AP Calculus AB

In this unit, students will evaluate, interpret, and apply the concept of limits to understand the behavior and continuity of functions. Students will use limit notation, numerical information, and graphical information to estimate and describe limits. They will determine limits using basic theorems, algebraic rules and manipulation, substitution, and the Squeeze Theorem. Students will also use limits to explain unbounded and asymptotic behavior, define continuity at a point and over an interval, and apply the Intermediate Value Theorem to describe the behavior of functions on intervals.

In this unit, students will develop an understanding of how the derivative is defined through limits and how it connects to average and instantaneous rates of change. Students will use derivative notation, estimate derivatives at a point, and analyze the relationship between differentiability and continuity, including when derivatives do and do not exist. They will also apply derivative rules to find derivatives of sums, differences, products, and quotients of functions, as well as derivatives of trigonometric, logarithmic, and exponential functions.

In this unit, students will extend their understanding of differentiation to more complex functions, including composite, implicit, and inverse functions. Students will apply the chain rule to find derivatives of composite functions and use implicit differentiation to analyze relationships that are not written explicitly as functions. They will also differentiate inverse functions and inverse trigonometric functions, and calculate higher-order derivatives to describe how rates of change themselves change.

In this unit, students will apply differentiation as a tool for understanding change in a variety of real-world and mathematical contexts. Students will interpret the meaning of derivatives and rates of change in context, including straight-line motion by connecting position, velocity, and acceleration. They will solve related rates problems, approximate values of functions using local linearity and linearization, and apply L'Hopital's Rule to determine limits involving indeterminate forms.

In this unit, students will use theorems, definitions, and derivative information to make analytic justifications about functions and their applications. Students will apply the Mean Value Theorem and Extreme Value Theorem, identify critical points, and determine local and global extrema. They will analyze intervals where functions are increasing or decreasing, determine concavity, and make connections between graphs of functions and their derivatives. Students will also use differentiation to solve optimization problems in mathematical and contextual situations.

In this unit, students will explore integration as a way to describe accumulation and establish the relationship between differentiation and integration through the Fundamental Theorem of Calculus. Students will approximate areas using Riemann sums, interpret summation notation and definite integral notation, and use accumulation functions to represent change over an interval. They will evaluate definite integrals, find antiderivatives and indefinite integrals using basic rules and notation, and apply substitution as a method of integration.

In this unit, students will set up, analyze, and solve separable differential equations. Students will find solutions using separation of variables and reason about differential equations through slope fields, including sketching slope fields to represent families of solutions. They will also apply differential equations to exponential models, using them to describe growth and decay in contextual situations.

In this unit, students will apply integration to solve problems involving average value, particle motion, net change, area, and volume. Students will find the average value of a function on an interval and connect position, velocity, and acceleration using integrals. They will use accumulation functions and definite integrals in applied contexts, find the area between curves expressed as functions, and determine volumes using cross sections as well as the disc and washer methods.