## MATHEMATICS AND STATISTICS MATEMATICA CON ELEMENTI DI STATISTICA

A.Y. Credits
2015/2016 10
Lecturer Email Office hours for students
Margherita Carletti One hour before and after lessons

### Assigned to the Degree Course

Giorno Orario Aula

### Learning Objectives

The course is meant to provide students with the fundamentals of differential and integral calculus, necessary to the biological/biotechnological disciplines.

### Program

1 Numerical sets
1.1 Integer, razional, irrational and real numbers
1.2 Absolute value
1.3 Intervals and neighbourhoods
1.4 Internal, external, boundary, isolated and accumulation points
1.5 Infimum and supremum, minimum and maximum of a set of real numbers
1.6 Complex numbers
2 Real functions of real variable
2.1 Definition of real funcion of real variable
2.2 Symmetric functions
2.3 Increasing and decreasing functions
2.4 Periodic functions
2.5 Domain and codomain of a function; rational and irrational algebraic functions; absolute value functions; exponential and logarithmic functions. Hints of trigonometric functions.
3 Limits and continuity
3.1 Definition of limit of a function
3.2 Theorem of uniqueness of the limit , theorem sign permanence ; theorem of comparison
3.3 Weierstrass' theorem and the intermediate zero theorem
3.4 Definition of continuous function in a point and in an interval
3.5 Discontinuity points
4 Derivation
4.1 Definition of first derivative of a function in a point and its geometric interpretation
4.2 Higher order derivatives
4.3 Lagrange's and Rolle's theorems and de l'Hospital's rule
4.4 Derivatives and increasing and decreasing functions
4.5 Maximum and minimum points
4.6 Convexity and concavity
4.7 Inflection points and inflection tangents
4.8 Graphic of a function
5 Integration
5.1 Primitive of a function
5.2 Indefinite integral and its properties
5.3 Integration by decomposition, by decomposition in simple fractions, by substitution and by parts
5.4 Area of trapezoid
5.5 Definite integral and its properties
5.6 Mean value theorem
5.7 Fundamental theorem of integral calculus ( Torricelli)
5.8 Improper integrals (hints).
6 Differential equations
6.1 I and II order differential equations; Cauchy's problem
6.2 Integration by separation of variables
6.3 Linear equations
6.4 II order differential equations with costant coefficients; Cauchy's problem
7 Elements of linear algebra
7.1 Vector spaces and subspaces
7.2 Linearly independent vectors
7.3 Bases and dimensions
7.4 Matrices and matrix operations
7.5 Inverse matrices
7.6 Rank of a matrix
7.7 Eigenvalues and eigenvectors of a matrix
7.8 Linear algebraic systems; Cramer's rule, Rouché-Capelli's theorem; homogeneous systems.

8 Probability and Statistics

8.1 Events
8.2 Definition of probability of an event
8.3 Axiomatic theory of probabylity by Kolmogorov
8.4 Independent evenys, conditional probability
8.5 Theorem of the sum, the product, of total probabilities, Bayes' theorem.
8.6 Elements of combinatorics
8.7 Representation of experimental data
8.8 Absolute and relative frequences
8.9 Double entry tables
8.10 Discrete and continuous random variables (r.vs.)
8.11 Cumulative distribution function
8.12 Probability distribution functions
8.13 Probability density functions
8.14 Mean, mode and median of a r.v.
8.15 Variance and standard deviation of a r.v.
8.16 Discrete r.vs.: Bernoulli, binomial and Poisson

8.17 Continuous r.vs.: uniform, exponenzial  and Gaussian

8.18 Central Limit Theorem

8.19 Population and samples

8.20 Introduction to inferential statistics

None

### Learning Achievements (Dublin Descriptors)

Students will have to demonstrate:

-  Knowledge and comprehension of the treated topics

-  Capability of application of known concepts in unknown contexts

-  Ability in switching from the analytical concepts to the corresponding geometric interpretation

-  Good linguistic exposition of the treated topics

-  Capability of linking the main treated topics one another.

### Teaching Material

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

### Didactics, Attendance, Course Books and Assessment

Didactics

Frontal lessons

Attendance

Not compulsory, though strongly recommended

Course books

M. Abate, Matematica e Statisica  (2nd edition), Le basi per le scienze della vita, McGrawHill, 2013

Assessment

The examination consists of a written and an oral part.

The written examination is 90 minutes long.

Students are admitted to the oral examination if they have passed the written one with the minimum mark of 12/30.

### Additional Information for Non-Attending Students

Didactics

None

Course books

M. Abate, Matematica e Statisica (2nd edition), Le basi per le scienze della vita, McGrawHill, 2013

Assessment

The examination consists of a written and an oral part.

The written examination is 90 minutes long.

Students are admitted to the oral examination if they have passed the written one with the minimum mark of 12/30.

 « back Last update: 14/06/16

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