Mathematics
A.Y. 2023/2024
Learning objectives
The goal of the course is to introduce some mathematical concepts and tools with particular reference to the topics which can be useful for applications to Agricultural and Food Sciences. The course aims at helping students to gain an adequate theoretical understanding of the matter, as well as good computational skills. At the end of the course students should be able to exploit their math knowledge in order to set and solve simple applied problem in a rigorous way.
Expected learning outcomes
Knowledge and understanding concepts of basic mathematics and elementary Mathematical Analysis. In particular, with regard to basic mathematics, the student will be able to manipulate formulas containing algebraic expressions, percentages and proportions, radicals, logarithms and exponentials, to solve equations and inequalities, to use the main tools and techniques of analytical geometry, plane and solid geometry and trigonometry. As far as elementary Mathematical Analysis is concerned, the student will be able to draw and use graphics of real functions of one variable in many different frameworks, to calculate limits, derivatives and integrals and to use these concepts for describing and solving real problems. Moreover, students will be able to understand and execute autonomously simple mathematical steps commonly used in the scientific literature of his own sector.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1. The integers, the rational numbers, the real numbers and operations. Percentage and propotions and their use in resolution of real problems. Absolute value, nth roots, logarithms and exponential: definitions and properties. Definitions and primary properties of goniometry.
2. Equations and disequations: algebric of various degrees, fractional, irrational, with absolute values, systems of disequations. Exponential, logarithmic, trigonometric equations and disequations.
3. Review of analytic geometry. Cartesian coordinates; equations of the line; perpendicular and parallel lines; distance between 2 points and between line and point; midpoints formula and axis of a segment. Proper and improper bundles of straight lines. Intersections of lines and systems of equations. Linear functions and their applications in real problems. Systems of disequations in 2 variables. Equation and graphic of parabola and applications to equation and disequation of second degree. Condition of tangency. Equilateral hyperbola and homographic function. Short review of trigonometry: sines and Carnot theorems.
4. Functions: Definition. Domain, codomain. Graph of a function. Injective functions and inverse functions. Composition of functions. Increasing , decreasing functions, convex and concave functions. Limited and unlimited functions. Maximum and minimum points in functions. Zeros an sign of functions. Even and odd functions. Review of polynomials, exponential, logarithmic, trigonometric and inverse functions. Operations on functions (translations, symmetry, absolute value). Disequations with graphic method.
5. Open and closed interval. Accumulation points. Definition of limits. Right and left limits. Horizontal and vertical asymptotes. Existence and uniqueness of the limit. The permanence of the sign theorem, comparison theorem and others. Continuity of a function. Computing limits and algebra of limits. Limits of compound functions. Special limits Indeterminate forms. Comparison of limits. Slant asymptote. Continuity of a function and properties. Weierstrass theorem and zero theorem.
6. Differential calculus and qualitative study of a function. Derivatives: definition, right and left derivates.Geometrical meaning, rules for differentiating functions (sum, difference, product, quotient, chain rules.. Differentiability and continuity. Tangent line at a point of a function. Rolle's theorem, Lagrange's theorem and De L'Hopital's theorem. Higher derivates. Relative maxima and minima. Fermat's theorem. Derivability and monotonia. Critical points. Twice differentiable functions and concavities. Concavity test and inflection points. The study of a function of one real variable.
7. Integration: primitives of a function, the indefinite integral. Integrability conditions. Methods of integration: by parts, substitutions, partial fractions . The definite integral. The fundamental Theorem of Integral Calculus. integral Mean Value Theorem. The calculus of plane areas.
2. Equations and disequations: algebric of various degrees, fractional, irrational, with absolute values, systems of disequations. Exponential, logarithmic, trigonometric equations and disequations.
