AP® Calculus AB Course And Exam Description
At first glance, the AP® Calculus AB course might seem vast and challenging, but this overview will help you build a strong foundation in Calculus AB and set you up for success at the college level. We’ve broken down the AP Calculus AB course curriculum into simple chunks for an easy read. This page provides a well-rounded AP Calculus AB course outline, from prerequisites to content, including key concepts and topics.
AP Calculus AB Units, Topics, and Key Concepts
The AP Calc AB course comprises two primary components — Course Content and Mathematical Practices. As you progress through the course, you will learn the essential mathematical practices through the course content. Combined, both of these components prepare you to build a solid foundation in Calculus AB and help you succeed in the exam. The course content is further divided into units. Each unit is based on the three big ideas that provide the foundation upon which the course is structured.
Before you embark on your journey through the AP Calculus AB course, it is essential to understand the prerequisites needed to help you grasp the course successfully. Having these requirements met beforehand eases your way through the course. It also lets you decide if you are ready to take a math-heavy course like AP Calculus AB.
As far as prerequisites are concerned, your high school classes should provide you with courses that include Algebra, Geometry, Trigonometry, and Precalculus. These courses will help you build a strong foundation in reasoning with algebraic symbols and working with algebraic structures that are essential to learning Calculus.
AP Calculus AB’s Three Big Ideas
The curriculum for this course revolves around three key AP Calculus AB concepts or big ideas that provide the foundation upon which the course is structured. As you journey through the Cal AB course, you’ll find that each of these key concepts is interwoven into the course units. Let’s take a look at what these big ideas include:
- BIG IDEA 1: Change (CHA)
Using derivatives to describe rates of change of one variable with respect to another or using definite integrals to describe the net change in one variable over an interval of another allows students to understand the concept of ‘change’ in a variety of contexts. The first big idea of Change (CHA) allows you to understand the relationship between integration and differentiation as expressed in the Fundamental Theorem of Calculus. - BIG IDEA 2: Limits (LIM)
The second idea of Limits (LIM) teaches you to understand essential ideas, definitions, formulae, and theorems in calculus: for example, continuity, differentiation, and integration.- Differentiation: Defining the derivative of a function, estimating derivatives at a point, connecting differentiability and continuity, determining derivatives of constants, sums, differences, and constant multiples and trigonometric functions. You’ll also need to learn about composite, Implicit, and inverse Functions.
- Integration: Finding the average value of a function, applying accumulation functions, finding the area between curves of functions, and finding volumes from cross-sections and revolutions. You’ll also need to study the Fundamental Theorem of Calculus, finding anti-derivatives and indefinite integrals, and integrating using substitutions.
- BIG IDEA 3: Analysis Of Functions (FUN)
Understanding and analyzing the behaviors of functions by relating limits to differentiation, integration, and infinite series and relating each of these concepts to the others.
As we’ll soon discuss, these three big ideas are distributed across eight units to help students understand each concept efficiently. In doing so, both the big ideas and the course units create a template of the Calculus course, which is similar to many college courses and textbooks. Now let’s walk through each of the eight course units to develop a clearer understanding of how these big ideas and the units intertwine to build a solid foundation in Calculus.
The Eight Units of AP Calculus AB and Their Topics
The units contain the course material you’ll learn throughout your AP classes. Based on what you’ve studied in these units, the AP Calculus AB exam will test you with 45 multiple-choice questions (MCQs) and six free-response questions (FRQs).
Remember the big ideas that we talked about earlier on this page? As we outline each AP Calculus AB unit, we will also see which of those big ideas spiral across the course units. Each of the eight units comes with specific topics you’ll learn during your course.
Understanding how these topics are categorized will help you focus on individual topics in detail. It will also help you assess your target areas and weaknesses to know precisely which unit and topic to work on during your revision period. We’ve also included the relative exam weightage for every unit so that you get all the info in one place. If you’re curious to find detailed information about any particular AP Calculus AB unit, click on the unit widgets below to take you there!
