Class Calculus I

The curricular unit of Calculus I introduces the essential concepts and tools for differential and integral calculus of real-valued functions of a real variable. It is intended to extend the mathematical training acquired in secondary education by developing the capacity for abstraction and logical reasoning in formulating and solving problems, formalizing them with rigor and precision, but without neglecting the virtues of intuitive thinking. The main objective is to provide an introduction to Mathematical Analysis, showing the rigorous side of Calculus and establishing the theoretical foundations for further studies in areas of Science and Engineering.

Not applicable

1. Sets: intersection, union, difference and Cartesian product 2. Functions: domain, range, graph, injectivity, surjectivity, composition, inverse function 3. Real numbers: algebraic and order properties, intervals 4. Module function and polynomials 5. Equations and inequalities 6. Summation symbol and mathematical induction method 7. Real functions of a real variable: domain, range, graph, monotony, algebraic operations, composition, inverse function 8. Rational functions, exponential function, logarithm function, hyperbolic functions, trigonometric functions and their inverses 9. Limits, continuity, sandwich theorem, Bolzano theorem, Weierstrass theorem 10. Differentiability: derivation rules, chain rule and the inverse function theorem, Rolle, Lagrange and Cauchy theorems, monotony, concavity, extremes, asymptotes 11. Primitives: immediate, by parts, by substitution, of rational functions 12. Integrability: fundamental theorem of Calculus and Barrow's rule

In the theoretical lectures, the underlying principles and motivations behind the chosen approaches and directions for the development of the course content are explained. The relevant theorems are then presented, together with formal or heuristic proofs of their statements. The meaning and implications of these theorems and results are illustrated through examples and counterexamples. Weekly exercise sheets are assigned, designed to provide comprehensive understanding while remaining compatible with students' study workload. These exercises are thoroughly discussed and solved during the tutorial sessions, allowing students to ask questions, address gaps in their understanding, and clarify any doubts. Short videos covering topics related to the course content are also created and made available, along with historical insights, recreational mathematical problems, and real-world engineering applications.

Sá, A. A. e Louro, B. (2022) Cálculo Diferencial e Integral em R. Escolar Editora. Stewart, J., Clegg, D., & Watson, S. (2020). Calculus (9th ed.). Cengage. Anton, H., Bivens, I., & Davis, S. (2012). Calculus (10th ed.). John Wiley & Sons.