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Advanced Placement Calculus AB ’11 – ’12 School Year Mrs. Jami Gore jgore@nthurston.k12.wa.us The Hawk Way Hawks are: House A Respectful Room 113 Responsible Honorable (360) 412-4820 Successful COURSE DESCRIPTION This course introduces students to the four major concepts found within a typical first semester of calculus: limits, derivatives, and definite and indefinite integrals. We will use a variety of ways to approach and solve problems: numerical analysis (using known data points), graphical analysis (using a known graph), algebraic analysis (using an equation and variables), and verbal/written methods of representing problems (including justification of one’s thinking). A detailed list of topics can be found in the AP Calculus Course Description, which is available at the school or online at http://apcentral.collegeboard.com. This class is designed to prepare you for the AP Calculus Test! The test date is Wednesday, May 9th at 8:00 a.m.. PRIMARY TEXT BOOK Foerster, Paul. Calculus: Concepts and Applications. Berkeley, CA: Key Curriculum Press, 1998. CLASSROOM PROCEDURES & POLICIES Good attendance is critical for success in Mathematics. The school’s Attendance and Tardy Policy is strictly enforced. When absent, it is the student’s responsibility to obtain and complete the missed assignments and notes. NO electronic devices will be allowed in class. This includes cell phones, ipods, etc. Academic dishonesty will NOT be tolerated. Any student who is caught cheating on any individual test, quiz, or assignment will receive an automatic zero for that assignment with no chance to retake. TECHNOLOGY The AP Calculus curriculum demands the use of a graphing calculator. Because of this requirement, each student should have their own TI-83 or TI-84 graphing calculator. Other graphing calculators could be used, but support of those calculators will not be offered in this course. Calculators for student use may be checked out in the library for the school year. We will use graphing calculators to help solve a variety of problems, including: estimate limits, determine asymptotic behavior, estimate roots, find points of intersection, calculate numerical differentiation and perform numeric integration. We will also use graphing calculators to experiment with and investigate. This includes investigating different functions, exploring the numerical approach to limits, derivatives and integrals, and exploring the continuity of a function. Finally we will use graphing calculators to interpret results, support conclusions, and check solutions. Half of the AP Calculus exam requires the use of a graphing calculator so you must be comfortable and confident in using one. GRADING The overall grade will be determined (approximately) as follows: Grading scale: Free-Response Quizzes: 20% 85% - A 70 - 84% B Homework/Class Participation: 30% 55 - 69% C 50 – 54% D Tests/Projects: 50% Below 50% F *I will follow the River Ridge grade incentive program. This states that if you pass the official AP exam with a 3 you can move one semester grade up by one letter grade. If you earn a 4 or 5 on the AP exam you can change one semester grade to an A. To take advantage of this you must bring proof to the main office (score reports usually arrive in July.)* Projects We will have one major project each quarter of both semesters. All projects will include numeric, graphic, and algebraic analysis as well as written justification or explanation. They will each be weighted the same as one unit test. Late/Make-Up Work: One school day will be allowed for each day missed upon return to school. Make-up work is accepted for excused absences only. No late assignments or projects will be accepted for credit. Course Outline By successfully completing this course, you will be able to: Work with functions represented in a variety of ways: graphically, numerically, analytically, or verbally, and understand the connections among these four representations. Understand the meaning of the derivative in terms of a rate of change and local linear approximation and be able to use derivatives to solve problems. Understand the meaning of the definite integral as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve problems. Understand the relationship between the derivative and the definite integral as expressed in the Fundamental Theorem of Calculus. Communicate mathematics and explain solution to problems verbally and in written sentences. Work effectively as part of a group or team of students to solve problems and present their solutions. Model a written description of physical situation with a function, differential equation, or an integral. Use technology to help solve problems, experiment, interpret results, and support conclusions. Determine the reasonableness of solutions. Develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment! Units and Pacing: (Sections with an * indicate that they are BC topics only.) All other sections are covered on both the AB and BC exam. Chapter 1: Introduction to Limits, Derivatives, and Integrals (7 days) Section 1: Instantaneous rate of change as the limit of average rate of change