AP CALCULUS AB
SYLLABUS
FOR
SCHOOL YEAR: 2007 –2008
INSTRUCTOR: ANN MILSTEAD
CENTRAL HIGH SCHOOL
POLLOK, TEXAS
Course Overview and Brief Description
AP Calculus AB is an enriched mathematics course and curriculum that is designed
to help students in their understanding of the calculus curriculum and to provide
and prepare them for the mathematics needed to be successful in post secondary
studies. Students are introduced to the wonderful and exciting world of higher
mathematics through a comprehensive study of all of the objectives outlined in the
AP Calculus Course Description. In addition, students are encouraged to take the AP
Calculus AB exam.
Goals from the AP Calculus Course Description

Students should be able to work with functions numerically, graphically,
analytically, and verbally…

The derivative should be understood as the instantaneous rate of change of a
function and as the local linear approximation of the function…

The definite integral should be understood as the limit of a Riemann sum and
as the net accumulation of a rate of change…

The relationship between derivatives and the definite integral should be
understood in terms of both parts of the Fundamental Theorem of
Calculus…

Students learn to communicate about mathematics verbally and in writing…

Students should be able to model a written description of a physical situation
with a function, a differential equation, or an integral…

Students learn to use technology to analyze problems, experiment, and verify
and interpret results…
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
Students are expected to learn to judge the reasonableness of their
solutions…

