AP Calculus - AB Syllabus Course Overview This is a one year course designed to give students the necessary knowledge and skills to do well on the AP Calculus AB exam. Hopefully this course will also enable students to appreciate the beauty and power of not just calculus, but mathematics as well. Textbook(s): 1.) Finney, Demana, Waits and Kennedy. Calculus Graphical, Numerical, Algebraic. Third edition. Pearson, Prentice Hall, 2007. This textbook will be our primary resource. 2.) Schwartz, Stu. AP Calculus AB. www.mastermathmentor.com 2005 Selected sections of this online textbook will be used throughout the course. Technology Requirement The use of a graphing calculator is a required part of this course. I will use the Texas Instruments TI-Smartview software emulating the 84 Plus graphing calculator in class regularly. Although not required, I suggest that students get their own graphing calculator for use in this class. I recommend the TI-84 or the TI-89. I will have a classroom set of TI-84 Plus calculators, and some are available for use in the learning center. If students choose to use any other calculator they may do so, as long as it can perform the four required functionalities of graphing technology : 1. Finding a root 2. Sketching a function in a specified window 3. Approximating the derivative at a point using numerical methods 4. Approximating the value of a definite integral using numerical methods A Balanced Approach Current mathematical education emphasizes a Rule of Four. There are a variety of ways to approach and solve problems. The four branches of the problem-solving tree of mathematics are: Numerical analysis (where data points are known, but not an equation) Graphical analysis (where a graph is known, but again, not an equation) Analytic/algebraic analysis (traditional equation and variable manipulation) Verbal/written methods of representing problems (classic story problems as well as written justification of one s thinking in solving a problem) These four methods will be stressed throughout the course. Students will often be given a problem in a verbal format and asked to translate it to correct algebraic notation or sketch a graph. Likewise, students may be given an equation, expression or graph and asked to explain in complete sentences the meaning of what they were given.
Course Outline Introduction to Calculus (1 week) A. An intuitive approach to calculus will be developed using the classic motion problem viewed from several approaches. 1. Finding speed from the position (the derivative) 2. Finding distanced traveled from the speed (the integral) 3. Understanding the relationship between the two processes (Fundamental Theorem of Calculus) Unit 1: Limits and Continuity (3-4 weeks) A. Rates of Change 1. Average Speed 2. Instantaneous Speed Before beginning a formal look at limits, students will develop an intuitive sense of limits by exploring discontinuities and asymptotes of functions using the graphing calculator. Students will use the zoom feature or tables of values to verbally explain the behavior of the graph as it approaches certain points. B. Limits at a Point 1. 1-sided Limits 2. 2-sided Limits 3. Sandwich Theorem C. Limits involving infinity 1. Asymptotic behavior (horizontal and vertical) 2. End behavior models 3. Properties of limits (algebraic analysis) 4. Visualizing limits (graphic analysis) D. Continuity 1. Continuity at a point 2. Continuous functions 3. Discontinuous functions a. Removable discontinuity (0/0 form) b. Jump discontinuity (We look at y = int(x).) c. Infinite discontinuity E. Rates of Change and Tangent Lines 1. Average rate of change 2. Tangent line to a curve 3. Slope of a curve (algebraically and graphically) 4. Normal line to a curve (algebraically and graphically) 5. Instantaneous rate of change Unit 2: The Derivative (5-6 weeks) A. Derivative of a Function 1. Definition of the derivative (difference quotient) 2. Derivative at a Point 3. Relationships between the graphs of f and f 4. Graphing a derivative from data
