AP CALCULUS AB
COURSE OVERVIEW
All topics and concepts included in the AP Course Description are covered in this course with a focus
on the historical story of Calculus. The primary textbook used is Calculus with Analytic Geometry,
Fifth Edition by Larson, Hostetler and Edwards (as approved by the state’s adoption). The course is
divided into two sections: differential calculus and integral calculus. Students who take this course
have the objective of scoring a 4 or 5 on the AP exam as well as the ability to gain college credit by
taking any university-specific exams.
COURSE PLAN
Below is a list of topics in the order covered, along with additional projects and presentations. Exams
and quizzes have been left out. Assessment techniques are explained in a later section. All italics
represent sections apart from the main text.
DIFFERENTIAL CALCULUS
TEXT SECTION
N/A1
N/A2
P.4
P.4
P.5
2.13
1.1
1.1
1.2
1.3
1.4
1
TOPICS
Discussion and Project with chapters
from A Tour of the Calculus
Family of Functions Overview & An
introduction to the “rule of four”
Lines in the Plane (linear and constant
functions)
Functions (polynomial, linear, constant,
cubic, piece-wise, radical and rational
functions)
Review of Trigonometric Functions and
Graphs
Zeno’s Paradox and the Grapefruit
Experiment
An introduction to Limits & The Tangent
Line problem
The Delta-Epsilon Definition of Limits
Properties of Limits
Techniques for Evaluating Limits (using
Algebra and Geometry to evaluate limits)
Continuity & The Intermediate Value
Theorem & The Extreme value Theorem
TIMELINE
3 days
5 days
1.5 days
1.5 days
1.5 days
1 day
1.5 days
1.5 days
1 day
4 days
3 days
Discussion is outlined below
Worksheets and discussion are used. Family of Functions include: linear, exponential, constant, trigonometric, logarithmic,
quadratic, cubic, radical, rational, absolute value, and greatest integer
3
This section is from the text Calculus by Deborah Hughes-Hallett and Andrew M. Gleason
2
1.5
3.5
2.1
Approved by the College Board © June, 6, 2007
Infinite Limits
2 days
Limits at Infinity
2 days
4 days
Grapefruit Experiment Revisited &
Tangent Line Problem Revisited &
The Definition of the Derivative
N/A4
2.2
2.3
4.95
2.4
2.5
2.6
3.1
3.2
N/A
3.3
3.4
3.6
3.7
N/A6
3.8
3.9
5.3
5.6
7.7
Qualitative Graphing Project
Basic Differentiation Rules (definition of
differentiability)
Product and Quotient Rules and HigherOrder Derivatives
Tangent Line Approximation and Local
Linearity
Chain Rule
Implicit Differentiation
Related Rate Word Problems
Extrema of a Function (critical numbers)
Rolle’s Theorem and The Mean Value
Theorem
Further Discussion from A Tour of the
Calculus and corresponding project
Increasing & Decreasing Functions and
the First Derivative Test
Concavity & the Second Derivative Test
(points of inflection)
Curve Sketching (including from f to f’
and vice versa)
Optimization Problems
Optimization Problems Project
Presentations
Newton’s Method
Differentials
Inverse Functions
Inverse Trigonometric Functions and
Differentiation
L’Hopital’s Rule and Indeterminate
Forms
TOTAL:
2 days
2 days
2.5 days
2 days
3 days
3 days
3 days
2 days
2 days
2 days
3 days
3 days
4 days
3 days
2 days
1.5 days
2 days
2 days
2 days
2.5 days
81 days
INTEGRAL CALCULUS
4
This project deals with using the TI-83 graphing calculator and corresponding CBR software (and hardware). The project is
explained in further detail below.
5
This section is from the text Calculus by Deborah Hughes-Hallett and Andrew M. Gleason
6
This project deals with students verbally explaining a chosen optimization problem. This project is outlined below.
Approved by the College Board © June, 6, 2007
TEXTS
TOPICS
N/A
Discussion and Project with chapters
from A Tour of the Calculus
Antiderivatives and Indefinite Integration
Grapefruit Experiment Revisited: How
do we measure distance traveled?