Students develop an appreciation of the wonderful world of calculus and for
their personal accomplishment in learning calculus…
Teaching Strategies
Connections in mathematics are stressed frequently. For instance: not all students
realize at the beginning of the study of limits that the definition relates back to the
study of slope in Algebra I. For comprehension of calculus concepts, students must
make the mathematical connections to previous learning in order to have a true
understanding of new calculus concepts and applications. Solutions to problems are
found graphically, numerically, analytically, and verbally in order to demonstrate
knowledge of the calculus curriculum being studied. In addition, proper vocabulary
and symbolism are used in the classroom and expected of the students.
Students jump right into Calculus, Chapter 1, Section 1, the first day of school.
Precalculus is reviewed as needed. Students are taught proper form in putting their
work on paper, justifying their solutions, and how to state their solutions in written
form.
Students are encouraged to ask questions immediately during lecture. No hands
raised in this math class. Consequently, problems are cleared up quickly and no
classmates are left behind and in a quandary due to a lack of understanding.
Students are made comfortable early in the year with going to the white board,
asking questions of their teacher, and working with their classmates. Students learn
the first week of school to give their classmates “put-ups” and not “put-downs”.
Study groups are formed early in the school year and employ the use of cooperative
learning techniques for daily assignments with access to the instructor as needed.
The instructor strives for a positive learning environment in the classroom.
Students practice on questions from old AP exams on a weekly basis. A set is due
each week. In addition, students build a notebook (which includes handouts, lab
sheets, notes, charts, projects, and homework) to take to college with them to use as
a study aid in future math courses.
Examples of some (but not all) homework are illustrated by the instructor. Students
are expected to extend their knowledge to problems that are different from the
homework examples.
Graphing Calculators and Technology
Graphing calculators are used on a daily basis to reinforce calculus concepts and
interpret results. Students are provided with a TI-83+ and a TI-89 Titanium by the
school which they may take with them and use at home for the school year.
Demonstrations are done on occasion with the TI-200 calculator. Our students are
very comfortable with the TI-83+, which they have been using since Algebra I. The
TI-89 is not used until the spring semester. Students are expected to find solutions
with the calculator and without the calculator.
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Students will be able to do the following with their graphing calculators:
1.
2.
3.
4.
Plot the graph of a function with an arbitrary viewing window
Find the zeros of functions (solve equations numerically)
Numerically calculate the derivative of a function
Numerically calculate the value of a definite integral1
Early in the course students use their calculators to approximate and arrive at a
reasonable conclusion numerically of what the slope of a tangent line is to some
quadratic function at a particular point on that function. This activity then leads to
further investigation by the student doing the same thing graphically and
analytically.
In addition, a computer projector is used to demonstrate calculus concepts and a TICBL unit is used for labs, demonstrations, and to collect data to further enhance
studies. Some of the software used in this class is Geometer’s Sketchpad and
Calculus in Motion. Also, students have access to the Internet in the classroom for
research. And, the class has access to a computer lab (on request) in order to work
on the APCD Calculus AB2 software for which we have a site license.
Primary Textbook
Larson, Ron, et al. Calculus with Analytic Geometry, Eighth and Advanced
Placement Edition, Boston: Houghton Mifflin Company, 2006.
Each student is issued a copy of the primary text.
Some Examples of Textbook Assignments
Section Problems
Chapter 1: Limits and Their Properties
1.1
1.2
1.3
1.4
1.5
3.53
1, 2, 5, 7, 8, 9 and set up notebook with handouts
1-25 odd, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 59, 63, 65, 67
5-61 odd, 67, 69, 71, 73, 75, 77, 78, 83, 84, 85, 86, 87, 101,103, 113, 115
1-19 odd, 25, 29-51 odd, 57, 59, 61, 63, 69, 71, 75, 77, 83, 85, 87, 91, 105
1-47 odd, 53, 55, 57, 58, 59, 61, 62, 63
1, 3, 5, 7, 15-33 odd, 41, 45, 51, 85, 87
Course Planner, Pacing Guide, and Topic Outline
The pacing guide has 142 teaching days including 10 days of formal assessment.
Precalculus is reviewed as needed throughout the course. The sections listed fulfill
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The College Board. AP Calculus AB Course Description
The College Board. APCD Calculus AB
3
Deliberately out of order
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3
the requirements of content demanded by the AP Course Description Guide. A twoweek review period precedes the AP Calculus AB exam.
Section
1.1
1.2
1.3
1.4
1.5
3.54
2.1
2.2
2.3
2.4
2.5
2.6
3.1
3.2
3.3
3.4
3.6
3.7
3.9
4.1
4.2
4.3
4.4
4.5
4.6
Topic
Chapter 1: Limits and Their Properties
A Preview of Calculus
Finding Limits Graphically and Numerically
Evaluating Limits Analytically
Continuity and One-Sided Limits
Infinite Limits
Limits at Infinity
Review and Assessment
Chapter 2: Differentiation
The Derivative and Tangent Line Problem
Basic Differentiation Rules and Rates of Change
Product and Quotient Rules and Higher-Order
Derivatives
The Chain Rule
Implicit Differentiation
Related Rates
Review and Assessment
Chapter 3: Applications of Differentiation
Extrema on an Interval
Rolle’s Theorem and the Mean Value Theorem
Increasing and Decreasing Functions and
