Mathematics I

Code	12914
Year	1
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	none
Mode of delivery	Attendance
Work placements	Not applicable
Learning outcomes	Students should acquire Calculus concepts (both in R and R^n) in order to formulate and solve Economics and Management problems. Differentiate basic Topology concepts in R Characterize and interpret real functions of one real variable Determine and interpret composition and inverse functions Define, determine and interpret limits, continuity and differentiability of real functions of one real variable Compute and interpret derivatives of composition and inverse functions Determine extreme values of real functions of one real variable Determine primitives: immediate, by parts, by substitution and of rational functions Geometrically interpret the Fundamental Theorem of Calculus and Barrow’s rule Graphically represent domains of real functions of two real variables Define, determine and interpret limits, continuity and differentiability of real functions of several real variables Determine and interpret partial and directional derivatives and also the Jacobian matrix Apply and interpret the chain rule and the implicit function theorem Determine and interpret second order derivatives and the Hessian matrix
Syllabus	1. Real functions of one real variable 1.1 Topology basic concepts in R 1.2 Basic concepts of real functions of one real variable 1.3 Examples 1.4 Composition and inverse functions 1.5 Limits and continuity 1.6 Derivatives: definition and geometric interpretation. Differentiability 1.7 Derivatives of composition and inverse functions 1.8 Optimization 2. Functions from R^n to R^m 2.1 Topology basic concepts in R^n 2.2 Domains and geometrical representation 2.3 Limits and continuity 2.4 Partial and directional; Jacobian matrix 2.5 Composition function derivative. Chain rule 2.6 Implicit function theorem 2.7 Differentiability and tangent plane 2.8 Second order derivatives. Schwarz’s theorem. Hessian matrix 3. Primitives and Integral Calculus in R 3.1 Immediate primitives 3.2 Primitives by parts 3.3 Primitives by substitution 3.4 Rational function primitives 3.5 Riemann’s integral geometric interpretation 3.6 Integral Calculus Fundamental Theorem; Barrow’s rule 3.7 Applications
Main Bibliography	Recommended Pires, Cesaltina, Cálculo para Economistas, McGraw-Hill, 2001; Stewart, James, Cálculo, Volumes I e II, Thomson Learning, 2001; Additional references Abreu, Miguel, Fernandes, Rui L. e Ricou, Manuel, Folhas de Cálculo Diferencial e Integral I, Departamento de Matemática do IST, 2008; e Silva, Jaime C., Princípios de Análise Matemática Aplicada, McGraw-Hill, 1994; Dias Agudo, F. R., Análise Real, Escolar Editora, 2ª edição, 1994; Ferreira, J. C., Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1997;
Language	Portuguese. Tutorial support is available in English.