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AP Calculus BC Syllabus PHILOSOPHY: My main objective in teaching AP Calculus BC is to provide students with a rich and thorough understanding of calculus that will enable them to succeed at higher levels of mathematics. Students will be working in cooperative learning groups throughout the school year. Within these groups students will be asked to discover the mathematical concepts and relationships numerically, graphically, and algebraically. A graphing calculator (TI 89) will be used to help students understand these mathematical ideas. Students will also be asked to communicate their discoveries using the proper mathematical vocabulary. TIMELINE: I. Students are to complete a summer packet prior to the opening of school. The packet consists of problems from chapter one of Hughes – Hallet’s 4th edition and Larson’s 7th edition. II. Review (4 weeks) A. Linear, quadratic, exponential, logarithmic, and sinusoidal functions B. Compositions and Inverses C. Parametric Equations ACTIVITY: I use the old College Board Pacesetter activity “Ships in the Fog” to introduce the concept of parametric equations. Students are asked to use the TI-89 graphing calculator to explore the concept of parametric equations numerically, graphically, and algebraically. D. Sequences and Series E. Polar II. LIMITS and CONTINUITY (2 weeks) A. Limits at a point 1. Properties of limits 2. Two sided 3. One-sided B. Limits involving infinity 1. Asymptotic behavior 2. Properties of limits 3. Visualizing limits C. Continuity 1. Continuous 2. Discontinuous functions ACTIVITY: I use the old College Board Pacesetter activity “Rational Functions” to introduce the concept of limits and continuity. Students are asked to graph various rational functions. The TI 89 calculator is used to verify their graphs as well as help interpret the concepts of limits and continuity. The students are then asked to report their findings in a short essay. The TI 89 helps students visualize the concepts of limits and continuity both numerically (table of values) and graphically. III. DERIVATIVES (5 weeks) A. Definition 1. Secant line 2. Tangent line B. Using the definition to find a derivative 1. Algebraic interpretation 2. Geometric interpretation ACTIVITY: We play a derivative game. Each student is handed a card at random. The card either contains a graph or a function. The card is also labeled either FUNCTION or DERIVATIVE. The students are then to find their correct partners. I then use these pairings as my cooperative learning groups. C. Rates of change – To help students understand the concept of derivatives, I use a CBL experiment. Students toss a ball in the air and examine the height versus time graph that is generated. Students then use a quadratic equation to fit the data and determine how high the ball went and how long the ball was in the air. Students can then calculate the average velocity over the time interval. They are then asked to determine the velocity at exactly 0.5 seconds. We can then use the CBL to graph the acceleration. Great classroom discussions take place during and after this activity. 1. Average rate of change (slopes of secant lines) 2. Instantaneous rate of change (slopes of tangent lines) D. Mean Value Theorem (MVT) ACTIVITY: The Door Problem—Instantaneous Rate of Change of a Function (taken from Calculus Concepts and Applications Instructor’s Resource Book by Paul A. Foerster) This activity helps students further understand the relationship between average and instantaneous rates of change. E. The second derivative F. Continuity and differentiability IV. SHORTCUTS TO DIFFERENTIATION (4 weeks) A. The power rule and polynomial functions B. The exponential and logarithmic functions C. Product and quotient rules D. The chain rule E. Inverses and Trigonometric functions F. Related rates G. Implicit differentiation H. Local linearity, linear approximations, and Newton’s Method ACTIVITY: “Investigating the Accuracy of the Tangent Line Approximation” (ancillary materials from Hughes-Hallet) V. USING THE DERIVATIVE (4 weeks) A. First and second derivative tests B. Curve sketching and family of curves – The TI 89 calculator is used to help verify results graphically. C. Optimization ACTIVITY: “Packaging: The Optimal Form” (Larson, 7th ed., p.158) MIDTERM: I review with the students for the midterm examination. The midterm consists of: 15 M/C without the calculator, 10 M/C with the calculator, and 2 Free Response (no calculator). Approximately 5 class periods prior to the exam, students are tested. I give them old AP free response that covers limits and derivatives as well as M/C from various sources. VI. INTEGRATION (5 weeks) A. Riemann Sums, Trapezoidal Rule and Simpson’s Rule B. The Definite Integral C. Interpretation of the definite integral D. Fundamental Theorem of Calculus ACTIVITY: “The Wankel Rotary Engine and Area” (Larson, 7th ed., p.240) PROJECT: Students are to take a ride in a car with a parent. They are to record the odometer reading on the car prior to the trip and after the trip is completed. During the trip they are to use the speedometer and record the car’s speed every minute. The students are then to use their data to create a speed versus time graph. From their graph, students are to use integration techniques to calculate the distance traveled. The students then compare their results with the actual distance traveled. E. Antiderivatives and the Substitution Method F. Integration by parts, trigonometric substitution, and partial fractions G. Equations of Motion H. The Second Fundamental Theorem of Calculus I. Area between Curves J. Volumes 1. Disc Method 2. Washer Method 3. Cylindrical Shells Method 4. Known Cross-sections a. Squares b. Semicircles c. Equilateral Triangles d. Isosceles Right Triangle VII. DIFFERENTIAL EQUATIONS (3 weeks) A. Separation of Variables B. Slope Fields – I use a TI -89 graphing calculator to project a grid on the chalkboard. Each student is then assigned a point and comes to the board and draws the appropriate segment based on the differential equation we are working with. The class then completes the slope field. C. Euler’s Method VIII. PARAMETRIC, POLAR, and VECTOR FUNCTIONS (3 weeks) A. Length of parametrically defined curves B. Vectors and vector-valued functions C. Calculus of vector functions D. Calculus of polar functions, including slope, length, and area IX. SEQUENCES (2 weeks) A. Arithmetic, harmonic, alternating harmonic, and geometric B. Convergence and divergence C. Bounded, monotonic, oscillating sequences D. Limit properties of sequences E. L’HÔpital’s Rule and indeterminate forms F. Relative rates of growth of functions G. Improper integrals and the comparison test X. SERIES (4 weeks) ACTIVITY: An Investigation into the Accuracy of Polynomial Approximations to Transcendental Functions A. Geometric B. Power - Term by term differentiation and integration C. Taylor’s /Maclaurin D. Tests of convergence/divergence The month prior to the AP exam, we review. The students are administered the 1997, 1998, 2003, and released practice exams in their entirety. The ’97 exam is giving as a Spring Break assignment. The ’98, ’03, and released practice exams are administered on Saturdays. All exams are timed and graded like an actual AP exam. ASSESSMENTS: Do Now: Students are administered a two question M/C quiz for the first 10 minutes of class every day beginning in October Exams are administered after each unit 2 to 3 projects are assigned throughout the course Students are required to keep a Calculus Journal. Once every two weeks students are giving a “Calculus Prompt.” They must respond to this prompt using the appropriate calculus to support their solution. Textbooks: Hughes-Hallett, Deborah, et al. Calculus Single Variable. 4th ed. New York: Wiley & Sons 2005. Larson, R., Hostetler, R., and Edwards, B. Calculus with Analytic Geometry. 7th ed. Boston: Houghton Mifflin 2002.