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กก CATALOGUE ENTRY: Calculus
with the two branches of calculus: differential calculus and integral calculus. indefinite
or definite integrals. matrices and their applications, systems of
Linear Equations, Cramer's rule, determinants, eigenvalues and eigenvectors,
diagonalization, ordinary
differential equations, phase plane
analysis, Laplace transforms. MAIN OBJECTIVE: It is a mistake to
think of mathematics in general as primarily a tool for finding answers (although it is
also a mistake to think, as many graduate students do, that calculating is an inferior,
unworthy aspect of mathematics). The primary importance of calculus in the hard sciences
is that it provides a language, a conceptual framework for describing relationships that
would be difficult to discuss in any other language. Therefore, the purpose of learning
differential calculus is not to be able to compute derivatives. In fact, computing
derivatives is usually exactly the opposite of what one needs to do in real life or
science. In a calculus course, one starts with a formula for a function, and then computes
the rate of change of that function. But in the real world, you usually don't have a
formula. The formula, in fact, is what you would like to have: the formula is the unknown.
What you do have is some information, given by the laws of science, about the way in which
the function changes. Therefore, the overall objective of the course is to provided
students with some fundamental knowledge in the theory and techniques of differentiation
and integration of algebraic and trigonometric functions, of ordinary differential
equations and of matrix algebra. Main topics covered include limits, optimization, curve
sketching, areas, and application of invertible matrices such as coding and solution of
linear equations. Probability theory is the mathematical underpinning of Statistics, as
well as of many areas of physics, finance, and other disciplines. The mathematics of
probability will be the topic of this course. The course can be followed by other courses
in statistics or application areas, or the mathematics can be pursued further, through
more advanced courses in stochastic processes or probability theory. Students
contemplating taking actuarial examinations are strongly advised to take this course.
Moreover, The course will introduce the basic mathematical model of randomness, and will
examine the fundamental notions of independence and conditional probability. Calculations
will be based both on combinatorial methods and on integral calculus. A variety of
concrete distributions will be studied (Normal, Binomial, Poisson, etc, together with
their multivariate generalizations), using density functions, distribution functions, and
moment-generating functions. Prior exposure to statistics or combinatorics would be
useful, but is not assumed. In addition, it is very important to train the students to
think logically. SYLLABUS: (1) Algebra Matrix operations---Groups and
subgroups---homomorphisms--- equivalence relations and partitions---quotient groups---
vector spaces---bases and dimension---linear transformations---the matrix of a linear
transformation---linear operators and eigen values---the characteristic
polynomial---orthogonal matrices and rotations---diagonalization. (2) Calculus Functions of one variable: Real
number system---Limits and continuity---differentiation ---chain rule---mean value
theorems and applications---Taylor expansion--- L'Hospital's
rule---integration---fundamental theorem of calculus---change of variable and
applications---exponential, logarithmic, trigonometric and inverse trigonometric
functions---applications of integration. Functions of several variables:
Limits and continuity---partial derivatives and applications---chain rule---directional
derivatives and the gradient vector---Lagrange multiplier---Taylor expansion---implicit
and inverse function theorems---double integrals, iterated integrals and
applications---multiple integrals---change of variables. Differential equations:
Homogeneous equations---first order linear equations---second order linear equations---non
homogeneous linear equations---applications. (3) Probabilty:
Borel's normal number theorem---simple random variables ---the law of large
numbers---Binomial, Poisson and normal distributions--- central limit theorem---random
variables--- expectations and moments. LEARNING OUTCOMES: At the
end of the module, students should be able to: (1) think logically some
information given by the laws of science in real life or science; (2) demonstrate understanding of the
mathematical theory underlying the topics listed in the outlined syllabus above; (3)
express a given mathematical relationship in the form of an integral and gain enough
insight that they can set up formulas using integrals with a fair amount of confidence;
(4) compute determinant, inverse, eigenvalues and eigenvectors, diagonalization of a
matrix; (5) demonstrate understanding of methods for dealing with financial data; (6)
explain how and when some mathematical theory can be used to solve optimizations problems;
(7) apply calculus to solve problems in economics and Physics; (8) strive for a set of
equations which describes the physical system approximately and adequately.(9) use
appropriate statistical techniques to organize and summarize numerical data; (10)
calculate probabilities for the binomial and normal distributions; (11) formulate and
solve linear programming problems graphically. LEARNING MATERIALS: (1)
Abelson, Harold; Fellman, Leonard; Rudolph, Lee. Calculus of Elementary Functions. New
York: Harcourt, 1970; (2) Adams, Robert A. Calculus: A Complete Course. Reading, MA:
Addison-Wesley, 1994; (3) Courant, Richard and John, Fritz. Introduction to Calculus and
Analysis, Vol. 1 & 2. New York: Springer-Verlag; (4) Marsden, Jerrold E. and
Weinstein, Alan. Calculus I, II & III, 2nd ed. New York: Springer-Verlag, 1985; (5)
John B. Hahn, Terry J. D, Hurricane Calculus: The New Approach to First Year Calculus, Prometheus
Enterprises Inc, 1996; (6) Aitken, Alexander Craig. Determinants and Matrices, 8th ed.
