SM/0025 - MATHEMATICAL ANALYSIS 1
Academic Year 2017/2018
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LUCIO CADEDDU (Tit.)
- Teaching style
- Lingua Insegnamento
|[60/64] MATHEMATICS||[64/00 - Ord. 2017] PERCORSO COMUNE||12||96|
KNOWLEDGE AND SKILLS 'UNDERSTANDING: learning the basic concepts of
Mathematics: functions of one variable, continuity, differentiability, the graphic studio, study
limits, the Taylor development. Sequences and series. Riemann integral and calculation methods
CAPACITY 'APPLICATION: the student must be able to apply all the general knowledge
mathematical analysis necessary for the understanding of the study of function related concepts (design
chart), the calculation of the integrals, to the study of the convergence of series and numerical sequences, also with
JUDGEMENT: The course aims to stimulate objective teaching evaluation
constantly offering students a comparison between the theoretical contents proposed while they were lessons
front and obtain them through self-study using the recommended texts and
provided educational materials.
IN COMMUNICATION SKILLS: ability to express with the appropriate mathematical terminology
fundamental analysis concepts with particular reference to theorems and proofs, while promoting and
differentiating theses and hypotheses, showing a good grasp of the various demonstration techniques
(Constructive, by contradiction, by induction, etc.)
TO LEARN SKILLS: The student will develop a methodology of study and analysis that will allow him
to interpret and investigate the issues that will arise in the continuation of the study and
college / career.
SKILLS EXPECTED: development of professional communication skills in math,
thanks to the use of a proper terminology and description of a mode of organized and comprehensible helpful
not only to pass the exam, but also in view of a preliminary exam preparation
Subsequent (in particular of analysis).
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 2-dimensional Euclidean space)
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.)). Bolzano-Weirstrass theorem (w.p.). Compact and connected sets, Heine-Borel theorem (w/p).
3.Functions between sets: injection, surjection, inverse. Composition of functions. Definitions and examples (pg. 24-29, 32-35). 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. Heine-Cantor 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 R-integrable functions. Integrability fundamental theorem, integral mean value theorem (all w.p.). Methods for calculating integrals. Generalized integrals and convergence.
Traditional lessons on the blackboard, exercises in collaboration with students.
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 30 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/
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"
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/
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