## 21001991 - FUNDAMENTALS OF MATHEMATICS 1

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 the physical-mathematical models.

Course topics are: the differential and integral calculus in one variable; its concepts, tools and modeling instances; linear algebra analyzed from a geometrical point of view; abstract theory and its geometric interpretation in two and three dimensions.

Canali

teacher profile | teaching materials

Programme

Quantifiers. Numbers: natural, integers, rational and real. Axioms of real numbers.
Cartesian coordinates in the plane. Points and vectors. Distance: formal definition. Absolute value. Density of Q in R.
Linear algebra: vector sum, scalar product. Matrices. Matrix operations of sum and product, determinant, rank of a matrix.
Matrix representation of linear transformations. Geometric meaning of the determinant.
Rotation matrices and omotethy. Parametric equation of the line. Orthogonality conditions.
Introduction to real functions. Graphs.
Working with graphics, absolute value of a graph. Exponential, logarithm of a function for which you know the plot.
Accumulation points. Limits. Operations with limits. Comparison theorem. Continuous functions. Theorems on continuous functions.
Asymptotes. Derivatives: definition, geometric meaning. Operations: sum, product, quotient, scalar product. Main rules of derivation. 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.
Second derivatives, concavity, inflections. Plotting graphs of functions. Theorems of Cauchy and De l'Hopital. Word problems.
Taylor polynomial. Formula of the rest of Lagrange. Hyperbolic functions, conic sections as geometric loci. Classification of conic sections.
Introduction to the problem of calculating the area of a flat region. The fundamental theorem of calculus, definite integrals.
The theorem of the average. Integration by parts and substitution. Integration of rational functions.
Definition of parametric curve. From parametric to cartesian equations and viceversa.
Examples: circumference cycloid, conical. Vector and unit vector tangent vector and the unit vector normal. Length of a curve. Curvature

Core Documentation

G.B. THOMAS, R.L. FINNEY ELEMENTI DI ANALISI MATEMATICA E GEOMETRIA ANALITICA ED. ZANICHELLI

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

Naldi, Pareschi, Aletti “calcolo differenziale e algebra lineare”, Ed. Mc Graw-Hill

ROBERT A. ADAMS CALCOLO DIFFERENZIALE IED. CEA (CASA EDITRICE AMBROSIANA)

COURANT, ROBBINS "CHE COS' È LA MATEMATICA?" ED. BORINGHIERI

Reference Bibliography

G.B. THOMAS, R.L. FINNEY ELEMENTI DI ANALISI MATEMATICA E GEOMETRIA ANALITICA ED. ZANICHELLI Bramanti, Pagani, Salsa “Analisi Matematica 1. Con elementi di geometria e algebra lineare”, Zanichelli Naldi, Pareschi, Aletti “calcolo differenziale e algebra lineare”, Ed. Mc Graw-Hill ROBERT A. ADAMS CALCOLO DIFFERENZIALE IED. CEA (CASA EDITRICE AMBROSIANA) 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

The frequency is mandatory for 75% of the lessons.

Type of evaluation

The oral exham will be online and it will consist of questions and argumentation talk. It will be given on the platform Teams. The exham will be scheduled at the beginning of the first day of the session, when all the candidates will be invited to connect.

teacher profile | teaching materials

Programme

Quantifiers. Numbers: natural, integers, rational and real. Axioms of real numbers.
Cartesian coordinates in the plane. Points and vectors. Distance: formal definition. Absolute value. Density of Q in R.
Linear algebra: vector sum, scalar product. Matrices. Matrix operations of sum and product, determinant, rank of a matrix.
Matrix representation of linear transformations. Geometric meaning of the determinant.
Rotation matrices and omotethy. Parametric equation of the line. Orthogonality conditions.

Introduction to real functions. Graphs.
Working with graphics, absolute value of a graph. Exponential, logarithm of a function for which you know the plot.
Accumulation points. Limits. Operations with limits. Comparison theorem. Continuous functions. Theorems on continuous functions.

Asymptotes. Derivatives: definition, geometric meaning. Operations: sum, product, quotient, scalar product. Main rules of derivation. 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.

Second derivatives, concavity, inflections. Plotting graphs of functions. Theorems of Cauchy and De l'Hopital. Word problems.

Taylor polynomial. Formula of the rest of Lagrange. Hyperbolic functions, conic sections as geometric loci. Classification of conic sections.

Introduction to the problem of calculating the area of a flat region. The fundamental theorem of calculus, definite integrals.
The theorem of the average. Integration by parts and substitution. Integration of rational functions.
Definition of parametric curve. From parametric to cartesian equations and viceversa.
Examples: circumference cycloid, conical. Vector and unit vector tangent vector and the unit vector normal. Length of a curve. Curvature

Core Documentation

ROBERT A. ADAMS CALCOLO DIFFERENZIALE I ED. CEA (CASA EDITRICE AMBROSIANA)
G.B. THOMAS, R.L. FINNEY ELEMENTI DI ANALISI MATEMATICA E GEOMETRIA ANALITICA ED. ZANICHELLI
Bramanti, Pagani, Salsa “Analisi Matematica 1. Con elementi di geometria e algebra lineare”, Zanichelli
Marsden, Jerrold E. and Weinstein, Alan J. (1985) Calculus I. Springer-Verlag , New York.

Reference Bibliography

G.B. THOMAS, R.L. FINNEY ELEMENTI DI ANALISI MATEMATICA E GEOMETRIA ANALITICA ED. ZANICHELLI Bramanti, Pagani, Salsa “Analisi Matematica 1. Con elementi di geometria e algebra lineare”, Zanichelli ROBERT A. ADAMS CALCOLO DIFFERENZIALE I ED. CEA (CASA EDITRICE AMBROSIANA) Marsden, Jerrold E. and Weinstein, Alan J. (1985) Calculus I. Springer-Verlag , New York. Further readings 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

it is compulsory to attend the 75% of lessons

Type of evaluation

Oral exam concerning the program of the course