Unit CALCULUS
 Course
 Business administration
 Studyunit Code
 2002109
 Location
 PERUGIA
 Curriculum
 In all curricula
 CFU
 9
 Course Regulation
 Coorte 2020
 Offered
 2020/21
 Learning activities
 Base
 Area
 Statisticomatematico
 Academic discipline
 SECSS/06
 Type of studyunit
 Obbligatorio (Required)
 Type of learning activities
 Attività formativa monodisciplinare
CALCULUS  Cognomi AL
Code  2002109 

Location  PERUGIA 
CFU  9 
Teacher  Mauro Pagliacci 
Teachers 

Hours 

Learning activities  Base 
Area  Statisticomatematico 
Academic discipline  SECSS/06 
Type of studyunit  Obbligatorio (Required) 
Language of instruction  Italian 
Contents  1. Introductory part 2. Elementary functions 3. Limits of functions and of infinite sequences, continuous functions 4. Elements of differential calculus and optimization 5. Elements of integration 6. Linear algebra 7. Elements of functions of several variables 
Reference texts  P. BOIERI, G. CHITI, Precorso di matematica, Zanichelli, Bologna, 1994 (for points 1. and 2. of the programme). L. PECCATI, S. SALSA, A. SQUELLATI, Matematica per l'Economia e l'Azienda, Egea, 2004 (for points 3., 4., 5., 6., 7. of the programme). 
Educational objectives  At the end of the course students will possess and will be able to use the main mathematical tools, suitable to business, economic, financial and statistical applications. 
Prerequisites  Basic mathematical notions, being already known to students since faced during every upper secondary school, are assumed to be preparatory to the course. For a sake of completeness and uniformity, preparatory topics are again considered during lectures. 
Teaching methods  The course is organized in facetoface lectures and exercise sessions. Some topics, being already known to students since faced during every upper secondary school, are assumed to be preparatory to the course. For a sake of completeness and uniformity, preparatory topics are again considered during lectures. 
Other information  Besides teaching and exercise hours, there will be extra hours of teaching support – mainly finalized to training for the written exam and to filling possible knowledge gaps – given either in lecture form or as individual meetings. Such activity will tentatively start at the beginning of October. 
Learning verification modality  The exam can be taken in two alternative modalities (a) and (b) explained below. In every case, the student must bring his university id card during the exam. (a) Standard exam: The exam is organized in a written test, including also theoretical questions, and in an oral test. A student is admitted to the oral test in case his mark in the written test is greater or equal to 15/30. The evaluation of the written test can be confirmed, after a discussion on the correction of the written test, if the mark is between 18/30 and 26/30. Students with a mark in the written test between 15/30 and 17/30 and those with a mark greater or equal to 27/30 must take the oral test. Every student passing the written test can anyway take the oral test. The discussion of the written test or the oral test must be taken before February, if the written test has been taken during the winter term (JanuaryFebruary) and before September if the written test has been taken during the summer term (JuneJulySeptember). A student not passing the written or oral test can take the next examination in the same term. (b) Exam in the form of an intermediate written test and a completion written test: For interested students, there is the opportunity to take an intermediate written test, during the teaching break, covering the first part of the programme. Passing the intermediate written test guarantees the access to a completion written test that will take place during the first examination date of the winter term in January. The intermediate and completion written tests are unique. Such tests are passed if the mark in both of them is greater or equal to 15/30. Students not passing either the intermediate or completion written