Unit 1 – Limits and Continuity
Exam Weightage: 10-12 % ｜ Class periods ~ 22-23
This unit introduces you to the idea of change, which is why we study calculus. Calculus allows us to generalize knowledge about motion to diverse problems involving change. Using the concept of limits, you’ll learn the subtle distinction between evaluating a function at a point and considering what value the function is approaching after a point in time. As you progress through this unit, you’ll learn how to determine change by justifying claims about limits and continuity through definitions, theorems, and properties.
The big ideas explored in this unit:
- Big Idea 1: Change – Can change occur in an instant?
- Big Idea 2: Limits – How does knowing the value of a limit, or that a limit does not exist, help you to make sense of interesting features of functions and their graphs?
- Big Idea 3: Analysis Of Functions – How do we close loopholes so that a conclusion about a function is always true?
In this unit, you will learn:
- How the average rate of change of a function can closely approximate the rate of change at an instant: Topic 1.1.
- What a limit is, how to express it, and how to calculate or approximate it from a table, graph, or function: Topics 1.2 – 1.4.
- Properties of limits and how to simplify them with algebra and trigonometry: Topics 1.5 – 1.7, 1.9.
- How limits of known functions can provide information about an unknown function using the squeeze theorem: Topic 1.8.
- What continuity is and how to identify when a function or graph is discontinuous at a point or over an interval: Topics 1.10 – 1.13.
- How to find infinite limits and limits at infinity, and what information these limits can provide about a function’s asymptotes: Topics 1.14 – 1.15.
- How continuity can prove the existence of a function value using the intermediate value theorem: Topic 1.16.
Unit 2 – Differentiation: Definition and Fundamental Properties
Exam Weightage: 10-12 % ｜ Class periods ~ 13-14
Unit 2 will develop your concept of differentiation, and how it is used to determine instantaneous rates of change. You will also learn to use a graphing calculator and explore how its various operations affect slopes of tangent lines. This exercise will help you develop a sense of basic rules and properties of differentiation.
The big ideas explored in this unit:
- Big Idea 1: Change – How can a state determine the rate of change in high school graduates at a particular level of public investment in education (in graduates per dollar) based on a model for the number of graduates as a function of the state’s education budget?
- Big Idea 2: Limits – Why do mathematical properties and rules for simplifying and evaluating limits apply to differentiation?
- Big Idea 3: Analysis Of Functions – If you knew that the rate of change in high school graduates at a particular level of public investment in education (in graduates per dollar) was a positive number, what might that tell you about the number of graduates at that level of investment?
In this unit, you will learn about the following AP calculus AB topics:
- How to find an average rate of change using a difference quotient: Topic 2.1.
- What a derivative is, its relationship to an instantaneous rate of change, and how to calculate or estimate it using the limit of a difference quotient or a table: Topics 2.2 – 2.3.
- How continuity and differentiability are related: Topic 2.4.
- How to calculate the derivative of different families of functions: Topics 2.5, 2.7, 2.10.
- Properties of derivatives and advanced differentiation techniques: Topics 2.6, 2.8 – 2.9.
Unit 3 – Differentiation: Composite, Implicit, and Inverse Functions
Exam Weightage: 9-13% ｜ Class periods ~ 10-11
In this unit, you’ll learn how to differentiate composite functions using the chain rule and apply that understanding to determine derivatives of implicit and inverse functions. The unit revolves around the principle that for composite functions, y is a function of u while u is a function of x. This unit is one of the most important units in the AP Calculus course, as mastering the chain rule is essential to your success in all future units.
The big ideas explored in this unit:
- Big Idea 3: Analysis Of Functions – If pressure experienced by a diver is a function of depth and depth is a function of time, how might we find the rate of change in pressure with respect to time?
In this unit, you will learn:
- More advanced differentiation techniques: Topics 3.1 – 3.5.
- How to differentiate a function multiple times: Topic 3.6.
Unit 4 – Contextual Applications of Differentiation
Exam Weightage: 10–15% ｜ Class periods ~ 10-11 Unit 4 begins by understanding average and instantaneous rates of change in problems involving motion. The unit then identifies differentiation as a common underlying structure to build an understanding of change in various contexts. You’ll also learn to apply L’Hopital’s Rule to determine limits of indeterminate forms. The big ideas explored in this unit:- Big Idea 1: Change – How are problems about position, velocity, and acceleration of a particle in motion over time structurally similar to problems about the volume of a rising balloon over an interval of heights, the population of London over the 14th century, or the metabolism of a dose of medicine over time?
