AP CALCULUS AB Course Overview My main objective in teaching AP Calculus is to have my students experience and understand the beauty and elegance of calculus and receive the necessary knowledge and skills that will help them succeed in future mathematics courses. It is a journey that invites them to think and work hard. Students study the following major conceptual concepts: limits, derivatives, indefinite integrals and definite integrals, and the application of these principles in modeling real world scenarios and situations. Course Goals (as stated in the College Board Course Description): 1. Students should be able to work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations. 2. Students should understand the meaning of the derivative in terms of a rate of change and local linear approximation and should be able to use derivatives to solve a variety of problems. 3. Students should understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve a variety of problems. 4. Students should understand the relationship between the derivative and the indefinite integral as expressed in both parts of the Fundamental Theorem of Calculus. 5. Students should be able to communicate mathematics both orally and in wellwritten sentences and should be able to explain solutions to problems. 6. Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral. 7. Students should be able to use technology to help solve problems, experiment, interpret results, and verify conclusions. 8. Students should be able to determine the reasonableness of solutions of their classmates, including sign, size, relative accuracy, and units of measurement. 9. Students should develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment. Teaching Strategies Every day the students will be provided with the opportunity to learn through exploration and discovery. They will communicate and share their findings with their peers within group settings. Students are expected to analytically present problem solving strategies in a logical and organized fashion, articulate their mathematical reasoning using technology to support their findings both numerically and graphically (whenever it is possible), and apply their understanding of calculus to real life problems. I have students present and explain solutions on the board to, and/ or they are grouped in pairs and justify answers to each other. I believe that students learn more by 1
presenting Calculus material to each other. These approaches help me find out which students need extra help and if their understanding of the presented material is correct. Socratic dialog will be part of our daily activities in questioning our understanding of the concepts of Calculus. Grades will be computed based on homework/projects (30%), classroom participation (10%), and tests and quizzes (60%). For homework students are allowed to use calculators ((TI-84 or TI-89), but must analytically support their findings and solutions of problems with appropriate mathematical symbolism, equations, and wellwritten sentences. Free- response questions as well as multiple-choice practice questions will be incorporated in the curriculum. The midterms and final tests will be based on a mock AP exam. On a regular basis we will be using technology, i.e. graphing calculators (TI-84 or TI-89) in the classroom and at home, to investigate and conduct science activities between the related functions, ideas and concepts presented in Calculus AB. An audiovisual projector will be used to display activities from the computer software Calculus in Motion as well as related website information. Students have a better understanding of calculus concepts when they learn them through what they see and experience. Technology Requirement I have a classroom set of TI-84 Plus calculators, so all students have access to TI 84 Plus calculators while in class. Most of the students have their own graphing calculator (TI-84 or TI-89) and for the ones that are not able to buy one on their own there are some available for extended checkout. We will use the calculator to: 1. Conduct explorations 2. Plot the graph of a function within a specified window 3. Finding the zeros of functions 4. Analyze and interpret results of graphs and equations 5. Approximate a derivative at a given point using numerical methods 6. Approximate a definite integral at a given point using numerical methods. AP CALCULUS AB COURSE OUTLINE Unit Name and Timeframe Unit 1: Prerequisites for Calculus (5 hours) (C1) A. Lines 1. Increments 2. Slope as rate of change 3. Parallel and perpendicular lines 4. Equations of lines B. Functions and graphs 1. Functions 2. Domain and range 3. Even functions and odd functions-symmetry 4. Piecewise functions 2
5. Composition of functions C. Exponential and logarithmic functions 1. Exponential growth and decay 2. Logarithmic functions 3. Properties of logarithms 4. Inverse functions D. Parametric Equations E. Trigonometric functions 1. Graphs of basic trigonometric functions 2. Transformations of trigonometric graphs 3. Inverse trigonometric functions F. Section projects 1. Parametrizing circles, ellipses, line segments 2. Finding the frequency of a musical note. Unit 2: Limits and Continuity (20 hours) (C2) & (C5) A. Rates of Change 1. Average speed and instantaneous speed B. Limits at a point 1. 1-sided Limits 2. 2-sided Limits 3. Sandwich Theorem C. Limits involving infinity 1. Asymptotic behavior (vertical and non-vertical) 2. End behavior 3. Properties of limits (algebraic approach) 4. Visualizing limits (graphic investigation) D. Continuity 1. Continuity at a point 2. Continuous functions 3. Properties of continuous functions 4. Discontinuous functions a. Removable discontinuity b. Jump discontinuity c. Infinite discontinuity 5. The Intermediate Value Theorem for continuous functions 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 to a curve (algebraically and graphically) 5. Instantaneous rate of change F. Section Projects: 1. Graphs and limits of trigonometric functions 2. Removing a discontinuity 3
