Many students opt to take a math AP exam, and Calculus AB is a popular choice since it's slightly easier than Calculus BC. But to get a high score on it, you'll need to do very well on the free-response section, which requires you to write out your answers to multi-step problems.
In this complete guide, we go over the AP Calculus AB Free Response section structure, provide examples to show you how it's graded and what a correct answer looks like, and give you some tips for acing this tricky section of the exam.
What to Expect on the AP Calculus AB Exam
The AP Calculus AB exam is one of two Calculus tests you can take. (The other is AP Calculus BC, which covers a slightly bigger and harder array of high-level math concepts.)
Calculus AB tests your knowledge of various calculus concepts, including derivatives, limits, and differential equations. You can learn more about what it tests by referring to the AP Calculus AB and BC Course and Exam Description.
Format-wise, the AP Calculus AB test is divided into two sections—a multiple-choice section and a free-response section—each of which makes up 50% of your score. You'll have three hours and 15 minutes for the exam.
Here's a brief overview of the format of the AP Calculus AB test:
| AP Calc AB Section | % of Score | Time | # of Questions | Calculator? |
| 1. Multiple Choice | Part A: 33.3% | Part A: 60 mins | Part A: 30 | Part A: No |
| Part B: 16.7% | Part B: 45 mins | Part B: 15 | Part B: Yes | |
| Total: 50% | Total: 1 hr 45 mins | Total: 45 | — | |
| 2. Free Response | Part A: 16.7% | Part A: 30 mins | Part A: 2 | Part A: Yes |
| Part B: 33.3% | Part B: 60 mins | Part B: 4 | Part B: No | |
| Total: 50% | Total: 1 hr 30 mins | Total: 6 | — | |
| TOTAL | 100% | 3 hrs 15 mins | 51 | — |
Both the multiple-choice and free-response sections are divided up into a Part A and a Part B. Note too that you'll have questions on which you may not use a calculator and some on which you are required to use a graphing calculator (required on Part B in Section 1, Part A in Section 2).
Now then, let's take an even closer look at the AP Calculus AB Free Response section.
AP Calculus AB Free Response Section Overview
As explained above, the free-response section is the second section on the AP Calculus AB exam, after the multiple-choice section. You'll have a total of 90 minutes to complete this section, which consists of six questions divided into two parts: Part A and Part B.
According to the College Board, these questions "include various types of functions and function representations and a roughly equal mix of procedural and conceptual tasks." There will also be "at least 2 questions that incorporate a real-world context or scenario into the question" (bold emphasis mine).
Here's a quick recap of what you can expect on Parts A and B in the free-response section:
| AP Calc AB Free Response | Time | # of Questions | Time per Question | % of Total Score | Calculator Policy |
| Part A | 30 mins | 2 | 15 mins | 16.7% | Required |
| Part B | 60 mins | 4 | 15 mins | 33.3% | Not permitted |
The AP Calculus AB FRQ section requires you to use four "Mathematical Practices," or skills, as defined by the College Board. Here's how these practices are weighted on this section of the test:
| Mathematical Practice | Description | % of Free-Response Section |
| Practice 1: Implementing Mathematical Processes | Determine expressions and values using mathematical procedures and rules | 37-55% |
| Practice 2: Connecting Representations | Translate mathematical information from a single representation or across multiple representations | 9-16% |
| Practice 3: Justification | Justify reasoning and solutions | 37-55% |
| Practice 4: Communication and Notation | Use correct notation, language, and mathematical conventions to communicate results or solutions | 13-24% |
Each AP Calc AB FRQ is graded on a scale of 0-9 points. Questions have multiple parts to them, labeled A-C or A-D, and you'll get a certain number of raw points (typically 1-3) for each part.
The maximum number of points you can get on one FRQ is 9 points, meaning you've correctly and fully answered every part of the question. The points you earn for the six FRQs are combined with your multiple-choice score and converted into a final AP score on a scale of 1-5.
AP Calculus AB FRQ Samples and Solutions
Here, we show you two examples of real Calculus AB Free Response questions taken from the 2020 Course and Exam Description. We'll go over how to solve each question to earn full credit. All answers come from the official scoring guidelines.
