Calculus II


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





Responsible teacher

Patrícia Xufre; Pedro Chaves


Weekly - Available soon

Total - Available soon

Teaching language

Portuguese and English


Mandatory Precedence:

- 1301. Calculus I

- 1303. Linear Algebra (highly recommended)


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 (, 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.


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


Programs where the course is taught: