The following are the blank copies of the notes and assignments that are given out in the AP Calculus AB/BC classroom at Belton High School.
Chapter section numbers and homework assignments are from:
Larson, Hostetler, Edwards (2006). Calculus of a Single Variable (8th Ed.). Boston, MA: Houghton Mifflin Company
*****You can find the problems for the homework at the following site:
http://college.cengage.com/mathematics/blackboard/content/larson/calc8e/calc8e_solution_main.html?CH=00&SECT=a&TYPE=se
Here is a file that contains an emulator for the TI-83 and TI-89 calculator that you can download.
Calculator Emulator
Instructions for emulator:
Click on link and download files
Right click on downloaded folder and select "extract all ..."
Open the new folder and find the icon with a computer on a desk that is named "vti"
In order to change between the TI-83 and TI-89, press F12 and select the appropriate tab.
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. The Delta-Epsilon definition of a limit can be discussed in this chapter. I choose to exclude it because it is not part of the AP curriculum.
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. The Delta-Epsilon definition of a limit can be discussed in this chapter. I choose to exclude it because it is not part of the AP curriculum.
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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.
Chapter 3 Applications of Differentiation
The third chapter include further applications of derivatives. Relative and absolute extrema, attributes of functions based on the first and second derivative, the Mean Value Theorem (MVT), the 1st and 2nd derivative tests, and optimization are found here. Linearization, differentials, and Newton's Method for finding zeroes are extension lessons for teacher with additional time. I will often skip them because I cover both AB and BC topics in the same school year.
The third chapter include further applications of derivatives. Relative and absolute extrema, attributes of functions based on the first and second derivative, the Mean Value Theorem (MVT), the 1st and 2nd derivative tests, and optimization are found here. Linearization, differentials, and Newton's Method for finding zeroes are extension lessons for teacher with additional time. I will often skip them because I cover both AB and BC topics in the same school year.
Chapter 4 Integration
Chapter 4 is another big chapter: integration. The chapter begins with the idea of an antiderivative and indefinite integrals. Finding the area under a curve using Riemann sums, midpoint sums, trapezoidal sums, and geometric shapes follows. Then, antiderivatives and the area under the curve are connect using definite integrals and the 1st Fundamental Theorem of Calculus (or just THE Fundamental Theorem of Calculus). Connecting area and accumulating area with the integral is a very important topic in a calculus course. The chapter continues with the 2nd Fundamental Theorem of Calculus and the Mean Value Theorem for integrals. The antiderivatives at the beginning of the chapter were for the power rule and trig functions. This chapter closes with u-substitution, the technique used to find the antiderivative of composite functions.
Chapter 4 is another big chapter: integration. The chapter begins with the idea of an antiderivative and indefinite integrals. Finding the area under a curve using Riemann sums, midpoint sums, trapezoidal sums, and geometric shapes follows. Then, antiderivatives and the area under the curve are connect using definite integrals and the 1st Fundamental Theorem of Calculus (or just THE Fundamental Theorem of Calculus). Connecting area and accumulating area with the integral is a very important topic in a calculus course. The chapter continues with the 2nd Fundamental Theorem of Calculus and the Mean Value Theorem for integrals. The antiderivatives at the beginning of the chapter were for the power rule and trig functions. This chapter closes with u-substitution, the technique used to find the antiderivative of composite functions.
Solving Differential Equations II 4.1 |
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Riemann Sums Day I |
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Riemann Sums Day II (Midpoint/Trapezoids) |
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AP Questions 4.1-4.3 |
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Application of 1st Fundamental and Properties of Integrals |
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AP Questions Ch. 4 Free Response |
Chapter 5 Transcendental Functions
Chapter 5 is the first foray into transcendental functions: logarithms, exponential functions, inverse functions, and inverse trig. There is a lot of memorization in this chapter; it cannot be helped. There are so many derivative and integration rules/formulas that must be memorized. The unit on compound interest is an extension for derivatives of exponential functions that I have skipped in recent years due to time restraints.
Chapter 5 is the first foray into transcendental functions: logarithms, exponential functions, inverse functions, and inverse trig. There is a lot of memorization in this chapter; it cannot be helped. There are so many derivative and integration rules/formulas that must be memorized. The unit on compound interest is an extension for derivatives of exponential functions that I have skipped in recent years due to time restraints.
Inverse Function Derivatives |
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Logarithmic Differentiation |
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Chapter 6 Differential Equations
Chapter 6 is the first chapter where we start to see the split between Calculus AB and Calculus BC. Calculus BC covers all the same material as AB in this chapter with the addition of Euler's Method and logistic growth. Separation of variables is almost always found in the free response of the AP test because of the way the free response questions on the Calculus BC AP test are written (3 shared exactly with AB, 2 unique to BC, 1 split between AB and BC concepts).
Chapter 6 is the first chapter where we start to see the split between Calculus AB and Calculus BC. Calculus BC covers all the same material as AB in this chapter with the addition of Euler's Method and logistic growth. Separation of variables is almost always found in the free response of the AP test because of the way the free response questions on the Calculus BC AP test are written (3 shared exactly with AB, 2 unique to BC, 1 split between AB and BC concepts).
