Course SyllabusDepartment of Mathematics and Computer Science, Lincoln University
Gordon, Russell. Real Analysis, A First Course. Pearson: Boston. 2002
***To be provided for each section***
This is the first semester in a one-year sequence that is designed as a rigorous development of the fundamentals of analysis. The following topics will be covered in this course: analytic and algebraic structure of the set of real numbers, sequences and series of real numbers, limits and continuity of functions.PREREQUISITE: MAT-220 (Set Theory & Logic) and MAT-221 (Calculus III)
COURSE GOALS- STUDENT LEARNER OUTCOMES:
To prepare students with the theoretical foundation needed for further study in higher mathematics.
At a minimum, upon completion of this course, students will be able to present accurately mathematical definitions and concepts, and provide rigorous mathematical proofs, including the use of epsilon-delta arguments, of basic facts in the following topics:
Proving that a given number is rational or irrational,
Proving a given inequality using the properties of the absolute value of a real number, basic inequalities such as the triangle inequality, the Cauchy-Schwarz inequality,
Proving that a given set of real numbers is bounded or unbounded, the existence of the supremum or infimum of a given set, the existence of a solution of a given equation, using the notion of bounded and unbounded sets of real numbers, their upper bounds and lower bounds, their suprema and infima, and the completeness axiom for the set of real numbers and its consequences,
Proving the boundedness, the convergence, or the divergence of a given sequence using various techniques including monotone sequences, Cauchy sequence,
Finding or proving the existence of the limit inferior, the limit superior of a given sequence using the notion of subsequences, the Bolzano-Weierstrass theorem and some of its consequences,
Finding or proving the existence of limits of functions,
Proving the continuity of a given function, the existence of real numbers satisfying preassigned properties or conditions using the intermediate-value and extreme-value theorems and their consequences,
Proving the uniform continuity of a given function.
TENTATIVE SCHEDULE OF WEEKLY ASSIGNMENTS:
CH 1: REAL NUMBERS (Weeks 1 to 5 )
Section 1.1: What is a Real Number ?
Section 1.2: Absolute Value, Intervals, and Inequalities.
Section 1.3: The Completeness Axiom.
Section 1.4: Countable and Uncountable Sets.
Section 1.5: Real-Valued Functions.
CH 2: Sequences (Week 6 to 10)
Section 2.1: Convergent Sequences.
Section 2.2: Monotone Sequences and Cauchy Sequences.
Section 2.3: Subsequences.
CH 3: Limits and Continuity (Weeks 11 to 16)
Section 3.1: The Limit of a Function.
Section 3.2: Continuous Functions.
Section 3.3: Intermediate and Extreme Values.
Section 3.4: Uniform Continuity.
Section 3.5: Monotone Functions.
COURSE ASSESSMENT- LEARNING OPPORTUNITIES:*
At least 4 collections of home assignments will be given The students may discuss the assignments with each other only before they start their assignments. While they are doing their assignments the university rule on academic integrity applies.
GRADING STANDARDS- ASSESSMENT TOOLS:
The grading scale guideline: **
Lincoln University uses the class method of teaching, which assumes that each student has something to contribute and something to gain by attending class. It further assumes that there is much more instruction absorbed in the classroom than can be tested on examinations. Therefore, students are expected to attend all regularly scheduled class meetings and should exhibit good faith in this regard. For the control of absences, the faculty adopted the following regulations:
Students are responsible for proper conduct and integrity in all of their scholastic work. They must follow a professor's instructions when completing tests, homework, and laboratory reports, and must ask for clarification if the instructions are not clear. In general, students should not give or receive aid when taking exams, or exceed the time limitations specified by the professor. In seeking the truth, in learning to think critically, and in preparing for a life of constructive service, honesty is imperative. Honesty in the classroom and in the preparation of papers is therefore expected of all students. Each student has the responsibility to submit work that is uniquely his or her own. All of this work must be done in accordance with established principles of academic integrity.An act of academic dishonesty or plagiarism may result in failure for a project or in a course. Plagiarism involves representing another person's ideas or scholarship, including material from the Internet, as your own. Cheating or acts of academic dishonesty include (but are not limited to) fabricating data, tampering with grades, copying, and offering or receiving unauthorized assistance or information.
3) The Student Conduct Code:
Students will be held to the rules and regulations of the Student Conduct Code as described in the Lincoln University Student Handbook. In particular, excessive talking, leaving and reentering class, phones or pagers, or other means of disrupting the class will not be tolerated and students may be asked to leave. Students who constantly disrupt class may be asked to leave permanently and will receive an F.4) The Core Curriculum Learner Competencies:
All courses offered through the Department of Mathematics and Computer Science require students to meet at least the following out of the 8 Core Curriculum Learner Competencies:(1) Listen and effectively communicate ideas through written, spoken, and visual means;
(2) Think critically via classifying, analyzing, comparing, contrasting, hypothesizing, synthesizing, extrapolating, and evaluating ideas;
(6) Apply and evaluate quantitative reasoning through the disciplines of mathematics, computational science, laboratory science, selected social sciences and other like-minded approaches that require precision of thought;
(8) Demonstrate positive interpersonal skills by adhering to the principles of freedom, justice, equality, fairness, tolerance, open dialogue and concern for the common good.
* The instructor of a given section of the course may make some modifications to the evaluation as well as to the rest of the syllabi including but not limited to; the grade weights, number of tests, and test total points.
**The grading scale guideline includes a 2-point flexibility.
Please consult with the department chairperson for any program updates or corrections which may not be yet reflected on this page _ last updated 9/10/2007.