AP® Calculus Syllabus AB (Form ID: 34156)
Course Description
Calculus is the study of change. Pre-requisite to this course was the study of linear,
quadratic, polynomial, rational, exponential, logarithmic, trigonometric functions.
But their study was static. You did not look at how the function was changing, only at
what the function was, but now we will begin to look at how these functions change.
This course will search for this change through a multi-representational approach -numerically, graphically, analytically, and verbally.
As outlined by the College Board, the course will cover: limits, continuity,
differentiation, fundamental theorem of calculus, integration of algebraic,
trigonometric, and transcendental functions, as well as numerous applications.
Primary Text
Thomas, George B., Finney, Ross L. Calculus and Analytic Geometry. 8th ed. Reading:
Addison-Wesley Publishing Company, 1993
Graphing Calculator [C5]
All students will be provided a TI-83 Plus graphing calculator to use for the entire school year.
The calculator must be returned on the day of the final exam. Students may decline the use of
a school issued calculator if they are in possession of their own TI-83 or higher or any other
graphing calculator with derivative and integration capabilities.
C2-The course teaches
Course Timeline [C2]
Week (days) sections
assignments
September week 1 1.1, 1.2
Slopes, lines, & functions
p. 10-11, 1-25 odd, 29-39 odd, 40,41-47 odd, 48
p. 22-24, 1-31 odd, 33-38, 41-47 odd, 49-53
September week 2 1.7, 1.8
Limits of function Values
p.69-72, 1-33 odd, 34,35,37,41-44
p.78, 1-51 odd, no 47.
September week 3 1.9, 1.10
p.83, 1-19 odd
Limits involving infinity
p.93, 1-6, 11-33 odd
Continuous Functions
Test 1.1 – 1.2 & 1.7 - 1.10
*Finding limits numerically lab (see activity 1) [C3][C5]
September week 4 1.6, 2.1
p.56-57, 1,3,5,6,8,10, 11-17 odd, 18,23,24
Slopes, Tangent lines, & Derivatives p.111-112, 1-53 odd
Differentiation Rules
*Slopes experiment (see activity 2) [C3][C5]
October week 1
2.2
p.121-124, 1-19 odd, 23-26
Velocity & Rates of Change
*Ball drop experiment (see activity 3) [C3][C5]
.
all topics associated with
Functions, Graphs, and
Limits Derivatives; and
Integrals as delineated in
the Calculus AB Topic
Outline in the AP
Calculus Course
Description.
C3-The course provides
students with the opportunity
to work with functions
represented in a variety of
ways -- graphically,
numerically, analytically, and
verbally -- and emphasizes
the connections among
these representations.
C4-The course teaches
students how to
communicate
mathematics and explain
solutions to problems
both verbally and in
written sentences.
C5-The course teaches
students how to use
graphing calculators to
help solve problems,
experiment, interpret
results, and support
conclusions.
1
October week 2
2.3
p.131-132, 1-41 odd, 42-46
Derivatives of Trigonometric
Test 1.6 & 2.1 - 2.3
Functions
*Finding trig derivatives activity (see activity 4) [C4]
October week 3
2.4, 2.5
TheChain Rule &
Implicit Differentiation
p.139-141, 1,5,9,11,19-27 odd,
35,37,41,45,46,49,51
p.146-148, 9-47 odd, no 17,19,43, or 45
October week 4
2.6
Linearization & Differentials
p.158-160, 1,5,9,11,13,15,25,27,31-41 odd, 45,
November week 1 3.1
Related Rates of Change
p.181-183, 1-29
C3-The course provides
students with the opportunity
to work with functions
represented in a variety of
ways -- graphically,
numerically, analytically, and
verbally -- and emphasizes
the connections among
these representations.
Test 2.4 – 2.6
November week 2 3.2
p.191-192, 1-7 odd, 11,12,15,19,25
Maxima, Minima, & The Mean Value Theorem
November week 3 3.2, 3.3
Graphing with y’ & y’’
p.192, 17,25
p.199, 1-35 odd, 36,37
November week 4 3.3
Graphing with y’ & y’’
p.200, 38-42, 44-50, 53,54
December week 1
3.3
Graphing with y’ & y”
Test 3.1 – 3.3
C4-The course teaches
students how to
communicate
mathematics and explain
solutions to problems
both verbally and in
written sentences.
December week 2
3.4, 3.5
p.206, 1-19 odd, 23
Asymptotes & Dominant Terms
p.215, 1-9 odd
Optimization
*Group Optimation Activity (see activity 5) [C3][C4]
December week 3
3.5,3.6
Optimization & L’Hopital’s Rule
p.216-217, 11-25 odd
p.225, 1-21 odd
Test chapter 3
December week 4
Indefinite Integrals
4.1
p.245-246, 1-57 odd
12/25
Happy Holidays
January week 1
4.2
Initial Value Problems
p.252, 1-21 odd, 22,23,25
C5-The course teaches
students how to use
graphing calculators to
help solve problems,
experiment, interpret
results, and support
conclusions.
