AP CALCULUS AB
2012/2013
School: South Gibson County High School
Instructor: Beth Davis
Brief Description of Course: This course is designed to cover one semester of college
calculus material. It covers an extensive study of function, graphs, limits, derivatives,
definite integrals and applications of all of the above. Each of these topics is approached
via the “Rule of Four”, with activities that emphasize expressing mathematics from
graphical, numerical, analytical and verbal representations. Students must become
familiar with functions written not only as equations, but also shown as a graph, table or
in words. Students must become adept with the concepts of limits, derivatives and
integrals given these various forms of function representation. Students must learn that
many calculus concepts can be written with symbols or words that they must become
familiar with.
Some examples of verbal representations:
Function is always increasing = first derivative is always positive
Function is increasing at a decreasing rate =
increasing and concave down (first derivative positive and second derivative negative)
Find where function is increasing fastest = maximize the first derivative
Find where average rate of change is equal to the instantaneous rate of change =
apply the Mean Value Theorem ( or find where algebra slope = calculus slope )
These are just some of the examples of what students will learn in this class about the
language of calculus. This will help students become familiar with the language of
calculus and how this language applies to the real world.
Course Overview: My main objective in teaching AP Calculus AB is to provide students
with an opportunity to explore higher levels of mathematics. Through this exploration
and interaction with mathematics I hope to enable students to appreciate the higher
intricacies of problems and develop a solid foundation in the Calculus AB topic outline as
it appears in the AP Calculus Course Description, which they can take with them to their
higher level classes. I expect a lot from my students, whether it is in class in discussion
and group time, or at home working on homework and AP sample problems.
In order to reach all students, I strive to present all topics in many different ways.
Among these are graphical, numerical, analytical and verbal approaches to almost all
topics.
I use a TI-84 calculator to help students use the table feature, or math menu (zeros,
derivative at a point or numerical integral) in graphical mode, to get a more numerical
approach to problems. All students are required to have a graphing calculator, with about
half of the class using a TI-83+ or TI-84+ and half purchasing a TI-89. Graphing
calculators will be used daily to explore, discover, and reinforce the concepts of calculus.
Graphing calculators will be used to solve problems, complete experiments, interpret
results, and support conclusions throughout the course.
Java applets, PowerPoint presentations, Geometer’s Sketchpad sketches, WinPlot,
Graphmatica, and other technology-based visual aids will be used in the delivery of
instruction in this AP Calculus course.
Assessment: Students are assessed in my class daily, weekly and once a unit. Each day
students will have an assignment addressing the topics we covered in class. Each week
students will work on problems from released AP exams, both multiple choice and free
response designed to target the subject from the week before. On both of these
assignments (homework and class assignments) students are encouraged to work together
to discuss and complete the problems, but are required to write their own solutions to the
problems. At the end of each unit students are given an exam covering all material in the
unit. These exams are generally written in two parts, one calculator active, and one
without calculators. Problems are generally open ended, and students are required to
show work to get full credit. Often there is at least one released Free Response question
from previous AP exams, which is scored in the same manner as they will be on the AP
exam. We will also have some multiple-choice questions from released AP exams for
practice.
It is important to me to convey to students the importance of knowing how to do a
problem and also for students to know what their solutions actually mean in the context
of the problem. Students will be asked to explain work and justify answers whenever
they are working with open-ended questions. They will be shown that it is equally if not
more important to know how they are finding the answer they are finding and what their
answer means than to actually get the correct answer. They will be asked to explain their
work and answers using complete sentences using correct calculus language and symbols
and to use no calculator syntax. This is a great way to show students how calculus relates
to the real world.
Course Format: Students enrolled in AP Calculus AB have successfully completed
Honors Precalculus. Our class will meet for 90 minutes a day, 5 days a week all year
long. Each student receives 5 points added to each grading period’s final grade. If the
student receives a 3, 4 or 5 on the Calculus AB exam, our school district will pay the
students their AP exam fee back.
Textbooks: Larson, Ron, Hostetler, Robert P., and Edwards, Bruce H. Calculus of a
Single Variable 7TH edition. Boston, Houghton Mifflin Company. 2002
Rogawski, Jon. Single Variable Calculus: Early Transcendentals. New
York, W. H. Freeman and Company. 2008
Other Resources Used:
These are some of the many other resources that I use for Calculus AB.
- AP Calculus AB Teacher’s Manual by Duke University Talent Identification
Program 2nd Edition
-Be Prepared for the AP Calculus Exam by Skylight Publishing 2005
-Multiple Choice and Free Response Questions in Preparation for the AP
Calculus Exam by Lin McMullin. D and S Marketing.
-Preparing for the AP Calculus Examination by George Best. Venture Publishing.
-AP Calculus Teachers Guide by The College Board.
