Base Knowledge
Knowledge of the subject of Mathematics in secondary education.
Teaching Methodologies
The curricular unit has theoretical, theoretical-practical and practical classes. Theoretical classes take place in an essentially expository way, approaching the themes foreseen in the program, prevailing a strong interaction between the concepts and their concrete application. The theoretical-practical classes will be aimed at solving problems and practical cases under the guidance of the teacher. Practical classes are for solving problems such as practical cases. The exercises will be performed individually or in small groups. The teaching of the curricular unit is complemented by student service periods.
Learning Results
The teaching of Mathematics in general should facilitate mathematical communication, reflective thinking, the application of mathematical techniques to problem solving, critical analysis of the results obtained, and finally, interdisciplinarity. One of the teaching objectives of the 1st year Calculus I discipline is to provide the basic foundations of mathematical methods, usually applied in the areas of Engineering, used by the various disciplines of the Degree in Electrical Engineering.
It is intended that students develop abilities (skills) of algebraic manipulation and independent and analytical reasoning and the ability to apply mathematical concepts in solving practical problems.
Program
1. Trigonometry and real functions of real variable
Inverse trigonometric functions. Elementary Functions: Exponential, Logarithmic and Hyperbolic.
2. Differential Calculus
Limits and continuity. Derivative and its geometric interpretation. Rolle, Lagrange and Cauchy theorems. Cauchy’s rule. Derivation rules. Successive derivatives. Derived from the implicit function.
3. Primitivation
Definition and properties. Immediate Primitivation. Primitivation Methods: Primitive by decomposition, primitive by parts, primitive of rational functions and primitive by substitution.
4. Integral Calculus
Definite integral: Definitions and properties. Fundamental theorem of calculus; Fundamental results. Definite integral applications · Calculation of areas, volumes and arc lengths of curves. Indefinite integral: properties. Improper integrals: Integrals on unbounded intervals and integrals of unbounded functions. Application of Integral Calculus in solving some problems usually associated with Electrical Engineering.
5. ODE (1st order)
Definition of differential equation. Cauchy problem. Differential equations of separable and linear variables of order 1.
6. Numerical analysis component
Error Theory. Taylor formula with Lagrangian remainder · Approximation of functions by the Taylor Polynomial. Differential of a function. Linear approximation. Nonlinear equations. Bisection Method · Newton-Raphson method. Numerical Integration · Newton-Cotes formulas · Trapezoidal rule · Simpson’s rule.
Curricular Unit Teachers
Internship(s)
NAO
Bibliography
Cristina M.R. Caridade, Class notes, DFM, ISEC, 2019.
Cristina M.R. Caridade, Class slides, DFM, ISEC, 2019.
e-MAIO (Módulos de Aprendizagem Interativa online) – https://dfmoodle.isec.pt/
Moodle ISEC – Análise Matemática I – http://moodle.isec.pt/
J. Stewart, Cálculo, Vol.I, 4ª ed. Pioneira, Thomson Learning, 2001.
A. Howard, Cálculo: um novo horizonte, 6 ª ed., Porto Alegre, Bookman, 2000.
R. Larson, R. P. Hostetler, B. H. Edwards, Cálculo, Vol. I, 8ª ed., McGraw Hill, 2006.
Pré-Cálculo e Introdução ao Cálculo, Departamento de Física e Matemática, Secção de texto do ISEC
E. W. Swokowsky, Cálculo com Geometria Analítica, Vol. 1, 2ª ed., Rio de Janeiro Makron Books, cop. 1995.
M. A. Saraiva, M. A. Silva, Primitivação, Edições Asa, 1993.
B. Demidovitch, Problemas e Exercícios de Análise Matemática, McGraw-Hill, 1993.
NEW bibliography – Cálculo I. Conceitos, Exercícios e Aplicações, Matlab. Ana Cristina Meira da Silva e Castro, Ana Júlia Viamonte, António Augusto Varejão Teixeira de Sousa. PUBLINDUSTRIA. 2013. ISBN: 9789897230547