4.1
N/A7
4.2
4.3
4.4
4.5
4.6
5.1
5.2
5.6
5.8
N/A
7.108
6.1
6.2
6.3
7.1
7.2
7.3
TIMELINE
2 days
3 days
2 days
3 days
The Definite Integral and Riemann Sums 4 days
The Fundamental Theorem of Calculus
4 days
Integration by Substitution
3 days
Numerical Integration (Trapezoidal Rule) 2 days
2 days
The Natural Logarithm Function and
Area & Upper and Lower Sums
Integration
Logarithmic, Exponential and
Transcendental Integration
Differential Equations and Integration
with corresponding Logarithmic,
Exponential and Transcendental
Applications
Inverse Trigonometric Functions:
Integration and Completing the Square
Discussion and Project with chapters
from A Tour of the Calculus
Slope Fields
Area of Region Between Two Curves
Volume of Revolution: The Disc Method
Volume of Revolution: The Shell Method
Basic Integration Rules
Integration by Parts
Trigonometric Integrals
TOTAL
3 days
3 days
3 days
2 days
2 days
3 days
4 days
4 days
3 days
3 days
3 days
58 days
This allows for 20-30 days of pure assessment and practicing for the AP exam. There are two Saturday
practice examinations held at the school as well.
TEACHING STRATEGIES
At our school, those students who take Calculus AB (AP) have taken PreCalculus (Pre-AP) by the
same teacher. Expectations for students taking this course are considerably high, as with the
PreCalculus course they took the previous year. Teaching strategies include explaining all main topics
algebraically (analytically), geometrically (graphically), numerically and verbally and ultimately
7
8
This section is from the text Calculus by Deborah Hughes-Hallett and Andrew M. Gleason
This section is from the text Calculus by Deborah Hughes-Hallett and Andrew M. Gleason
Approved by the College Board © June, 6, 2007
assessing students using that four-pronged approach. We refer to this as the “rule of four.” A
concentration on the history of Calculus is threaded through the course to give the students a historical
prospective as well as a deep appreciation for mathematical processes as an evolution of human
thought. Many concepts are presented in lecture format, with group and single projects interspersed.
Most lectures are presented using a student inquiry based approach, as the philosophy behind the
course is that the main concepts in Calculus are best explained through exploration (eg how do we
measure speed? How do we determine instantaneous velocity? How do we find the area under a curve
using the area of a rectangle and limits?)
A concentration on the reason behind the mathematics drives parts of the curriculum. Students are
expected to derive most formulas and use algebra combined with certain theorems to prove certain
concepts. For example, students are expected to derive Reimann Sums as well as use them to calculate
area before the Fundamental Theorem of Calculus is taught. Moreover, students must use the limit
definition of a derivative to create certain short-cuts (eg power rule). Using the “rule of four,” students
understand that all four aspects must be presented before a proof is considered valid. A basic
introduction to mathematical rigor is also introduced throughout the course.
At our relatively small school, only 4-7 students on average take Calculus either in their junior or
senior year. This allows for more 1-on-1 communications between teacher and student, as well as
assessing student understanding.
The “Grapefruit Experiment” in the course outline refers to an experiment involving throwing a
grapefruit directly up in the air and data collected as the grapefruit returns to the ground. Deborah
Hughes-Hallett created this example in her text, Calculus. Average vs. Instantaneous velocity is
introduced (with links to Zeno’s paradox) as well as answering the question: how do we measure
speed? This inevitably lends itself to the idea of limits and the definition of a derivative at a point
(proving that although instantaneous velocity is an oxymoron, by using limits we can state what
happens at an exact point). This approach is also used to answer the question: how do we measure
distance traveled? Going from the data collected with the grapefruit experiment, students are expected
to add up all of the individual distances (using velocity times time), and then realizing that by taking
infinitely small intervals, we approach the best guess as to the distance traveled which leads directly
into Reimann Sums and ultimately, the definite integral. Students are expected to compute the average
velocity over a certain interval to figure out the limit and thus, instantaneous velocity. Using this
experiment as a thread helps students realize the power and importance of Calculus. Lectures often
refer back to this experiment. CBR activities are also used to gather more data by dropping other
objects, such as a ball. Students then look at the graph to determine a general trend as well as using the
table function to look at the numerical component. Students are then asked to extrapolate on this data.
(eg at what time was it as its maximum height? When was the velocity zero? at its peak?)
