AP Calculus AB Course Syllabus

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1 AP Calculus AB Course Syllabus Course Overview Calculus AB is a course in single-variable calculus that includes techniques and applications of the derivative, techniques and applications of the definite integral, and the Fundamental Theorem of Calculus. It is equivalent to at least a semester of calculus at most colleges and universities, perhaps to a year of calculus at some. Algebraic, numerical, and graphical representations are emphasized throughout the course. Students taking this course have completed a solid foundation of mathematical courses that include algebra, geometry, trigonometry, analytic geometry, and elementary functions. Students are familiar with the properties, graphs, algebra, and language of linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piece-wise defined functions. With Georgia s Integrated Math program, students will normally have taken Accelerated Math I through Accelerated Math III; or Math I through Math IV before taking AP Calculus. Attitude prerequisites include a willingness to work both in and out of class, a willingness to collaborate with classmates to foster mutual understanding, and a sincere intent to place out of the first semester of college calculus rather than repeat it. This AP Calculus AB class covers all the topics in the Calculus AB topic outline as it appears in the AP Calculus Course Description. This syllabus follows the presentation order of topics in the class and the part of the course description when it is first addressed. Topics are revisited continually throughout the year as the course continues to build on all previous material. All topics in the course are taught with the Rule of Four in mind. The source of understanding of each concept begins with a picture and a simple numerical solution or estimation of the solution to a problem. Then students are asked to complete an accurate solution analytically using proven mathematical and calculus techniques. Mastery is developed verbally through homework presentations and test essay questions. Instruction is developed in accordance with Common Core and standards-based instructional techniques. The connections between numerical, graphical, analytical, and verbal communication of Calculus topics are a major theme in the class along with the overall conceptual understanding of the Calculus topics covered. Chapter 1: Preparation for Calculus (2 weeks) We will review the following Pre-Calculus and Trigonometry topics: Graphs & Models: Analysis of graphs: with the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function Linear Models & Rates of Change Functions and their Graphs Fitting Models to Data: Mathematical models are discussed so students realize that is the way most problems in life and careers are presented. This is the first introduction of the real life word problem format that will become quite common throughout the course. Trigonometric Functions, Unit Circle Values, & Identities Chapter 2: Limits and Their Properties (3-4 weeks) Limits: Rates of Change and Tangent Lines: Pre-Cal versus Calculus. Compare the secant line and tangent line. Use numerical tables to calculate the slope of the secant line as the change in x gets smaller on each side. The graphing calculator makes this less tedious, and Geogebra examples make the graphs of the secant lines interactive and easier to comprehend.

2 Limits: A Numerical and Graphical Approach: Gain an intuitive understanding of the limiting data; Calculator is used to explore what happens to a function as a point is approached from each side. Basic Limit Laws Limits and Continuity: Continuity & One-Sided Limits: Intuitive understanding of continuity; Understanding continuity in terms of limits; Calculator is used to gain concept of local linearity; Infinite Limits: Understanding asymptotes in terms of graphical behavior; Describing asymptotic behavior in terms of limits involving infinity Evaluating Limits Analytically: Calculate limits algebraically; Trigonometric Limits Intermediate Value Theorem: Geometric understanding of graphs of continuous functions and Extreme Value Theorem Chapter 3: Basic Differentiation Rules and Rates of Change (4-5 weeks) Definition of a Derivative: Derivative defined as the limit of the difference quotient; graphically as the slope of the tangent line at a point on the graph. Derivative presented graphically, numerically, and analytically; Instantaneous rate of change as the limit of average rate of change; Derivative interpreted as an instantaneous rate of change; The Derivative as a Function: Relationship between differentiability and continuity; Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents; Tangent line to a curve at a point and local linear approximation; Knowledge of derivatives of basic functions including power and sums; Product, Quotient Rules: Basic rules for the derivative of products and quotients of functions; Rates of Change: Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed; Higher Order Derivatives: Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration; Trigonometric Functions: Knowledge of derivatives of trigonometric and inverse trigonometric functions; Chain Rule: Chain rule and more evaluation of Leibniz notation; Implicit Differentiation Derivatives of Inverses Functions: Use of implicit differentiation to find the derivative of an inverse function; Derivatives of General Exponential and Logarithmic Functions: Knowledge of derivatives of basic functions, including power, exponential, and logarithmic functions; Related Rates: Modeling rates of change, including related rates problems; Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. Chapter 4: Applications of the Derivative: (4-5 weeks) Linear Approximation and Applications: Linear approximation using the derivative at a point. Extreme Values: Absolute extrema on an open or closed interval; relative extrema on an interval; Critical points of a graph and based on the derivative; Corresponding characteristics of graphs of f and f ; Rolle s Theorem; Mean Value Theorem: The Mean Value Theorem and its geometric consequences; Increasing & Decreasing Functions & First Derivative Test: Relationship between the increasing and decreasing behavior of f and the sign of f ;

