#
Calculus

A.Y. 2015/2016

Learning objectives

The course is splitted in three units with the aim of providing the basic elements of Mathematics, Statistics and Computer Science.

Expected learning outcomes

Undefined

**Lesson period:** year
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### A-L

Responsible

Lesson period

year

**modulo: Matematica generale**

**Course syllabus**

Unit: Calculus:

Numbers: natural, rational and real numbers. The field R and its operations. The simbols of infinity. Inequalities in R.

Functions: 1:1, onto and bijective functions; composition and invertibility. Real functions of one real variable, graphics, monotonicity. Elementary functions (powers, logarithms, exponentials, ...).

Linear Algebra: vectors, matrices and their operations. Determinants, inversion, and ranks. Linear systems. Cramer's and Rouché's theorems.

Limits: definitions and basic properties; uniqueness; monotonicity. Limits for elementary functions. Uncertainties. Asymptotics and comparison results. Eulero's number "e" and some fundamental limits. Divergent and vanishing functions. Continuity and related proprties: Darboux and Weierstrass' results.

Differential calculus: first derivative, geometric and mechanical meaning, tangent line. Operations with derivatives: chain rule. Differentiation of elementary functions. Optimization and the classical theorems (Fermat, Rolle, Lagrange), monotonicity. Convexity. Hopital's rule. Qualitative study for the graph of a real valued function. Antiderivatives and some methods to compute indefinite integrals.

Integral calculus: definite integrals and their main properties. The Fundamental Theorem of Integral Calculus. Area of plane regions.

Formal prerequisites

None

Suggested textbook:

P. Marcellini, C. Sbordone, Calcolo - edizione aggiornata per i nuovi corsi di laurea, Liguori editore (2004).

Online course "Matematica Assistita", http://ariel.unimi.it/User/

Previous knowledge and exam

Knowledge of elementary algebra and plane trigonometry, use of logarithms in the real field.

Written and oral exam, being possible to replace the written final exam with medium term exams. Attendance is highly recommended.

Teaching methods:

Traditional

Teaching language

Italian

Web page http://users.mat.unimi.it/users/paleari

Numbers: natural, rational and real numbers. The field R and its operations. The simbols of infinity. Inequalities in R.

Functions: 1:1, onto and bijective functions; composition and invertibility. Real functions of one real variable, graphics, monotonicity. Elementary functions (powers, logarithms, exponentials, ...).

Linear Algebra: vectors, matrices and their operations. Determinants, inversion, and ranks. Linear systems. Cramer's and Rouché's theorems.

Limits: definitions and basic properties; uniqueness; monotonicity. Limits for elementary functions. Uncertainties. Asymptotics and comparison results. Eulero's number "e" and some fundamental limits. Divergent and vanishing functions. Continuity and related proprties: Darboux and Weierstrass' results.

Differential calculus: first derivative, geometric and mechanical meaning, tangent line. Operations with derivatives: chain rule. Differentiation of elementary functions. Optimization and the classical theorems (Fermat, Rolle, Lagrange), monotonicity. Convexity. Hopital's rule. Qualitative study for the graph of a real valued function. Antiderivatives and some methods to compute indefinite integrals.

Integral calculus: definite integrals and their main properties. The Fundamental Theorem of Integral Calculus. Area of plane regions.

Formal prerequisites

None

Suggested textbook:

P. Marcellini, C. Sbordone, Calcolo - edizione aggiornata per i nuovi corsi di laurea, Liguori editore (2004).

Online course "Matematica Assistita", http://ariel.unimi.it/User/

Previous knowledge and exam

Knowledge of elementary algebra and plane trigonometry, use of logarithms in the real field.

Written and oral exam, being possible to replace the written final exam with medium term exams. Attendance is highly recommended.

Teaching methods:

Traditional

Teaching language

Italian

Web page http://users.mat.unimi.it/users/paleari

**modulo: Laboratorio di Metodi Matematici e Statistici**

**Course syllabus**

Unit: Mathematical and Statistical methods

Descriptive Statistics.

18) Population, sample, parameter, statistics. Types of data and variables. Sampling.

19) Graphs and tables. Frequency tables. Histograms/bar graphs.

20) Mean, modal value, median, midrange and their relations. Range, standard deviation, variance and their relations. Percentiles, quartiles and outliers. Boxplot. Weighted mean.

Probability and random variables.

21) Introduction.Events and space of events; probability o fan event.

22) Probability of the union and the intersection. Complemento of an event. Independence. Conditional probability. Bayes Theorem.

