AP Calculus Audit: AP Calculus
Advanced Placement Calculus AB Syllabus
2008-2009
Introduction
Our high school is a small, rural high-school in the northeastern corner of the state, with approximately
800 students. The math department consists of 6½ teachers, and offers classes in Algebra, Geometry,
Algebra II, Advanced Functions & Modeling (online course through VPS), Pre-Calculus, Discrete Math and
AP Calculus AB.
Classes are 90 minutes long, and students take 4 per semester. (This is referred to as a “4 x 4 Block.”)
All AP courses are year-long courses. A school year is 180 days in our state.
All AP classes follow the (state) Standard Course of Study, as well as the College Board AP Syllabus.
Advanced Placement Calculus develops the student’s understanding of the concepts of calculus
(functions, graphs, limits, derivatives and integrals) and provides experience with its methods and
applications. The course encourages the geometric, numerical, analytical, and verbal expression of
concepts, results, and problems. Appropriate technology, from manipulatives to calculators and
application software, is used regularly for instruction and assessment. (from the DPI Standard course of
study.) The Curriculum Guide is attached at the end of this document.
Class Structure & Technology
The classroom is arranged into a grouped seating arrangement, as I believe in cooperative learning and
peer assistance. Students are encouraged to ask questions as they arise; our class is informal and
modeled after college courses. The teacher is viewed as a facilitator for the class – an instructor, but
also a mentor or a coach. Students are encouraged to try new approaches, and to problem solve using
their own abilities, as well as those of their peers. Problems and ideas are often presented in a “what
if...” manner in order to stimulate the students into discovering some concepts on their own, in a guided
“behind-the-scenes” manner.
Quizzes are given 3-4 days a week, homework is given daily, and tests are given after each unit of study.
“AP Opportunities” are given every 5-7 days. These are 20 point quizzes which contain questions
worded in the same format as AP Free Response Questions, or may actually be released questions. One
multiple-choice question is usually included. Students are required to write detailed responses,
explaining their work and answering the questions fully. Answers to these questions are discussed in
class, and complete explanations are developed together with the students. A Midterm Exam and a
Final Exam are given, as well as chapter tests.
Out-of-class projects are assigned during each grading period that tie together research, technology and
calculus topics. Students are required to research topics of interest, use calculus ideas and techniques
to solve problems, and to write summary papers of 3 – 5 pages. Some examples of these projects
include College Tuition Trends, Optimization of Cat Food Cans, Normal Curve of a Distribution, and
Fitting Lines to Data. Writing in a mathematical manner is emphasized at all times, and technology is
used in scanning graphs, submitting papers in an electronic format, and for research and presentation
AP Calculus Audit: AP Calculus
techniques. Students are required to journal electronically with student blogs to discuss math in current
events, and to provide explanations for additional classroom extensions. A final project involves the
continued development of a class webpage with resources, tutorials and explanations included for the
succeeding class.
Multimedia tools and technology are utilized as much as possible. All students have personal calculators
(either TI-83/84+ and/or TI-89) and we frequently use both. Students learn the four required skills
needed for the AP exam (graphing in an arbitrary window, solving an equation, finding numerical
derivatives and integrals), but may not use other features (e.g. min/max finder) to perform tasks that
can be done with calculus methods. Calculators are used to verify answers, to support solutions
graphically, and to “jump start” a problem, but students are discouraged from reliance upon calculators.
Some elementary programs are written for familiarity with the concept, but not for use on calculus
problems. Software such as Geometer’s Sketchpad and application applets are frequently used for
demonstrations, as well as tangible manipulatives. Laboratory activities are scattered throughout the
course. Guided practice, online drills and quizzes, and “Catch-Up Days” are used for reinforcement and
remediation. The teacher’s website is used for resources, access to quizzes and drills, class notes and
homework assignments (if a student was absent), additional tutorials, and for homework submissions.
Students are taught the proper technique for using Moodle and Blackboard, (using the teacher’s
courses) in preparation for their college experience.
Resources
Primary Textbook
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Larson, Roland, Hostetler, Robert and Edwards, Bruce. Calculus of a Single
Variable, Sixth Edition. Massachusetts: Houghton Mifflin Company, 1998.
Larson, Ron, Hostetler, Robert and Edwards, Bruce. Calculus of a Single
Variable, Eighth Edition. Massachusetts: Houghton Mifflin Company, 2006.
Supplementary Textbooks & Resources
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Anton, Howard, Bivens, Irl, and Davis, Stephen. Calculus, Seventh Edition. New
York. John Wiley and Sons, 2002.
