Math 401/501 - Advanced Calculus I

MATH 401/501 - Avanced Calculus I
Spring 2020

This class is cross-listed as:

Practice Problems for Exam 1
Review and Practice Problems for Exam 2

Textbook (required): Analysis I - Third Edition, by Terence Tao. Text and Readings in Mathematics 37, Springer 2016 (editions one and two work equally well). Here you will find the first four chapter of Tao's book in pdf format pdf file. An here are scanned copies of Chapter A an appendix on the basics of mathematical logic.
Terence Tao is an exceptional mathematician who has earned almost all possible awards a mathematician can get, including the 2006 Fields Medal (the equivalent of a Nobel Price in Mathematics), the 2014 Breakthrough Prize in Mathematics, and most recently he recieved the 2019 Riemann Prize. He also cares deeply about teaching, and our textbook is an example of that. Here are links to his webpage and his wikipedia page, see also a 2015 article in the New York Times article about Tao.

There are many other excellent introductory analysis books. Reading from other sources is always very valuable. For example: Elementary Analysis by Kenneth A. Ross and Analysis with an Introduction to Proofs By Steven Lay used recently by other faculty. Other good books are Calculus by Spivak, 4th edition Publish or Perish, 2008, or The Way of Analysis by Robert Strichartz, Jones & Bartlett Publishers, Revised Edition, 2000. For a non-standard book, but a very lively one, full of historical anecdotes, see A radical approach to real analysis by D. M. Bressoud.
Here is a resource for learning how to write proofs Book of Proofs by Richard Hammack. This book has been used as a companion to the main advanced calculus text in previous semesters. The author kindly makes it available on his webpage.

Course Structure: Tuesdays and Thursdays will be devoted to lecturing on new material and occasional quizes. The recitation hour on Wednesdays will be used for problem solving and review of the material.

Course content: This is a first course in analysis. We will cover the fundamentals of calculus in one variable, starting with the construction of the real numbers, sequences of numbers and working our way through the concepts of limits, functions, continuity and basic properties of functions, we will then study carefully the theory of differentiation and integration. Basic calculus is a prerequisite, it provides you with computational skills and some intuition. Prior experience with mathematical abstractions and proofs will be helpful (for example exposure to at least one of Math 306, 317, 318, 319, 321, 322 or 327 will provide such experience). However we do not expect the students to be able to read, understand, and actually construct mathematical proofs at the begining of the course, although most of you have probably been already exposed to proofs. A great amount of time will be devoted to learn and practice logical thinking. At the end of the course we expect the students to have adquired the basic skills of mathematical reasoning, and a deeper understanding of calculus.

Homework: The problems and exercises in the textbook are an integral part of the course. You should attempt most of them. Homework will be assigned periodically, the problems in the homework will be carefully graded, and returned to you with feedback that will help you correct any errors. The highlights of the solutions will be discussed in the Wednesday session. You are encouraged to discuss the homework with each other, but you should do the writing separately. You learn mathematics by doing, and there is no way around this. It is not enough to see your teacher or your friends solving problems, you have to try it yourself. Difficult as it may seem at the beginning, if you persist you will learn how to write a proper mathematical proof, you will learn how to read and understand other's proofs, and you will learn to appreciate and enjoy the beauty of an elegant argument.

Exams: There will be two midterms during weeks 6 and 12, and a final exam.

Grades: The final grade will be determined by your performance on homeworks, midterms, and a final exam. The grading policies will be discussed in class.

Prerequisites: Math 264 and two MATH courses 300-level or above (or permission from the instructor).

Accomodation Statement: Accessibility Services (Mesa Vista Hall 20121, 277-3506) provides academic support to students who have dissabilities. If you think you need alternative accesible formats for undertaking and completing coursework, you should contact this service right away to assure your needs are met in a timely manner. If you need local assistance in contacting Accessibility Services, see the Bachelor and Graduate Programs office.

Academic Integrity: The University of New Mexico believes that academic honesty is a foundation principle for personal and academic development. All university policies regarding academic honesty apply to this course. Academic dishonesty includes, but is not limited to, cheating or copying, plagiarism (claiming credit for the words or works of another from any type of source such as print, Internet or electronic database, or failing to cite the source), fabricating information or citations, facilitating acts of academic dishonesty by others, having unauthorized possession of examinations, submitting work of another person or work previously used without informing the instructor, or tampering with the academic work of other students. The University's full statement on academic honesty and the consequences for failure to comply is available in the college catalog and in the Pathfinder.

Gender Discrimination: In an effort to meet obligations under Title IX, UNM faculty, Teaching Assistants, and Graduate Assistants are considered "responsible employee" by the Department of Education (see pg 15 - ). This designation requires that any report of gender discrimination which includes sexual harassment, sexual misconduct and sexual violence made to a faculty member, TA, or GA must be reported to the Title IX Coordinator at the Office of Equal Opportunity ( For more information on the campus policy regarding sexual misconduct, see:

Citizenship and/or Immigration Status: All students are welcome in this class regardless of citizenship, residency, or immigration status. Your professor will respect your privacy if you choose to disclose your status. As for all students in the class, family emergency-related absences are normally excused with reasonable notice to the professor, as noted in the attendance guidelines above. UNM as an institution has made a core commitment to the success of all our students, including members of our undocumented community. The Administration's welcome is found on the website

Return to: Department of Mathematics and Statistics, University of New Mexico

Last updated: January 25, 2020