Syllabus – AP Calculus BC

Course overview:

Calculus requires students to expand on many of the concepts that they have learned from previous math courses. The relevance of these concepts is discovered through real world application problems where science and math come together to explain the wonders of our physical world. I want students to have a deeper understanding of how these concepts are applied. That means students will need to stretch beyond number crunching and symbol manipulation. Topics addressed in class include Limits, Continuity, Derivative Mechanics, Derivative Applications, Integration Mechanics, Integration Applications, Polynomial Approximations, Polynomial Series, and other related topics. Students are required to use precise language to describe these concepts and the relationships between ideas. [C2]

Expectations:

Students are required to be active participants in the learning process. In order for students to be successful in AP Calculus they should expect to work and study hard each day.

• Students can expect to take notes on a daily basis.
• Ahomework assignment will be given every class period.
• Students can expect a daily quiz to check for understanding of homework.
• Each unit will be tested with both multiple choice and short answer response questions.
• It is the students’ responsibility to seek extra help when they do not understand a concept or skill. I suggest doing so immediately; do not wait until after the test to seek help. I am available before school from 8:15 to 8:50 and after school from 3:50 to 4:15.
• I also suggest that students form study groups so as questions arise after school hours they have a support network.

Functions from Multiple Representations

My goal is to show students the concepts of calculusare expressed in multiple ways. Many real life situations use functions that can be expressed analytically, verbally, graphically or numerically. We will work problems that represent all four of these representations throughout the year. Students will be able to move with ease from one representation to another. [C3]

Justification of answers

Students are required to justify their answers on homework, quizzes and tests. Justifications should be in the form of complete sentences. We will discuss the amount of work that it is necessary to show and the correct way to justify your answer in class.[C4]

Technology

Students will be using a graphing calculator regularly during class. In addition, students can bring their own device to school. Students are also encouraged to find apps for the iPad (or other device) that are useful for explaining the concepts of Calculus. Although being in class for the lesson is preferred, sometimes absences are unavoidable. The iPad apps can be very useful to explain lessons that were missed or provideextra instruction as needed.

Graphing Calculator

It is highly recommended that students have their own graphing calculator. Students will be required to bring your own calculator to the AP Exam. We have a class set of TI–84 plus calculators that students can use during class, but homework assignments often require the use of a graphing calculator. If you wish to purchase a calculator, I recommendthe TI-84 or a TI-83. Do not buy a calculator with a full QUARTY keyboard as they are not allowed on the AP Exam.

Students are expected to know basic calculator skills and be able to graph a function using an appropriate window. During the review of elementary functions, we will explore how to use the calculator to produce an equation using the regression equations from the STAT menu. The TABLE feature will initiate our classroom discussions of the limit of a function. Many calculator dependent solutions are based on manipulating graphs or understanding how several graphs are related to each other. We will practice using the calculator to interpret results that are solved analytically. When a function is expressed graphically, we can either describe its behavior in broad terms or provide details of the function at specific points. When we use technology to graph and describe an analytical function, we need to be sure that some of the behaviors are not being masked.[C5]

Assessments:

Students’ grades are based on the traditional methods of assessment such as tests, homework and quizzes. Homework, quizzes and other daily classroom assignments will count as 30% of their average, exams 70%. Exams not only address the concepts of the given unit, but they also are designed to prepare students to know how to express their thinking in preparation for the AP exam. I will include both AP style multiple choice and short answer questions on all exams throughout the year. Students are required to answer some questions with a calculator and some without. All questions will not be given full credit unless all the work is shown and answers are written in a complete sentence. Students will also be assessed on their level of participation during class. We will take turns presenting answers and being class leaders for the day. Also, due to time constraints, and the necessity to keep all of the material fresh in our minds for the AP Exam, there will be cumulative take home tests (in addition to the regular tests taken during class). Students areencouraged to talk with each other, but expected to produce their own work; again credit is notawarded to answers that have not been properly justified.

Review for AP Exam:

My goal is to leave some time before the exam to spend reviewing for the AP exam. We will go through AP free response questions together in class to get an understanding of how the AP exam is scored. Students will take mini tests in class with questions from released tests. Some questions allow the use of a calculator and some do not. These mini tests will have the same time constraints as the AP exam. We will then use the scoring guide to score the test. Also, I will hold after school review sessions once a week during the month of April for any student that wishes to participate. I also recommend students purchase an AP test prep book of some sort (new or used) to help in preparation for the AP Exam.

Primary Textbook:

• Anton, Howard, IrlBivens, and Stephen Davis. Calculus: Early Transcendentals. 8th Edition. John Wiley and Sons, 2005.