- Big Idea 2: Limits – Since certain indeterminate forms seem to actually approach a limit, how can we determine that limit, provided it exists?
- What the derivative means in a real-world context: Topic 4.1 – 4.3.
- How to calculate a rate of change when two or more rates are related: Topic 4.4 – 4.5.
- How to approximate values of a function using a line tangent to the function at a nearby x-value: Topic 4.6.
- How to efficiently calculate difficult limits with L’Hopital’s Rule: Topic 4.7.
Unit 5 – Analytical Applications of Differentiation
Exam Weightage: 15-18% ｜ Class periods ~ 15-16
This unit will focus on the analytical application of differentiation to obtain formal conclusions through reasoning, definitions, and theorems. You’ll learn to present justifications for their conclusions about the behavior of functions over certain intervals or the locations of extreme values or points of inflection. The unit concludes this study of differentiation by applying abstract reasoning skills to justify solutions for realistic optimization problems.
The big ideas explored in this unit:
- Big Idea 3: Analysis Of Functions – How might the Mean Value Theorem be used to justify a conclusion that you were speeding at some point on a certain stretch of highway, even without knowing the exact time you were speeding?
- What additional information is included in a sound mathematical argument about optimization that a simple description of an equivalent answer lacks?
In this unit, you will learn about the following AP Calculus AB topics:
- How differentiability can prove the existence of a derivative value in an interval using the mean value theorem: Topic 5.1.
- What information the derivative can provide about the graph of a function like extreme values, intervals of increasing/decreasing, points of inflection, and intervals of concavity: Topics 5.2 – 5.9.
- How to use the derivative to optimize a quantity within a context: Topics 5.10 – 5.11.
- How implicit differentiation can provide information about a relation: Topic 5.12.
Unit 6 – Integration and Accumulation of Change
Exam Weightage: 17-20% ｜ Class periods ~ 18-20
Unit 6 is one of the most important units in the AP Calculus AB course. In this unit, you’ll learn the relationship between differentiation and integration using the Fundamental Theorem of Calculus. Just as differentiation determines instantaneous rate of change at a point, integration determines the accumulation of change over an interval. This unit forms the foundation for future units in this course because you’ll have to deploy the concept of Integration to various realistic and geometric applications.
The big ideas explored in this unit:
- Big Idea 1: Change – Given information about a rate of population growth over time, how can we determine how much the population changed over a given interval of time?
- Big Idea 2: Limits – If compounding more often increases the amount in an account with a given rate of return and term, why doesn’t compounding continuously result in an infinite account balance, all other things being equal?
- Big Idea 3: Analysis Of Functions – How is integrating to find areas related to differentiating to find slopes?
In this unit, you will learn:
- What a definite integral is, its relationship with accumulation of change and area, and how to approximate the area under a curve with Riemann sums: Topics 6.1 – 6.5.
- Properties of definite integrals: Topics 6.6 – 6.7.
- What an antiderivative is and its relationship with indefinite integrals: Topic 6.8.
- Advanced integration techniques: Topics 6.9 – 6.10, 6.14.
Note: Topics 6.11- 6.13 are taught in the Calculus BC course.
Unit 7 – Differential Equations
Exam Weightage: 6-12% ｜ Class periods ~ 8-9
In this unit, you’ll learn to set up and solve separable differential equations. This unit will teach you to translate mathematical information from one representation to another by matching equations and slope fields, rewriting verbal statements as differential equations, and sketching slope fields that match their symbolic representations.
The big ideas explored in this unit:
- Big Idea 3: Analysis Of Functions – How can we derive a model for the number of computers, C, infected by a virus, given a model for how fast the computers are being infected, dC dt , at a particular time?
In this unit, you will learn:
- How to model a context with a differential equation: Topics 7.1 – 7.2.