Section 2: Approximate rates of change from equations, graphs, and tables Section 3: Estimating definite integrals Section 4: Using the Trapezoidal Rule to estimate definite integrals from equations and tables Section 5: Derivative as an instantaneous rate of change Chapter 2: Properties of Limits (14 days) Section 1/2: Exploring the Definition of Limit by graphical, numerical, and algebraic techniques Section 3: Using the Limit Theorems Section 4: Using the Definition of Continuity to tell whether a function is or is not continuous Section 4: Exploring piecewise functions and continuity Section 5: Limits involving infinity Section 6: The Intermediate Value and Extreme Value Theorems *Our first unit assessment will be after Chapter 2 and will cover Limits* Chapter 3: Derivatives, Antiderivatives, and Indefinite Integrals (14 days) Section 1: Using difference quotients to estimate the slope of a tangent line and understanding the graphical implications Section 2: Using the Definition of Derivative (at a point) to find derivatives Section 3: Comparing functions and their derivatives numerically and graphically Section 4: Using the (general) Definition of Derivative to find derivatives of Power Functions Section 5: Displacement, Velocity and Acceleration relationships. Section 6/8: The derivative of sine and cosine. Section 7: Chain Rule Section 9: Exponential and Logarithmic derivatives. *Our second unit assessment will be after Chapter 3 and will cover Basic Derivatives* Chapter 4: Products, Quotients, and Parametric Functions (20 days) Section 1: Making conjectures to find the derivative of combinations of functions. Section 2: Using the Product Rules to find derivatives Section 3: Using the Quotient Rules to find derivatives Section 4: Finding the derivatives of Trigonometric Functions Section 5: Finding the derivatives of Inverse Trigonometric Functions Section 6: Relationship between differentiability and continuity. *Section 7: Derivatives of a Parametric Function* Section 8: Differentiating implicit relations Section 9: Related rates. *Our third unit assessment will be after Chapter 4 and will cover Differentiation* Chapter 5: Definite and Indefinite Integrals (20 days) Section 1: Exploring the definite integral Section 2: Linear Approximations and Differentials Section 3: Formal Definition of the Indefinite Integral Section 4: Riemann Sums and Definite integrals Section 5: The Mean Value Theorem Section 6: The Fundamental Theorem of Calculus Section 7: Definite integral practice Section 8: Applying definite integrals to area Section 9: Volume of a Solid by Plane Slicing Section 10: The integration feature of you graphing calculator *Our fourth unit assessment will be after Chapter 5 and will cover Integrals* *This should be the LAST assessment for 1st semester* Chapter 6: The Calculus of Exponential and Logarithmic Functions (12 days) Section 1: Integral of the reciprocal function Section 2: Antidifferentiating the Reciprocal Function Section 3: Properties of Logarithmic Functions Section 4: Exploring e and the change of base theorem. Section 5: Limits of Indeterminate Forms: I’ Hospital’s Rule Section 6: Derivative and integral practice. *Unit Assessment* Chapter 7: The Calculus of Growth and Decay (10 days) Section 1: Expressing exponential growth and decay as a direct proportion Section 2: Separating and integrating differential equations Section 3: Applications for Differential Equations Section 4: Slope Fields *Section 5: Euler’s Method* *Section 6: Logistic differential equations and modeling* *Unit Assessment* Chapter 8: The Calculus of Plane and Solid Figures and Other Topics (16 days) Section 1: Exploring cubic functions and their derivatives. Section 2: Finding critical points from derivatives. Section 3: Using derivatives to solve Maximum & Minimum problems Section 4: Finding the area of a plane region Section 5: Finding volume by plane slicing Section 6: Finding the volume of a solid of revolution by using cylindrical shells *Section 7: Lengths and Areas for Polar Coordinates* *Unit Assessment* *Chapter 9: Techniques of Antidifferentiation* (15 days) *Section 2: Integration by Parts* *Section 3: Rapid Repeated Integration by Parts* *Section 6: Integration by Trigonometric Substitution* *Section 7: Integration of Rational Functions by Partial Fractions* *Section 10: Improper Integrals* Chapter 10: The Calculus of Motion (10 days) Section 1: Distance and Displacement for Motion along a Line Section 2: Distance, Displacement and Acceleration Section 3: Average Value of a Function Section 4: Related Rates Section 5: Maximum and Minimum Problems in Motion *Section 6: Vector Functions for Motion in a Plane* *Unit Assessment* *Chapter 12: The Calculus of Functions Defined by the Power Series* (20 days) *Section 1: Introduction to Power Series* *Section 2: Geometric Sequences and Series as Mathematical Models* *Section 3: Power Series for an Exponential Function* *Section 4: Power Series for Other Elementary Functions* *Section 5: Taylor and Maclaurin Series* *Section 6: The Ratio Technique* *Section 7: Convergence of Series (including harmonic series and the integral test)* *Section 8: The Lagrange Error Bound*