28622 - Mathematical Analysis T-A

Learning outcomes

Methodological and operational aspects of differential and integral calculus for functions of a real variable.

Course contents

1. LOGIC. Propositions, operations, sets, relations, functions, quantifiers, principle of induction.
2. REAL NUMBERS. Basics of topology, intervals, inf sup max and min
3. FUNCTIONS. Functions of one real variable. Definition, composition, inversion, domain, image, injectivit, surjectivity, bijectivity; elementary functions of real variable: power, exponential, logarithm, trigonometric functions and their inverses (possibly: hyperbolic functions and their inverses)
4. SUCCESSIONS. Real-valued successions. Definition, limits of successions, operations on limits, monotone successions and their limits; theorems: permanence of sign and comparison; boundedness and extremes of subsets of R; the Nepero number e, some limits of successions.
5. LIMITS. Limits of real functions. Extensions of results established for successions, limit of composed functions, left and right limits; monotone functions and their limits, algebra of limits, theorems on limits of functions.
6. CONTINUITY. Definition, operations on continuous functions, theorem of zeros, intermediate values and Weierstrass theorem, continuity of composition, theorem of variable change, points of discontinuity.
7. DIFFERENTIAL CALCULUS. Derivatives of a function in one real variable. Definition, rules of derivation, derivatives of elementary functions, Rolle's and Lagrange's theorem, their consequences, de l'Hôpital's theorem; results on monotone functions, relative maxima and minima, Fermat's theorem; higher-order derivatives, Taylor's formula and its applications, convexity and concavity, inflection points, asymptotes, function study.
8. INTEGRAL CALCULUS. Integrals of functions of one real variable. Definition, primitive of a function, integral of continuous functions, properties of integrals (linearity, additivity, monotonicity), integral mean theorem, fundamental theorems of integral calculus; integration by parts, integration by substitution, integration of rational functions (possibly: generalized integrals).

Readings/Bibliography

Lecture notes, available on the Virtuale online platform.

Suggested readings:

• G.C. Barozzi, G. Dore, E. Obrecht. Elementi di Analisi Matematica - Volume 1, Zanichelli (2009).
• M. Bramanti. Esercitazioni di Analisi Matematica 1, Esculapio (2011).

Teaching methods

Lectures aimed at illustrating basic concepts, examples and counterexamples.

Exercises solved by the teacher for better understanding of the basics.

Proposal of supplementary exercises to be used as an outline for individual study.

Weekly reception (in the teacher's office)

Assessment methods

The final exam for this course consists of a written test and an oral test, both of which are mandatory and must be taken in that order.

The written test aims to test the ability to apply theory to solving exercises of the type of those proposed during the course. Passages should be reported and justified. No books, notes, calculators, cell phones or computers are allowed; paper and pen only. It lasts 3 and 1/2 hours.

The evaluation of the written test has a max result of 30, and the test is considered passed with a grade greater or equal to 18/30.

If the written test is passed, one can enter the oral test. This aims to test knowledge and understanding of the theory developed during the course.

The oral test is to be taken within the same session (winter or summer) as the passed written test.

Teaching tools

Tutoring (to be determined) and student reception.

Additional course materials will be made available on the course page on the Virtuale online platform.

Office hours

See the website of Mattia Francesco Galeotti