Barton, Ray, John R. Brunstring, John J. Diehl, Greg Hill, Karyl Tyler, and Steven L.
Wilson. Preparing for the Calculus AP Exam with Calculus: Graphical,
Numerical, Algebraic. New York: Pearson, Addison-Wesley, 2006.
DeTemple, Duane, and Robertson, Jack. The C.A.L.C. Handbook: Conceptual
Activities for Learning the Calculus. New Jersey: Dale Seymore Publications,
1991.
Finney, Ross L., Franklin Demana, Bert Waits, and Daniel Kennedy. Calculus:
Graphical, Numerical, Algebraic. New Jersey: Prentice Hall, 2003.
Forester, Paul A. Calculus Explorations. California: Key Curriculum Press, 1998.
Hughes-Hallett, Deborah, Gleason, Andrew et al. Calculus: Single Variable,
Fourth Edition. New Jersey: John Wiley & Sons, 2005.
Kamischke, Ellen. A Watched Cup Never Cools: Lab Activities for Calculus
and Precalculus. California: Key Curriculum Press, 1999.
AP Calculus Audit: AP Calculus
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Rogawski, Jon. Single Variable Calculus: Early Transcendentals. New York:
W.H. Freeman & Co, 2008.
Stewart, James. Single Variable Calculus, Fifth Edition. California: Thomson
Learning, Brooks/Cole Publishing, 2003.
College Board. Released AP Problems/Exams. College Board, 1993 - 2007.
Assorted websites & webpages.
Grading Policy
The following grading scale is used:
93 – 100 A
85 – 92
B
77 - 84
C
70 - 76
D
0 - 69
F
All papers, quizzes, tests and projects are graded “AP Style” with full annotations required of students,
and scoring is consistent with College Board rubrics and scoring guides.
Student assessments include lab experiments, handouts, homework, participation, Free Response
Question packets (FRQ’s), quizzes, AP Opportunities, projects, tests, blogs, journals and discussion
board participations. Students will also sit for a practice AP exam during the spring. Before school
tutoring is available daily from 7:00 to 7:45 am and after school tutoring is available from 3:00 to 4:00
pm. Many students seek help during lunch periods and during the instructor’s planning period.
Additional tutoring is available by appointment and whenever the instructor is free. Students are also
encouraged to form their own study groups, and to assist others whenever possible.
No “bald answers” are accepted. All work is expected to be shown, to be annotated when appropriate,
and to be legible. Answers are boxed for clarity. Extensive notebooks are kept by each student, as a
resource for future math courses in college.
Summer work is given at the beginning of August, in order to start students thinking mathematically.
This work is chosen from a variety of Precalculus topics, including functions (their properties, their
graphs and their attributes) and trigonometry. A solid review of Algebra I and Algebra II skills is also
included.
Goals (adapted from the AP Calculus Course Description)
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Students should be able to work with functions numerically, graphically, analytically, and verbally
and understand the connections between these methods;
The derivative should be understood as the instantaneous rate of change of a function and as the
local linear approximation of the function;
The definite integral should be understood as the limit of a Riemann sum and as the net
accumulation of a rate of change;
The relationship between derivatives and the definite integral should be understood in terms of
both parts of the Fundamental Theorem of Calculus;
AP Calculus Audit: AP Calculus
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Students learn to communicate about mathematics verbally and in writing;
Students should be able to model a written description of a physical situation with a function, a
differential equation, or an integral;
Students learn to use technology to analyze problems, experiment, and verify and interpret results;
Students are expected to learn to judge the reasonableness of their solutions;
Students develop an appreciation of calculus and its applications, as well as for their personal
accomplishment in learning calculus.