3. Review of analytic geometry. Cartesian coordinates; equations of the line; perpendicular and parallel lines; distance between 2 points and between line and point; midpoints formula and axis of a segment. Proper and improper bundles of straight lines. Intersections of lines and systems of equations. Linear functions and their applications in real problems. Systems of disequations in 2 variables. Equation and graphic of parabola and applications to equation and disequation of second degree. Condition of tangency. Equilateral hyperbola and homographic function. Short review of trigonometry: sines and Carnot theorems.
4. Functions: Definition. Domain, codomain. Graph of a function. Injective functions and inverse functions. Composition of functions. Increasing , decreasing functions, convex and concave functions. Limited and unlimited functions. Maximum and minimum points in functions. Zeros an sign of functions. Even and odd functions. Review of polynomials, exponential, logarithmic, trigonometric and inverse functions. Operations on functions (translations, symmetry, absolute value). Disequations with graphic method.
5. Open and closed interval. Accumulation points. Definition of limits. Right and left limits. Horizontal and vertical asymptotes. Existence and uniqueness of the limit. The permanence of the sign theorem, comparison theorem and others. Continuity of a function. Computing limits and algebra of limits. Limits of compound functions. Special limits Indeterminate forms. Comparison of limits. Slant asymptote. Continuity of a function and properties. Weierstrass theorem and zero theorem.
6. Differential calculus and qualitative study of a function. Derivatives: definition, right and left derivates.Geometrical meaning, rules for differentiating functions (sum, difference, product, quotient, chain rules.. Differentiability and continuity. Tangent line at a point of a function. Rolle's theorem, Lagrange's theorem and De L'Hopital's theorem. Higher derivates. Relative maxima and minima. Fermat's theorem. Derivability and monotonia. Critical points. Twice differentiable functions and concavities. Concavity test and inflection points. The study of a function of one real variable.
7. Integration: primitives of a function, the indefinite integral. Integrability conditions. Methods of integration: by parts, substitutions, partial fractions . The definite integral. The fundamental Theorem of Integral Calculus. integral Mean Value Theorem. The calculus of plane areas.
Prerequisites for admission
As a first semester course in the first year, there are no specific prerequisites other than those required for entrance to the degree course. High school mathematical formalism are used.
Teaching methods
Frontal lessons, exercises, use of e-learning platform associated with the textbook and test simulation.
The course uses the Ariel platform, on which are loaded weekly sheets of exercises and other teaching materials related to the topics covered in the lesson.
A tutor will integrate the course lessons with additional exercise sessions, in order to enhance the learning of some important subjects and the preparation for the exam (written test).
As far as smart lessons (distance learning) are concerned, please check for the most updated university provisions.
For the time being, lessons and exercises will be carried out in presence.
Attendance at the course, although not compulsory, is strongly recommended.
The course uses the Ariel platform, on which are loaded weekly sheets of exercises and other teaching materials related to the topics covered in the lesson.
A tutor will integrate the course lessons with additional exercise sessions, in order to enhance the learning of some important subjects and the preparation for the exam (written test).
As far as smart lessons (distance learning) are concerned, please check for the most updated university provisions.
For the time being, lessons and exercises will be carried out in presence.
Attendance at the course, although not compulsory, is strongly recommended.
Teaching Resources
Lesson notes, other eventually advised textbooks and exam topics (available on web portal Ariel).
Suggested textbook:
Annaratone S. "Matematica sul Campo: metodi ed esempi per le scienze della vita". Pearson.
Abate M. "Matematica e Statistica" / 4 Ed con Connect_McGraw Hill
(only part "Matematica")
Any other textbook dealing with the subjects of the program can be equally used.
Teacher availability:
The teacher is available for explanations or clarifications just after the end of each lesson or later on by mail at the university mail address.
Suggested textbook:
Annaratone S. "Matematica sul Campo: metodi ed esempi per le scienze della vita". Pearson.
Abate M. "Matematica e Statistica" / 4 Ed con Connect_McGraw Hill
(only part "Matematica")
Any other textbook dealing with the subjects of the program can be equally used.