Part A Sample Question (Calculator Required)
In this problem, parts A-C are each worth 2 points, while part D is worth 3 points. You'll need to understand differential functions to be able to solve this correctly. You will also need to show your work clearly at each step in order to earn full points.
Here are model answers for each part of this AP Calc AB FRQ.
(A) Model Solution
Here, you will get a point for approximating using values from the table and another for correctly interpreting with units.
(B) Model Solution
You'll get a point for doing the midpoint sum setup correctly and another point for approximating using values from the table with units.
(C) Model Solution
Here, you will get a point for using the definite integral and another for finding the correct answer with supporting work.
(D) Model Solution
You'll earn a point for using the slope, another point for finding that L(t) = 2,000, and a final point for getting the right answer (14.759) and showing your work.
Wave goodbye to your calculator for this next question!
Part B Sample Question (No Calculator)
For this AP Calculus AB FRQ, you are not allowed to use a calculator, so be prepared to solve everything by hand and show your work!
You can earn up to 2 points for parts, A, B, and D, and up to 3 points for part C. You must be able to understand differential functions and derivatives, as well as what intervals are, to get this right.
Below are model answers to each of the four parts of this question.
(A) Model Solution
The graph of f is decreasing and concave down on the intervals (1, 1.6) and (3, 3.5) because f′ is negative and decreasing on these intervals.
Here, you will get a point for providing the correct answer and another point for explaining your reasoning.
(B) Model Solution
You will get a point for proposing x = 2 as a candidate and another point for justifying your answer with the math.
(C) Model Solution
You will get a point for finding the antiderivative of the form a[f(x)]2, another point for showing that a = 1/2, and a third point for finding the right answer (-40).
(D) Model Solution
Here, you'll get a point for using the product rule and another for finding the correct answer (60).
AP Calculus AB Free Response Section: 6 Essential Tips
Here are some tips to keep in mind as you prepare for the AP Calc AB Free Response section.
#1: Know How to Use Your Calculator
As you know, there are two parts on the AP Calculus AB Free Response section.
With Part A, you get two questions, which together count for 16.7% of your total AP test score. A graphing calculator is required for this section—not merely permitted!—so it is extremely important that you know how to use it effectively to solve those tricky, multi-step problems, which typically require you to graph complicated functions.
If you are not that comfortable with your calculator, you'll likely struggle with figuring out how to even start a problem!
Below are some examples of how you might be required to use your graphing calculator on the AP Calc exam, according to the College Board:
- Zooming to reveal local linearity
- Constructing a table of values to conjecture a limit
- Developing a visual representation of Riemann sums approaching a definite integral
- Graphing Taylor polynomials to understand intervals of convergence for Taylor series
- Drawing a slope field and investigating how the choice of initial condition affects the solution to a differential equation
Make sure you can do all these major functions (and more!) well before you take the AP Calculus AB exam.
#2: Memorize Key Formulas
Aside from knowing how to use your calculator, you should take care to really get down critical formulas you'll need to know, especially for Part B (the no-calculator part) of the free-response section.
Unfortunately, the College Board does not give out any formula sheets for the exam, but your AP Calculus teacher will probably give you a list of some to study for the test.
As a general rule, any formula you learn in class will likely be an important one to know for test day.
#3: Learn What the Task Verbs Mean
Every part of each AP Calculus AB FRQ contains a task verb that tells you what to do. It's a good idea to familiarize yourself with these different task verbs so that you can know right away what the question wants to see and how you can earn full credit for your answer.
Here are the most commonly used task verbs on AP Calc, according to the College Board:
- Approximate: Use rounded decimal values or other estimates in calculations, which require writing an expression to show work.
- Calculate/Write an expression: Write an appropriate expression or equation to answer a question. Unless otherwise directed, calculations also require evaluating an expression or solving an equation, but the expression or equation must also be presented to show work. "Calculate" tasks might also be formulated as "How many?" or "What is the value?"
- Determine: Apply an appropriate definition, theorem, or test to identify values, intervals, or solutions whose existence or uniqueness can be established. "Determine" tasks may also be phrased as "Find."
- Estimate: Use models or representations to find approximate values for functions.
