# AP® Calculus AB Study Guide

The AP Calculus AB exam tends to be one of the harder AP® exams, with about 51% of students achieving a score of 3 or above in 2021. That’s good for the sixth-lowest mark of all the AP course offerings that year. However, the test can become substantially more manageable with the right preparation and resources. Whether you’re aiming for a 3 to get some college credit or a 5 to stand out from your college applications, UWorld has you covered! This study guide gives you all the essential knowledge and preparation tips you need to crush the AP Calculus AB exam and attain your target score.

## How To Study 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:

• Begin preparing the summer before the course.
• Review College Board’s® AP Calculus AB course and exam description.
• If possible, talk to your AP Calculus AB teacher about the expectations of the class. It might also be a good idea to speak with your guidance counselor about the rigor of the course.
• Brush up on algebra and pre-calculus concepts. Specifically, focus on functions and graphs, rational functions, limits, trigonometry, and the unit circle, factoring polynomials and completing the square, exponent and logarithm rules, and working with e and natural logarithms.
• Review formulas from geometry, like 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 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.
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### How to pass an 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. For more advanced practice with these concepts, try Units 4 and 8 to apply the derivative and integral to context.

Here are some tips for passing an AP Calculus AB class:

• Practice at least 30 minutes a day, every day.
• Try hard on your homework, even if you don’t fully get it yet; that’s where you build your skills. Besides, teachers are much more likely to work with a student and help them improve if they give consistent effort.
• Make flashcards for basic derivative and integral rules and use them regularly until you have them down.
• Get additional practice with UWorld MCQs. Emphasize practicing derivatives and integrals, especially chain rule and u-substitution. Only when you have the basics down should you spend some time on the other units.
• Practice FRQs with 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.
• Spend some time to revisit early units; the material from the beginning of the year might be somewhat rusty by now.
• Two weeks before the exam, increase your study time to 1 hour per day, reviewing two units each day.
• Ask the teacher for help and suggestions on how to improve; they will be happy to assist you and may be able to identify specific growth areas.

### How to do well on the AP Calculus AB exam

If you’re confident you can master the basics and want to shoot for a 4 or 5, you’ll want to spend much time working with contexts and word problems. Units 4, 5, 7, and 8 should be your focus. Practice FRQs frequently with College Board’s previous years’ questions. They use a lot of similar question types in their FRQs, so pay attention to patterns.

For example, every test since 2012 (with the possible exception of 2020, for which College Board has not released FRQs) has had one FRQ with a table of data asking questions like approximating derivatives with AROC, approximating integrals with Riemann sums, and interpreting the meanings of those things within the given context. Practice those kinds of questions so you know how to handle them because you will most likely see one on the test.

Here are some tips to improve your score from 3 to 4:

• Increase practice time to 45 minutes every day.
• Try hard on your homework.
• Everyday after school, review your notes from that day and the previous day’s class. Concepts often connect, so reviewing the previous class’ concept can help you understand the next one.
• Around Spring Break, create a study plan for reviewing the course content (see below).
• Continue to use UWorld's AP Calculus AB question bank to practice MCQs.
• Continue to practice FRQs with AP Classroom and the ones the College Board provides. Keep an eye out for commonly asked questions and emphasize them.
• One month before the exam, increase study time to 1.5 hours a day.
• Ask the teacher for help and suggestions on how to improve; they will be happy to assist you and may even identify specific growth areas.

### How to score a perfect 5 in the AP Calculus AB exam

So you want to score a 5? The most important thing to understand if you want a 5 is that you do not need to get every single point on the exam. You don’t even need to get 90%⁠—the requirement for an A on an exam in the US school system. You typically need just under two-thirds of the total possible points to achieve a 5. Mistakes are inevitable on the AP Calculus AB exam, but that’s ok.

According to The Washington Post, only 12 students worldwide, or 0.004% of AB test-takers, got 100% of the points in 2015.

