Mathematical Analysis I

Mathematics knowledge from secondary education.

The course includes lectures, theoretical-practical and practical classes, with interactive presentation of the contents to encourage active student participation. Exercises are solved individually or in small groups, using digital resources and specific software. The use of a personal computer is recommended.

Study of differential and integral calculus of real-valued functions of one real variable and its applications. Introduction to the study of differential equations.
Development of the ability to interpret and apply the concepts studied in solving exercises and problems, using, whenever appropriate, software such as GeoGebra, WolframAlpha and MATLAB. The course also aims to promote rigour in the interpretation, use and description of mathematical concepts, as well as the ability to analyse and solve problems using computational tools.

1. Real-valued functions of one real variable
Properties of real-valued functions of one real variable; trigonometric and inverse trigonometric functions; exponential and logarithmic function; hyperbolic functions.

2. Differential calculus
Properties of the derivative of real-valued functions of one real variable; derivative of composite and inverse functions; indeterminate forms and Cauchy’s rule; polynomial approximation: increments and differentials and Taylor polynomials.

3. Primitives of real-valued functions of one real variable
Direct integration and integration by decomposition; Integration methods: integration by parts, integration of rational functions and integration by substitution.

4. Integral calculus
Definite Integrals of real-valued functions of one real variable; properties; the fundamental theorem of calculus; integration by parts and by substitution; applications of the definite integral: area of a plane region, volume of a solid of revolution and arc length; improper integrals over unbounded intervals.

5. Introduction to the study of ordinary differential equations
First-order ordinary differential equations. First-order linear equations and separable equations.

6. Numerical analysis component
Approximation and error in numerical computation; numerical methods for solving nonlinear equations: the bisection method and Newton-Raphson method; polynomial interpolation; numerical integration methods methods: the trapezoid rule and Simpson´s rule; numerical methods for differential equations: Euler’s method.

Assessment by examination: consists of a single examination, worth 20 marks, to be taken during either the normal examination period or the resit examination period.

Marks equal to or higher than 16 may require the completion of a supplementary written examination to defend the mark awarded.

Complementary oral examinations may be used as an additional assessment tool whenever considered necessary to clarify doubts or investigate situations that may give rise to suspicions of fraud or plagiarism.

The assessment methods described also apply to students with working-student status.

NAO