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AP Calculus AB

AP Calculus AB

It’s been said that change is the only true constant. Calculus helps make sense of change by grappling with questions that inspire thinkers from around the globe, across time, and in many disciplines. Can change occur in an instant? When is the next solar eclipse or the turning point for an economy? In AP Calculus AB, you’ll develop a deeper understanding of mathematical principles that can help you answer questions such as these.

College Board

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 using Riemann Sums
Accumulation functions, the Fundamental Theorem of Calculus, and definite integrals
Antiderivatives and indefinite integrals
Properties of integrals and integration techniques

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.

Interpreting verbal descriptions of change as separable differential equations
Sketching slope fields and families of solution curves
Solving separable differential equations to find general and particular solutions
Deriving and applying a model for exponential growth and decay

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 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