Calculus II

MAT039M2: Calculus II

2025 – Second Semester 

Schedule: Monday and Wednesday – 1:00 PM–2:40 PM

Room: 2006 - MAT   

Course Start: August 13, 2025  

Syllabus:  Numerical Sequences and Series, Power Series, Taylor's Formula, Conic Sections and Polar Coordinates, Differentiability of Multivariable Functions.  

Program Outline:  

1. Numerical Sequences:  

   - Definition, finite and infinite limits.  

   - Bounded monotonic sequences.  

   - Properties of limits.  

2. Numerical Series:

   - Definition, convergence, and divergence.  

   - Necessary condition for convergence.  

   - Geometric and harmonic series.  

   - Convergence tests for positive-term and alternating series.  

   - Absolute and conditional convergence.  

3. Power Series:  

   - Definition, pointwise convergence.  

   - Radius and interval of convergence.  

   - Taylor series.  

   - Continuity, differentiation, and integration of power series.  

   - Series expansions for sin(x), cos(x),  e^x, log(1+x), (1+x)^s.  

4. Taylor's Formula for Single-Variable Functions:  

   - Polynomial approximation.  

   - Taylor's formula: derivation, uniqueness, remainder term.  

   - Applications.  

5. Conic Sections:  

   - Introduction: circles, parabolas, ellipses, hyperbolas.  

6. Polar Coordinates: 

   - Definition, curve equations.  

   - Area and arc length in polar coordinates.  

7. Multivariable Functions:  

   - Domain sketching, level surfaces.  

   - Quadric and cylindrical surfaces.  

   - Limits and continuity.  

8. Differentiation of Multivariable Functions:  

   - Partial derivatives, geometric interpretation.  

   - Differentiability, linear approximation.  

   - Tangent planes, normal lines.  

   - Higher-order derivatives, Clairaut's theorem.  

   - Chain rule, implicit differentiation.  

   - Directional derivatives, gradient.  

   - Local/absolute maxima and minima.  

   - Constrained optimization.  

Links: 

• Seqüências de Fibonacci 1 

• Seqüências  de Fibonacci 2 

Assessment System:  

- Exam 1 (P1): September 17, 2025 (Wednesday) – 30 points  

- Exam 2 (P2): October 29, 2025 (Wednesday) – 30 points  

- Exam 3 (P3): December 3, 2025 (Wednesday) – 30 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: STEWART, J. – Calculus (Vol. 2), Pioneira, 2010.  

Supplementary Reference:  

1.   LEITHOLD, L. – Calculus with Analytic Geometry (Vol. 2), Harbra.  

2.   APOSTOL, T. M. – Calculus (Vols. 1–2), Reverté.