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Calculus I
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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.


(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.
   Homework will be issued in class and submitted one week later. On test days all homework associated with that test will be collected and graded.    Homework should be corrected and added, as the material is better understood.   Completeness, accuracy, and effort will all be considered. This homework will consist, typically, of exercises from the textbook that cover the major concepts that were introduced in class. It is important to complete the homework on time for the day it is due. If you do this, you will have reviewed the ideas covered in class and be ready for the new ideas to come. Remember that new mathematical ideas build on old ones. Only by staying up to date in the homework will you avoid getting lost.
    A student may earn extra homework points by demonstrating problems at the board.

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.
    If a student is absent for an excused reason, he/she will be given an "OMIT" grade which means this grade will not be averaged in as part of his/her grade.  It is difficult to make up group work by one's self. If the absence is unexcused, a zero will be recorded for his/her work.
    Individual (not with a group) problems will also be assigned periodically.

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.
  Small quizzes may be given as deemed necessary, but it is not necessary to make these up if absent for an excused reason.  If the absence is unexcused, a zero will be recorded for the quiz.

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.