test can take the exam according to the standard modality (a), during the first available examination date. Both the intermediate and completion written tests are not limited to students attending the first year. Students passing both tests with an average mark (rounded by ceiling) which is sufficient (= 18/30), can skip the oral test, getting as final evaluation the average of the two marks. Every student passing the written tests (intermediate and completion), can anyway take the oral exam. In any case, students must take the oral exam in the following two cases: • If the average of the marks in the intermediate and completion written tests is = 27/30; • If the average of the marks in the intermediate and completion written tests is 15, 16 or 17. The registration to the intermediate written test and the standard written test must be through the site: https://unipg.esse3.cineca.it/Home.do Teaching material, old written tests and other information on the course are available in the UniStudium page of the course. Information on facilities for special needs students are available at the link https://www.unipg.it/en/internationalstudents/generalinformation/facilitiesforspecialneedsstudents The above modalities are subject to possible changes due to University guidelines following the COVID19 emergency. 
Extended program  1. Introductory part – Mathematics as method and tool. Recall on set theory. Operations on sets. Functions between sets. Sets of numbers: natural, integer, rational and real numbers. Algebraic and order structure of R. Dense and complete sets. Infimum and supremum. Neighborhoods. Accumulation points. Isolated, interior, external and border points. Nondenumerability of R. Finite and infinite sets. Hints on complex numbers and on the fundamental theorem of Algebra. 2. Elementary functions – Real functions of a real variable and their graph. Infinite sequences. Equations and inequalities. The Cartesian equation of the line and of the circumference. Graphs of elementary functions and their transformations on the plane: the line, the parabola, the hyperbola, the square and cubic root, the power function, the exponential function and the logarithmic function, hints on trigonometric functions: properties and relations; solutions of right triangles. Even and odd functions. Bounded functions. Compound functions. Inverse function. Monotonic functions. Maxima and minima. Convex and concave functions. Epigraph of a function. 3. Limits of functions and of infinite sequences, continuous functions – Intuitive definition of limit for functions and infinite sequences. Continuous functions. Computation of limits. Indeterminate forms. Remarkable limits. Discontinuity of first and second type. Removable discontinuity. Theorems on continuous functions. Infinitesimals and infinites. 4. Elements of differential calculus and optimization – Definition of derivative. Right and left derivative. Geometric meaning. Connections between continuity and derivability. Derivatives of elementary functions. Derivation rules. Logarithmic derivative. Derivative of higher order. Differentiable functions. Elasticity of a function. Rolle, Lagrange and Cauchy’s theorems. de l'Hôpital’s theorems. Taylor’s formula. Increasing and decreasing functions. Maximum and minimum points. Convex and concave functions. Inflection points. Asymptotes. Study of functions. 5. Elements of integration – Definite integral and its properties. Mean value theorem. Primitives of a function. Fundamental theorem of calculus and its consequences. Indefinite integral. 6. Linear algebra – The vector space Rn. Operations on vectors. Linear dependent and independent vectors. Matrices. Operations on matrices. Determinant of a square matrix. Rank of a matrix. Linear systems. 7. Elements of functions of several variables – Functions of several variables. Graph of functions of several variables. Contour lines. Partial derivatives and their geometric meaning. Free and constrained maxima and minima. 
CALCULUS  Cognomi MZ
Code  2002109 