Teacher availability:
The teacher is available for explanations or clarifications just after the end of each lesson or later on by mail at the university mail address.
Assessment methods and Criteria
The learning verification goes normally through an exam consisting in a written test about the entire course program of the semester. In particular cases hereafter specified, an additional oral test could also be taken into consideration.
The students can get through the exam whether by means of two intermediate tests (so called "prove in itinere", that is, taking place during the course) or by attending the exam sessions (so called "appelli d'esame", taking place after the end of the course).
Intermediate tests ("prove in itinere")
Students have an option (not mandatory) to get through the exam by means of two written intermediate tests, each one lasting 90 minutes and consisting in three problems / exercises and one theoretical question. The first test will take place at the end of the first half-semester and will deal with the subjects taught in the first part of the course; the second test will take place just at the end of the semester and will deal with the subjects taught in the second part of the course.
Particularly, the following scenarios could occur:
- In the first intermediate test, the student gets a mark equal to or higher than 18/30.
In such first case, the student can attend the second intermediate test at the end of the semester. If, in the second test, the student gets a mark equal to or higher than 18/30, then the exam will be considered got through, with a final mark calculated as the average of the marks in the two intermediate tests.
- In the first intermediate test, the student gets a mark equal to or higher than 18/30, yet in the second intermediate test the student gets a mark lower than 18/30 or he/she turns out to be absent. In such second case, the student must repeat the written test entirely in one of the foreseen exam sessions.
- In the first intermediate test the student gets a mark lower than 18/30 or he/she turns out to be absent. In such third case, the student will not be admitted to the second intermediate test and will have to get through the exam by means of one of the foreseen exam sessions.
Please remind that intermediate tests are not mandatory. However, for organization reasons, those students who, for any reason, are not going to attend the second intermediate test, are kindly invited to inform the teacher by mail at least within one week before the date planned for the second intermediate test.
Exam sessions
An exam session entails a written test, lasting 120 minutes and consisting in five problems / exercises and one theoretical question, concerning the entire course program. If the student gets a mark lower than 18/30, he/she needs to repeat the test during one of the next exam sessions.
Exam sessions take place after the end of the course. Normally the first exam session (so called "preappello") is organized in the same day as the second intermediate test and can be attended also by those students who have gone through the first intermediate test: be aware that a positive mark obtained in the latter would be automatically invalidated.
Rules
In order to attend any type of written test, the students should correctly register using the Online Services (former SIFA) web application. They should get to the classroom at least 15 minutes before the beginning of the written test, provided with university badge, passport and protocol sheets. Dates, hours and rooms of the tests will be published on the Online Services (former SIFA) web application.
During any type of written test, students are not allowed to look up in textbooks, notes, forms and alike and to use pocket calculators, PC's and mobile phones of any kind.
Further, they are not allowed to talk with each other: should such a circumstance occur, students would be promptly banned out of the room.
Further, during the written test the students cannot leave the room. After the first hour, they can submit the test or withdraw.
The results of any type of written test are published on the web portal Ariel, along with the date planned for the review and comment of the corrected tests.
The students are not obliged to attend the test review. Anyway, should a student decide to refuse a positive mark (> or = 18/30) and to repeat the written test, he/she needs to inform expressly the teacher not later than the end of the test review.
Oral test
Those students who get a final mark equal to or higher than 25/30 in the written test (both after intermediate tests and exam session) can ask for taking an additional oral test, in order to integrate the evaluation. For organization reasons the request should be done within 48 hours before the review of the corrected tests.
Oral tests, when relevant, are arranged after reviewing the written tests.
During the oral test, the teacher will evaluate the ability of the student in properly using terms and symbols, effectively solving a problem by means of the most suitable algebric and graphic tools and analysing and interpreting the problem results.
In case of oral test, students should be aware that the final mark could be lower than the one obtained after the written test, since it will need to be calculated as the average of the marks coming from written and oral tests.
