Mechanical Engineering

MICOL AMAR

The lectures are mainly addressed to first year students who need to recall and deepen the knowledge of fundamental mathematical notions.
It aims to fill the gaps often encountered in the mathematical background of incoming students, in order to give them the mastery needed to successfully face the courses of Mathematical Analysis and Geometry and, in general, all the following university path.
During the lessons, the basic notions on the properties of powers, logarithms and exponentials, absolute value, trigonometry and conic curves will be recalled, in order to bring students to operate with appropriate self confidence when dealing such mathematical notions.

At the end of this course, the student is expected to have reached the necessary skills in basic mathematics, in order to be able to successfully attend the courses of Mathematical Analysis and Geometry.

Fundamental mathematical topics of the secondary school.

Elementary operations
Decimal numbers, fractions
Literal calculation
Monomials and polynomials
First and second degree algebraic equations and inequalities

The program includes a review of the fundamental notions of basic mathematics, studied in the last three years of the high school.
More precisely:

Powers, polynomials and their properties (3 hours).
Review of the power properties. Operations with powers: product, ratio and powers of powers. Ratio between polynomials and monomials and ratio between polynomials and polynomials.

Equations, inequalities and systems of algebraic type (5 hours).
Equations and inequalities of first and second order or related to them (biquadratic or bicubic equations). Systems of equations and inequalities of first and second order or related to them. Some examples of systems of generic nonlinear equations. Geometric interpretation of equations and inequalities of second order according to the discriminant sign.

Equations and inequalities of Irrational type (5 hours).
Equations and inequalities in which radicals are present, with particular attention to radicals with even index.

Logarithmic properties: equations and inequalities of exponential and logarithmic type (5 hours).
Review of the properties of logarithms. Operations with logarithms: sum and difference of logarithms, change of base. Equations and inequalities containing logarithms or exponentials, with particular attention to the case in which the base is less than one.

Absolute value properties and applications to equations and inequalities (5 hours).
Review of the definition and properties of the absolute value. Geometrical meaning. Triangular inequality. Equations and inequations containing the absolute value.

Fundaments of trigonometry. Equations and inequalities of trigonometric type (6 hours).
Definition of the main trigonometric functions (sine, cosine, tangent) and their geometrical meaning. Trigonometric functions on the unit circle. Periodicity of the trigonometric functions. List and use of the main trigonometric formulas (fundamental identity, sum of angles formulas, double-angle formulas, half-angle formulas, prosthaphaeresis and Werner formulas, parametric formulas). Validation of rigonometric identities. Trigonometric equations and inequalities.

Analitical geometry in two-dimension (4 hours).
Straight lines in the plane. Non rotated conic curves: parabola, circle, ellipse, hyperbola. Recognition and reduction to canonical form.

M. Amar - A.M. Bersani
Analisi Matematica I: esercizi e richiami di teoria
edizione AMAZON (Codice ASIN: B0BCRXJM2M)
(Chapter 1)

G. Anichini - G. Carbone - A. Chiarelli - P. Conte
Precorso di Matematica
Pearson (2018)


F. G. Alessio - C. de Fabritiis - C. Marcelli - P. Montecchiari
Matematica Zero
Pearson

Material available on-line at the web-page
http://www.dmmm.uniroma1.it/~micol.amar/Laboratorio.html

Lessons and exercises conducted at the blackboard.

Theoretical lessons are aimed at recalling the main notions relating to the topics of the classic mathematics programs of the first 3/4 years of high school.

Exercises aim to enable the student to acquire the necessary skills on the above mentioned topics, according to the prerequisites required in the courses of Mathematical Analysis and Geometry.

The attendance is optional, but strongly recommended.

The exam (over a period of half an hour) is made by a test with multiple-choice containing 15 questions. Students have 30 minutes to complete the exam. It is necessary to answer correctly to at least 12 questions.

See the following link:
https://www.sbai.uniroma1.it/~micol.amar/Laboratorio.html