AP Calculus BC
Course Syllabus
Course Overview: The main goal of this course is for students to learn all topics
included within the first two semesters of college level Calculus. However, nested within
this goal is the idea that students will become more proficient thinkers when they finish
the entire Calculus BC syllabus. Students in this course will also be a part of a dynamic
community of analytic thinkers, constantly challenging their preconceived notions of
what it means to really understand high level mathematics. For many of my students, this
class is the crescendo of their own personal symphony of mathematics that they have
encountered in their lives. The actual music that is made is provided in the daily
interactions and true learning that occurs in the classroom.
The class itself is constantly evolving with regards to the ebb and flow of the daily
syllabus and in the manner that the lessons are presented to the students. I always stick to
my general guidelines for how many days to stick to the material, but rarely stick to an
already prepared path for how I proceed through each topic. Over the past few years,
there have been many lessons that I have presented in entirely different ways. For
example, just this past year we developed the concept of a derivative based solely on a
classroom discussion of finding how fast a NASCAR driver was going the instant that he
crossed the finish line. In the past I have presented this topic more numerically and
graphically with much success, but this year the richness of our discussion added a layer
of depth to the material that has been invaluable throughout this year.
As a result of my flexibility and willingness to adapt with the material, I attempt to
promote an atmosphere of unparalleled enthusiasm, investigation, and inquiry in the
classroom. Most of my lessons, such as the aforementioned NASCAR lesson above, are
designed with the intent that students are going to choose the direction of their own
learning through their questions. As a result, at the end of each unit each student feels a
sense of accomplishment for their ability to tackle the hard concepts that they may have
previously been scared to learn about.
Primary Textbook:
Hughes-Hallett, Deborah, et al. Calculus: Single Variable, 3rd Edition, New York: John
Wiley and Sons Inc.
Secondary Textbook:
Forester, Paul A. Calculus: Concepts and Applications. Emeryville, CA: Key
Curriculum Press.
Required Components:
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Use of Technology: Almost every lesson requires the incorporation of graphing
calculator technology. The preferred technology is TI-83 or TI-84 graphing
calculators. The goal is for the calculators to serve as an aid in understanding the
main concepts of Calculus, and never to be used as a crutch by the students. As a
result of this main goal, all tests are half calculator and half non-calculator. This
allows students to show off their analysis of the material by both traditional and
new analytic approaches.
Students also use other computer technologies such as Power Point, Excel, or
even apple’s iMovie to help explain calculus concepts. Extensive use of the
internet is encouraged, however from only a few trusted sources that are
recommended to the students. One such trusted source is the encyclopedia of
mathematics online (http://mathworld.wolfram.com).
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Communication: Almost every day students must communicate verbally their
understanding of the material in the midst of whole class discussions. We
develop a “big concept” every lesson that has been expanded from both teacher
led and student driven questioning. Also at the end of each semester, each student
participates in a big project where students split up into small groups to present
one of our “big concepts” in a unique way to the entire class. I have had very
unique presentations ranging from a puppet show demonstrating derivatives, to a
15 minute short film developing the concept of various integration techniques.
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“Rule of 4” Our course goals are accomplished by almost daily study of calculus
through graphical, numerical, analytic, and verbal techniques. Most topics are
developed at first verbally through the standard student-teacher and big group
interactions. Topics then become clearer through the other three approaches.
Extra Classroom Components:
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Calculus movies: Each quarter students attend a movie in the evening at a local
theatre where they view such movies as: Stand and Deliver, Proof, A Beautiful
Mind, and Good Will Hunting. Students then are asked for their unique
reflections on each film by writing a short essay and discussing the movie with
each other the next day in class.
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Calculus Projects: At the end of each semester, students form into small groups
to create an original, creative presentation of a “big” Calculus BC concept. These
projects are creative in nature and provide “hooks” for the students to further
remember the material.
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“Calculus Parties” During each unit, we meet as a class at least two to three
times outside of school to provide further enrichment of the material and to help
students that are struggling with the material. We call these help sessions that we
name “Calculus Parties” because we want the students to think of the material as
being fun in nature to learn.
Content: All AP Calculus BC topics are covered in this course. This can be seen on the
unit an lesson syllabi given on the following pages.
Course Syllabus
Unit 1 – A Library of Functions (5 days)
This unit reviews all of the functions introduced in the Precalculus Course. We have our
Calculus BC students study this material over the summer, and send them home with a
listing of homework problems to review the material. When school starts we spend about
2-3 days reviewing the material they saw before, and introduce continuity in Lesson 7 of
the book as an intuitive concept that has continued from their studies. The test for this
unit is an open ended take home test, the only such take home test provided throughout
the entire course. The true-false conceptual “Check Your Understanding” is added as a
point of review for the end of every chapter.