Examples of previous written tests are available on web portal Ariel.
The students can get through the exam whether by means of two intermediate tests (so called "prove in itinere", that is, taking place during the course) or by attending the exam sessions (so called "appelli d'esame", taking place after the end of the course).
Intermediate tests ("prove in itinere")
Students have an option (not mandatory) to get through the exam by means of two written intermediate tests, each one lasting 90 minutes and consisting in three problems / exercises and one theoretical question. The first test will take place at the end of the first half-semester and will deal with the subjects taught in the first part of the course; the second test will take place just at the end of the semester and will deal with the subjects taught in the second part of the course.
Particularly, the following scenarios could occur:
- In the first intermediate test, the student gets a mark equal to or higher than 18/30.
In such first case, the student can attend the second intermediate test at the end of the semester. If, in the second test, the student gets a mark equal to or higher than 18/30, then the exam will be considered got through, with a final mark calculated as the average of the marks in the two intermediate tests.
- In the first intermediate test, the student gets a mark equal to or higher than 18/30, yet in the second intermediate test the student gets a mark lower than 18/30 or he/she turns out to be absent. In such second case, the student must repeat the written test entirely in one of the foreseen exam sessions.
- In the first intermediate test the student gets a mark lower than 18/30 or he/she turns out to be absent. In such third case, the student will not be admitted to the second intermediate test and will have to get through the exam by means of one of the foreseen exam sessions.
Please remind that intermediate tests are not mandatory. However, for organization reasons, those students who, for any reason, are not going to attend the second intermediate test, are kindly invited to inform the teacher by mail at least within one week before the date planned for the second intermediate test.
Exam sessions
An exam session entails a written test, lasting 120 minutes and consisting in five problems / exercises and one theoretical question, concerning the entire course program. If the student gets a mark lower than 18/30, he/she needs to repeat the test during one of the next exam sessions.
Exam sessions take place after the end of the course. Normally the first exam session (so called "preappello") is organized in the same day as the second intermediate test and can be attended also by those students who have gone through the first intermediate test: be aware that a positive mark obtained in the latter would be automatically invalidated.
Rules
In order to attend any type of written test, the students should correctly register using the Online Services (former SIFA) web application. They should get to the classroom at least 15 minutes before the beginning of the written test, provided with university badge, passport and protocol sheets. Dates, hours and rooms of the tests will be published on the Online Services (former SIFA) web application.
During any type of written test, students are not allowed to look up in textbooks, notes, forms and alike and to use pocket calculators, PC's and mobile phones of any kind.
Further, they are not allowed to talk with each other: should such a circumstance occur, students would be promptly banned out of the room.
Further, during the written test the students cannot leave the room. After the first hour, they can submit the test or withdraw.
The results of any type of written test are published on the web portal Ariel, along with the date planned for the review and comment of the corrected tests.
The students are not obliged to attend the test review. Anyway, should a student decide to refuse a positive mark (> or = 18/30) and to repeat the written test, he/she needs to inform expressly the teacher not later than the end of the test review.
Oral test
Those students who get a final mark equal to or higher than 25/30 in the written test (both after intermediate tests and exam session) can ask for taking an additional oral test, in order to integrate the evaluation. For organization reasons the request should be done within 48 hours before the review of the corrected tests.
Oral tests, when relevant, are arranged after reviewing the written tests.
During the oral test, the teacher will evaluate the ability of the student in properly using terms and symbols, effectively solving a problem by means of the most suitable algebric and graphic tools and analysing and interpreting the problem results.
In case of oral test, students should be aware that the final mark could be lower than the one obtained after the written test, since it will need to be calculated as the average of the marks coming from written and oral tests.
Examples of previous written tests are available on web portal Ariel.
MAT/02 - ALGEBRA - University credits: 6
Practicals: 40 hours
Lessons: 28 hours
Lessons: 28 hours
Professor:
Genco Immacolata
Professor(s)