Location  PERUGIA 
CFU  9 
Teacher  Davide Petturiti 
Teachers 

Hours 

Learning activities  Base 
Area  Statisticomatematico 
Academic discipline  SECSS/06 
Type of studyunit  Obbligatorio (Required) 
Language of instruction  Italian 
Contents  1. Introductory part 2. Elementary functions 3. Limits of functions and of infinite sequences, continuous functions 4. Elements of differential calculus and optimization 5. Elements of integration 6. Linear algebra 7. Elements of functions of several variables 
Reference texts  P. BOIERI, G. CHITI, Precorso di matematica, Zanichelli, Bologna, 1994 (for points 1. and 2. of the programme). L. PECCATI, S. SALSA, A. SQUELLATI, Matematica per l'Economia e l'Azienda, Egea, 2004 (for points 3., 4., 5., 6., 7. of the programme). 
Educational objectives  At the end of the course students will possess and will be able to use the main mathematical tools, suitable to business, economic, financial and statistical applications. 
Prerequisites  Basic mathematical notions, being already known to students since faced during every upper secondary school, are assumed to be preparatory to the course. For a sake of completeness and uniformity, preparatory topics are again considered during lectures. 
Teaching methods  The course is organized in facetoface lectures and exercise sessions. Some topics, being already known to students since faced during every upper secondary school, are assumed to be preparatory to the course. For a sake of completeness and uniformity, preparatory topics are again considered during lectures. 
Other information  Besides teaching and exercise hours, there will be extra hours of teaching support – mainly finalized to training for the written exam and to filling possible knowledge gaps – given either in lecture form or as individual meetings. Such activity will tentatively start at the beginning of October. 
Learning verification modality  The exam can be taken in two alternative modalities (a) and (b) explained below. In every case, the student must bring his university id card during the exam. (a) Standard exam: The exam is organized in a written test, including also theoretical questions, and in an oral test. A student is admitted to the oral test in case his mark in the written test is greater or equal to 15/30. The evaluation of the written test can be confirmed, after a discussion on the correction of the written test, if the mark is between 18/30 and 26/30. Students with a mark in the written test between 15/30 and 17/30 and those with a mark greater or equal to 27/30 must take the oral test. Every student passing the written test can anyway take the oral test. The discussion of the written test or the oral test must be taken before February, if the written test has been taken during the winter term (JanuaryFebruary) and before September if the written test has been taken during the summer term (JuneJulySeptember). A student not passing the written or oral test can take the next examination in the same term. (b) Exam in the form of an intermediate written test and a completion written test: For interested students, there is the opportunity to take an intermediate written test, during the teaching break, covering the first part of the programme. Passing the intermediate written test guarantees the access to a completion written test that will take place during the first examination date of the winter term in January. The intermediate and completion written tests are unique. Such tests are passed if the mark in both of them is greater or equal to 15/30. Students not passing either the intermediate or completion written test can take the exam according to the standard modality (a), during the first available examination date. Both the intermediate and completion written tests are not limited to students attending the first year. Students passing both tests with an average mark (rounded by ceiling) which is sufficient (= 18/30), can skip the oral test, getting as final evaluation the average of the two marks. Every student passing the written tests (intermediate and completion), can anyway take the oral exam. In any case, students must take the oral exam in the following two cases: • If the average of the marks in the intermediate and completion written tests is = 27/30; • If the average of the marks in the intermediate and completion written tests is 15, 16 or 17. The registration to the intermediate written test and the standard written test must be through the site: https://unipg.esse3.cineca.it/Home.do Teaching material, old written tests and other information on the course are available in the UniStudium page of the course. Information on facilities for special needs students are available at the link https://www.unipg.it/en/internationalstudents/generalinformation/facilitiesforspecialneedsstudents The above modalities are subject to possible changes due to University guidelines following the COVID19 emergency. 
Extended program  1. Introductory part – Mathematics as method and tool. Recall on set theory. Operations on sets. Functions between sets. Sets of numbers: natural, integer, rational and real numbers. Algebraic and order structure of R. Dense and complete sets. Infimum and supremum. Neighborhoods. Accumulation points. Isolated, interior, external and border points. Nondenumerability of R. Finite and infinite sets. Hints on complex numbers and on the fundamental theorem of Algebra. 2. Elementary functions – Real functions of a real variable and their graph. Infinite sequences. Equations and inequalities. The Cartesian equation of the line and of the circumference. Graphs of elementary functions and their transformations on the plane: the line, the parabola, the hyperbola, the square and cubic root, the power function, the exponential function and the logarithmic function, hints on trigonometric functions: properties and relations; solutions of right triangles. Even and odd functions. Bounded functions. Compound functions. Inverse function. Monotonic functions. Maxima and minima. Convex and concave functions. Epigraph of a function. 3. Limits of functions and of infinite sequences, continuous functions – Intuitive definition of limit for functions and infinite sequences. Continuous functions. Computation of limits. Indeterminate forms. Remarkable limits. Discontinuity of first and second type. Removable discontinuity. Theorems on continuous functions. Infinitesimals and infinites. 4. Elements of differential calculus and optimization – Definition of derivative. Right and left derivative. Geometric meaning. Connections between continuity and derivability. Derivatives of elementary functions. Derivation rules. Logarithmic derivative. Derivative of higher order. Differentiable functions. Elasticity of a function. Rolle, Lagrange and Cauchy’s theorems. de l'Hôpital’s theorems. Taylor’s formula. Increasing and decreasing functions. Maximum and minimum points. Convex and concave functions. Inflection points. Asymptotes. Study of functions. 5. Elements of integration – Definite integral and its properties. Mean value theorem. Primitives of a function. Fundamental theorem of calculus and its consequences. Indefinite integral. 6. Linear algebra – The vector space Rn. Operations on vectors. Linear dependent and independent vectors. Matrices. Operations on matrices. Determinant of a square matrix. Rank of a matrix. Linear systems. 7. Elements of functions of several variables – Functions of several variables. Graph of functions of several variables. Contour lines. Partial derivatives and their geometric meaning. Free and constrained maxima and minima. 