6 ECTS; 1º Ano, 1º Semestre, 75,0 TP

**Lecturer**

- Ana Cristina Becerra Nata dos Santos

**Prerequisites**

Not applicable.

**Objectives**

1. Learn/consolidate mathematical skills in the area of: 1.1. Mathematical analysis; 1.2. Financial mathematics; 1.3. Numerical methods 3. Develop mathematical, logical, analytical and critical reasoning. 4. Identify, interpret, formulate, solve problems and make decisions.

**Program**

1. Functions and differential calculus

1.1. An overview of real value functions of real variable.

1.2. Brief mention to the limits calculation of a function in one variable. The derivative function. Differentiable functions. Geometric interpretation. Derivation rules. Higher derivatives.

1.3. Application of derivatives to the study of functions and to the study of some Economic's problems.

2. INTEGRAL CALCULUS IN ONE VARIABLE

2.1. Definition and generalities. Properties of indefinite integrals (also known as antiderivative or primitive).

2.2. Immediate and almost-immediate primitives.

2.3. Techniques of primitivation

2.4. Riemann's integral definition and its geometrical interpretation. Conditions of integrability and properties of the integrals.

2.5. The fundamental theorem of integral calculus. 2.6. The mean-value theorem and its applications.

2.7. Integral calculation methods

2.8. Improper integrals with infinite integration limits.

2.9. Geometric applications of the integral.

3. THE BASIC OF FINANCIAL MATHEMATICS

3.1. Simple, compound and continuously interest compound.

3.2. Geometric progressions: definition and expression of the sum of the first n terms. Applications to the savings plan (compound and continuously compound interest) and loans (compound interest).

4. FUNCTIONS AND DIFFERENTIAL CALCULUS IN IRn

4.1. Real value functions of several real variables.

4.1.1. Structure of Rn.

4.1.2. Definition of real valued functions of several real variables. Domain computation and its graphical representation.

4.2. Concept of limit of functions of several variables. Partial derivatives. Higher order partial derivatives.

4.3. Homogeneous functions: definition and Euler's theorem.

4.4. Taylor's formula and its application to the extreme values computation of functions in Rn.

4.5. Lagrange multipliers and constrained optimization.

5.LINEAR DIFFERENTIAL EQUATIONS OF 1ST ORDER

5.1. Basic concepts: definition of differential equation, order of a differential equation,

general solution, particular solutions, initial

and boundary value problems. Motivating examples.

5.2. Linear differential equations of 1st order.

5.2.1. Homogeneous equation.

4.2.2. Non-homogeneous (or complete) equation: Lagrange's method or method of the arbitrary constant variation.

6. NUMERICAL ANALYSIS

6.1. Auxiliary of numerical calculation.

6.2. Numerical solution of non-linear equations in one variable (bisection method and false position method).

6.3. Polynomial interpolation: table of finite differences and Newton's polynomial interpolation.

6.4. Numerical differentiation.

6.5. Numerical integration: trapezoidal and Simpson rules. Error analysis.

**Evaluation Methodology**

Continuous assessment: one work (minimum mark: 5/20 each) and three tests.

Final grade = 0.2T+0.25M1+0.25M2+0.30M3 if the minimum mark in T.

Final assessment: closed-book summative test. Minimum pass mark: 10/20

**Bibliography**

- Armstrong, B. e Davis, D. (2002). *College mathematics: Solving problems in finite mathematics and calculus*. USA: Pearson Education

- Ferreira, M. e Amaral, I. (2006). *Primitivas e Integrais*. (pp. 1-184). Lisboa, Portugal: Edições Sílabo

- Santos, C. (2002). *Fundamentos de análise numérica*. Lisboa: Edições Sílabo

- Edwards, B. e Larson, R. e Hostetler, R. (2006). *Cálculo*. (Vol. I). USA: McGraw-Hill

**Teaching Method**

Lectures and practical sessions.

**Software used in class**

Not applicable.

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