Universidade Nova de Lisboa

Universidade Nova de Lisboa

Faculdade de Ciências e Tecnologia

Mathematical Analysis I

Code

11504

Academic unit

Faculdade de Ciências e Tecnologia

Department

Departamento de Matemática

Credits

6.0

Teacher in charge

José Maria Nunes de Almeida Gonçalves Gomes

Weekly hours

6

Total hours

72

Teaching language

Português

Objectives

Students should acquire the necessary knowledge and skills to pursue their learning on subsequent Analysis courses, Probability Theory, Numerical Analysis as well as to the specific disciplines of their academic studies.

Prerequisites

Students should have a solid background on Mathematics taught at high-school level within the European Union.

Subject matter

1. Basic topology of the real line.


1.1 Neighborhood of a Point. Interior, exterior, boundary, isolated, limit point and limit point of a set.
1.2 Open, closed, bounded and compact sets.

2. Mathematical induction and sequences

2.1 Mathematical Induction.

2.2 Limit of a sequence. Algebra of  limits. Subsequences and sublimits. Squeezed sequence convergence theorem. Bolzano-Weierstrass theorem and other fundamental theorems. Cauchy''s sequences.

3. Limits and Continuity in R

3.1 Definition of limit according to Cauchy and to Heine. Algebra of  limits.

3.2 Continuity of a function at a point and over a set. Continuous extension of a function. Bolzano''s theorem and Weierstrass''s theorem. Continuity of the composite function and continuity of the inverse function. Inverse trignometric functions.

4. Differential Calculus in R


4.1 Definition of the derivative of a function at a point. Physical and geometrical meaning. Differentiability. Algebra of derivatives. Derivative of the composite function and derivative of the inverse function. Derivatives of inverse trignometric functions. Rolle''s Theorem, Lagrange''s theorem. Derivative and monotony. Darboux''s theorem and Cauchy''s theorem. Cauchy’s mean theorem.

4.2 Taylor''s theorem and applications to the study of extrema and concavities.


5. Integral Calculus in R

5.1 Primitive of a function. Primitivation by parts. Primitives by substitution. Primitives of rational functions. Primitives of irrational functions and transcendent functions.

5.2 Riemann integral. Mean value theorem for integrals. Fundamental Theorem of  Calculus. Barrow''s rule. Integration by parts and integration by substitution. Application to the calculation of areas.

Teaching method

Teaching Method bases on conferences and problems solving sessions complemented by an individual attending schedule.

Evaluation method

Important:

In order to be evaluated, the student must attend at least 2/3 of the solving sessions.

 

Evaluation Methods:

1-Continuous evaluation

The continuos evaluation consists on three tests during the semester. One of the tests may be improved in the final examination date. The final grade is the average of the grades of the three tests. The student is aproved if the final grade is greater or equal than 9,5.

Each test lasts for 1h 30min. 

2-Final exam evaluation.

The student is aproved if the grade of the final exam is greater or equal than 9,5.

The final exam lasts for 3h.

 

Calculatory devices including mobile phones are not allowed in tests or exams.

Courses