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1.1 Functions and Change
1.2 Exponential Functions
1.3 Transformation of Functions
1.4 Logarithmic Functions
1.5 Trigonometric Functions
1.6 Power, Polynomial, and Rational Functions
1.7 Continuity of Functions
Review, Check Your Understanding, Chapter Test
Unit 2 – Key Concept: The Derivative (10 days)
This unit introduces the concept of the derivative as students’ transition from the concept
of an “average rate of change” to “instantaneous rate of change.” Slope as a way of
measuring rate of change is introduced. Students estimate derivatives graphically,
numerically, and study them analytically towards the end of the chapter. The notion of a
limit is introduced this chapter, and all of their properties are studied. The graphing
calculator is used to introduce the numerical derivative function in further analysis of
derivatives. Students should be able to use their calculators to graph a functions
derivative without even knowing the actual function that is the derivative. They will also
be able to understand derivatives at the limit of average derivatives. Every test,
beginning with chapter 2, consists of four parts divided into calculator and non-calculator
as well as multiple choice and free-response. This unit also starts our look at AP level
Free Response problems. We review a few sample problems together as a class, and then
analyze how they would be scored on the 9-point scale. Students are assessed on a similar
9 point scale on their test free response problems.
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2.1 Measuring Speed
2.2 Limits
2.3 The Derivative at a Point
2.4 The Derivative Function
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2.5 Interpretation of the Derivative
2.6 The Second Derivative
2.7 Continuity and Differentiability
Review, Check Your Understanding, Chapter Test
Unit 3 – Short-Cuts to Differentiation (21 days)
This unit allows students to expand their conceptual knowledge of derivatives to include
rules and algebraic procedures for differentiation. These rules are discovered through
using graphical, analytic, and numerical methods throughout the chapter. These include
finding the derivatives of implicitly defined functions, estimating the derivative from
tables and graphs, formulas for all relevant functions, learning about and finding
derivatives for parametric equations, and estimating derivatives by taking limits of
difference quotients. Graphing calculators are used to illustrate local linearity and the
closeness of the fit of the tangent line near the point of tangency.
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3.1 Derivatives of Polynomials
3.2 Exponential Derivatives
3.3 Derivatives of Products and Quotients
3.4 Derivatives by the Chain Rule
3.5 Derivatives of Trig Functions
3.6 Applications of the Chain Rule
3.7 Implicit Functions
3.8 Parametric Equations
3.8 Linear Approximation and Derivatives
3.10 Local Linearity and Limits
Review, Check Your Understanding, Chapter Test
Unit 4 – Using the Derivative (8 days)
This unit provides students with practice finding derivatives as they apply them to
various problems. Students will learn various tests on how to find local and global
extrema, critical values, and concavity and points of inflection. Newton’s method will be
introduced, but then highlighted again during the AP review part of the course. This
lesson is presented in the Appendix of our textbook. Students will be able to use these
techniques for modeling and optimization problems, particle motion problems, position,
velocity, acceleration functions, and to solve related rate problems. We also will study
generic families of curves, but in the intuitive and analytical sense. The Mean Value
Theorem, Extreme Value Theorem, and others are discussed in section seven of this
chapter.
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4.1 Using Derivatives
4.2 Families of Curves
4.3 Optimization
4.5 Optimization and Modeling
4.7 Theorems about Continuous and Differentiable Functions
Review, Check Your Understanding, Chapter Test
Unit 5 – Key Concept: The Integral (10 days)
Students are introduced to the definite integral by revisiting velocity-time graphs in order
to find total distance traveled. Dimensional analysis is sued to connect integrals and
areas of rectangles. Students work with Riemann Sums as a way of estimating definite
integrals. We therefore introduce summation notation and take a limit of these Riemann
sums as the width of each subdivision approaches zero. We take a look at Riemann sums
in the numerical, analytic and algebraic contexts. The students are introduced to the
definite integration function of the graphing calculator as a way to estimate the total
accumulation or change of a function. Lesson 5.4 introduces the Fundamental Theorem
of Calculus which connects derivatives with integrals. We also learn the difference
between a definite and indefinite integral.
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5.1 Measuring Distance
5.2 The Definite Integral
5.3 Interpret the Definite Integral
5.4 Theorems about Integrals
Review, Check Your Understanding, Chapter Test
Unit 6 – Constructing Antiderivatives (8 days)
Integrals are approached in this unit as the “undoing” process of differentiation. Students
however will understand that there may be a vertical shift in the final result, this occurs in
adding a constant to the result of undoing a derivative. Accumulation functions are
studied in this chapter as a way to predict any generic output of an antiderivative.
Differential equations are introduced as well as basic integration rules for basic functions,
including inverse trig functions. We discuss the intuitive notion of the how the derivative
of an antiderivative is just the original function itself. The equations of motion are
highlighted again as a framework tool for understanding the interplay between
derivatives and antiderivatives.
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6.1 Antiderivatives
6.2 Antiderivatives Analytically
6.3 Differential Equations
6.4 Second Fundamental Theorem of Calculus
Review, Check Your Understanding, Chapter Test
First Semester Chapter Projects (7 days)
Students form into small groups to create an original, creative presentation of a “big”
Calculus BC concept. These projects are creative in nature and provide “hooks” for the
students to further remember the material.
