AP CALCULUS AB SYLLABUS COURSE OVERVIEW Course Description We start the year with an intensive review of precalculus material. Since our textbook does not include trigonometry in this review, I supplement this material. Then we proceed with the calculus curriculum. Throughout the year, for all chapters, the rule of four is emphasized. Students find that their daily lessons, homework assignments, tests, and exam preparation include problems that they must be able to consider and solve graphically, numerically, analytically, or verbally. Students learn the importance of understanding all aspects of a concept and not relying too heavily on just one rule. Analysis of graphs (on the calculators, in the text, drawn by hand, etc.) helps the students acquire good visual comprehension of what they are learning. For instance, they compare and contrast graphs of equations (such as functions with their first and second derivatives), analyze limits, identify slopes of tangent lines, find relative extrema and points of inflections, and determine the areas of regions between functions. Students work problems both numerically and analytically, and I find that students need to understand the distinctions between the two. They learn that one method does not replace the other, but offers additional justification for a solution. Many exercises in the textbook reinforce analytical solving of problems, but I make sure to include problems that emphasize the other three rules. Students learn to use table capability of the calculators to assist with numerical approaches to problems and supporting conclusions. For verbal explanations, instead of only having students provide written explanations and justifications, a common practice during daily lessons in my class is discussion time where students discuss, explain, and justify theorems, problems, graphs and concepts to each other. Limits is one of several topics in calculus that lends itself nicely to a rule of four approach to learning. We start by considering limits visually, analyzing graphs of functions (on the graphing calculators) to determine what a limit is (starting with the informal definition of a limit) and how to find it as x approaches a particular number or positive/negative infinity. Then we look at a table of values (also on the calculators) to verify this numerically. After this, the formal definition of a limit is introduced. Analytical techniques for finding limits follow this as students learn numerous ways to calculate a limit when direct substitution is not an option. Students are given discussion questions as well as homework problems that require verbal/written explanation, such as
In your own words, describe three situations that would create nonexistence of a limit, or Find the limit of f(x) as x increases without bound and explain how this relates to the asymptote. Classroom Routine Students are taught a daily lesson, which includes notes, concept exploration, and discussion. With each lesson, homework exercises are assigned, some of which require a graphing calculator. When students return the next day, the first part of the period is reserved for giving students time to look at solutions to the homework, compare their solutions with other students, or ask me for assistance or explanation. We have one or two tests for every unit, a comprehensive final exam at the end of first semester, and several practice tests in the spring during the month before the AP exam. Calculator Use Graphing calculators are required for this course. Keeping with the AP exam format, I allow the students to use calculators for some problems, and not others. I have the students use their calculators for discovery learning, analysis of graphs (functions, limits, derivates, integrals, area, etc.), problem solving, data interpretation, and checking solutions. In addition to this, students are required to know how to use the calculator to find zeros and intersection points, and how to evaluate a derivative at a point and how to evaluate a definite integral. Assessment Students are given daily homework assignments correlating with the day s lesson. Quizzes are periodically used for both formative and summative evaluations. One to two tests are given for every unit. Many test questions are completion style, where the students are required to show the work that justifies their answer. Written explanation is also required for some problems. Other types of test items include short discussion questions, and multiple choice questions. Free response questions from previous AP exams are included on some tests. A comprehensive final is given at the end of first semester. A released AP exam is given for a test grade at the end of second semester right before the students take the AP exam. COURSE CURRICULUM Unit 1: Prerequisite Material Review 1) Trigonometry Review 2 weeks
Reciprocal Identities, Quotient and Pythagorean Identities Special Right Triangles Trig. Table for π/6, π/4, π/3, π/2 and Their Multiples Degree and Radian Circles Trig. Function Graphs 2) Graphs and Models 3) Linear Models and Rates of Change 4) Functions and Their Graphs 5) Fitting Models to Data Unit 2: Limits 2 weeks 1) Introduction to Limits The Tangent Line Problem Informal Definition of a Limit 2) Formal Definition of a Limit 3) Evaluating Limits Analytically Properties of Limits Simplifying Techniques The Squeeze Theorem Special Limits 4) Continuity Definition Removable and Nonremovable Discontinuities Intermediate Value Theorem 5) One Sided Limits 6) Infinite Limits and Their Properties Unit 3: Differentiation 4 weeks 1) The Tangent Line and The Derivative Finding the derivative using the definition (limit) Local linearity 2) Power Rule, Constant Multiple Rule, Sum/Difference Rule 3) Derivatives of Sine and Cosine Functions 4) Rates of Change Average Velocity Instantaneous Velocity 5) Product Rule, Quotient Rule 6) Derivatives of Trig. Functions and Higher Order Derivatives
