Mathematical analysis (2006/2007)

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Course code
4S00006
Name of lecturer
Marco Squassina
Number of ECTS credits allocated
6
Academic sector
MAT/05 - MATHEMATICAL ANALYSIS
Language of instruction
Italian
Location
VERONA
Period
1st quadrimester (only for 1st year of 3-year degree courses), 2nd quadrimester
Web page
http://analisimatematica.blogspot.com
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Lesson timetable

Learning outcomes

The aim of this course is that of showing the core notions of mathematical analysis of real functions of a single real variable. The content can be splitted into four main parts: the system of real numbers, limits and continuity, differential calculus and, finally, integral calculus.

Syllabus

The system of real numbers. Algebraic and ordering properties. Sets and functions. Injective, onto, and bijective mappings. Extension of the real numbers and the related algebraic and ordering properties. Real numbers and carthesian coordinates. Maximum and minimum of a set. Infimum and supremum of a set. Natural, integers, and rational numbers. Density of the rational number in the set of the real numbers. Newton binomial formula. Complex numbers. Limits and continuity. Topology of the real line.Limits over restrictions. Classification of the points of discontinuity. Maximum and minimum limit. Sequences and subsequences. Bolzano Weierstrass's theorem. Cauchy convergence criterion for sequences. Elementary functions, exponential and circular functions. Uniformly continuous functions. Main properties. Series of real numbers. Series with positive terms. Absolutely continuous series. Comparison, root, and quotience criterion for series with positive terms. Series with variable sign. Leibniz criterion. Differentiable functions. Classifications of points where the differentiability fails. Classical theorems of differentiable functions: Rolle, Lagrange, Cauchy. L'Hopital's theorems. Derivatives of higher order. Taylor formulas with Peano and Lagrange rest. Convex functions and convexity criteria for functions with are differentiable once or twice. Upper and lower sums. Upper and lower integrals. Integrable functions. Integrability criteria. Integrability of monotone and continuous functions. Fundamental theorem of calculus. Integral calculus computation rules. Integration by substitution. Integration by parts. Improper integrals.

Assessment methods and criteria

Final written examination with multiple choice answers. Optional oral examination.

Teaching aids

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