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