- How to use a slope field to estimate solutions to differential equations: Topics 7.3 – 7.4.
- How to find a general or particular solution to a differential equation: Topics 7.6 – 7.7.
- How to model exponential growth or decay with a differential equation and how to solve it: Topic 7.8.
Note: Topics 7.5 and 7.9 are taught in the Calculus BC course.
Unit 8 – Applications of Integration
Exam Weightage: 10–15% ｜ Class periods ~ 19-20
Unit 8 teaches you to find the average value of a function, model particle motion and net change, and determine areas and volumes defined by graphs and functions. This unit also involves geometric applications of integration. The aim of this unit is to develop an understanding of Integration that can be transferred across these and many other applications.
The big ideas explored in this unit:
- Big Idea 1: Change – How is finding the number of visitors to a museum over an interval of time based on information about the rate of entry similar to finding the area of a region between a curve and the x-axis?
In this unit, you will learn:
- How to use a definite integral to find the average value of a function on an interval: Topic 8.1.
- What the definite integral means in a context: Topics 8.2 – 8.3.
- How to use a definite integral to find the area between curves: Topics 8.4 – 8.6.
- How to use a definite integral to find the volume of a solid with a cross-section of a particular geometric shape: Topics 8.7 – 8.8.
- How to use a definite integral to find the volume of a solid of revolution: Topics 8.9 – 8.12.
Note: Topic 8.13 is taught in the Calculus BC course.
Mathematical Practices for AP Calculus AB
The AP Calculus AB Mathematical Practices describe the skills you should acquire while exploring the Cal AB course units. There are four Mathematical Practices, and they form the core skills that you need to develop for succeeding in the exam. The AP Calculus AB course should enable you to integrate content with the practices mentioned below. With sufficient repetition and revision, you’ll be able to transfer these skills to the AP exam. Now, let’s look at how the College Board® categorizes these core Mathematical Practices:
- Implementing Mathematical Processes
This is the first practice you will learn during the AP Calculus AB course. As the name suggests, this Mathematical Practice will help you determine expressions and values using mathematical procedures and rules and teach you how to implement those rules to solve problems. - Connecting Representations
This Mathematical Practice will teach you to translate mathematical information from a single representation or across multiple representations. - Justification
Justifying how you deduce the solution to a mathematical problem is crucial for the Free-Response Section of the AP Calculus AB exam. This Mathematical practice will equip you with reasoning skills to logically establish the steps required to solve problems. - Communication and Notation
Understanding and solving a problem are not enough. You also need to know the correct way of communicating it. With the help of this Mathematical Practice, you’ll learn to use correct notation, language, and mathematical conventions to communicate results or solutions.
“Remember to apply the core skills and Mathematical Practices you learned during your course content. Developing a clear understanding of the concepts and theorems and mastering the skill to apply those concepts effortlessly is the key to unlocking that 5 on your AP Calculus AB exam!”
As you journey through each AP Calculus AB concept and topic, remember to go back and review the big ideas that each of these units encompasses. Knowing the fundamentals of a subject is the core of a solid learning process.
If you are preparing for your AP Calculus AB exam, we can help! Challenging practice questions, detailed explanations for correct and incorrect answers, and digital performance tracking are just a part of what makes our online courses for AP second to none. You’ll love your increased understanding in the classroom and the higher test scores you’ll achieve with UWorld. Take our free AP Calculus AB practice exam today and feel the difference!
A Few FAQs to help you prepare for the AP Calculus AB Exam
What are the most important topics in AP Calculus AB?
- (17-20%) Unit 6: Integration and Accumulation of Change
- (15-18%) Unit 5: Analytical Applications of Differentiation
- (10-15%) Unit 4: Contextual Applications of Differentiation
- (10-15%) Unit 8: Applications of Integration
- (10-12%) Unit 1: Limits and Continuity
- (10-12%) Unit 2: Differentiation: Definition and Fundamental Properties
What is the hardest unit topic in AP Calculus AB?
According to the 2021 AP exam results, the most challenging topic in AP Calculus AB is Unit 10: Infinite Sequences and Series.
References
The College Board®
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