# BASIC MATHEMATICS

## Learning outcomes of the course unit

Formative purposes:

The aim of the course is to provide basic mathematical knowledge of the programmes of primary and secondary schools, necessary to follow all of the courses of the first year. During the formative activity the student can fill possible gaps or consolidate his knowledge.

At the end of the course the student is expected to be able:

Knowledge and understanding:

- to know the different sets of numbers and their properties

- to remember all the properties of equations, inequalities and systems

- to know function theory

- to know the basics of trigonometry

- to have understood the concepts of exponential and logarithm

- to know basic figures of analytical geometry and their equations

- to have understood logic of propositions and set theory

Applying knowledge and understanding:

- to order numbers, to factorize a polynomial

- to perform calculations with fractions, radicals, exponentials and logarithms

- to calculate sine, cosine and tangent of a known angle

- to solve equations, inequalities and systems of first and second degree, of degree larger than two, irrational, trigonometric, exponential and logarithmic

- to determine the domain, the range and the inverse image of a function whose graph is given; to prove that a function is injective, surjective, increasing or decreasing

- to draw the graph of an elementary function or of a piecewise defined function based on transformations of elementary functions

- to analyse and draw straight lines, parabolas, circumferences, ellipses, hyperbolas

- to analyse and to deny a proposition, to prove set’s properties

Making judgments:

- to be able to face an autonomous analysis of possible applications during the following courses

Communication skills:

- to have acquired the ability to work both autonomously and in groups

Learning skills:

- to be able to carry on with his scientific studies autonomously.

Formative purposes:

Knowledge and understanding:

At the end of the course the student will have filled possible gaps and will have consolidated his knowledge of the programs of primary and secondary schools.

He will be able to best follow all the courses of the first year.

Moreover he will have learnt the first notions of Algebra and Mathematical analysis and tested himself on precise mathematical writing and easy proofs.

Applying knowledge:

At the end of the course the student should be able to resolve all the problems on basic mathematical knowledge such as operations, calculus of measurements, equations and inequalities in one variable, set theory, polynomials, trigonometry, equations and inequalities like irrational, exponential and logarithmic.

Besides he should be able to apply his theoretical knowledge to make use of propositions and mathematical induction, to analyse the graph of a function and to develop easy proofs.

Making judgments:

On getting through the exam the student should have acquired the logic ability necessary to deal with an easy problem and the skill to plan the solution.

At the same time he should have developed the precision in organising his work and the ability to check the credibility of the results.

Learning skills:

After getting through the exam, the student should have consolidated his basic mathematical knowledge to face, later, an autonomous analysis of possible applications during following courses.

## Prerequisites

Mandatory prerequisites: elementary mathematics.

All further mathematical knowledge from primary and secondary schools is useful.

Prerequisites:

None

## Course contents summary

Topics:

1. NUMBERS

2. LOGIC OF PROPOSITIONS, SET THEORY

3. EQUATIONS, INEQUALITIES, POLYNOMIALS

4. FUNCTIONS

5. ANALYTICAL GEOMETRY, TRIGONOMETRY

6. EXPONENTIAL FUNCTIONS, LOGARITHMS

Topics:

1. NUMBERS

2. EQUATIONS, INEQUALITIES, POLYNOMIALS

3. LOGIC OF PROPOSITIONS, SET THEORY

4. FUNCTIONS

5. ANALYTIC GEOMETRY, TRIGONOMETRY

6. EXPONENTIAL FUNCTIONS, LOGARITHMS

## Course contents

Topics:

1. NUMBERS

2. LOGIC OF PROPOSITIONS, SET THEORY

3. EQUATIONS, INEQUALITIES, POLYNOMIALS

4. FUNCTIONS

5. ANALYTICAL GEOMETRY, TRIGONOMETRY

6. EXPONENTIAL FUNCTIONS, LOGARITHMS

Topics:

1. NUMBERS

2. EQUATIONS, INEQUALITIES, POLYNOMIALS

3. LOGIC OF PROPOSITIONS, SET THEORY

4. FUNCTIONS

5. ANALYTIC GEOMETRY, TRIGONOMETRY

6. EXPONENTIAL FUNCTIONS, LOGARITHMS

## Recommended readings

Reference books:

E. Acerbi, G. Buttazzo: Matematica Preuniversitaria di Base. Pitagora Editrice, Bologna (2003).

Additional material (on the platform ELLY):

Lectures 2017-18.

Exercises with solution.

Examinations with solution (2017-2018).

