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Overview of AP Calculus BC

AP Calculus BC is a college-level course offered by the College Board to high school students who are interested in studying biology at an advanced level. It was opted by 136,000 students in 2023 as it is one of the preferred AP courses. The exam’s difficulty rating stands at 5.5 out of 10 by class alumnae on a reddit survey. AP Calculus BC is known for its challenging curriculum and rigorous exams.The course covers a broad range of topics such as integration techniques, infinite series, and parametric equations.

Course content

Unit 1: Limits and Continuity

You’ll start to explore how limits will allow you to solve problems involving change and to better understand mathematical reasoning about functions.

How limits help us to handle change at an instant
Definition and properties of limits in various representations
Definitions of continuity of a function at a point and over a domain
Asymptotes and limits at infinity
Reasoning using the Squeeze theorem and the Intermediate Value Theorem

Unit 2: Differentiation: Definition and Fundamental Properties

You’ll apply limits to define the derivative, become skillful at determining derivatives, and continue to develop mathematical reasoning skills.

Defining the derivative of a function at a point and as a function
Connecting differentiability and continuity
Determining derivatives for elementary functions
Applying differentiation rules

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

You’ll master using the chain rule, develop new differentiation techniques, and be introduced to higher-order derivatives.

The chain rule for differentiating composite functions
Implicit differentiation
Differentiation of general and particular inverse functions
Determining higher-order derivatives of functions

Unit 4: Contextual Applications of Differentiation

You’ll apply derivatives to set up and solve real-world problems involving instantaneous rates of change and use mathematical reasoning to determine limits of certain indeterminate forms.

Identifying relevant mathematical information in verbal representations of real-world problems involving rates of change
Applying understandings of differentiation to problems involving motion
Generalizing understandings of motion problems to other situations involving rates of change
Solving related rates problems
Local linearity and approximation
L’Hospital’s rule

Unit 5: Analytical Applications of Differentiation

After exploring relationships among the graphs of a function and its derivatives, you'll learn to apply calculus to solve optimization problems.

Mean Value Theorem and Extreme Value Theorem
Derivatives and properties of functions
How to use the first derivative test, second derivative test, and candidates test
Sketching graphs of functions and their derivatives
How to solve optimization problems
Behaviors of Implicit relations

Unit 6: Integration and Accumulation of Change

You’ll learn to apply limits to define definite integrals and how the Fundamental Theorem connects integration and differentiation. You’ll apply properties of integrals and practice useful integration techniques.

Using definite integrals to determine accumulated change over an interval
Approximating integrals with Riemann Sums
Accumulation functions, the Fundamental Theorem of Calculus, and definite integrals
Antiderivatives and indefinite integrals
Properties of integrals and integration techniques, extended
Determining improper integrals

Unit 7: Differential Equations

You’ll learn how to solve certain differential equations and apply that knowledge to deepen your understanding of exponential growth and decay and logistic models.

Interpreting verbal descriptions of change as separable differential equations
Sketching slope fields and families of solution curves
Using Euler’s method to approximate values on a particular solution curve
Solving separable differential equations to find general and particular solutions
Deriving and applying exponential and logistic models

Unit 8: Applications of Integration

You’ll make mathematical connections that will allow you to solve a wide range of problems involving net change over an interval of time and to find lengths of curves, areas of regions, or volumes of solids defined using functions.

Determining the average value of a function using definite integrals
Modeling particle motion
Solving accumulation problems
Finding the area between curves
Determining volume with cross-sections, the disc method, and the washer method
Determining the length of a planar curve using a definite integral

Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

You’ll solve parametrically defined functions, vector-valued functions, and polar curves using applied knowledge of differentiation and integration. You’ll also deepen your understanding of straight-line motion to solve problems involving curves.

Finding derivatives of parametric functions and vector-valued functions
Calculating the accumulation of change in length over an interval using a definite integral
Determining the position of a particle moving in a plane
Calculating velocity, speed, and acceleration of a particle moving along a curve
Finding derivatives of functions written in polar coordinates
Finding the area of regions bounded by polar curves

Unit 10: Infinite Sequences and Series

You’ll explore convergence and divergence behaviors of infinite series and learn how to represent familiar functions as infinite series. You’ll also learn how to determine the largest possible error associated with certain approximations involving series.

Applying limits to understand convergence of infinite series
Types of series: Geometric, harmonic, and p-series
A test for divergence and several tests for convergence
Approximating sums of convergent infinite series and associated error bounds
Determining the radius and interval of convergence for a series
Representing a function as a Taylor series or a Maclaurin series on an appropriate interval

Exam Date

AP Calculus BC Exam

Will be conducted on

Mon, May 13, 2024 | 12 PM Local

AP Calculus BC Exam Guide

AP Calculus BC is known for its challenging curriculum and rigorous exams. In the AP Calculus BC exam guide, we provide you with the exam structure, topics, study resources, and student performance in AP Calculus BC exams to put together the comprehensive AP Calculus exam guide.

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AP Calculus BC Credit Policy

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AP Calculus BC Practice Tests

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AP Calculus BC Formula Sheet

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AP Calculus BC Course Resources

AP Calculus BC Course

The College Board website offers a variety of extra resources to help AP Calculus BC students succeed on the exam. These resources include the AP Calculus Course at a Glance, the AP Calculus Course and Exam Description, the AP Calculus Scoring Guidelines, and more. To help students prepare for the AP Calculus BC exam, we have provided a list of extra resources below.

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AP Calculus Course at a Glance

Excerpted from the AP Calculus AB and BC Course and Exam Description, the Course at a Glance document outlines the topics and skills covered in the AP Calculus courses, along with suggestions for sequencing.

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AP Calculus AB and BC Course and Exam Description

This is the core document for this course. Unit guides clearly lay out the course content and skills and recommend sequencing and pacing for them throughout the year. The CED was updated in the summer of 2020 to include scoring guidelines for the example questions. 

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