AP Calculus BC Syllabus
Course Overview
Advanced Placement Calculus is a course that requires a student to learn the fundamental
concepts and mathematics of calculus and to recognize and formulate connections between
topics. It is expected from this course that students will gain mathematical skill, understanding
and use of technology to help them be successful in further mathematics classes and in their
future careers. Students are expected to think hard, try different approaches to problems
(graphical, numerical, analytical or verbal), and enjoy seeing their understanding of mathematics
grow.
Course Outline
Below is an outline of the topics covered in AP Calculus BC.
First Semester
UNIT 1
Section
1.7
1.8
2.1
2.2
2.3
2.4
2.5
2.6
UNIT 2
Section
3.1
Topics
Continuity
Intermediate Value Theorem
Limits and Continuity
Properties of Limits
Evaluating Limits
One-Sided Limits
Infinite Limits
Quiz
Average and Instantaneous Velocity
Derivative at a Point
Average Rate of Change
Definition of Derivative
Derivative Function
Increasing/Decreasing
Derivatives of Constants, Linear Functions, Power Functions
Quiz
Interpretations of the Derivative
Alternative Notations (Newton vs. Leibniz)
Second Derivative
Concavity
Average and Instantaneous Acceleration
Differentiability
Relationship with Continuity
Review
Test Chapter 1/2
Timeline (days)
Topics
Derivatives
Properties
Timeline (days)
1
2
1
1
2
0.5
1
1
2
3.2
3.3
3.4
3.5
3.6
3.7
3.9
3.10
UNIT 3
Section
4.1
4.2
4.3
4.5
4.6
4.7
4.8
Power Rule
Polynomials
Derivatives
Exponential Functions (y = ax and y = ex)
Product and Quotient Rules
Quiz
Chain Rule
Derivatives
Trigonometric Functions
Chain Rule and Inverse Functions
Natural Log Function
Inverse Trigonometric Functions
General Inverse Function
Quiz
Implicit Differentiation
Linear Approximation
Local Linearization
Theorems about Differentiable Functions
Mean Value Theorem
Increasing Function Theorem
Constant Function Theorem
Racetrack Principle
Review
Test Chapter 3
Topics
Using First and Second Derivatives
Critical Points
Inflections Points
Local Extrema
First and Second Derivative Tests
Families of Functions
The Effect of Changing Parameters
Optimization
Global Maximum/Minimum
Extreme Value Theorem
Upper/Lower Bounds
Quiz
Optimization and Modeling
Rates and Related Rates
L’Hopital’s Rule
Quiz
Parametric Equations
Direction of Motion
Equations for a Straight Line
Instantaneous Speed
Slope and Concavity
2
2
2
2
2
1
1
1
Timeline (days)
2
1
2
2
2
1
2
Review
Test Chapter 4
UNIT 4
Section
5.1
5.2
5.3
5.4
UNIT 5
Section
6.1
6.2
6.3
6.4
Topics
Measuring Distance Traveled
Velocity vs. Time
Area under a Curve
Distance Traveled vs. Displacement
Left and Right Sums
Definite Integral
Area under a Curve
Riemann Sums
Fundamental Theorem of Calculus
Applications
Average Value of a Function
Quiz
Definite Integral Theorems
Properties on Limits of Integration
Properties of the Integrand
Area between Curves
Even and Odd Functions and Integrals
Review
Test Chapter 5
Timeline (days)
Topics
Antiderivatives (Graphically and Numerically)
Graphs
Calculate using the Fundamental Theorem
Finding Antiderivatives
Constants
Power Functions
Hyperbolas
Sine and Cosine Functions
Properties: Sums and Constant Multiples
Definite Integrals
Quiz
Differential Equations and Initial Value Problems
Second Fundamental Theorem of Calculus
Construction Theorem
Review
Test Chapter 6
Timeline (days)
Review
Mid-Term Exam
1
1
2
2
1
2
1
1
Also included in the first semester are several days for review, multiple choice and free response
practice (questions from past AP exams), group work and presentation, quizzes and tests for
assessment.
