A.P. Calculus BC
Course Syllabus for Advanced Placement Calculus BC
Course Overview/Objective
Every concept will be taught starting with a theoretical approach followed with a series of
examples relevant to the theory. It is absolutely imperative that every student understand this
process and review the lecture notes as soon as possible. It is necessary for students to master
the precise terminology used in the text and in class to know how to apply concepts and
understand the relationships between concepts. [C3] [C4]
In this rigorous course I plan to teach students to help one another to succeed as a group, rather
than as an individual.
Course Planner [C2]
Primary Textbook
AP Calculus will be taught using the Fifth Edition of the Stewart textbook “ Calculus,” Early
Transcendentals. There will be some supplemental material used regularly, including previous
AP Exams from the College Board and other sources.
Chapter 11, Infinite Sequences and Series, will be the last chapter we cover in
class.
Chapter 1: Functions and Models (summer)
Students complete this review of Pre - Calculus material over summer.
Chapter 2: Limits and Derivatives (8 days)
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Limits
o Point, infinite, function
o Properties of limits
o Horizontal Asymptotes
Continuity
The Tangent
Rates of Change and Slope
Velocity and other Rates of Change
Chapter 3: Derivatives (15 days)
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Definition of the Derivative
The Derivative as a Function
The Product and Quotient Rules
Derivatives of Transcendental Functions
The Chain Rule
Implicit Differentiation
Related Rates
Chapter 4: Applications of Derivatives (13 days)
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Mean Value Theorem
Extreme Values
o critical points
o monotonic functions
o points of inflection
o concavity
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Tangent line approximation
L’ Hospital’ s Rule
Curve Sketching
Optimization
Newton’ s Method
Antiderivatives
Chapter 5: Integrals (14 days)
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Areas and Distances
Riemann Sums
Definite integrals
Area Accumulator Functions
Fundamental Theorem of Calculus
Indefinite Integrals/Net Change Theorem
The Substitution Rule
Trapezoidal Rule
Chapter 6: Applications of Integration (12 days)
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Areas between curves
Volumes by Cross – sections
Volumes
o Washers
o Shells
Average Value of a Function
Chapter 7: Techniques of Integration (12 days)
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Integration by Parts
Trig. Integrals
Partial Fractions
Strategy for Integration
Improper Integrals
Chapter 8: Further Applications of Integration (4 days)
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Arc Length
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Surface Area
Chapter 9: Differential Equations (12 days)
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Slope Fields
Slope Fields on the Grapher
Euler’ s Method of Approximation
Separable Equations
Exponential Growth/Decay
Logistic Growth Models/Differential Equations
Chapter 10: Parametric Equations and Polar Coordinates (14 days)
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Parametric Curves
Calculus of Parametric Curves
Polar Coordinates
Areas and Lengths in Polar Coordinates
Chapter 12 & 13: Vectors and Vector Functions (8 days)
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Review Vectors
Vector Functions
Derivatives and Integrals
Arc Length and Curvature
Motion in Space
o Position, velocity, acceleration vectors
o Speed
o Total Distance Traveled
Chapter 11: Infinite Sequences and Series (15 days)
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Sequences
Series
Geometric Series
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Tests for Convergence/Divergence
o Nth root test
o P – series test
o Integral test
o Comparison test/Limit comparison test
o Alternating series test
o Ratio and Root test
Radius and Interval of Convergence
Power Series
o Differentiation/Integration
Representations of Functions as Power Series
Taylor Series
Maclaurin Series
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Taylor Polynomials/Maclaurin Polynomials
Lagrange Error Bounds
Applications of Taylor Polynomials
Teaching Strategies
The outline shown above is a list of all the topics from the Stewart textbook. We also cover topics
that aren’ t necessarily mentioned in the AP Calculus Course Description. The reason to cover
these additional topics is to give students alternate approaches that they can rely upon if needed.
The Course Planner provides students with a general list of the sections covered in this advanced
placement course consistent with the primary textbook. The course also provides students with a
variety of representations of functions as the AP course requirements claim.