January week 2
4.3, 4.4
p.267-268, 1-25 odd, 27-37
Definite Integrals
p.276-277, 1-4, 7-27 odd, 33-36
*Area under a curve experiment (see activity 6) [C3][C4][C5]
2
January week 3
4.5
The Fundamental Theorem
of Integral Calculus
p.285-287, 1-29 odd, 31-34,37,39,49,50
January week 4
4.6, 4.7
Integration by Substitution &
Numerical Integration
p.295, 1-51 odd
p.305, 1,3,5,9,11
Test 4.1 – 4.4
January week 5
6.1, 6.2
p.401, 1-19 odd
Inverse Functions &
p.410-411, 1,3,7,11-17 odd
Natural Logarithms
Test 4.5 – 4.7
*Discovery of the inverse functions’ theory of derivative (see activity 7) [C3][C4][C5]
February week 1
6.2 – 6.3
Natural Logarithms & Exponential
Function
p.410-411, 27-37 odd, 41-61 odd, 65
p.418-419, 1-53 odd
February week 2
6.4, 6.5
Natural Logarithms & Exponential
Functions; Growth & Decay
p.428-429, 1-65 odd
p.437-438, 1-10
February week 3
6.7, 6.8
Inverse of Trigonometric
Functions & Their Derivatives
p.450. 1-37 odd
p.456, 1-43 odd
February week 4
7.1, 7.2
Basic Integration Formulas &
Integration by Parts
p.484-485, 1,7,11-21 odd, 31-53 odd
p.491, 1-31 odd
March week 1
7.3, 7.4
Trigonometric Integrals &
Trigonometric Substitutions
p.499-500,1-17 odd,21–27 odd, 33-43 odd, 51,53,57
p.506-507, 1-15 odd
March week 2
7.4,7.5
Trigonometric Substitutions &
Partial Fractions
p.506-507, 17-39 odd
p.511-512, 1-21 odd, 25,31,33,37,39
March week 3
7.6
Using Integral Tables &
Reduction Formulas
p.518-519, 1-17 odd, 27-37 odd, 41,43
April week 1
Areas of Regions
Between Curves
p.323-324, 1-45 odd
p.335-337, 1-31 odd
5.1, 5.2
Test chapter 6
Test chapter 7
3
April week 2
5.3
p. 343-344, 1- 15 odd
Volume: Discs, Washers
Test 5.1 - 5.2
& Shells
Website used for visual help
Visual Calculus. University of Tennessee Mathematics Department.
http://archives.math.utk.edu/visual.calculus/
April week 3
Slope Fields
Separation of Variables
Handouts provided by Nancy Stephenson provided by
collegeboard.com
Test Slope Fields
April week 3
AP Exam Review
AP Exam Review consists of practice exams provided by AP Exam preparation books and a
thorough review of all free response problems from the past few years. A list of AP Exam
preparations books is provided at the end.
April week 4
AP Exam Review
May week 1
AP Exam Review
May week 2
AP Exam (then a couple days rest)
May week 3
8.1, 8.2
Limits of Sequences of Numbers
Infinite Series
p.549-550, 1-61 odd, no #51
p.561, 1-35 odd, 49–55 odd
May week 4
8.3, 8.4
Infinite Series &
The Comparison & Integral Tests
The Ratio & Root Test
p.570-571, 1-29 odd
p.577, 1-37 odd
June week 1
Start end of year project
June week 2
Finish end of year project
Begin review for final
June week 3
Review for final
June week 4
Congratulations, Seniors!
Test Sections 8.1 – 8.4
The calculus is the greatest aid we have to the appreciation of physical truth in the broadest
sense of the word.
-William F. Osgood
4
Student Evaluation
Grading is based strictly on students’ performance on tests and quizzes. While a few of these
tests and quizzes may be completed in group work or at home as a take-home assessment, the
majority of assessments will be in class in a test setting resembling the AP Exam itself. Some
tests will require the use of a calculator, while for others; calculator use will not be permitted.
All quizzes are worth 100 points and all tests are worth 200 points. There are anywhere from 2
to 4 tests per marking period and 4 to 6 quizzes per marking period for a total of approximately
1200 points. Each marking period average is the calculated by point out of points. C3-The course provides
Activities
students with the opportunity to
work with functions
represented in a variety of
ways -- graphically,
numerically, analytically, and
verbally -- and emphasizes the
connections among these
representations.
Activity 1 [C3][C5] Finding limits numerically lab
Students are given several functions and asked to find function values using their
calculators at values that get approach a particular value. For example, a student may be asked
2
to find function values of f(x) =
at x = 3.1, 3.01, 3.001, etc. and draw conclusions about
x3
how the function values are behaving as the x values approach 3 from the right hand side.