Course Outline
(order differs slightly from textbook)
Preparation for Calculus (Summer Work)
1. Graphs and Models
2. Linear Models and Rates of Change
3. Functions and Their Graphs
4. Fitting Models to Data
Topic One: Limits and Their Properties (15 days)
1. A Preview of Calculus
2. Finding Limits Graphically and Numerically
3. Evaluating Limits Analytically
4. Continuity and One-Sided Limits
5. Infinite Limits
Additional Activities: Lines & Circles & Limits, Oh My!; Limits Bingo
Topic Two: Differentiation (25 days)
1. The Derivative and the Tangent Line Problem
2. Basic Differentiation Rules and Rates of Change
3. The Product and Quotient Rules and Higher-Order Derivatives
4. The Chain Rule
5. Implicit Differentiation
6. Related Rates
Additional Activities: How Many Licks?; Derivative Gateway Quizzes; Flashcards Relay;
Derivatives Dominoes
Topic Three: Applications of Differentiation (24 days)
1. Extrema on an Interval
2. Rolle’s Theorem and the Mean Value Theorem
3. Increasing and Decreasing Functions and the First Derivative Test
4. Concavity and the Second Derivative Test
5. A Summary of Curve Sketching
6. Optimization Problems
7. Differentials
Additional Activities: More Ice Cream, Please; Prism Pop; Derivative Gateway Quizzes;
Derivatives Dominoes; Derivatives Jeopardy
AP Calculus Audit: AP Calculus
Topic Four: Integration (19 days)
1. Antiderivatives and Indefinite Integration
2. Area
3. Reimann Sums and Definite Integrals
4. The Fundamental Theorem of Calculus
5. Integration by Substitution
6. Numerical Integration
Additional Activities: A River Runs Through It; It Averages Out in the End; Square Pegs in
Round Holes; Antiderivatives Puzzles; Integration Relays; Play-doh Constructions
Topic Five: Differential Equations and Mathematical Modeling (22 days)
(Transcendentals introduced during Derivatives)
1. The Natural Logarithmic Function and Differentiation
2. The Natural Logarithmic Function and Integration
3. Inverse Functions
4. Exponential Functions: Differentiation and Integration
5. Bases Other Than e and Applications
6. Inverse Trigonometric Functions and Differentiation
7. Inverse Trigonometric Functions and Integration
Additional Activities: Derivatives & Integrals Gateway Quizzes; Derivative & Integral
BINGO and puzzles,
Topic Six: Applications of Integration (14 days)
1. Slope Fields
2. Differential Equations: Growth and Decay
3. Separation of Variables and the Logistic Equation
Additional Activities: Slope Field Constructions; It Averages Out in the End; Square Pegs
In Round Holes; Jeopardy!; Puzzles
Topic Seven: Integration Techniques & Applications (17 days)
1. Area of a Region Between Two Curves
2. Integration and Volume: Discs and Washers
Additional Activities: Home in the Dome; Play-doh Constructions; As Easy as 𝜋; Heads
Or Tails, Disks or Shells; Let the Games Begin!, Jeopardy!, Puzzles
Topic Eight: Integration Techniques, L’Hopital’s Rule and Improper Integrals (5 days)
1. Basic Integration Rules
Additional Activities: Jeopardy!; Puzzles & Dominoes; The Cup; Why Do We Have to
Learn This?
Topic Nine: Review for AP Exams (14 days)
AP Calculus Audit: AP Calculus
Topic Ten: After the AP Exam
1. Research topics (including some BC Topics)
2. Projects
3. End of Course Tutoring Assistance for EOC Students
Teaching Strategies
Using the Rule of Four, (Solving & analyzing using four approaches: graphically, numerically,
analytically, and verbally), the teacher will be the guide for the students in discovering many calculus
concepts from these multiple representations. Students use the Rule of Four approach in their
PreCalculus course, and use the Demana, Waits, Foley and Kennedy textbook.
Continuity is
emphasized within the courses, due to the same instructor being available to teach both courses.
Communication in mathematics is stressed continuously – both oral and written communication.
Clear and concise explanations are modeled and expected of both the instructor and the students.
Proper vocabulary and symbolism are required.
(State) Standard Course of Study
ADVANCED PLACEMENT CALCULUS
Advanced Placement Calculus develops the student’s understanding of the concepts of calculus (functions, graphs, limits,
derivatives and integrals) and provides experience with its methods and applications. The course encourages the geometric,
numerical, analytical, and verbal expression of concepts, results, and problems. Appropriate technology, from manipulatives
to calculators and application software, should be used regularly for instruction and assessment.
Prerequisites
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Use circle, trigonometric, and inverse trigonometric functions to solve problems. Use the trigonometric ratios and
the laws of sines and cosines to solve problems.
Translate among graphic, algebraic, numeric, tabular, and verbal representations of relations.
Use functions (linear, polynomial, exponential, logarithmic, rational, power, piecewise) to model and solve
problems.
Use the composition and inverse of functions to model and solve problems.
Transform relations in two and three dimensions; describe algebraically and/or geometrically the results.
Use the conic relations to model and solve problems.
Write equivalent forms of algebraic expressions.
Find special points (zeros, intercepts, asymptotes, local maximum, local minimum, etc.) of relations and describe in
the context of the problem.
Number and Operations
Competency
Goal 1
Objectives:
The learner will demonstrate an understanding of the behavior of functions.
