Calculus
A.Y. 2023/2024
Learning objectives
The educational objective of the course is to provide students with theoretical and practical matematical tools for their application in the basic and the characteristics courses of the CdS.
Expected learning outcomes
At the end of the course the students will be able to apply the main concepts of infinitesimal calculus to the resolution of exercises related to the topics covered in class.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
Integer numbers, rational numbers and real numbers. Ordering of the real line, the symbols of ±∞, absolute value, nth roots, logarithms and exponentials: definitions and properties. Percentages, averages and proportions and their use in solving real problems. (0.5 CFU)
Equations and inequalities: I and II degrees, fractional, irrational, logarithmic and exponential, with absolute value, inequalities systems of two variables . (0.5 CFU)
Elementary functions and their graphics: exponential, logarithmic functions. Domain, range, injective function, surjective function, even function, odd function, increasing and decreasing functions, inverse function, composition of functions. (1 CFU)
Cartesian coordinates system, element of analytic geometry, distance betwee two points, midpoint, straight lines, axis of a segment.
Elementary functions and their graphs. Zeroes of a function. Positivity and negativity of a function. Geometric transformations and functions. (1 CFU)
Real functions of one real variable: definition of relations and function
Limits and continuity: limits of functions. Continuity of functions Existence of limits for monotone functions. Limits and algebraic operations. Indeterminate forms of limits. Asymptotes. Discontinuities. Weierstrass Theorem. Intermediate value theorem and applications. (1 CFU)
Differential calculus for real functions of one real variable.
Differentiability and the definition of derivative. Geometrical meaning. Derivatives of elementary functions. Computation of derivatives: algebraic operations, composition of functions. The theorems of Rolle, Lagrange. Higher order derivatives.
Applications of differential calculus to the study of functions: monotonicity, local and global optimisation. De L'Hospital Theorem. Asymptotes. The study of a function of one real variable. (1.5 CFU)
Integration: primitives of a function, the indefinite integral, the definite integral and the calculus of plane areas -the fundamental Theorem of Integral Calculus. (1.5 CFU)
Equations and inequalities: I and II degrees, fractional, irrational, logarithmic and exponential, with absolute value, inequalities systems of two variables . (0.5 CFU)
Elementary functions and their graphics: exponential, logarithmic functions. Domain, range, injective function, surjective function, even function, odd function, increasing and decreasing functions, inverse function, composition of functions. (1 CFU)
Cartesian coordinates system, element of analytic geometry, distance betwee two points, midpoint, straight lines, axis of a segment.
Elementary functions and their graphs. Zeroes of a function. Positivity and negativity of a function. Geometric transformations and functions. (1 CFU)
Real functions of one real variable: definition of relations and function
Limits and continuity: limits of functions. Continuity of functions Existence of limits for monotone functions. Limits and algebraic operations. Indeterminate forms of limits. Asymptotes. Discontinuities. Weierstrass Theorem. Intermediate value theorem and applications. (1 CFU)
Differential calculus for real functions of one real variable.
Differentiability and the definition of derivative. Geometrical meaning. Derivatives of elementary functions. Computation of derivatives: algebraic operations, composition of functions. The theorems of Rolle, Lagrange. Higher order derivatives.
Applications of differential calculus to the study of functions: monotonicity, local and global optimisation. De L'Hospital Theorem. Asymptotes. The study of a function of one real variable. (1.5 CFU)
Integration: primitives of a function, the indefinite integral, the definite integral and the calculus of plane areas -the fundamental Theorem of Integral Calculus. (1.5 CFU)
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.
Teaching methods
Frontal lessons, exercises, applications of examples to concrete cases, use of e-learning platform associated with the textbook, use of educational software, group work, use of didactic games as a motivational lever for the learning of the subject and as a tool of verification and self-evaluation on curricular themes. The course uses the MyAriel platform, on which are loaded weekly sheets of exercises and other teaching materials related to the topics covered in the lesson. Attendance at the course, although not compulsory, is strongly recommended.
Teaching Resources
Silvia Annaratone, Matematica sul campo. Metodi ed esempi per le scienze della vita 2/Ed. con MyLab
(ISBN 9788891910615, Euro 29,00)
(ISBN 9788891910615, Euro 29,00)
Assessment methods and Criteria
The examination consists of a written test and an oral test. The written test is organized in two parts:
· Part A, lasting 30 minutes, consists of 10 open questions concerning the prerequisites to the course. The extremely simple questions are intended to assess whether the student has the minimum skills to approach a university course of mathematics. Part A will be passed answering correctly at least 8 out of 10 questions. Passing Part A is a necessary (but not sufficient!) condition for passing the written test
· Part B, which lasts 90 minutes, consists of six exercises related to topics of the course. The written test is passed if and only if both parts A and B are passed. The score obtained in Part A does not contribute to the final score of the written test.