23) Random Variables. Expected value, variance and deviation standard of discrete r.v.s.

24) Discrete r.v.s: Binomial and Poisson. Continuous r.v.s: Uniform and Normal.

25) Sample distributions. Centrale Limit Theorem. Normal approximation of the binomial distribution.

Confidence intervals and Hypothesis tests.

26) Confidence interval for a proportion.

27) Confidence interval for the mean, and known/unknown variance. T-Student Distribution.

28) Confidence interval for the variance of a population normally distributed. Chi-square distribution.

29) Hypothesis tests:general concepts. Null and alternative hypothesis, test statistic, critical region, level of significance, critical values, one/two tails test, P-value, errors of the first/second kind, power of a test.

30) Hypothesis test for a proportion. Hypothesis test for one sample: test on the mean (known/unknown variance), test on the variance or on the standard deviation.

31) Inference for two independent samples: inference on two proportions. Inference on two means, either for independent samples or for coupled samples.

Linear dependence.

32) Linear correlation and hypothesi test on the correlation coefficient.

33) Linear regression.

34) Test of independence.

35)

Teaching resources:

1) Book: Triola M.M. e Triola M.F., Statistica per le discipline biosanitarie, Pearson, 2009.

2) Self-evaluation test: can be found on Ariel.

Previous knowledge and exam

Elements of algebra and basic use of the scientific calculator.

Written multiple choice exam.

Teaching methods: frontal lectures, attendante is hgihly suggested

Language: italian

Fromal prerequisites: none

Further information and WEB page: all the information concerning the course will be published on the Ariel webpage

Descriptive Statistics.

18) Population, sample, parameter, statistics. Types of data and variables. Sampling.

19) Graphs and tables. Frequency tables. Histograms/bar graphs.

20) Mean, modal value, median, midrange and their relations. Range, standard deviation, variance and their relations. Percentiles, quartiles and outliers. Boxplot. Weighted mean.

Probability and random variables.

21) Introduction.Events and space of events; probability o fan event.

22) Probability of the union and the intersection. Complemento of an event. Independence. Conditional probability. Bayes Theorem.

23) Random Variables. Expected value, variance and deviation standard of discrete r.v.s.

24) Discrete r.v.s: Binomial and Poisson. Continuous r.v.s: Uniform and Normal.

25) Sample distributions. Centrale Limit Theorem. Normal approximation of the binomial distribution.

Confidence intervals and Hypothesis tests.

26) Confidence interval for a proportion.

27) Confidence interval for the mean, and known/unknown variance. T-Student Distribution.

28) Confidence interval for the variance of a population normally distributed. Chi-square distribution.

29) Hypothesis tests:general concepts. Null and alternative hypothesis, test statistic, critical region, level of significance, critical values, one/two tails test, P-value, errors of the first/second kind, power of a test.

30) Hypothesis test for a proportion. Hypothesis test for one sample: test on the mean (known/unknown variance), test on the variance or on the standard deviation.

31) Inference for two independent samples: inference on two proportions. Inference on two means, either for independent samples or for coupled samples.

Linear dependence.

32) Linear correlation and hypothesi test on the correlation coefficient.

33) Linear regression.

34) Test of independence.

35)

Teaching resources:

1) Book: Triola M.M. e Triola M.F., Statistica per le discipline biosanitarie, Pearson, 2009.

2) Self-evaluation test: can be found on Ariel.

Previous knowledge and exam

Elements of algebra and basic use of the scientific calculator.

Written multiple choice exam.

Teaching methods: frontal lectures, attendante is hgihly suggested

Language: italian

Fromal prerequisites: none

Further information and WEB page: all the information concerning the course will be published on the Ariel webpage

**modulo: Laboratorio di informatica**

**Course syllabus**

PART I - Introduction to Computer Science

G.1. Introduction to Computer Science

G.2. Data representation

G.3. Computer hardware

G.4. Software

G.5. Computer networks

PART II - Data analysis using spreadsheets

F.1. Spreadsheets

F.2. Mathematical functions in Excel

F.3. Statistical functions in Excel

F.4. Graphics in Excel

PART III - Data management and databases

B.1. Data management

B.2. Storing data in databases

B.3. Data models

B.4. Relational databases

B.5. Creation of databases using Access

B.6. Query in Access

B.7. Databases on the web

PART IV - Internet and the Web

I.1. Internet

I.2. Web architecture

I.3. Standard for the Web

I.4. Markup languages

I.5. Client-side applications

I.6. Search engines

PART V - Computer Science and Biology

1. Use of PubMed and UNIMI Library Network for bibliographic search

2. Biological databases

3. Notions of Bioinformatics: sequence alignment, protein folding, molecular docking, bio-inspired algorithms

Teaching resources

Teaching resources consist in lecture notes, slides and spreadsheets exercises, all available through the online software platform.

