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IV. “Integration” BC. Chapter 4. 15 days. A. Topics 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. B. Anti-derivative Constant of Integration, General Solution Differential Equation Anti-differentiation Indefinite Integration, Integrand, Variable of Integration, Indefinite Integral Integration Rules General Solutions Initial Condition, Particular Solution Integration using u-substitution vs Integration using Anti-differentiation General Power Rule for Integration Sigma Notation, Index of Summation Area—Upper Sums, Lower Sums Circumscribed Rectangle, Inscribed Rectangle Limit of the Lower and Upper Sums Definition of the Area of a Region in the Plane Riemann Sums – left, right, and midpoint sums Definite Integrals and Riemann Sums Continuity and Integrability The Definite Integral as the Area of a Region Properties of Definite Integrals The Fundamental Theorem of Calculus U Substitution and the Change of the Variables and the Intervals of Integration. The Mean Value Theorem for Integrals Average Value of a Function The Definite Integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval. The Second Fundamental Theorem of Calculus Integration of Even and Odd Functions Trapezoidal Rule, Simpson’s Rule Accumulation of Rate of Changes. Numerical Approximation of the Definite Integral Integral applications to finding distance. Teaching Methods and Evaluations. 1. Each topic presented through lecture, student and group interaction, 2. 3. 4. 5. graphing calculator usage, projects, supplements, Power Point presentations, and assignments. Integration presented through anti-differentiation. Project to solve differential equations to a general solution, then a particular solution. Discussion of the role of the Constant of Integration. Riemann Sums Project. Upper Sums, Lower Sums, Right Hand Sums, Left Hand Sums and Midpoint Sums explored graphically and algebraically. The definite integral developed as a limit of Riemann sums. Student involvement in the teacher led proof of the Fundamental Theorem of Calculus. 6. 7. 8. 9. 10. 11. 12. 13. Student development of skill in integration through anti-differentiation and u-substitution in assignment problems. Integration explored as a net accumulation of change. The definite integral used to illustrate the Fundamental Theorem of Calculus in the role of solving functional values from graphs. Understanding the Second Fundamental Theorem of Calculus through its proof, and its utilization in algebraic and geometric problem solving. Teacher led and student interaction in the development of the Mean Value Theorem for Integrals and the Average Value of a Function. Application of these concepts in AP generated problems of the text and the College Board. Trapezoidal and Simpson’s Rules illustrated with applications to data analysis in charts and graphs. Distance applications and its role with definite integrals. Supplements of Chapter 4. a. b. c. d. e. f. 14. Evaluations. a. C. Limit of a Riemann Sum as a definite integral. Riemann sum using left, right, and midpoint evaluations. Proofs related to Integration topics of Chapter 4. Indefinite and Definite Integration. Integration Overview. Selected problems of AP Test Preparation questions of section XI to provide: 1. Integral of a rate of change to give accumulated change. 2. Integration to find the area of a region. 3. Integration to find total distance or displacement. 4. Use of anti-derivatives to discuss motion along a line. 5. Using tables or graphs of velocity to answer questions concerning distance, accumulation, or lower and upper estimate of total accumulations. 6. First and Second Fundamental Theorem of Calculus Discovery 7. Problems concerning area, Second Fundamental Theorem of Calculus, and average rate of change. 8. Using graphs to find definite integrals given the graph of the derivative. 9. Trapezoidal and Simpson Rules to approximate definite integrals (algebraically, graphically, tables). Teacher created individual and group tests, projectsb. Multiple Choice and Free Response questions of the College Board and Fast Track to a 5 listed in XI of the syllabus. Assignments. Text: Larson, Ron., Edwards, Bruce and Robert P.Hostetler. Calculus Boston: Houghton Mifflin, 2006. 1. 2. 3. 4. 5. 4.1: Antiderivatives and Indefinite Integration. 21, 25, 27, 35-71o, 72, 73, 77, 78, 81.* E: 81, 82, 85, 94, 95, 97. 4.2: Area – Summation. 3, 9, 11, 15, 19, 23, 25, 27, 29, 33, 35, 41, 47, 51, 53, 55, 59, 63, 65. *E: 84. 75, 81 in class discussion. 4.3: Riemann Sums and Definite Integrals. 5, 9-12, 13-43o, 47-49.51-54, 55, 57. Supplement. 4.4: Fundamental Theorem of Calculus. 11, 15, 21, 23, 25, 29, 31, 37, 39, 43, 45, 47, 49, 51 53-60, 63, 67-91o. Supplement – Proofs. *E: 97, 98. 4.5: Integration by Substitution. 9, 11, 19, 25, 29, 31-39o, 40, 45-97o, 101105, 107, 111, 113. Supplement. . * E: 135, 136. 6. 7. 8. 9. 4.6: Numerical Integration. 3, 7, 13, 17, 23, 25, 27, 49, 51, 52. Supplement – Proofs. Teacher and student presentations of AP Questions. Selected problems of Section VIII of AP Test Preparation. Fast Track to 5. “We are what we repeatedly do. Excellence, then, is not an act, but a habit…” Aristotle. Parents and Students: Grading. Major grades: (70%) Daily grades: (20%). Fast Track; Class Presentations; Supplements; in Class Activities; Quizes; Free Response; Group work. Homework grades: (10%). Assignments. (view Calendar) Parents and students are to be aware that the student receives use of a solution key to the homework exercises and receives solutions to all supplements in class. Grade Sheets issued 2 times per 6 weeks. Gradespeed is updated within 48 hours of an assignment and or test. Teacher Website updated per week. Calendar of pace on the website/Calendar. Teacher Contact: [email protected] Use email, NOT Voice Mail please. 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