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Calculus I B

Objectives

At the end of this course, the student will have acquired the knowledge, skills, and competencies to:

(i) Understand and apply the principle of mathematical induction. (ii) Solidify the concepts of continuity and differentiability of real functions of one real variable. Understand the theorems of Rolle, Lagrange and Cauchy and their applications; (iii) Understand Taylor expansion and its applications; (iv) Understand the concept of indefinite integral and be able to calculate it using appropriate calculus techniques. (v) Understand the notion of Riemann integral, its computation techniques and applications; (vi) Be able to analyse the convergence of improper integrals; (vii) Analyse  the convergence of numerical series; (viii) Develop logical reasoning, critical thinking, and autonomy; (ix) Acquire mathematical modelling skills and apply the acquired concepts in various fields, such as Physics, Engineering and Risk Management.

General characterization

Code

13491

Credits

6.0

Responsible teacher

Paula Cristiana Costa Garcia Silva Patrício

Hours

Weekly - 5

Total - 70

Teaching language

Português

Prerequisites

The student must master the mathematical knowledge lectured until the end of Portuguese high school teaching.

Bibliography

Claudio Canuto and Anita Tabacco, Mathematical Analysis I, 1st Edition,  Springer, 2008.

 M. Thamban Nair, and Nair. Calculus of one variable. Springer International Publishing, 2021.

 Anton, Howard, Irl C. Bivens, and Stephen L. Davis. Cálculo-Volume I-8. Bookman, 2007.

 Ana Sá, Bento Louro. Sucessões e Séries, Teoria e Prática. Escolar Editora, 2009.

Teaching method

The teaching method follows the Theoretical-Practical Classes (TP) academic model. In TP classes, the course content is presented through concepts and results, appropriately supplemented with examples and application cases when relevant. Some results are rigorously proven, while for others, an outline of the proof is provided.

A list of exercises will be made available, with some exercises being solved during TP classes. However, the majority should be completed independently by students and later discussed in TP classes or during individual sessions scheduled with the instructors.

Additionally, students may have access to the MATH∑RIA digital platform, a valuable study support resource that enables guided learning beyond the classroom. MATH∑RIA will offer various features, including progressively challenging exercise lists accompanied by hints, explanations, and solution tips, ensuring continuous support. Furthermore, it may provide explanatory videos illustrating the practical application of the taught content, making learning more dynamic and contextualized. To enhance students'' autonomy, MATH∑RIA may also offer interactive challenges, diagnostic assessments, and personalized learning paths tailored to each student''s specific needs.

Evaluation method

Except for cases provided by law, attendance is a necessary requirement for passing the course. Attendance is granted to students who attend at least two-thirds of the taught theoretical-practical classes.

Assessment is carried out through two tests taken during the semester or through a final exam. The final grade for the continuous assessment component will be the average of the two test scores.

Final grades higher than 17 points may require supplementary examinations (written or oral) for grade confirmation.

Subject matter

1. Mathematical Induction.

2. Generalities about functions of real variable. Inverse trigonometric functions.

3. Review of Some Concepts on Continuity:  Convergence according to Cauchy. Removable discontinuity. Continuous extension. discontinuity of the first kind.  Continuity and reciprocal bijections.

4.Differenciability: Theorems of Rolle, Lagrange and Cauchy. Calculus of limits. Taylor formula and applications.

5.Indefinite Integration: Introduction. Indefinite integration by parts and by substitution. Indefinite integration of rational functions.

6.Riemann Integration: Fundamental theorem of calculus. Integration by parts and by substitution. Applications, namely for calculating areas.

7.Improper integration: Improper Integrals of the first and second kind: Definition and convergence criteria.

8.Numerical Series: Geometric and telescoping series. Series with non-negative terms. Convergence criteria for series with non-negative terms. Absolute convergence. Alternating series.