AP Calculus Units: Full List, Explained

The AP Calculus course is structured as several units, each covering a set of closely related topics. This article will give you an overview of which topics and concepts are included in each unit to help you organize your studies.

Unit 1: Limits and Continuity

This is one of the most fundamental sections of content for the AP Calculus course. It covers the rigorous definitions for limits and continuity, which have far-reaching consequences for the units which follow this. Unit 1: Limits and Continuity covers:

• Definition of a limit

• Infinite limits and limits at infinity

• Properties of limits

• Evaluating limits analytically, numerically, and graphically

• Squeeze theorem and trigonometric limits

• Selecting procedures for determining limits

• Continuity and continuous functions

• Removable and non-removable discontinuities

• Intermediate value theorem

Unit 2: Differentiation: Definition and Basic Derivative Rules

This unit introduces the basic concepts of differentiation and how the derivative operates as a rate of change. The unit also covers the essential rules used when handling a variety of different functions, including products and quotients of functions. Unit 2: Differentiation: Definition and Basic Derivative Rules covers:

• Average rate of change

• Instantaneous rate of change

• Derivatives and tangents

• Estimating the derivative at a point

• Differentiability and continuity

• Derivative rules

• Derivatives of exponentials and logarithms

• Derivatives of sine and cosine functions

• The product rule

• The quotient rule

• Derivatives of tangent and reciprocal trigonometric functions

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

This unit introduces further methods for finding derivatives, including how to differentiate composite functions and deal with inverse functions. Unit 3: Differentiation: Composite, Implicit, and Inverse Functions covers:

• The chain rule

• The inverse function theorem

• Derivatives of inverse trigonometric functions

• Higher-order derivatives

• Implicit differentiation

• Selecting procedures for calculating derivatives

Unit 4: Contextual Applications of Differentiation

Unit 4 focuses on how average and instantaneous rates of change can be applied to problems such as straight line motion. This unit also considers how the rates of change of different variables are linked, and introduces, with L'Hospital's rule, a new way of calculating limits. Unit 4: Contextual Applications of Differentiation covers:

• Meaning of a derivative in context

• Motion in a straight line

• Related rates

• Approximating values of a function

• L'Hospital's rule

Unit 5: Applying Derivatives to Analyze Functions

In unit 5, further applications of differentiation are explored. These topics consider using differentiation and higher order derivatives to find critical points of graphs and functions, which has applications for optimization problems. Unit 5: Applying Derivatives to Analyze Functions covers:

• Mean value theorem

• Extreme value theorem

• Critical points

• Increasing and decreasing functions

• First derivative test for local extrema

• Candidates test for global extrema

• Concavity of functions

• Second derivative test for local extrema

• Graphs of f, f' & f''

• Optimization problems

• Second derivatives of implicit functions

• Critical points of implicit relations

Unit 6: Integration & Accumulation of Change

In this unit, integration is introduced as the inverse of differentiation. It covers a large range of concepts from the basic rules of integration for different types of functions, to using numerical approaches like Riemann and trapezoidal sums. Unit 6 also includes some very important methods for finding integrals such as integration by substitution, along with important theoretical results like the fundamental theorem of calculus. Unit 6: Integration & Accumulation of Change covers:

• Indefinite integrals

• Derivatives & antiderivatives

• Indefinite integral rules

• Constant of integration

• Accumulation of change

• Riemann sums

• Trapezoidal sums

• Accumulation functions

• Fundamental theorem of calculus

• Properties of definite integrals

• Evaluating definite integrals

• Integrals of composite functions

• Integration using substitution

• Integration using completing the square

• Integration using long division

• Selecting techniques for integration

• Integration using integration by parts (BC only)

• Integration using partial fractions (BC only)

• Evaluating improper integrals (BC only)

Unit 7: Differential Equations

This unit focuses on differential equations, covering exactly what they mean in terms of derivatives. Unit 7 also covers how to solve differential equations using the method of separation of variables, as well as finding particular solutions and utilizing mathematical models involving exponential functions. Unit 7: Differential Equations covers:

• Introduction to differential equations

• Slope fields

• Separation of variables

• Finding particular solutions

• Exponential models

• Approximating solutions using Euler’s method (BC only)

• Logistic models (BC only)

Unit 8: Applications of Integration

Unit 8 introduces further applications of integration, both as the inverse of a derivative, and as a tool to find areas and calculate other accumulations of change. This includes finding the area between curves and axes, and extending this to find volumes of solids.