PROJECT DESCRIPTIONS
Throughout their study of Calculus, students are required to complete some projects dealing with
various concepts. Three main projects are described below:
Approved by the College Board © June, 6, 2007
A Tour of the Calculus Project
Students are required to read the book A Tour of the Calculus by David Berlinski throughout the year.
The book is divided into 4-6 sections. For each section students (in groups) present their thoughts (1)
about the book itself (2) about the math topics presented and (3) how the book relates directly to the
understanding of Calculus. Papers are submitted for approval and presentations are delivered for
assessment purposes. The students are also required to lead the discussion of each section, but are
graded on the three components mentioned above. An understanding of Berlinski’s approach to
Calculus is used to further concretize the evolution of Calculus.
Problem Explanations at the Board
Often times, students are instructed to explain a problem on the board in front of the class to their
peers. This assessment strategy allows students to fully grasp a concept as the teacher often acts as
another student and asks probing questions.
Optimization Problems Project
Students are given 1 particularly difficult optimization problem to solve. They generally have a
weekend to gather information (an important component of the project) in order to present their
solution to the problem (a visual aide is required). Students are also graded on their questions during
the presentation. The use of a graphing calculator is necessary for their presentation, as they must
answer the problem using the “rule of four.”
TECHNOLOGY COMPONENTS
Throughout their study of mathematics, students are taught how to use a graphing calculator beginning
with Algebra I. By the time they approach Calculus, students are already aware of the basic functions
(graphing, intercepts, maximums, minimums, correct use of tables, how to set your tables, how to
restrict domain, how to use the window properly, etc.).
CBR activities are also used to re-enforce Calculus concepts. At the beginning of the year, students use
CBR technology to gather data about the dropping of a ball and connect that to the grapefruit
experiment. To completely grasp the connection between acceleration, velocity and distance traveled,
the CBR and calculator are used to recreate various qualitative graphs.
With their projects, a calculator component is required. The scientific idea of gathering evidence,
analyzing data and creating hypotheses is definitely required by all students. Generally, during group
work, each student is assigned a prong of the “rule of four” and must present their findings (or
solutions to problems) only using their specified prong. The students labeled “graphically” must
present the solution to the problem completely on the calculator (using the overhead component of the
TI-83) to the class.
STUDENT EVALUATIONS
Approved by the College Board © June, 6, 2007
As stated before, the “rule of four” serves as a primary basis for student evaluation in all aspects.
Student grades are broken up into three components: (a) Tests (b) Extensions and (c) Scrolls. Tests are
timed exams that focus primarily on the skills sets needed for Calculus. Every test contains problems
selected from past AP exams as well as problems from the assigned homework. Extensions are
worksheets that focus on one concept presented in class. The questions on the extensions extend the
concept and go further into detail. Students are assigned an extension each week on Monday and have
the entire week (including the weekend) to complete it. Scrolls are worksheets that contain 10
problems (that are not part of the homework) that can span any concept from the beginning of the year
until the present. The philosophy behind the scroll is that it always re-emphasizes ideas and problems
that have been covered in the weeks prior to the scroll being due. This helps to keep “old” concepts
fresh as well as basic algebraic skills in constant exercise.
Homework for each section is assigned (around 15-20 problems per night), but the homework is only
used as a guide to facilitate student discussion. Homework is viewed as needed practice, but is not
explicitly graded.
TEACHER RESOURCES
Primary Resource
Larson, Roland E. and Hostetler, Robert P. Calculus with Analytic Geometry. 5th ed.
Lexington, MA: Heath, 1994.
(An updated 2006 edition of the book has been adopted for next school year. This course
description is based on the 1994 edition.)
Secondary Resources
Hughes-Hallet, Deborah and Gleason, Andrew M. Calculus. New York: John Wiley &
Sons, Inc., 1994
(This is used as indicated on the course outline for certain concepts)
Berlinski, David. A Tour of the Calculus. New York: Vintage Books, 1995
(This is used as a supplemental text for a yearlong continuous project)
Cracking the AP Calculus AB & BC Exams. New York: Random House, Inc., 2006
(This is used as a supplemental text for constant practice for the AP exam)
Salas, Satunino L. and Hille, Einar and Etgen, Garret J. Calculus: One and Several
Variables. 8th ed. New York: John Wiley & Sons, Inc., 1998
(This is used as a supplemental text for additional homework problems)