AP Calculus BC

AP Calculus BC is a college-level calculus course designed to meet the Advanced Placement curricular requirements for Calculus BC (equivalent to one year of college calculus). The major topics of this course are limits, derivatives, integrals, the Fundamental Theorem of Calculus, and series.

The course provides students from all backgrounds with the tools to develop a deep conceptual understanding of calculus, reason with definitions and theorems, make connections, implement algebraic and/or computational processes, build notational fluency, use multiple representations and communicate effectively. The course adopts a student-centered approach and provides students with multiple learning pathways. Students regularly use technology to explore concepts, model real-life situations, support reasoning and access personalized learning opportunities.

Learning Modules

A balance of algebraic, graphical, numerical, tabular and verbal methods is integrated in the following learning modules in order to reinforce comprehension, problem solving and critical thinking skills.

Unit 1 - Limits

Many calculus concepts are developed by first considering a discrete model and then the consequences of a limiting case. Therefore, the idea of limits is essential for discovering and developing important ideas, definitions, formulas, and theorems in calculus. Students must have a solid, intuitive understanding of limits and be able to compute various limits, including one-sided limits, limits at infinity, the limit of a sequence, and infinite limits. They should be able to work with tables and graphs in order to estimate the limit of a function at a point. Students should know the algebraic properties of limits and techniques for finding limits of indeterminate forms, and they should be able to apply limits to understand the behavior of a function near a point. Students must also understand how limits are used to determine continuity, a fundamental property of functions.

Unit 2 - Differential Calculus

Using derivatives to describe the rate of change of one variable with respect to another variable allows students to understand change in a variety of contexts. In AP Calculus, students build the derivative using the concept of limits and use the derivative primarily to compute the instantaneous rate of change of a function. Applications of the derivative include finding the slope of a tangent line to a graph at a point, analyzing the graph of a function (for example, determining whether a function is increasing or decreasing and finding concavity and extreme values), and solving problems involving rectilinear motion. Students should be able to use different definitions of the derivative, estimate derivatives from tables and graphs, and apply various derivative rules and properties. In addition, students should be able to solve separable differential equations, understand and be able to apply the Mean Value Theorem, and be familiar with a variety of real-world applications, including related rates, optimization, and growth and decay models.

Unit 3 - The Integral and the Fundamental Theorem of Calculus

Integrals are used in a wide variety of practical and theoretical applications. AP Calculus students should understand the definition of a definite integral involving a Riemann sum, be able to approximate a definite integral using different methods, and be able to compute definite integrals using geometry. They should be familiar with basic techniques of integration and properties of integrals. The interpretation of a definite integral is an important skill, and students should be familiar with area, volume, and motion applications, as well as with the use of the definite integral as an accumulation function. It is critical that students grasp the relationship between integration and differentiation as expressed in the Fundamental Theorem of Calculus — a central idea in AP Calculus. Students should be able to work with and analyze functions defined by an integral.

Unit 4 - Series

The AP Calculus BC curriculum includes the study of series of numbers, power series, and various methods to determine convergence or divergence of a series. Students should be familiar with Maclaurin series for common functions and general Taylor series representations. Other topics include the radius and interval of convergence and operations on power series. The technique of using power series to approximate an arbitrary function near a specific value allows for an important connection to the tangent-line problem and is a natural extension that helps achieve a better approximation. The concept of approximation is a common theme throughout AP Calculus, and power series provide a unifying, comprehensive conclusion.

Trig Refresher

In this unit, you will explore how the trigonometric ratios from high school Geometry relate to circles. First, you will explore the height of a rider on a Ferris Wheel and develop precise definitions of sine and cosine using transformational geometry. This precision will lead us to a discussion of a mathematically natural unit of rotational measure, called the radian. You will also explore the graphs of periodic functions, and use them to model new situations and discover important properties of trigonometric functions.

Calculus Resources

Additional resouices for the study of Calculus include:

Video - 10 minute Intro to Calculus
Calculus Cheat Sheet - from Indiana Dept. of Education
Khan Academy - Online Calculus class

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