Advanced Placement Calculus BC Syllabus
Introduction
In this course, we cover the topics that are listed in The Calculus BC Course Description
posted on the AP Central website. We divide our studies into three major topics: limits,
differential calculus, and integral calculus. An understanding of limits is the basis for future
concepts in calculus. Differential calculus enables us to calculate rates of change, to find the
slope of a curve, and to calculate velocities and accelerations of moving bodies. Integral
calculus is used to find the area of an irregular region in a plane and volumes of irregular solids.
In this course students acquire knowledge of the basic mechanics of limits, derivatives,
and antiderivatives. Additionally students develop an understanding of the theory of calculus.
Each student then uses these concepts in applying calculus to multiple applications.
Problems are presented and solved in four distinct ways: analytically, numerically,
graphically, and verbally. Students will use a graphing calculator to examine continuity and
local linearity, to create tables of values for functions and relations, to solve an equation, to
graph a function in any window, to determine the value of a derivative at a point, and to find the
value of a definite integral. Each student is required to have his or her own graphing calculator.
A TI-84 or TI-89 is required. An overhead graphing calculator (TI-84 and TI-89) is available for
use.
The first seven months of the class are devoted to studying the topics covered in a typical
college Calculus I and II course. The next month is review and preparation for the AP exam.
Throughout the year, information concerning the administration, scoring, and content of the
exam are discussed and examples are provided for student analysis. After taking the exam in
May, students will complete a project and study additional topics in calculus not included in
Calculus BC.
The school operates on a 4-day rotation of classes. Each class is held for 45-minutes on
two of the days, 90-minutes on one of the days, and not at all on the last day of the cycle. This
amounts to an average of 45 minutes per day, but gives the flexibility to complete learning
activities that require a longer amount of time occasionally.
Topics Covered:
Review of Pre-Calculus Math
5 days
Students are given a packet of exercises to complete during the summer. This packet is
reviewed in class and assessed as a test grade. The topics include: graphing equations, intercepts,
intersection points, symmetry, mathematical models, equations of lines, parallel and
perpendicular, domain and range, trigonometric functions, line of best fit, transformations of
functions, and parametrically defined functions.
Limits and Their Properties
• General properties of limits
10 days
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Finding limits algebraically, analytically and graphically
Formal definition of limits (ε - δ) including proofs of limits of linear functions
Horizontal and vertical asymptotes
Continuity
Removable, jump, and infinite discontinuities
Infinite limits
Theorems involving limits: Squeeze Theorem, Extreme Value Theorem, and the
Intermediate Value Theorem.
Differentiation
• Definition of the derivative
• Relating the graph of a function with the derivative
• Non-differentiability
• Basic Differentiation Rules
• Rates of change
• The Product Rule ad Quotient Rule
• Derivatives of trigonometric functions
• Higher order derivatives
• The Chain Rule
• Implicit Differentiation
• Related Rates
• Rectilinear motion
15 days
Applications of Differentiation
20 days
• Extrema on a closed interval and critical points
• Rolle’s Theorem
• The Mean Value Theorem
• Increasing and decreasing functions and the First Derivative Test
• Concavity and the Second Derivative Test
• Limits at infinity (horizontal asymptotes, slant asymptotes and other end behavior)
• Summary of curve sketching
• Optimization problems
• Newton’s Method
• Differentials
Integration
15 days
• Definition of antiderivative
• Basic integration rules
• Initial conditions and particular solutions
• Area approximation using rectangles
• Riemann Sums and definite integrals
• The Fundamental Theorem of Calculus
• Integration by Substitution (both definite and indefinite integrals)
• Numerical Integration
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Integration by Partial Fractions
Integration by Parts
Logarithmic, Exponential, and Other Transcendental Functions
• The Natural Logarithmic Function: Differentiation
• The Natural Logarithmic Function: Integration
• Inverse Functions
• Exponential Functions: Differentiation and Integration
• Bases other than e
• Applications of logarithmic and exponential functions
• Inverse Trig functions: Differentiation
• Inverse Trig Functions: Integration
• Hyperbolic Functions
10 days
Differential Equations
• Slope Fields
• Euler’s Method
• Growth and Decay Models
• Separation of Variables
• Logistic Growth
• First-Order Linear Differential Equations
8 days
Applications of Integration
• Area of region between two curves
• Volume: The Disk and Washer Methods
• Volume: The Shell Method
• Volumes of solids with known cross sections
• Arc Length and Surfaces of Revolution
• Work
• Moments, Centers of Mass, and Centroids
• Fluid Pressure and Fluid Force
15 days
Integration Techniques, L’Hôpital’s Rule, and Improper Integrals
• Review of basic integration rules
• Integration by parts
• Trigonometric Integrals
• Integration using trigonometric substitution
• Integration using partial fractions
• Integration by tables and other integration techniques
• Applying L’Hôpital’s rule to indeterminate forms of limits
• Improper integrals
15 days
Infinite Series
• Sequences
• Series and Convergence
• The Integral test and p-series
• Comparisons of series
• Alternating series
