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

A. Regular Season

 10%: Average grade of the 4 best Quizzes

30%: Midterm Test

60%: Exam (minimum grade of 8.5)

B. Regular Season with justified absence from the Midterm Test: 100% Exam (different from the Regular Exam)

C. Resit Season

Choosing continuous assessment before seeing the exam

10%: Average grade of the 4 best Quizzes

30%: Midterm Test

60%: Exam (minimum grade of 8.5)

Choosing only exam before seeing the exam: 100% Exam

D. Grade improvement from previous to current semester

Attending the Midterm Test

10%: Average grade of the 4 best Quizzes

30%: Midterm Test

60%: Exam (minimum grade of 8.5)

Not attending the Midterm Test: 100% Exam

E. Grade improvement from Regular to Resit Season

Choosing continuous assessment before seeing the exam

10%: Average grade of the 4 best Quizzes

30%: Midterm Test

60%: Exam (minimum grade of 8.5)

Choosing only exam before seeing the exam: 100% Exam

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

Programs

Programs where the course is taught: