AP® Calculus AB Course Syllabus
Course Overview
The Calculus AB course objective is that students do well on their AP Exam, and
in subsequent courses that they may take in the future. Consequently, the syllabus is an
attempt to balance understanding, skills, and technology. As the class proceeds from
limits to derivatives to integrals, and their relationships, they are applying the Rule of
Four: solving problems analytically, numerically, graphically, and verbally.
The students prepare themselves during the summer working on a Pre-Calculus
review packet prior to the first day of their Calculus class. During the year, the class does
weekly, released, free-response questions where students are required to explain and/or
justify their solutions to problems in well-written sentences. As the time approaches to
their AP exam, the students take two practice released AP exams. They learn to score
their own tests, and discover where they are in the learning and understanding of
Calculus. The class addresses their strengths and weaknesses with the help of their
teacher. If re-teaching or reinforcement is necessary, students themselves address the
issues with classmates. Communication is stressed as a major goal. Students are expected
to explain problems using proper vocabulary, and are asked to explain solutions to the
class through both verbal and written means.
Use of Technology
Every student has a TI-83 Plus, TI-84 or TI-89 graphing calculator. The class
learns how to use the calculator on a need to know basis. This means that the class does
not spend a few days getting to know how to use their calculators but rather the students
learn to use the various calculator functions at the same moment in time when they will
need that calculator function to work with that day’s Calculus concept. For instance,
when learning limits at infinity, students will evaluate the limit numerically using the
table function of their calculator and then observe the asymptotic behavior of the graph of
the function. Students will also learn different ways to do functions on their calculator
like performing numerical integration from both the home screen and the graph screen. In
this way, the students can choose the method that would be most appropriate for the
given problem as well as taking their own personal preference into consideration.
Students will be expected to use their calculator to justify mathematical results obtained
through analytical means. For instance, if a function is obtained analytically that is
supposed to have a relative maximum, point of inflection, and relative minimum at
specific points, the students will be able to graph the function on their calculator and be
able to check to see if the required traits exist in the graph of the function.
I.
Limits and Their Properties (13 Days)
A.
B.
An Introduction to Limits
1. The Tangent Line Problem
2. An Introduction to Limits
3. Limits that Fail to Exist
4. A Formal Definition of a Limit
Properties of Limits
C.
D.
E.
F.
II.
Differentiation (23 Days)
A.
B.
C.
D.
E.
F.
G.
III.
Techniques for Evaluating Limits
1. Strategies for Finding Limits
2. Cancellation and Rationalization Techniques
3. The Squeeze Theorem
Continuity and One-Sided Limits
1. Continuity at a Point and on an Open Interval
2. One-Side Limits and Continuity of a Closed Interval
3. Properties of Continuity
4. The Intermediate Value Theorem
Infinite Limits
1. Infinite Limits
2. Vertical Asymptotes
Review and Assessment
The Derivative and the Tangent Line Problem
1. The Tangent Line Problem
2. Instantaneous Rate of Change
3. Differentiability and Continuity
Basic Differentiation Rule and Rates of Change
1. The Constant, Power and Constant Multiple Rules
2. The Sum and Difference Rules
3. Derivative of Sine and Cosine Functions
4. Rates of Change, Average and Instantaneous Velocity
The Product and Quotient Rules and Higher-Order Derivatives
1. The Product and Quotient Rules
2. Derivatives of Trigonometric Functions
3. Higher-Order Derivatives
4. Relationships between Position, Velocity and Acceleration
The Chain Rule
1. The chain Rule
2. The General Power Rule
3. Trigonometric Functions and the Chain Rule
Implicit Differentiation
1. Implicit and Explicit Functions
2. Implicit Differentiation
Related Rates
Review and Assessment
Applications of Differentiation (24 Days)
A.
Extrema on an Interval
1. Extrema of a Function
2. Relative Extrema and Critical Numbers
3. Finding Extrema on a Closed Interval
B.
C.
D.
E.
F.
G.
H.
I.
IV.
4. Extreme Value Theorem
Rolle’s Theorem and the Mean Value Theorem
Increasing and Decreasing and the First Derivative Test
Concavity, Points of Inflection and the Second Derivative Test
Limits at Infinity and Horizontal Asymptotes
Summary of Curve Sketching
Optimization Problems
1. Applied Minimum and Maximum Problems
Differentials
1. Linear Approximations
2. Differentials
Review and Assessment
Integration (18 Days)
A.
B.
C.
D.
E.
F.
G.
Antiderivatives and Indefinite Integration
1. Antiderivatives
2. Basic Integration Rules
3. Initial Conditions and Particular Solutions
Area
1. Sigma Notation
2. The Area of a Plane Region
Riemann Sums and Definite Integrals
1. Riemann Sums, Left, Right and Mid-Point
2. Properties of Definite Integrals
The Fundamental Theorem of Calculus
1. The Fundamental Theorem of Calculus
2. The Mean Value Theorem for Integrals
3. Average Value of a Function
4. The Second Fundamental Theorem of Calculus
Integration by Substitution
1. Pattern Recognition
2. Change of Variables
3. The General Power Rule for Integration
4. Change of Variables of Definite Integrals
5. Integration of Even and Odd Functions
Numerical Integration
1. The Trapezoidal Rule
2. Error Analysis
Review and Assessment
Logarithmic, Exponential, and Other
Transcendental Functions (20 Days)
V.
A.
B.
The Natural Logarithmic Function and Differentiation
The Natural Logarithmic Function and Integration
C.
D.
E.
1.
2.
3.
F.
1.
2.
G.
H.
I.
Inverse Functions
Exponential Functions: Differentiation and Integration
Bases Other that e and Applications
Bases Other than e
Differentiation and Integration
Applications of Exponential Functions
Differential Equations: Growth and Decay
Differential Equations
Growth and Decay Model
Inverse Trigonometric Functions and Differentiation
Inverse Trigonometric Functions: Integration and Completing the Square
Review and Assessment
VI. Applications of Integration (18 Days)
A.
B.
1.
2.
3.
C.
D.
1.
2.
E.
F.
Area of a Region between Two Curves
Volume
The Disc Method
The Washer Method
Solids with Known Cross Sections
Arc Length and Surfaces of Revolution
Work
Work Done by a Constant Force
Work Done by a Variable Force
Fluid Pressure and Fluid Force
Review and Assessment
VII. Slope Fields (3 Days)
A.
Slope Fields
Main Text
Larson and Edwards, Calculus AP Edition, 9th Edition, Belmont, CA:
Brooks/Cole, Cengage Learning, 2010, 2006.