# Mathematics II

**Auditing and Taxation**

6 ECTS; 1º Ano, 2º Semestre, 30,0 T + 45,0 PL

**Lecturer**

- José Manuel Borges Henriques Faria Paixão

- Luís Miguel Merca Fernandes

- 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. INTEGRAL CALCULUS IN ONE VARIABLE

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

1.2. Immediate and almost-immediate primitives.

1.3. Techniques of primitivation

1.4. Primitivation of rational functions.

1.5. Riemann's integral definition and its geometrical interpretation.

1.6. Conditions of integrability and properties of the integrals.

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

1.8. Integral calculation methods

1.9. Improper integrals with infinite integration limits.

1.10. Geometric applications of the integral.

2. THE BASIC OF FINANCIAL MATHEMATICS

2.1. Simple, compound and continuously interest compound.

2.2. 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).

3. FUNCTIONS AND DIFFERENTIAL CALCULUS IN IRn

3.1. Real value functions of several real variables.

3.1.1. Structure of Rn.

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

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

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

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

3.5. Lagrange multipliers and constrained optimization.

4. NUMERICAL ANALYSIS

4.1. Auxiliary of numerical calculation.

4.2. Numerical solution of non-linear equations in one variable (bisection method, Newton method, secant method and false position method).

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

4.4. Numerical differentiation.

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

**Evaluation Methodology**

Continuous assessment: two written tests without consultation (50% each and minimum score of 5 val. in each).

Examination assessment: written test without consult and including all syllabus contents. Approval: at least 10 (out of 20).

**Bibliography**

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

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

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

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

**Method of interaction**

Attendance-based classes including problem-solving and discussion.

**Software used in class**

Not applicable.