Base Knowledge
All the mathematical knowledge from the 10th, 11th and 12th grades.
Teaching Methodologies
The theoretical classes are essentially expository and cover the first five chapters of the unit, while the practical classes are intended for students to discuss and solve exercises on topics taught in theory, under the guidance of the teacher.
Learning Results
The unit of Mathematical Analysis I aims to provide students with knowledge of differential calculus and integral calculus in IR that will be essential to understand the subjects taught in the remaining units of the degree, in particular, it is intended that students clearly assimilate and interpret the concepts of derivative and integral and apply them in understanding, solving and analyzing the results of problems in engineering.
Program
1. Real functions of one real variable – Elementary functions: exponentials; logarithms; trigonometric functions and their inverses. Hyperbolic functions and their inverses.
2. Differential calculus in IR – Derivatives. Rolle, Lagrange and Cauchy theorems. Indeterminations and Cauchy’s rule. Additions and differentials. Taylor polynomial and rest of Lagrange.
3. Primitivation (antiderivative) – Definition of primitive and properties. Primitivation techniques.
4. Integral calculus in IR – Definition of Riemann integral and properties. Fundamental theorem of integral calculus. Applications to the calculation of areas, length of curves and volumes of revolution solids. Indefinite integral and improper integrals.
5. Introduction to the study of ordinary differential equations – Definition of ordinary differential equation. Cauchy’s problem. Methods for solving ordinary differential equations: separable variables; linear of 1st order; homogeneous of degree zero; Bernoulli.
6. Component of numerical methods – Introduction to error theory. Numerical methods for solving nonlinear equations. Polynomial interpolation. Numerical integration. Numerical methods for solving ordinary differential equations.
Curricular Unit Teachers
Internship(s)
NAO
Bibliography
SANTOS J. P. (2016). Cálculo numa variável real. Lisboa: IST Press (available in the library of ISEC: 3-2-90).
MURTEIRA J. & SARAIVA P. (2010). Equações diferenciais ordinárias: introdução teórica, exercícios e aplicações. Coimbra: Almedina (available in the library of ISEC: 3-11-70).
LARSON, R. E., HOSTETLER, R. P. & EDWARDS, B. H. (1998). Cálculo com geometria analítica – Volume 1. Rio de Janeiro: Livros Técnicos e Científicos (available in the library of ISEC: 3-2-244/5/8/9).
CHAPRA S. C. & CANALE R. C. (2008). Métodos numéricos para engenharia. São Paulo: McGraw-Hill (available in the library of ISEC: 3-4-118).