# MATHEMATICAL METHODS

## Learning outcomes of the course unit

Give the bases of calculus for functions of one real variable, in such a way that the students are able to solve simple problems in the field. Students should be able to draw and read graphs of functions of one variable, to study functions of one real variable and to compute elementary integrals and to solve some differential equations.

Give the bases of calculus for functions of one real variable, in such a way that the students are able to solve simple problems in the field. Students should be able to draw and read graphs of functions of one variable, to study functions of one real variable and to compute elementary integrals.

## Prerequisites

Mathematics taught in high schools

Mathematics taught in high schools

## Course contents summary

Real numbers, elementary algebra, equations and inequalities. Functions and their graphs, elementary functions. The principle of induction. Maximum, minimum, supremum and infimum of sets of real numbers. Limits of sequences and of functions of a real variable. Continuity, derivatives, primitives and their properties. Integrals of continuous functions over intervals. Study of the graphs of functions of one real variable.

Real numbers, elementary algebra, equations and inequalities. Functions and their graphs, elementary functions. The principle of induction. Maximum, minimum, supremum and infimum of sets of real numbers. Limits of sequences and of functions of a real variable. Continuity, derivatives, primitives and their properties. Integrals of continuous functions over intervals. Study of the graphs of functions of one real variable.

## Course contents

1.1. Real numbers.

1.2. Functions and their graphs.

1.3. Operations between functions.1.4. Polynomials, rational and trogonometric functions.

2.1. Limits.

2.2. Continuos functions.

2.3. Theorems on continuos functions.

3.1. derivatives and tangent lines.

3.2. Rules of differentiation.

3.3. Derivatives of elementary functions

3.4. Exponential and logarithm functions.

3.5. Graph of functions.

4.1. Riemann integrals.

4.2. Fundamental Theorem od calculus. Definite integrals.

4.3. Indefinite integrals and methods of integrations.

5.1. Ordinary differential equations: generalities.

5.2. Linear first order differential equations.

5.3. Linear differential equations with constant coefficients.

1.1. Real numbers.

1.2. Functions and their graphs.

1.3. Operations between functions.1.4. Polynomials, rational and trogonometric functions.

2.1. Limits.

2.2. Continuos functions.

2.3. Theorems on continuos functions.

3.1. derivatives and tangent lines.

3.2. Rules of differentiation.

3.3. Derivatives of elementary functions

3.4. Exponential and logarithm functions.

3.5. Graph of functions.

4.1. Riemann integrals.

4.2. Fundamental Theorem od calculus. Definite integrals.

4.3. Indefinite integrals and methods of integrations.

5.1. Ordinary differential equations: generalities.

5.2. Linear first order differential equations.

5.3. Linear differential equations with constant coefficients.

## Recommended readings

P. Marcellini, C. Sbordone: Calcolo, Liguori Editore

A. Nannicini, L. Verdi, S. Vessella: Note ed esercizi svolti di Calcolo 1, Pitagora Editrice.

M. Bramanti, C.D. Pagani, S. Salsa, Matematica Calcolo infinitesimale ed

algebra lineare, seconda edizione, Ed. Zanichelli, ISBN: 8808075478.

S. Salsa, A. Squellati, Esercizi di Matematica, calcolo infinitesimale e algebra

lineare, Vol. I, Ed. Zanichelli, ISBN: 9788808224880.

P. Marcellini, C. Sbordone: Calcolo, Liguori Editore

A. Nannicini, L. Verdi, S. Vessella: Note ed esercizi svolti di Calcolo 1, Pitagora Editrice

## Teaching methods

Lectures and exercises in the classroom

Lectures and exercises in the classroom

## Assessment methods and criteria

The final exam consists of a written test and, on request, in an oral exam.

The written test is divided into two parts:

• in the first part a text containing 8 multiple choice questions is delivered.

After 20 minutes, the correct answers are given. Only those who have correctly answered all the questions, with the possible exception of one, access the second part of the written test.

• in the second part a text containing exercises is delivered and, if necessary,

theoretical questions.

The evaluation of the second part of the written test takes place in thirtieths. The second part of the written test and, consequently, the exam of '' Mathematical Methods '', yes

intends to pass, when at least 18. The final mark is based on the mark obtained in the written test. Any oral examination, to be held after the written test and in the same session of the latter, is carried out at the request of the student and consists of the demonstration of significant theorems and / or exposure of topics, definitions, treated in the lessons . In the latter case, the final grade is the average between the written test and the oral exam.

In place of the written exam, students can take three intermediate tests. The evaluation of the intermediate tests and of the written test is thirty.

To register for the intermediate tests: connect to http://elly.scvsa.unipr.it/

enter the BIOTECHNOLOGIES section, course in "Mathematical Methods" and carry out

registration for the intermediate exam.

The evaluation of each intermediate written exam is as follows:

• The marks of each intermediate test are attributed out of thirty.

• Students who, at the end of the 3 intermediate tests, achieve a higher average

or equal to 18/30 and who scored a score greater than or equal to 18/30 in

at least two intermediate tests, pass the exam of '' Mathematical Methods '', with the final mark equal to the average of the 3 intermediate tests as above.

The oral examination, to be held after the 3 intermediate exams, by 30 September 2019, is carried out at the request of the student and consists in the demonstration of significant theorems and / or the exposition of topics, definitions, treated in the lessons. In the latter case the final grade is the average of the average of the 3 written intermediate tests and the oral exam.

The examination consists of a written test followed by an oral one. Students with sufficient marks in the written test are allowed to the oral part. The final mark will be the mean value between the marks of the written and of the oral tests.

## Other informations

The course consists of lectures, exercises, interviews with students, articulated

according to the official calendar and timetable; in particular, interviews with students are

fixed by appointment at the Department of Mathematical, Physical and

Informatiche, Plexus of Mathematics, University Campus, Park Area of Sciences

53 / A.

During the lectures, in traditional mode, i

topics will be presented in a formal and rigorous manner. The course will give particular emphasis to the application and calculation aspects, while not neglecting

the theoretical aspect. For this purpose the exercises carried out in the classroom in which the student learns to apply the application are particularly important

theory seen in class to solve concrete problems. The course notes in PDF format and all the material used during the lessons and exercises are made available to the students on the Elly educational platform.