AP Calculus AB Syllabus Overview An Advanced Placement course in calculus consists of a full high school academic year of work that is comparable to calculus courses offered in colleges and universities. It is expected that students who take an AP course will seek college credit by taking the AP exam. AP Calculus is concerned with developing the student s understanding of the concepts of calculus and providing experiences with its methods and applications. The course emphasizes a multirepresentational approach to calculus, with concepts, results, and problems being expressed graphically, numerically, analytically, and verbally. The connections between these representations are important. Broad concepts and methods are emphasized, not memorization and manipulation. Calculus is fundamentally different from the mathematics studied previously. Calculus is less static and more dynamic. It is concerned with change and deals with quantities that approach other quantities. This is a very useful form of mathematics and is utilized daily in activities as varied as determining satellite orbits, forecasting weather, and calculating interest rates and earnings. In this course, along with learning the basics of calculus, students will explore many of these applications. Teaching Strategies Learning by Discovery: Students learn best by doing problems and working activities. Exploration and discovery are effective ways for students learning because the students retain greater ownership of the material covered than from the more traditional lecture approach. A number of calculator-based lab activities are completed throughout the year in calculus. Students will better understand the concepts of calculus when they experience concrete applications. Graphing Calculator: The graphing calculator helps students develop a visual understanding of the material presented that they would not otherwise have. Calculators are used as a tool to illustrate ideas and make discoveries about functions. Students make connections between the graphs of functions and the mathematical analysis of these same functions. Based on their observations, students can form their own conclusions and make predictions about the behavior of functions.
The Texas Instruments 83 graphing calculator will be regularly utilized in class. Each student should have their own graphing calculator as well. The graphing calculator will be used in a variety of ways including: - Conducting explorations - Graphing functions within arbitrary windows - Solving equations numerically - Analyzing and interpreting results - Justifying and explaining results of graphs and equations Rule of Four: Students are given the opportunity to work problems in four ways: graphically, numerically, analytically, and verbally. They are expected to relate the various representations to each other. Graphs and tables are not sufficient to prove an idea. Verification always requires an analytic argument. Students are asked to work problems in front of the class and explain solutions to their classmates. Students are asked to justify their answers, in well written sentences, on homework, quizzes, and tests. They are expected to explain problems using proper vocabulary and terms. Teaching Resources: Primary Text: Larson, Ron, Robert P Hostetler, and Bruce H. Edwards. Calculus of a Single Variable. 8 th ed. Boston: Houghton Mifflin, 2006 Supplemental Materials: The primary text is supplemented extensively with materials from other calculus textbooks, AP Released Exams, and free response questions from AP Central. The supplemental texts are listed below. Smith, Robert T and Minton, Roland B. Calculus. 3 rd ed. New York, NY: McGraw-Hill Publishing. 2008 Stewart, James. Calculus. 3 rd ed. Pacific Grove, CA: Brooks/Cole Publishing. 2005 Best, George and Fischbeck, Sally. AP Calculus with the TI-83/84, Venture Publishing. 2005
AP Calculus AB Course Outline Unit 1: Precalculus Review (2-3 weeks) A. Lines 1. Slope as rate of change 2. Parallel and perpendicular lines 3. Equations of lines B. Functions and graphs 1. Functions 2. Domain and range 3. Families of functions 4. Piecewise functions 5. Composition of Functions C. Exponential and logarithmic functions 1. Exponential growth and decay 2. Inverse functions 3. Logarithmic functions 4. Properties of logarithms D. Trigonometric functions 1. Graphs of basic trigonometric functions a) Domain and range b) Transformations c) Inverse trigonometric functions 2. Applications Unit 2: Limits and Continuity (3 weeks) A. Rates of Change B. Limits at a point 1. Properties of limits 2. Two-sided 3. One-sided C. Limits involving infinity 1. Asymptotic behavior 2. End behavior 3. Properties of limits 4. Visualizing limits D. Continuity 1. Continuous functions 2. Discontinuous functions a) Removable discontinuity b) Jump discontinuity c) Infinite discontinuity E. Instantaneous rates of change
Unit 3: The Derivative (5 weeks) A. Definition of the derivative B. Differentiablility 1. Local linearity 2. Numeric derivatives using the calculator 3. Differentiability and continuity C. Derivatives of algebraic functions D. Derivative rules when combining functions E. Applications to velocity and acceleration F. Derivatives of trigonometric functions G. The chain rule H. Implicit derivative 1. Differential method 2. y method I. Derivatives of inverse trigonometric functions J. Derivatives of logarithmic and exponential functions Unit 4: Applications of the Derivative (4 weeks) A. Extreme values 1. Local (relative) extrema 2. Global (absolute) extrema B. Using the derivative 1. Mean value theorem 2. Rolle s theorem 3. Increasing and decreasing functions C. Analysis of graphs using the first and second derivatives 1. Critical values 2. First derivative test for extrema 3. Concavity and points of inflection 4. Second derivative test for extrema D. Optimization problems E. Linearization models F. Related Rates Unit 5: The Definite Integral (3 weeks) A. Approximating areas 1. Riemann sums 2. Trapezoidal rule 3. Definite integrals B. The Fundamental Theorem of Calculus C. Definite integrals and antiderivatives
Unit 6: Differential Equations and Modeling ( 3 weeks) A. Antiderivatives B. Integration using u-substitution C. Separable differential equation 1. Growth and decay 2. Slope fields 3. General differential equations Unit 7: Applications of Definite Integrals (3 weeks) A. Summing rates of change B. Particle motion C. Areas in the plane D. Volumes 1. Volumes of solids with known cross sections 2. Volumes of solids of revolution a) Disk method b) Shell method