Supplemental Textbooks:

• Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus with Analytic Geometry. 8th Edition. Boston: Houghton Mifflin, 2006
• Stewart, James. Single Variable Calculus: Concepts and Contexts. 4thedition. Brooks/Cole, Belmont: Cengage Learning, 2010, 2005

Web/Other Resources:

• Published AP Tests
• AP Central (apcentral.collegeboard.com)
• Teacher created materials
• AP Calculus Electronic Discussion Group

Contents of the Course

The following course contents are approximate; the number of days per unit may have to be adjusted due to school wide testing days, weather conditions and other unforeseen events. [C2]

Unit 1 – Review, Limits & Continuity (appr. 8 days) – 1 Test

1.1-1.7 Selected Topics

2.1Limits (An Intuitive Approach)

2.2Computing Limits

2.3Limits at Infinity; End Behavior of a Function

2.4Limits (Discussed More Rigorously)

2.5Continuity

Unit 2 – Derivative Mechanics (appr. 12 days) – 2 Tests

3.1Tangent Lines, Velocity, and General Rates of Change

3.2The Derivative Function

3.3Techniques of Differentiation

3.4The Product Rule and Quotient Rules

3.5Derivatives of Trigonometric Functions

3.6The Chain Rule

4.1Implicit Differentiation

4.2Derivatives of Logarithmic Functions

4.3Derivatives of Exponential and Inverse Trigonometric Functions

4.4L’Hopital’s Rule; Indeterminate Forms

Unit 3 – Derivative Applications (appr. 12 days) – 1 Test (2 days)

5.1Analysis of Functions I: Increase, Decrease, and Concavity

5.2Analysis of Functions II: Relative Extrema; Graphing Polynomials

5.3More on Curve Sketching: Rational Functions; Curves with Cusps and Vertical Tangent Lines

5.4Absolute Maxima and Minima

5.5Applied Maximum and Minimum Problems

4.7Related Rates

4.8Local Linear Approximation; Differentials

5.7Rolle’s Theorem; Mean-Value Theorem

5.8Rectilinear Motion

Limits, Continuity & Derivatives Final – 2 day Test

Semester Final – AP style Multiple Choice Questions

Unit 4 – Integration Mechanics (appr. 9 days) – 2 Tests (1 Test overlaps Unit 4 & 5)

(We actually start this unit during the first semester, between the Derivatives Final and Semester Final)

6.1An Overview of the Area Problem

6.4The Definition of Area as a Limit; Sigma Notation

6.5The Definite Integral

6.2The Indefinite Integral

6.6The Fundamental Theorem of Calculus

6.3Integration by Substitution

6.7Rectilinear Motion Revisited Using Integration

6.8Evaluating Definite Integrals by Substitution

6.9Logarithmic Functions from the Integral Point of view

Unit 5 – Integration Applications (appr. 8 days) – 1 Test

7.1Area Between Two Curves

7.2Volumes by Slicing; Disks and Washers

7.3Volumes by Cylindrical Shells

7.4Length of a Plane Curve

7.5Area of a Surface of Revolution

7.6Average Value of a Function and its Applications

7.7Work

Unit 6 – Integration Techniques & Topics (appr. 6 days) – 1 Test

8.1An Overview of Integration Methods

8.2Integration by Parts

8.5Integrating Rational Functions by Partial Fractions

8.6Using Computer Algebra Systems and Tables of Integrals

8.7Numerical Integration; Simpson’s Rule

8.8Improper Integrals

Unit 7 – Series (appr. 10 days) – 1 Test

10.1Sequences

10.2Monotone Sequences

10.3Infinite Series

10.4Convergence Tests

10.5The Comparison, Ration, and Root Tests

10.6Alternating Series; Conditional Convergence

10.7Maclaurin and Taylor Polynomials

10.8Maclaurin and Taylor Series; Power Series

10.9Convergence of Taylor Series

10.10Differentiating and integrating Power Series; Modeling with Taylor Series

Unit 8 – Topics in Calculus (appr. 7 days) – 1 Test

9.1First-Order Differential Equations and Applications

9.2Slope Fields; Euler’s Method

9.3Modeling with First-Order Differential Equations

11.1Polar Coordinates

11.2Tangent Lines and Arc Length for Parametric and Polar Curves

11.3Area in Polar Coordinates

AP Exam review is incorporated throughout the year in course work and activities,

as well as any time left after Unit 8.

[C2] – The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals; and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description.

[C3] – The course provides students with the opportunity to work with functions represented in a variety of ways – graphically, numerically, analytically, and verbally – and emphasizes the connections among these representations.

[C4] – The course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.

[C5] – The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions.