AP Calculus Instructional Syllabus
Calculus is “the mathematics of change”.
Summary:
AP Calculus is equivalent to a FIRST SEMESTER college calculus course. This course
will feature most major topics pertaining to derivatives, integrals and their applications. On
Wednesday, May 6th, 2009 students will have the opportunity to take the AP Calculus AB
Exam. A student earning a 3, 4, or a 5 on the exam can possibly earn college credit for this
course. It is highly recommended that ALL students take the AP exam. Review for the AP
exam will begin approximately three weeks before the exam date. The final exam will take place
at least one week before the exam date. The exam will be a “real” AP test. The results of the
final and the test itself will be used as the last study aide for the 2009 AP Calculus Exam.
Grades:
This course will be graded on four areas:
∙ Homework (20%) which includes assignments and in-class activities
∙ Weekly AP Free Response Questions (15%)
∙ Quizzes (30%)
∙ Exams and/or projects (35%).
There will be 6 chapter exams, 2 semester exams and approximately 15 quizzes
throughout the year. There will be a MINIMUM of 3 hours of homework a week. The
weekly AP problem will be assigned Tuesday and DUE the following Tuesday of each
week.
A
AB+
B
BC+
Grading Scale
 90%
85%
80%
75%
70%
67%
C
CD+
D
DF
63%
60%
57%
53%
50%
Not an Option
Your grade is a running total through the semester. Quarter Grades will be assigned
based on your scores at that given time. Semester grades will be assessed based on your
semester total (Quarter A&B cumulative) and your semester final.
Student Expectations:
▪ Students are expected to complete all homework assignments.
▪ Students are to make up missed work in a timely fashion.
▪ Students are to get help when it is needed.
▪ Students are expected come in once a week outside of class to check in
Note:
You are here because you belong here. You will be successful if you are willing to make
good decisions. Expect to work harder than ever before. Expect homework and expect to study
outside of class. YOU CAN SUCCEED IN THIS CLASS but you must first commit yourself to
doing well.
Contact:
If you or your parent needs to contact me, I can be reached through the following:
▪ Stop in and see me, before school or after school in Room B111
▪ Call me by phone: 333-6188 (classroom) before 8:00am and after 3:05pm.
▪ Email me (the best option) at: gkortuem@faribault.k12.mn.us
Please review this with your Parent/Guardian and have them sign below.
Parent/Guardian Use Below
Parent/Guardian’s Name(s): ________________________________
Parent’s First and Last Name
I have seen the course syllabus and have discussed it with my son/daughter, ________________.
Student’s First and Last Name
If I have any questions or concerns I am aware that I can reach you at the previous contacts.
A good number/email to reach me is _______________________________________.
Parent’s Email, Phone or Both
X____________________
Parent’s Signature
Course Design:
AP Calculus AB is equivalent to a first semester calculus course at the college level.
This rigorous year-long course meets 5 days a week with an average of 3 hours homework each
week. There are 172 student days, with nearly 150 of these days occurring before the AP exam.
Much of the time is devoted to derivatives, integrals and their applications. Other topics relating
to functions are also discussed but as a supporting role for the two essential topics stated before.
The course is broken down into three parts: Functions, Derivatives and Integrals. Below
is a brief description of each part of the course. The course outline gives a more detailed list of
topics and approximate time spent on each item and can be found on pages 4-6.
The first topic explored is functions. Function types, Domain, Range, Limits, Symmetry
and Asymptotes are all used to graph functions. Roots are found graphically and analytically.
Asymptotes are determined using domain restrictions and by exploring end behavior. Many of
these concepts are used again when discussing continuity and later, derivatives. Transcendental
functions along with their derivatives and inverses are covered later in the year.
Derivatives and their applications are the next topics covered. Students first explore the
idea of local linearity, delve into tangential lines and then define the derivative both analytically
and conceptually. After students are comfortable using derivatives, these ideas are used in:
Related Rates problems, Implicit Differentiation, the 1st and 2nd Derivative Tests and other
applied topics. Connections between a function and its derivatives are emphasized through inclass activities and calculator explorations.
The last topic covered in depth is Antiderivatvies and the Integral. The concept of area
under the curve and its meaning is first established using estimation techniques. Following
estimation is the definite integral and properties associated with it. The application of the
integral is the culmination of the course, beginning with the Fundamental Theorem of Calculus
and finishing with areas of regions and volumes of solids.
A varied approach to class time is used to establish the connections between concepts and
explore new topics. Lecture, guided notes, labs, book work, projects and in-class activities all
aid in establishing these connections in calculus. The calculator (TI-83/84/89) also plays a key
role in the modern calculus course as it can be used for more than calculations. Calculator
explorations and activities are integrated into the class. Explorations using graphical capabilities
demonstrate the relationships between functions as seen in the First and Second Derivative Tests.
Other explorations aid in establishing the Fundamental Theorem of Calculus. The services of the
calculator are also employed in many other topics ranging from limits and continuity to slopefields and when dealing with “difficult” functions such as in the case of differential equations.
Assessments ranging from small check-ups and chapter exams to group projects are all
utilized to identify student strengths and weaknesses. Homework is required and graded as it is
essential practice for topics explored in class. Chapter exams are either In-class or Take-Homes.
Most of these exams consist of different portions to reflect the AP Exam. The use of an AP
problem of the week also aids in demonstrating the level of proficiency expected from all
students.