Unit 3: Derivatives (20 hours) (C2),(C3),(C4),(C5) A. Derivative of a function 1. Definition of the derivative 2. Derivative at a point 3. Relationships between the graphs of f and f 4. Graphing the derivative from data 5. One-sided derivatives B. Differentiability 1. How f (a) might fail to exist 2. Local linearity 3. Numeric derivatives on the calculator 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, Constant Multiple, Sum and Difference, Product and Quotient rules 2. Second and higher order derivatives D. Applications of the derivatives 1. Position, velocity, acceleration, and jerk 2. Recognize f, f and f when given only the graph 3. Particle motion 4. Horizontal motion 5. Derivatives in Economics a) Marginal cost b) Marginal revenue c) Marginal profit E. Derivatives of trigonometric functions F. Chain rule G. Implicit derivatives 1. Differential method 2. y' method H. Derivatives of inverse trigonometric functions I. Derivatives of logarithmic and exponential functions Unit 4: Applications of Derivatives (25 hours) (C2) & (C5) A. Extreme values of functions 1. Absolute (global) extreme values 2. Local (relative) extreme values 3. The Extreme Value Theorem 4. Definition of a critical point B. Using the derivatives 1. Rolle s Theorem 2. Mean Value Theorem 3. Increasing and decreasing functions C. Connecting f and f with the graph of f 4
1. First derivative test for local extrema 2. Concavity 3. Points of inflection 4. Second derivative test for local extrema D. Modeling and optimization problems E. Linearization models F. Related rate problems G. Section Projects 1. Constructing cones 2. The sliding ladder Unit 5: The Definite Integral (20 hours) (C2) & (C5) A. Estimating with finite sums 1. Riemann sums a) Left sums b) Right sums c) Midpoint sums B. Definite integrals 1. The definite integral as a limit of Riemann Sums C. Definite integrals and antiderivatives 1. Properties of definite integrals 2. Mean Value Theorem for definite integrals D. Fundamental Theorem of Calculus 1. Part 1 2. Part 2 E. Trapezoidal Rule F. Section Projects 1. Finding the derivative of an integral 2. Area under a parabolic arc Unit 6: Differential Equations and Mathematical Modeling (15 hours) (C2) A. Slope fields and Euler s Method B. Antidifferentiation by Substitution 1. Leibniz notation and antiderivatives 2. Substitution in indefinite integrals 3. Substitution in definite integrals C. Antidifferentiation by Parts 1. Integration by Parts formula 2. Solving for the unknown integral 3. Tabular integration 4. Inverse trigonometric and logarithmic functions D. Exponential Growth and Decay 1. Separable differential equations 2. The law of exponential change 3. Continuously compounded interest 4. Radioactivity 5
5. Newton s Law of Cooling E. Logistic Growth F. Section Projects: 1. Exploration; Integration by Parts: Choosing the right u and dv 2. Problems using Newton s Law of Cooling Unit 7: Applications of Definite Integrals (15 hours) (C2) A. Integral as Net Change 1. Linear motion 2. Consumption over time 3. Net change from data B. Areas in the Plane 1. Area between curves 2. Area enclosed by intersecting curves 3. Boundaries with changing functions 4. Integrating with respect to y C. Volumes 1. Cross sections 2. Disc method 3. Shell method D. Section Project 1. Finding the surface area of a solid of revolution Unit 8: Review and Test Preparation (10 hours) (C4) A. Multiple-choice and response free practice questions released from the AP Central, AP Test Prep Series, and items from the past tests B. Rubrics review C. General strategies for AP examination preparation and test taking strategies (C2) The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals; and Polynomial Approximations and series as delineated in the Calculus AB Topic Outline in the AP Calculus Course Description. (C3) The course provides students with the opportunity to work with functions represented in a variety of ways graphically, numerically, analytically, and verbally and emphasizes the connections among these representations. (C4) The course teaches students how to communicate mathematics and explain solutions to problems both verbally, and in written sentences. (C5) The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. Textbook *Finney, Demana, Waits and Kennedy. Calculus---Graphical, Numerical, Algebraic. Third edition. Pearson, Prentice Hall, 2007. *AP Test Prep Series---AP Calculus; supplement to Calculus: Graphical, Numerical, Algebraic by Finney, Demana, Waits, and Kennedy 6
These materials will be our primary resources. The book invites students to explore many calculus topics before they get familiar to theoretical approaches. The workbook will help students identify and understand essential calculus concepts as well as enhance their ability to communicate their thinking so they could be successful in class. It is a useful review that improves student learning through practice. 7