HONORS CALCULUS
COURSE OF STUDY
POMPTON LAKES PUBLIC SCHOOLS
JUNE 2010
Submitted by
The Mathematics Department
Dr. Terrance Brennan, Superintendent
Vincent Przybylinski, Principal
Anthony Mattera, Vice Principal
Frances J. Macdonald Mathematics Supervisor K-12
Mary Curran, Board of Ed President
Mr. Ray Keating, III, Board of Ed Vice President
Board Members
Mr. William Baig, Mr. Joel Bernstock, Mrs. Catherine Brolsma,
Mrs. Joyce Colfax, Mr. Scott Croonquist, Mr. Tom Salus, Mrs. Stephanie Shaw
Honors
2
Calculus
I.
RATIONALE
This year long course consists of a full academic year of work in Honors
Calculus and related topics comparable to courses in colleges and universities.
II.
DESCRIPTION
This year long course, a full academic year of work in Honors 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
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
All students will develop the ability to pose and solve mathematical
problems in mathematics, other disciplines, and everyday
experiences.
All students will communicate mathematically through written, oral,
symbolic, and visual forms of expression.
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.
All students will develop reasoning ability and will become selfreliant, independent mathematical thinkers.
All students will regularly and routinely use calculators, computers,
manipulatives, and other tools to enhance mathematical thinking,
understanding and power.
All students will develop number sense and an ability to represent
numbers in a variety of forms and use numbers in diverse
situations.
All students will develop spatial sense and an ability to represent
geometric properties and relationships to solve problems in
mathematics and in everyday life.
All students will understand, select, and apply various methods of
performing numerical operations.
All students will develop an understanding of and will use
measurement to describe and analyze phenomena.
All students will use a variety of estimation strategies and recognize
situations in which estimation is appropriate.
All students will develop an understanding of patterns,
relationships, and functions and will use them to represent and
explain real world phenomena.
All students will develop an understanding of statistics and
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
Honors Calculus
4.14
4.15
4.16
IV.
3
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
Punctuality
Time management
Organization
Decision making
Honors Calculus
5.
4
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.
PRE-CALCULUS REVIEW
TIME LINE 10_days CCCS 4.1,4.2,4.4,4.3,4.5,4.6,4.8,4.15
1.
Objectives
a.
Review the real number system, inequalities, intervals, and
absolute values
b.
Use distance formula to determine the distances between
points
c.
Find the equations of parallel and perpendicular lines
d.
Identify domain and range of various functions
e.
Identify the graphs of various functions
f.
Evaluate the composition of functions
g.
Recognize symmetry of functions and odd and even
functions
h.
Change the functions to shift graphs up, down, left and right
i.
Identify the equations of circles and parabolas
j.
Identify and use the six basic trigonometric functions
2.
Subject Matter
a.
Real number system
b.
Inequalities
c.
Intervals
d.
Absolute values
e.
Distance formula
f.
Equations of parallel and perpendicular lines
g.
Composition of functions
h.
Even and odd functions
i.
Function symmetry
j.
Parabolas and circles
k.
Six trigonometric functions
3.
Assessments
A.P. Calculus 5
a.
b.
B.
Consumer Products. Citydog Screen Printers of Staten
Island, New York makes special-order T-shirts. Recently
Citydog received two orders for a special T-shirt designed
for a mathematical symposium. The first order was for 40 Tshirts at a total cost of $295, and the second order was for
as additional 80 T-shirts at a total of $565. Each order
included a standard shipping and handling charge.
i.
Write a linear equation to represent the situation,
ii.
What is the cost per T-shirt?
iii.
What is the standard shipping and handling charge?
Shalonda hit a home run that traveled in the path whose
height is described by the function h(x)= -0.003x2 +x +4,
where x represents the number of feet the call has traveled
from the plate and h(x0 represents the height of the ball. If
the scoreboard is 30 feet high and 410 feet from the plate,
show that Shalonda’s home run ball did not hit the
scoreboard,
LIMITS AND CONTINUITY
TIME LINE 15 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
s.
Find the slope of functions at a given point
A.P. Calculus 6
2.
3.
C.
t.
Find equation of tangent lines of a curve at given points
u.
