# FUNDAMENTALS OF CALCULUS AND GEOMETRY

## Learning outcomes of the course unit

The objectives of the course are not limited to the simple acquisition of mathematical tools, but we want to emphasize a deeper critical understanding of the ideas and of the thinking attitude. At the end of the course the student must therefore have acquired basic knowledge and skills in mathematics, starting from the structure of the Euclidean space, up to the differential and integral calculus for functions of one real variable. At the same time he will be able to apply such knowledge in a critical way to various concrete problems, in a solid way, and to handle them easily in relation to other areas of knowledge. In particular, the student must be able to:

be familiar with the basic theories of Linear Algebra and of Geometry and apply them to the manipulation of vectors and matrices in Euclidean space, to the computation of determinants, to the resolution of linear systems and simple exercises of linear geometry in space that in particular concern plans and lines;

be familiar with the structure of sets of real and rational numbers and with basic concepts of integro-differential calculus for functions of one variable (limits, derivatives, definite and indefinite integrals);

be able to qualitatively study problems such as the behavior of a function for certain values of the independent variable; (knowledge and ability to understand)

through the exercises carried out in class on the topics of the program, learn how to apply the abstract knowledge acquired to simple and concrete cases and, only in the second part of the course, be able to connect different concepts in order to solve complex exercises in an independent way;

use the mathematical method to break down complex problems into more easily attackable sub-problems; (ability to apply knowledge and understanding)

evaluate the consistency and correctness of the results obtained and analyze the appropriate resolution strategies for the proposed exercises; (autonomy of judgment)

learn how to use a formally correct language allowing to communicate both the contents of the program and the logical steps used in the resolution of the exercises, showing clarity of exposure and thought. Frontal lectures and direct comparisons with the teacher will ease the acquisition of a specific and appropriate scientific vocabulary; (communication skills)

autonomously deepen their knowledge; starting from the basic tools provided in the course, learn how to appropriately and effectively use additional tools and mathematical concepts. These will be important in the remaining courses of the Degree. (learning ability)

## Prerequisites

There are no prerequisites; a certain familiarity with basic pre-university mathematical concepts (operations, equations and algebraic inequalities, properties of powers, trigonometry) is necessary, but will be taken up during the course.

## Course contents summary

The course aims at providing the students with the basic elements of mathematics they can use in the subsequent technical/scientific courses. In particular we want to present an introduction to different basic aspects of linear Algebra, Euclidean Geometry and Mathematical Analysis. The first part will be used to review different concepts of pre-university mathematics and to introduce the real numbers, the numerical structure on which the rest of the course is based. The second part will focus on Euclidean Geometry in space (conic curves, vectors, lines and planes), matrices and linear systems. Moreover, we will study, with particular attention to the graphic aspect, vector subspaces of R^3 and linear applications. The third and last part will introduce the basic concepts of Mathematical Analysis, with particular emphasis on continuous and differentiable functions, qualitative study of their graphs and integral calculus.

## Course contents

Introductory part:

Elementary set theory. Real numbers with operations. Structure of R: intervals, minorants, majorants, infimum and supremum. Functions: injectivity, surjectivity, invertibility; composition. Functions of one real variable: monotonicity; definition and properties of the absolute value function; triangle inequality.

Linear geometry part:

Cartesian plane. Lines and conics in the plane.

Vectors in space, coordinates. Operations between vectors, scalar product. Length, distance, orthogonality, projection of a vector. Cauchy-Schwarz inequality; triangle inequality. Angle between vectors. Vector product in R^3. Hints of similar properties in the n-dimensional space R^n. Lines and plans. Orthogonality between lines and planes. Membership. Parallelism. Cartesian equations of a line. Matrices and operations (sum and product), with properties. Invertible matrices and inverse matrix. Transposed of a matrix. The determinant of a square matrix. Properties of the determinant. Rank of a matrix. Linear systems and matrices. Matrices and reduced systems. Solutions of a reduced system. Solutions of linear systems: Rouché-Capelli theorem and Gauss method. Vector subspaces (in R^n). Linear combinations and spanned spaces. Linear dependence and independence. Dimension of a subspace. Vector subspaces of R^3. Linear applications. Image and kernel of a linear application.

Analysis part:

Infimum and supremum, maximum and minimum of a function.Limits for functions: heuristic motivation, rigorous definition. Algebraic properties: theorems of sum, product and ratio. Indeterminate forms and speed of infinities.Continuous functions. Sum, product, ratio and composition of continuous functions. Zeros, mean value and Weierstrass Theorems.Derivative of a function, right and left. Relation between continuity and derivability; examples of non-derivable functions. Derivatives of elementary functions. Rules of derivation. Seek for local extremals of a function: Fermat's theorem. Theorems of Rolle and Lagrange.Relation between the monotonicity of a function and the sign of its derivative. Convexity of a function. Studies of graphs.Notion of primitive of a function and indefinite integral. Elementary integrals. Integration by substitution. Definite integral. Fundamental Theorem of the integral calculus and relation with the indefinite integral.

## Recommended readings

The material from the lectures will be largely sufficient to face the exam with complete success. As a complement to the course, we recommend the following books:

E. Acerbi, G. Buttazzo: Matematica preuniversitaria di base, Pitagora Editrice, Bologna.

L. Alessandrini, L. Nicolodi: Geometria A, Ed. Uninova, Parma.

A. Guerraggio: Matematica per le Scienze, Pearson Editore.

E. Acerbi, G. Buttazzo: Analisi Matematica ABC, Pitagora Editrice, Bologna.

Each of them deals with different parts of the program; please contact the teacher for suggestions and advice.

## Teaching methods

The course consists of 8 credits corresponding to 80 hours of frontal teaching. The preferred method is the frontal lesson at the blackboard, in which the topics will be proposed from a formal point of view, alternating them with significant examples, applications and exercises. The course will give particular emphasis to the application and calculation aspects, while not neglecting a rigorous theoretical treatment that will not be an end in itself, but it will be aimed at a deeper understanding of the phenomena involved. In order to promote the systematic, deep and concrete understanding of the topics, booklets will be distributed on the elly portal with exercises to be carried out by the student in parallel with the study of the theoretical arguments. On a weekly basis, the detailed program of the topics presented in the classroom will also be uploaded, in support of both attending and non-attending students. This program will eventually constitute the contents index in preparation for the final exam.

## Assessment methods and criteria

The verification of the knowledge will take place through a written test. In order to ease the student, at the end of the geometry part a partial written test will be held and its vote will contribute, through a weighted average, to the final grade of the exam. The analysis part will be verified in a second partial written test in the occasion of the dates of the exams of the spring session. Partial tests will be considered passed if the student has obtained a grade larger or equal than 16. In this written test, through the proposed exercises and simple theoretical questions, the student must show to possess the basic theoretical and practical knowledge related to the course.