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.
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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, chain rule, and implicit differentiation. 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.
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, chain rule, and implicit differentiation. 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.
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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.
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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.
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Chapter 5 Transcendental Functions
Transcendental functions include logarithms, exponential, and inverse function. The properties of each type of function are discussed in this chapter in addition to finding the derivative and integral for each type of function. Inverse functions include polynomial and rational function as well as inverse trigonometric functions.
Transcendental functions include logarithms, exponential, and inverse function. The properties of each type of function are discussed in this chapter in addition to finding the derivative and integral for each type of function. Inverse functions include polynomial and rational function as well as inverse trigonometric functions.
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Chapter 6 Differential Equations
Chapter 6 delves into the study of differential equations. Differential equations can be solved multiple ways including analytically, graphically, and by approximating the solution numerically. Each method is examined through the use of separation of variables, slope fields, and Euler's Method, respectively. Additional topics of study include exponential growth and decay and logistic growth for populations with limiting factors.
Chapter 6 delves into the study of differential equations. Differential equations can be solved multiple ways including analytically, graphically, and by approximating the solution numerically. Each method is examined through the use of separation of variables, slope fields, and Euler's Method, respectively. Additional topics of study include exponential growth and decay and logistic growth for populations with limiting factors.
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Chapter 7 Area and Volume
Chapter 7 introduces more applications of the integral; namely finding the area between two curves and finding the volume of a threedimensional figure. Volume of a threedimensional figure is found by using crosssections and rotation. Additionally, volume by shells is included although it is not found on the AP test. Also not found on the AP test but found in this unit is the surface area of a curved plane. Including surface area provides the opportunity to inspect interesting characteristics of Gabriel's Horn. Unique to the BC curriculum is length of a curve.
Chapter 7 introduces more applications of the integral; namely finding the area between two curves and finding the volume of a threedimensional figure. Volume of a threedimensional figure is found by using crosssections and rotation. Additionally, volume by shells is included although it is not found on the AP test. Also not found on the AP test but found in this unit is the surface area of a curved plane. Including surface area provides the opportunity to inspect interesting characteristics of Gabriel's Horn. Unique to the BC curriculum is length of a curve.
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Review of AP Calculus AB Material
We now pause at the conclusion of the Calculus 1 material to review the concepts that are found on the AP Calculus AB Examination. Not every topic will be hit, but the majority of them will be covered. The review will consist of 3 days separated by concepts. The first day will cover limits, derivatives, and applications of derivatives. The second day focuses on integrals and the fundamental theorems of calculus. The final day concludes with transcendental functions and area/volume. My students will take a fulllength AB test at the conclusion of the review to provide a capstone for the first calculus course and a checkpoint of their progression.
We now pause at the conclusion of the Calculus 1 material to review the concepts that are found on the AP Calculus AB Examination. Not every topic will be hit, but the majority of them will be covered. The review will consist of 3 days separated by concepts. The first day will cover limits, derivatives, and applications of derivatives. The second day focuses on integrals and the fundamental theorems of calculus. The final day concludes with transcendental functions and area/volume. My students will take a fulllength AB test at the conclusion of the review to provide a capstone for the first calculus course and a checkpoint of their progression.
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Chapter 8 Advanced Integration Techniques
Chapter 8 takes us into the advanced integration techniques. This chapter is almost entirely exclusive to the BC classroom. The lone exception being L'Hopital's Rule for evaluating indeterminate limits. The advanced integration techniques include: integration by parts, partial fractions, integrals with infinite limits of integration, and integrals with vertical asymptotes. Partial fractions on the AP only includes nonrepeating linear factors. We cover the other types in class regardless.
Chapter 8 takes us into the advanced integration techniques. This chapter is almost entirely exclusive to the BC classroom. The lone exception being L'Hopital's Rule for evaluating indeterminate limits. The advanced integration techniques include: integration by parts, partial fractions, integrals with infinite limits of integration, and integrals with vertical asymptotes. Partial fractions on the AP only includes nonrepeating linear factors. We cover the other types in class regardless.
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Chapter 9 Infinite Series
Chapter 9 opens with information about sequences. The remainder of the chapter details series: the summation of the infinite terms of a sequence. The first dealings with series involves convergence tests to determine whether a series will converge or diverge. These tests will be important for checking the boundaries of an interval of convergence for a power series. Power series, Taylor series, and Maclaurin series follow the convergence tests. These series, in conjunction with power series manipulation and Lagrange Error Bound, allow mathematicians to approximate definite integrals of nonintegrable functions with an acceptable amount of error.
Chapter 9 opens with information about sequences. The remainder of the chapter details series: the summation of the infinite terms of a sequence. The first dealings with series involves convergence tests to determine whether a series will converge or diverge. These tests will be important for checking the boundaries of an interval of convergence for a power series. Power series, Taylor series, and Maclaurin series follow the convergence tests. These series, in conjunction with power series manipulation and Lagrange Error Bound, allow mathematicians to approximate definite integrals of nonintegrable functions with an acceptable amount of error.
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Chapter 10 Parametrics, Vectors, and Polar
The final chapter looks at different types of functions where calculus can be applied: parametric equations, vectors, and polar equations. Keep calculus concepts of derivatives, integrals, and area are reviewed in a new light with these functions.
The final chapter looks at different types of functions where calculus can be applied: parametric equations, vectors, and polar equations. Keep calculus concepts of derivatives, integrals, and area are reviewed in a new light with these functions.
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