Subject: Mathematics II

Scientific Area:



80 Hours

Number of ECTS:




Overall objectives:

0 - Equip the students with theoretical and practical knowledge in Mathematics which are necessary for general and academic purposes, thus giving continuity to the work developed at curricular unit of Mathematics I.
1 - Determine conditioned extremes of real functions of vector variable using the method of Lagrange Multipliers and the method of Karush-Kuhn-Tucker.
2 - Learning how to evaluate integrals of functions of one real variable using techniques such as immediate (or almost immediate) integration, integration by parts, by substitution and the simplification of rational functions. Apply the concept of integral in order to solve some practical problems (e.g. calculus of areas). Learning how to solve some improper integrals.
3 - Identify and solve different types of ordinary differential equations applying different methods (e.g. variable separation, variable change, variation of parameters). Know how to solve some systems of linear differential equations.


1 - Constrained optimization
1.1 - The Method of Lagrange Multipliers
1.2 - The Method of Karush-Kuhn-Tucker
2 - Primitives of real functions of one real variable
2.1 - Immediate (or almost immediate) integration
2.2 - Integration by parts
2.3 - Integration by substitution
2.4 - Integration of rational functions
2.5 - Definite integral
2.6 - The Fundamental Theorem of Calculus
2.7 - Applications of Integrals
2.8 - Improper Integrals
3 - Ordinary Differential Equations
3.1 - First-order differential equations
3.2 - Linear differential equations of order 2 and higher
3.3 - Systems of linear differential equations of 1st order


Apostol Tom M. , 1983 , Cálculo , Editora Reverté Ltda
Ferreira, J. Campos , 1999 , Introdução à Análise Matemática , Fundação Calouste Gulbenkian
Piskounov, N. , 1990 , Cálculo Diferencial e Integral , Editora Lopes da Silva
McCann, Roger C. , 1982 , Introduction to ordinary differential equations , Harcourt Brace Jovanovich, New York
Stewart J. , 2008 , Calculus , Thomson Brooks/Cole

Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
Use of the whiteboard for presentation, explanation of subject material and problem solving. Exposed material should be provided to students, in detail, through a textbook or other supporting documents. Evaluation: two tests (one at the middle and one at the end of the semester), each one with a weight of 50% on the final grade.