MATH 481: Advanced Calculus

Course Description

This course provides a rigorous treatment of the foundations of analysis, building upon concepts from elementary calculus. We will explore the theoretical underpinnings of calculus through the lens of real analysis, focusing on proofs and mathematical reasoning rather than computational techniques. Topics will include real number properties, sequences and series, limits, continuity, differentiation, Riemann integration, and functions of several variables.

Prerequisites

MATH 281 (Multivariable Calculus) with a minimum grade of C, or consent of instructor. A solid understanding of single-variable calculus (limits, derivatives, and integrals) is essential for success in this course.

Course Objectives

Upon successful completion of this course, students will:

• Understand and be able to apply rigorous definitions of limit, continuity, derivative, and integral
• Construct clear, logically sound mathematical proofs
• Analyze convergence of sequences and series using appropriate tests and theorems
• Understand the properties of real numbers and how they form the foundation of calculus
• Apply theoretical results to solve problems in analysis
• Communicate mathematical concepts precisely using appropriate terminology and notation

Required Textbook

Understanding Analysis by Stephen Abbott, 2nd edition, Springer. (ISBN: 978-1493927111)

Additional resources will be provided via the course website.

Course Policies

Attendance

Regular attendance is essential for understanding the course material. While not strictly required, attendance will be taken regularly. Students missing more than three classes without university-approved excuses may face grade penalties.

Academic Integrity

All work submitted must be your own. Collaboration on homework is permitted and encouraged, but you must write up your own solutions and credit any collaborators or sources used. Violations of the academic integrity policy will result in a score of zero on the assignment and possible further disciplinary action.

Late Work

Homework submitted after the due date will receive a 20% penalty per day late. No homework will be accepted more than three days after the deadline. Make-up exams will only be given for documented emergencies or with prior approval from the instructor.

Grading Scheme

Component	Weight
Homework Assignments	25%
Midterm Exam 1	20%
Midterm Exam 2	20%
Final Exam	30%
Participation	5%

Grades will be assigned according to the following scale:

• A: 93-100%
• A-: 90-92%
• B+: 87-89%
• B: 83-86%
• B-: 80-82%
• C+: 77-79%
• C: 73-76%
• C-: 70-72%
• D: 60-69%
• F: Below 60%

Course Schedule

Unit 1: The Real Numbers (Weeks 1-3)

• Properties of real numbers
• Axioms of the real number system
• Completeness property
• Countable and uncountable sets

Unit 2: Sequences and Series (Weeks 4-6)

• Convergence and divergence of sequences
• Monotone convergence theorem
• Subsequences and Bolzano-Weierstrass theorem
• Cauchy sequences
• Convergence tests for series

Midterm Exam 1: Friday, February 26

Unit 3: Limits and Continuity (Weeks 7-9)

• - definition of limit
• Limit theorems
• Continuity
• Intermediate Value Theorem
• Extreme Value Theorem
• Uniform continuity

Unit 4: Differentiation (Weeks 10-11)

• Derivative as a limit
• Differentiability implies continuity
• Mean Value Theorem
• L'Hpital's Rule

Midterm Exam 2: Friday, April 2

Unit 5: Integration (Weeks 12-14)

• Riemann sums and the Riemann integral
• Properties of the integral
• Fundamental Theorem of Calculus
• Integrability of continuous functions

Review Week (Week 15)

Final Exam: Monday, May 10, 9:00-11:30am

Homework Policy

Homework assignments will be posted weekly on the course website and are typically due on Fridays at the beginning of class. Submitted solutions should be clear, well-organized, and mathematically rigorous. For proof-based problems, provide complete arguments with proper logical structure. The lowest homework score will be dropped at the end of the semester.

Academic Resources

If you are struggling with the course material, please come to office hours early in the semester. Additional resources include:

• Mathematics Department Tutoring Center (Math Building, Room 301)
• University Writing Center (for help with mathematical writing)
• Supplementary textbooks available in the library

Accommodations for Students with Disabilities

It is the policy and practice of this university to provide inclusive education. If you have a documented disability and need accommodations, you should contact the Disability Services Office at the beginning of the semester. Once approved, they will provide you with an accommodation letter to share with your instructors. Accommodations cannot be made retroactively.

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