7) Chain Rule 8) Implicit Differentiation 9) Related Rates Unit 4: Applications of Differentiation 5 weeks 1) Extrema On An Interval 2) Rolle s Theorem, Mean Value Theorem 3) Increasing/Decreasing Functions and the First Derivative Test 4) Concavity and the Second Derivative Test 5) Limits At Infinity 6) Curve Sketching for Rational, Radical, Polynomial, and Trig. Functions 7) Optimization Problems 8) Differentials Unit 5: Integration 5 weeks 1) Antiderivatives and Indefinite Integrals 2) Integrating Polynomial Functions 3) Area Under a Curve, Visual Exploration 4) Riemann Sums Upper and Lower Sums Left and Right Hand Sums Midpoint Sums Subintervals of Unequal Widths 5) Definite Integrals and Limits of Sums, Sigma Notation 6) Fundamental Theorem of Calculus Conceptual Understanding (Graphical, Analytical) Operational Procedures Applications 7) Mean Value Theorem for Integrals, Average Value of a Function on an Interval 8) Second Fundamental Theorem of Calculus 9) Introduction to Slope Fields 10) Integration By Substitution, Change of Variables, Change of Limits 11) Solving Differential Equations 12) Numerical Integration Trapezoidal Rule Simpson s Rule Unit 6: Logarithm, Exponential, and 5 weeks
Other Transcendental Functions 1) Definition of Natural Logarithm Function and e 2) Properties of the Natural Logarithm Function 3) Natural Log Functions 4) Inverse Functions 5) Exponential Functions 6) Bases other than e Definitions Rules/Operations Applications 7) Inverse Trigonometric Functions Graphs and Properties Unit 7: Differential Equations, Area, Volume, and L Hopital s Rule 4 Weeks 1) Differential Equations (General and Particular Solutions) 2) Slope Fields 3) Compound Interest 4) Separation of Variables for Solving Differential Equations 5) Differential Equations, Growth and Decay Models 6) Area of Regions Between Two Curves 7) Volume of Solids of Revolution Disk Method Washer Method Shell Method 8) Volume of Solids With Known Cross Sections 9) Indeterminate Forms and L Hopital s Rule Unit 8: Course Review 4 Weeks
SPECIAL PROJECTS 1) Balloon Inflation Experiment Related Rates This experiment allows students to experience a hands on demonstration of a related rates problem. Students seek to answer the following question: If the rate of change of the volume of a balloon increases at a constant rate, will the rate of change of the radius of their balloon also increase at a constant rate? Using consistent breaths, students record (through observation and calculations on graphing calculators): breath number, balloon circumference, radius, and increase in radius. As they interpret their results they find that the answer is no. Work they have done on the graphing calculator supports their conclusion and they also justify this by using calculus (determining dr/dt at different times). They are able to demonstrate how the rate of change of the radius varies as the rate of change of the volume remains constant. 2) Volumes of Solids With Known Cross Sections Accompanied By 3D Model Students are given the task of finding the volume of the solid whose base is bounded by two equations and a choice of four different cross sections. Then students actually construct a model to show what this 3D figure looks like. Students choose their own construction materials for the framework and skin. (To make the project reasonable the model only has to look like a solid the interior may be hollow). Size guidelines are given to the students. The computations are displayed with their model and a graphing calculator is permitted for definite integral calculations. Results for this project are quite impressive, both in creativity and accuracy (of the model and the calculus required for computing the volume). REFERENCES Textbook: Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus with Analytic Geometry, eighth edition. Boston: Houghton Mifflin, 2006. Support Materials: Cade, Sharon, Rhea Caldwell, and Jeff Lucia. Fast Track to a 5: Preparing for the AP Calculus AB and Calculus BC Examinations. Boston: Houghton Mifflin, 2006.
Lederman, David. Multiple Choice and Free Response Questions in Preparation for the AP Calculus (AB) Examination, eighth edition. Brooklyn, NY: D&S Marketing Systems, Inc., 2003. Santulli, Thomas V. A Deeper Look at Related Rates in Calculus. Mathematics Teacher. Vol. 100, No. 2, September 2006. Advanced Placement Program Professional Development books: 2003 2004 AP Calculus AB and Calculus BC, Special Topic: Differential Equations 2004 2005 Workshop Materials, Special Topic: Differential Equations AP Central Website: apcentral.collegeboard.com A Gallery of Volume Visualizations http://mathdemos.gcsu.edu/mathdemos/solids/gallery.html Visual Calculus Volumes Disk Method http://archives.math.utk.edu/visual.calculus/5/volumes.4/index.html AP Calculus Electronic Discussion Group for Advanced Placement Calculus Courses