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.

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

Creation and availability of short videos on topics related to the subject taught, as well as historical curiosities, playful problems and applications to the real world.

- SANTOS, J. P. (2012). Cálculo Numa Variável Real. IST Press.

- SÁ, A. A., & LOURO, B. (2020). Cálculo Diferencial e Integral em R (2ª edição). Matcubo.