POMPTON LAKES PUBLIC SCHOOLS
ADVANCED PLACEMENT CALCULUS
June 2012
COURSE OF STUDY
Submitted By
The Mathematic Department
Dr. Paul Amoroso, Superintendent
Mr. Vincent Przybylinski, Principal
Mr. Anthony Mattera, Vice Principal
Frances J. Macdonald, Mathematics Supervisor K-12
BOARD MEMBERS
Mr. Jose A. Arroyo, Mrs. Catherine Brolsma, Mr. Shawn Dougherty,
Mrs. Nancy Lohse-Schwartz, Mr. Garry Luciani, Mr. Carl Padula,
Mr. Tom Salus, Mrs. Stephanie Shaw, Mr. Timothy Troast, Jr.
AP Calculus
I.
RATIONALE
This year long course consists of a full academic year of work in Advanced
Placement Calculus and related topics comparable to courses in colleges and
universities.
II.
DESCRIPTION
This year long course, a full academic year of work in Advanced Placement
Calculus, consists of the following topics: coordinates, graphs and lines;
functions and limits; differentiation and applications of differentiation; Integration;
applications of the definite integral; logarithmic and exponential functions;
Inverse trigonometric functions; techniques of integration; L'Hopital's Rule;
extensive use of TI-84 Plus and TI-89 Graphing Calculators.
THE CORE CURRICULUM CONTENT STANDARDS
4.1
All students will develop the ability to pose and solve mathematical
problems in mathematics, other disciplines, and everyday
experiences.
4.2
All students will communicate mathematically through written, oral,
symbolic, and visual forms of expression.
4.3
All students will connect mathematics to other learning by
understanding the interrelationships of mathematical ideas and the
roles that mathematics and mathematical modeling play in other
disciplines and in life.
4.4
All students will develop reasoning ability and will become selfreliant, independent mathematical thinkers.
4.5
All students will regularly and routinely use calculators, computers,
manipulatives and other tools to enhance mathematical thinking,
understanding and power.
4.6
All students will develop number sense and an ability to represent
numbers in a variety of forms and use numbers in diverse
situations.
4.7
All students will develop spatial sense and an ability to represent
geometric properties and relationships to solve problems in
mathematics and in everyday life.
4.8
All students will understand, select, and apply various methods of
performing numerical operations.
4.9
All students will develop an understanding of and will use
measurement to describe and analyze phenomena.
4.10 All students will use a variety of estimation strategies and recognize
situations in which estimation is appropriate.
4.11 All students will develop an understanding of patterns,
relationships, and functions and will use them to represent and
explain real world phenomena.
4.12 All students will develop an understanding of statistics and
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4.13
4.14
4.15
4.16
IV.
probability and will use them to describe sets of data, model
situations, and support appropriate inferences and arguments.
All students will develop algebraic concepts and processes and will
use them to represent and analyze relationships among variable
quantities and to solve problems.
All students will apply the concepts and methods of discrete
mathematics to model and explore a variety of practical situations.
All students will develop an understanding of the conceptual
building blocks of calculus and will use them to model and analyze
natural phenomena.
All students will demonstrate high levels of mathematical thought
through experiences which extend beyond traditional computation,
algebra, and geometry.
STANDARD 9.1 (Career and Technical Education)
All students will develop career awareness and planning, employment skills, and
foundational knowledge necessary for success in the workplace.
Strands and Cumulative Progress Indicators
Building knowledge and skills gained in preceding grades, by the end of Grade
12, students will:
A.
Career Awareness/Preparation
1.
Re-evaluate personal interests, ability, and skills through various
measures including self assessments.
2.
Evaluate academic and career skills needed in various career
clusters.
3.
Analyze factors that can impact an individual’s career
4.
Review and update their career plan and include plan in portfolio.
5.
Research current advances in technology that apply to a sector
occupational career cluster.
B.
Employment skills
1.
Assess personal qualities that are needed to obtain and retain a job
related to career clusters.
2.
Communicate and comprehend written and verbal thoughts, ideas,
directions and information relative educational and occupational
settings.
3.
Select and utilize appropriate technology in the design and
implementation of teacher-approved projects relevant to
occupational and /or higher educational settings
4.