Course Outline:
Chapter 2: Functions, Limits and Continuity
A. Function Types
i.
ii.
iii.
iv.
v.
vi.
vii.
viii.
B. Limits
i.
ii.
iii.
17-days*
Piecewise
Composite
Trigonometric
Exponential
Power/Inverse Power Models
Linear Equations (3 forms)
Tangent/Secant Lines
Parametric Equations (Post AP-Exam)
Concept
One-sided Limits
Computations
a. Algebraically/Analytically
b. Substitution/Numeric
c. Graphically
iv. Limits as x   and Rational Functions
C. Graphing Functions
i. Domain and Range
ii. Symmetry-Even/Odd
iii. Intercepts and Roots
iv. Asymptotes and End Behavior
D. Continuity
i. Conceptually
ii. Three Properties
iii. End Points
iv. Three Theorems
a. Maximum Value
b. Minimum Value
c. Intermediate Value
E. Review of Trigonometry
i. Identities
ii. Properties of Sine and Cosine
iii. Unit Circle
iv. Radian Measure
Chapter 3: The Derivative
22-days*
A. The Concept
i. Instantaneous Slope
ii. Rate of Change
B. Mathematical Definition (Difference Quotient using Limits)
C. Notation
D. Implication of Continuity
E. Basic Derivative Rules
i. Power Rule
ii. Sum/Difference Rules
iii. Product Rule
iv. Quotient Rule
v. Other properties
F. Chain Rule and Composite Functions
G. Trigonometric Derivatives
H. Basic Antiderivatives
i. Power Rule
ii. Trig Derivatives
Chapter 4: Applications of the Derivative
35-days*
A. Theorems
i. Rolle’s Theorem
ii. Mean Value Theorem for Derivatives
a. Average Rate of Change
b. Instantaneous Rate of Change
c. Revisit Local Linearity and Tangent Lines
B. Derivative Tests
i. The First Derivative Test
i. Increasing/Decreasing Functions
ii. Critical Values/Points
1. Maximum/Minimums
2. Stationary Points
3. Importance of Sign changes
iii. Open vs Closed Intervals
ii. The Second Derivative Test
i. Concavity
ii. Inflection Numbers/Points
1. Sign changes
iii. Maximum/Minumum Values
C. Motion and the Second Derivative
i. Relationship between functions
i. Position
ii. Velocity
iii. Acceleration
ii. Initial Condition problems (working backwards)
i. Particle Problems
ii. Gravity Problems
iii. Other
D. Related Rates
i. First Derivative Problems
ii. Second Derivative Problems
E. Applied Maximum and Miniumum Problems
i. Geometric Concepts
i. Surface Area/Volumes (e.g.)
ii. Other Concepts
i. Cost/Time/Traffic (e.g.)
F. Implicit Differentiation
i. First Derivatives
ii. Second Derivatives
iii. Finding Extrema
G. The Differential and Linearization
i. Local linearity
ii. Estimation using the differential
H. Growth of a Function and the Second Derivative
I. Newton’s Method (Post AP-Exam)
Chapter 5: The Definite Integral
A. Physical Interpretation of Area Under a Curve
a. Signed Area
i. Quantities (Distance, Volume, Charge)
b. Estimation Techniques
i. Riemann Sums
1. Rectangles
ii. Trapezoids
B. Summation Notation
C. The Definite Integral
22-days*
a. Limits of Integration
b. Integrand
c. Properties
D. Antiderivatives
a. Properties
E. Mean-Value Theorem for Definite Integrals
a. Average Value of a Function
F. The Fundamental Theorem of Calculus
a. First FTC
i. Composite Functions
b. Second FTC
i. Evaluation of Integrals
Chapter 5
Continued
Chapter 6: Topics in Differential Calculus
24-days*
A. Logarithms
i. Properties
ii. Growth
iii. Derivatives
iv. Antiderivatives and Integration
v. Inverses
B. The Number e
C. Inverse Functions
i. One to One
ii. Role of the Derivative
iii. Graphing Inverses
D. Exponential Functions, e x , b x
i. Derivatives
ii. Inverses
iii. Growth
E. Inverse Trigonometric Functions
i. Derivatives
F. Differential Equations
i. Separable Differential Equations
ii. Natural Growth
1. Doubling Time
iii. Natural Decay
1. Half-life
iv. Slope Fields
Chapter 7: Computing Antiderivatives
4-days*
A. Short Cuts
a. Integral Tables
b. Geometric Areas
B. U-Substitution
a. Equivalent Definite Integrals
b. Composite Functions
c. Functions as Limits of Integration
C. Integration by Parts (Post AP-Exam)
Chapter 8: Applications of the Definite Integral
A. Area between Curves
a. Not Signed, Geometric
b. Division of Areas by linear Function
c. Horizontal Cross-sections
d. Vertical Cross-sections
B. Volumes of Solids
8-days*
Chapter 8
Continued
a. Perpendicular Plane Regions
b. Discs and Washers
i. Rotations about x-axis
ii. Rotations about y-axis
iii. Rotations about a generic line
c. Shell Method (Post AP-Exam)
Review for the AP Exam
17-days*
Total time spent on material 149 days.
Textbook
Stein, Sherman K. and Anthony Barcellos. Calculus and Analytic Geometry. 5th Edition
McGraw-Hill, Inc. 1992