Verbal descriptions of functions representing rates of change and area accumulator functions are
used often, especially during lessons and reviews. Students should know how to interpret rates
and use appropriate terms to describe how these rates change. The instructor will clearly define
these skills in class. [C4]
AP Exam Review
The class meets regularly after school for AP reviews. In these sessions, students receive
packets of Multiple – Choice questions and Free – Response questions pertinent to the topic
being discussed in class at that time. During these sessions, students work in groups and
individually to discuss solution approaches to the problems. [C4]
As we approach the actual test date, students are encouraged to begin individual preparation by
purchasing an AP preparation guide to assist in preparing for the exam. These guides provide
sample problems that students need to practice.
Depending on the available time in class, students are provided with drills consisting of Multiple –
Choice and Free – Response questions to determine their progress, or status for the AP exam.
These drills maybe graded. Drills are at times graded by students to give them the opportunity to
see exactly how the college – board has graded free – response questions. [C4]
Representations of Functions
The graphing calculator allows students to analyze graphs of functions, derivatives of functions
and antiderivatives of functions. The TABLE feature in the grapher can also be used to analyze
change in behavior of derivative functions and/or integral functions by simply analyzing table
values. This seems to be an important skill necessary for the AP exam. [C3]
Student Activity
We will have several student activities in the class, depending on time. One student activity is
“ Slope Fields.” This activity is done in groups and takes a whole block period. Students are
given graphs of slope fields and asked to analyze the slopes and predict the possible solutions of
the direction fields. This introduces Initial Value Problems, which is an important type of
questions for Differential Equations. [C3] [C4] [C5]
Graphing Calculator
The use of a graphing calculator in AP Calculus is considered an integral part of the course.
Students should be using this technology regularly to become familiar with all of the relevant
functions it contains. Remember, the graphing calculator is required for the AP Exam. . All
students will be required to bring a TI-83, TI-83 Plus, or a TI-89 EVERY day to class.
The TABLE feature in the calculator also allows students to view functions described numerically.
The slope field program provides students with different viewing windows to analyze slopes and
make conjectures regarding change in slopes. The most interesting feature of the graphing
calculator is the feature that allows students to graph the derivative and anti-derivative. This
feature can assist students in analyzing slopes, concavity, points of inflection, and extrema. [C3]
[C5]
The graphing calculator can also be used to support an answer. The graphing calculator gives
students confidence with their answers. It seems like when students use their calculator to find
answers they feel more comfortable with their results. [C4]
Student Evaluation
All exams and quizzes will be announced. Some exams are taken in groups, and all quizzes will
be taken individually. All exams will contain authentic multiple – choice problems and free
response problems, consistent with the actual AP exam. Every exam will be graded on the curve.
There will be a First Quarter Exam, Semester Exam, Final Exam (Actual A.P. Exam). All
these exams are taken individually. All exams are comprehensive exams. [C4]
There will be three take – home quizzes, one each quarter. Students are encouraged to discuss
approaches to solving the problems on the quizzes, but must submit their own solutions. [C4]
Web Resources:
o www.calculusinmotion.com/Sketchpad
o www.collegeboard.com, and www.apcentral.collegeboard.com
Curricular Requirements
These five Curricular Requirements are referenced in this syllabus.
[C1] The teacher has read the most recent AP Calculus Course Description, available to
download on the AP Calculus BC Course Home Page.
[C2] The course teaches all topics associated with Functions, Graphs, and Limits;
Derivatives;
Integrals; and Polynomial Approximations and Series as delineated in the
Calculus BC Topics Outline in the AP Calculus Course Description.
[C3] Evidence of Curricular Requirement: The course provides students with the
opportunity to work with functions represented in a variety of ways – graphically,
numerically, analytically, and verbally – and emphasizes the connections among these
representations.
[C4] The course teaches students how to communicate mathematics and explain
solutions to problems both verbally and in written sentences.
[C5] The course teaches students how to use graphing calculators to help solve
problems, experiment, interpret results, and support conclusions.