Calculus BC Honors
This course includes four Big Ideas: 1) limits; 2) derivatives; 3) integrals and the Fundamental Theorem of Calculus; 4) series:
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 the theorems in calculus.
2. Derivatives: 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. 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 derivatives include finding the slope of a tangent line to a graph at a point, analyzing the graph of a function (concavity, extrema), and solving problems involving rectilinear motion.
3. Integrals and the Fundamental Theorem of Calculus: Integrals are used in a wide variety of practical and theoretical applications. Students should understand the definition of a definite integral involving Riemann sum, be able to approximate a definite integral using different methods and be able to compute definite integrals using geometry. 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.
4. Series: This 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 techniques 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.