1.01 Demonstrate an understanding of limits both local and global.
a.
b.
Calculate limits, including one-sided, using algebra.
Estimate limits from graphs or tables of data.
AP Calculus Audit: AP Calculus
1.02 Recognize and describe the nature of aberrant behavior caused by asymptotes and unboundedness.
a.
b.
c.
Understand asymptotes in terms of graphical behavior.
Describe asymptotic behavior in terms of limits involving infinity.
Compare relative magnitudes of functions and their rates of change.
1.03 Identify and demonstrate an understanding of continuity of functions.
a.
b.
c.
Develop an intuitive understanding of continuity. (Close values of the domain lead to close values
of the range.)
Understand continuity in terms of limits.
Develop a geometric understanding of graphs of continuous functions. (Intermediate Value
Theorem and Extreme Value Theorem).
Geometry and Measurement
Competency
Goal 2
Objectives:
The learner will use derivatives to solve problems.
2.01 Explore and interpret the concept of the derivative graphically, numerically, analytically and verbally.
a.
b.
c.
Interpret derivative as an instantaneous rate of change.
Define derivative as the limit of the difference quotient.
Identify the relationship between differentiability and continuity.
2.02 Apply the concept of the derivative at a point.
a.
b.
c.
d.
Find the slope of a curve at a point. Examples are emphasized, including points at which there are
vertical tangents and points at which there are no tangents.
Find the tangent line to a curve at a point and local linear approximation.
Find the instantaneous rate of change as the limit of average rate of change.
Approximate a rate of change from graphs and tables of values.
2.03 Interpret the derivative as a function.
a.
b.
c.
d.
Identify corresponding characteristics of graphs of ƒ and ƒ'.
Identify relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ'.
Investigate the Mean Value Theorem and its geometric consequences.
Translate between verbal and algebraic descriptions of equations involving derivatives.
2.04 Demonstrate fluency and accuracy in the computation of derivatives.
a.
b.
c.
Find the derivatives of basic functions, including power, exponential, logarithmic, trigonometric,
and inverse trigonometric functions.
Use the basic rules for the derivative of sums, products, and quotients of functions.
Use the chain rule and implicit differentiation.
2.05 Interpret the second derivative.
a.
b.
c.
Identify the corresponding characteristics of the graphs of ƒ, ƒ', and ƒ".
Identify the relationship between the concavity of ƒ and the sign of ƒ".
Identify points of inflection as places where concavity changes.
AP Calculus Audit: AP Calculus
2.06 Apply the derivative in graphing and modeling contexts.
a.
b.
c.
d.
e.
f.
Analyze curves, with attention to monotonicity and concavity.
Optimize with both absolute (global) and relative (local) extrema.
Model rates of change, including related rates problems.
Use implicit differentiation to find the derivative of an inverse function.
Interpret the derivative as a rate of change in varied applied contexts, including velocity, speed,
and acceleration.
Interpret differential equations geometrically via slope fields and the relationship between slope
fields and solution curves for differential equations.
Data Analysis and Probability
Competency
Goal 3
Objectives:
The learner will use integrals to solve problems.
3.01 Explore and interpret the concept of the definite integral.
a.
b.
c.
Compute Riemann sums using left, right, and midpoint evaluation points.
Find the definite integral as a limit of Riemann sums over equal subdivisions.
Find the definite integral of the rate of change of a quantity over an interval interpreted as the
change of the quantity over the interval:
d.
Identify basic properties of definite integrals.
3.02 Apply standard techniques of anti-differentiation.
a.
b.
Find anti-derivatives following directly from derivatives of basic functions.
Find anti-derivatives by substitution of variables. (including change of limits for definite
integrals).
3.03 Apply and interpret the Fundamental Theorem of Calculus.
a.
b.
Use the Fundamental Theorem to evaluate definite integrals.
Use the Fundamental Theorem to represent a particular anti-derivative, and the analytical and
graphical analysis of functions so defined.
3.04 Define and use appropriate integrals in a variety of applications.
a.
b.
c.
d.
e.
f.
g.
h.
Interpret the integral of a rate of change to give accumulated change.
Find specific anti-derivatives using initial conditions.
Set up and use an approximating Riemann sum or trapezoidal sum and represent its limit as a
definite integral.
Find the area of a region.
Find the volume of a solid with known cross sections.
Find the average value of a function.
Find the distance traveled by a particle along a line.
Solve separable differential equations and use them in modeling. In particular, study the
equation y' = ky and exponential growth.