The First Derivative Test
Concavity and the Second Derivative Test
A Summary of Curve Sketching
Review and Assessment
Optimization Problems
Differentials
Review and Assessment
Chapter 4: Integration
Antiderivatives and Indefinite Integration
Area
Riemann Sums and Definite Integrals
The Fundamental Theorem of Calculus
Integration by Substitution
Numerical Integration
Review and Assessment
Number of
Days
2
2
2
2
2
2
3
5
4
3
3
3
3
3
3
3
2
2
2
3
5
3
3
3
3
3
3
3
1
3
Chapter 5: Logarithmic, Exponential,
and Other Transcendental Functions
4
Deliberately out of order
4
5.1
5.2
5.3
5.4
5.5
5.6
5.7
6.1
6.2
6.3
7.1
7.2
8.1
The Natural Logarithmic Function: Differentiation
The Natural Logarithmic Function: Integration
Inverse Functions
Exponential Functions: Differentiation and Integration
Bases Other than e and Applications
Review and Assessment
Inverse Trigonometric Functions: Differentiation
Inverse Trigonometric Functions: Integration
Review and Assessment
3
2
3
3
2
3
2
1
3
Chapter 6: Differential Equations
Slope Fields
Differential Equations: Growth and Decay
Separation of Variables and the Logistic Equation
Review and Assessment
3
5
3
3
Chapter 7: Applications of Integration
Area of a Region Between Two Curves
The Integral as Net Change Over a Specific Period of
Time
(4.5 Exercise 115, Ch. 4 Review Exercises 93 and 94)
Volume: The Disk Method (Includes disks, washers
and volumes of solids with known cross sections)
Review and Assessment
Chapter 8: Integration Techniques, L’Hopital’s
Rule and Improper Integrals
Basic Integration Rules
Review and Assessment
3
6
5
3
2
3
AP Calculus Exam
May 2008
After the AP Exam
Research topics on calculus applications
Selected topics from AP Calculus BC
Student Activities
Students review parent functions, domain, and range early in the school year. In
addition, Precalculus is reviewed throughout the course as needed. Students
approach their study of calculus with a multi-representational view (i.e. graphically,
numerically, analytically, and verbally).
On a daily basis students are working in and adding to their notebooks.
Cooperative learning groups are frequently used in the classroom on assignments.
In addition, students often work at the board or at the overhead projector desk to
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demonstrate calculus solutions to their classmates. Students participate and work
together on lab assignments.
During the teacher’s lecture and modeling of example problems, students are
encouraged to jump in, ask questions, and participate in a class discussion of the
day’s lesson. Students are comfortable and free to learn in this math class.
Students use technology on a daily basis; however, practice with the TI-89
calculator is not done until the spring semester. Students also have a weekly practice
on questions from old AP exams.
Students participate in a number of lab activities. For example: “What Goes Down,
Must Come Up” from the book A Watched Cup Never Cools is a lab activity in which
students use their calculators along with the TI-CBL unit and motion detector to
investigate average velocity as a numeric derivative of position, average acceleration
as a numeric derivative of velocity, derivative as a slope of a tangent to a curve, and
differentiability of a curve. Another lab activity in which students participate is “ As
Easy as Pie” in which graphing calculators and cake pans are used to find the
volume of a cylinder. 5
Students are directed to carefully read all sections in their textbooks that are
assigned by their instructor. In addition, they will have worksheets, lab activities,
experiments, reviews, and projects.
Students may attend a morning tutorial session beginning at 7 a.m. if further
individual help is needed.
Student Evaluation
The pacing guide listed above provides 10 days of assessments. In addition, students
will be assessed with a number of other types of evaluation. For example:
homework assignments, daily quizzes, pop test, three week tests, nine week tests,
midterm exam, final exam, AP free-response questions, AP multiple-choice
questions, and cooperative learning projects. Students will take a full-length
practice exam prior to sitting for the AP Calculus AB Exam.
A high level of expectation is maintained at all times. Frequent daily assessments
keep students ever mindful of keeping up with their work and staying current in
their studies.
On all work, solutions alone will not be given credit. Answers must be accompanied
by the appropriate work. Scrambled versions of tests are administered in order to
maintain honor and integrity in the classroom.
Test questions may include any material that the instructor has taught from the first
day of school. Likewise, students are also held accountable for math concepts taught
in previous grade levels.
No extra credit work will be extended to students.
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Kamischke, Ellen. A Watched Cup Never Cools
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Some Examples of Important Resources
The College Board. AP Calculus AB Course Description
The College Board. Released Exams for AP Calculus AB
Kamischke, Ellen. A Watched Cup Never Cools: Lab Activities for Calculus
and Precalculus. Emeryville, CA. Key Curriculum Press
Roberts, A. Wayne. Resources for Calculus Collection, 5 volumes. Published
by MAA
Vol. 1: Learning by Discovery
Vol. 2: Calculus Problems for a New Century
Vol. 3. Applications of Calculus
Vol. 4: Problems for Student Investigation
Vol. 5: Readings for Calculus
Anderson, Frank. Review for the AP Calculus AB Examination: The Two
Week Difference. Atlanta, GA. Andco Educational Service
Audrey Weeks. Calculus in Motion. Software
The College Board. Advanced Placement APCD Calculus AB. New York.
Software
Website Resources
(Quicker to search by Google to get to these websites)
AP Central
Handley Math Page
Dr. Math
Math Forum
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