Evaluate the following academic and career skills as they relate to
home, school, community, and employment.
Communication
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5.
Punctuality
Time management
Organization
Decision making
Goal setting
Resources allocation
Fair and equitable competition
Safety
Employment application
Teamwork
Demonstrate teamwork and leadership skills that include student
participation in real world applications of career and technical
educational skills.
All students electing further study in career and technical education will
also: participate in a structural learning experience that demonstrates
interpersonal communication, teamwork, and leadership skills.
V.
UNITS
A.
LIMITS AND CONTINUITY
TIME LINE 10 days CCCS 4.1,4.2,4.4,4.3,4.5,4.6,4.8,4.15
1.
Objectives
a.
Define function domain and range
b.
Find limits of functions using graphs
c.
Determine existence/non-existence of a limit
d.
Find limits using tables
e.
Understand asymptotes in terms of graphical behavior
f.
Describe asymptote behavior in terms of limits involving
infinity
g.
Find limits using algebraic substitutions
h.
Compute average rates of change for functions over a given
interval
i.
Compare relative magnitudes of functions and their rates of
change
j.
Apply rules of limits
k.
Calculate limits of average rates of change
I.
Interpret one-sided limits using graphs
m.
Interpret one-sided limits algebraically
n.
Calculate infinite limits
o
Determine continuity from graphs
p.
Apply the continuity test
q.
Find limits of composite functions
r.
Estimate slopes of tangent lines via graphs
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2.
3.
B.
s.
Find the slope of functions at a given point
t.
Find equation of tangent lines of a curve at given points
u.
Verify all material using TI-84 plus graphing calculator
v.
Complete AP Exam question on topics in this unit.
Content
a.
Functions, domain and range
b.
Graphs of functions
c.
Definition of limits of functions
d.
Limit notation
e.
Average rates of change
f.
Rules of limits
g.
Right and left hand limits
h.
Infinite limits
I.
Continuity and points of discontinuity
j.
Continuity tests
k.
Composite functions
l.
Slopes of tangent lines and secant lines
m.
TI-84 Plus graphing calculator verification
Assessment
1981 AB 5 a, b
1986 AB 4 a, b, c
1986 BC 6 b
DIFFERENTIATION
TIME LINE 25 days CCCS 4.1,4.2,4.4,4.3,4.5,4.7,4.15
1.
Objectives
a.
Calculate derivatives using definition
b.
Derivatives presented graphically, numerically and
analytically
c.
Difference between differentiability and continuity
d.
Use derivatives to find the slope of tangent lines
e.
Use derivatives to find the equations of tangent lines
f.
Apply rules of differentiation
g.
Determine first n derivatives of a function
h.
Find displacement, speed, and acceleration of a particle in
linear motion
i.
Find velocity, acceleration, and duration of flight for free-fall
applications
j.
Interpret conclusions about motion from graphs
k.
Find derivatives of trig functions
l.
Find limits of trig functions
m.
Find equations of tangent lines to trig functions
n.
Apply chain rule
o.
Find numerical values of derivatives
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AP
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p.
q.
r.
s.
t.
C.
Find derivatives of functions with rational powers
Use implicit differentiation
Compute related rates of change problems
Local linear approximation
Instantaneous rate of change as a limit of average rate of
change
u.
Approximate rate of change from graphs and tables of
values
v.
Equations involving derivatives; verbal descriptions are
translated into equations involving derivatives
q.
Complete AP Exam question on topics in this unit.
2.
Contenta.
First derivatives
b.
Derivative notation
c.
Slope of tangent and secant lines
d.
Equation of tangent lines
e.
Rules of differentiation
f.
Higher order derivatives
g.
Applications of rates of change
h.
Graphs of motion
I.
Chain rule
k.
Numerical values of derivatives
l.
Implicit differentiation
m.
Related rates of change problems
n.
TI-84 Plus graphing calculators
3.
Assessments
1993 AB 6 a
1985 AB 6 a
1993 AB 6 a
APPLICATIONS OF DIFFERENTIATION
TIME LINE 25 days CCCS 4.1,4.2,4.3,4.4,4.5,4.7, 4.13, 4.15
1.
Objectives
a.
Find extreme values using graphs
b.
Find absolute extreme values on closed intervals
c.
