MATHEMATICAL ANALYSIS A

Degree course: 
Corso di First cycle degree in ENGINEERING FOR WORK AND ENVIRONMENT SAFETY
Academic year when starting the degree: 
2021/2022
Year: 
1
Academic year in which the course will be held: 
2021/2022
Course type: 
Basic compulsory subjects
Seat of the course: 
Varese - Università degli Studi dell'Insubria
Language: 
Italian
Credits: 
9
Period: 
First Semester
Standard lectures hours: 
88
Detail of lecture’s hours: 
Lesson (56 hours), Exercise (32 hours)
Requirements: 

Arithmetic, elementary geometry, trigonometry.

Final Examination: 
Orale

Final exam with written (4 problems and 2 questions in 2 hours, open questions) and oral examination.
The final score will be made of the score obtained in the written examination out of 30 eventually averaged with the score of the oral examination always out of 30.

Assessment: 
Voto Finale

AIM OF THE COURSE
Learning basic tools in one variable Differential and Integral Calculus and Ordinary Differential Equations. In particular we study numerical series, local and global properties of real functions of one variable, among which: continuity, differentiability, approximation through numerical series. We introduce the Riemann integral, the fundamental theorem of calculus and develop methods for calculating integrals with applications to area problems both for definite and indefinite integrals. Special attention is devoted to the study of integral functions. We introduce ordinary differential equations of the first and second order, linear and non linear with separated variables. We introduce qualitative studies for non integrable equations.

LEARNING OUTCOMES
At the end of the course, the student will be able to:

1. mastering basic tools of Analysis of functions one variable;

2. applying theoretical results to more applied topics in the Physics and Engineering within the modelling as well calculus environment;

3. studying problems with both qualitative and quantitative approach;

4. adapting and implementing basic approximation tools in applications.

Sets of numbers: natural numbers and induction principle, Bernoulli inequality, integers and rational numbers, square root of two is not rational, axiomatic introduction of real numbers. Introduction to real functions of one variable: domain, range, injectivity and surjectivity, invertibility, monotone and bounded functions. Numerical sequences, convergent, divergent, and irregular sequences, theorem on monotone sequence, Nepier's number, operations with limits, permanence of sign. Subsequences and uniqueness of the limit, sandwich theorem, Landau symbols, order of infinite and infinitesimal quantities, asymptotic comparison theorem. Ratio test for sequences. Numerical series and necessary condition for the convergence, geometric series, Mengoli series and telescopic series. Comparison criteria, harmonic series. Root and ratio test for series. Changing sign series and the Leibniz theorem. Completeness of real numbers and the real powers. Elementary functions, graphs, symmetry, compositions, inverse, transformations in the plane and deduction of the graph following elementary transformations. Limits for functions: extending the results proved for sequences, Landau symbols and infinite, infinitesimal functions. Continuity and global properties of continuous functions: zeros theorem, Weierstrass' theorem, Darboux theorem, invertibility of continuous functions, continuous extensions. Introduction to derivatives, relationship between continuity and derivability, operations with derivatives. The chain rule and derivative of the inverse function. Local optimization, Fermat's theorem, Lagrange's theorem, monotonicity criterium and characterizing functions with zero derivative. The Hospital theorem, higher order derivatives, Taylor expansion (with Lagrange and Peano remainders). Introduction to the Riemann integral, Riemann summations and geometric interpretation, integrability of continuous functions. Properties of the Riemann integral, primitives (antiderivatives). The mean value theorem and the fundamental theorem of Calculus, generalized integral and the study of integral functions. First order Ordinary Differential Equations, population dynamics, separated variables equations, Cauchy problems, linear equations and the variation of constants formula. Second order equations, Cauchy problems, the theorem of structure of solutions, homogeneous equations with constant coefficients, variation of constants formula and similarity methods, Euler's equation.

Sets of numbers: natural numbers and induction principle, Bernoulli inequality, integers and rational numbers, square root of two is not rational, axiomatic introduction of real numbers (4 h.). Introduction to real functions of one variable: domain, range, injectivity and surjectivity, invertibility, monotone and bounded functions (2 h.). Numerical sequences, convergent, divergent, and irregular sequences, theorem on monotone sequence, Nepier's number, operations with limits, permanence of sign (4 h.). Subsequences and uniqueness of the limit, sandwich theorem, Landau symbols, order of infinite and infinitesimal quantities, asymptotic comparison theorem. Ratio test for sequences (2 h.). Numerical series and necessary condition for the convergence, geometric series, Mengoli series and telescopic series. Comparison criteria, harmonic series. Root and ratio test for series. Changing sign series and the Leibniz theorem (4 h.). Completeness of real numbers and the real powers. Elementary functions, graphs, symmetry, compositions, inverse (4 h.). Transformations in the plane and deduction of the graph following elementary transformations (2 h.). Limits for functions: extending the results proved for sequences, Landau symbols and infinite, infinitesimal functions. Continuity and global properties of continuous functions: zeros theorem, Weierstrass' theorem, Darboux theorem, invertibility of continuous functions, continuous extensions (4 h.). Introduction to derivatives, relationship between continuity and derivability, operations with derivatives. The chain rule and derivative of the inverse function (4 h.). Local optimization, Fermat's theorem, Lagrange's theorem, monotonicity criterium and characterizing functions with zero derivative. The Hospital theorem (4 h.). Higher order derivatives, Taylor expansion (with Lagrange and Peano remainders) (4 h.).
Introduction to the Riemann integral, Riemann summations and geometric interpretation, integrability of continuous functions (2 ore). Properties of the Riemann integral, primitives (antiderivatives). The mean value theorem and the fundamental theorem of Calculus (4 h.). Generalized integral and the study of integral functions (4 h.). First order Ordinary Differential Equations, population dynamics, separated variables equations, Cauchy problems, linear equations and the variation of constants formula (4 h.). Second order equations, Cauchy problems, the theorem of structure of solutions, homogeneous equations with constant coefficients, variation of constants formula and similarity methods, Euler's equation (4 h.).

Book: R. Adams, Calculus: A Complete Course (7th Edition), Pearson.

Classical lectures in presence and on-line through Microsoft Teams. Slides and further teaching files available through e-learning.

Professor is available to meeting students before and after lectures and in his office upon email appointment: daniele.cassani@uninsubria.it

Professors