The use of calculator if forbidden during the written test.
Particular cases: Laboratorio di Matematica di base (only if it is activated)
Students who took part to "Laboratorio di Matematica di base", performing at least 80% of the scheduled activities in the allotted time and passing the final test are exempted from Part A of the written test for the first 5 sessions.
To attend the written test, students must be enrolled regularly through SIFA and must be in front of the classroom 15 minutes before the beginning of the written test, with photo ID and protocol sheets.
The total duration of the written test is 2 hours. During the written test it is forbidden to consult books, notes, use calculators of any kind, computers and mobile phones. It is also forbidden to communicate with the companions. During all the written test it is also forbidden to leave the classroom: in particular, during the first hour of part B it will not be possible to leave the classroom for any reason. At the end of the first hour, students who wish to do so can either finish or withdraw. The oral test may be taken only if the written test has been passed with a score of 18/30 or more, and only at the same session of the written test. Students who, after passing the written test, did not show up for the oral part will fail the exam.
The oral test will cover all the topics covered in the course.
The final exam mark will be the arithmetic mean between the written mark and the oral mark and will be expressed over 30.
During the exam the student must demonstrate knowledge of the contents of the course syllabus and be able to:
- operate with mathematical symbolism and formulas
- formalize the solution of a problem through algebraic and graphical models
- use mathematical methods, tools, and models in different situations
- identify apprpriate strategies for problem solving
- use appropriate language and symbology
- analyze an interpret the results obtained
In particular, with regard to basic mathematics, the student will have to demonstrate to 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.
As far as elementary Mathematical Analysis is concerned, the student will have to demonstrate to be able to draw and use, both qualitatively and quantitatively, 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.
Examples of written tests from past years are available on the Ariel course website.
· Part A, lasting 30 minutes, consists of 10 open questions concerning the prerequisites to the course. The extremely simple questions are intended to assess whether the student has the minimum skills to approach a university course of mathematics. Part A will be passed answering correctly at least 8 out of 10 questions. Passing Part A is a necessary (but not sufficient!) condition for passing the written test
· Part B, which lasts 90 minutes, consists of six exercises related to topics of the course. The written test is passed if and only if both parts A and B are passed. The score obtained in Part A does not contribute to the final score of the written test.
The use of calculator if forbidden during the written test.
Particular cases: Laboratorio di Matematica di base (only if it is activated)
Students who took part to "Laboratorio di Matematica di base", performing at least 80% of the scheduled activities in the allotted time and passing the final test are exempted from Part A of the written test for the first 5 sessions.
To attend the written test, students must be enrolled regularly through SIFA and must be in front of the classroom 15 minutes before the beginning of the written test, with photo ID and protocol sheets.
The total duration of the written test is 2 hours. During the written test it is forbidden to consult books, notes, use calculators of any kind, computers and mobile phones. It is also forbidden to communicate with the companions. During all the written test it is also forbidden to leave the classroom: in particular, during the first hour of part B it will not be possible to leave the classroom for any reason. At the end of the first hour, students who wish to do so can either finish or withdraw. The oral test may be taken only if the written test has been passed with a score of 18/30 or more, and only at the same session of the written test. Students who, after passing the written test, did not show up for the oral part will fail the exam.
The oral test will cover all the topics covered in the course.
The final exam mark will be the arithmetic mean between the written mark and the oral mark and will be expressed over 30.
During the exam the student must demonstrate knowledge of the contents of the course syllabus and be able to:
- operate with mathematical symbolism and formulas
- formalize the solution of a problem through algebraic and graphical models
- use mathematical methods, tools, and models in different situations
- identify apprpriate strategies for problem solving
- use appropriate language and symbology
- analyze an interpret the results obtained
In particular, with regard to basic mathematics, the student will have to demonstrate to 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.
As far as elementary Mathematical Analysis is concerned, the student will have to demonstrate to be able to draw and use, both qualitatively and quantitatively, 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.
Examples of written tests from past years are available on the Ariel course website.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 8
Practicals: 48 hours
Lessons: 40 hours
Lessons: 40 hours
Professor:
Scappini Nadia
Educational website(s)
Professor(s)