# PRINCIPLES OF MATHEMATICS

## Learning outcomes of the course unit

Knowledge and understanding of the basic concepts and tools of calculus and statistics.

Ability to use such tools to study graphs of functions and solve problems related with the study of functions.

Ability to apply such tools to the study of experiments (to create models and/or provide the correct interpretations of the data).

## Prerequisites

Knowledge of basic mathematics.

## Course contents summary

Basic notions of set theory and

mathematical logic. Real numbers. Real

functions of a real variable and their

properties. Limits, continuity,

differentiability, and Riemann integrability.

ODE. Brief introduction to Probability and

Statistics.

## Course contents

Naturals, integers, rationals and reals: : definition, operations and their properties, powers with rational exponent. Intervals and neighbourhoods in the real line. Equations, inequalities and absolute value.

Funtions: definition, domain, codomain and image. Injectivity, surjectivity, bijectivity. Inverse of a bijective function. Composition of functions: definition and properties, properties and unicity of the inverse function. Graph of a function. Algebraic (polynomial) functions): linear, quadratic, monomial functions, data interpolation, even and odd functions. Monotonic funstions: (not) increasing, (not) decreasing, inveritibility of a monotonic function. Supremum, infimum, maximum and minimum of a subset of R, bounded functions, (local and global) maxima and minima of a real function.

Limits and continuity: definition, right and left limit, properties and limit calculus, indeterminated forms. Continuity of a real function: definition, continuity of polynomial and rational functions, vertical and horizontal asimptotes. Theorems on limits and continuity: continuity of composition, Weierstrass, existence of zeroes, permanence of sign, mid-value theorem, invertibility of continue functions, continuity of inverse function.

Trascendental functions: exponential, logaritmic and trigonometric functions: definition, properties (injectivity, surjectivity, continuity, invertibility), data interpolation.

Differential calculus: derivative of a function, right and left derivative, derivability and continuity, derivatives of algebraic and trascendental functions. Derivative calculus: derivative for sum, product, composition and quotient and for the inverse function. Theorems of Fermat, Rolle, Cauchy and Lagrange. Applications: rule of de l'Hopital and limit calculus, Taylor development, qualitative study of the graph of a function, monotonicity, maxima and minima.

Antiderivative calculus: definition, properties, additivity and linearity. Antiderivatives formulas for algebraic and trascendental functions. Fundamental theorems of calculus, mean value theorem. Integration methods: by parts and by substitution.

Statistics: mean, variance, distributions., regression line, Pearson coefficient, T test and chi-square test.

## Recommended readings

Abate M. "Matematica e statistica - Le basi per le scienze della vita" Mc Graw Hill Ed.

## Teaching methods

During the lectures we shall describe the basic objects and tools of calculus and statistics,

together with numerous examples and exercise to improve the understanding and to show how to apply them to the resolution of numerical exercises. There will be weekly exercise sessions to recall some basic notions and to provide more examples and exercises.

Students with difficulties (OFA) will have additional lectures.

## Assessment methods and criteria

Test to check basic knowledge: 25m, 6 multiple choice questions.

Written exam to check the ability to apply the basic tools of calculus to the solution of concrete numerical exercises: 2h, exercises.

(Non-compulsory) oral exam to verify the knowledge and understanding of the basic definitions and theorems of calculus.