3 Equations involving derivatives: Verbal descriptions are translated into equations involving derivatives and vice versa; The Shape of a Graph: Concavity & Second Derivative Test: Relationship between the concavity of f and the sign of f ; Points of inflections as places where concavity changes; Graph Sketching and Asymptotes: f, f, f : Graph matching of functions and derivatives; exploring relationships between the graphs of f, f, and f ; Limits at Infinity: Understanding asymptotes in terms of graphical behavior; Describing asymptotic behavior in terms of limits involving infinity; Curve Sketching Summary: Analysis of curves including the notions of monotonicity and concavity; Corresponding characteristics of graphs of f, f and f ; Assimilating all the tools developed to be able to graph a complicated function before checking it with the calculator. Applied Optimization: Optimization, both absolute (global) and relative (local) extrema; Compare difference between related rates and optimization; Differentials: explore meaning of differentials, compare dy and delta y and tie into local linearity; Slope Fields: Develop an intuitive understanding of what integration will do by drawing slope fields and understanding the shape for the solution to a differential equation has infinite number of solutions before integration is ever broached. Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations; Numerical solution of differential equations using Euler s method; Antidervatives & Indefinite Integration: Antiderivatives following directly from derivatives of basic functions; Finding specific antiderivatives using initial conditions, including applications to motion along a line; Chapter 5: The Integral (6-7 weeks) Approximating and Computing Area: Sigma Notation and left and right sums used to approximate area; Riemann Sums & Definite Integrals: Numerical approximations to definite integrals: Setting up and approximating Riemann sum and representing its limit as a definite integral; Using the integral of a rate of change to give accumulated change; Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values Fundamental Theorem of Calculus: Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval; Use of the Fundamental Theorem to evaluate definite integrals; Use the FTC to represent a particular antiderivative, include the calculator to explore graphically and algebraically; Net or Total Change as the Integral of a Rate: displacement compared to total distance traveled over an interval of time; marginal costs. Relating Integration to real world applications in economics and physical settings; Integration by Substitution: Antiderivatives by substitution of variables (including change of limits for definite integrals); Further Transcendental Functions: Trigonometric, exponential, and logarithmic functions; Differential Equations: Growth & Decay: Solving separable differential equations and using them in modeling (in particular, studying the equation y = ky and exponential growth); Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations;

4 Chapter 6: Applications of Integration (2-3 weeks) Applications of integrals: Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations Area of a Region Between Two Curves: Basic properties of definite integrals; Numerical approximations to definite integrals; Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations; finding the area of a region Volume: Disk & Washer Method: finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections; Volume: with known Cross Sections: finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections Review for AP Exam (4-6 weeks) This is a chance to review the concepts learned throughout the year and tie everything together in the big picture. Time is spent practicing released AP Free Responses and Multiple Choice Questions, many of which have been seen throughout the year in the problem sets and as part of classroom tests. Technology Use All students are encouraged to use a TI-89 or TI-Inspire, but a TI 84+ calculator will also suffice. The school has some TI-84+ and TI-Inspire calculators that students may check out throughout the year for use at home and at school. Students are introduced to calculator functions and procedures periodically throughout the year, focusing on the skills required for the AP exam including graphing equations, finding roots, and finding numerical derivatives and integrals. Regardless of whether a calculator is used or not, answers alone are not acceptable. Students must demonstrate and communicate their mathematical understanding of the concepts. Tests and assignments have both calculator active and non-calculator sections designed to assist students in solving problems, experimenting with ideas, interpreting results, and supporting conclusions. Student Evaluation Grades are determined through the evaluation of homework, class work, tests, and presentations. Homework is evaluated by assessing that a student has made a legitimate attempt at each problem. Students are provided access to the Instructor s Solutions Manuel, which contains detailed solution to problems. Each day during class, students present their own solutions to selected problems by writing their solutions on the board and then explaining their work to the class orally. Tests are set up in the AP format, but scaled down to 50 minutes with calculator and non-calculator sections for multiple-choice, free-response, and essay questions. These questions are selected to correspond closely with material covered in class for each unit. Questions for each unit s problem set/study guide and the actual test are selected from released AP exams in order to simulate the length, format, content, and difficulty of the actual AP exam. Essay questions tend to address the theoretical aspects of the course such as explaining the Fundamental Theorem of Calculus, or the difference between Rolle s Theorem and the Mean Value Theorem, or they allow students to explain the meaning behind their own answers, such as justifying why a specific interval is concave up. Primary Text The primary textbook is listed below. All students have access to this book through school provision. Rogaawski, Jon. Single Variable Calculus Early Transcendentals, New York: W. H. Freeman and Company

5 Resource Materials: Larson Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus of a Single Variable. Ninth Edition. Boston: Houghton Mifflin Company, 2010 Finney, Ross, Franklin Demanna, Bert Waits, Daniel Kennedy. Calculus: graphical, Numerical, Algebraic, Fourth Edition. Needham, MA: Pearson/Prentice Hall, 2012 Hockett, Shirley, David Bock. Barron s How to Prepare for the AP Calculus Advanced Placement Examination, Hauppauge, NY: Barron s Educational Series, 2002 APSI Workshop Materials AP Central Materials and Released Exams