MATHS
(objectives)
Learn basic contents and techniques from Mathematical Analysis, which are needed to study functions, to solve problems relying on integral calculus and to solve simple differential equations. When possible, these themes will be related to applied problems, mainly in Biology.
Students will learn basic concepts: limit, differentiability, study of functions; integral and its applications; differential equations.
This concepts will be used to solve concrete problems and to face simple mathematical models.
Knowledge and understanding (Dublin descriptor 1) Understand the concepts of function, limiti, differentiability of functions of 1 variable and all notions needed to study a function; integral, methods of integration and basic applications of integral calculus; differential equation and some resolution methods. Applying knowledge and understanding (Dublin descriptor 2) To be able to use the studied tools to • solve equations and inequalities; • calcolate limits, derivates, integrals and study functions; • solve differential equations. Making judgements (Dublin descriptor 3) • To be able to detect the rules needed to solve new problems, analogous to the ones faced in lessons. Communication skills (Dublin descriptor 4) • Stimulate students to intervene, reason and discuss on questions raised in lessons. Learning skills (Dublin descriptor 5) • To be able to discuss some scientific topics with easy mathematical models.
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Code
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118542 |
Language
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ITA |
Type of certificate
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Profit certificate
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Credits
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7
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Scientific Disciplinary Sector Code
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MAT/05
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Contact Hours
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48
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Exercise Hours
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8
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Type of Activity
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Basic compulsory activities
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Group: 1
Teacher
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MUGNAI Dimitri
(syllabus)
Functions and number sets Introduction: operations among sets. Function; domain, co-domain, image and graph of functions. Injective, surjective, inverse function and composition. Increasing and decreasing, odd and even functions, Number sets N, Z, Q, R.
Elementary functions Review on lines, parabolas, exponential, logarithmic and trigoniometric functions. Absolute value. Neighborhood of a real number.
Sequences Limit of sequences. Uniqueness of the limit. Extended algebra of limits. Monotone sequences. Nepero's number. Sign permanence theorem. Comparison theorem. Squeeze theorem (Theorem of the two Cabinieri). Special limits. Hierarchy of functions.
Limit and continuity Finite and infinite limit; sign permanence theorem. Right-side and left-side limit. Existence and uniqueness of the limit. Comparison theorem. Algebra of limits and indeterminate forms. Infinite and infinitesimal. Vertical, horizontal and oblique asymptote. Continuous functions. Weierstrass theorem. Intermediate value theorem. Intermediate zero theorem.
Derivatives Definition of derivative and its geometric interpretation. Calculation of derivatives. Differentiability and continuity. Point of non differentiability. Higher derivatives. Rolle's and Lagrange's theorem. De L’Hôpital's theorems. Taylor's theorem and McLaurin's expansion. Fermat's theorem. Maximum and minimum points. Convexity and concavity. Inflection point. Study of a function.
Integral Definition of indefinite integral and its properties. Straightforward anti-derivatives. Integration by parts. Integration by substitution. Definite integral and its properties. The fundamental theorem of calculus. generalized integral. Area. Volume of solids of revolution. Measure of the ball and of the sphere. Generalized integrals. Comparison and absolute convergence theorem. Integrability of negative powers at 0 and at infinity.
Differential Equations Differential Equations: an introduction. Differential Equations of first and second order and Cauchy problems. Separate variables differential equations. Malthus model; bacterial growth; epidemic diffusion; radioactive decay. Logistic growth. Time of the crime.
(reference books)
"Elementi di Calcolo. Versione semplificata per i nuovi corsi di laurea"
di Paolo Marcellini e Carlo Sbordone
Liguori Editore.
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Dates of beginning and end of teaching activities
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From to |
Delivery mode
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Traditional
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Attendance
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not mandatory
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Evaluation methods
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Written test
Oral exam
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Group: Nuovo canale 2
Teacher
|
MUGNAI Dimitri
(syllabus)
Functions and number sets Introduction: operations among sets. Function; domain, co-domain, image and graph of functions. Injective, surjective, inverse function and composition. Increasing and decreasing, odd and even functions, Number sets N, Z, Q, R.
Elementary functions Review on lines, parabolas, exponential, logarithmic and trigoniometric functions. Absolute value. Neighborhood of a real number.
Sequences Limit of sequences. Uniqueness of the limit. Extended algebra of limits. Monotone sequences. Nepero's number. Sign permanence theorem. Comparison theorem. Squeeze theorem (Theorem of the two Cabinieri). Special limits. Hierarchy of functions.
Limit and continuity Finite and infinite limit; sign permanence theorem. Right-side and left-side limit. Existence and uniqueness of the limit. Comparison theorem. Algebra of limits and indeterminate forms. Infinite and infinitesimal. Vertical, horizontal and oblique asymptote. Continuous functions. Weierstrass theorem. Intermediate value theorem. Intermediate zero theorem.
Derivatives Definition of derivative and its geometric interpretation. Calculation of derivatives. Differentiability and continuity. Point of non differentiability. Higher derivatives. Rolle's and Lagrange's theorem. De L’Hôpital's theorems. Taylor's theorem and McLaurin's expansion. Fermat's theorem. Maximum and minimum points. Convexity and concavity. Inflection point. Study of a function.
Integral Definition of indefinite integral and its properties. Straightforward anti-derivatives. Integration by parts. Integration by substitution. Definite integral and its properties. The fundamental theorem of calculus. generalized integral. Area. Volume of solids of revolution. Measure of the ball and of the sphere. Generalized integrals. Comparison and absolute convergence theorem. Integrability of negative powers at 0 and at infinity.
Differential Equations Differential Equations: an introduction. Differential Equations of first and second order and Cauchy problems. Separate variables differential equations. Malthus model; bacterial growth; epidemic diffusion; radioactive decay. Logistic growth. Time of the crime.
(reference books)
"Elementi di Calcolo. Versione semplificata per i nuovi corsi di laurea"
by Paolo Marcellini and Carlo Sbordone
Liguori Editore.
|
Dates of beginning and end of teaching activities
|
From to |
Delivery mode
|
Traditional
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Attendance
|
not mandatory
|
Evaluation methods
|
Written test
Oral exam
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