## 21010183 - FUNDAMENTALS OF MATHEMATICS

To provide the conceptual and methodological tools for finding information transmitted by the formalized and deductive language of mathematics. To provide the fundamentals of mathematical analysis and plane geometry oriented towards the understanding of mathematical-physical models. Course topics are: the differential calculus in one variable and first hints of integral calculus; its concepts, tools and modeling instances; linear algebra analyzed from a geometrical point of view.

Canali

teacher profile | teaching materials

Programme

Quantifiers. Natural, integer, rational and real numbers. Axioms of real numbers. The square root of 2 is irrational.

Cartesian coordinates in the plane and in the space; coordinate planes. Points and vectors. Distance: formal definition. Absolute value. Density of Q in R. Distance in the plane and in the space; equations of circumference and sphere.

Linear algebra (in 2 and 3 dimensions): slope of a segment, vector sum, scalar and vector product. Equivalence of geometric and coordinate definitions of vectors. Orthogonality and parallelism conditions.

Introduction to functions. Graphic of a function in the coordinate planes. Operations with graphics.

Open and closed sets, accumulation points, definitions and examples. Limits. Operations with limits, examples of limits of quotients of polynomials. Asymptotes. Comparison theorem. List of relevant limits.

Continuous functions (continuity at a point and in an interval). Theorems on continuous functions: existence of maximum and minimum values, intermediate values. Exponential and logarithmic functions.

Derivatives: definition, geometric meaning. Operations with derivatives: sum, product, quotient, multiplication by a number. Main rules of derivation, derivatives of relevant functions. Equation of the tangent line at a point to the graph. Derivative of a composite function and inverse functions. Stationary points.

Fermat's theorem. Theorems of Rolle and Lagrange. Monotony and sign of the first derivative. Linear approximation.

Second derivatives, concavity, inflection points. Plotting graphs of functions. Theorems of Cauchy and De l'Hopital. Modeling problems and optimizations. Taylor polynomial. Lagrange form of the remainder for n=2.

Introduction to indefinite and definite integrals. The problem of calculating the area of a flat region. The theorem of the integral average. The fundamental theorem of integral calculus. Integration by parts and substitution.

Introduction to the use of computer software for plotting functions.

Core Documentation

James Stewart, Calcolo. Funzioni di una variabile, Apogeo Education - Maggioli Editore.

Robert A. Adams Calcolo Differenziale I ed. CEA (Casa Editrice Ambrosiana)

Bramanti, Pagani, Salsa “Analisi Matematica 1. Con elementi di geometria e algebra lineare”, Zanichelli

Reference Bibliography

James Stewart, Calcolo. Funzioni di una variabile, Apogeo Education - Maggioli Editore. Robert A. Adams Calcolo Differenziale I ed. CEA (Casa Editrice Ambrosiana) Bramanti, Pagani, Salsa “Analisi Matematica 1. Con elementi di geometria e algebra lineare”, Zanichelli Courant, Robbins "Che Cos' è La Matematica?" Ed. Boringhieri

Type of delivery of the course

The course is organized in lectures and exercise class. Lectures on different subjects starts with several examples then general cases followed by definitions, theorems and proofs. The subject is presented from a geometrical and analytical point of view with a modeling description. Part of the lectures is on exercises made by singular students or divided in small groups. During exercise classes we give some exercises and problems, let the students try to solve them then we discuss the solution and, if necessary, we give the full solution at the blackboard. Some of the exercise classes are dedicated to hands-on activities with the use of paper and other materials or a computer for visualization.

Attendance

Students must attend at the least the 75% of lessons to be admitted to the final exam

Type of evaluation

The student assessment involves a written and an oral exam. Some tests during the course are also planned. The written exam (1,5-2 hours) consists of 5 or 6 questions to assess students’ understanding of concepts and their autonomous application. Some past written exams are available at the course web page http://www.formulas.it/sito/corsi/istituzioni-di-matematiche-i/

teacher profile | teaching materials

Programme

Quantifiers. Natural, integers, rational and real numbers. Axioms of real numbers. The square root of 2 is irrational.

Cartesian coordinates in the plane and in the space; coordinate planes. Points and vectors. Distance: formal definition. Absolute value. Density of Q in R. Distance in the plane and in the space; equations of circumference and sphere.

Linear algebra (in 2 and 3 dimensions): slope of a segment, vector sum, scalar and vector product. Equivalence of geometric and coordinate definitions of vectors. Orthogonality and parallelism conditions.

Introduction to functions. Graphic of a function in the coordinate planes. Operations with graphics.

Open and closed sets, accumulation points, definitions and examples. Limits. Operations with limits, examples of limits of quotients of polynomials. Asymptotes. Comparison theorem. List of relevant limits.

Continuous functions (continuity in a point and in an interval). Theorems on continuous functions: existence of maximum and minimum values, intermediate values. Exponential and logarithmic functions.

Derivatives: definition, geometric meaning. Operations with derivatives: sum, product, quotient, multiplication by a number. Main rules of derivation, derivatives of relevant functions. Equation of the tangent line at a point to the graph. Derivative of a composite function and inverse functions. Stationary points.
Fermat's theorem. Theorems of Rolle and Lagrange. Monotony and sign of the first derivative. Linear approximation.
Second derivatives, concavity, inflections. Plotting graphs of functions. Theorems of Cauchy and De l'Hopital. Applicative problems and optimizations. Taylor polynomial. Formula of the rest of Lagrange for n=2.

