Mathematical Analysis I

High school mathematics knowledge.

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

Differential and integral calculus of real functions of a real variable and its applications. Introduction to the study of differential equations. Topics are presented and discussed by solving exercises and using the following software: GeoGebra, WolframAlpha and MATLAB. Autonomous learning and rigor in the interpretation, use and description of mathematical concepts.

1. Real function of a real variable
Properties of real functions of a real variable, trigonometric functions and inverse trigonometric functions, exponential function and logarithmic function, hyperbolic functions.

2. Differential calculus
Properties of the derivative of real functions of a real variable, derivative of the composition function and the inverse function, undetermined forms and Cauchy’s rule, polynomial approximation: differentials and Taylor polynomial.

3. Primitive of real functions of real variable
Techniques for calculating the primitive function.

4. Integral calculus
Definite Integral of real functions of a real variable, properties, 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 length of the arc of a curve; improper integrals at unlimited intervals.

5. Introduction to the study of ordinary differential equations
First order ordinary differential equations, first order linear equation and equation of separate variables.

6. Component of numerical analysis
Approximation and error in numerical calculation, numerical methods for solving non-linear equations: bisection method and Newton-Raphson method; polynomial interpolation, numerical integration methods: trapezoid rule and Simpson rule; numerical methods for differential equations: Euler’s method.

NAO