Calculus AB Syllabus
Overview
AP Calculus is focused on the course description as written by the College Board. Particular
attention is paid to the “Rule of Four” method of quantitative analysis in setting up and solving
problems throughout the course. Students will continually have opportunities to make
connections between prior knowledge and current topics. In order to keep up to date on topics
and methods of instruction, the AP Calculus teacher will attend the AP Summer Institute each
year. Throughout the course students will explore and experience a variety of concepts,
working both independently and collaboratively in groups. The students will investigate
mathematical problems and situations from several perspectives, utilizing numerical, graphical,
and algebraic methods and employing technology such as computers and graphing calculators
where necessary. The course teaches students how to explain solutions to problems in written
sentences.
Goals
Upon completion of this course, the students should be able to:
1. Understand the major topics of functions, limits, derivatives, and integrals
2. Incorporate multiple representations of functions using graphic, numeric, algebraic
and verbal descriptions
3. Understand the connections among graphical, numerical, algebraic, and verbal
representations
4. Understand continuity and graphical properties of continuous functions
5. Understand derivatives as an instantaneous rate of change
6. Understand definite integrals as a limit of Riemann sums
7. Apply derivatives and integrals to a variety of problem solving situations
8. Understand the Fundamental Theorem of Calculus as a relationship between
derivatives and definite integrals
9. Determine if solutions or results are reasonable
10. Communicate results and connections verbally and in writing
11. Develop an appreciation for the history of calculus
Text
Single Variable Calculus
Early Transcendentals
Sixth Edition
James Stewart
Timeline
1. Functions and Models – approx. 2 weeks

Representing functions numerically, algebraically, and graphically


Function families
Diagnostic tests in algebra, analytic geometry, functions, and trigonometry

Example activities: Students investigate graph behaviors of a variety of functions and
relations, some of which they have seen before, such as exponential, polynomial,
logarithmic functions, and conic sections. Others are new to them, such as y 
or y 
1 e
1
1 e
1
x
. The students investigate graphs by looking at individual values or
3  x2
tables of values or by comparing them to the parent functions or other less
complicated or familiar functions, utilizing paper and pencil graphs and the graphing
calculator. The students share their results in writing, verbally with a “pair share”
partner, or with the whole class.
2. Limits and Continuity – approx. 3 weeks


Finding limits graphically and numerically
Evaluating limits analytically

Rates of change


Continuity and one-sided limits
Infinite limits

Example activities: Students explore the limit of a function at values close to some x
value, using the zoom, trace, and table features of the calculator. They explore the
algebra that leads to calculating the limit. They write about and discuss how the
question “How close is close?” applies to a function and its tangent lines. The
students use the calculator to then explore the tangent lines and how the graph is
“locally linear.”
3. Differentiation – approx. 6 weeks


Basic differentiation rules
Chain rule


Product and quotient rules
Derivatives of trigonometric, logarithmic, and exponential functions
x
x


Implicit differentiation
Related rates

Differentials

Example activities: Students use CBL’s to conduct dropped and tossed ball
experiments. The calculators are used to model various situations with quadratic
functions of time vs. height, average velocities are calculated, and instantaneous
velocities are estimated. The students also use the CBL’s to model behavior of a
moving body by graphing time vs. position, time vs. velocity, and time vs.
acceleration.
4. Applications of Differentiation – approx. 4 weeks


Extrema values
Rolle’s Theorem and the Mean Value Theorem


Increasing and Decreasing Functions and the First Derivative Test
Concavity and the Second Derivative Test


L’Hospital’s Rule
Summary of curve sketching

Optimization problems

Example activities: Given measurements of various hallways, students work in pairs
to determine the longest ladder that can be carried horizontally around a corner. They
work with scale drawings and analytic methods.
5. Integration – approx. 5 weeks
 Areas and distance


Definite integrals
Riemann sums


The Fundamental Theorem of Calculus
Integration by substitution


Indefinite integration
Slope fields

Example activities: Students apply integration properties and rules to a wide variety of
functions, comparing the results they get with geometric methods, Reimann sums,
and analytic methods and then verifying them with definite integrals obtained using a
calculator. They also investigate situations involving data or a graph without an
equation, and calculate or estimate such quantities as water flow, marginal cost,
velocity.
6. Applications of Integration – approx. 3 weeks

Areas between curves

Volumes by shells


Work problems
Average value of a function

Example activities: Students work in groups with springs and weights to investigate
situations involving work and force and Newton’s second law of motion.
7. Integration Techniques and Applications – approx. 3 weeks

Integration by parts

Trigonometric integrals


Integration of rational functions and partial fractions
Volumes by discs

Example activities: Student use modeling clay to investigate volumes created by
various functions rotated about axes and then dental floss to create “slices.” In a
similar activity, students work in groups to set up multiple ways to calculate the
volume of Legg’s eggs.
8. Comprehensive Review for AP Exam – approx. 3 weeks
 Students practice multiple choice and free-response questions

Students share their answers with partners, in small groups, and in whole class
discussions

Example activities: CPS student response systems are used to tally and track
responses for multiple choice questions. An activity that emphasizes the importance
of full explanations is for the students to analyze sample responses (both good and
not-so-good) for content, accuracy, and communication of process and results.
Assessment
Students will complete summative assessments for each unit of study. There will also be
summative semester assessments. In addition, there will be weekly quizzes and problem sets
and many formative assessments. All answers to free response questions must appear in a
form which is consistent with AP Calculus AB criteria.
Problem Solving
Throughout the course, students will be required to solve many different types of problems on a
daily basis. They will learn how to approach those problems from various perspectives, including
graphing, algebraic, numeric, and verbal perspectives. The students will be required to show
connections between the different approaches. They will demonstrate and communicate their
methods and solutions in a variety of manners, including discussion in small groups and large
groups, written assignments, oral presentations, tests, and quizzes.
Technology
Calculators and computers will be used often to explore, discover, and reinforce the concepts of
calculus. Working with the concept of change is at the foundation of this course. To assist in
their understanding of change, students enrolled in AB Calculus are required to have a TI–83/84
graphing calculator. For example, students will investigate rates of change in such topics as
population growth, interest income, productions costs, physics, volume and distance. When
students obtain solutions analytically, they will support, confirm, and interpret their results using
the graphing calculators in the following ways:
1. Perform arithmetic calculations
2. Model and analyze data using the statistics feature of the calculator
3. Define functions
4. Evaluate functions at a particular value
5. Use the trace feature to explore functions, limits, derivatives, and integrals
6. Use the table feature to explore functions, limits, derivatives, and integrals
7. Graph functions and change the viewing window in meaningful ways
8. Graph relations using a calculator or implicit grapher
9. Determine the zeros of a function
10. Determine the intersection point of the graphs of two functions
11. Investigate “local behavior” and “end behavior” of a function
12. Determine maximum and minimum values of a function
13. Examine secant and tangent lines to a variety of types of functions
14. Determine the derivative of a function at a point
15. Explore the graphs of derivatives
16. Calculate Riemann sums
17. Determine the definite integral of a function over an interval
18. Graph slope fields