60/60/136  CALCULUS I
Academic Year 2022/2023
Free text for the University
 Professor

LUCIO CADEDDU (Tit.)
 Period

First Semester
 Teaching style

Convenzionale
 Lingua Insegnamento

ITALIANO
Informazioni aggiuntive
Course  Curriculum  CFU  Length(h) 

[60/60] PHYSICS  [60/00  Ord. 2012] PERCORSO COMUNE  12  96 
Objectives
1. Knowledge and understanding
Knowledge and understanding of the terms and the notation in the theory of limits and differential and integral calculus for functions of one real variable. Knowledge of the basic notions of the theory of numerical sequences and infinite series, as well as the theory of functions of one real variable.
2. Applying knowledge and understanding
Ability in the usage of the algorithms for computing limits of numerical sequences as well as of functions of one real variable, and for the symbolical computation of derivatives and the integration by quadratures of such functions. Usage of integration theory to compute the area of simple plain regions. Usage of some criteria to establish the convergence of an infinite series.
3. Making judgements
Apply the differential calculus to discover the qualitative properties of the graph of a given function, and to determine its maxima and minima. Bring together and connect the information arising from different algorithms concerning a given function, and intelligently use such information to establish the nature of the function. Critically evaluate if an alleged proof, or, more generally, a given argument is correct, as well as if a given definition is well posed. Locate and correct one's own mistakes, without getting lost.
4. Communication skills
Ability in communicating information, ideas, problems and solutions by means of the current terminology, in order to effectively interact with other students and teachers, in the present, and prospectively with colleagues or in a possible teaching activity. Ability in supporting a claim by means of a logical argument (a proof), making a correct usage of terms as "if", "then", "that is to say".
5. Learning skills
Being able to consult and interpret textbooks to build and expand one's own knowledge. Being able to use the sources of information as a tool in problemsolving activities. Being able, if necessary, to use the sources of information in order to go slightly beyond a strict interpretation of the course boundaries.
Prerequisites
Basic notions of set theory and numbers sets. Basic algebraic and symbolic calculus skills. 2nd degree polinomial equations and inequalities. Systems of equations and inequalities. Trigonometry. Basic notions of analytic geometry (lines and curves in the 2dimensional Euclidean space)
Contents
Contents (w.p. means “with proof”, w/p means “without proof”)
1.Preliminary concepts on sets and sets of real numbers. Real numbers properties: maximum, minimum, supremum, infimum.
2.Real line topology: different nature of the points of the real line, definitions and examples. Subsets of the real line: open sets, closed sets, bounded sets,, definitions and examples. Fundamental properties of open and closed sets (thm. 2.1, 2.2 e 2.3 (w.p.)). BolzanoWeirstrass theorem (w.p.). Compact and connected sets, HeineBorel theorem (w/p).
3.Functions between sets: injection, surjection, inverse. Composition of functions. Definitions and examples (pg. 2429, 3235). The principle of induction. Application (sum of the first N numbers).
4.Real functions: sign and symmetries, absolute value of f, even and odd functions. Bounded functions, local and global maxima and minima. Monotone functions. Examples: the elementary functions (sin(x), cos(x), log(x), ex, etc.).
5.Sequences and series: limit of a sequence, comparison theorem (due carabinieri), special limits (Nepero number e included), max and min lim. Sequences and topology, Cauchy principle, Weierstrass theorem.
Infinite sums: definitions and convergence. Method for establishing the convergence of a series (ratio, comparison, square, Cauchy), generalized harmonic series, absolute convergence, alternate series, Leibnitz rule.
6.Limits for functions: uniqueness, right and left limits. Definitions for limits at infinity. Invariance of the sign theorem (w.p.). Comparison theorem (w.p.). Special cases (e.g. sin(x)/x). Limits properties and operations (w.p.). Non existence of the limit. Indeterminate forms. Composed function limit (w/p). Monotone functions limit theorem (w/p). Limits of elementary functions. Hyperbolic functions, definitions and graphs.
7.Asymptotes. Comparison principles for infinite and infinitesimal functions. Landau symbols.
8.Continuous functions. Continuity for the composition of functions (w/p). Discontinuity, classification of three cases. Examples. Fundamental theorems for continuous functions (zero's, sign, Weirestrass, Darboux, all w.p.). Uniform continuity. HeineCantor theorem (w/p). Continuous functions on connected sets (all properties w/p).
9.Differential calculus: derivative of a function. Definition, geometric and physical meaning. Properties of the derivatives (w.p.). Composed and inverse functions. Elementary functions derivatives. Higher order derivatives. Differential df and dx.
10.Methods for finding maxima and minima using derivatives. Fermat, Rolle, Cauchy and Lagrange theorems (all w.p.). Monotone functions and derivatives (rate of growth). Monotonicity test. De L'Hopital theorems (w.p.). Concavity, convexity.
11.Scheme for approaching the analysis of the graph of a function. Taylor approximation formula (w.p.). Peano and Lagrange error formulas (w.p.) Fundamental properties, examples and applications.
12.Integral calculus. Riemann integral, definition and properties. Class of Rintegrable functions. Integrability fundamental theorem, integral mean value theorem (all w.p.). Methods for calculating integrals. Generalized integrals and convergence.
13. Ordinary differential equations, by separation of variables, first order linear, second order linear with constant coefficients. The Cauchy problem, boundary value problems, initial conditions.
Teaching Methods
Traditional lessons on the blackboard and, eventually, with online streaming. Didactic activities (for a total of 96 hours) consist of theoretic explaination of mathematical concepts (in terms of definitions, properties and theorems with proofs), closely linked to examples and applications.
Verification of learning
First half: written tests (6 possibilities per year: January, 2 in February, June, July, September) with exercises. With a result of 18/30 one can apply for the second part of the exam (theory, definitions, theorems and proofs). The written test “lifespan” is, normally, 60 days. If a student fails to get a positive result during the second half of the exam he must apply again for the first part.
For previous exams tests see http://people.unica.it/luciocadeddu/
Texts
C. D. Pagani, S. Salsa – “Analisi Matematica, Vol. 1” – zanichelli Editore.
Esercizi: P. Marcelllini e C. Sbordone, “Esercitazioni di Matematica, vol. 1”, parte
prima e parte seconda, Liguori Editore.
F. Buzzetti, E. Grassini Raffaglio, e A. Vasconi Ajroldi, “Esercitazioni di Analisi"
More Information
On my personal website a collection of previous examination tests are available, sometimes they include traces and suggestions for the solution of the exercises. http://people.unica.it/luciocadeddu/
Visit this link for more infos on LD:
http://corsi.unica.it/fisica/infodsa/