Publication in the Diário da República: Despacho nº 10366/2022 - 24/08/2022 (Parceria ESTT/ESAI)
6 ECTS; 1º Ano, Anual, 70,0 TP , Cód. 908940.
Lecturer
- Eugénio Manuel Carvalho Pina de Almeida (2)
- Maria Cristina Oliveira da Costa (1)(2)
(1) Docente Responsável
(2) Docente que lecciona
Prerequisites
Not applicable.
Objectives
a) Consolidate and expand Mathematics knowledge acquired in secondary school, in particular those involving the study of real functions of one real variable;
b) To know and apply basic fundamentals of the mathematical procedures used in the other curricular units of the degree course in Civil Engineering;
c) Interpret data, formulate and solve problems involving differential calculus or integral calculus of real functions of one real variable, as well as their applications.
Program
1. Real Numbers
1.1. Sets.
1.2. First properties of real numbers.
1.3. Limited sets. Brief notions of topology in IR.
1.4. Powers and logarithms.
1.5. Trigonometry rectilinear.
2. Real Functions of a Real Variable
2.1. Definition.
2.2. Graph.
2.3. Injective and surjective functions.
2.4. Composition of functions
2.5. Inverse functions.
2.6. Supreme and infimum of a function.
2.7. Monotonous functions.
2.8. Limited functions.
2.9. Odd and even functions.
2.10. Periodic functions.
2.11. Some classes of functions:
2.11.1. Polynomial, rational and irrational functions;
2.11.2. Trigonometric functions;
2.11.3. Exponential function and logarithmic function;
2.11.4. Functions f(x)^g(x).
3. Limits and Continuity
3.1. Notion of limit.
3.2. Definition of limit.
3.3. Lateral limits.
3.4. Theorems on the calculation of limits.
3.5. Indeterminations in the calculation of limits.
3.6. Definition of continuity.
3.7. Theorems on continuity.
4. Differential Calculus
4.1. Definition of derivative.
4.2. Geometric interpretation of the definition of derivative.
4.3. Differentiability and Continuity.
4.4. Derivation rules.
Derivative of the implicit function.
4.6. Derivative of functions defined in parametric form.
4.7. Derivative of the inverse function.
4.8. Derivative of the composite function.
4.9. Successive derivatives.
4.10. Properties of continuous and derivable functions: Bolzano's theorem, Weierstrass's theorem, Rolle's theorem, Lagrange's theorem and their corollaries.
4.11. Cauchy's Theorem.
4.12. Cauchy's rule and L'Hôpital's rule.
4.13. Indeterminations in the calculation of limits.
4.14. Applications of derivatives to the graphical study of functions.
4.15. Maximum and minimum.
4.16. Concavity and convexity of a function.
4.17. Inflection points.
4.18. Vertical, horizontal and oblique asymptotes.
4.19. Complete study of a function.
4.20. Differentials. Geometric definition and interpretation.
5. Integral Calculation
5.1. Primitives.
5.2. Rules of integration.
5.3. Integration by parts.
5.4. Integration by substitution.
5.5. Integration of rational functions.
5.6. Integration of powers of trigonometric functions.
5.7. Defined integral.
5.8. Fundamental theorem of calculus.
5.9. Properties of the defined integral.
5.10. Applications of integral calculus: areas and volumes.
5.11. Improper integrals.
Evaluation Methodology
Continuous assessment: two written tests marked from 0 to 10. A minimum mark of 3 in each test and an overall mark of 10 exempts students from exam.
Exam-based assessment: an exam (worth 0-20) or resit covering all the material taught. Minimum pass mark:10
NOTES:
In any form of assessment, a mark of 17 or higher requires the student to take an extraordinary exam.
Bibliography
- Howard, A. (2007). Cálculo um novo horizonte. (Vol. 1). São Paulo: Bookman
- Monteiro, H. (2023). Apontamentos de Análise Matemática I . . Acedido em 12 de setembro de 2024 em https://doctrino.ipt.pt/
- Silva, J. (1999). Princípios de Análise Matemática Aplicada. (Vol. 1). Lisboa: McGraw-Hill
- Stewart, J. (2012). Cálculo. (Vol. 1). USA : Brooks/Cole, Cengage Learning. Belmont
- Swokowsi, E. (1995). Cálculo com Geometria Analítica. (Vol. 1). São Paulo: Makron Books
Teaching Method
Theoretical lectures, with presentation and illustration of the proposed subjects. Theoretical-practical lectures in which exercises are proposed and solved.
Software used in class
