Prerequisite: MATH 352, or equivalent, with grade C- or better

Course Learning Outcomes: By the end of this course, students will be able to:

• Understand and describe the properties of real numbers.
• Find limits of sequences by definition.
• Prove limit theorems.
• Prove convergence of sequences by monotone convergence theorem and Cauchy’s criterion.
• Prove the Bolzano-Weierstrass theorem.
• Find limits of functions by definition.
• Verify the continuity of functions.
• Prove the maximum-minimum theorem and the intermediate value theorem.
• Understand the definitions of uniform continuous functions and Lipschitz functions.
• Prove the chain rule and use it to calculate derivatives.
• Prove the mean value theorem.
• Prove L’Hospital’s rules and use them to evaluate limits.
• Use definition to prove Riemann integrability of functions.
• Use Cauchy’s criterion and squeeze theorem to verify Riemann integrability.

Program Learning Outcomes and Mission Statement of the Department of Mathematics

Course Requirements

• Textbook (required): R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis, 4th ed, Wiley 2000.
• Lecture Notes
• Goals and Objectives of the Course: The purpose of this course, is to bridge the gap between the ``undergraduate'' calculus and the graduate level analysis courses. We will study sequences, limits, continuity, differentiation and integration with the goal of developing a rigorous foundation for analysis. Our primary interest will not be how to compute the derivatives and integrals, but rather how to prove theorems. The course will cover the material in the textbook from Chapter 1 through Chapter 7.
• Class Procedures: The majority of each class period will be lecture oriented. You are expected to:
  • attend the lectures and to take your own lecture notes,
  • read the textbook and work out problems from the section discussed in class,
  • do the homework problems.
• Attendance Requirements: It is important that you attend every class scheduled for this course. You are responsible for all announcements and material covered in this class as well as material from the text which is assigned but not covered in class.
• Computers: Any electronic devices such as laptops, tablets and cell phones cannot be used during the class. Please set your cell phones on vibrate.

Grading

• Homework: There will be weekly homework assignments that will count for 25% of the grade.
• Tests: There will be two midterms tests and a final exam. Each test will count for 25% of the total grade.
• Make-ups: Make-up exams will be given only in cases of extremely properly documented emergencies. Late homework will not be accepted.
• Evaluation: Grades will be determined on a 100 point scale. Those point totals near cutoff will be individually considered for the next higher grade. I do reserve the right to add some extra points depending on the class participation and outstanding exam performance.

Attendance Requirements: It is important that you attend every class scheduled for this course. You are responsible for all announcements and material covered in this class as well as material from the text which is assigned but not covered in class.

Changes: The course plan may be modified during the semester. Such modifications will be announced in advance during class periods. You are responsible for keeping abreast of such changes.

Remarks: It is important to note that this course is cumulative (builds on itself) and it is assumed that all earlier parts covered are liable for any exam. It is strongly advised that you start working on this course from the very beginning. The course moves at a fast place, so it is very hard to catch up if you get behind.

Counseling and Disability Services. Reasonable Accommodations: New Mexico Tech is committed to protecting the rights of individuals with disabilities. Qualified individuals who require reasonable accommodations are invited to make their needs known to the Office of Counseling and Disability Services (OCDS) as soon as possible. In addition, New Mexico Tech offers mental health and substance abuse counseling through the Office of Counseling and Disability Services. The confidential services are provided free of charge by licensed professionals. To schedule an appointment, please call 835-6619.