AP® Calculus BC is a full-year course for students wishing to pursue a career focused on mathematics or allied subjects requiring heavy math. This course is equivalent to Calculus I and II in college, and scoring well on the exam can help you skip both the college courses and move on to advanced ones. Since it is one of the most extensive AP courses, preparing for it requires a detailed study plan and thorough knowledge of the course content and materials.
This guide will give you an overview of the AP Calculus BC units, topics, and key concepts taught during the course. In addition to these, we will also discuss the essential prerequisites to excel in this course. By the end of this guide, you’ll understand how to begin preparing for your upcoming Calculus BC exams and what skills to focus on.
AP Calculus BC Units, Topics, and Key Concepts
Before you learn about the AP Calculus BC course curriculum, it is vital to know about the prerequisites required to approach this course. Having these requirements set up 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 BC. We’ve listed the prerequisites for the Calculus BC course below:
- Algebra I
- Geometry
- Algebra II
- Pre-calculus
The concepts in these courses help you build a solid foundation in reasoning with algebraic symbols and working with algebraic structures. This aspect is crucial to mastering AP Calculus BC. Make sure your high school offers courses that cover all or some of these concepts to help you prepare for the challenges of AP Calculus BC.
With the prerequisites set up, let’s explore the course component. Like AP Calculus AB, the AP Calculus BC exam tests you on two components—the course content and mathematical practices. The course content combines two aspects: units and big ideas. The ten units in the AP Calculus BC curriculum revolve around three recurring themes. The College Board® labels these themes as “big ideas”. In the following section, let’s dive deeper into these big ideas in the following section.
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AP Calculus BC’s Three Big Ideas
The AP Calculus BC units rely on three key elements, or big ideas, that form the basis for this course. Each of these elements is weaved into the course units as you advance through Calculus BC. Let’s take a look at the following big ideas:
- Big Idea 1: Change (CHA)Students can understand the concept of ‘change’ in a variety of contexts by using derivatives to describe rates of change of one variable with respect to another or definite integrals to describe the net change in one variable over an interval of another. It is essential to understand the link between integration and differentiation as represented in the Fundamental Theorem of Calculus. This forms the essence of the first Big Idea of Change (CHA).
- Big Idea 2: Limits (LIM)Understanding essential calculus ideas, definitions, formulae, and theorems such as continuity, differentiation, and integration constitutes the second big idea of Limits (LIM).
- 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, find anti-derivatives and indefinite integrals, and integrate using substitutions.
- Big Idea 3: Analysis Of Functions (FUN)The third big idea of Analysis of Functions (FUN) enables you to understand and evaluate 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 discuss in the following section, these three big ideas are distributed across ten units to help students understand each concept efficiently. Both the big ideas and the course units constitute a framework for the calculus course similar to many college courses and textbooks. Now let’s discuss each of the ten course units to develop a clearer understanding of how these big ideas and the units work together to build a solid foundation for Calculus BC.
The Ten Units of AP Calculus BC and Their Topics
The units feature the study material you will learn in your AP Calculus BC course. Although the curriculum for AP Calculus BC is similar to that of the AP Calculus AB course curriculum, BC contains two additional units (Units 9 and 10), plus some extra topics in Units 6 to 8. We’ll discuss these extra units in detail, but below is a table to give you an idea:
| Unit | Additional Topics in AP Calculus BC |
|---|---|
| Unit 6 | Additional techniques of Integration |
| Unit 7 | Euler’s method and logistic models with differential equations |
| Unit 8 | Arc length and distance traveled along a smooth curve |
| Unit 9 | Parametric equations, polar coordinates, and vector-valued functions |
| Unit 10 | Infinite sequences and series |
As we outline each unit, we will also look into the big ideas that spiral across the ten course units and topics. Understanding how these topics are categorized will help you assess your target areas and know which unit and topic you need to work on during your revision. We’ve also included the relative weights for each unit, so you can gauge the importance of these units in the AP Calculus BC exam. If you’re interested in learning more about any specific unit, click on the below unit sections.
Unit1
Unit2
Unit3
Unit4
Unit5
Unit6
Unit7
Unit8
Unit9
Unit10
Unit1
Unit2
Unit3
Unit4
Unit5
Unit6
Unit7
Unit8
Unit9
Unit10
AP Calculus BC Mathematical Practices
As you progress through the course content, you’ll be taught specific skills to approach and solve problems, which are combined under mathematical practices by the College Board. There are four mathematical practices, which are composed of the core skills to help you succeed in the AP Calculus BC exam. The AP course is designed to integrate these practices within the course content so that by the end of the course, you’ll be able to transfer these skills into the AP exam. Now, let’s take a look at these mathematical practices:
Practice 1: Implementing Mathematical Processes
This is the first mathematical practice you will learn during the AP Calculus BC course. As the name suggests, this mathematical practice will teach you how to solve problems by determining expressions and values by implementing mathematical processes and rules.
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Practice 1: Implementing Mathematical Processes
This is the first mathematical practice you will learn during the AP Calculus BC course. As the name suggests, this mathematical practice will teach you how to solve problems by determining expressions and values by implementing mathematical processes and rules.
Practice 2: Connecting Representations
In the second mathematical practice, you’ll learn to translate mathematical information from a single representation or across multiple representations by understanding the common underlying structures of mathematical problems.
Practice 3: Justification
The Free-Response Section of the AP Calculus BC exam requires you to justify how you solved a mathematical problem. This mathematical practice will help you develop reasoning skills to establish the steps required to solve problems logically.
Practice 4: Communication and Notation
Understanding and resolving a problem are insufficient. You must also understand how to communicate it correctly. You will learn to communicate findings or answers using correct notation, language, and mathematical conventions with the help of this mathematical practice.
Remember to use the core skills and mathematical practices that you learned during your course content. Developing a clear understanding of the concepts and theorems and mastering the ability to apply those concepts effortlessly is the key to achieving a 5 on your AP Calculus BC exam!
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Frequently Asked Questions
What are the most important topics in AP Calculus BC?
The four most important topics in AP Calculus BC are
- Unit 10: Infinite Sequences and Series
- Unit 6: Integration and Accumulation of Change
- Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
- Unit 5: Analytical Applications of Differentiation
What is the hardest unit topic in AP Calculus BC?
According to the 2021 AP exam results, the most challenging topic in AP Calculus BC was Unit 10: Infinite Sequences and Series.
What AP Calculus BC unit topics aren't in Calculus AB?
The following topics are not included in the AP Calculus AB curriculum:
- Unit 6: Additional techniques of integration
- Unit 7: Euler’s method and logistic models with differential equations
- Unit 8: Arc length and distance traveled along a smooth curve
- Unit 9: Parametric equations, polar coordinates, and vector-valued functions
- Unit 10: Infinite sequences and series