Previous knowledge and exam

Before taking the examination, students are required to compile the self-assessment questionnaires that are available through the online software platform. The examination consists in a multiple answer test.

G.1. Introduction to Computer Science

G.2. Data representation

G.3. Computer hardware

G.4. Software

G.5. Computer networks

PART II - Data analysis using spreadsheets

F.1. Spreadsheets

F.2. Mathematical functions in Excel

F.3. Statistical functions in Excel

F.4. Graphics in Excel

PART III - Data management and databases

B.1. Data management

B.2. Storing data in databases

B.3. Data models

B.4. Relational databases

B.5. Creation of databases using Access

B.6. Query in Access

B.7. Databases on the web

PART IV - Internet and the Web

I.1. Internet

I.2. Web architecture

I.3. Standard for the Web

I.4. Markup languages

I.5. Client-side applications

I.6. Search engines

PART V - Computer Science and Biology

1. Use of PubMed and UNIMI Library Network for bibliographic search

2. Biological databases

3. Notions of Bioinformatics: sequence alignment, protein folding, molecular docking, bio-inspired algorithms

Teaching resources

Teaching resources consist in lecture notes, slides and spreadsheets exercises, all available through the online software platform.

Previous knowledge and exam

Before taking the examination, students are required to compile the self-assessment questionnaires that are available through the online software platform. The examination consists in a multiple answer test.

modulo: Laboratorio di informatica

INF/01 - INFORMATICS - University credits: 3

Basic computer skills: 18 hours

Professor:
Lanzarotti Raffaella

modulo: Laboratorio di Metodi Matematici e Statistici

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

SECS-S/02 - STATISTICS FOR EXPERIMENTAL AND TECHNOLOGICAL RESEARCH - University credits: 0

SECS-S/02 - STATISTICS FOR EXPERIMENTAL AND TECHNOLOGICAL RESEARCH - University credits: 0

Laboratories: 32 hours

Lessons: 8 hours

Lessons: 8 hours

Professors:
Cogliati Alberto, Maggis Marco

modulo: Matematica generale

MAT/01 - MATHEMATICAL LOGIC - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

Practicals: 48 hours

Lessons: 24 hours

Lessons: 24 hours

Professors:
Alzati Alberto, Paleari Simone

### M-Z

Responsible

Lesson period

year

**modulo: Matematica generale**

**Course syllabus**

Numbers: natural, rational and real numbers. The field R and its operations. The simbols of infinity. Inequalities in R.

Functions: 1:1, onto and bijective functions; composition and invertibility. Real functions of one real variable, graphics, monotonicity. Elementary functions (powers, logarithms, exponentials, ...).

Linear Algebra: vectors, matrices and their operations. Determinants, inversion, and ranks. Linear systems. Cramer's and Rouché's theorems.

Limits: definitions and basic properties; uniqueness; monotonicity. Limits for elementary functions. Uncertainties. Asymptotics and comparison results. Eulero's number "e" and some fundamental limits. Divergent and vanishing functions. Continuity and related proprties: Darboux and Weierstrass' results.

Differential calculus: first derivative, geometric and mechanical meaning, tangent line. Operations with derivatives: chain rule. Differentiation of elementary functions. Optimization and the classical theorems (Fermat, Rolle, Lagrange), monotonicity. Convexity. Hopital's rule. Qualitative study for the graph of a real valued function. Antiderivatives and some methods to compute indefinite integrals.

Integral calculus: definite integrals and their main properties. The Fundamental Theorem of Integral Calculus. Area of plane regions.

Formal prerequisites

None

Suggested textbook:

P. Marcellini, C. Sbordone, Calcolo - edizione aggiornata per i nuovi corsi di laurea, Liguori editore (2004).

Online course "Matematica Assistita", http://ariel.unimi.it/User/

Previous knowledge and exam

Knowledge of elementary algebra and plane trigonometry, use of logarithms in the real field.

Written and oral exam, being possible to replace the written final exam with medium term exams. Attendance is highly recommended.

Teaching methods:

Traditional

Teaching language

Italian

Web page http://users.mat.unimi.it/users/zanco

Functions: 1:1, onto and bijective functions; composition and invertibility. Real functions of one real variable, graphics, monotonicity. Elementary functions (powers, logarithms, exponentials, ...).

