# Mathematical Analysis I

**Electronics and Computer Engineering**, Publication in the Diário da República - Despacho nº 10766/2011 - 30/08/2011

6 ECTS; 1º Ano, 1º Semestre, 28,0 T + 14,0 PL + 28,0 TP + 5,0 OT

**Lecturer**

- Maria Manuela Morgado Fernandes Oliveira

**Prerequisites**

Not applicable.

**Objectives**

a) Provide students with the basic foundations of mathematical methods normally used by the various curricular units of the Degree in Chemical Technology course.

b) Give students the ability to use the concepts and methods of differential and integral calculus of real functions of a real variable.

The proposed program was elaborated taking as a reference base the knowledge acquired by the students, in the courses that precede their entry into this Degree.

**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. Straight trigonometry.

2. Real functions of a real variable

2.1. Definition.

2.2. Graphic.

2.3. Injective and surjective functions.

2.4. Role composition

2.5. Reverse functions.

2.6. Supreme and lowest 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. Direct and inverse 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. Limit definition.

3.3. Lateral limits.

3.4. Limit calculation theorems.

3.5. Indeterminacy in the calculation of limits.

3.6. Definition of continuity.

3.7. Continuity theorems.

4. Differential Calculus

4.1. Definition of derivative.

4.2. Geometric interpretation of derivative definition.

4.3. Differentiability and Continuity.

4.4. Derivation rules.

4.5. Derived from the implicit function.

4.6. Derived from functions defined in parametric form.

4.7. Derived from the inverse function.

4.8. Derived from the composite function.

4.9. Successive derivatives.

4.10. Properties of continuous and derivable functions: Bolzano's theorem, Weierstrass' theorem, Rolle's theorem, Lagrange's theorem and its corollaries.

4.11. Cauchy's Theorem.

4.12. Cauchy's rule and L'Hôpital's rule.

4.13. Indeterminacy in the calculation of limits.

4.14. Applications of derivatives to the graphical study of functions.

4.15. Maximums and Minimums.

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. Additions and differentials. Definition and geometric interpretation.

5. Integral Calculus

5.1. Primitives.

5.2. Integration rules.

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. Definite integral.

5.8. Fundamental theorem of calculus.

5.9. Properties of the definite integral.

5.10. Applications of integral calculus: areas and volumes.

5.11. Improper integrals.

**Evaluation Methodology**

Continuous assessment: two written tests. Exam assessment: one written test.

**Bibliography**

(2013). *Cálculo*. (Vol. 1). São Paulo: Thomson Pioneira

(2007). *Cálculo um novo horizonte*. (Vol. 1). São Paulo: Bookman

(1995). *Cálculo com Geometria Analítica*. (Vol. 1). São Paulo: Makron Books

(1999). *Princípios de Análise Matemática Aplicada*. (Vol. 1). Lisboa: McGraw-Hill

**Method of interaction**

Theoretical lectures, with presentation and illustration of the proposed subjects. Theoretical-practical lectures in which exercises are proposed and solved.

**Software used in class**

Wolfram Alpha; Symbolab.