AP Calculus AB
This is a rigorous course for the mathematically advanced student capable of college-level work. In order to enroll in the course, students must have successfully completed Precalculus. This course prepares students to take the Advanced Placement Calculus AB Examination administered by the College Entrance Examination Board (CEEB) in the spring.
Jean Adams (Flamingo Math) Materials
Calc-Medic Lesson Plans
Stewart, James. Calculus, 7th edition. Belmont, California: Brooks/Cole, 2012. (And all ancillary materials)
Rogawski, Jon and Ray Cannon. Rogawski's Calculus Early Transcendentals for AP. New York: W. H. Freeman, 2012.
Larson, Ron and Bruce Edwards. Calculus of a Single Variable (AP Edition) 9th edition. New York: Brooks/Cole. 2010
McMullin, Lin, Teaching AP Calculus 2nd edition. New York: D&S Marketing. 2005.
College Board. AP Calculus Free Response Questions.
Students are required to bring a graphing calculator to each class meeting. Students are expected to have a calculator available for homework assignments, as well. Students who do not own a graphing calculator are provided one for use in class and able to check one out for at home assignments. The TI-83/84 series is recommended calculator. A class set of TI-Nspire calculators will also be used for investigations and activities in class. The calculator will be used as a tool to illustrate ideas and make discoveries about functions. The four required functionalities of a graphing calculator for this course are:
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
Students are also required to make connections between the graphs of functions and their analysis, and conclusions about the behavior of functions when using a graphing calculator.
Students taking AP Calculus are expected to have had a strong Algebra and Precalculus foundation. Students will participate in exploration and discovery activities. Instruction will often be facilitative, rather than informative. Students will frequently work in pairs or small groups to create an understanding of calculus concepts and methods as they participate in teacher-guided activities. A strong emphasis is placed on the student’s knowledge of how all course topics are connected and interrelated and to see calculus as a whole.
Current mathematical education emphasizes a “Rule of Four,” a multirepresentational approach to problem solving. The topics addressed in this course will be presented in and approached from all four representations:
1) Numerical analysis
2) Graphical analysis
3) Analytic/algebraic analysis
4) Verbal/written methods of representing problems and solutions
There will also be a strong emphasis in making connections among the various representations.
Grades will include major tests, quizzes, cumulative quizzes, free response assignments, and projects/activities.
Assignments will fall into one of two categories: major (tests, major projects) or minor (quizzes, free-response, minor projects). Major assessments will account for 80% of a student’s grade, and minor assessments will account for the remaining 20%.
Matching Functions and Derivatives (adapted from “Teaching AP Calculus” by Lin McMullin)
In this activity, students will be grouped in pairs and given several sets of cards printed on cardstock. Each set of cards will consist of the following five cards:
1. the equation of a function
2. a written description of its graph
3. the graph of the function
4. the graph of its first derivative
5. the graph of the second derivative
The students will receive several sets of cards that are already cut apart and mixed up. Without the use of a graphing calculator, students will match each equation with is description and the three graphs that correspond to it (f, f’, and f’’).
Slope Fields Forever (a TI-Nspire Activity from TImath.com)
In this activity, students will dynamically explore a particular solution to a differential equation for different initial conditions. Slope fields are investigated. Self-check multiple-choice questions help students build the foundation for the second part of the activity, a Differential Equation (DE) matching activity. In the DE matching activity students will use the strategies they have developed from the self-check question to identifying the slope field that corresponds to a differential equation.
Unit 1: Limits and Continuity (4 weeks)
Students will discover several methods of finding a limit of a function. They will investigate limits and their properties.
A. Rates of change
B. Limits at a point
1. Properties of limits
2. Two-sided
3. One-sided
C. Limits involving infinity
1. Asymptotic behavior
2. End behavior
3. Properties of limits
4. Visualizing limits
D. Continuity
1. Continuous functions
2. Discontinuous functions
a. Removable discontinuity
b. Jump discontinuity
c. Infinite discontinuity
Unit 2: The Derivative (3 weeks)
Students will discover the concept of the derivative graphically, numerically, and analytically.
A. Instantaneous rates of change
B. Definition of the derivative
C. Differentiability
1. Local linearity
2. Numeric derivatives using the calculator
3. Differentiability and continuity
D. Derivatives of algebraic functions
E. Derivative rules when combining functions
F. Applications to velocity and acceleration
G. Derivatives of trigonometric functions
H. Derivatives of logarithmic and exponential functions
Unit 3: Differentiation: Composite, Implicit and Inverse Functions (2 weeks)
Students learn how to differentiate composite functions using the chain rule and apply that understanding to determine derivatives of implicit and inverse functions.
A. The chain rule
B. Implicit derivatives
1. Differential method
2. yʹ method
C. Derivatives of inverse trigonometric functions
Unit 4: Contextual Applications of the Derivative (2 weeks)
Students will investigate applications of the derivative, including curve sketching, optimization and related rates.
A. Interpreting the mean of the derivative in context
B. Straight-line motion
C. Rates of change in applied contexts (other that motion)
D. Related Rates
E. Linearization models
F. L’Hospital’s Rule
Unit 5: Analytical Applications of Differentiation (3 weeks)
Students will focus on abstract structures and formal conclusions (reasoning with definitions and theorems).
Unit 6: Integration and Accumulation of Change (4 weeks)
Students will understand the concept of integration as the antiderivative, and representation of the limit of an approximating Riemann sum. Students will establish the relationship between differentiation and integration using the Fundamental Theorem of Calculus.
Unit 7: Differential Equations (2 weeks)
Students will learn to set up and solve separable differential equations
Unit 8: Applications of Integration (4 weeks)
Students will learn how to find the average value of a function, model particle motion and net change, and determine areas and volumes defined by the graphs of functions
A. The Average Value Theorem
B. Connecting position, velocity and acceleration
C. Using accumulation and definite integrals in applied contexts
D. Areas between curves
E. Volumes
1. Volumes of solids with known cross sections.
2. Volumes of solids of revolution (disk method)
Remaining time prior to the administration of the AP examination is spent reviewing past AP multiple choice and free response questions and taking practice exams. These are assigned as homework as well as assessed through quizzes and tests.