Linear Algebra: vectors, matrices and their operations. Determinants, inversion, and ranks. Linear systems. Cramer's and Rouché's theorems.

Limits: definitions and basic properties; uniqueness; monotonicity. Limits for elementary functions. Uncertainties. Asymptotics and comparison results. Eulero's number "e" and some fundamental limits. Divergent and vanishing functions. Continuity and related proprties: Darboux and Weierstrass' results.

Differential calculus: first derivative, geometric and mechanical meaning, tangent line. Operations with derivatives: chain rule. Differentiation of elementary functions. Optimization and the classical theorems (Fermat, Rolle, Lagrange), monotonicity. Convexity. Hopital's rule. Qualitative study for the graph of a real valued function. Antiderivatives and some methods to compute indefinite integrals.

Integral calculus: definite integrals and their main properties. The Fundamental Theorem of Integral Calculus. Area of plane regions.

Formal prerequisites

None

Suggested textbook:

P. Marcellini, C. Sbordone, Calcolo - edizione aggiornata per i nuovi corsi di laurea, Liguori editore (2004).

Online course "Matematica Assistita", http://ariel.unimi.it/User/

Previous knowledge and exam

Knowledge of elementary algebra and plane trigonometry, use of logarithms in the real field.

Written and oral exam, being possible to replace the written final exam with medium term exams. Attendance is highly recommended.

Teaching methods:

Traditional

Teaching language

Italian

Web page http://users.mat.unimi.it/users/zanco

**modulo: Laboratorio di Metodi Matematici e Statistici**

**Course syllabus**

Descriptive Statistics.

18) Population, sample, parameter, statistics. Types of data and variables. Sampling.

19) Graphs and tables. Frequency tables. Histograms/bar graphs.

20) Mean, modal value, median, midrange and their relations. Range, standard deviation, variance and their relations. Percentiles, quartiles and outliers. Boxplot. Weighted mean.

Probability and random variables.

21) Introduction.Events and space of events; probability o fan event.

22) Probability of the union and the intersection. Complemento of an event. Independence. Conditional probability. Bayes Theorem.

23) Random Variables. Expected value, variance and deviation standard of discrete r.v.s.

24) Discrete r.v.s: Binomial and Poisson. Continuous r.v.s: Uniform and Normal.

25) Sample distributions. Centrale Limit Theorem. Normal approximation of the binomial distribution.

Confidence intervals and Hypothesis tests.

26) Confidence interval for a proportion.

27) Confidence interval for the mean, and known/unknown variance. T-Student Distribution.

28) Confidence interval for the variance of a population normally distributed. Chi-square distribution.

29) Hypothesis tests:general concepts. Null and alternative hypothesis, test statistic, critical region, level of significance, critical values, one/two tails test, P-value, errors of the first/second kind, power of a test.

30) Hypothesis test for a proportion. Hypothesis test for one sample: test on the mean (known/unknown variance), test on the variance or on the standard deviation.

31) Inference for two independent samples: inference on two proportions. Inference on two means, either for independent samples or for coupled samples.

Linear dependence.

32) Linear correlation and hypothesi test on the correlation coefficient.

33) Linear regression.

34) Test of independence.

35)

Teaching resources:

1) Book: Triola M.M. e Triola M.F., Statistica per le discipline biosanitarie, Pearson, 2009.

2) Self-evaluation test: can be found on Ariel.

Previous knowledge and exam

Elements of algebra and basic use of the scientific calculator

Written multiple choice exam.

Teaching methods

Frontal lectures, attendante is hgihly suggested

Language

Italian

Fromal prerequisites

None

Further information and WEB page

All the information concerning the course will be published on the Ariel webpage.

18) Population, sample, parameter, statistics. Types of data and variables. Sampling.

19) Graphs and tables. Frequency tables. Histograms/bar graphs.

20) Mean, modal value, median, midrange and their relations. Range, standard deviation, variance and their relations. Percentiles, quartiles and outliers. Boxplot. Weighted mean.

Probability and random variables.

21) Introduction.Events and space of events; probability o fan event.

22) Probability of the union and the intersection. Complemento of an event. Independence. Conditional probability. Bayes Theorem.

23) Random Variables. Expected value, variance and deviation standard of discrete r.v.s.

24) Discrete r.v.s: Binomial and Poisson. Continuous r.v.s: Uniform and Normal.

25) Sample distributions. Centrale Limit Theorem. Normal approximation of the binomial distribution.

Confidence intervals and Hypothesis tests.

26) Confidence interval for a proportion.

27) Confidence interval for the mean, and known/unknown variance. T-Student Distribution.