Activity 2 [C3][C5] Slopes experiment
Students will be given a couple of quadratic functions. For each function they are asked to find
function values at x = 0 and x = 2 and to find the slope of the secant line passing thru these 2
points. Then the students are asked to repeat the process for x = 0 and x = 1, then for x = 0 and
x = .5, then for x = 0 and x = .1, etc. We then discuss how the successive slopes we are finding
are related to the slope of a tangent line at x = 0. I use this activity to help the students build an
understanding of the difference quotient and how the limit of it as h approaches zero will give
the exact value of the slope of a curve at a point.
Activity 3 [C3][C5] Ball Drop Experiment
Students will be put into groups and each given a tennis ball. Using the position
function for free fall, students are asked to find the velocity at which the ball
hits the ground from varying heights. Students are also asked to record the time it
takes for the ball to hit the ground and solve for the time analytically to check for
accuracy. Students are then given several scenarios of projectile motion to analyze
using derivatives.
C4-The course teaches
students how to
communicate
mathematics and explain
solutions to problems
both verbally and in
written sentences.
Activity 4 [C4] Finding Trigonometric Derivatives
Students will be given the derivatives of sin x and cos x and be asked to find the derivatives of
the other 4 trig functions. The students will be put into groups and not be told how to find the
other derivatives. As each group thinks they have the answers, they are to call me over
C5-The course teaches
so I can check to see if they are correct.
students how to use
graphing calculators to
help solve problems,
experiment, interpret
results, and support
conclusions.
5
Activity 5 [C3][C4] Group Optimization Activity
Students are put into groups and given a 9 by 12 piece of paper. Students are then
asked to optimize 3 things: 1. The volume of the largest open top box they can make
by cutting congruent squares off of each corner and folding up the sides, 2. The area of the
largest triangle that can be enclosed by folding the upper left hand corner of the paper down to
the bottom the page, and 3. The volume of the largest cylinder that can be made by rolling up a
piece of paper of same perimeter, not necessarily 9 by 12,
C3-The course provides
students with the opportunity to
work with functions
represented in a variety of
ways -- graphically,
numerically, analytically, and
verbally -- and emphasizes the
connections among these
representations.
Activity 6 [C3][C4][C5] Area under a curve experiment.
Prior to them learning the definite integral, the students are put into groups and asked to find
the area under the curve y = 4x – x2 from 0 to 4. Prizes are awarded to the group who comes
closest. If any students think to insert rectangles or trapezoids under the curve and find the
sum of their areas, we use that group as a springboard into a lesson on area estimations using
left hand, right hand, and mid- points of rectangles and trapezoids, and the notion of inscribed
and circumscribed.
C4-The course teaches
students how to
communicate
Activity 7 [C3][C4][C5] Discovering the inverse theory of derivatives.
mathematics and explain
solutions to problems
Students are put into groups and given 5 functions. Students are asked to find the
both verbally and in
inverse of each function, and graph each function, its inverse, and the line y = x in the
written sentences.
first quadrant. Students are then asked to describe the relationship between a function and its
inverse graphically. Once students realize that one function is a reflection of the other over the
line y = x, students are then asked to find 3 pairs of reflected points. They are then asked to
calculate the derivates at each of the respective points. (3 derivatives for each function).
Lastly, students are asked to describe the relationship that they see and write their own theorem
using correct mathematical symbols.
C5-The course teaches
students how to use
graphing calculators to
help solve problems,
experiment, interpret
results, and support
conclusions.
In this course, students will frequently present solutions to the class or to me, work in groups,
collaborate on projects, and generally communicate their understanding of the Calculus as it is
learned. A complete mastery of the processes of Calculus will be expected so that students will
be able to intelligently analyze the concepts, theory and connections imbedded at the core of
Calculus.
6
Supplemental Materials
In this course I try to provide examples of sample multiple choice questions as frequently as
possible. Below is a list of the AP Preparation books the students are provided for a majority
of the sample problems presented in this course particularly in the weeks leading up to the
exam.
AP Calculus AB. New York: Simon & Schuster (Kaplan Publishing), 2001.
AP Mathematics: Calculus AB & Calculus BC. 4th ed. New York: Macmillan Publishing,
1998.
AP Success: Calculus AB/BC. Lawrenceville: Peterson’s, 2001.
How to Prepare For The Advanced Placement Exam Calculus. 8th ed. New York: Barrons,
2005.
Kahn, David S. Cracking the AP Calculus AB & BC Exams. 2002-2003 ed. New York:
Princeton Review Publishing, 2002.
King, Kerry J., Johnson, Dale W. Cliffs AP Calculus AB and BC. 3rd ed. New York: Hungry
Minds, Inc., 2001.
The Best Test Preperation For The Advanced Placement Examination: Mathematics Calculus
AB. Piscataway: 2001.
7