AP Calculus BC

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

In this unit, students will explore differentiation by connecting the definition of the derivative to limits and rates of change. Students will define average and instantaneous rates of change, use derivative notation, and estimate derivatives of functions at specific points. They will examine the relationship between differentiability and continuity, including when derivatives do and do not exist. Students 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 composite, implicit, and inverse functions. Students will use the chain rule to find derivatives of composite functions and apply that understanding to differentiate functions written implicitly. They will also differentiate inverse functions, including inverse trigonometric functions, and calculate higher-order derivatives to analyze how functions change over time or across repeated rates of 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'Hospital's Rule to determine limits involving indeterminate forms.

In this unit, students will use differentiation, theorems, and definitions to make analytical conclusions about functions and applications. Students will apply the Mean Value Theorem and the Extreme Value Theorem to reason about function behavior, identify critical points, and determine global and local extrema. They will analyze intervals where functions are increasing or decreasing, determine concavity over a function’s domain, and make connections between graphs of functions and their derivatives. Students will also use derivatives to solve optimization problems in context.

In this unit, students will explore integration as a way to measure accumulation and connect it to differentiation through the Fundamental Theorem of Calculus. Students will approximate areas using Riemann sums, work with summation and definite integral notation, and interpret accumulation functions and definite integrals. They will find antiderivatives and indefinite integrals using basic rules and substitution, and extend their integration techniques to include integration by parts and partial-fraction decomposition with linear factors. Students will also evaluate improper integrals or determine when an improper integral diverges.

In this unit, students will set up, analyze, and solve differential equations to model change. Students will find solutions to differential equations using separation of variables and use slope fields to reason about and sketch solution behavior. They will explore exponential models written as differential equations, estimate solutions using Euler’s Method, and interpret the meaning of the logistic growth model in context.

In this unit, students will apply integration to find average values, model motion and net change, and determine areas and volumes defined by graphs of functions. Students will calculate the average value of a function on an interval and use integrals to connect position, velocity, and acceleration. They will apply accumulation functions and definite integrals in real-world contexts, find areas between curves, and determine volumes using cross sections as well as the disc and washer methods. Students will also determine the length of a curve in the plane using a definite integral.

In this unit, students will develop an understanding of motion along curves in the plane using parametric equations, polar coordinates, and vector-valued functions. Students will calculate derivatives of parametric functions and vector-valued functions, determine arc length using definite integrals, and find particular solutions given a rate vector and initial conditions. They will determine positions and rates of change in planar motion problems, calculate derivatives of functions written in polar coordinates, and use definite integrals to find areas of regions defined by polar curves.

In this unit, students will explore infinite sequences and series and develop an understanding that a sum of infinitely many terms may converge to a finite value. Students will determine whether a series converges or diverges, approximate the sum of a series, and represent functions using Taylor polynomials. They will use Taylor polynomials to approximate function values and determine error bounds for those approximations. Students will also determine the radius and interval of convergence for power series and represent and interpret functions as Taylor, Maclaurin, or power series.