Mathematical Analysis
The following basic knowledges are required: equations and inequalities, polynomial division, trigonometry, analytic geometry (line and parabola).
The exam is written and consists of both exercises and theoretical questions on all topics presented during the course. The exercises will concern the study of a function and the behavior of a series, as well as the computation of limits, derivatives and integrals. The theoretical part will be tested through a number of open questions that concern definitions, theorems and short proofs.
The students have 3 hours to complete the test and they are not allowed to use any material. The use of a calculator is also forbidden. Those students that get marks between 15/30 and 17/30 must also take an oral exam aimed at reaching the passing grade (18/30). In all other cases, an oral examination will follow only if the written exam leaves doubts on the real preparation of the student.
The criteria that will guide the exam assessment concern the correctness of the steps taken in solving the exercises and not only the exactness of the provided solution. Regarding the open questions, the formalism and mathematical presentation skills as well as the effective knowledge acquired by the student will be assessed.
During the semester the students have the possibility to take one middle test and to accrue up to 2 marks that will be summed to the grade obtained in the written test, whenever the exam is taken during June-July session (afterwards those marks will be lost). Such middle test should be intended by the students as a chance to self-evaluate their own preparation and then to identify, if any, those topics that ask for a deeper study.
The course has the purpose to provide the students with both theoretical and practical tools of basic mathematical analysis with special attention to the study of a real function in one real variable, to numerical sequences and series and to the theory of Riemann integrable functions. At the end of this course the students will be able to:
1. identify and properly define the provided theoretical concepts;
2. understand and present with an adequate formalism the given results and the introduced techniques;
3. replicate short proofs and provide examples and counterexamples;
4. use the fundamental calculus techniques for solving exercises that involve the study of a function or the behavior of a series, as well as the computation of limits, derivatives and integrals.
Moreover, at the end of this course the students will have acquired a mathematical formalism and a rigorous scientific methodology that they will be able to apply to the other subjects encountered during their studies.
The contents of the lectures and related exercises sessions are:
Preliminaries on sets and functions (10 hours, learning objectives 1,3)
- Recap on the rational and real numerical sets and their properties
- Notion of infimum/supremum, minimum/maximum of a set
- Notion of function and its properties
- Elementary functions: modulus, power function, exponential function, logarithmic function, inverse trigonometric functions
Limit of a sequence (8 hours, learning objectives 1-4)
- Convergent, divergent, oscillating, bounded sequences and uniqueness of the limit
- Operations with limits and indeterminate forms
- Main theorems about the limit of sequences (e.g., squeeze theorem, permanence of sign, limit of monotone sequences)
- The number of Nepero as limit of a certain sequence
- Infinite/Infinitesimal sequences and use of the asymptotic
Limit of a function (12 hours, learning objectives 1-4)
- Definition of limit of a function and relation with the limit of a sequence
- Extension of the main theorems on the limit of sequences to the case of a generic function
- Order of infinite/infinitesimal of a function, asymptotic analysis and “little-o” notation
- List of limits for common functions and their formulation as asymptotic relations or by using the “little-o” notation
- Use of the limits to compute the asymptotes of a function
Continuous functions of one real variable (6 hours, learning objectives 1-4)
- Continuity at one point and classification of the discontinuity
- Continuity vs algebraic operations, composition, inversion and monotonicity
- Main theorems concerning continuous functions defined over an interval
Differential calculus (16 hours, learning objectives 1-4)
- Notion of derivative given by its geometrical interpretation and classification of non-differentiability points
- Differentiability and continuity
- Differentiability vs algebraic operations, composition, inversion and monotonicity
- Critical points and search for local extrema
- Main theorems concerning differentiable functions defined over an interval
- De L'Hospital’s rule and its usage to compute limits and derivatives
- Convex/concave functions and their geometric interpretation, inflection points and relation with the sign of the second derivative
- Recap of the steps that characterize the study of a function
- Taylor and MacLaurin polynomials as application of higher-order derivatives and as tool for computing limits of functions
Integration (12 hours, learning objectives 1-4)
- Definition of the Riemann integral by means of inferior and superior sums and its properties
- The fundamental theorem of calculus as a tool for computing integrals through the notion of antiderivative
- Methods of integration by parts and by substitution
Numerical series (6 hours, learning objectives 1-4)
- Series as the limit of the sequence of the partial sums
- Some known series: Mengoli, geometrical, harmonic series
- Necessary condition of convergence
- Series with nonnegative terms and related sufficient convergence criteria
- Series with non-constant sign addends and notion of absolute convergence
- Alternating series and Leibniz criterion
Complex numbers (4 hours, learning objectives 1-3)
- Algebraic and trigonometric form of a complex number
- Operations with complex numbers, modulus, power and conjugate of a complex number
- Roots of a complex number and mention to the fundamental theorem of algebra
Lectures (64 hours) and exercises sessions (12 hours).
All lectures are theoretical and conducted using a slides projector. The lectures are dedicated to the study of a real function in one real variable, to numerical sequences and series and to the theory of Riemann integrable functions. Each lecture is started with an intuitive presentation of the considered topic, and it is continued by formalizing the given ideas in terms of definitions, theorems and short proofs. Examples and counterexamples that clarify the use of the treated subjects are also given.
All exercises sessions are aimed at applying the theoretical tools acquired during the aforementioned lectures. Each session consists in the guided solution of problems and exercises concerning the overall set of topics treated in the course.
Office hours are booked on demand by email writing at mariarosa.mazza@uninsubria.it