Second Semester
UNIT 6
Section
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
UNIT 7
Section
8.1
8.2
8.3
Topics
Integration
Guess and Check
Substitution
Integration
By Parts
Quiz
Tables of Integrals
Integration
Partial Fractions
Trigonometric Substitutions
Approximation of Definite Integrals
Left/Right Rules
Midpoint Rule
Trapezoid Rule
Approximation of Definite Integrals
Simpson’s Rule
Quiz
Improper Integrals
Limit(s) is Infinite
Integrand becomes Infinite
Comparison of Improper Integrals
Comparison Test where the Limit is Infinite
Comparison Test where the Integrand becomes Infinite
Review
Test Chapter 7
Timeline (days)
Topics
Finding Areas and Volumes
Horizontal and Vertical Slicing
Applications to Geometry
Volumes of Revolution (disks and disks with holes)
Volumes of Regions with Known Cross-Sections
Arc Length of Curves defined by Functions and
Parametrically
Quiz
Area and Arc Length in Polar Functions
Relationship between Cartesian and Polar Coordinates
Graphing Polar Equations
Area in Polar Coordinates
Arc Length in Polar Coordinates
Timeline (days)
2
2
1
1
1
1
1
1
2
1
2
8.4
8.5
UNIT 8
Section
9.1
9.2
9.3
9.4
9.5
UNIT 9
Section
10.1
10.2
Density and Center of Mass
Point Masses
Thin Rods
Two- and Three-Dimensional Regions
Applications to Physics
Work
Mass vs. Weight
Force and Pressure
Review
Test Chapter 8
1
1
Topics
Sequences
Numerical, Algebraic, and Graphical Representations
Recursive Definition
Convergence
Monotone
Geometric Series
Finite and Infinite Series
Quiz
Convergence of Series
Partial Sums
Convergence Properties (Sums and Constant Multiples)
The Integral Test
1/np Convergence
Tests for Convergence
Comparison Test
Limit Comparison Test
Absolute Value Convergence
Ratio Test
Alternating Series Test
Absolute and Conditional Convergence
Quiz
Power Series
Numerical and Graphical Representation of Convergence
Intervals of Convergence
Radius of Convergence
Review
Test Chapter 8
Timeline (days)
Topics
Taylor Polynomials
Degree n polynomials for x near 0 and for x near a
Taylor Series
Series about x = 0 and about x = a
Intervals of Convergence
Binomial Series
Timeline (days)
1
2
2
2
1
2
2
10.3
10.4
UNIT 10
Section
11.1
11.2
11.3
11.4
11.5
11.6
11.7
Quiz
Finding and Using Taylor Series
New Series by Substitution
New Series by Differentiation and Integration
Taylor Polynomial Errors
Lagrange Error Bound
Convergence of Taylor Series for cos x
Review
Test Chapter 8
Topics
Differential Equations
First and Second Order
Slope Fields
Euler’s Method
Quiz
Separation of Variables
Growth and Decay
Applications and Modeling
Models of Population Growth
Logistic Equations
Review
Test Chapter 11
1
1
Timeline (days)
1
1
1
1
1
1
1
Review for AP Exam
AP Exam
11.8
11.10
11.11
Systems of Differential Equations
Second-Order Differential Equations: Oscillations
Linear Second-Order Differential Equations
2
3
3
Final Second Semester Exam
Also included in the second semester are several days for review, multiple choice and free
response practice (questions from past AP exams), group work and presentation, quizzes and
tests for assessment.
Primary Textbook
Hughes-Hallet, Deborah, et al. Calculus: Single Variable. 4th ed. New York: Wiley, 2005.
In addition to the textbook, additional materials include multiple choice and free response
questions from previous AP Calculus AB and AP Calculus BC exams and Hughes-Hallet
supplemental materials.
Curricular Requirements
The following is a list of the curricular requirements for AP Calculus BC along with evidence of
how these requirements are fulfilled within this course.
Curricular Requirement 1
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.
The topics listed above in the course outline are aligned with the topics outlined in the
AP Calculus Course Description.
Curricular Requirement 2
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.
In each class, a graphical representation of functions that follow from a table is shown.
It is important to be able to explain what is happening and also to be able to
algebraically solve the problems.
In a specific example, students use graphing calculators to examine the effects of
changing parameters on the functions. The students are given a function like y =
Asin(Bx) from which they must decide what happens to the graph of the function when
the parameters of A and B are changed. Each group of students will have to examine
different parameters using their graphing calculators to determine how the function’s
graph characteristics change. They must analyze the graphs of the functions and how
they transform based on the parameters. Each group then presents their results and
conclusions to the class.
Curricular Requirement 3
The course teaches students how to communicate mathematics and explain solutions to problems
both verbally and in written sentences.
The students solve problems in small groups and give presentations to the class.
Graded tests and homework assignments include components (such as sample freeresponse questions from past AP exams accompanied by text assignments and
instructor generated assignments) where students are required to explain and/or justify
their solutions to problems in well-written sentences.
Students often come to the whiteboard or overhead projector to explain their solutions
to the class. They must learn how to speak in a way that communicates the
mathematics using the correct vocabulary and showing each step. By having they
students teach each other, they begin to learn what is important to say and show in
their work in order to clearly express the mathematics so that others understand.
Curricular Requirement 4
The course teaches students how to use graphing calculators to help solve problems, experiment,
interpret results, and support conclusions.
Graphing calculators will be used to investigate the limits of functions, confirm
characteristics of functions (i.e. critical and inflection points), find areas below curves,
find numerical derivatives, compute Riemann sums, create slope fields, show how a
particular solution fits within a slope field, use Euler’s method, and determine how
certain functions are related to each other. Students who are somewhat unfamiliar with
their graphing calculator become proficient at using their calculator for many different
purposes.
Calculator technology is used to find numerical approximations of integrals with
Riemann sum approximations. In one lesson, students enter a program on their
calculator which allows them to approximate integrals based on Left, Right, Midpoint,
Trapezoidal, and Simpson sums. From this, they can see the effect of the number of
subintervals, determine which sums are over- or under-estimates, and make a best
estimate of the area under a curve.
Calculators are used to find solutions to equations using numerical methods. For
example, if an equation cannot be solved using any known method (factoring,
quadratic formula, logarithms), the students become adept at using the graph or table
on their calculator to find the solution(s) to the equation.
In general, TI-83+, TI-84, and TI-89 calculators are utilized by the students. For
presentation and demonstration, the TI-84 is used most often.
Student Evaluation
Students are assessed on the basis of tests (80 points), quizzes (30 points), and homework
(variable 1-4 points per graded assignment). The total points per semester ranges from 500-600
points. The students are required sometimes to complete homework or assessments with and
without the use of calculators. The two quarter grades per semester are worth 40 percent of the
final grade. The midterm or final exam completes the final 20 percent of the grade.