Edinburgh: Oliver and Boyd, 1954; (7) Anton, Howard. Elementary Linear Algebra. New York:
Wiley, 2000; (8) Anton, Howard and Rorres, Chris. Elementary Linear Algebra: Applications
Version, 8th ed. New York: Wiley, 2000; (9) Curtis, Charles W. Linear Algebra: An
Introductory Approach. New York: Springer-Verlag, 1984; (10) Lax, Peter D. Linear Algebra.
New York: Wiley, 1997; (11) Arnold, Vladimir Igorevich. Ordinary Differential Equations,
3rd ed. Berlin: Springer-Verlag, 1992; (12) Betz, Herman; Burcham, Paul B.; and Ewing,
George M. Differential Equations with Applications, 2nd ed. New York: Harper and Row,
1964; (13) Birkhoff, Garrett and Rota, Gian-Carlo. Ordinary Differential Equations, 3rd
ed. New York: Wiley, 1978; (14) Coddington, Earl A. An Introduction to Ordinary
Differential Equations. New York: Dover, 1989; (15) Klain, Daniel A. and Rota, G.-C.
Introduction to Geometric Probability. New York: Cambridge University Press, 1997; (16)
Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, 1978; (17) Cramér,
Harald. Mathematical Methods of Statistics. Princeton, NJ: Princeton University Press,
1946; (18) DeGroot, Morris H. Probability and Statistics, 3rd ed. Reading, MA:
Addison-Wesley, 1991. EQUIPMENT NEEDED: You will
be expected to bring your textbook, graphing calculator, notebook, graphing paper, dry
erase marker, and pencil every day. TIME COMMITMENT: Homework
should take anywhere from 30 minutes to an hour, depending on the type of
assignment. Since there is a lot of material to learn in a short time, tudents must
be aware of this and plan to be in class at every opportunity. In April and
May, some activities may have to be curtailed in order to make room for the calculus.
Please be aware of this and plan accordingly. Overextended students will have
difficulties succeeding. HOMEWORK: Homework is an
important part of learning mathematics and will be assigned daily. Every assigned problem
should be tried and the answer checked. It is permissible to discuss problems with
other students or relatives. It is not permissible to copy another student's work.
Do your best to think through the problems and understand why things work the way
they do. Homework should take anywhere from 30 minutes to an hour, depending on the type
of assignment. GROUP WORK: Several times
each six weeks problems will be given to groups of students to work. These may be
general problems, calculator/computer labs, or projects. All group members will work
on the problem. Sometimes one paper, agreed upon by everyone, will be collected and
graded and everyone will receive the same grade. If a group cannot come to a
consensus, then each member may turn in his/her own solution. Other times each
student will be required to turn in a paper with his/her own work on it. MEETING WITH INSTUCTORS: For
additional assistance, for discussion of specific problems that you are encountering, or
for any other matter that can help you learn calculus better; you should not hesitate to
contact your instructor. TESTS: Periodically, unit
tests will be given. If a student is absent on test day, he/she must make it up or a
zero will be recorded. Make up tests will not be the same test the other students
took. Tests will be cumulative in nature. GRADES: Grades will be
figured on a total point basis. Each activity will be assigned a certain number of
points. At the end of the term, add up the points earned and divide by the number
possible in each category. Then figure 30*Mid-Term Test + 40*Final Exam + 30*Daily Work. MAKE -UP WORK: If you are
absent you must make up the work within 3 days. However, you must keep up with the current
work at the same time. If instruction is necessary, make an appointment with me to
go over the material. Class time will not be used to help one individual catch up.
It is the student's responsibility to see me about make-up work when he/she returns
from an absence. If arrangements have not been made within three days, zeros will be
recorded for all work missed. Make-up tests will be given during your study hall or
after school whenever possible to avoid having to miss another class discussion.
Good attendance is of utmost importance in a mathematics class. We realize
that sometimes absences are necessary, but we encourage you to keep them at a minimum.
We must sign an excused "admit to class" on the day you return from an
absence or an unexcused absence will be recorded and zeroes will be recorded for any work
missed, including tests. CLASS RULES: Follow directions: (1) Be in your seat when the bell rings; (2) Do not work on other subjects; (3) No eating, drinking, or chewing; (4) Most of the time the consequences for not following rules will be remaining after class. If it becomes more of a problem, the student will be asked to stay a short time after school for a discussion. |