Unit 7 – Integration (25 days)
Techniques for integration are introduced and practiced. Substitution, integration by
parts, rapid repeated integration by parts (Forester), Partial Fraction Decomposition,
Heaviside’s method (Forester), as well as special methods for powers of trig functions are
discussed in this chapter. We rely heavily on the Forester text for this chapter for the
extra supplemental practice. The Midpoint and Trapezoid rules for estimating Definite
Integrals are discussed in further detail. Improper integrals are introduced as our first
idea of an infinite region having an area that limits to a finite answer. Graphing
calculators are used as a way to back up a very conceptual concept with hard data results.
The table feature is used extensively with improper integration. Comparison testing is
provided as a foreshadowing of the analysis that will occur when the students get to the
topic of sequences and series.
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7.1 Integration by Substitution
7.2 Integration by Parts
7.4 Algebraic Identities
7.5 Approximating Definite Integrals
7.7 Improper Integrals
7.8 Comparison of Improper Integrals
Forester 9.3 Rapid Repeated Integration By Parts
Forester 9.5 Integrating Special Powers of Trig Functions
Forester 9.7 Integration of Rational Functions by Partial Fractions
Forester 9.8 Integration of Inverse Trig Functions
Review, Check Your Understanding, Chapter Test
Unit 8 – Using the Definite Integral (18 days)
The main focus of this unit is finding volumes of revolution and arc length of curves.
The approach taken is to find the volume of a small representative region of the shape,
write a Riemann sum representing all slices, and convert it to integral notation. We also
study measuring volume of solids of known cross sections, and arc length of curves in
parametric, rectangular, and polar form. Polar coordinates are re-introduced in this
chapter as well as areas of regions presented in polar coordinates. We again heavily lean
on the Forester text for further practice with the material.
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8.1 Area and Volume by Integration
8.2 Volume of Revolution and Arc Length
8.3 Density using Integrals
8.4 Applications to Physics
Appendix B Polar Coordinates
Forester 8.4 Area of a Plane Region
Forester 8.5 Volume of a Solid by Plane Slicing
Forester 8.7 Length of a Plane Curve – Arc Length
Forester 8.9 Lengths and Areas for Polar Coordinates
Review, Check Your Understanding, Chapter Test
Unit 9 – Series (15 days)
Students start this chapter by looking at sequences of partial sums for geometric series to
determine the sum of an infinite geometric series. Students then move to general
sequences and series of constants and what it means for such series to converge. We
learn a series of test to allow us to determine such a convergence. Power series are
introduced in which the terms are constants multiplied by powers of x. The graphing
calculator is used extensively in this unit to demonstrate infinite sums. This works with
integral tests as well as in determining limits of nth terms in general to help with
convergence. We learn about telescoping, geometric, harmonic, alternating series, and
very special power series. We go into definitions of convergence and divergence, and
talk about bounded sequences from above and relations to convergence. Radius of
convergence of Power Series is discussed in section four.
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9.1 Geometric Series
9.2 Convergence of Sequences and Series
9.3 Tests for Convergence
9.4 Power Series
Review, Check Your Understanding, Chapter Test
Unit 10 – Approximating Functions (10 days)
In this chapter we start to take a look at McLaurine and Taylor series as a continuation of
our series studies. We learn about the main McLaurine polynomials (for sine, cosine,
exponential functions, ln(x), and 1/1-x), functions defined by series, alternating series
error bounds, nth term testing, direct comparison testing, absolute and conditional
convergence, ratio testing, interval of convergence and testing for endpoints, limit
comparison test, and the alternating series test. Fourier Series are presented as a topic
after the AP test.
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10.1 Taylor Polynomials
10.2 Taylor Series
10.3 Finding and Using Taylor Series
10.4 The Error in Taylor Polynomial Approximations
Review, Check Your Understanding, Chapter Test
Unit 11 – Differential Equations (12 days)
Students are introduced to more involved differential equations with separable variables.
Slope fields are introduced as a graphical solution to a differential equation. Big slope
fields are introduced on the board as a beginning tool. A slope field graphing calculator
program is then given to each student as a beginning tool for analysis of the general
solution of differential equations. The general and particular solutions for differential
equations are discussed. We also discuss and learn Newton’s Method in more detail, and
learn about application and modeling problems with regards to the usefulness of
differential equations. We go in depth to the logistic model for growth as well as
exponential growth.
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11.1 Differential Equations
11.2 Slope Fields
11.3 Euler’s Method
11.4 Separation of Variables
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11.5 Growth and Decay
11.6 Applications and Modeling
11.7 Models of Population Growth
Review, Check Your Understanding, Chapter Test
Review for the AP Test (15 days)
Students spend this time working through at least 5 released versions of old AP Calculus
BC tests. These are all graded for completion, but the last two are also graded on the AP
scale for correctness. A schedule of many “calculus parties” is set that students must
attend for participation points leading up to the test. Students get together in class and
out of class to review these practice problems, and also make an outline of all of the “big
concepts” that they have encountered throughout the year. Test results are analyzed from
the last two practice tests to pinpoint strengths and weaknesses and determine focus areas
for the week before the test. This is basically the final push towards the AP test.
Second Semester Chapter Projects (7 days)
Students form into small groups to create an original, creative presentation of a “big”
Calculus BC concept. These projects are creative in nature and provide “hooks” for the
students to further remember the material.