• Average value of a function

• Definite integrals as accumulated change

• Position, velocity and acceleration

• Distance and speed

• Area between a curve and the x-axis or y-axis

• Area between two curves and multiple areas

• Volumes from areas of known cross sections

• Volumes from cross sections of various shapes (Squares, rectangles, triangles, and semicircles)

• Volumes of Revolution: Disc method and washer method

Unit 9: Parametric Equations, Polar Coordinates & Vector-Valued Functions (BC Only)

Unit 9 is only examined as part of the Calculus BC course. It considers parametric curves and equations and how to differentiate them, as well as differentiating and integrating vector-valued functions. Polar coordinates are also introduced, leading to differentiating functions expressed in polar form, and finding the areas of regions bounded by polar curves. Unit 9: Parametric Equations, Polar Coordinates & Vector-Valued Functions covers:

• Differentiating parametric equations

• Arc lengths of curves given by parametric equations

• Differentiating vector-valued functions

• Integrating vector-valued functions

• Solving motion problems using parametric and vector-valued functions

• Polar coordinates and differentiating in polar form

• Areas of polar regions and areas bounded by polar curves

Unit 10: Infinite Sequences & Series (BC Only)

Unit 10 is only examined as part of the Calculus BC course. It introduces several key concepts for studying infinite series and focuses on how some functions can be written as infinite series. The unit also covers determining the behavior of infinite sequences and series, as well as specific important series such as Taylor series. Unit 10: Infinite Sequences & Series covers:

• Convergent and divergent infinite series

• Geometric series

• nth term test for divergence

• Integral test for convergence

• Harmonic series and p-series

• Comparison tests for convergence

• Alternating series test for convergence 

• Ratio test for convergence

• Determining absolute or conditional convergence

• Alternating series error bound

• Taylor polynomial approximations of functions

• Lagrange error bound

• Radius and interval of convergence of power series

• Taylor or Maclaurin series for a function

• Representing functions as power series

Frequently Asked Questions

How many units are in AP Calculus?

There are 10 units in total for AP Calculus BC, while AP Calculus AB contains 8 units.

What are the differences between Calculus AB and BC?

Calculus BC includes all units 1-8 from Calculus AB, plus 2 extra (units 9 and 10). Calculus BC also has a couple of extra concepts added into units 6 and 7.

All of unit 9 (Parametric Equations, Polar Coordinates & Vector-Valued Functions) and all of unit 10 (Infinite Sequences & Series) are exclusively in Calculus BC. 

In unit 6 (Integration & Accumulation of Change) there are three extra concepts included for the BC course:

• Integration using integration by parts

• Integration using partial fractions

• Evaluating improper integrals

In unit 7 (Differential Equations) there are two extra concepts included for the BC course:

• Approximating solutions using Euler’s method

• Logistic models

Both Calculus AB and Calculus BC follow the exact same exam format; students sit 1 exam paper which is 3 hours and 15 minutes in length.

What are the hardest AP Calculus units?

The units exclusive to Calculus BC, unit 9 (Parametric Equations, Polar Coordinates, and Vector-Valued Functions) and unit 10 (Infinite Sequences and Series), are generally regarded as more difficult as BC is designed as a higher-level course.

Within the first 8 units, common to both Calculus AB and BC, individuals will find different units harder than others based on personal preferences and prior knowledge. However, some units focus on higher-order problem solving skills and applications to contexts, rather than focusing on the simple application of processes. This means it is common for students to find Unit 5: Applying Derivatives to Analyze Functions, and Unit 8: Applications of Integration, harder than others.

What units are most common on the AP Calculus exam?

Each unit for Calculus AB and Calculus BC has predefined weightings in the exam paper. These weightings are summarized in the table below.

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