Locate extreme values in a given domain
d.
Apply the mean value theorem
e.
Find critical points of a function
f.
Determine the intervals on which a function is increasing or
decreasing
g.
Find local extreme values of a function using the derivative
h.
Identify points of inflection
i.
Identify intervals on which functions are concave up and
concave down using graphs
j.
Use properties of first and second derivatives to graph
functions
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AP
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Calculate limits as x approaches →∞
Solve optimization problems
Modeling rates of change, including related rate problems
Use of implicit differentiation to find the derivative of an
inverse function.
o.
Geometric interpretation of the relationship of slope fields
and solution curves for differential equations
p.
Complete AP Exam question on topics in this unit.
2.
Contenta.
Extreme values
b.
Mean value theorem
c.
Critical points
d.
Increasing/decreasing functions
e.
Points of inflection
f.
Concavity of functions
g.
Graphing functions using derivatives
h.
Limits as x →∞
I.
Optimization
j.
Verification of all material via TI-84 Plus graphing calculator
Assessments
1992 AB 2 a
1993 AB 2 a
1992 AB 3 c
k.
l.
m.
n.
3.
D.
INTEGRATION
TIME LINE as days CCCS 4.1,4.2,4.4,4.3,4.4,4.5,4.6,4.15,4.16
1.
Objectives
a.
Find anti-derivatives of functions
b.
Evaluate integrals
c.
Verify integral formulas by differentiation
d.
Find anti-derivatives of trig functions
e.
Solve initial value problems
f.
Find position from velocity and acceleration
g.
Evaluate indefinite integrals by substitution
h.
Estimate using finite sums
i.
Use summation notation
j.
Find values of finite sums
k.
Use Riemann sums to determine the numerical
approximations of definite integrals.
l.
Definite integral as a limit of Riemann sums
m.
Properties and known values to find integrals
n.
Find area under a curve
o.
Find the average value of a function over a given interval
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AP
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p.
q.
r.
s.
t.
2.
3.
E.
Apply the Fundamental Theorem of Calculus
Evaluate definite integrals
Evaluate definite integrals using substitution
Find area between curves
Apply the trapezoidal rule to approximately definite integrals
of functions
u.
Using the integral of rate of change to give an accumulative
change
v.
Distance traveled as a particle along a line
w.
Complete AP Exam question on topics in this unit.
Content
a.
Anti-derivatives
b.
Anti-derivatives of trig functions
c.
Initial value problems
d.
Position, velocity, and acceleration
e.
Evaluating indefinite integrals by substitution
f.
Finite sums
g.
Summation notation
h.
Riemann sums
I.
Area under a curve
j.
Average value of a function
k.
Fundamental Theorem of Calculus
l.
Definite integrals
m.
Evaluating definite integrals by substitution
n.
Area between curves
o.
Trapezoidal rule
p.
Verification of all material via TI-84 Plus graphing calculator
Assessments
1980 AB 1 a
1980 AB 2
1981 AB 2 c
1988 BC 2 a
APPLICATIONS OF INTEGRALS
TIME LINE 30 days CCCS 4.1,4.2,4.4,4.3,4.4,4.5,4.6,4.15,4.16
1.
Objectives:
a.
Find the area between two curves y=f(x) and y=g(x)
b.
Find the volume by slicing method: by cross-sections
perpendicular to x-axis and perpendicular to y-axis
c.
Find volumes of solids of revolution using disk and washer
method
d.
Find the volume by cylindrical shells centered on the y-axis
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AP
Calculus
and variations
Areas of surfaces of revolution
Solving separable differential equations and using them in
models
g.
TI-84 Plus graphing calculator verification
h.
Complete AP Exam question on topics in this unit.
Content
a.
Area between two curves y=f(x) and y=g(x)
b.
Volume by slicing; by cross sections perpendicular to x-axis
and perpendicular to y-axis
c.
Volume of solids of revolution using disk and washer method
d.
Volume by cylindrical shells centered on the y-axis and
variations
e.
Areas of surfaces of revolution
f.
TI-84 Plus graphing calculator
Assessments
1990 AB 3 b
1990 AB 1 b
1999 BC 2 b
e.
f.
2.
3.
F.
