AP Calculus BC Syllabus Course Overview AP Calculus BC is the study of the topics covered in college-level Calculus I and Calculus II. This course includes instruction and student assignments on all of the topics as listed in the AP Course Description: Topic Outline for Calculus AB. AP Calculus BC is primarily concerned with developing the students understanding of the concepts of Calculus and providing experience with its methods and applications. The course is to help students see and interpret the world through the lens of integral and differential calculus. To that end, a focus is placed on providing a strong conceptual foundation including the concepts of a limit, a derivative and an integral. With a strong foundation and extensive practice with applications and problems, students become prepared for the AP Calculus Exam as well as additional coursework in Calculus. Rule of Four The course emphasizes an approach to Calculus, with concepts, results, and problems being expressed geometrically, numerically, analytically, and verbally. This course gives equal emphasis to all four methods of representing functions and their rates of change. Students are encouraged to be open-minded when approaching problems and to keep the Rule of Four in mind. Whenever possible, concepts are developed and applied using all of these representations. Additionally, emphasis is placed upon the connections among the representations. Technology is used regularly by students to reinforce the relationships among the multiple representations of functions, to confirm written work, to implement experimentation, and to assist in interpreting results. Through the use of the unifying themes of derivatives, integrals, limits, approximations and applications and modeling, the course becomes a cohesive whole, rather than a collection of unrelated topics. Assessment With each lesson, problems and exercises are assigned from WebWork (homework system) as well as the textbook. Released AP questions are used throughout the course as assessment items, homework and launching points for discussion. Students solve both calculator active and non-calculator problems, and they are required to provide appropriate written presentation of solutions, similar to the requirements of the Free Response section of the AP Calculus Exam. Written justification of calculator solutions are taught and learned so that there is a clear and logical link leading from the mathematics to the technology and then supporting the result. Explaining the why is stressed as much as the how. Examinations are designed to be experiences that allow students to make connections beyond merely learning procedures. Students are required to maintain a Calculus notebook to summarize their learning and provide a valuable resource for preparing for the AP exam. AP Calculus BC Syllabus Page 1
Calculus Classroom Team In order to facilitate student learning and ownership of content, the students are placed into teams. The typical class begins each day with students articulating previously covered topics and discussing homework assignments within their respective team and in whole-class discussion. Homework assignments are designed to reinforce new topics covered. The emphasis for the class is discussion among team partners and fellow teams as opposed to a more traditional direct-lesson approach. Students are expected to take an active part in daily discussions and activities. Technology Instruction will be given using primarily the TI-83/84. This graphing calculator will be used daily in the class. The chapter tests are divided in two parts: one without the use of any calculator and the other part requiring the use of a graphing calculator. The graphing calculator allows the student to support their work graphically, and to make conjectures regarding the behavior of functions, limits, and other topics. This allows students to view problems in a variety of ways. The most basic skills on the calculator: graphing a function with an appropriate window, finding roots and points of intersection, finding numerical derivatives and approximating definite integrals, are mastered by all students. Students have their own calculator and programs, such as Riemann sums, slope fields, and Newton s method, to name a few. The homework delivery system of WebWork will provide a bulk of the practice and grading of the daily exercises. This system provides instant feedback on correctness. Key Curriculum s SketchPad program Calculus in Motion, web based Visual Calculus, and Winplot graphing utility (for student presentations) are also incorporated. Principle Classroom Text Smith, Robert and Minton, Roland; Calculus with Early Transcendental Functions 3E. McGraw Hill Publishing AP Calculus BC Syllabus Page 2
