphing calculator not permitted (33.3% of score)
• Questions include various types of functions and function representations and a roughly equal mix of procedural and conceptual tasks.
• Questions include at least 2 questions that incorporate a real-world context or scenario into the question.

Exam Essentials

Exam preparation, ap classroom resources.

Once you join your AP class section online, you’ll be able to access AP Daily videos, any assignments from your teacher, and your assignment results in AP Classroom. Sign in to access them.

• Go to AP Classroom

Free-Response Questions and Scoring Information

Go to the Exam Questions and Scoring Information section of the AP Calculus AB Exam on AP Central to review the latest released free-response questions and scoring information.

Past Exam Free-Response Questions and Scoring Information

Go to AP Central to review free-response questions and scoring information from past AP Calculus AB exams.

AP Calculus AB and BC Course and Exam Description

This is the core document for the course. It clearly lays out the course content and describes the exam and AP Program in general.

Services for Students with Disabilities

Students with documented disabilities may be eligible for accommodations for the through-course assessment and the end-of-course exam. If you’re using assistive technology and need help accessing the PDFs in this section in another format, contact Services for Students with Disabilities at 212-713-8333 or by email at [email protected] . For information about taking AP Exams, or other College Board assessments, with accommodations, visit the Services for Students with Disabilities website.

Credit and Placement

Search AP Credit Policies

Find colleges that grant credit and/or placement for AP Exam scores in this and other AP courses.

Additional Information

How To Study For The AP® Calculus AB Exam

The AP® Calculus AB exam tends to be one of the more challenging AP® exams, with about 58% of students achieving a score of 3 or above in 2023. That's good for the sixth-lowest mark of all the AP course offerings that year. However, the right preparation and resources can make the test substantially more manageable. Whether you’re aiming for a 3 to get some college credit or a 5 to stand out in your college applications, UWorld has you covered! This AP Calculus AB study plan gives you all the essential knowledge and preparation tips you need to crush the exam and attain your target score.

How To Prepare Effectively for the AP Calculus AB Exam

In this guide, we will help you with every stage of the process. From the moment you sign up for an AP Calculus AB course to exam day, you can take the following steps to prepare and improve your chances of getting the score you want.

How to prepare for an AP Calculus AB class

It's spring, and you've signed up to take an AP Calculus AB class next year. What now? Here are some things you can do before the school year starts to hit the ground running:

• Get a head start by preparing over the summer before school starts in the fall.
• Review College Board®'s AP Calculus AB course and exam description .
• If possible, talk to your AP Calculus AB teacher about the class expectations. You might also want to discuss the course rigor with your guidance counselor.
• Brush up on algebra and pre-calculus concepts. Specifically, focus on functions and graphs, rational functions, limits, trigonometry, and the unit circle, factoring polynomials, completing the square, exponent, and logarithm rules, and working with e and natural logarithms.
• Review formulas from geometry , such as the areas of circles, triangles, rectangles, and trapezoids. Volume and surface area of common 3D shapes like cubes, spheres, cylinders, and cones also show up occasionally, but the College Board will often provide formulas in a question stem when applicable.
• Study Unit 1 concepts and practice UWorld AP Calculus AB questions on limits. Paul’s online math notes are an excellent, free resource for getting started with calculus and reviewing some pre-calculus topics. Khan Academy is another free resource if your learning style is geared more toward watching videos than reading.
• Identify the ideal learning strategy for you : how do you best absorb information? Reading? Reading while taking notes? Watching videos? Practicing problems? A combination of these? The summer is a great time to figure this out.

How to pass the AP Calculus AB class

Most of the concepts in AB Calculus stem from two things: the derivative and the integral. Knowing how to differentiate and integrate various kinds of functions is key to doing well in AP Calculus. Practice with UWorld questions in Topics 2.5 - 2.10 and 3.1 - 3.2 for derivatives and Unit 6 for integrals. Try Units 4 and 8 to apply the derivative and integral to context for more advanced practice with these concepts.

