Sumários

Aula 29

27 Abril 2023, 08:00 • Rita Sousa

- Realização do MT3

PRÁTICA

Resolução dos seguintes exercícios da PUC - Semana 10:

- Ex. 1 e 3 (alínea 12)

BIBLIOGRAFIA

Secção 14.7 do livro James Stewart, "Cálculo, Volume 2", Tradução da 7a Edição Norte Americana, 2013, Cengage Learning.

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27 Abril 2023, 08:00 • Rita Sousa

Mini-test 3.
Maximum and minimum values of functions of two variables.
Second derivatives test.
Resolution of exercise 2 from the CUP (week 10).

Aula TP 28

26 Abril 2023, 09:30 • Rita Sousa

Otimização em R^n.
Valores máximo e mínimo de funções reais de duas variáveis reais.
Exemplos.
Teste da segunda derivada.
Exemplos e resolução do exercício 1 da PUC (semana 10).

Lecture 28

26 Abril 2023, 09:30 • Rita Sousa

THEORY

- Introduction to Optimization in IR^2 (unconstrained extrema):

- Definition of local maximum, local minimum, global maximum and global minimum of a real function of two real variables at a point;

- Definition of local maximum value, local minimum value, global maximum value and global minimum value;

- Definition of local extremum of a real function of two real variables at a point;

- Fermat's theorem for local extremum;

- A cautionary example where the reciprocal implication of Fermat's theorem fails: definition of saddle point;

- Critical points of a real function of two real variables;

- The Hessian matrix (at a point) of a real function of two real variables;

- The Hessian matrix as a symmetric matrix when the function and the point are both under Clairaut's theorem;

- The classification of critical points using the determinant of the Hessian matrix at those points: the Second Derivatives Test;

PRACTICE

The following exercises from the CUP - Week 10 were discussed:

- Ex. 7, items (a), (b) and (c) (discussing the general idea)

BIBLIOGRAPHY

Section 14.7 of James Stewart, "Calculus, Early Transcendentals", International Metric Edition, 8th Edition, 2017, Cengage Learning.

Complete the resolution of exercise 8 from the CUP of week 9.
Otimization in R^n.
Maximum and minimum values of functions of two variables.
Examples.
Second derivatives test.
Examples and resolution of exercise 1 from the CUP (week 10).