Course Outline Unit 1 A Library of Functions (13 days) (Unit tests for all units are included in days indicated.) 1 st Semester Functions as models of change o Representing functions using the Rule of Four o Domain & range, increasing & decreasing, even & odd, concavity of graphs, zeros, end behavior, asymptotic behavior graphically and in terms of limits involving infinity Linear functions o Slope as a rate of change Exponential functions o Applications Logarithmic functions Trigonometric functions Power functions, polynomials and rational functions Transformation of functions (calculator activity) o Inverse functions o Composition of functions o Shifts, stretches, compressions Working with functions in verbal, graphical, algebraic and tabular depictions Comparing behavior of functions and dominance (calculator activity) o Local and global behavior of functions o Comparing relative magnitudes and their rates of change Introduction to the concept of continuity o Intuitive meaning o Graphical interpretation o Numerical interpretation o Intermediate Value Theorem Calculator Refresher (activity) o Plotting graphs, finding roots, window manipulation, finding values of functions, using tables, lies my calculator told me, dangers of intermediate rounding Unit 2 The Derivative (10 days) Development of the derivative of a function at a point using the derivation of instantaneous speed from average speed o Introduction of the concept of instantaneous rate of change of a function at a point as the slope of the curve of the graph of the function at that point Introduction to limits AP Calculus BC Syllabus Page 3
o Intuitive concept of the limiting process o Calculating limits from numerical data o Calculating limits using algebra o Calculating limits from graphs of functions o Formal definition of limit o Properties of limits o One- and two-sided limits o Proving limits exist o Limits at infinity and end behavior Concept of the derivative o Instantaneous rate of change from average rate of change o Definition of the derivative as the limit of the difference quotient analytical depiction o Graphical depiction of limiting process secant line to tangent line o Determining the derivative of a function numerically o Left- and right-hand derivatives and proving differentiability Derivative of a function at a point o Slope of tangent line to graph of a function at a point including cases where there are vertical or infinitely many tangents o Slope of curve at a point o Approximating rates of change of functions from graphs and tables of data o Finding derivatives of a function at a point using calculators The derivative function o Definition of first and second derivative functions o What derivatives tells us graphically Increasing/decreasing behavior, concavity (signs of f and f ), inflection points o What derivatives tell us about rates of change o Working with derivative functions graphically, analytically, verbally and numerically Understanding the corresponding characteristics between the graphs of f, f and f Interpreting the derivative o Leibniz notation o Dimension analysis o Equations of motion o As rates of change in various applications o Interpreting equations involving derivatives AP Calculus BC Syllabus Page 4
Unit 3 Differentiation Techniques (14 days) Unit 4 Using the Derivative (11 days) verbally and vice versa Continuity o Definition of continuity (limits) o Proving continuity o Continuity of sums, products, and quotients of functions o Continuity of composite functions o Recognizing continuous functions (graphically, algebraically, numerically) o Differentiability implies continuity Finding derivatives for basic functions power, exponential, logarithmic, trigonometric, and inverse trigonometric functions Finding derivatives of sums, products, and quotients of functions o Product and quotient rules Finding derivatives of composite functions using the chain rule Finding derivatives of implicitly defined functions o Using implicit differentiation to find the derivative of an inverse function Linear approximation o Tangent line approximation o Differentiability and local linearity o Using local linearity to find limits L Hopital s Rule Indeterminate forms Demonstrating dominance of functions with L Hopital s Rule Analysis of curves of functions o Monotonicity and concavity/inflection points Local extrema o Critical points o First derivative test o Second derivative test Global extrema and upper and lower bounds of functions Optimization and modeling rates of change o Applications to marginality o Related rate problems (student project) Theorems about continuous and differentiable functions and their geometric consequences o Extreme Value Theorem o Mean Value Theorem o Rolle s Theorem AP Calculus BC Syllabus Page 5
Unit 5 The Definite Integral (6 days) o Increasing Function Theorem o The Racetrack Principle Development of the definite integral as the total accumulated change of a function on an interval using velocity to find distance traveled Graphical development of the definite integral as a limit of Riemann sums o Left- and right-hand sums and limits o Calculating Riemann Sums Interpretation of the definite integral of the rate of change of a quantity over an interval as the change in quantity over the interval: b a f '( x) dx f ( b) f ( a) Interpretation of the definite integral as an area Average value of a function on an interval The Fundamental Theorem of Calculus Computing definite integrals using the Fundamental Theorem, graphically, and numerically Properties of definite integrals o Even and odd functions o Additivity, linearity, multiplying by a constant Comparing definite integrals Using the calculator to evaluate a definite integral Unit 6 Constructing Antiderivatives (8 days) Unit 7 Integration Techniques (13 days) Families of antiderivatives o Visualizing antiderivatives using slopes the graph of f from the graph of f Constructing antiderivatives analytically o Properties of antiderivatives: sums and constant multiples o Antiderivatives of power functions Second Fundamental Theorem o Finding a particular antiderivative with the Fundamental Theorem and analyzing it analytically and graphically Finding specific antiderivatives using initial conditions o Various applications including rectilinear motion Integration following from derivatives of basic functions Integration by substitution o Change of limits for definite integrals Integration by parts Integration using partial fractions Using tables of integrals Approximating definite integrals represented algebraically, graphically, and by tables of values AP Calculus BC Syllabus Page 6
