Chapter 1 Limits and Their Properties
This first chapter involves the fundamental calculus elements of limits. While limits are not typically found on the AP test, they are essential in developing and understanding the major concepts of calculus: derivatives & integrals. These notes cover the properties of limits including: how to evaluate limits numerically, algebraically, and graphically. An important characteristic of functions, continuity, is also discussed in greater detail than in previous math classes. This chapter also contains two major theorems: The Intermediate Value Theorem (IVT) and the Squeeze Theorem. While neither are prominent on the AP test, the IVT has applications with derivative tests found in the third chapter. Lastly, the idea of infinity is discussed in greater detail.
This first chapter involves the fundamental calculus elements of limits. While limits are not typically found on the AP test, they are essential in developing and understanding the major concepts of calculus: derivatives & integrals. These notes cover the properties of limits including: how to evaluate limits numerically, algebraically, and graphically. An important characteristic of functions, continuity, is also discussed in greater detail than in previous math classes. This chapter also contains two major theorems: The Intermediate Value Theorem (IVT) and the Squeeze Theorem. While neither are prominent on the AP test, the IVT has applications with derivative tests found in the third chapter. Lastly, the idea of infinity is discussed in greater detail.
Chapter 2 Derivatives
The second chapter concerns derivatives. The beginning lesson establishes the meaning of a derivative and how it is developed from limits. The limit definition of a derivative is almost always found as a multiple choice question on the AP test. Mechanically finding the derivative of a multitude of functions follows once the definition of a derivative is understood. Derivative rules include: power rule, product rule, higher order derivatives (2nd derivative, 3rd derivative, etc.), quotient rule, and chain rule. Together the derivative rules cover how to find derivatives for all types of mathematical operations: addition, subtraction, multiplication, division, and composition.
The second chapter concerns derivatives. The beginning lesson establishes the meaning of a derivative and how it is developed from limits. The limit definition of a derivative is almost always found as a multiple choice question on the AP test. Mechanically finding the derivative of a multitude of functions follows once the definition of a derivative is understood. Derivative rules include: power rule, product rule, higher order derivatives (2nd derivative, 3rd derivative, etc.), quotient rule, and chain rule. Together the derivative rules cover how to find derivatives for all types of mathematical operations: addition, subtraction, multiplication, division, and composition.
Incorporated into this chapter are some applications of derivatives: position/velocity/acceleration, tangent lines, and related rates. Related rates can be a challenging section, so I devote several days to it in class. Related rates are also a favorite of the AP writers. Expect to see one in the multiple choice and one in the free response every couple of years. Additionally, I have included the lyrics to a song about the quotient rule that I sing to my students. Encores are requested each year without fail, but more importantly help them remember how to find the derivative of a quotient.
Chapter 3 Applications of Derivatives
This third chapter gives applications of derivatives ranging from the shape of a graph to finding the maximum or minimum values that would optimize a given scenario. The major theorem for this chapter is the Mean Value Theorem for Derivatives; make sure the initial conditions are met before you can apply the theorem. The 1st Derivative Test also plays a major role in this chapter and beyond.
This third chapter gives applications of derivatives ranging from the shape of a graph to finding the maximum or minimum values that would optimize a given scenario. The major theorem for this chapter is the Mean Value Theorem for Derivatives; make sure the initial conditions are met before you can apply the theorem. The 1st Derivative Test also plays a major role in this chapter and beyond.
Chapter 4 Antiderivatives and Integration
Chapter 4 introduces the next big concept: integration. Integration is the process of finding the antiderivative of a function. There are many applications of integrals that are studied in this section. Three big theorems are found in this chapter: 1st Fundamental Theorem of Calculus, 2nd Fundamental Theorem of Calculus, and the Mean Value Theorem for Integrals. The 1st Fundamental Theorem of Calculus is an extremely important theorem that allows us to find the area under a curve over an interval. Riemann Sums are also part of chapter 4 and are included to find the area under a curve before the introduction of the 1st Fundamental Theorem of Calculus.
Chapter 4 introduces the next big concept: integration. Integration is the process of finding the antiderivative of a function. There are many applications of integrals that are studied in this section. Three big theorems are found in this chapter: 1st Fundamental Theorem of Calculus, 2nd Fundamental Theorem of Calculus, and the Mean Value Theorem for Integrals. The 1st Fundamental Theorem of Calculus is an extremely important theorem that allows us to find the area under a curve over an interval. Riemann Sums are also part of chapter 4 and are included to find the area under a curve before the introduction of the 1st Fundamental Theorem of Calculus.