Below are some AP Calculus AB study tips for passing your exam:

• Keep working on your homework, even if it feels tough at first—that's how you get better. If you make mistakes, it's a great chance to go back and see how to do it right next time.
• Make flashcards for basic derivative and integral rules and review them regularly until you have them down.
• Try getting some extra practice with UWorld MCQs. Focus on working through derivatives and integrals, particularly chain rule and u -substitution. Ensure you've got these basics solid before moving on to other topics.
• Practice FRQs using the ones the College Board provides from past years’ tests . Analyze the scoring guidelines to understand what the College Board expects. If you have access to AP Classroom, it is also a good source of FRQs.
• Take a moment to brush up on the early units; by the end of the year, the material from the beginning might be somewhat rusty.

How to do well on the AP Calculus AB exam

If you're aiming for a top score and really want to nail the basics, focus a lot on word problems and different contexts. Concentrate on Units 4, 5, 7, and 8. It's also a good idea to practice Free Response Questions (FRQs) using past exams from the College Board. They often use similar types of questions, so spotting patterns can really help.

For instance, every test since 2012 (except possibly for 2020, as those FRQs haven’t been released) has included an FRQ that features a table of data and asks questions about approximating derivatives with average rates of change or estimating integrals using Riemann sums, all within a specific context. Get comfortable with these kinds of questions to boost your confidence for the exam.

Here are AP Calculus AB study tips for improving your score from a 3 to a 4:

Start Early

Regular Practice

Focus on FRQs

How to score a 5 on the AP Calculus AB exam

Want to score a 5 on the AP Calculus AB exam ? Here's a comforting fact: you don't need to ace it to earn that top score. You don’t even need to hit 90%—the usual A-grade mark in most US schools. In fact, you generally need just under two-thirds of the total points available to score a 5. Remember, everyone makes mistakes, especially in a challenging exam like this, and that’s perfectly fine.

According to the College Board, 22.4% of 273,987 test-takers scored a 5 on the 2023 AP Calculus AB exam.

Aiming for perfection isn't the goal; focus on achieving what you know best. If you're stuck, try to rule out wrong answers and make an educated guess instead of spending too much time on it.

It's crucial to know where to focus your study time. Create a systematic AP Calculus AB study plan that allocates enough time for practicing core concepts. For instance, the derivative of inverse functions usually appears in just one multiple-choice question per test and isn't typically featured in the free-response section. If this formula trips you up, review it with a few UWorld questions, and maybe make a flashcard to test yourself now and then. But remember, spending time on more important concepts is more beneficial.

Study strategy for scoring a 5 on the AP Calculus AB exam:

Practice Regularly

Work on Free Response Questions (FRQs)

Understand the Scoring

What units are most difficult to learn or require focus due to complexity?

According to the College Board, the 2023 AP Calculus AB test-takers struggled with Unit 5 the most on MCQs. Slope fields tend to be an area where students struggle, but they don't show up very often on the test. Focus more attention on separating variables and solving differential equations, where algebra, exponent, and logarithm rules will be valuable. Brush up on previous years' math concepts. 

Also, pay close attention to the constant of integration , as it may simplify in unexpected ways. Practice is the key here. Try all of UWorld's questions in 7.6-7.7 to see several examples of how these questions may appear.

One concept that UWorld's AP Calculus team identified that gives AB students a lot of trouble is volume in Unit 8 at the end of the course (8.7-8.12). Know the formulas for the disk and washer methods for volumes of revolution and when to apply each. Also, strive to understand how to find cross-sectional volume; area formulas for semicircles and other shapes are also important here.

Generally, word problems and contextual questions tend to be difficult for many AP Calculus AB students. Specifically, related rates questions in Unit 4 give students a lot of trouble (4.4-4.5), and they consistently appear on the exam as both MCQs and FRQs. The key to these questions is organizing information. Pay close attention to the specific quantity the question asks and what information it provides (formulas, values of quantities or derivatives/rates, etc.). Scrutinize whether a quantity is “increasing” or “decreasing," as this determines whether the derivative is positive or negative, respectively. This is also where many geometric area and volume formulas come in handy.

