Syllabus: AP® Calculus BC
Course Description/Objective
This course covers all topics included in the Calculus BC topic outline as it appears in the
AP Calculus Course Description.
This course is designed and taught to give students a complete and sound understanding of
calculus concepts, all related mathematical skills and the necessary use of technology to achieve
a high level of success on the Advanced Placement Exam and in subsequent college mathematics
courses.
®
All students are expected to prepare for and take the AP exam at the conclusion of the
school year.
The textbook used for this course is: Finney, Ross L., Franklin D. Demana, Bert K. Waits,
and Daniel Kennedy. Calculus—Graphical, Numerical, Algebraic. 3rd ed. Pearson Prentice Hall,
Boston. 2007.
Supplemental Resources:
David Lederman. Multiple-Choice & Free-Response Questions in Preparation for the AP
Calculus BC Examination, 8th edition. D & S Marketing Systems, Inc. Brooklyn, NY.
www.dsmarketing.com
Tamara Lefcourt Ruby, James Sellers, Lisa Korf, Jeremy Van Horn, and Mike Munn. AP
Calculus AB & BC, 2006 edition. Simon & Schuster, New York. ISBN 13:978-0-7432-6575-1/
®
Course Outline/Timeline [# of class periods are approximate]
Chapter 2: Limits and Continuity [5 class periods]
A. Rates of Change and Limits
1. Average and Instantaneous Speed
2. Definition of Limit
3. Sandwich Theorem
B. Limits Involving Infinity
1. Finite Limits
2. Infinite Limits
3. End Behavior Models
C. Continuity
1. Continuity at a Point
2. Continuous Functions
3. Algebraic Combinations
4. Composites
5. Intermediate Value Theorem
D. Rates of Change and Tangent Lines
1. Average Rates of Change
2. Tangent to a Curve
3. Slope of a Curve
4. Normal to a Curve
1
Chapter 3: Derivatives [14 class periods]
A. Derivative of a Function
1. Definition
2. Notation
3. Relationship Between the graphs of f and f’
4. Graphing the Derivative from Data
5. One-sided Derivative
B. Differentiability
1. How f’(a) Might Fail to Exist
2. Differentiability Implies Local Linearity
3. Derivatives on a Calculator (Numerical and Graphical)
4. Differentiability Implies Continuity
5. Intermediate Value Theorem for Derivatives
C. Differentiation Rules
1. Positive Integer Powers, Multiples, Sums, and Differences
2. Products and Quotients
3. Negative Integer Powers of x
4. Second and Higher Order Derivatives
D. Velocity and Other Rates of Change
1. Instantaneous Rates of Change
2. Motion along a Line (position, velocity, acceleration)
3. Slopes of parametric curves vs. velocity, two-dimensional particle motion
and vector forms (position, velocity, acceleration )
4. Application of Derivatives in Economics
E. Derivatives of Trigonometric Functions
1. Sine and Cosine Functions
2. Simple Harmonic Motion
3. Other Basic Trigonometric Functions
F. Chain Rule
1. Composite Functions
2. Power Chain Rule
G. Implicit Differentiation
1. Implicitly Defined Functions
2. Lenses, Tangents, and Normal Lines
3. Derivatives of Higher Order
4. Rational Powers of Differentiable Functions
H. Derivatives of Inverse Trigonometric Functions
I. Derivatives of Exponential and Logarithmic Functions
Chapter 4: Applications of Derivatives [8 class periods]
A. Extreme Values of Functions
1. Absolute Extreme Value
2. Local Extreme Value
B. Mean Value Theorem
1. Increasing and Decreasing Functions
2. Other Consequences
2
C. Connecting f’ and f’’ with the Graph of f
1. First Derivative Test
2. Concavity
3. Points of Inflection
4. Second Derivative Test
D. Modeling and Optimization
E. Linearization and Newton’s Method
F. Related Rates of Change (Various Word Problems)
G. Introduction to L’Hopital’s Rule for some indeterminate forms
Chapter 5: Definite Integrals [7 class periods]
A. Estimating with Finite Sums
B. Definite Integrals
1. Riemann Sum
2. Notation of Integration
C. Definite Integrals and Antiderivatives
1. Properties
2. Average Value of a Function
D. The Fundamental Theorem of Calculus
1. Part I
2. Part II
3. Analyzing Antiderivatives Graphically
E. The Trapezoidal Rule
Chapter 6: Differential Equations and Mathematical Modeling [7 class periods]
A. Slope Fields and Particular Solutions given an Initial Condition
B. Integration by Substitution with change of limits
C. Integration by Parts
D. Exponential Growth and Decay
E. Numerical Methods
F. Slope Fields, again, with Euler’s Method to determine a solution curve to a differential
equation
G. Anti-differentiation by Partial Fractions (simple non-repeating linear factors)
H. Logistic Differential Equations
Chapter 7: Applications of Definite Integrals [9 class periods]
A. Integral as a Net Change
B. Areas in Planes
1. Area Between Curves
2. Area Enclosed by Intersecting Curves
3. Integration with Respect to y
C. Volumes
1. Volumes of Solids with Known Cross Sections
2. Volumes of Solids of Revolution
3
D. Lengths of Curves including Parametrically Defined Curves
E. Area of a Region Bounded by Polar Curves
F. One Dimensional Particle Motion Revisited to include Distance Traveled and
Displacement
G. Integrals as Accumulators of Change; Definite Integral of the Rate of Change over an
Interval as a Measure of Accumulated Change (as many “real world” applications as
possible)
H. Analysis of Functions Defined by Integrals
Chapter 8 & 9: Preparation for Series [6 class periods]
A. L’Hopital’s Rule Revisited – More Indeterminate Forms
B. Relative Magnitudes and Rates of Growth of Various Functions
C. Sequences: Notation, Limits, Arithmetic & Geometric Sequences
D. Improper Integrals (using limits of definite integrals to solve and incorporating
L’Hopital’s Rule)
E. Convergence and Divergence of Improper Integrals
