# 00674 - Mathematics

## Learning outcomes

At the end of the course, the student is familiar with the basic tools of differential and integral calculus for functions of one variable, knows the most elementary methods for the solution of differential equations of first order and uses elementary tools from three-dimensional analytical geometry. In particular, the student will be able to complete standard tasks related to differential and integral calculus (e.g., draw a quantitative graph of a function, calculate the area of a planar domain, etc.), solve simple first-order differential equations and solve basic problems of analytical geometry in 3-space (e.g., study lines and places).

## Course contents

PART I
Survey of fundamentals notions. Basic symbols and notions of set theory; logical quantifiers. Terminology for subsets of the real numbers. Definition of function. One-to-one, onto, and one-to-one & onto functions. Inverse function. Domain of a function. Cartesian product of two sets. Correspondence between R^2 and the points of the Cartesian plane.

Real-valued functions. Domain, range, and graph of a function. Criteria of the vertical lines and of the horizontal lines. Translation of graphs.

Trigonometric functions. Radians. Definition of sine, cosine, and tangent. Even and odd functions.

Trigonometric identities. Addiction and subtraction formula. Inverse trigonometric functions. Arcsine, arcosine, arctangent.

Exponential and logarithms. Powers with real exponents. Natural exponential and logarithm. Infinitesimal calculus. Definition of limit. Neighborhood of a point.

Limits. Indefinite forms. Techniques for the calculations of limits of rational functions. Definition of continuity.

Compositions of functions. Compositions of functions and compositions of continuous functions.

PART II
Derivatives.
Definition and geometric interpretations; tangent line to a curve; discussion on the meaning and usefulness of the derivative. Derivatives of the most common functions; linearity of the derivative operator; derivatives of products and quotients of functions; derivative of composite functions. Introduction to implicit differentiation. Derivative of trigonometric functions and of their inverse.

Graphs of functions.
Definition of local maximum and minimum; test of the first derivative; inflection points; concavity and convexity; test of the second derivative. Guidelines for sketching the qualitative graph of a function.

Applications of the derivatives.
L'Hopital rule; mean and instant rate of change of a quantity; introduction to the Newton's method for the calculation of the root of a function f(x)=0. Rolle's theorem. Mean values theorem.

Antiderivatives.
Definition of antiderivatives and main techniques for their calculation. Introduction to ordinary differential equations with the example of the exponential growth.

Indefinite and definite integrals.
Fundamental theorem of integral calculus. Techniques of integration: integration by substitution and integration by parts.

Introduction to analytical geometry.
Vectors in two dimensions. Three-dimensional coordinate systems. Vectors in three dimensions. Scalar and cross products. Equations of lines and planes in the three dimensional space. Distance of a point from a line and from a plane.

Linear systems of equations. Matrices and matrix approach to the solution of a linear system. Fundamentals of matrix algebra; determinants and inverse matrix. Cramer's rule.

A good reference book, which however cannot replace a good set of notes from the class, is:

• M. Bramanti, C. D. Pagani, S. Salsa, "Matematica: calcolo infinitesimale e algebra lineare", 2nd ed., Zanichelli, 2004

A useful companion book for the exercises is:

• S. Salsa, A. Squellati, "Esercizi di matematica: calcolo infinitesimale e algebra lineare. VOLUME 1", Zanichelli, 2001 (ISBN: 8808088871)

## Teaching methods

Classroom lectures, to be attended in person or through the web (subject to modification based on the University guiding lines).

## Assessment methods

The assessment method relies on a written test based on eight questions, five of which are numerical exercises and three of which are theory-oriented questions. The numerical exercises potentially span all the topics of the class; they are similar in nature, length and complexity to the exercises discussed during the class, assigned as homeworks and discussed during the extra exercise hours of which the tutor is in charge.

The theory-oriented questions concern theorems, definitions and theoric discussion covered in class; all the relevant material is available on the University web platform.

The total weight of the numerical exercises amounts to 20 points (each exercise has a weight between 3 and 5 points; the weight of each exercise is notified together with the exercise); the total weight of the theory-oriented questions is 12 points (also in this case, the weight of each exercise is between 3 and 5 points and the weight of each exercise is notified together with the exercise). The total maximum achievable points is 32 (20+12). In order to successfully take the exam, it is necessary to obtain at least 8 points in the numerical exercises and at least 4 points in the theory-oriented questions. If the total achieved points exceed 30, the final evaluation is 30/30 cum laude.

In order to take the exam, it is mandatory to subscribe to the Almaaesami website in due time.

## Teaching tools

• Class notes (to be studied together with the textbook).
• Transcript of teacher's tablet notes, or other electronic resources (slides) used in class
• Edited notes written by teacher on some topics.
• Homework problems, taken from the exercise book, from the list of sample problems written by the teacher and from the previous exams. Some of these problems are later solved and discussed in class.
• Depending upon the economic resources of the School, elective problem-solving meetings coordinated by a Teaching Assistant.

## Office hours

See the website of Andrea Mentrelli

See the website of Marco Lenci