Course Introduction
Calculus AB is primarily concerned with developing your understanding of the concepts of calculus and providing you with its methods and applications. The course emphasizes a multi-representational approach to calculus, with concepts, results, and problems being expressed graphically, numerically, analytically, and verbally. Broad concepts and widely applicable methods are also emphasized. The focus of the course is neither manipulation nor memorization of an extensive taxonomy of functions, curves, theorems, or problem types, but rather, the course uses the unifying themes of derivatives, integrals, limits, approximation, and applications and modeling to become a cohesive whole. The course is a yearlong high school mathematics course designed to prepare you to write and pass the AP Calculus AB test in May. Passing the test can result in one semester of college credit in mathematics.
Unit 1: Functions, Graphs, and Limits
This unit begins with a review of functions that should have been learned in a previous course. Specifically, we will examine the relationship between formulas and graphs of functions, as well as general properties of their graphs. After completing the first part of this unit, you should feel comfortable explaining general patterns that occur in certain types of functions, as well as be able to give specific information about a graph given its formula. After studying functions, we will examine the first "big" topic in calculus, the limit. As you will see, much of calculus is based on this concept, so it is important to have a good understanding of its fundamental meaning now. We will study limits as they relate to graphs and formulas, and we will use formal limit notation to explain certain behaviors of graphs that you have studied before (such as continuity), but without the concept of limits. This is a very exciting unit and it is important to understand limits well before moving on to the next part of the course.
Completing this unit should take you approximately 15 hours.
Unit 2: Derivatives
In this unit, you will learn first the concept of a derivative by explaining it through limits. You will see that the limit definition and computation are drawn out, and often difficult, so you will learn shortcuts to compute derivatives. You will learn to compute derivatives for all different types of functions, as well as for combinations of functions. After you learn how to compute derivatives, you will look at what derivatives tell you specifically at any point in a function, as well as what they tell you about the function as a whole. You will finish this unit by looking at the computation and meaning of second derivatives, as well as real-life applications of derivatives. The main applications to be reviewed in this unit will be related rates problems, optimization problems, and problems involving displacement, velocity, and acceleration.
Completing this unit should take you approximately 54 hours.
Unit 3: Integrals
Integrals are the final unit in this course. In a similar way to derivatives, you will approach integrals first by learning their formal definition, then by learning techniques by which to compute them, and then lastly learning about their applications. In this unit, you will also learn the FTC (fundamental theorem of calculus), learn to solve basic differential equations, and also learn some cool approximation techniques. One of the reasons this unit is really interesting is because of all the cool images that can be generated using integrals. You will see what this means very soon!
Completing this unit should take you approximately 65 hours.