AP Questions Ch. 6 |
Chapter 7 Applications of Integration
Chapter 7 is the final chapter for the Calculus AB course. The BC course continues on past this chapter and includes arc length while AB does not. Volume by cross section is an extension of volume not covered on the AP test. Surface area of curves can also be found in the unit (but not on the AP test). Be careful with volume by washers as students will often have difficulty setting up the integrals when the shaded region or axis of rotation are moved around.
Chapter 7 is the final chapter for the Calculus AB course. The BC course continues on past this chapter and includes arc length while AB does not. Volume by cross section is an extension of volume not covered on the AP test. Surface area of curves can also be found in the unit (but not on the AP test). Be careful with volume by washers as students will often have difficulty setting up the integrals when the shaded region or axis of rotation are moved around.
Arc Length 7.4 |
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Calculus 1 Review
These review days are something that I sometimes will cover with my class and sometimes will not. It depends on the students that I have and the amount of time left in the school year. I like to give my students are full-length practice AP exam during class once chapter 7 is completed. It takes 4 days to take in class and the 5th day is used to grade the free response. My students and I find the mock exam to be a valuable exercise whether a review is given beforehand or not.
These review days are something that I sometimes will cover with my class and sometimes will not. It depends on the students that I have and the amount of time left in the school year. I like to give my students are full-length practice AP exam during class once chapter 7 is completed. It takes 4 days to take in class and the 5th day is used to grade the free response. My students and I find the mock exam to be a valuable exercise whether a review is given beforehand or not.
AP Multiple Choice Questions |
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AP Free Response |
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Chapter 8 Integration Techniques
Now we begin the topics (almost) completely unique to the Calculus BC classroom. These advanced integration techniques finally allow us to find an antiderivative that matches up with the product rule for derivatives. Only non-repeating partial fractions are covered on the AP exam, so you may choose to skip the quadratic factors. The only AB topic found in this chapter is L'Hopital's Rule for evaluating indeterminate limits. It was formerly just a BC topic, but was added to the AB curriculum with the latest update to AP Calculus framework.
Now we begin the topics (almost) completely unique to the Calculus BC classroom. These advanced integration techniques finally allow us to find an antiderivative that matches up with the product rule for derivatives. Only non-repeating partial fractions are covered on the AP exam, so you may choose to skip the quadratic factors. The only AB topic found in this chapter is L'Hopital's Rule for evaluating indeterminate limits. It was formerly just a BC topic, but was added to the AB curriculum with the latest update to AP Calculus framework.
Trig Substitution 8.4 |
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AP Questions Ch. 8 |
Chapter 9 Infinite Series
Chapter 9: "And now for something completely different." The final big topic for Calculus BC (Limits, Derivatives, Integrals, The Fundamental Theorem of Calculus) is Series. The chapter on series is extremely long. It takes about a month to cover the whole chapter. I split the chapter into two parts: infinite series including convergence/divergence tests, and power series. My students often ask why the first part of the chapter (convergence/divergence) is covered. I tell them we are working toward answering a specific question: How do you numerically find the definite integral for a function that has no integral (sin(2x))? These convergence/divergence tests, if asked on the AP test, will be found in the multiple choice section and may be required for justifying an interval of convergence in the free response. If there is a multiple choice question, it will likely be of the roman numeral format.
Chapter 9: "And now for something completely different." The final big topic for Calculus BC (Limits, Derivatives, Integrals, The Fundamental Theorem of Calculus) is Series. The chapter on series is extremely long. It takes about a month to cover the whole chapter. I split the chapter into two parts: infinite series including convergence/divergence tests, and power series. My students often ask why the first part of the chapter (convergence/divergence) is covered. I tell them we are working toward answering a specific question: How do you numerically find the definite integral for a function that has no integral (sin(2x))? These convergence/divergence tests, if asked on the AP test, will be found in the multiple choice section and may be required for justifying an interval of convergence in the free response. If there is a multiple choice question, it will likely be of the roman numeral format.
Chapter 9 Power, Taylor & Maclaurin Series
The second portion of chapter 9 concerns power series. The schedule I now follow covers Taylor/Maclaurin first and then goes back to cover general power series and geometric power series. The 6th and final free response question of the AP test always covers power series in some capacity. Typically that involves finding an interval of convergence and determining the error bound for an approximation of n terms.
The second portion of chapter 9 concerns power series. The schedule I now follow covers Taylor/Maclaurin first and then goes back to cover general power series and geometric power series. The 6th and final free response question of the AP test always covers power series in some capacity. Typically that involves finding an interval of convergence and determining the error bound for an approximation of n terms.
Taylor/Maclaurin Series Part II 9.10 |
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AP Questions Day 1 |
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AP Questions Day 2 |
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AP Questions Day 3 |
Chapter 10 Polar & Parametrics
The final chapter returns to using and applying derivatives and integrals. Most of this chapter is pretty straightforward until the polar portion. Finding the areas of complex polar regions can be, well, complex. Be sure your students are familiar with converting between polar and rectangular coordinates as that will usually be part of free response question that is easy to forget how to do but very easy to get.
The final chapter returns to using and applying derivatives and integrals. Most of this chapter is pretty straightforward until the polar portion. Finding the areas of complex polar regions can be, well, complex. Be sure your students are familiar with converting between polar and rectangular coordinates as that will usually be part of free response question that is easy to forget how to do but very easy to get.
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AP Questions |
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AP Questions #2 |
AP Material is in the AP Test Review Section