• The Ratio and Root tests
• Taylor Polynomials and approximations
• Power series
• Representation of functions by Power Series
• Taylor and Maclaurin Series
20 days
Parametric Equations and Polar Coordinates
8 days
• Defining a function parametrically and eliminating the parameter
• Find the first and second derivative (with respect to x) of a parametrically defined
function
• Finding arc length and the area of a surface of revolution using paramterics
• Polar equations and graphs
• Tangent lines to polar equations, area of polar regions and arc length of polar graphs
Vectors
• Component form of vectors
• Basic vector operations
• Space coordinates and vectors in space
• Vector valued functions
• Differentiation and integration of vector valued functions
• Velocity and acceleration vectors
• Tangent and normal vectors
• Arc length of plane and space curves
10 days
Review for AP Exam:
During the course of the school year, students work individually, in pairs or in small groups to
complete many released free-response questions as they pertain to the topics that have already
been taught. They are required to show a “complete” solution to each of the problems and
evaluate their responses in comparison to the solutions provided and example student solutions,
where applicable. On occasion, students are required to present problems to the rest of the class
and use precise mathematical language in their presentations. It is important that students
understand all of the components that must be included when completing the free-response
questions. As the AP Exam date draws closer, the extended time block periods in our schedule
are used to take simulated parts of the AP Exam and the shorter periods are used to review
important concepts and common mistakes that are made.
Student Evaluation:
Weekly assessments are given; some allow the use of a graphing calculator, while some do not.
Quizzes are approximately 20% of the marking period grade, approximately 20% are
assignments, the remaining 60% is test grades. In each assessment students are expected to
analyze problems graphically, analytically, verbally, and numerically. Students are also often
required to provide written explanation of their solutions, explaining the process used to solve
various problems. Some class assessments are given that require students to work in pairs or
small groups to solve problems. They are encouraged to discuss the problems in depth and come
to consensus on the process used and the solution. These small groups are then required to
present their problem back to the whole class, using precise mathematical language in their
explanations. All unit assessments are written in the same format as the AP Exam, just in a
shorter form for classroom use. This helps to prepare students for the rigor that is expected on the
AP Exam. Each marking period grade is 20% of the final grade, the midterm exam grade is 10%
and the final exam grade is 10%.
Teaching Strategies:
Most units consist of a day of lecture followed by 1 or 2 days of clarification. Students are
encouraged to ask questions at any time. The students are divided into small study groups. They
sit together in class and work together out of class to increase understanding. The members are
responsible for each other and work well together. In class, material is presented algebraically,
numerically, graphically and verbally. Students are expected to be able to work with equations,
graphs, and tables equally well. In addition, they are expected to be able to verbalize the
information presented and solutions given. Students are often expected to submit written
explanations of their problem solutions and/or present their problem solutions to the class.
Technology:
Each student owns his/her own graphing calculator and is expected to be familiar with its general
use. Most own a TI-84 or TI-89. In addition, a classroom set of TI-84’s are available. Overhead
calculators (TI-84 and TI-89) are available. Students work with their calculators on a daily basis
and must be able to explain mathematically how the calculator was used to assist in solving a
problem, without using “calculator speak.” They are able to graph functions in an arbitrary
window, find roots and zeros of a function, produce tables of functions fro various domains, find
the derivative at a point, and find the definite integral of a function. Students are expected to be
able to interpret the results obtained from their graphing calculator and are able to support, in
writing or verbally, the conclusions that were drawn. Often times, the graphing calculator allows
us to investigate concepts in more depth and lead to other discoveries or questions; these
teachable moments are valuable in the daily instruction process.
Student Activities:
Many of the Section Projects provided in the Larson text are used in class with students working
in small groups. These projects and activities help students to gain a better understanding of the
topics covered and allow students to see where the topics have application and relevancy to
various real-world situations. Students are required to apply newly gained knowledge to a variety
of problems. These activities also provide opportunities for students to demonstrate their
knowledge through written explanations of the processes used in solving a particular problem.
Resources:
Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus of a Single Variable.
Boston: Houghton Mifflin, 2006, 8th edition.
Lederman, David. Multiple-Choice and Free Response Questions in Preparation for the AP
Calculus (AB) Examination. D & S Marketing, 2011, 9th edition.
McMullin, Lin. Visualizing Differential Equations: Slope Fields. College Entrance
Examination Board, 2004.
Fraga, Robert. Calculus Problems for a New Century. MAA Notes Volume 28, 1999
AP Examination Released Problems. AP Central