Analysis I

MAT243N1: Analysis I  

2025 – Second Semester  

Schedule: Monday, Wednesday, and Friday – 2:55 PM–4:35 PM  

Room: 2006 - MAT 

Course Start: August 13, 2025  

Syllabus: Real Numbers, Introduction to Topology of the Real Line, Continuous Functions, Differentiable and Integrable Functions.  

Program Outline:

1. Real Numbers:  

   - Axiomatic approach, emphasis on proof techniques.  

   - Supremum and Infimum (elementary applications).  

2. Topology of the Real Line:  

   - Sequences (limits, monotonicity, subsequences), Cauchy sequences.  

   - Bolzano-Weierstrass Theorem.  

   - Open, closed, and compact sets on the real line. 

3. Continuous Functions:  

   - Limits and continuity.  

   - Intermediate Value Theorem.  

   - Weierstrass Theorem (extrema of continuous functions on compact intervals).  

4. Derivatives:  

   - Definition and derivation rules.  

   - Mean Value Theorem and consequences.  

   - Relationship between continuous and differentiable functions.  

5. Integration: 

   - Upper and lower integrals, integrable functions.  

   - Integral as a limit of Riemann sums.  

   - Antiderivatives, Fundamental Theorem of Calculus.  

   - Mean Value Theorem for Integrals.  

   - Taylor’s Formula with integral remainder and other remainders.  

6. Additional Topics:  

   - L’Hôpital’s Rule.  

   - Logarithmic and Exponential Functions.  

   - Trigonometric Functions.  

Assessment System:

The semester will include three exams and problem sets:  

- Exam 1 (P1): September 19, 2025 (Friday) – 20 points  

- Exam 2 (P2): October 31, 2025 (Friday) – 35 points  

- Exam 3 (P3): December 5, 2025 (Friday) – 35 points  

- Problem Sets (L): 10 points  

- Special Exam (EE): December 10, 2025 (Wednesday)  

Grading Criteria:  

- Direct Approval:  

  Students with N = P1 + P2 + P3 + L ≥ 60 pass with final grade N.  

- Special Exam (EE):  

  If 40 ≤ N < 60, the student may take the special exam. The final grade will be:  

  Final Grade = max{(N + EE)/2, N}  

Notes: 

- Passing requires Final Grade ≥ 60.  

- If (N + EE)/2 ≥ 60, the student passes.  

- Otherwise, the original grade N is retained (if higher).  

Primary Reference: LIMA, E. L. – Análise Real – Coleção Matemática Universitária, IMPA, 2001.  

Supplementary References: 

1.   APOSTOL, T. M. – Mathematical Analysis – Addison-Wesley.  

2.   SPIVAK, M. – Calculus – NY, WA, Benjamin.  

3.   FIGUEIREDO, D. G. – Análise I – R. J., LTC.