Advanced Calculus I
Mathematics 301 | Course Outline & Syllabus
Course Information
Course Code: MATH 301
Credits: 4
Prerequisites: Calculus II (MATH 202) and Introduction to Proof (MATH 200) or consent of the instructor.
Instructor: Dr. A. Mathematician | Office: Hamilton Hall 304 | Email: amath@university.edu
Course Description
This course provides a rigorous foundation in single-variable calculus. It covers the real number system, sequences and series, limits, continuity, differentiation, and Riemann integration. Emphasis is placed on mathematical proofs and the logical structure of analysis rather than merely computational techniques.
Learning Objectives
Upon successful completion of this course, students will be able to:
- Demonstrate a thorough understanding of the axiomatic structure of the real number system, including the Completeness Axiom.
- Formulate and write rigorous proofs using epsilon-delta definitions for limits and continuity.
- Analyze the convergence of sequences and series using various tests (e.g., Ratio Test, Root Test, Integral Test).
- Apply the Mean Value Theorem and Taylor's Theorem to approximate functions and analyze their behavior.
- Define the Riemann integral and prove fundamental theorems of calculus.
- Communicate mathematical ideas clearly and precisely in both oral and written form.
Required Materials
- Primary Textbook: "Principles of Mathematical Analysis" (3rd Edition) by Walter Rudin.
- Supplementary Text: "Understanding Analysis" by Stephen Abbott.
- Tools: A scientific calculator is allowed but rarely necessary. Assignments requiring computational assistance will be specified.
Weekly Schedule of Topics
The following outline is tentative and may be adjusted to meet the pace of the class.
Part I: Foundations and Sequences
| Week | Topic | Reading |
|---|---|---|
| 1 | The Real and Complex Number Systems: Ordered sets, fields, the Completeness Axiom. | Ch. 1 |
| 2 | Basic Topology: Euclidean spaces, finite, countable, and uncountable sets, compact sets. | Ch. 2 |
| 3 | Sequences and Series: Convergent sequences, subsequences, Cauchy sequences. | Ch. 3 |
| 4 | Series: Absolute and conditional convergence, rearrangement of series. | Ch. 3 |
Part II: Continuity and Differentiation
| Week | Topic | Reading |
|---|---|---|
| 5 | Limits of Functions: Epsilon-delta definition, limits at infinity. | Ch. 4 |
| 6 | Continuous Functions: Continuity on compact sets, discontinuities, the Intermediate Value Theorem. | Ch. 4 |
| 7 | Differentiation: The derivative of a real function, the Mean Value Theorem. | Ch. 5 |
| 8 | L'Hospital's Rule, Taylor's Theorem, and differentiation of vector-valued functions. | Ch. 5 |
Part III: Integration
| Week | Topic | Reading |
|---|---|---|
| 9 | The Riemann-Stieltjes Integral: Definition and existence of the integral. | Ch. 6 |
| 10 | Properties of the Integral: Linearity, integration by parts, change of variable. | Ch. 6 |
| 11 | The Fundamental Theorem of Calculus and integration of vector-valued functions. | Ch. 6 |
| 12 | Sequences and Series of Functions: Pointwise vs. uniform convergence. | Ch. 7 |
| 13 | Uniform Convergence and Continuity/Differentiation. The Weierstrass Approximation Theorem. | Ch. 7 |
| 14 | Power Series: Radius of convergence, exponential and logarithmic functions. | Ch. 8 |
| 15 | Review and Final Examination preparations. | N/A |
Assessment and Grading
Grading will be based on a combination of problem sets, quizzes, a midterm exam, and a cumulative final exam.
Weekly problem sets will be assigned. Students are encouraged to collaborate on ideas but must write up their solutions independently. Late submissions will incur a 10% penalty per day.
Short, 15-minute quizzes will be given on Thursdays of non-exam weeks to test basic comprehension of recent definitions and theorems.
Scheduled for Week 8. Covers topics from Weeks 1 through 7.
Cumulative exam covering all course material. Date and time to be determined by the registrar's office.
Grading Scale
- A: 93 - 100%
- A-: 90 - 92%
- B+: 87 - 89%
- B: 83 - 86%
- B-: 80 - 82%
- C+: 77 - 79%
- C: 70 - 76%
- D: 60 - 69%
- F: Below 60%
Course Policies
Attendance
Attendance is mandatory. While no specific grade is assigned for attendance, active participation is crucial for success in proof-based courses. Excessive absences may result in a grade reduction.
Academic Integrity
The University takes academic honesty very seriously.
Any act of plagiarism, cheating, or assisting others in cheating will be reported to the Dean of Students. In this course, copying homework solutions from external sources (e.g., solution manuals, online forums) without citation is considered plagiarism. You must cite any sources you consult other than the textbook and lecture notes.
Accommodations
Students with disabilities who require reasonable accommodations must contact the Office of Disability Services. Please provide the instructor with your accommodation letter within the first two weeks of the semester.
Electronic Devices
The use of laptops, tablets, and cell phones is generally prohibited during lectures to minimize distractions. Exceptions will be made for students requiring assistive technology.
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