Many AP Calculus AB students make a lot of small mechanical errors that add up throughout the exam. Practice is the key to avoiding these mistakes. Complete UWorld AP Calculus AB practice questions, and note any common mistakes you make. Review your notes and the explanations for those questions, and then practice more UWorld questions on those topics, keeping a careful eye out for your typical pitfalls. The College Board knows a lot of common errors students make and structures their answer choices accordingly (and so do we!), so just because the answer you arrived at happens to be an answer choice doesn’t mean it’s correct.

One common source of such errors is u -substitution in Unit 6. This advanced integration technique has many steps and parts to consider, each of which could be the source of a minor mistake. To avoid such mistakes, recall that differentiation and integration are inverse operations. If you have time, check your answer by differentiating the result of your integration and making sure it matches the integrand you started with. If it doesn't, you may have made a mistake along the way.

How to self-study for the AP Calculus AB exam

If you plan to self-study for the AP Calculus AB exam without taking an AP course, you may have a few more obstacles and challenges ahead. However, it is definitely doable. The biggest challenge will be not having a teacher introduce concepts and help you improve. So, your first step is to find what learning style works best for you:

Do you learn best from watching videos or visual presentations?

Khan Academy is a great place to start. Their videos introduce the concepts at a great pace, and they provide good base-level questions for building your skills. Another popular set of videos is Professor Leonard on YouTube. He breaks down concepts well and is passionate about learning. However, his videos are lengthy, so you might need to break them up and spend multiple days on each one.

Do you learn best from reading a textbook and taking notes?

A good free resource is Paul's Online Math Notes. As a college professor's notes on calculus, they aren’t specifically geared toward AP, but they are nonetheless a good starting point. If your budget permits, buy AP Calculus AB study material from Amazon or a secondhand bookstore. Any (single-variable) calculus textbook will work but try to find one specifically for AP Calculus so you know it’s written for high school students and follows the flow of the AP coursework. Otherwise, follow along with the AP Calculus AB Course and Exam Description to ensure that each topic is necessary for the exam.

Do you learn best from practice problems?

Subscribe to a question bank. Our UWorld AP Calculus AB QBank is specifically geared towards helping you learn from mistakes with in-depth explanations. Take similar questions as ones you’ve previously made mistakes to show your improvement in preparation for the AP Calculus AB exam. If you’ve purchased a textbook, you can practice problems inside or practice some problems in Paul’s Online Math Notes.

Most students learn from a combination of these approaches, so try different things and see what works best for you. Our recommendation would be to incorporate all of them. Here's a general flow you can use to facilitate your learning process:

Watch a video on a topic (Khan, Leonard, or other) and take notes.

• If you use Khan Academy, the videos are short enough to watch two or three in one session. They are also very well organized for proper pacing.
• If you use Professor Leonard, the videos are lengthy, so segment them into multiple viewing sessions. He stitches together multiple lectures into one video, so maybe watch until it cuts to the next lecture (his clothes will be different).

If the topic still isn't clear, rewatch parts or the full video or watch a similar video from another source . Sometimes, hearing or seeing a concept presented in multiple ways can help clear up confusion.

Read a text explanation of the topic (textbook, Paul’s, or other), and add it to your notes.

Work practice problems on the topic (UWorld, Khan, textbook, Paul’s, or other).

• If you use Khan Academy for videos, take their progress checks and quizzes along the way to help cement the ideas. They are generally not AP-level questions but are great when first learning a topic.
• If you use UWorld, read through our explanations, especially on questions you answer incorrectly. We also include hyperlinks to general explanations of concepts or alternate/more detailed solutions, so we encourage you to explore those as well.

Review your notes at the end of your study session.

This flow may or may not work for you. Experiment and figure out what elements to incorporate into your study plan. The next section provides tips on creating a study plan that’s right for you.

