Academic Year 2022-2023



Guglielmo Feltrin
Aleks Jevnikar
Simone De Reggi
Course Year
Unit Credits
Teaching Period
Course Type
Prerequisites. Good knowledge of spoken and written Italian language. Ability to understand and synthesize a text. Logical reasoning skills.

Knowledge of numerical sets (natural, integer, and rational numbers). Knowledge of power functions with natural, integer, rational exponent, logarithms, absolute value (modulus), square root function. Knowledge of solving methods for equations and polynomial inequalities of degrees one and two, exponential and logarithmic. Familiarity with the use of the sign product rule and the cancellation property to solve more complex equations and inequalities. Knowledge of the basics of analytic geometry: Cartesian coordinate system, equations of lines, parabolas, circumferences, ellipses, and hyperbolas. Knowledge of elementary functions and of their properties. Knowledge of the basis of trigonometry and of the main formulas associated with trigonometric functions.

Teaching Methods. The course includes theoretical lessons and classroom exercises, with an explanation of how to deal with various problems and exercises related

to the topics presented. The learning material includes a large number of training exercises, that the student has to solve independently. Finally,

every week there are various hours available for clarification meetings with the Professor and with teaching collaborators.

Verification of Learning. In order to verify the knowledge and skills acquired by the student during the course (which include theoretical knowledge and the ability to apply the theory in solving the exercises), with reference to the educational objectives, the exam consists of a written test and an oral test.

The written part consists of answering questions of a theoretical nature and solving exercises. The written test is not a test to solve the exercises in a mechanical way, but it requires knowing how to correctly and effectively apply the tools of Mathematical Analysis studied during the course, being aware of how the methods are applied in solving some problems. If the result of the written test is greater than 16/30, the student is admitted to the oral test, mainly of a theoretical nature, where the correction of the written test is also discussed.

The student can take the exam by dividing it into two tests related to the program of each of the two semesters of the course.

More Information. As a support of the learning activity, in the e-learning web platform the detailed lesson program, several exercises, and quiz sheets are available.
The course aims to provide a solid basic preparation, to cultivate and develop knowledge and skills in differential and integral calculus for the real valued functions of a real variable, in the theory of limits of functions and sequences, in series. The topics covered are essential to be able to easily face the subsequent courses. The theory is presented with considerable rigor, always starting from ideas and intuition, and at the same time accompanied by numerous examples and exercises, when possible with a concrete meaning.

For more detailed information, see also:

Basic concepts of mathematical logic and set theory. Examples of sets and ways of correctly defining a set. Set operations. Ordered pairs and

Cartesian product of two sets. Relations and functions between two sets. Fundamental concepts related to functions: domain, codomain, image

set, injective, surjective, invertible functions, inverse function, the composition of two functions, with examples. Binary order relations in a

set. Numerical sets: N, Z, Q, and induction principle with examples and applications. The complete ordered field R of real numbers (axiomatic

introduction) and its main properties.

Maximum and minimum of a set in R, infimum and supremum, and related properties. Absolute value function and floor function, graphs of a

function, maximum function and minimum function between two assigned functions. Sequences in R. Distance between points of R and

between points of the plane. Cartesian coordinate system, abscissa/ordinate. Intervals of the real line and neighbourhood of a point.

Properties of intervals and neighbourhoods. Isolated points and accumulation points of a set. Extension of the real number line with the

points at infinity.

Limits and continuity. Definitions and main theorems. Monotone functions (increasing/decreasing). Theorems on continuous functions: intermediate value theorem, Weierstrass theorem on maxima and minima. Continuous functions defined on an interval. Theorems about limits: sum, product, quotient, the composition of functions; squeeze theorem (comparison on limits). List of important limits. Elementary functions and their graphs: powers, polynomials, rational, exponential functions, logarithms, trigonometric functions and their inverse. Limits concerning these

functions. Local comparison of functions. Landau symbols. Infinite and infinitesimal functions. Order of an infinity and of an infinitesimal, principal part with respect to a test function. Asymptotes.

Derivatives, derivability and continuity, successive derivatives, theorems on derivatives. Critical points (stationary points), Rolle, Cauchy and

Lagrange theorems. Maxima and minima, monotonicity on intervals using the first derivative. Taylor’s and Taylor-MacLaurin’s formula. Property of

the remainder and estimation of the error in the use of Taylor’s formula. Convex functions: equivalent definitions and criterion of the second

derivative for the maximum/minimum points. Inflection points. The concept of primitive of a function and the problem of searching for primitives.

Integrals and area computations. Lower and upper sums, lower and upper integral of a limited function over an interval. Integrable (and nonintegrable) functions according to Riemann. The integral mean value theorem and the fundamental theorem of calculus. The Newton-Leibniz theorem. Examples of integration techniques: by parts, by decomposition, by substitution, with significant examples. Computation of areas and function study using the fundamental theorem of calculus and its corollaries. Volume of solids of revolution.

Numerical series and main properties, with particular regard to generalising harmonic series and power series. Series with positive terms and comparison theorems for series. Convergence criteria for series with positive terms. Absolutely converging series and alternating series.

Improper integrals and main properties. Relationships between improper integrals and series.

Introduction to ordinary differential equations, first-order equations with separable variables, and related integration techniques.

Suggested text (but not mandatory):

– E. Acerbi, G. Buttazzo, Primo corso di analisi matematica

For those wishing to study with an English textbook, the manuscript “Introduction to Real Analysis” by Prof. William Trench (available at: is suggested.