Enrico Giusti, Elementi di Analisi Matematica, 2008, Bollati Boringhieri.
Learning Objectives - Last names A-C
The students will learn the basic elements of the scientific method and of calculus: limits, continuity, and derivatives.
The aim of this course is to provide the necessary tools for developing and studying mathmatical methods using functions of a single variable.
The students will be able to develop a complicated logical-mathematical argument through elementary steps.
Students will be able to appreciate the importance of assumptions in theorems.
Students will be able to define and apply the basic ideas of calculus as limits and derivatives.
Students will be able to solve equations approximately, using appropriate algorithms.
Students will be ablve to descrive how the problems of calcolusm arise from economic, or social, or political, or environments phenomema.
Students will be able to model these problems in mathematical terms.
Students will be able to solve these problems and obtain results which will improve their understanding of reality.
Learning Objectives - Last names M-P
The goal of this course is to provide mathematical tools which allow to build and understand simple economic models which rely on real functions of one real variable.
Learning Objectives - Last names Q-Z
The goal of this course is to provide mathematical tools which allow to build and understand simple economic models which rely on real functions of one real variable.
Prerequisites - Last names M-P
The natural numbers. The integers. The rationals. An intuitive idea of real numbers. Arithmetic operations and their properties. Percentages. Prime numbers. Factorization of a natural number. Greatest common divisor and least common multiple. Inequalities and manipulation of inequalities. Absolute value. Powers, roots and their properties. Simplifying expressions.
Polynomials. Special products. How to factor polynomial into irreducible terms in simple cases. Polynomial identity. Ratios of polynomials. Identities and equations. Solutions for an equation. Equations of degree one or degree two. Solutions (roots) of a polynomial. Equations containing ratios of polynomials. Equations with radicals. Inequalities. Solving inequalities of degree one and degree two, inequalities, with ratios of polynomials, with radicals.
Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems with two equations and two unknowns. Parallel lines and perpendicular lines. Equation of the parabola. Equation of a circle.
Prerequisites - Last names Q-Z
The natural numbers. The integers. The rationals. An intuitive idea of real numbers. Arithmetic operations and their properties. Prime numbers. Factorization of a natural number. Greatest common divisor and least common multiple. Inequalities and manipulation of inequalities. Absolute value. Powers, roots and their properties. Simplifying expressions.
Polynomials. Special products. How to factor polynomial into irreducible terms in simple cases. Polynomial identity. Ratios of polynomials. Identities and equations. Solutions for an equation. Equations of degree one or degree two. Solutions (roots) of a polynomial. Equations containing ratios of polynomials. Equations with radicals. Inequalities. Solving inequalities of degree one and degree two, inequalities, with ratios of polynomials, with radicals.
Cartesian coordinates in the plane. Pythagorean theorem. Distance between two points. Equation of the line. Linear systems with two equations and two unknowns. Parallel lines and perpendicular lines. Equation of the parabola. Equation of a circle.
Teaching Methods - Last names M-P
Class lectures. The course lenght is 12 weeks with three classes a week.
Teaching Methods - Last names Q-Z
Class lectures. The course lenght is 12 weeks with three classes a week.
Further information - Last names M-P
The course has an internet page on the platform Moodle, which provides further information on the course.
Further information - Last names Q-Z
The course has an internet page on the platform Moodle, which provides further information on the course.
Type of Assessment - Last names M-P
In order to pass the exam, the student needs to pass a written exam and then an oral exam.
Further information on the exam rules are available in the Regolamento on the Moodle page of the course.
Type of Assessment - Last names Q-Z
In order to pass the exam, the student needs to pass a written exam and then an oral exam.
Further information on the exam rules are available in the Regolamento on the Moodle page of the course.
Course program - Last names M-P
Set theory and introduction to mathematical reasoning.
Real numbers. Algebraic and order properties. Geometric representation of real numbers. The real line. Topology of the real line. Neighborhood of a point. Interior points and accumulation points. Bounded sets. Supremum and infimum of a set.
Functions. Domain, image and graph of a function. Injective and inverse functions. Composition of functions. Domain of the composite function. Monotone functions. Strict monotonicity and invertibility. Supremum and Infimum of a function. Maximum/minimum of a function and maximizers/minimizers. The exponential function. The logarithmic function. Power functions, absolute value, sine, cosine, tangent and piecewise defined functions.
Limit of a function. Uniqueness of the limit. Sign invariance theorem. Comparison or squeeze theorem. Left and right limits. Operations with limits: sum, product, quotient. Limit of composite functions (change of variables). Limits at infinity, infinite limits. Limits of rational functions, of logarithmic functions, and of the exponential functions. Indeterminate forms. Special limits.
Continuous functions. Definition and examples. Continuity of elementary functions. Operations with functions and continuity. The intermediate value theorem. Local maxima and minima and local minimizer/maximizer.
Weierstrass theorem.
Definition of derivative. Tangent line. Rules of differentiation for sum, product, quotient and composition. Fermat Theorem, Rolle and Lagrange Theorem, de l'Hospital Theorem. Sign of derivative and monotonicity. Concave and convex functions. Sign of second derivative and convexity. Sign of the second derivative as a sufficient condition for local maxima and minima. Taylor formula.
Course program - Last names Q-Z
Set theory and introduction to mathematical reasoning.
Real numbers. Algebraic and order properties. Geometric representation of real numbers. The real line. Topology of the real line. Neighborhood of a point. Interior points and accumulation points. Bounded sets. Supremum and infimum of a set.
Functions. Domain, image and graph of a function. Injective and inverse functions. Composition of functions. Domain of the composite function. Monotone functions. Strict monotonicity and invertibility. Supremum and Infimum of a function. Maximum/minimum of a function and maximizers/minimizers. The exponential function. The logarithmic function. Power functions, absolute value, sine, cosine, tangent and piecewise defined functions.
Limit of a function. Uniqueness of the limit. Sign invariance theorem. Comparison or squeeze theorem. Left and right limits. Operations with limits: sum, product, quotient. Limit of composite functions (change of variables). Limits at infinity, infinite limits. Limits of rational functions, of logarithmic functions, and of the exponential functions. Indeterminate forms. Special limits.
Continuous functions. Definition and examples. Continuity of elementary functions. Operations with functions and continuity. The intermediate value theorem. Local maxima and minima and local minimizer/maximizer.
Weierstrass theorem.
Definition of derivative. Tangent line. Rules of differentiation for sum, product, quotient and composition. Fermat Theorem, Rolle and Lagrange Theorem, de l'Hospital Theorem. Sign of derivative and monotonicity. Concave and convex functions. Sign of second derivative and convexity. Sign of the second derivative as a sufficient condition for local maxima and minima. Taylor formula.