Unit 8 Using the Definite Integral (8 days) o Riemann sums (left, right, midpoint) o Trapezoidal sums o Simpson s Rule o Errors in approximations including graphical interpretation Improper integrals o Convergence and divergence o Graphical interpretation o Evaluating improper integrals as limits of definite integrals o Comparing improper integrals comparison tests Integrals are used to model various real-world phenomena. Emphasis is placed on visualizing models as Riemann sums and representing them with its limit as a definite integral. Applications include: o Areas of regions o Volumes o Arc length, distance traveled, average value of a function on an interval o Volumes of solids of revolution o Volumes of regions of known cross-section o Density and center of mass o Other various applications 6 additional days allotted for 6-week reviews, midterm examinations and semester final examination. AP Calculus BC Syllabus Page 7
Unit 9 Series (10 days) Unit 10 Approximating Functions (8 days) 2 nd Semester Sequences Series defined as a sequence of partial sums Concept of convergence of a series as the limit of the sequence of partial sums. Use of graphic calculator to demonstrate convergence/divergence of sequences and series Examples of applications of series: decimal expansion, etc. Geometric series and computation of sums Applications of geometric series Alternating series, the alternating series test and the alternating series remainder (error bound) Terms of series as areas of rectangles and their relationship to improper integrals (calculator exploration) Development of behavior of p-series. (convergence/divergence) using improper integrals Harmonic series Development of the Integral test Comparison tests (Direct Comparison & Limit Comparison) n-th Term test for divergence Absolute and conditional convergence Ratio & Root tests Telescoping series and partial fractions Functions defined by power series Radius and interval of convergence of power series (includes calculator exploration) Interpretation of radius and interval of convergence Development of the concept of the Taylor polynomial from linear approximation (includes calculator exploration using e x and sine x) through higher order polynomials Application of Taylor polynomial approximations Development of Maclaurin series and the Taylor series centered at x = a from the Taylor polynomial Derivation of Maclaurin series for e x, sin x, cos x, 1/(1-x), etc. Manipulation of known Taylor series including substitution, differentiation, antidifferentiation, multiplication, and division Formation of new series from known series Using the Ratio test for determination of radius and interval of convergence Use of L Hopital s Rule in determining convergence behavior AP Calculus BC Syllabus Page 8
Unit 11 Differential Equations (8 days) Unit 12 Parametric, Vector & Polar Functions (10 days) AP Exam Review & AP Exam (25 days) Post AP Exam Activities (19 days) Lagrange error bound for Taylor polynomials The meaning of differential equations and their solutions o General solutions o Initial value problems, particular solutions Solution of differential equations: o Graphical/geometric interpretation using slope fields to find solution curves given initial conditions o Numerically (Euler s Method) o Analytically (Separation of Variables) Application of differential equations to model real-world phenomenon o Exponential (y =ky) & Logistic models o Models of population growth o Newton s Law of Heating & Cooling o Mock murder trial activity applying Newton s Law Calculator activities for slope fields and developing Euler s Method Review of parametric, polar and vector-valued functions o Applications o Use of calculator to graph functions Derivatives of parametric, polar and vector-valued functions and their interpretations o Includes velocity and acceleration o Modeling projectile motion Integration of functions given in parametric, polar, vectorvalued form o Distance traveled, arc length o Areas of regions bounded by polar curves Students continue to work in teams using previous AP exams and sample tests. A plan, do, study, act (PDSA) approach is used to continually assess progress during the review in order to adapt to students needs. Introduction to Multivariable Calculus o Partial derivatives o Multiple integrals o Applications A brief topical survey of post-calculus mathematics with examples of applications. 6 additional days allotted for 6-week reviews, midterm examinations and semester final examination. AP Calculus BC Syllabus Page 9