- Evaluate: Apply mathematical processes, including the use of appropriate rounding procedures, to find the value of an expression at a given point or over a given interval.
- Explain: Use appropriate definitions or theorems to provide reasons or rationales for solutions and conclusions. "Explain" tasks may also be phrased as "Give a reason for..."
- Identify/Indicate: Indicate or provide information about a specified topic, without elaboration or explanation.
- Interpret: Describe the connection between a mathematical expression or solution and its meaning within the realistic context of a problem, often including consideration of units.
- Interpret (when given a representation): Identify mathematical information represented graphically, symbolically, verbally, and/or numerically, with and without technology.
- Justify: Identify a logical sequence of mathematical definitions, theorems, or tests to support an argument or conclusion, explain why these apply, and then apply them.
- Represent: Use appropriate graphs, symbols, words, and/or tables of numerical values to describe mathematical concepts, characteristics, and/ or relationships.
- Verify: Confirm that the conditions of a mathematical definition, theorem, or test are met in order to explain why it applies in a given situation. Alternately, confirm that solutions are accurate and appropriate.
#4: Use Realistic Practice Questions
By far the best way to prepare for the AP Calc AB exam—any AP exam, really!—is to use official questions and practice tests. Doing this will help you quickly get used to the difficulty, content, and pacing of both the free-response section and the test as a whole.
It's best to begin your prep by taking an official full-length practice test so you can see what areas you struggle with the most and then zero in on those in your study sessions. Make sure that you time yourself accordingly and tweak any older tests you use for practice so that they reflect the current format of the test.
You can get started with our collection of AP Calculus AB practice tests.
For free-response questions specifically, the College Board offers tons of sample questions in its Exam Description and on its Calculus AB exam page, which has an impressive collection of student sample responses as well.
#5: Get Used to Showing Your Work
To earn full points on free-response questions, you'll need to show all your work, from the very first step all the way to the last.
As you saw above with the sample FRQs, you'll often earn a point or two simply for justifying your response and showing how you got your answer. As you practice with official questions, make sure that you're writing out everything it takes to get to the right answer.
Note that showing your work doesn't just mean finding the correct answer, but also indicating the setup and intermediate steps needed to get there. Even if the steps for solving a problem seem really obvious to you, remember that this is the stuff AP graders want to see!
#6: Practice Pacing Yourself
Our final tip is to focus on pacing yourself in your prep so you're not spending too much time on any one AP Calc AB FRQ. As a reminder, you'll have about 15 minutes to spend on each free-response question, so try your best not to go over this "limit" in your studying—especially when you take full-length practice tests.
If you find that certain FRQs are taking you longer than 15 minutes, that's a sign that you need to target those question types in your prep and review their content more.
Recap: Acing the AP Calculus AB Free Response Section
The AP Calc AB Free Response section might sound intimidating, but as long as you know what to expect on test day and how to prep effectively, you're more than capable of walking away with an amazing (or even perfect!) AP score.
The Calculus AB free-response section is split into two parts:
- Part A: 2 questions, 30 minutes, graphing calculator required
- Part B: 4 questions, 60 minutes, calculator not permitted
Altogether, this section counts for 50% of your AP Calc AB score (the other half is your multiple-choice section score). FRQs are divided into three or four parts (labeled A-C/D) and are worth up to 9 raw points each.
To prepare for this tricky section of the AP Calc exam, be sure to do the following:
- Know how to use your calculator
- Memorize key formulas
- Learn what the task verbs mean
- Use realistic practice questions
- Get used to showing your work
- Practice pacing yourself
Heed these six tips and you're sure to ace the AP Calculus AB FRQ section in no time!
What's Next?
Got more questions about the AP Calculus AB exam as a whole? Then read our expert guide to the AP Calculus AB exam.
Want resources you can use for your AP Calculus AB test prep? We've got you covered with our collection of Calculus AB practice tests and questions.
Not sure whether to take Calculus AB or BC? Let us help you figure out which AP test is the ideal choice for you. And if you decide to opt for Calc BC instead of AB, you'll definitely want to check out our comprehensive guide to the Calculus BC free-response questions!