5. One-sided derivatives B. Differentiability 1. Cases where f (x) might fail to exist 2. Local linearity 3. Derivatives on the calculator (Numerical derivatives using NDERIV) 4. Symmetric difference quotient 5. Relationship between differentiability and continuity 6. Intermediate Value Theorem for Derivatives C. Rules for Differentiation 1. Constant, Power, Sum, Difference, Product, Quotient Rules 2. Higher order derivatives D. Derivatives of trigonometric functions E. Chain Rule F. Implicit Differentiation 1. Differential method 2. y method G. Derivatives of inverse trigonometric functions H. Derivatives of Exponential and Logarithmic Functions Unit 3: Applications of the Derivative (5-6 weeks) A. Extreme Values 1. Relative Extrema 2. Absolute Extrema 3. Extreme Value Theorem 4. Definition of a critical point B. Implications of the Derivative 1. Rolle s Theorem 2. Mean Value Theorem 3. Increasing and decreasing functions C. Connecting f and f with the graph of f(x) 1. First derivative test for relative max/min 2. Second derivative a. Concavity b. Inflection points c. Second derivative test for relative max/min D. Optimization problems E. Linearization models We will be begin this section with a discussion of how students use local linearization every day (i.e. What does it mean to walk a straight line?)students will then explore the concept of local linearization by graphing fuctions on a calculator and zooming in until the curve
"straightens out". They will then be asked to explain the significance of this. 1. Local linearization 2. Tangent line approximation 3. Differentials F. Particle motion, Position, velocity, acceleration, and jerk. We will explore this concept using a CBL (Calculator Based Labratory) attached to a graphing calculator, and have the students be the particle. G. Economics 1. Marginal cost 2. Marginal revenue 3. Marginal profit H. Related Rates MIDTERM EXAM Unit 4: The Definite Integral (3 4 weeks) A. Approximating areas Students will a Riemann Sums program on the calculator to visualize exactly what a Riemann sum is. The will then be asked to explain the differences between the four types. 1. Riemann sums a. Left sums b. Right sums c. Midpoint sums d. Trapezoidal sums 2. Definite integrals B. Properties of Definite Integrals 1. Power rule 2. Mean value theorem for definite integrals C. The Fundamental Theorem of Calculus 1. Part 1 2. Part 2 Unit 5: Differential Equations and Mathematical Modeling (4 weeks) A. Slope Fields B. Antiderivatives 1. Indefinite integrals 2. Power formulas 3. Trigonometric formulas 4. Exponential and Logarithmic formulas C. Separable Differential Equations 1. Growth and decay 2. Slope fields (Resources from the AP Calculus website will be liberally used.) 3. General differential equations 4. Newton s law of cooling D. Logistic Growth
Unit 6: Applications of Definite Integrals (3 weeks) A. Integral as net change 1. Calculating distance traveled (particle motion) 2. Consumption over time 3. Net change from data B. Area between curves 1. Area between a curve and an axis a. Integrating with respect to x b. Integrating with respect to y 2. Area between intersecting curves a. Integrating with respect to x b. Integrating with respect to y C. Calculating volume 1. Cross sections 2. Disc method 3. Shell method Unit 7: Review/Test Preparation (Approximately 2 4weeks) A. Multiple-choice practice (Items from past exams 1997, 1998, and 2003 are used as well as items from review books.) 1. Test taking strategies are emphasized 2. Individual and group practice are both used B. Free-response practice (Released items from the AP Central website are used liberally.) 1. Rubrics are reviewed so students see the need for complete answers, written in full sentences. 2. Students collaborate to formulate team responses 3. Individually written responses are crafted. Attention to full explanations is emphasized. Unit 8: After the exam A. Review of the AP exam and try to identify points that students felt confident with or points that students felt they needed more work with. This will help strengthen those points before students go to college, as well as help me adjust the course for next year. B. Students will be required to do and present projects designed to incorporate this year s learning in applied ways. Projects can be of the students own choosing with teacher approval or chosen from one of the following sources. MAA Notes. Resources for Calculus Collection., D.C.: MAA, 1993. Volume 3: Applications of Calculus, edited by Philip Straffin. MAA Notes Number 29. Volume 4: Problems for Student Investigation, edited by Michael B. Jackson and John R. Ramsay. MAA Notes Number 30. Volume 5: Readings for Calculus, edited by Underwood Dudley. MAA Notes Number 31.
Evaluation Students will be assigned homework on a daily basis. Each week a cumulative assignment will be given that will collected and graded. Homework will count as 30% of the students grade. Once a week a short quiz will be given. Quizzes will count as 30% of the students grade. Two exams will be given each quarter, one after each unit. Tests will count 40% of the students grade. All graded homework assignments and exams will include questions where students must explain their answers in complete sentences using correct units as well. Students will also be asked to explain the thought process they used to solve a problem.