# Mathematics II

**Auditing and Taxation**

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

**Lecturer**

- José Manuel Borges Henriques Faria Paixão

- 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. Formulation, resolution and extrapolation of mathematical problems to other realities.

**Program**

1. FUNCTIONS AND DIFFERENTIAL CALCULUS IN IR

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

2.3.1. Primitivation by decomposition.

2.3.2. primitivation by parts.

2.3.3. Brief mention to the primitivation by substitution.

2.4. Primitivation of some kinds of rational functions.

2.5. Riemann?s integral definition and its geometrical interpretation (Darboux's sums). Conditions of integrability and properties of the integrals.

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

2.7. Integral calculation methods

2.7.1. Integration by decomposition.

2.7.2. Integration by parts.

2.7.3. Brief mention to the integration by substitution.

2.8. Improper integrals with infinite integration limits.

2.9. Geometric applications of the integral: calculation of plane regions areas in cartesian coordinates.

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. Brief mention to the 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.

4. LINEAR DIFFERENTIAL EQUATIONS OF 1ST AND 2ND ORDERS

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

general solution, particular solutions, initial

and boundary value problems. Motivating examples.

4.2. Linear differential equations of 1st order.

4.2.1. Homogeneous equation.

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

4.3. Linear differential equations of 2nd order with constant coefficients.

4.3.1. Homogeneous equation and corresponding characteristic equation.

4.3.2. Non-homogeneous (or complete) equation : Lagrange's method or method of variation of arbitrary constants.

5. THE BASIC OF FINANCIAL MATHEMATICS

5.1. Simple, compound and continuously interest compound.

5.2. 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).

6. NUMERICAL ANALYSIS

6.1. Auxiliary of numerical calculation.

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

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

6.4. Numerical differentiation.

6.5. Numerical integration: trapezoidal rule. 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

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

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

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

**Method of interaction**

The theoretical lectures are expository and it prevail a strong interaction between concepts and their practical application. The practical classes are designed to consolidate the knowledge acquired by means of resolution and discussion exercises.

**Software used in class**

Not applicable.