# MATHEMATICAL ANALYSIS I.A

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Versione italiana
2022/2023
Teacher
CHIARA BOITI
Credits
6
Didactic period
Primo Semestre
SSD
MAT/05

#### Training objectives

Aim of the course is to provide the basic tools of mathematical analysis, especially for what concerns the differential calculus for functions of one real variable and its applications to the resolution of problems based on mathematical models.
At the end of the unit, students will know the theoretical contents and the methods proper of mathematical analysis calculus.
They should be able to properly apply the concepts learnt to the resolution of different kinds of problems and to identify the most appropriate approach to do so.
Students will have to master the mathematical language and the logical-deductive approach, demonstrating the ability to illustrate their problem solving strategies in a logical, effective, pertinent and synthetic way.

#### Prerequisites

Notion of mathematics usually teached in the secondary school: equations and inequalities (of first and second degree, with absolute values, roots, rational), logarithms, exponentials, trigonometry. Such contents are handled during the pre-course, that will be held before the beginning of the lessons, and will be quickly retrieved during the first lessons.

#### Course programme

The total duration of the course is 60 hours, divided into 24 lessons of 2 hours and half. The lessons also include exercises
on the treated topics.
(1) Reminder of set theory, logic, rational and irrational equations and inequalities, or with absolute value. Numeric sets: N; Z; Q; R.
(2) Maximum and minimum. Infimum and supremum. Remind of powers, exponentials and logarithms. Review of trigonometry.
(3) Factorial and Induction Principle. Arithmetic and geometric progression. Bernuoilli inequality. Binomial coefficients and Newton's binomial.
(4) Functions: first definitions and examples, restrictions and extensions. Real functions of a real variable. Sequences. Injective, surjective and bijective functions. Composition of functions.
(5) Inverse function. Inverse trigonometric functions. Monotone functions. Special functions, like integer part function, fractional part function, Heaviside's function, sign function. Operations with functions.
(6) Elementary graphs and graph transformations (translations, homotheties, symmetries). Hyperbolic functions and their inverse functions. Bounded functions. Infimum and supremum. Global maximum and minimum.
(7) Local properties, neighborhoods. Extended real line. Accumulation points and isolated points. Bolzano-Weierstrass theorem. Open and closed sets, border points. Properties valid locally and definitively.
(8) Limits of functions. Uniqueness of the limit. Left and right limits. Above and below limits. Limits of monotone functions.
(9) Continuous functions. Permanence of the sign theorem. Algebra of the limits. Operations with continuous functions. Sandwich theorem.
(10) Indeterminate forms. Limite of the composite function. Continuity of the composite function. Important limits.
(11) The number of Nepero. Other important limits. Infinitesimals, infinites and comparisons (little o, asymptotic relations). Important limits for sequences. Stirling's formula.
(12) Non-existence of limits. Subsequences. Arithmetic mean and geometric mean. Cesaro's convergence criteria.
(13) Recursive sequences, Fibonacci sequence. Continuous functions in a set. Points of discontinuity.
(14) Continuous functions in an interval: Theorem of zeros and Theorem of intermediate values. Weierstrass theorem. Continuity and monotony. Invertibility and monotony.
(15) Incremental ratio and derivative (geometric and physical meanings). Tangent line. Continuity of a differentiable function. Angular points and cusps.
(16) Algebraic operations with derivatives. Derivative of the composite function. Derivative of the inverse function.
(17) Higher order derivatives. Leibniz's formula. Relative extremes. Fermat's theorem. Search for maxima and minima.
(18) Rolle's theorem. Cauchy's theorem. Lagrange's theorem (of the mean value). Monotony criteria. Optimization problems. Relation between relative extremes and second derivative.
(19) Convexity and concavity. Convexity criteria. Points of flex. Rule of De l'Hôpital.
(20) Continuity and discontinuity of the first derivative. Taylor's formula with Peano remainder and Lagrange remainder. Notable developments. Problems of approximation.
(21) Calculation of limits using the Taylor approximation.
(22) Horizontal, vertical and oblique asymptotes. Graphs of functions.
(23) Exercises on graphs of functions. Summary exercises.
(24) Summary Exercises.

#### Didactic methods

There will be theoretical lectures and exercises.
In performing the exercises we will also try to engage students. There will be assigned exercises to do at home.

#### Learning assessment procedures

The exam is composed by a written test and an oral interview.
- In the written test the student is required to solve problems and exercises related to the course topics. It is not permitted to consult neither texts nor electronic calculator devices. However, the student may consult a personal sheet (A4, double-sided) in which he or she can write anything. At the end of the written test, a score of at most 30 points is awarded. To pass the test and gain access to the oral interview the student must get a score of at least 15 points.
- In the oral interview the student will be required to present some aspects of the course topics, illustrating some definitions, examples, exercises, properties, formulas, theorems, proofs (optional), or applications. More than the mnemonic knowledge of the topics, we want to evaluate the logical understanding of concepts, the accuracy and rigor of the mathematical language used to describe them, and the ability to grasp the relationship between abstract aspects and concrete applications.
The final vote is proposed at the end of the oral interview and will take into account all the elements that allow the teacher to evaluate the student's preparation: active participation during lectures, the delivery of the exercises carried out during the year, correctness and completeness in the written test, quality of exposure in the oral interview.