TRANSCENDENTAL FUNCTION
TIME LINE 15 days CCCS 4.1,4.2,4.4,4.3,4.4,4.5,4.6,4.15,4.16
1.
Objectives:
a.
Identify one-to-one graphs
b.
Graph inverse functions
c.
Find the formulas for inverse function
d.
Take derivatives of inverse functions
e.
Apply properties of logarithms
f.
Take derivatives of logarithms
g.
Use logarithms differentiation to find derivatives of
composite functions
h.
Evaluate integrals containing logarithms
I.
Perform algebraic calculations with the exponential and
logarithms
j.
Solve equations with logarithmic or exponential terms
k.
Differentiate functions containing exponential and logarithms
terms
l.
Integrate functions containing exponential and logarithmic
terms
m.
Solve initial values containing exponential and logarithmic
terms
n.
Rewrite logarithmic expressions as an expression containing
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AP
Calculus
exponents
Apply properties of logarithms in problems containing growth
and decay
p.
Taking derivatives of inverse trigonometric functions
q.
Complete AP Exam question on topics in this unit.
Content
a.
One-to-one graphs
b.
Inverse functions
c.
Derivatives of inverse functions
d.
Properties of logarithms
e.
Derivatives of logarithms
f.
Logarithmic differentiation
g.
Integrals containing logarithms
h.
Initial value problems containing exponential and logarithmic
expressions
i.
TI-84 Plus graphing calculator verification.
Assessments
1989 BC 3 a
1990 AB 2 a
1990 AB 4 a
o.
2.
3.
G.
TECHNIQUES OF INTEGRATION
TIME LINE 10 days CCCS 4.1,4.2,4.4,4.3,4.4,4.5,4.6,4.15,4.16
1.
Objectives
a.
Review integration techniques
b.
Define integration by parts formula
c.
Apply integration by parts formula
d.
TI-84 Plus graphing calculator verification
g.
Complete AP Exam question on topics in this unit.
2.
Content
a.
Integration techniques
b.
Integration by parts formula
c.
Application of Integration by parts formula
d.
TI-84 Plus graphing calculator verification3.
Assessments
Sample AP Exam
H.
CALCULATORS
1.
Objective
a.
Know the calculus functions of the TI-84 Plus graphing
calculator.
b.
Know how to utilize Computer and graphing calculator
programs for volume, slope fields, etc.
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c.
2.
IV
EVALUATIONS
A.
B.
C.
D.
E.
F.
G.
V.
Know how to use the graphing calculator to solve word
problems
Content
a.
Demonstration of calculator use
b.
Demonstration of TI-84 Plus Graphing Calculator
c.
Key strokes
Presentations which require students to demonstrate their knowledge of
the different methods used to solve calculus problems. Students will
present solutions verbally and in written form to demonstrate their ability
to communicate their methods and rationale in arriving at solutions. The
remainder of the class will be provided time to discuss alternate solutions
and ask questions of the presenter.
Tests using AP exam based problems.
Quizzes
Semester exams
Homework
Classwork
Assessments which require students to use calculator functions
appropriate to calculus topics
AFFIRMATIVE ACTION
Evidence of:
A-1 minorities and females incorporated into plans
A-2 human relations concepts being taught
A-3 teaching plans to change ethnic and racial stereotypes
VI.
BIBLIOGRAPHY
Albert, B.H. Teacher's Guide to Advanced Placement Courses in Mathematics:
Calculus AB and Calculus BC, College Board and Educational
Testing Service, New York, NY 1988.
Anton, Howard, Calculus, John Wiley and Sons. New York, NY 1984.
Anton, Howard, Calculus, John Wiley and Sons. New York, NY 1995.
Broadwin, J. Lenchner, G. Solutions A.P. Calculus Problems Part II AB and
BC 1977-1993. Mathematical Olympiad for Elementary Schools,
1993.
Finney, R.L. Thomas, G.B. Demana, F.D. Waits, B.K. Calculus, Addison-Westley
Publishing Co., New York, NY 1994.
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Calculus Tool Kit, Cedo Publishing co., Orem, UT 1994
Thomas, G. B., Finney.R.L.,Calculus and Analytic Geometry, Addison-Westley
Publishing Co., New York, NY 1996 (Course Text)
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