Download AP Calculus BC

®
AP Calculus BC
Syllabus 1
Course Overview
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The main objective in teaching AP Calculus is to enable students to appreciate the
strength of calculus and receive a thorough foundation that will give them the tools to
succeed in future mathematics courses. Calculus with Analytic Geometry II, includes the
study of real numbers, limits, continuity, series, sequences, vector, parametric equations,
polar equations, differential and integral calculus of functions of one variable.
Textbook: Calculus, 2nd Edition, Robert T. Smith and Roland B. Minton,
McGraw Hill
Technology:
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All students have a TI-83/TI-83+/TI84 graphing calculator for use in class, at
home, and on the AP exam. Students will use their graphing calculator
extensively throughout the course.
Various applets on the Internet
APCD software
Course Planner [C2]
A Library of Calculus AB
1st Quarter – Review of AB Concepts – They include:
 Concept of Limits
 Computation of Limits
 Continuity and its consequences
 Limits involving infinity
 Formal Definition of a limit
 Tangent Lines and Velocity
 The Derivative
 The Derivative at a point
 Powers and Polynomials
 The Exponential/logarithmic Function
 The Product and Quotient Rules
 The Chain Rule
 The Trigonometric Functions
 Applications of the Chain Rule and Related Rates
 Implicit Functions
 Mean Value Theorem
 Linear Approximation and the Derivative
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L’Hopital’s Rule
Newton’s Method
Using Local Linearity to Find Limits
Using First and Second Derivatives
Families of Curves
Optimization and Modeling
How Do We Measure Distance Traveled?
Antiderivatives Graphically and Numerically
Constructing Antiderivatives Analytically
Sums and Sigma Notation
Reimann Sums
Fundamental Theorem of Calculus
The Definite Integral
Interpretations of the Definite Integral
Theorems About Definite Integrals
Integration by Substitution
Differential Equations
Second Fundamental Theorem of Calculus
The Equations of Motion
Numerical Integration
Area and Volumes
Arc Lengths and Surface Areas
Projectile Motion
Work, Moments and Hydrostatic Force
Probability
Natural Logarithm
Inverse Functions
The Exponential Function Revisited
Growth and Decay Problems
Separable Differential Equations
Euler’s Method
Slope Fields
Inverse Trig
Key Concept: Integration Techniques (Chapter 7) [C2] [C5] [C4] [C3]
1st Quarter
1) 1. Review of Formulas and Techniques
2) 2. Integration by Parts
3) 3. Trigonometric Techniques of Integration
4) 4. Partial Fractions
5) 5. Indeterminate Forms and L’Hopital’s Rule
6) 6. Improper Integrals
Key Concept: Infinite Series (Chapter 8) [C2] [C5] [C4] [C3]
2nd Quarter
1. Sequences of Real Numbers
2. Infinite Series
3. The Integral Test and Comparison Test
4. Alternating Series
5. Absolute Convergence and the Ratio Test
6. Root Test
7. Power Series
8. Taylor Series
9. Applications of Taylor Series
LAB EXAMPLE: The Leaning Tower Project – The objective of this project is to
have the students stack objects so that the top block is one length away from the
bottom block without having the stack fall over. The main idea to get across is the
idea of centers of mass. The stack balances if the center of mass is over the table,
otherwise it falls. I have the students prepare a general problem with n things to stack
and have them determine the farthest that they can go. This allows them to sum the
harmonic series. The close of this lab has the students write out their response to the
following question: If you started adding up the harmonic series in the year 4000 BC,
and could add up ten terms per second, what would be the total today?
Midterm Exam
Key Concept: Parametric Equations and Polar Coordinates [C2] [C5] [C4] [C3]
(Chapter 9)
3rd Quarter
1. Plane Curves and Parametric Equations
2. Calculus and Parametric Equations
3. Arc Length and Surface Area in Parametric Equations
4. Polar Coordinates
5. Calculus and Polar Coordinates
6. Conic Sections
7. Conic Sections in Polar Coordinates
LAB EXAMPLE: I give the students selected problems with parametric and polar
functions and have them work in small groups. Once they have completed all the
problems I randomly select groups to present their findings. We use white boards for
them to illustrate their work.
Key Concept: Vector and the Geometry of Space (Chapter 10) [C2] [C5] [C4]
[C3]
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3 Quarter/4 Quarter
1. Vectors in the Plane
2. Vectors in Space
3. Dot Product
4. Cross Product
5. Lines and Plane in Space
6. Surfaces in Space
LAB EXAMPLE: The students are given real world applications and work in groups to
solve their particular problem. Two students from each group then rotate from group to
group to explain their problem. After the students have had time to complete this
exercise, I have them research other real world applications involving vectors. They are
required to write about their research and present their findings to the class.
Key Concept: Vector Valued Functions (Chapter 11) [C2] [C5] [C4] [C3]
4th Quarter
1. Vector-Valued Functions
2. The Calculus of Vector-Valued Functions
3. Motion in Space
4. Curvature
5. Tangent and Normal Vectors
Key Concept: Functions of Several Variables and Partial [C2] [C5] [C4] [C3]
Differentiation (Chapter 12)
4th Quarter
1. Functions of Several Variables
2. Limits and Continuity
3. Partial Derivatives
4. Tangents Planes and Linear Approximations
Final Exam
Teaching Strategies
During the first nine weeks of school I start with review of all the AB topics that these
students have studied in the previous year. This requires students to use their prior
knowledge about all types of functions to discuss derivatives, anti derivatives, parametric
relations, limits, etc. The students work on review packets for a time and then I have
them work together in groups to discuss their findings. This is how I structure the class
for the rest of the year. I incorporate the use of technology throughout the year so that
students have multiple means of interpreting a problem. I encourage students to explore
and discover throughout the course.
I like to incorporate whiteboard activities while learning new topics. Sometimes I will
give the entire class the same problem but have them work on it in groups of two. They
then have to write up their findings on the whiteboard. Once all are done I call for a
“board meeting”. All the students hold up their board and we then discuss the different
findings. The students enjoy this atmosphere. I use technology throughout the year.
These include TI series calculators (this varies depending on which one the student
owns), TI InterActive!, APCD, CBL’s, and LabPro. I use four simple rules when
presenting topics. They are graphically, numerically, algebraically, and verbally. This
allows the students to gain an in-depth understanding of the material.
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 Topic Outline in the AP Calculus Course Description.
C5—The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support
conclusions.
C4—The course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written
sentences.
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.