-Handley, PA math department website.
-AP Calculus in a Nutshell website Franklin Road Academy, TN.
-1997, 1998, 2003 Released Exams from The College Board.
-1989-2010 Free Response Question Collection .pdf file from The College Board.
-Teaching AP Calculus by Lin McMullin. D and S Marketing.
-2011 AP Summer Institute Manual by Phyllis Hillis. Oak Ridge High School,
Oak Ridge, TN
Expectations: To succeed in this class a student must
1. BE IN CLASS EVERY DAY
2. BE PREPARED FOR CLASS EVERY DAY
3. DO HOMEWORK ASSIGNMENTS
4. PAY ATTENTION IN CLASS
5. ASK QUESTIONS
6. PARTICIPATE IN GROUP WORK/DISCUSSIONS
7. TAKE GOOD NOTES
8. TAKE THE AP CALCULUS EXAM ON May 8, 2013
9. TRY THEIR BEST TO MAKE A 3,4 or 5 ON THE AP EXAM
Topic Timeline:
AB TOPICS
I.
Functions, Graphs, and Limits --6 weeks
II.
Derivatives and Applications of the Derivative – 8 weeks
III.
Integrals and Applications of Integrals -- 8 weeks
IV.
Differential Equations and Slope Fields – 6 weeks
V.
Review for the AP Exam
AB COURSE OUTLINE
Unit 1 – Limits
Larson 1.2 - tangent line problem/intro to limits – table, graph and algebra
Larson 1.3 – properties of limits
Larson 1.3 – techniques for evaluating limits
Larson 1.4 – continuity and intermediate value theorem
Larson 1.4 – continuity of piecewise functions
Larson 1.5 – infinite limits and vertical asymptotes
Larson 3.5 – limits at infinity and horizontal asymptotes
Unit 2 – Concept of the Derivative
Larson 2.1 – rate of change by equation, graph and table
Larson 2.1 – graphical interpretation of the derivative
Larson 2.1 – difference quotient, derivative at c
Larson 2.1 – derivative at x, properties of derivatives
Various - displacement, velocity and acceleration
Larson 2.4 – chain rule
Unit 3 – Derivative Formulas
Larson 2.3 – product rule
Larson 2.4 – quotient rule
Larson 2.2, 2.3, 2.4 – trigonometric functions
Larson 5.3, 5.8, 5.9 – inverse functions and inverse trig functions
Larson 2.1 – differentiability and continuity of piecewise functions
Larson 2.5 – implicit differentiation
Larson 2.6 – related rates
Unit 4 – Graphical Analysis
Larson 3.1 – extrema and extreme value theorem
Larson 3.2 – Rolle’s Theorem, Mean Value Theorem
Larson 3.3 – increasing/decreasing, first derivative test
Larson 3.4 – concavity, point of inflection, second derivative test
Larson 3.6 – curve-sketching
Larson 3.7 – optimization
Unit 5 – Integrals
Larson 4.1 – intro to definite integrals
Larson 4.1 – antiderivatives and indefinite integrals
Larson 3.9 – linear approximations and differentials
Larson 4.5 – antiderivatives and indefinite integrals, u-substitution
Larson 4.6 – trapezoid rule (including unequal subdivisions)
Larson 4.4 – mean value theorem for integrals
Larson 4.4 - Fundamental Theorem of Calculus
Larson 4.3 – properties of definite integrals
Various -- functions defined by integrals, second fundamental theorem, accumulation
function
Unit 6 – Exponential and Logarithmic Equations
Larson 5.1 – natural logarithmic function and definition of e
Larson 5.1 – derivative of natural log function, logarithmic differentiation
Larson 5.2 – antiderivatives of reciprocal functions and trig functions
Larson 5.3 – inverse functions
Larson 5.4 – derivative of natural exponential function, antiderivatives of natural
exponential functions
Larson 5.5 – exponential functions with other bases
Unit 7 – Applications of Integrals
Larson 6.1 – area between two curves
Larson 6.2 – volume by disks, washers and known cross-sections
Larson 4.4 – average value
Various -- distance vs. displacement, velocity vs. speed
Unit 8 – Differential Equations
Differential equations and slope fields will be covered using other books in my arsenal.
Topics Covered:
Solutions (general and particular) to separable differential equations
Slope Fields
Exponential Growth and Decay
Unit 9 – AP EXAM REVIEW
We will spend a couple of days reviewing each unit above. We will cover many released
free response and multiple choice questions. We will also fill out study cards with all
formulas that will need to be memorized. Students will come up with sample problems
that will be worked on in group setting and as a class. The students will sit for at one
least full length practice exam. We will cover the testing procedure and how their score
will be determined. We will review all major Theorems and definitions and how they are
usually presented on the AP exam.