AP Calculus AB Study Plan 

It's the spring semester and it's crunch time! No matter how much time you have left, we've got some handy tips to help you get ready for the AP Calculus AB exam. Here’s a straightforward study plan you can follow:

Early Start

Review Strategy

Basic Concepts

Refine Your MCQ Skills

FRQ Practice

Essential Breaks

• Begin in February to ensure ample time to cover all units.
• Unit 1 – Limits: Spend 1 week.
• Units 2-5 – Derivatives: Allocate 2 weeks.
• Units 6-8 – Integrals: Dedicate 2 weeks.
• Refresh your understanding using your textbook or UWorld’s prep book.
• Set aside two to three short sessions each week to review MCQ-style practice questions.
• Consistently revisit challenging concepts.
• Start timing your MCQs to match exam conditions: aim for 2 minutes per non-calculator question and 3 minutes per calculator question.
• Review past FRQs from the College Board to familiarize yourself with potential essay questions.
• Outline and practice writing responses at least twice a month.
• Time yourself to adapt to the exam pace.
• Create and review flashcards daily for key formulas and rules.
• Regularly schedule breaks to stay refreshed—plan a day off each week or take an afternoon off when necessary.

Initial Review

Practice Focus

Timeframe Breakdown

Final Week Preparation

• Start with a general review of course topics, unit-by-unit at a quicker pace.
• Utilize resources such as textbooks, class notes, and course materials for concept refreshers.
• Prioritize completing textbook practice problems and using the UWorld QBank two to three times per week for 20-30 minutes per study session. 
• With UWorld:  To save time, focus on explanations for missed questions reviewing the concepts that need the most improvement.
• Limits and Derivatives (Units 1–5): Allocate 2 weeks, approximately one unit every two days.
• Integrals (Units 6–8): Allocate 1 week, approximately one unit every two days.
• Use any remaining time to review concepts from each unit which you missed previously to reinforce learning.

First Week: Review Growth Areas

Second Week: Focus on FRQs

Kick off your study by tackling a few UWorld AP Calculus AB MCQs in each topic to spot the ones that are harder to you. Dive into the explanations for any questions you miss. If you’re still feeling shaky, throw in a video or review your notes on those tough topics. Then, circle back with more UWorld questions to see how much you’ve improved!

Spend most of your week diving into FRQs—they’re a great way to cover lots of topics at once and really efficient for review. Plus, they make you show your work, which helps you get a solid handle on the whole problem-solving process.

AP Calculus AB Review/Study Materials

Finally, here is a collection of AB Calculus AB study materials you can use to facilitate your exam prep. Most of these links are sprinkled throughout this guide, but we've listed them in one section for your convenience.

Question Banks and Practice Problems

• UWorld : AP-level MCQs with in-depth explanations that help you learn from mistakes.
• College Board FRQs : FRQs from past exams that provide excellent practice for those sections.

Video Content

• Khan Academy: Great introductory-level videos to learn concepts with skill-building questions.
• Professor Leonard: A set of college lecture videos where he breaks down the concepts and emphasizes the core ideas behind calculus.

Now that you know how to study for AP Calculus AB, it's time to begin your exam prep. Good luck, and happy studying!

• Student Score Distributions* AP Exams - May 2022 . (2022). apcentral.collegeboard.org. Retrieved February 14, 2024, from https://apcentral.collegeboard.org/media/pdf/ap-score-distributions-by-subject-2022.pdf  
• Student Score Distributions* AP Exams - May 2023 . (2023). apcentral.collegeboard.org. Retrieved February 14, 2024, from https://apcentral.collegeboard.org/media/pdf/ap-score-distributions-by-subject-2023.pdf  
• Packer. (2023, June 30). The 2023 AP Calculus AB scores . twitter.com. Retrieved February 14, 2024, from https://twitter.com/AP_Trevor/status/1674500852122255360?lang=en  
• 2021 AP Exam Administration Scoring Guidelines - AP Calculus AB . (2021). apcentral.collegeboard.org. Retrieved February 14, 2024, from https://apcentral.collegeboard.org/media/pdf/ap21-sg-calculus-ab.pdf

AP Calculus AB Related Topics

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Every AP Calculus AB Practice Test Available: Free and Official

Advanced Placement (AP)

One of the best ways to prepare for the AP Calculus AB exam, as well as stay on top of lessons 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 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. 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 those problems 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.

• 1988 AP Calculus AB Released Exam
• Answer Key for 1998 Multiple-Choice Questions
• Answer Key for 1998 Free-Response Questions

Because these exams are from a while back, they both have some format differences compared to the current AP Calculus AB exam. But looking through these old exams can give you a sense of the test format, and you can work some of the questions as practice, too.