Don’t try to be perfect; maximize your points on the things you know well, and on anything else, try to eliminate answer choices and make an educated guess if it’s taking too long.

Knowing where to spend your study time is vital. Make sure you spend a lot of time with major concepts and know which concepts aren’t essential. For example, the derivative of inverse functions shows up consistently in one MCQ per test but never (as far as we know) in an FRQ. One MCQ is not worth a lot, so if you’re struggling to remember the formula, practice a few UWorld questions and make a flashcard with the formula to quiz yourself occasionally, but don’t spend too much time on it. You’ll get more out of focusing on more critical concepts.

• Be prepared to practice 45 to 60 minutes every day.
• Try hard on your homework.
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• Make flashcards of derivative and integral rules, common formulas, conditions and conclusions of theorems, etc.
• Around Spring Break, create a study plan for reviewing the course content (see below).
• Continue to use UWorld's AP Calculus AB question bank to practice MCQs.
• Continue to practice FRQs with AP Classroom and the ones the College Board provides. Keep an eye out for commonly asked questions and emphasize them.
• Identify and keep track of concepts you struggle with. Consider asking your teacher for extra clarification on those topics, and practice more problems in those areas.
• Ask your teacher for help and suggestions on how to improve; they will be happy to assist you and may even identify specific growth areas.

By Spring Break:

• Make a study plan for yourself (see below).
• Increase study time to 1.5 to 2 hours per day.
• Time yourself on FRQs based on the timing of each section on the exam. If you find yourself running out of time, identify concepts and question types that take you the longest and focus on those. If you find yourself with extra time, get in the habit of checking your work. See our AP Calculus AB FRQ guide for more tips.
• Study the FRQ scoring guidelines on College Board’s website to understand what they look for when grading. Pay particular attention to the 2021 guidelines as they detail exactly what the graders are looking for.
• Consider purchasing an exam prep book with more sample problems if you’ve completed UWorld’s AP Calculus AB question bank or all of the FRQs College Board provides.

### What units are most difficult to learn or do I need to focus due to complexity?

According to College Board, 2021 AP Calculus AB test-takers struggled with Unit 7 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, so brush up on previous years’ math concepts.

Also, pay close attention to the constant of integration as it may simplify in ways you don’t expect. 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 very end of the course (8.7-8.12). This aligns with the College Board data linked above, which identified an area-volume FRQ as the most challenging in 2021. 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 a textbook 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:

1. Watch a video on a topic (Khan, Leonard, or other) and take notes.
• If you use Khan Academy, the videos are short enough that you can watch 2 or 3 at once. They are 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).
2. Rewatch parts of or the full video if the topic still isn’t clear, or watch a similar video from another source. Sometimes, hearing or seeing a concept presented in multiple ways can help clear up confusion.
3. Read a text explanation of the topic (textbook, Paul’s, or other), and add it to your notes.
4. 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.
6. This flow may or may not work for you. Experiment and figure out what elements to incorporate into your study plan. See the next section for tips on creating a study plan that’s right for you.

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## AP Calculus AB Study Exam Tips

Now it’s Spring Break (or later), and it’s time for the full-court press. No matter your time frame, we’ve got some tips for you to prepare for the AP Calculus AB exam.

## AP Calculus AB Review/Study Materials

Finally, here is a collection of materials you can use to facilitate your study. 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

• 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.

Good luck, and happy studying!

## AP Calculus AB Related Topics

Check out our tips and tricks for free response questions on the AP Calculus AB exam.

Here, we provide helpful information and tips for multiple choice questions on the AP Calculus AB exam.

There are a lot of formulas to remember for the AP Calculus AB exam, so we put them all together for you as a study tool.

The UWorld AP Calculus AB QBank will give you lots of practice to prepare for the exam.
The Course and Exam Description for AP Calculus AB is a huge document, so we’ve broken down the essential information you need to know.