# MAT 401 Advanced Analysis I

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### Course Description

Elements of set theory, numerical sequences and series, continuity and differentiability of functions of one and several variables.

3 units credit.

### Prerequisites

MAT 211 and MAT 271 or equivalent with grade C or better.

### Text

Texts are chosen by the instructor. For example:

Advanced Calculus, by R. Creighton Buck. McGraw-Hill, 1978.

### Course Requirements, Tentative Schedule of Class Meetings and Topics, Readings, Assignments and Due dates, Exams

A schedule of class meetings, topics, assignments, due dates, exam dates, etc. will be provided by instructor. See your class syllabus.

Here is an example course outline, based on the above text.

• 1. Sets and Functions
• 1.1. Introduction
• 1.2. R and R^n
• 1.3. Distance
• 1.4. Functions
• 1.5. Topological Terminology
• 1.6. Sequences
• 1.7. Consequences of the Monotonic-Sequence Property
• 1.8. Compact Sets
• 2. Continuity
• 2.1. Preview
• 2.2. Basic Definitions
• 2.3. Uniform Continuity
• 2.4. Implications of Continuity
• 2.5. Limits of Functions
• 2.6. Discontinuities
• 2.7. Inverses for Functions of One Variable
• 3. Differentiation
• 3.1. Preview.
• 3.2. Mean Value Theorems and L'Hospital's Rule
• 3.3. Derivatives for Functions on R^n

The final exam is given at the date and time announced in the Schedule of Classes.

### Learning Objectives

After completing MAT 401 the student will

• use and understand techniques and concepts from elementary topology and real analysis: open, closed, and compact sets, least-upper-bounds, Euclidean norm, triangle and Schwartz inequalities, convergence, limits, Cauchy sequences, epsilon-delta proof techniques, uniform convergence, continuity, differentiability.
• prove basic results about functions on R^n and sets and sequences in R^n. These basic results include the following
• a sequence converges in R^n if and only if it is Cauchy
• bounded sequences have Cauchy subsequences
• closed bounded sets in R^n are compact
• sums, products, quotients of convergent sequences are convergent, sums, products, quotients, compositions of continuous, (uniformly continuous) functions on open sets are continuous, (uniformly continuous), where defined
• on compact sets, continuous functions are uniformly continuous and have maxima and minima
• differentiable functions are continuous
• method for prooving from first principles that a given function is continuous or differentiable at a given point
• solve concrete problems and write clear and coherent mathematical arguments incorporating these techniques and concepts.

### Computers and Calculators, Computer Literacy

Most instructors encourage the use of machines, calculators computers, phones etc., for analyzing data. The use of machines may be restricted during examinations or at certain other times. Ask your instructor for the policy in your class.

Students are not expected to be programmers or to know any particular computer language before starting this class. Some instructors may expect students to be able to access information on the internet, or to use calculators, or to learn to use particular software with instruction. Basic skill in algebra and the use of mathematical symbols, order of operations etc., and the willingness to read and follow instruction manuals and help files will suffice.

Students' grades are based on homework, class participation, short tests, and scheduled examinations covering students' understanding of the topics covered in this course. The instructor determines the relative weights of these factors and the grading scale. See the syllabus for your particular class.

### Location of Class Meetings

Classes meet on the dates and room announced in the official Schedule of Classes. This is a traditional, face-to-face class.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates, Make-Up Work, Missed Exams, and Extra-Credit Assignments

Due dates and policy regarding make-up work and missed exams are set by the instructor. Instructors may, or may not, choose to offer extra credit assignments. If extra credit assignments are offered they will be available to all students.

The mathematics department does not tolerate cheating. Students who have questions or concerns about academic integrity should ask their professors or the counselors in the Student Development Office, or refer to the University Catalog for more information. (Look in the index under "academic integrity".)

### Accomodations for Students with Disabilities

Cal State Dominguez Hills adheres to all applicable federal, state, and local laws, regulations, and guidelines with respect to providing reasonable accommodations for students with temporary and permanent disabilities. If you have a disability that may adversely affect your work in this class, I encourage you to register with Disabled Student Services (DSS) and to talk with me about how we best can help you. All disclosures of disabilities will be kept strictly confidential. Please note: you must register with DSS to arrange an no accommodation. For information call (310) 243-3660 or send an email message to dss@csudh.edu or visit the DSS website http://www4.csudh.edu/dss/contact-us/index or visit their office WH D-180

### Behavioral Expectations

We all are adults so behavior rarely is an issue. Just follow the Golden Rule: "do unto others as you would have them do unto you" then everything will be fine.

The university must maintain a classroom environment that is suitable for learning, so anyone who insists on disrupting that environment will be expelled from the class.

Revision history:

Prepared by G. Jennings 1/28/00. Revised 4.28/01, 7/25/06, 1/10/15 (G. Jennings).