Reference books:

E. Acerbi, G. Buttazzo: Matematica Preuniversitaria di Base. Pitagora Editrice, Bologna (2003).

Exercises with solution (available in TEACHING STUFF on ELLY)

Previous years' examinations with solution (available in TEACHING STUFF on ELLY)

## Teaching methods

Teaching methods:

Mathematics: 6 CFU, 56 hours in the lecture-hall (32 hours of lectures, 24 hours of practices).

Physics: 3 CFU, 28 hours in the lecture-hall (16 hours of lectures, 12 hours of practices).

The course takes place from the 17th of September to the 8th of October 2018, with 18 hours per week during the first three weeks and 2 hours on the 8th of October. Some hours between the 8th to the 12th of October will be devoted to reviewing and to the first exam (on the 12th of October).

The teaching activities consist of frontal lectures at the blackboard and practices. Some practices hours are devoted to guided exercises during which the students, autonomously or in small groups, solve the proposed exercises with the supervision of the teacher.

During the course three formative tests (each one hour long) are planned in order to point out gaps, evaluate the progress of the student’s learning and give a feedback to the students before the final test. The results of those tests will not affect the final test.

At the beginning of the course some additional material is uploaded on the platform Elly: lectures 2017-18, examinations 2017-18 with solution, exercises on the whole programme with solution. Moreover, every week worksheets of exercises will be uploaded on the platform in preparation of the formative tests and of the final exam (14 worksheets overall).

To download the teaching stuff the on-line registration is needed.

The uploaded lectures are an integral part of the teaching material.

Non-attending students are advised to check the available teaching material and the informations given by the teacher via the platform Elly.

Teaching methods:

The course is organised into a series of frontal lessons at the blackboard (1/2 of the course) and practical exercises (1/2 of the course). Each student individually has to do some exercises and the teacher will follow the progress using a series of revisions.

## Assessment methods and criteria

Method of testing learning:

The final evaluation on the learning consists of a 3 hours long written test; books, notes and electronic devices are not allowed.

The student must prove he has understood, and he is able to apply, the basic concepts of every topic in the programme.

The written test is splitted into two parts: a theoretical test only for students of Mathematics and an applying knowledge test for all students in Mathematics and Phisics.

The two tests are strucured as follow:

1) theoretical test (1 hour) with 4 questions on logic of propositions, set theory and function theory (0-31 points) and an optional short proof (0-2 points)

2) applying knowledge test (2 hours) with 5 exercises (0-33 points for Mathematics and 0-32 for Physics) and an optional exercise for Physics (0-2 points). A first preliminary exercise (0-15 points) contains eight simple questions on the whole programme, the second is an irrational or absolute value inequality (0-4 points), in the third it is asked to draw the graph of a piecewise defined function (0-7 points), the fourth consists of the analysis of a function whose graph is given (0-4 points for Mathematics, 0-3 points for Physics), while the fifth exercise concerns analytical geometry (0-3 points).

The final mark is calculated by adding the mark of the optional exercise to the points of the written test (0-32 points, for Mathematics the mean value of the two parts). The exam is passed with a final mark of minimum 18/30.

For Physics the written test programme does not contain logic of propositions, definitions and proofs.

The results of the exam will be published on the platform Elly within two weeks from the written test’s date.

The students can examine their written tests during the time specified by the teacher or by appointment.

Method of testing learning:

The final test of the course consists of a written test which is weighted as follows:

(10%) Theoretical questions (knowledge)

(90%) Practical exercises (applying knowledge)

The theoretical questions concern logic of propositions, set theory, functions and mathematical induction.

The exam is passed with a final mark of minimum 18/30.

## Other informations

Other information:

This course (6CFU) is mandatory for all students in Mathematics.

Part of this course (3CFU) is mandatory for all students in Physics.

The course of Basic Mathematics is a prerequisite to Algebra and Mathematical Analysis 1. Attending to the course is mandatory (75%) for all students who have not passed their self-evaluation test or are relieved from it, for instance with the certificate of the final test of the CORDA Project with two bonus points. For the full list of relieving conditions see the OFA (added formative obligations) in the Manifesto degli studi 2018-19 at the entry: ESONERO TEST VPI. Moreover for these students the course of Basic Mathematics is a prerequisite to all other exams in the degree course.

Attending to the course is strongly recommended to all students.

Other information:

This course(6CFU) is mandatory for all students in Mathematics.

A part of this course(3CFU) is mandatory for all students in Physics.

The course is concentrated in the first four weeks of the first semester, for 12 hours a week.

Attending to the course is strongly recommended.