Harold P. Boas

Math 409
Sections 200 and 501
Advanced Calculus I
Spring 2023

Course Information

Course Data

Instructor Details

Course Description

This course covers the axioms and the properties of the real number system; sets, sequences, and series of real numbers; continuity and uniform continuity of real functions; properties of derivatives of real functions; and the theory of the Riemann integral.

Course Prerequisites

The official prerequisites for this course are Foundations of Mathematics (Math 300) and Several Variable Calculus (Math 221 or Math 251 or Math 253). The essential background is some exposure to writing mathematical proofs (which Math 300 provides).

Special Course Designation

The regular section 501 and the honors section 200 of Math 409 constitute a “stacked” course, meeting in the same room at the same time. The assignments will differ between the two sections. The honors section will interact with the subject at a deeper level.

Course Objective

A main goal of the course, essential to mastering the indicated material, is to learn how to apply precise mathematical reasoning in reading, writing, understanding, and evaluating proofs of theorems in analysis.


The textbook is Basic Analysis: Introduction to Real Analysis (Volume I) by Jiří Lebl. This book is available as a free pdf download from the author’s website. There are links at the same site to read the book online in html format and optionally to purchase a paper copy.

Grading Policy

Course letter grades are based on the standard scale (60% is passing, 70% or higher earns a C, 80% or higher earns a B, 90% or higher earns an A).

The categories contributing to the course grade have the following weights.

Late Work Policy

The expectation is that you will meet announced deadlines for submission of assignments. I recognize that extraordinary circumstances may arise, so I will accept late work for partial credit. I have configured Canvas to apply a penalty of 10% per day for late submissions. (To handle partial days, Canvas uses the ceiling function \(\lceil\cdot\rceil\), the smallest integer greater than or equal to the number, so the grade penalty for a submission \(x\) days late is \(\lceil x\rceil \times{}\)10%, capped at 100%.)

Work submitted to make up for an excused absence is not considered late and is exempted from the late work policy. See Student Rule 7.

Course Schedule

The following schedule is subject to revision if circumstances change.

Week 1
Monday, January 16: Martin Luther King, Jr. Day. TAMU Holiday. Classes do not meet.
Wednesday, January 18: § 0.3, review of Math 300.
Friday, January 20: follow-up.
Week 2
Monday, January 23: § 1.1, basic properties of the real numbers.
Wednesday, January 25: § 1.2, deeper properties of the real numbers.
Friday, January 27: follow-up.
Week 3
Monday, January 30: § 1.3, absolute value.
Wednesday, February 1: § 1.4, intervals in \(\mathbb{R}\).
Friday, February 3: follow-up.
Week 4
Monday, February 6: § 1.5, decimal representation of real numbers.
Wednesday, February 8: § 2.1, sequences and limits.
Friday, February 10: follow-up.
Week 5
Monday, February 13: § 2.2, theorems about limits.
Wednesday, February 15: § 2.3, convergent subsequences.
Friday, February 17: follow-up.
Week 6
Monday, February 20: § 2.4, Cauchy sequences.
Wednesday, February 22: § 2.5, infinite series.
Friday, February 24: follow-up.
Week 7
Monday, February 27: § 2.6, more about infinite series.
Wednesday, March 1: review.
Friday, March 3: Midterm examination.
Week 8
Monday, March 6: § 3.1, limits of functions.
Wednesday, March 8: § 3.2, continuous functions.
Friday, March 10: follow-up.
Spring Break
Classes do not meet March 13–17.
Week 9
Monday, March 20: § 3.3, theorems about continuous functions on intervals.
Wednesday, March 22: § 3.4, uniform continuity.
Friday, March 24: follow-up.
Week 10
Monday, March 27: § 3.5, limits and \(\infty\).
Wednesday, March 29: § 3.6, monotone functions.
Friday, March 31: follow-up.
Week 11
Monday, April 3: § 4.1, the derivative.
Wednesday, April 5: § 4.2, the mean-value theorem.
Friday, April 7: Reading Day, classes do not meet.
Week 12
Monday, April 10: § 4.3, Taylor’s theorem.
Wednesday, April 12: § 4.4, the inverse-function theorem.
Friday, April 14: follow-up.
Week 13
Monday, April 17: § 5.1, the Riemann integral.
Wednesday, April 19: § 5.2, properties of the integral.
Friday, April 21: follow-up.
Week 14
Monday, April 24: § 5.3, the fundamental theorem of calculus.
Wednesday, April 26: § 5.4, the logarithm function and the exponential function.
Friday, April 28: follow-up.
Week 15
Monday, May 1: § 5.5, improper integrals.
Tuesday, May 2 (redefined as Friday): review for the final examination.
Wednesday, May 3: Reading Day, classes do not meet.
Final Examination
Monday, May 8, 8:00–10:00 a.m.