Verify all material using TI-84 plus graphing calculator
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
p.
Find derivatives of functions with rational powers
A.P. Calculus 7
q.
r.
s.
t.
3.
D.
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
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
Assessments
1993 AB 6 a
1985 AB 6 a
1993 AB 6 a
APPLICATIONS OF DIFFERENTIATION
TIME LINE 30 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
A.P. Calculus 8
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
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.
E.
INTEGRATION
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 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
p.
Apply the Fundamental Theorem of Calculus
q.
Evaluate definite integrals
A.P. Calculus 9
r.
s.
t.
2.
3.
F.
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 bt a particle along a line
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
and variations
e.
Areas of surfaces of revolution
f.
Solving separable differential equations and using them in
models
A.P. Calculus 10
2.
3.
G.
g.
TI-84 Plus graphing calculator verification
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
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
exponents
o.
Apply properties of logarithms in problems containing growth
and decay
p.
Taking derivatives of inverse trigonometric functions
2.
Content
a.
One-to-one graphs
A.P. Calculus 11
b.
c.
d.
e.
f.
g.
h.
3.
Inverse functions
Derivatives of inverse functions
Properties of logarithms
Derivatives of logarithms
Logarithmic differentiation
Integrals containing logarithms
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
H.
VI.
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.
c.
Know how to use the graphing calculator to solve word
problems
2.
Content
a.
Demonstration of calculator use
b.
Demonstration of TI-84 Plus Graphing Calculator
c.
Key strokes
EVALUATIONS
A.
VII.
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.
B.
Tests using AP exam based problems.
C.
Quizzes
D.
Semester exams
E.
Homework
F.
Classwork
G.
Assessments which require students to use calculator functions
appropriate to calculus topics
BENCHMARKS
A.
(Semester I Exam)
A.P. Calculus 12
B.
1.
Functions, domain and range
2.
Graphs of functions
3.
Definition of limits of functions
4.
Limit notation
5.
Average rates of change
6.
Rules of limits
7.
Right and left hand limits
8.
Infinite limits
9.
Continuity and points of discontinuity
10.
Continuity tests
11.
Composite functions
12.
Slopes of tangent lines and secant lines
13.
TI-84 Plus graphing calculator verification
14.
First derivatives
15.
Derivative notation
16
Slope of tangent and secant lines
17.
Equation of tangent lines
18
Rules of differentiation
19
Higher order derivatives
20
Applications of rates of change
21
Graphs of motion
22.
Chain rule
23.
Numerical values of derivatives
24
Implicit differentiation
25.
Related rates of change problems
26.
Extreme values
27.
Mean value theorem
28.
Critical points
29.
Increasing/decreasing functions
30.
Points of inflection
31.
Concavity of functions
32.
Graphing functions using derivatives
33.
Limits as x →∞
34.
Optimization
(Semester II Exam)
1.
Area between two curves y=f(x) and y=g(x)
2.
Volume by slicing; by cross sections perpendicular to x-axis and
perpendicular to y-axis
3.
Volume of solids of revolution using disk and washer method
4.
Volume by cylindrical shells centered on the y-axis and variations
5.
Areas of surfaces of revolution
6.
Anti-derivatives
7.
Anti-derivatives of trig functions
A.P. Calculus 13
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
VIII.
Initial value problems
Position, velocity, and acceleration
Evaluating indefinite integrals by substitution
Finite sums
Summation notation
Riemann sums
Area under a curve
Average value of a function
Fundamental Theorem of Calculus
Definite integrals
Evaluating definite integrals by substitution
Area between curves
Trapezoidal rule
Area between two curves y=f(x) and y=g(x)
Volume by slicing; by cross sections perpendicular to x-axis and
perpendicular to y-axis
Volume of solids of revolution using disk and washer method
Volume by cylindrical shells centered on the y-axis and variations
Areas of surfaces of revolution
One-to-one graphs
Inverse functions
Derivatives of inverse functions
Properties of logarithms
Derivatives of logarithms
Logarithmic differentiation
Integrals containing logarithms
Initial value problems containing exponential and logarithmic
expressions
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
IX.
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.
A.P. Calculus 14
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.
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)