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 exam score
• 30 questions
• 60 minutes long
• No calculator allowed
• 15 questions
• 45 minutes long
• Calculator permitted

Free-Response Section

• Six questions total
• 1 hour 30 minutes total
• Two questions
• 30 minutes long
• Four questions

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 1998-2021 , 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 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.

This site has a 50-question multiple-choice test. The questions typically are 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.

On this site there are 50+ quizzes, all multiple choice. It's a pretty standard example what most unofficial, free online practice AP resources are like. The questions are decent (though easier and simpler to work through than actual AP Calc questions), and explanations are pretty barebones. 

High School Test Prep

The ten quizzes and two full-length exams on this site are best for someone earlier on in the course/their AP studying. The quizzes are each about 15 questions long, and immediately after each question, you'll be told if you answered correctly or incorrectly and be given an answer explanation. They're not the greatest match for actual AP Calculus questions, but because the quizzes are organized by category, they can be helpful if you're looking to practice a specific topic.

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 resource also has practice questions or both the Calculus AB and BC exams, so you can get in a little additional practice, too.

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 a test out of either the Barron's or Princeton Review's study guide and saving the official practice exams for down the line. After you take this practice test, score the exam to see what you earned on the test.

Step 2: Analyze Your Score Results

After you've figured out your practice exam 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 skip this step!

Now is also 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. Getting a 4 or a 5 on the AP Calculus AB exam looks more impressive to colleges, and it can sometimes get you more college credit.

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/or answering multiple-choice and free-response questions that focus specifically on those topics.

If you're struggling with your test-taking techniques—like 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 your mistakes . If you don't understand why you got a question wrong, go back and review that particular skill! Also, when you're reviewing notes, pause every few minutes and mentally go over what you just learned to make sure you're 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 AP Calculus AB test, the question types you'll see, and how to best prepare for the exam .

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!

These recommendations are based solely on our knowledge and experience. If you purchase an item through one of our links, PrepScholar may receive a commission.

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Christine graduated from Michigan State University with degrees in Environmental Biology and Geography and received her Master's from Duke University. In high school she scored in the 99th percentile on the SAT and was named a National Merit Finalist. She has taught English and biology in several countries.

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College Board AP® Calculus AB

Tools to help you ace your exams, including: past papers, revision notes, and exam-style questions (organised by topic)

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Past Papers

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AP Calc AB Exam

Exam Description: Additional Information    Register for this exam

AP Calculus AB Class Schedule and Homework Assignments

* Lesson recordings are from 2020-21, so they may not completely match the lessons from this year.