Introduction to indefinite and definite integrals. The problem of calculating the area of a flat region. The theorem of the integral average. The fundamental theorem of integral calculus. Integration by parts and substitution.
Introduction to the use of computer software for plotting functions.

Core Documentation

James Stewart, Calcolo. Funzioni di una variabile, Apogeo Education - Maggioli Editore.
Robert A. Adams Calcolo Differenziale I ed. CEA (Casa Editrice Ambrosiana)
Bramanti, Pagani, Salsa “Analisi Matematica 1. Con elementi di geometria e algebra lineare”, Zanichelli

Reference Bibliography

James Stewart, Calcolo. Funzioni di una variabile, Apogeo Education - Maggioli Editore. Robert A. Adams Calcolo Differenziale I ed. CEA (Casa Editrice Ambrosiana) Bramanti, Pagani, Salsa “Analisi Matematica 1. Con elementi di geometria e algebra lineare”, Zanichelli Courant, Robbins "Che Cos' è La Matematica?" Ed. Boringhieri

Type of delivery of the course

The course is organized in lectures and exercise class. Lectures on different subjects starts with several examples then general cases followed by definitions, theorems and proofs. The subject is presented from a geometrical and analytical point of view with a modeling description. Part of the lectures is on exercises made by singular students or divided in small groups. During exercise classes we give some exercises and problems, let the students try to solve them then we discuss the solution and, if necessary, we give the full solution at the blackboard. Some of the exercise classes are dedicated to hands-on activities with the use of paper and other materials or a computer for visualization.

Attendance

Students must attend at the least the 75% of lessons to be admitted to the final exam

Type of evaluation

The student assessment involves a written and an oral exam. Some tests during the course are also planned. The written exam (2,5-3 hours) consists of 5 or 6 exercises to assess students understanding of concepts and their autonomous application. Some past written exams are available at the course web page http://www.formulas.it/sito/corsi/istituzioni-di-matematiche-i/

teacher profile | teaching materials

Programme

Quantifiers. Natural, integers, rational and real numbers. Axioms of real numbers. The square root of 2 is irrational.

Cartesian coordinates in the plane and in the space; coordinate planes. Points and vectors. Distance: formal definition. Absolute value. Density of Q in R. Distance in the plane and in the space; equations of circumference and sphere.

Linear algebra (in 2 and 3 dimensions): slope of a segment, vector sum, scalar and vector product. Equivalence of geometric and coordinate definitions of vectors. Orthogonality and parallelism conditions.

Introduction to functions. Graphic of a function in the coordinate planes. Operations with graphics.

Open and closed sets, accumulation points, definitions and examples. Limits. Operations with limits, examples of limits of quotients of polynomials. Asymptotes. Comparison theorem. List of relevant limits.

Continuous functions (continuity in a point and in an interval). Theorems on continuous functions: existence of maximum and minimum values, intermediate values. Exponential and logarithmic functions.

Derivatives: definition, geometric meaning. Operations with derivatives: sum, product, quotient, multiplication by a number. Main rules of derivation, derivatives of relevant functions. Equation of the tangent line at a point to the graph. Derivative of a composite function and inverse functions. Stationary points.

Fermat's theorem. Theorems of Rolle and Lagrange. Monotony and sign of the first derivative. Linear approximation.

Second derivatives, concavity, inflections. Plotting graphs of functions. Theorems of Cauchy and De l'Hopital. Applicative problems and optimizations. Taylor polynomial. Formula of the rest of Lagrange for n=2.

Introduction to indefinite and definite integrals. The problem of calculating the area of a flat region. The theorem of the integral average. The fundamental theorem of integral calculus. Integration by parts and substitution.

Introduction to the use of computer software for plotting functions.

Core Documentation

James Stewart, Calcolo. Funzioni di una variabile, Apogeo Education - Maggioli Editore.

Robert A. Adams Calcolo Differenziale I ed. CEA (Casa Editrice Ambrosiana)

Bramanti, Pagani, Salsa “Analisi Matematica 1. Con elementi di geometria e algebra lineare”, Zanichelli

Bibliography

James Stewart, Calcolo. Funzioni di una variabile, Apogeo Education - Maggioli Editore.

Robert A. Adams Calcolo Differenziale I ed. CEA (Casa Editrice Ambrosiana)

Bramanti, Pagani, Salsa “Analisi Matematica 1. Con elementi di geometria e algebra lineare”, Zanichelli

Courant, Robbins "Che Cos' è La Matematica?" Ed. Boringhieri

Type of delivery of the course

The course is organized in lectures and exercise class. Lectures on different subjects starts with several examples then general cases followed by definitions, theorems and proofs. The subject is presented from a geometrical and analytical point of view with a modeling description. Part of the lectures is on exercises made by singular students or divided in small groups. During exercise classes we give some exercises and problems, let the students try to solve them then we discuss the solution and, if necessary, we give the full solution at the blackboard. Some of the exercise classes are dedicated to hands-on activities with the use of paper and other materials or a computer for visualization.

Attendance

Students must attend at the least the 75% of lessons to be admitted to the final exam.

Type of evaluation

The student assessment involves a written and an oral exam. Some tests during the course are also planned. The written exam (2,5-3 hours) consists of 5 or 6 exercises to assess students understanding of concepts and their autonomous application. Some past written exams are available at the course web page http://www.formulas.it/sito/corsi/istituzioni-di-matematiche-i/ In case of an extension of the health emergency from COVID-19, all the provisions that regulate the methods of carrying out the teaching activities and student assessment will be implemented. In particular the student assessment will be organized through an oral exam concerning the program of the course.