(including the direct comparison test)
Chapter 9: Infinite Series [11 class periods]
A. Definition and Concept of a Series: Notation, Terminology: Common Use of Series
(decimal expansion)
B. Informal Techniques for Determining Convergence or Divergence (Graphical and
Numerical Techniques); Convergence Defined by a Limit
C. Geometric Series, including its Characteristics and Applications
D. Representing Functions by Power Series
E. Manipulation of Power Series by Substitution, Differentiation, and Integration to
Produce New Series from known ones
F. Maclaurin and Taylor Series for ex, sin x, cos x, ln(1+x)
G. Taylor Polynomial Approximation and its Convergence with the Function it
Approximates
H. General Rule for Producing Maclaurin Series and General Taylor Series Centered at
x=a
I. Remainder and Lagrange Error Bound for Taylor Polynomials
J. Radius and Interval of Convergence of Power Series
K. Tests for Convergence and Checking Endpoints (Direct Comparison Test, Ratio Test,
Integral Test and its Basis in Rectangular Areas and Approximations of Improper
Integrals, Limit Comparison Test)
L. Harmonic Series, p-series, and Alternating Series (including its error bound)
®
AP Exam Review (Remaining Classes are used on Review and Practice for AP
4
®
Exam)
Teaching Methods
At the beginning of the year, students are given a list of pre-calculus and calculus terms
formulas, and identities to be memorized and understood. They are also guided to reconstruct
the unit circle to re-familiarize themselves with key trigonometric values and relationships. As
the year progresses, they are repeatedly asked to recite from memory any and all of this valuable
information.
With few exceptions, most lessons are initially presented in a lecture/discussion format
using any or all of the following media tools: Smart Board, PowerPoint, TI-Smart View, TI
Navigator System and TI CBL units. In the days following each presentation/lesson, students are
directed to work either alone or collectively, through various problems which reinforce the
conceptual, mechanical, algebraic, and analytical skills needed to understand the concepts which
are being studied. Graphing technology is used extensively everyday to explore, investigate, and
reinforce concepts being studied. The school insures that each student has the use of a TI-83+ or
®
TI-84+ calculator for use at home, in class and on the AP exam. Students are instructed on how
to use these calculators for each application. All students are asked to model their understanding
of concepts and solutions through various methods including but not limited to: Numerical,
Analytical, Verbal, Graphical, and Written (In the form of complete sentences/paragraphs), and
Pictorial.
Various Approaches
Throughout the course, students are required to use multiple approaches to the
understanding of functions. Graphs are produced both with the calculator and by hand to assist
in the understanding of problems. Students use the graphing calculator to experimentally
determine solutions to problems and to interpret the results. Students also learn to use the
calculator to support answers and conclusions that they have developed analytically.
Numerical solutions are developed both manually and with the calculator. Students are
encouraged to check the reasonableness of their numerical solutions by using other approaches.
Students use analytical techniques to solve applied problems in differential and integral
calculus. They are encouraged to develop general analytical approaches that can by applied to
non-traditional problems.
Students are asked to explain calculus problems and techniques verbally and in writing.
Technology
Students usually purchase their own calculators however calculators are provided for
students for whom this is a financial burden. Most homework problems are clearly identified as
being “calculator allowed” or “non-calculator” problems. Students are encouraged to develop a
clear sense of when it is appropriate to use a calculator and when a calculator is not appropriate.
Tests are divided into calculator and non-calculator sections.
Clear examples of graphing calculator use are included in the lectures. In addition, the
teacher uses a TI View Screen in combination with an overhead projector to present additional
5
examples and to explore concepts of calculus using the graphing calculator. Appropriate
calculator software is provided to the students by the teacher.
Students are given a personal copy of the PowerPoint lectures. They are able to review
these lectures at home for use as a review tool, and to make-up lectures that they have missed
due to absences.
Student Evaluation
Our school is on an A/B block schedule. Each class meets every other day for 90 minutes.
The school year is divided into two semesters with 2 nine-week grading periods per semester.
Each nine-week grade is 40% of the semester average with a semester exam weighted at 20% of
the semester average. Each class typically meets 42-45 times per semester. Each nine-week
grade is calculated from daily homework assignments, quizzes, class activities, and major tests.
®
Each major test is patterned after the AP exam with both multiple choice and free response
questions. The free response questions require the student to explain and justify their solutions
in written form through the use of clear and concise sentences, and by showing written evidence
of all meaningful and necessary calculations. Both the multiple choice section as well as the free
response section are split between calculator and non-calculator use.
6