# Calculus II

## Objectives

The course develops the fundamental tools of differential calculus in R, that enable the mathematical formulation and study of models in Economics, Business and Finance.

## General characterization

##### Code

1302

##### Credits

7,5

##### Responsible teacher

Patrícia Xufre; Pedro Chaves

##### Hours

Weekly - Available soon

Total - Available soon

##### Teaching language

Portuguese and English

### Prerequisites

Mandatory Precedence:

- 1301. Calculus I

- 1303. Linear Algebra (highly recommended)

### Bibliography

Main references:

• Pires, C., Cálculo para Economia e Gestão, Escolar Editora, 2011. (PT)

• Simon, C.P., Blume, L., Mathematics for Economists, W.W. Norton & Company, Inc, 1994. (EN)

• Xufre, P., Silva, P., Mendes, D., Análise em ???, 1ª edição, Escolar Editora, 2017.

Other references:

• Dias Agudo, F.R., Análise Real, Livraria Escolar Editora, 2ª edição, 1994.

Excepto tópicos 6.5 e 6.6

• Azenha, A., Jerónimo, M.A., Elementos de Cálculo Diferencial e Integral em ? e

???, McGraw-Hill, 1995.

Excepto tópicos 6.5 e 6.6

• Campos Ferreira, J., Introdução à Análise em , Publicação electrónica (https://math.tecnico.ulisboa.pt/textos/iarn.pdf), DM, IST, 2003.

Excepto tópicos 6.5 e 6.6

• Sydsæter, K. et al., Further Mathematics for Economic Analysis, Prentice Hall, 2005.

### Teaching method

Theoretical classes; practical classes; resolution and proposal of problems and exercises; mini-tests, midterm test and final exam.

### Evaluation method

The final grade in normal season is calculated as follows:

Final Score = 0.4 × Average of the 2 best TI + 0.6 × EF

- Three Intermediate Tests (IT) - during the semester students will perform 3 intermediate tests of which only the grade of the 2 best will be considered for the calculation of the final grade*

- Final Exam (EF)- minimum score of 8.5 out of 20 values

In The Appeal/Special Season the final grade may correspond exclusively to the exam grade if the student expresses this interest in writing on the day of the exam. By default, it will be calculated identically to that used for Normal Season.

Improvements:

Situation 1 - Obtained approval of the discipline in another semester and regardless of the time of examination in which he takes the final exam:

- If the student performs any of the intermediate Tests, his final grade will be obtained through the following formula:

Final Score = 0.4 × Average of the 2 best TI + 0.6 × EF

- Otherwise, your final grade will match the grade obtained in the final exam:

Final Score = EF

Situation 2 - Obtains approval of the discipline in the normal season of this semester:

* For a student who does not justifiably take at least two of the three intermediate tests, the weighting assigned to the final exam grade will be 100%. The exam that these students will take will obviously be different from that of the other students, as it will include subjects not yet evaluated.

The final grade may correspond exclusively to the exam grade if the student expresses this interest in writing on the day of the exam. Otherwise, it will be calculated according to the formula:

Final Score = 0.4 × Average of the 2 best TI + 0.6 × EF

## Subject matter

a) The ??? Space

Notion of norm

Notion of distance

Short notions of topology

b) Functions from ?? ? ??? to ???

Examples in Economics/Management Domain

Particular case: ??: ?? ? ?2 ? ?; Grahical representation through level sets Limit of a function

Definition of limit

Limit of a function following a specific trajectory Some important properties

Continuity: main results

c) Derivation in ???

1st order partial derivatives (gradient vector and jacobian matrix) Higher order derivatives (hessean matrix)

Directional derivative Differentiability

Main properties of differentiable functions Symmetry of 2nd order derivatives

Derivative of the composite function Homogeneous function

Economic examples

Euler's theorem

Homogeneity of the partial derivatives

d) Taylor's Formula

Finite increment's theorem

Taylor's theorem and MacLaurin's formula

e) Inverse Function Theorem and Implicit Function Theorem

Functions invertibility in ???

Implicit functions; Economic examples

f) Optimization

Some basic concepts

Convex sets; Convex functions Unconstrained optimization

Optimization with equality constraints; Method of Lagrange multipliers Envelope's theorem

Optimization with inequality constraints; Karush-Kuhn-Tucker conditions