8/21	Policy Sheet First Day Survey Linear, Polynomial, and Exponential Functions (1.1, 1.2, 1.5)		p. 12:  17-18, 23-27, 39-42, 74, 76-77, 79, 82 p. 25:  32-33, 35, 43a p. 58:  33, 39-43 odd, 56-57, 61-62
8/23	Trigonometry and Combining Functions (1.3, 1.4)	Video	p. 37:  23-33 odd, 77-87 odd p. 46:  16, 57-71 odd, 76-78
8/27	Inverse Functions and Logarithms (1.6, 1.7)	Video	p. 73:  21-29 odd, 34-36, 39-43 odd, 53-63, 70-72, 77-79 odd, 94, 101-113 odd p. 78:  5-10, 13
8/29	Limits		p. 103:  25-36 p. 129:  17 (a-d), 25, 27, 29-35, 45-52
9/3	Limits, Asymptotes, and Continuity		p: 102:  7-8, 10, 11-12 (skip f), 14-15, 39-51 p. 130:  63-68, 80-82 p. 143:  17-18, 27-32, 35 (use calculator), 44-45, 50-51, 57-60
9/5	Review Chapter 1/2		Sample Test (answers)
	Test Chapter 1/2
9/11	Velocity and Tangents		p. 89:  8, 13-15 p. 174:  30-33, 63
9/13	The Derivative (3.1)		p. 173:  14-17 (NC on 16), 22-25 (NC on 22), 34-37, 41-42, 47-48 (NC), 57-58, 68-71
9/17	The Derivative Function (3.2)		p. 188:  7-9, 11-19, 25-27, 30-34, 53-56, 67-70, 74-76
9/19	Interpreting Derivatives and Higher Order Derivatives		p. 176:  73-78 p. 189:  20-24, 47-48, 52 p. 190: 41-42, 59-61, 63, 65-66
9/23	Derivatives of Polynomials and Exponents (3.3)		p. 203: 11-39 odd, 45-49 odd, 61-65, 70-76, 86-87, 91-92
9/25	Review Chapter 3A		Sample Test (answers)
9/27	Test Chapter 3A
10/1	Product and Quotient Rules (3.4)	Video	p. 213: 6-7, 11-41 odd, 42-45, 47-51 odd (only part a), 54-66, 75
10/3	Derivatives of Trig Functions (3.5)		p. 223:  5, 9-25 odd, 45-51 odd, 59-62, 67-68 p. 206:  96-102
10/8	Chain Rule (3.6)		p. 234: 7-8, 17-41 odd (skip 35), 61, 63, 68-78, 82-85, 87
10/10	Derivatives of Logs and Inverse Functions (3.7, 3.8)		p. 248:  41-51 odd p. 259:  7-31 odd, 44, 56-59, 62-66, 68-71
	Implicit Differentiation		p. 248: 9-16, 19-35 odd, 36-37, 55-61 odd, 63-70, 79-80, 106-112
	Derivative Quiz Motion Problems (3.9)	Video	p. 270: 1-10
10/18	Linear Approximation (3.11)		p. 285: 1-4, 7-15, 17-19, 21-22
10/22	Review Chapter 3B		Sample Test (answers)
10/24	Test Chapter 3B
	L’Hopital’s Rule		p. 347: 5, 12, 13-39 odd (skip 33), 43, 73, 81
10/30	Related Rates		p. 279: 1-11, 13-19
11/1	Practice Related Rates		p. 280: 20-21, 23-29, 31-32, 34-35, 38
11/5	Graphing		p. 304:  35-49 odd p. 331:  11-12, 14-15, 17-25 odd, 26-29, 45-53
11/7	Graphing, Part 2		p. 304:  21-33 odd, 53-67 odd, 69-72, 79-80 p. 334:  66, 70-71, 82, 89, 98 p. 363:  69-73
11/12	Optimization		p. 374: 6-8, 11-14, 20, 24, 48
11/14	Practice Optimization		p. 374: 18, 22, 23, 27, 34, 42, 49
	MVT, IVT, EVT		MVT, IVT, EVT Worksheet (solutions)
11/20	Review Chapter 4 Assign Mathematician Report		Sample Test (answers)
11/22	Test Chapter 4
12/3	Riemann Sums (5.2)		p. 411: 8-9, 11-12, 15-16, 21, 23, 32-33, 41-42
12/5	The Definite Integral (5.3)		p. 428:  10-17, 32, 34, 40-41, 43-53 odd, 58-62, 64-65, 67-68 p. 456:  68-69, 72-73, 76
	FTOC		p. 455:  58-66, 75-81 odd, 82 p. 506:  17-27
12/11	Review Chapter 5A		Sample Test (answers)
12/13	Test Chapter 5A
12/17	Mathematician Reports Due
12/19	Review		Semester 1 Final Review (answers)
	Review
1/8	Review
1/10	Review		Sample Final (answers)
1/14 1/16	Final Exam		Mathematician Reports Due

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Prepare for the AP Calculus AB exam with this college-level course that covers topics in single-variable differential and integral calculus. You’ll learn from your instructor, as well as from an engaging electronic textbook, videos, interactive lessons, and other online course materials. Along the way, you’ll complete challenging homework assignments, free-response questions, quizzes, projects, and cumulative exams, and participate in discussion forums as you move through this comprehensive course. You’ll solve problems by hand and with technology and will be expected to compose clear solutions for applied and abstract problems involving limits, derivatives, and integrals. As you progress through the course at your own pace, your instructor will be available for learning support and one-on-one review sessions.

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