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Mathematics I

Code 12078
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 Stewart, James, (2016), Calculus, (EUA: Cengage Learning)
Sydsaeter, Knut; Hammond, Peter & Strom, Arne, (2012), Essential Mathematics for Economic Analysis, (RU: Pearson Education Limited).
Bradley, Teresa & Patton, Paul (2003), Essential Mathematics for Economics and Business, (EUA: John Wiley and Sons)
Pires, Cesaltina, (2010), Cálculo para Economia e Gestão, Escolar Editora.
Teaching Methodologies and Assessment Criteria Final grade: arithmetic mean grade of two written partial tests.

Each partial written test: 0-20.

Dates of the assessment
1st test: November 22th
2nd test: to be announced

Language Portuguese. Tutorial support is available in English.
Last updated on: 2021-10-10

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