Base Knowledge
Differential and integral calculus of real functions of a real variable.
Teaching Methodologies
The lectures are intended to provide all the mathematical concepts described in the syllabus with examples where the concepts are applied. In the practical classes exercises are solved in order to discuss the topics presented in the lectures. The software Mathematica, Wolfram Alpha and Moodle are used for further treatment of the subjects studied. There is a final exam (100%).
Learning Results
Knowledge and interpretation of the concepts of ordinary differential equation, numerical series and power series, real function of several variables, directional derivative and multiple integral. Ability to apply this knowledge in solving problems. Development of critical thinking and reasoning skills.
Program
1. Introduction to ordinary differential equations: First-order differential equations – First-order linear differential equation, Bernoulli equation, separable equation and homogeneous equation. 2. Differential and integral calculus in IR^n: (a) Scalar fields – Notions of topology. Differentiable functions. A sufficient condition for differentiability. Directional derivatives and tangent plane. Chain rule. Geometric relation of the directional derivative to the gradient vector. Maxima, minima and saddle points. Extrema with constraints and Lagranges multipliers. (b) Integral calculus in Rn: Double integrals. 3. Infinite series: Sequences of real numbers. Infinite series. Geometric series and Telescoping series. Necessary condition for convergence. Series of nonnegative terms. Tests for convergence. 4. Real power series: Radius and interval of convergence. Properties of functions represented by power series. The Taylor series generated by a function. Power series expansions.
Curricular Unit Teachers
Grading Methods
- - Exame - 100.0%
Internship(s)
NAO
Bibliography
Guidorizzi , H.L., Um curso de cálculo, vol. 1 and vol. 2., , LTC, 2011
Anton, H., Cálculo – um novo horizonte, vol. 1 and vol. 2, Bookman, 2007
Azenha, A., Jerónimo, M.A., Cálculo diferencial e integral em R e Rn, McGraw-Hill, 1995
Apostol, T.M., Calculus, vol. I and vol. II, John Wiley, 1969
Ferreira, M.F., Equações diferenciais ordinárias, McGrawHill, 1995
Rodrigues, R.C., Notas teóricas e exercícios de análise matemática, DFM, ISEC
Abbott, S., Understanding analysis, Springer, 2002