# MATHEMATICS 1 AND EXERCISES

## Learning outcomes of the course unit

The aim of the course is to provide students with the basic tools of Mathematical Analysis and Linear Algebra.

## Course contents summary

1. Real numbers.

Maximum, minimum, least upper and greatest lower bound; integer part and modulus of real numbers; powers, roots, nth roots of non-negative numbers; rational and irrational numbers; intervals, distance. Complex numbers. The principle of induction.

2. An overview of linear algebra.

Vector spaces, linearly independent vecors, basis; matrix, determinant; linear operators; systems of linear equations. Lines and planes in the space.

3. Functions.

One to one, surjective and bijective functions; inverse functions; graphs; monotone functions; exponential and logarithmic functions; trigonometric functions.

4. Sequences and series.

Limits of sequences. Series with positive terms; criteria for their convergence.

5. Limits.

Limits of functions with real values, uniqueness of the limit, limits of the restrictions; limit of the sum, product, quotient of two functions; permanence of the sign, comparison theorems; right and left limit; limits of monotone functions.

6. Continuous functions.

Continuity of real functions of a real variable, restrictions of continuous functions, composition of continuous functions; sum, product, quotient of continuous functions; examples of continuous functions; discontinuity, examples of discontinuous functions; zeroes of continuous functions defined in an interval; continuity and monotonicity; continuity of inverse functions; Weierstrass theorem.

7. Differential calculus.

Incremental ratio, derivatives, right and left derivatives; geometrical significance of the derivative; derivation rules: derivatives of the sum, product, quotient of two functions; derivatives of composite functions and inverse functions; derivatives of elementary functions; delative maxima and minima; stationary points; connection between the monotonicity and the sign of the derivative; Rolle's theorem and Lagrange's theorem and their geometrical interpretation, Cauchy's theorem and de l'Hopital's theorem; convex functions, derivatives of convex functions, connection between the convexity and the sign of the second order derivative; study of local maxima and minima via the study of the derivatives.

8. Integrals.

Partitions of an interval; upper and lower integral, Integrability of continuous functions; geometrical interpretation of the integral; properties of integrals; mean of an integrable function; integrals on oriented intervals; fundamental theorem of integral calculus; primitives, indefinite integrals; integration by parts and by substitution; integrals of rational functions.

9. Ordinary differential equations.

Separable differential equations; first-order linear differential equations with variable coefficients; linear differential equations of order n with constant coefficients.

## Course contents

1. Real numbers.

Maximum, minimum, least upper and greatest lower bound; integer part and modulus of real numbers; powers, roots, nth roots of non-negative numbers; rational and irrational numbers; intervals, distance. Complex numbers. The principle of induction.

2. An overview of linear algebra.

Vector spaces, linearly independent vecors, basis; matrix, determinant; linear operators; systems of linear equations. Lines and planes in the space.

3. Functions.

One to one, surjective and bijective functions; inverse functions; graphs; monotone functions; exponential and logarithmic functions; trigonometric functions.

4. Sequences and series.

Limits of sequences. Series with positive terms; criteria for their convergence.

5. Limits.

Limits of functions with real values, uniqueness of the limit, limits of the restrictions; limit of the sum, product, quotient of two functions; permanence of the sign, comparison theorems; right and left limit; limits of monotone functions.

6. Continuous functions.

Continuity of real functions of a real variable, restrictions of continuous functions, composition of continuous functions; sum, product, quotient of continuous functions; examples of continuous functions; discontinuity, examples of discontinuous functions; zeroes of continuous functions defined in an interval; continuity and monotonicity; continuity of inverse functions; Weierstrass theorem.

7. Differential calculus.

Incremental ratio, derivatives, right and left derivatives; geometrical significance of the derivative; derivation rules: derivatives of the sum, product, quotient of two functions; derivatives of composite functions and inverse functions; derivatives of elementary functions; delative maxima and minima; stationary points; connection between the monotonicity and the sign of the derivative; Rolle's theorem and Lagrange's theorem and their geometrical interpretation, Cauchy's theorem and de l'Hopital's theorem; convex functions, derivatives of convex functions, connection between the convexity and the sign of the second order derivative; study of local maxima and minima via the study of the derivatives.

8. Integrals.

Partitions of an interval; upper and lower integral, Integrability of continuous functions; geometrical interpretation of the integral; properties of integrals; mean of an integrable function; integrals on oriented intervals; fundamental theorem of integral calculus; primitives, indefinite integrals; integration by parts and by substitution; integrals of rational functions.

9. Ordinary differential equations.

Separable differential equations; first-order linear differential equations with variable coefficients; linear differential equations of order n with constant coefficients.

## Recommended readings

M. Bramanti, C.D. Pagani, S. Salsa, Matematica: calcolo infinitesimale e algebra lineare. Seconda edizione. Zanichelli, 2004

## Teaching methods

Lectures and classroom exercises. During the lectures the basic results of the calculus for functions of one variable will be analyzed and discussed. Students will be provided also with the basic results on sequences of real numbers and numerical series and with the basic knowledges of ordinary differential equations.

The exercises class are aimed at showing how and where the abstract results can be applied to make the students understand better the relevance of what they are studying

## Assessment methods and criteria

The exam consists of two parts: a written part and an oral part in different dates. The written part is based on some exercises and it is aimed at evaluating the skills of the student

in applying the abstract results

proposed during the course to some concrete situations. The oral part is aimed at evaluating the knowledge of the abstract results seen during the course and their proofs.