University Policies

This section contains university-level policies. The TAMU Faculty Senate established the wording of these policies.

Attendance Policy

The university views class attendance and participation as an individual student responsibility. Students are expected to attend class and to complete all assignments.

Please refer to Student Rule 7 in its entirety for information about excused absences, including definitions, and related documentation and timelines.

Makeup Work Policy

Students will be excused from attending class on the day of a graded activity or when attendance contributes to a student’s grade, for the reasons stated in Student Rule 7, or other reason deemed appropriate by the instructor.

Please refer to Student Rule 7 in its entirety for information about makeup work, including definitions, and related documentation and timelines.

“Absences related to Title IX of the Education Amendments of 1972 may necessitate a period of more than 30 days for make-up work, and the timeframe for make-up work should be agreed upon by the student and instructor” (Student Rule 7, Section 7.4.1).

“The instructor is under no obligation to provide an opportunity for the student to make up work missed because of an unexcused absence” (Student Rule 7, Section 7.4.2).

Students who request an excused absence are expected to uphold the Aggie Honor Code and Student Conduct Code. (See Student Rule 24.)

Academic Integrity Statement and Policy

“An Aggie does not lie, cheat or steal, or tolerate those who do.”

“Texas A&M University students are responsible for authenticating all work submitted to an instructor. If asked, students must be able to produce proof that the item submitted is indeed the work of that student. Students must keep appropriate records at all times. The inability to authenticate one’s work, should the instructor request it, may be sufficient grounds to initiate an academic misconduct case” (Section, Student Rule 20).

You can learn more about the Aggie Honor System Office Rules and Procedures, academic integrity, and your rights and responsibilities at aggiehonor.tamu.edu.

Americans with Disabilities Act (ADA) Policy

Texas A&M University is committed to providing equitable access to learning opportunities for all students. If you experience barriers to your education due to a disability or think you may have a disability, please contact Disability Resources office. Disabilities may include, but are not limited to, attentional, learning, mental health, sensory, physical, or chronic health conditions. All students are encouraged to discuss their disability-related needs with Disability Resources and their instructors as soon as possible.

Disability Resources is located in the Student Services Building or at (979) 845-1637 or visit disability.tamu.edu.

Title IX and Statement on Limits to Confidentiality

Texas A&M University is committed to fostering a learning environment that is safe and productive for all. University policies and federal and state laws prohibit gender-based discrimination and sexual harassment, including sexual assault, sexual exploitation, domestic violence, dating violence, and stalking.

With the exception of some medical and mental health providers, all university employees (including full and part-time faculty, staff, paid graduate assistants, student workers, etc.) are Mandatory Reporters and must report to the Title IX Office if the employee experiences, observes, or becomes aware of an incident that meets the following conditions (see University Rule 08.01.01.M1):

Mandatory Reporters must file a report regardless of how the information comes to their attention — including but not limited to face-to-face conversations, a written class assignment or paper, class discussion, email, text, or social media post. Although Mandatory Reporters must file a report, in most instances, a person who is subjected to the alleged conduct will be able to control how the report is handled, including whether or not to pursue a formal investigation. The University’s goal is to make sure you are aware of the range of options available to you and to ensure access to the resources you need.

Students wishing to discuss concerns in a confidential setting are encouraged to make an appointment with Counseling and Psychological Services (CAPS).

Students can learn more about filing a report, accessing supportive resources, and navigating the Title IX investigation and resolution process on the University’s Title IX webpage.

Statement on Mental Health and Wellness

Texas A&M University recognizes that mental health and wellness are critical factors that influence a student’s academic success and overall wellbeing. Students are encouraged to engage in healthy self-care by utilizing available resources and services on your campus.

Students who need someone to talk to can contact Counseling & Psychological Services (CAPS) or call the TAMU Helpline (979-845-2700) from 4:00 p.m. to 8:00 a.m. weekdays and 24 hours on weekends. 24-hour emergency help is also available through the national 988 Suicide & Crisis Lifeline (988) or at https://988lifeline.org.