## MATHEMATICAL ANALYSIS

The idea of a set and the idea of a mapping between sets

Calculations with integers, with decimals, and with fractions

Solution of equations of first and second degree

Powers and the algebraic laws which they satisfy

Polynomial algebra

Graphs, axes, and coordinates

The graph of the linear function: y=mx+q, which is a line

Logarithms

Trigonometry for triangles

Written exam, at the end of the course, during the regular exam periods.

The exam will be held in presence or online, depending on the evolution of the pandemic.

The written exam consists of two parts: a multiple choice theory test and a practical paper with open questions. The two parts must be taken within a single day. Students are required to pass both parts in order to pass the exam.

Oral exam is not mandatory. Students willing to take the oral exam must apply to the instructor. Written exam and oral exam must be taken within the same session. The oral exam can cover any of the topics of the course, including proofs of theorems as seen during class.

Oral exam could be mandatory for some candidates, depending on the instructor's judgment.

Copying will be treated as a serious offence. Offenders could be denied admission to the next exam session.

Students will learn some calculational methods and some important mathematical concepts of mathematical analysis which are fundamental for science and technology in general and for computer engineering in particular.

Sets. Main examples of sets of numbers.

Real numbers, neighborhoods, absolute value.

Upper and lower bounds,

Sup, inf, maximum, minimum of a set. Completeness of R.

Functions. Injective (One-to-one) and surjective functions. Monotonic and invertible (bijective) functions. Graph of a function. Symmetries of graphs. Convex functions. Main examples of functions: powers,

exponential, logarithm, sine, cosine, tangent,

arctangent. Hyperbolic functions.

Limit of a sequence. Some theorems about limits. Algebraic properties of limits. Indeterminate forms. Series. Generalized harmonic series. Geometric series. Criteria for convergence of a series. Simple and absolute convergence.

Limit of a function. Continuous functions. Main theorems about limits. Algebraic properties of limits. Indeterminate forms.

Derivatives. Differentiation rules. Main theorems about derivatives. Maxima, minima and critical points. Monotonic and convex functions. Inflection points. Landau's Notation "Big O / little o".

Taylor expansions. Taylor series.

Definite integrals. Integral function. Main theorems about integration. Methods for the evaluation of integrals. Integration by parts and by substitution. Integration of some classes of fundamental functions. Improper integrals.

Ordinary differential equations. Cauchy Problem for for

explicit ordinary differential equations. Existence and uniqueness theorems. First order linear differential equations. Linear differential equations with constant coefficients. Systems of differential equations with constant coefficients.

The Laplace transform

Robert A. Adams, C. Essex: Calculus: A Complete Course

Publisher: Prentice-Hall Canada

This textbook should be good for this course, and also for the follow-up course, Analisi Matematica 2. A closer analog to the suggested Italian textbook would be Calculus: Single Variable, same authors as above, but that would cover only the present course, Analisi Matematica 1.

lectures and guided exercise sessions

Students can contact the instructor at the address:fgladiali@uniss.it