28) Confidence interval for the variance of a population normally distributed. Chi-square distribution.

29) Hypothesis tests:general concepts. Null and alternative hypothesis, test statistic, critical region, level of significance, critical values, one/two tails test, P-value, errors of the first/second kind, power of a test.

30) Hypothesis test for a proportion. Hypothesis test for one sample: test on the mean (known/unknown variance), test on the variance or on the standard deviation.

31) Inference for two independent samples: inference on two proportions. Inference on two means, either for independent samples or for coupled samples.

Linear dependence.

32) Linear correlation and hypothesi test on the correlation coefficient.

33) Linear regression.

34) Test of independence.

35)

Teaching resources:

1) Book: Triola M.M. e Triola M.F., Statistica per le discipline biosanitarie, Pearson, 2009.

2) Self-evaluation test: can be found on Ariel.

Previous knowledge and exam

Elements of algebra and basic use of the scientific calculator

Written multiple choice exam.

Teaching methods

Frontal lectures, attendante is hgihly suggested

Language

Italian

Fromal prerequisites

None

Further information and WEB page

All the information concerning the course will be published on the Ariel webpage.

**modulo: Laboratorio di informatica**

**Course syllabus**

PART I - Introduction to Computer Science

G.1. Introduction to Computer Science

G.2. Data representation

G.3. Computer hardware

G.4. Software

G.5. Computer networks

PART II - Data analysis using spreadsheets

F.1. Spreadsheets

F.2. Mathematical functions in Excel

F.3. Statistical functions in Excel

F.4. Graphics in Excel

PART III - Data management and databases

B.1. Data management

B.2. Storing data in databases

B.3. Data models

B.4. Relational databases

B.5. Creation of databases using Access

B.6. Query in Access

B.7. Databases on the web

PART IV - Internet and the Web

I.1. Internet

I.2. Web architecture

I.3. Standard for the Web

I.4. Markup languages

I.5. Client-side applications

I.6. Search engines

PART V - Computer Science and Biology

1. Use of PubMed and UNIMI Library Network for bibliographic search

2. Biological databases

3. Notions of Bioinformatics: sequence alignment, protein folding, molecular docking, bio-inspired algorithms

Teaching resources

Teaching resources consist in lecture notes, slides and spreadsheets exercises, all available through the online software platform.

Previous knowledge and exam

Before taking the examination, students are required to compile the self-assessment questionnaires that are available through the online software platform. The examination consists in a multiple answer test.

G.1. Introduction to Computer Science

G.2. Data representation

G.3. Computer hardware

G.4. Software

G.5. Computer networks

PART II - Data analysis using spreadsheets

F.1. Spreadsheets

F.2. Mathematical functions in Excel

F.3. Statistical functions in Excel

F.4. Graphics in Excel

PART III - Data management and databases

B.1. Data management

B.2. Storing data in databases

B.3. Data models

B.4. Relational databases

B.5. Creation of databases using Access

B.6. Query in Access

B.7. Databases on the web

PART IV - Internet and the Web

I.1. Internet

I.2. Web architecture

I.3. Standard for the Web

I.4. Markup languages

I.5. Client-side applications

I.6. Search engines

PART V - Computer Science and Biology

1. Use of PubMed and UNIMI Library Network for bibliographic search

2. Biological databases

3. Notions of Bioinformatics: sequence alignment, protein folding, molecular docking, bio-inspired algorithms

Teaching resources

Teaching resources consist in lecture notes, slides and spreadsheets exercises, all available through the online software platform.

Previous knowledge and exam

Before taking the examination, students are required to compile the self-assessment questionnaires that are available through the online software platform. The examination consists in a multiple answer test.

modulo: Laboratorio di informatica

INF/01 - INFORMATICS - University credits: 3

Basic computer skills: 18 hours

Professor:
Lanzarotti Raffaella

modulo: Laboratorio di Metodi Matematici e Statistici

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

SECS-S/02 - STATISTICS FOR EXPERIMENTAL AND TECHNOLOGICAL RESEARCH - University credits: 0

SECS-S/02 - STATISTICS FOR EXPERIMENTAL AND TECHNOLOGICAL RESEARCH - University credits: 0

Laboratories: 32 hours

Lessons: 8 hours

Lessons: 8 hours

Professors:
Aletti Giacomo, Villa Elena

modulo: Matematica generale

MAT/01 - MATHEMATICAL LOGIC - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

Practicals: 48 hours

Lessons: 24 hours

Lessons: 24 hours

Professors:
Gori Anna, Zanco Clemente