MATHEMATICAL ANALYSIS 1
Learning outcomes of the course unit
The course will introduce the basic concepts of Mathematical Analysis. Besides providing computing tools, classes will be aimed at stimulating a deeper understanding of the ideas and methods of mathematical thinking. The main goals are: 1) improving the skills of the student in learning new mathematical tools; 2) improving the ability of the students to apply the newly learned theoretical notions in the solution of problems and exercises;
3) improving to student's skill in communicating with clarity and precision the mathematical concepts acquired during the course.
Basic pre-university mathematics
Course contents summary
The course will introduce the basic concepts of Mathematica Analysis and will cover the following topics:
-Review of some basic notions of pre-university mathematics
-Limits of functions and connections with the notion of Continuity
-Derivatives and applications to the study of the qualitative behavior of functions
- Basic concepts of set theory
- Rational numbers
- Tha axiom of continuity and the properties of real numbers
- inf and sup of a set of real numbers
- Functions: definitions and examples.
- Limits of functions.
- Properties of the limits.
- Continuous functions and their application to the computation of limits.
- The fundamental properties of continuous functions: Weiestrass Theorem and the Intermediate Value Theorem.
- The concept of derivative: heuristics, geometric motivation and rigorous definition.
- Examples of non-differentiable functions.
- Differentiability implies continuity.
- Rules for the computation of derivatives.
- Search of minimum and maximum points of a differentiable function: Fermat's Theorem.
- Rolle's and Lagranges's Theorems and their important consequences on the study of the monotonicity properties of functions.
- -Convexity/concavity and second derivatives.
- Qualitative study of functions.
- The notion of antiderivative.
-Main rules for the computations of antiderivatives: change of variables and integration by parts.
- The notion of integral
- The Fundamental Theorem of Integral Calculus.
-Examples and applications.
1) E. Acerbi, G. Buttazzo: "Matematica preuniversitaria di base". Pitagora Editrice, Bologna.
2) E. Acerbi, G. Buttazzo: "Analisi Matematica ABC". Pitagora Editrice, Bologna.
3) A. Guerraggio: "Matematica per le Scienze". Editor: Pearson
Lectures at the blackboard and homework assignments aiming at stimulating "active" learning of the concepts taught in class.
Assessment methods and criteria